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PhysRevB.87.220401.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 87, 220401(R) (2013) Anomalous domain wall velocity and Walker breakdown in hybrid systems with anisotropic exchange Henrik Enoksen, Asle Sudbø, and Jacob Linder Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway (Received 16 April 2013; published 7 June 2013) It has recently been proposed that spin-transfer torques in magnetic systems with anisotropic exchange can be strongly enhanced, reducing the characteristic current density with up to four orders of magnitude compared toconventional setups. Motivated by this, we analytically solve the equations of motion in a collective-coordinateframework for this type of anisotropic exchange system, to investigate the domain wall dynamics in detail. Inparticular, we obtain analytical expressions for the maximum attainable domain wall velocity of such a setupand also for the occurrence of Walker breakdown. Surprisingly, we find that, in contrast to the standard casewith domain wall motion driven by the nonadiabatic torque, the maximum velocity obtained via the anisotropicexchange torque is completely independent of the nonadiabaticity parameter β, in spite of the torque itself being very large for small β. Moreover, the Walker breakdown threshold has an opposite dependence on βin these two cases; i.e., for the anisotropic exchange torque scenario, the threshold value decreases monotonically withβ. These findings are of importance to any practical application of the proposed giant spin-transfer torque in anisotropic exchange systems. DOI: 10.1103/PhysRevB.87.220401 PACS number(s): 75 .78.Fg, 75 .60.Ch, 75 .40.Gb, 75 .70.Cn Introduction . The concept of spin transport in magnetic structures has proven to be of much relevance in terms ofboth applications and fundamental physics. 1One particularly promising topic in the field of spintronics is electrical controlof domain wall motion in textured ferromagnets. 2The essential idea is that controllable domain wall motion via spin-transfertorque may be used to represent information. This forms thebasis for possible applications such as magnetic random accessmemory, magnetic racetrack technology, and various types ofmagnetic logic gates. 3–6 In order to make controllable domain wall motion a feasible technology, two key aspects7,8need to be addressed: (i) The required current density to induce high-speed domain wallmotion must be lowered as much as possible and (ii) thestructural deformation of the domain wall triggered at theWalker breakdown 9must be delayed as much as possible, allowing for a higher maximum wall velocity. An interestingproposition in the right direction concerning point (i) wasrecently made in Ref. 10. By considering an anisotropic exchange interaction of a bilayer system consisting of aferromagnetic insulator and a semiconducting quantum well,it was proposed that sending a current through the latter partof the system would induce a torque on the texture M(x,t) of the ferromagnetic insulator which would be four orders ofmagnitude stronger than in conventional spin-transfer torquesetups. The torque originating with anisotropic exchange wasfound to be proportional to 1 /βwithβbeing the so-called nonadiabaticity parameter. 11For the typical nonadiabatic torque term, the corresponding proportionality is β, which is seen to be much smaller than the torque in the presentsystem with anisotropic exchange, since β/lessmuch1. It should be noted that although the term “nonadiabatic” for this torqueis standard in the literature, it is somewhat misleading sincethis is a dissipative torque present in the adiabatic limit andnot a nonlocal torque. 12However, we will adhere to the established convention in what follows and refer to this termas nonadiabatic.The usefulness of this novel anisotropic exchange torque in applications relies on the resulting domain wall dynamics, inparticular the maximum wall velocity and the occurrence ofWalker breakdown. Here, we address these issues by solvinganalytically the Landau-Lifshitz-Gilbert (LLG) equation in acollective-coordinate framework for the domain wall. The twomain degrees of freedom in this treatment are the velocity ofthe domain wall center ˙Xand the tilt angle φdescribing the deformation of the domain wall as it propagates. While weconfirm the finding of a considerably lowered characteristiccurrent density proposed in Ref. 10, we find that in contrast to the conventional case with domain-wall motion drivenby the nonadiabatic spin-transfer torque, the maximum wallvelocity under the influence of an anisotropic exchange torqueis completely independent ofβ. Deriving analytical expressions both for the maximum velocity and the Walker breakdown threshold, we demonstratethat the latter also behaves differently from the conventionalnonadiabatic case. For a system with anisotropic exchangeinteraction, the breakdown threshold is proportional to βand thus decreases as β→0. Finally, we consider the properties of a new hybrid system where the total spin-transfer torqueacting on the domain wall has a contribution both fromthe conventional dissipative torque and the anisotropic ex-change torque by means of two separate currents. In thiscase, we show that the relative magnitude and direction of thecurrents flowing can be used to tune both the maximum domainwall velocity and the threshold value for Walker breakdown. Theory . The physical systems under consideration are shown in Fig. 1. In (a), the textured ferromagnet is electrically conducting and corresponds to the conventional case ofcurrent-induced domain wall motion. In (b), the ferromagnetis an insulator so that the domain wall motion is inducedexclusively via exchange interaction to the semiconductingquantum well when a current passes through the latter. In (c),the ferromagnet is no longer insulating and separate currentsmay flow in the two layers due to insertion of an electrical 220401-1 1098-0121/2013/87(22)/220401(4) ©2013 American Physical SocietyRAPID COMMUNICATIONS HENRIK ENOKSEN, ASLE SUDBØ, AND JACOB LINDER PHYSICAL REVIEW B 87, 220401(R) (2013) (a) (b) (c) FIG. 1. (Color online) (a) A conducting ferromagnet (FC) with a N´eel magnetic domain wall. (b) A bilayer consisting of a ferromag- netic insulator (FI) and a semiconducting quantum well (2DHG). A N´eel magnetic domain wall makes the ferromagnet textured, and the magnetic coupling between the two layers is strongly anisotropic.(c) The ferromagnet is again conducting and separated from the 2DHG by an insulating layer (I), allowing for separate currents to flow in the FC and 2DHG region. Here, 2DHG denotes a two-dimensionalhole gas. insulator between them. Our starting point is to consider a N´eel-type domain wall M(x,t) where the easy (hard) axis of magnetic anisotropy are taken to be along the ˆz(ˆy) direction,13 m(x,t)=[sinθ(x) cosφ(t),sinθ(x)s i nφ(t),σcosθ(x)], (1) where σdenotes the topological charge of the domain wall. Here, we have defined the magnetization unit vector as m(x,t)=M(x,t)/M 0withM0=|M(x,t)|being the satura- tion magnetization. The tilt angle φ(t) represents a deformation mode of the domain wall from its equilibrium configuration,while θ(x) represents the angle between the magnetization and the easy axis. The domain wall texture in the staticcase is determined by the exchange stiffness and magneticanisotropy axes present in the system. We emphasize that wehave also performed the calculations to be presented for aBloch-type domain wall profile (same easy axis, but hardaxis along the ˆxdirection) and find identical results. This equivalence between the wall profiles no longer holds in thepresence of spin-orbit interactions. 14–16We do not consider spin-orbit interactions here, as it is not central to the mainresults. The interactions in our system are taken into accountvia the effective field H eff: Heff=2Aex M0∇2m−H⊥myˆy+Hkmzˆz+Hext. (2) Here,Aexis the exchange coupling constant, HkandH⊥are the anisotropy fields for the easy and hard axes, respectively,andH extis an external magnetic field. The components of the magnetization vector depend on both space and time according to cos θ=tanh(x−X(t) λ),sinθ=sech(x−X(t) λ).Here, λ=√2Aex/M 0Hkis the domain wall width and X(t)i st h e position of the domain wall center. Assuming that the easyaxis anisotropy field H kis larger than its hard axis equivalent H⊥, i.e.,|Hk|/greatermuch|H⊥|, the domain wall may be treated as rigid considering φ(t) as the only relevant deformation mode of the wall. To account for the novel anisotropy exchange torque, one must add an extra term which was derived in Ref. 10to the LLG equation. The physical setup is as follows: Considerthe exchange interaction between the quantum well and theferromagnetic film in Fig. 1(b). Assuming a p-type hole gas, the exchange arises from the overlap of wave functionsbetween the holes in the quantum well and the atoms in theferromagnet. The point is now that this exchange interaction is strongly anisotropic if only the heavy-hole subband is filledand the splitting to the light-hole band exceeds the exchangesplitting J. 17In effect, we are considering an exchange interaction of the type −JSz(r)mz(r) where Szandmz are the zcomponents of the spin densities of heavy holes and the magnetization unit vector, respectively. One finds thatthe full equation which governs the magnetization dynamicsreads 10 ∂tm=− ˜γm×Heff+˜αm×∂tm+τQW, (3) τQW=vs,QW βQW(m׈z)(∂xmz). Here, ˜ γand ˜αare the renormalized gyromagnetic ratio and Gilbert damping constant, respectively, whereas βQWis the nonadiabaticity parameter proportional to the spin relaxationtimeτ −1 QWof itinerant holes in the quantum well. The origin and value of this parameter is debated in the literature, althoughthere appears to be consensus that spin-relaxation processes(which may be model-dependent) are essential. 12,13,18We have also introduced the spin velocity vs,QW of the current in the semiconducting layer, which is proportional to the currentdensity (see below). We will obtain the equations of motion forthe collective coordinates {X,φ}and solve these analytically, thus obtaining a description of how the domain wall velocityv DW≡˙Xand the Walker breakdown criterion ˙φ/negationslash=0 depend on the external parameters of the system, such as the appliedcurrent. The breakdown threshold is the critical current densitywhere the domain wall starts deforming from its original shaperather than simply being translated along the current direction. The theory up to now has been described with the setup in Fig. 1(b) in mind. For comparison, we will also consider the system depicted in Fig. 1(a). 11,19The current flowing through the textured ferromagnet generates two differenttypes of torque terms compared to the anisotropic exchangemechanism, in the literature often referred to as the adiabaticand nonadiabatic torque. Their combined effect is captured inat e r m τ FMwhich is added to the right-hand side of Eq. (3): τFM=vs,FM∂xm−βFMvs,FMm×∂xm. (4) This torque term is controlled by the spin current vs,FM flowing in the ferromagnetic conductor, and the nonadiabiticity parameter βFMis in general different from the one in the quantum well, βQW. Finally, in Fig. 1(c)we consider a scenario where both the ferromagnet and quantum well are conductingwith an insulator dividing the two regions and thus permittingtwo separate currents to flow in each layer. In this case, bothτ QWandτFMshould be considered simultaneously in the LLG equation. Results and discussion . For a system where all torque terms are present, like the one depicted in Fig. 1(c), the equations of motion for {X,φ}are σ(1+˜α2)˙˜X=sin 2φ−σ(1+˜αβFM)˜vs,FM+σ˜α˜vs,QW 2βQW, (5) (1+˜α2)˙˜φ=− ˜αsin 2φ+σ(˜α−βFM)˜vs,FM+σ˜vs,QW 2βQW. (6) 220401-2RAPID COMMUNICATIONS ANOMALOUS DOMAIN WALL VELOCITY AND WALKER ... PHYSICAL REVIEW B 87, 220401(R) (2013) We will consider the general form for the equations of motion to begin with, whence we obtain the cases shown in Figs. 1(a) and 1(b) by setting some parameters equal to zero in the final result for the domain wall velocity and Walker break-down threshold. Here, we have introduced the dimensionless parameters˙˜X=∂(X/λ)/∂˜t,˙˜φ=∂φ/∂ ˜t,˜t=(˜γH ⊥/2)t, and ˜v=(2/˜γλH ⊥)v.W eh a v eu s e d vs,FM=(¯h˜γ/2eM 0)Pjwhere eis the electronic charge, Pis the spin polarization of the current, and jis the current density. Parameter values areP=0.7,M0=5×105A/m,Aex=10−11J/m,H⊥= 0.04 T, and Hk=0.4 T. By solving Eq. (6)analytically and inserting the solution into Eq. (5), we obtain an expression for the average domain wall velocity: /angbracketleft˙˜X/angbracketright=−βFM ˜α˜vs,FM+1 2˜αβQW˜vs,QW−sgn( ˜J) 1+˜α2/radicalbig ˜J2−1, (7) where ˜J=(1−βFM ˜α)˜vs,FM+1 2˜αβQW˜vs,QW. Details on this pro- cedure can be found in, e.g., Ref. 20. As stated above, Walker breakdown occurs when ˙φ/negationslash=0. From Eq. (6)we obtain, for a constant tilt angle, the result sin(2φ)=σ/parenleftbigg 1−βFM ˜α/parenrightbigg ˜vs,FM+σ˜vs,QW 2˜αβQW, (8) which gives the domain wall velocity /angbracketleft˙˜X/angbracketright=−βFM ˜α˜vs,FM+1 2˜αβQW˜vs,QW. (9) This is the same as the first two terms in Eq. (7).W es e e that the domain wall velocity reaches a maximum when thetwo currents flow in opposite directions. However, when theright-hand side of Eq. (8)becomes greater than unity, domain wall deformation sets in. Consider first the results for the case shown in Fig. 1(a). 11,19 Setting ˜ vs,QW=0 in the above results, one finds that Walker breakdown sets in when ˜ vs,FMc=1 1−βFM/˜αand the maximum domain wall velocity attainable in this setup is /angbracketleft˙˜X/angbracketrightc= 1 |1−˜α/β FM|. We emphasize that both ˜ vs,FMcand/angbracketleft˙˜X/angbracketrightcare normalized and thus dimensionless quantities, to facilitate comparison between the different setups in Fig. 1. Importantly, the normalization constants are independent of ˜ αandβ.T h e above results show that when ˜ α=βFM, Walker breakdown is absent and the maximum domain wall velocity has no upperbound; it is simply proportional to the applied current density.In practice, however, these parameters cannot be controlledor tuned in a well-defined manner such that this limit isnot easily obtained. A notable feature is that for large β FM, the threshold current is lowered whereas the maximum wallvelocity increases. With this in mind, we analyze the results for domain wall dynamics in the anisotropic exchange system Fig. 1(b). By using Eqs. (7)–(9), we find after some calculations the following results for Walker breakdown and maximum domainwall velocity: ˜v s,QWc=2˜αβQW, (10) /angbracketleft˙˜X/angbracketrightc=1.We note that, despite the fact that the spin-transfer torque is increased by as much as four orders of magnitude inthe anisostropic exchange system, due to the coefficient1/β QW, the maximum domain wall velocity is completely independent of the value of βQW. This should be compared to the standard setup in Fig. 1(a), where the wall is driven by the nonadiabatic spin-transfer torque, yielding a maximumdomain wall velocity strongly dependent on the value of β FM. The above result may be understood as a result of competitionbetween the magnitude of the spin-transfer torque and theoccurrence of Walker breakdown. Looking at Eq. (9),i ti s seen that the domain wall velocity is related to the spin velocity ˜v s,QWvia a constant of proportionality ∼β−1 QW.A tt h es a m et i m e , the critical value of the spin velocity where the domain wallno longer is stable towards deformation is given by ˜ v s,QWcin Eq.(10) and is proportional to βQW. As a result, the maximum domain wall velocity (which occurs right at the critical valuefor ˜v s,QW) is independent of βQW, since the dependence on this parameter cancels out when multiplying the domain wallvelocity with the critical spin velocity. Physically, this meansthat although the very large magnitude of the torque givesrise to a rapid increase of wall velocity with current, thesame property of the torque also renders the wall unstabletowards deformation faster than in the conventional case. Thedependence on βfor these two effects is such that they exactly compensate. Another qualitatively new aspect of the anisotropic ex- change case is in the manifestation of Walker break-down. The situation is different from the standard setupin Fig. 1(a). Namely, the critical current where Walker breakdown takes place now increases withβ QW. Effectively, this means that the Walker breakdown threshold decreasesas the spin-transfer torque increases, yet the maximum wallvelocity is unaffected by any change in β QW.T h i si sa unique feature of the spin-transfer torque originating withanisotropic exchange. The various domain wall dynamicsis depicted in Fig. 2, comparing the setups in Figs. 1(a) and1(b). The domain-wall velocity resulting from τ QWreaches appreciable values at much lower current densities comparedto a domain wall driven by τ FM. At the same time, the maximum attainable wall velocity is independent of βQW, while the maximum velocity obtained via τFMcan be very high for large values of βFM. An intriguing scenario could be realized if one were to combine these two torques to actsimultaneously on the same domain wall. This can be realizedexperimentally as shown in Fig. 1(c), where two separate currents flowing in the FM and QW layers control the magni-tude of these torques. Introducing the ratio between the spincurrents in the ferromagnet and the semiconducting quantumwell, i.e., a=v s,QW/vs,FM, we can express ˜Jfrom Eq. (7) as ˜J=˜vs,FM/bracketleftbigg/parenleftbigg 1−βFM ˜α/parenrightbigg b+a 2˜αβQW/bracketrightbigg , (11) where bis a parameter which controls the direction of ˜ vs,FM: b=1(−1) for parallell (antiparallell) flow of the currents. Note that the currents in the QW and FM region can be set tozero respectively by a=0o rb=0. 220401-3RAPID COMMUNICATIONS HENRIK ENOKSEN, ASLE SUDBØ, AND JACOB LINDER PHYSICAL REVIEW B 87, 220401(R) (2013) 0 2 4 6 8 10 12 x 10701020304050607080˙X [m/s] js,FM [A/cm2] 0 0.5 1 1.5 2 2.5 x 10501020304050 js,QW [A/cm2]˙X [m/s]β→0 β=0.01 β=0.02 β=0.04 β=0.1 FIG. 2. (Color online) Average domain wall velocity obtained for the setups in Fig. 1(a) (corresponding to the top panel) and (b) (corresponding to the bottom panel), for several choices of β. We have fixed ˜ α=0.02. Note the difference in order of magnitude on the xaxes, demonstrating the vastly different current magnitudes required. For β→0 in the bottom panel, Walker breakdown occurs almost immediately (not seen) and the increasing domain wallvelocity is above threshold so that the wall is continuously deforming. The maximum wall velocity prior to Walker breakdown is seen to be independent of βin the bottom panel [Fig. 1(b)] in contrast to the top panel [Fig. 1(a)]. Since the Walker breakdown condition is ˜J2>1, we can now identify the critical spin current velocity and thecorresponding critical domain wall velocity for the setup in Fig.1(c)where both τQWandτFMare active: ˜vs,FMc=1/parenleftbig 1−βFM ˜α/parenrightbig b+a 2˜αβQW, (12) /angbracketleft˙˜X/angbracketrightc=1 1−˜αb βFMb−a/2βQW. (13) In this way, one can control both the maximum attainable domain wall velocity and the occurrence of Walker breakdownby the relative magnitude aand relative sign bof the currents flowing in the FM and QW layer. It should be kept in mindthat the analysis performed here is for an idealized domain wallsystem without any defects or pinning potentials. Nevertheless,our results demonstrate the qualitatively different role playedby the nonadiabaticity parameter βin the anisotropic exchange system. Summary . In summary, we have shown that although the spin-transfer torque in magnetic systems with anisotropicexchange can be made much larger than the conventionalnonadiabatic torque, due to the former being proportionaltoβ −1rather than β, the maximum domain wall velocity is independent ofβin contrast to what one might expect. The Walker breakdown threshold decreases monotonically withβ, which also differs from the conventional scenario. These findings are of practical relevance to any application of theproposed giant spin-transfer torque in anisotropic exchangesystems. Acknowledgments. H.E. acknowledges support from NTNU. J.L. and A.S. acknowledge support from the ResearchCouncil of Norway through Grants No. 205591/V20 andNo. 216700/F20. A.S. acknowledges useful discussions withF. S. Nogueira. 1I.ˇZutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 2For a recent review on this topic, see, e.g., J. Grollier,A. Chanthbouala, R. Matsumoto, V . Cros, F. Nguyen van Dau,and A. Fert, C. R. Phys. 12, 309 (2011). 3A. D. Kent, B. Ozyilmaz, and E. del Barco, Appl. Phys. Lett. 84, 3897 (2004). 4S. Matsunaga, J. Hayakawa, S. Ikeda, K. Miura, H. Hasegawa,T. Endoh, H. Ohno, and T. Hanyu, Appl. Phys. Exp. 1, 091301 (2008). 5S. S. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 6H. Liu, D. Bedau, D. Backes, J. A. Katine, J. Langer, and A. D. Kent, Appl. Phys. Lett. 97, 242510 (2010). 7M. Hayashi, L. Thomas, C. Rettner, R. Moriya, Y . B. Bazaliy, and S. S. P. Parkin, P h y s .R e v .L e t t . 98, 037204 (2007). 8S. Pizzini, V . Uhlir, J. V ogel, N. Rougemaille, S. Laribi, V . Cros, E. Jim ´enez, J. Camarero, C. Tieg, E. Bonet, M. Bonfim, R. Mattana, C. Deranlot, F. Petroff, C. Ulysse, G. Faini, and A. Fert, Appl. Phys. Exp.2, 023003 (2009). 9N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974).10V . L. Korenev, arXiv:1210.4306 . 11S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). 12I. Garate, K. Gilmore, M. D. Stiles, and A. H. MacDonald, Phys. Rev. B 79, 104416 (2009). 13H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn. 75, 113706 (2006). 14K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee, Phys. Rev. B85, 180404(R) (2012); J. Ryu, S.-M. Seo, K.-J. Lee, and H.-W. Lee, J. Magn. Mater. 324, 1449 (2012). 15J. Linder, Phys. Rev. B 87, 054434 (2013). 16A. V . Khvalkovskiy, V . Cros, D. Apalkov, V . Nikitin, M. Krounbi, K. A. Zvezdin, A. Anane, J. Grollier, and A. Fert, Phys. Rev. B 87, 020402(R) (2013). 17I. A. Merkulov and K. V . Kavokin, P h y s .R e v .B 52, 1751 (1995). 18Y . Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 74, 144405 (2006). 19A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki, Europhys. Lett. 69, 990 (2005). 20G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468, 213 (2008). 220401-4

PhysRevB.92.024426.pdf

PHYSICAL REVIEW B 92, 024426 (2015) Gyrational modes of benzenelike magnetic vortex molecules Christian F. Adolff,1,*Max H ¨anze,1Matthias Pues,1Markus Weigand,2and Guido Meier1,3,4 1Institut f ¨ur Angewandte Physik und Zentrum f ¨ur Mikrostrukturforschung, Universit ¨at Hamburg, 20355 Hamburg, Germany 2Max-Planck-Institut f ¨ur Intelligente Systeme, Heisenbergstr. 3, 70569 Stuttgart, Germany 3The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany 4Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany (Received 12 February 2015; revised manuscript received 7 July 2015; published 27 July 2015) With scanning transmission x-ray microscopy we study six magnetostatically coupled vortices arranged in a ring that resembles a benzene molecule. Each vortex is contained in a ferromagnetic microdisk. When excitingone vortex of the ring molecule with an alternating magnetic high-frequency field, all six vortices performgyrations around the equilibrium center positions in their disks. In a rigid particle model, we derive the dispersionrelation for these modes. In contrast to carbon atoms, magnetic vortices have a core polarization that stronglyinfluences the intervortex coupling. We make use of this state parameter to reprogram the dispersion relation ofthe vortex molecule experimentally by tuning a homogeneous and an alternating polarization pattern. In analogyto the benzene molecule, we observe motions that can be understood in terms of normal modes that are largelydetermined by the symmetry of the system. DOI: 10.1103/PhysRevB.92.024426 PACS number(s): 75 .70.Kw,02.20.−a,68.37.Yz,75.40.Gb I. INTRODUCTION In magnetic nanodisks of suitable geometry, the magne- tization curls in the plane around the center of the diskand turns out-of-plane in the center. This magnetic ground-state configuration is called magnetic vortex. Dynamically,a gyration of the vortex core around the center of the disk isinherent in magnetic vortices [ 1]. Therefore it can be compared to a harmonic oscillator [ 2]. The polarization p, i.e., the out-of-plane direction of the vortex core ( p=±1), determines the gyration direction. It gyrates anticlockwise for positivepolarization and clockwise for negative polarization. A secondstate parameter, the chirality ( C=±1), describes the sense of in-plane curling of the magnetization in the disk [ 3,4]. The gyrotropic mode can be resonantly excited in various ways,using magnetic fields or electric currents [ 5,6]. V ortices in coupled periodic arrangements feature properties that can bedescribed with common concepts of solid state physics, i.e.,group velocity, density of states, and band structure [ 7–9]. The coupling of vortices strongly depends on their relativepolarizations. Thus arrangements of vortices are expected tofeature a reprogrammable band structure depending on theirpolarization configuration [ 7,10–12]. The molecule benzene (C 6H6) is a ring of six carbon atoms that each binds a hydrogen atom. When excited, for example, with infrared light,small vibrations of the atoms with respect to the interatomicdistances emerge. Historically, the comprehension of theso-called normal modes and the relation to their excitationfrequencies was crucial for understanding the infrared andRaman spectra. In this article, we study six magnetostaticallycoupled vortices arranged in a ring that resembles the benzenemolecule. Scanning transmission x-ray microscopy is used todirectly observe the gyrational excitations. We find that inanalogy to the benzene molecule, normal modes explain themeasured dynamics that largely depend on the symmetry of the *cadolff@physnet.uni-hamburg.desystem. As for the actual benzene molecule, such symmetryconsiderations allow to understand the dynamics in a vividfashion. The normal modes are plane waves with wavelengthsthat are fractions of the circumference of the ring, such asa breathing mode of the molecule. Our approach allows fordeducing the dispersion relation of the vortex molecule in aconvenient way. This will be shown in the last section. Thedispersion relation depends on the tuned polarization states inthe molecule and is measured for two states exemplarily. II. NORMAL MODES OF VORTEX GYRATION A convenient and powerful model to describe the motions of coupled vortices is the Thiele model [ 2,13–15]. It describes the magnetic vortex as a quasiparticle that is exposed to a force /vectorF=−/vector∇Ethat acts in the plane of the disk. In our case, it reads /parenleftbig G2 0+D2 0α2 Gilbert/parenrightbig˙/vectorx=G0˜r90/vectorF−D0αGilbert/vectorF. (1) Here, /vectorxis the two-dimensional position vector of the vortex core within the disk and ˜r90is a 90◦rotation matrix. Two components add to the velocity of the vortex core. The firstterm describes the nature of the gyrotropic mode that moves the vortex perpendicular to the driving force /vectorF. The second term depends on the dimensionless Gilbert damping parameterα Gilbert and forces the vortex core back to its equilibrium position. The constants G0andD0depend on material parameters [ 2]. It can be challenging to determine the driving force /vectorFin a coupled system. For a single magnetic vortex, a harmonic confining potential can be assumed to approximatethe internal forces. A linear energy term is commonly usedto describe the influence of external magnetic fields. Recentapproaches for systems of vortices in coupled arrays employsurface charges that emerge when the vortex is deflected fromthe center of the disk to approximate the coupling mediatedby the stray field [ 11,14–16]. Even when neglecting the damping, for a number of Ncoupled vortices, the Thiele equation becomes a 2 N-dimensional system of differential 1098-0121/2015/92(2)/024426(5) 024426-1 ©2015 American Physical SocietyADOLFF, H ¨ANZE, PUES, WEIGAND, AND MEIER PHYSICAL REVIEW B 92, 024426 (2015) FIG. 1. (Color online) X-ray micrographs of six disks that con- tain a vortex each. The permalloy disks are 60-nm thick and have a2 -μm diameter, the minimal distance between the disks is 50 nm. (a) V ortex molecule with homogeneous core polarizations in all six disks. The magnetic contrast can be seen in the raw data of one time frame. The vortex cores appear as black dots. (b) V ortex moleculewith alternating polarizations. The static contrast is subtracted to emphasize the magnetic contrast even more prominently. Disks and stripline are colorized. In the captured movie (see Ref. [ 18], movie 1) black vortex cores gyrate clockwise ( p i=−1) and white cores counterclockwise ( pi=1). equations that can only be solved numerically. Following the ideas of Wigner [ 17], we show that for rings of Ncoupled magnetic vortices the solution can be deduced exclusivelyby symmetry considerations. Figure 1shows the investigated vortex molecule consisting of six permalloy (Ni 80Fe20)d i s k s . A stripline is fabricated on one disk in order to excite the gyrotropic mode with the unidirectional high-frequencymagnetic field generated by an alternating current sent throughthe stripline. The steady-state motions are directly observed byscanning transmission x-ray microscopy at the MAXYMUSmicroscope of the BESSY II synchrotron in Berlin, Germany.As can be seen in Fig. 1(b), the method provides magnetic contrast that allows to clearly see the vortex cores as whiteor black dots, corresponding to their polarization. The timeresolution provided by the third generation synchrotron of upto 40 ps allows to trace the vortex trajectories (see Ref. [ 18], movies 1 and 3). In accordance with our measurements, weassume that the excitation will lead to approximately circularmotions of the N=6 vortices: /vectorx i=aiCi/parenleftbigg cos(ωt+ϕi) pisin(ωt+ϕi)/parenrightbigg ,i∈{0,1,..., N −1}. (2) With given chiralities Ciand polarizations piam o t i o no ft h e molecule is fully determined by the Ngyration amplitudes aiand phases ϕi. In the experiment, the polarizations and the chiralities are measured [ 19]. Due to the N-fold rotational symmetry and the linearity of the system, there has to be a basisofNnormal modes, that fulfill this symmetry. In analogy to the description of a linear chain of harmonic oscillators withperiodic boundary conditions, we determine these modes tobe plane waves with wavelengths that are fractions of thecircumference of the ring. For a ring of an even number of N disks, the normal modes /vectorx i,κare given by /vectorxi,κ∈{ /vectorxi|ai=aκ,ϕi=ϕi,κ=(κ+pi)iα+φκ}.(3)(a) (b) FIG. 2. (Color online) (a) Pictograms for the form and the propa- gation direction of the normal modes of the ring (see Ref. [ 18], movie 2). (b) Experiments with a homogeneous core polarization pattern inthe ring ( p i=−1). Each graph shows the contribution of a normal mode to the overall motions in the molecule for different excitation frequencies. The data points are obtained by a fit to the trajectoriestraced via scanning transmission x-ray microscopy (see Fig. 1and Ref. [ 18], movie 3). The solid lines are Lorentzian fit curves. The vertical scale of each graph ranges from 0 to 34 nm /mT. The integer number κ∈[−N/2,..., N/ 2) indexes the normal mode and is analogous to the wave number k=2π/λ in a linear chain of oscillators. The angle α=2π/N corresponds to the lattice constant in a linear chain. Since a generalvibration of the molecule is given by a linear combinationof the normal modes /vectorx i=/summationtext κ/vectorxi,κ, the factor aκdescribes the contribution of the normal mode /vectorxi,κto the motion. The relative phases of the normal modes are given by φκ. Figure 2(a) depicts the form of the normal modes for equal chiralities andpolarizations ( c i=1,pi=−1) of all vortices. For each point in time, the vortex cores are located on geometric roulettes,i.e., epitrochoids and hypotrochoids. For wave numbers κwith |κ|>0, the form of the roulettes stays constant over time and they rotate around the center of the ring, whilst the vortexcores are always located on the curve (see Ref. [ 18], movie 2). For positive wave numbers κ> 0, the roulettes rotate in the same direction as the vortices (clockwise). In contrast, fornegative wave numbers, the roulettes rotate anti-clockwise,i.e., against the gyration direction of the vortices. Thus thesign of κdenotes the propagation direction of the waves. Forκ=0, the normal mode /vectorx i,0is called the breathing mode since the vortices lie on a circle that changes its sizeover time. It can be compared to the modes 1 and 2 ofthe actual benzene molecule in the seminal work of Wilson,see Ref. [ 20], when only the vibrations of the carbon atoms are considered. At the edge of the Brillouin zone κ=±3t h e 024426-2GYRATIONAL MODES OF BENZENELIKE MAGNETIC . . . PHYSICAL REVIEW B 92, 024426 (2015) waves can be understood as propagating in both directions. Figure 2(b)shows the experimental results for the investigated vortex benzene, when the homogeneous polarization patternp i=−1 is present. The steady-state motions of the vortices are traced for 24 different frequencies around the resonancefrequency of an isolated disk. The grey line in each of thesix graphs is a Lorentzian fit through the black data pointsthat are proportional to the absolute gyration amplitude |a κ| of one normal mode /vectorxi,κ. These data points are obtained by applying a curve fit with the linear combination of normalmodes given by Eqs. ( 2) and ( 3) to the vortex trajectories of the six vortices. In order to ensure a linear gyration regime,the amplitude of the excitation is adjusted to small vortextrajectories. The influence of different excitation strengths onthe core velocity is normalized out [ 6,21]. For each frequency, one global curve fit is performed that comprises the completemotion of the six vortices and thus yields one data point ineach of the six graphs. We point out that each eigenmodehas its maximal contribution at different frequencies that lieon a sinusoidal line (dashed blue). Thus, contrary to the actualbenzene molecule, the propagation of waves in the two possibledirections (sign of κ) is not degenerated. The global rotation direction of the vortices in the homogeneous polarization casehas no equivalent in the linear vibrations in benzene. Such kindof global gyration direction cannot be defined for an alternatingpolarization pattern since the vortices gyrate in differentdirections according to their polarization p i. The alternating polarization pattern is shown in Fig. 1(b) and can be adjusted when a strong alternating magnetic field with a frequencyof 224 MHz is applied via the stripline and is then reducedadiabatically. Although only one vortex is directly excited, thisprocess of self-organized state-formation [ 16,23,29] allows to tune the polarizations in the whole molecule. The symmetryof the ring changes due to the alternating polarization patternso that two normal modes /vectorx i,|κ|=/vectorxi,κ+/vectorxi,−κhave to be combined in order to get standing waves. The combination ofsuch standing waves is depicted in Fig. 3(a)(see also Ref. [ 18], movie 4). This time, all modes can be compared to the normalmodes of the actual benzene molecule when only the carbonatoms are regarded. Using the Wilson numbering [ 20], the normal mode with |κ|=1 corresponds to mode “Y” of the actual benzene, |κ|=2 corresponds to mode “6a” and |κ|=3 can be compared to normal mode “12.” The arrow pictogramsin Fig. 3(a) are identical to those used by Wilson for the motions of the actual benzene. This elucidates the strongsimilarity between the actual benzene and vortex benzene.The standing waves are fitted to the trajectories and yield theresults presented in Fig. 3(b) [28]. III. DISPERSION RELATION Until now, we showed that there are strong similarities between the very different physical systems of magneticvortices and bound carbon atoms with regard to their motionsduring a harmonic excitation. Both systems feature similarnormal modes that are largely determined by the symmetry ofthe system. In the following, we will show that the symmetryconsiderations can be used to determine the dispersion relationof a ringlike vortex-molecule of arbitrary number of vorticesin a very convenient way.(a) (b) FIG. 3. (Color online) (a) Pictograms of the composition of the normal modes to obtain standing waves. (b) Experiments withalternating polarization pattern. Each graph shows the contribution of a standing wave to the overall motions in the molecule for different excitation frequencies. The data points are obtained by a fit to thetrajectories traced via scanning transmission x-ray microscopy. The solid lines correspond to the fit with Eq. ( 10). The vertical scale is identical to that in Fig. 2. To calculate the dispersion relation, we temporarily neglect the damping ( αGilbert=0) in Eq. ( 1). All vortex trajec- tories are described by the 2 N-dimensional vector /vectoru:= (/vectorx0,/vectorx1,.../vectorxN−1)Tand each two-dimensional component of it follows Eq. ( 1). When a normal mode with circular trajectories /vectoruκ=(/vectorx0,κ,/vectorx1,κ,.../vectorxN−1,κ)Tis inserted, it simplifies to ωκp/vectoruκ=1 G0(/vectorF0,/vectorF1,...,/vectorFN−1)T. (4) /vectorFidescribes the sum of all driving forces of vortex i. Multiplying both sides of the equation with /vectoruκyields ωκ=1 pG 0/summationtextN−1 i=0/vectorFi/vectorxi,κ/summationtextN−1 i=0/vectorx2 i,κ=1 pG 0/summationtextN−1 i=0/vectorFi/vectorxi,κ Na2κ.(5) The driving forces /vectorFiare given by the total energy with respect to vortex i. We approximate the coupling between two vortices iandjby the most simple approximation, which is dipolar stray-field interaction [ 22]: Edipole ,i,j=μ0 4πr3/parenleftbigg /vectorμi/vectorμj−3 r2(/vectorμi/vectorr)(/vectorμj/vectorr)/parenrightbigg . (6) The dipole moment of a vortex is proportional to the deflection of the vortex rotated by 90 degrees ( /vectorμi= ˜r90Ci˜aκ/vectorxi,κ|/vectorxi,κ|[26]). The strength of the dipole moment is denoted as ˜aκsince it is proportional to the gyration amplitude aκ. For the given harmonic excitation, the chirality Cihas no influence on the dipole moment, since the change of sign is compensated by aphase shift of 180 ◦in time. The anchor points of the dipoles are assumed to be fixed at the centers of the disks. Thus, thevector /vectorr, that connects the dipoles, is constant. Separating the dipolar coupling from all other forces that act on the isolated 024426-3ADOLFF, H ¨ANZE, PUES, WEIGAND, AND MEIER PHYSICAL REVIEW B 92, 024426 (2015) FIG. 4. (Color online) Experimentally determined dispersion re- lations for the two polarization patterns and theoretical fit curves derived from the extended Thiele model. The dashed lines result from a global curve fit of Eq. ( 10). The data points result from individual Lorentzian curve fits to the experimental results. disks yields ωκ−ω0=−1 pG 01 Na2κN−1/summationdisplay i=0/summationdisplay j/negationslash=iEdipole ,i,j. (7) The resonance frequency ω0applies for noninteracting vortices in isolated disks. Only considering next-neighbor interactionyields the discrete dispersion relation [ 27] ω hom(κ)=ω0−1 2Bhomcos((κ+p)α) (8) for the homogeneous case. For the alternating polarization pattern, it is useful to integrate Eq. ( 7) over one period of gyration to separate ωκ. It then yields ωalt(κ)=ω0+1 2Baltcos(κα),B alt=3Bhom, (9) for the alternating polarization pattern. The bandwidth Bhom is a positive constant given by Bhom=−1 pG 0μ0 4πr3(˜aκ aκ)2. When comparing the two analytically calculated dispersion relations ωaltandωhom, one can see that the factor phas vanished in the cosine and that the prefactor is multiplied by ( −3). The different bandwidths are commonly explained by aweaker coupling between vortices of equal polarization thanof vortices with different polarizations [ 24]. For the borderline case of an infinite linear chain ( N→∞ ), the results are in concordance with previous results [ 11]. In the following, we include the effects of damping and the experimental results.For negligible damping, there are sharp resonances when theeigenfrequency of a normal mode is met. In the experiment,the damping allows to excite the system in between thoseresonances. The normal modes mix in the way shown inFig.2(also see Ref. [ 18], movie 3). The contributions a κ(ωexc) are fitted to the experimental data with Lorentzian functionsthat are shifted according to the analytically derived discretedispersion relation ω(κ): a κ(ωexc)=L/Gamma1(ωexc−ω(κ)),κ∈[−N/2,...,N/2).(10) This set of equations can be understood as the continuous dispersion relation of the damped system, where ωexcis the frequency of the exciting magnetic field and L/Gamma1(ω)t h e Lorentzian peak function with damping parameter /Gamma1.T h efit yields the three model parameters ω0,Bhom, and/Gamma1.T h e first two parameters determine the discrete dispersion relationsshown in Fig. 4. The dashed green [blue] curve corresponds to the dispersion relation ω alt(κ)[ωhom(κ)] measured for the alternating [homogeneous] polarization configuration of thevortex molecule. The data points correspond to peaks ofindividual Lorentzian fits as presented in Fig. 2(b).T h e parameters are determined to be ω 0/2π=(225.5±1.5) MHz andBalt=(96±6) MHz for the alternating case and ω0/2π= (227.6±0.8) MHz and Bhom=(31±2) MHz for the homo- geneous polarization pattern. Those are appropriate valuesfor the bandwidth when considering the low disk interdis-tance [ 30]. In both cases, the damping parameter has a value of/Gamma1=(29±3) MHz, which is in reasonable accordance with the relation /Gamma1=2α Gilbertω≈0.02ωexpected from other studies [ 32]. IV . CONCLUSION In conclusion, we have shown that there are strong similarities between the vibrational modes of benzene and thegyrational modes of a sixfold magnetic-vortex ring molecule.The symmetry of both systems determines the motions ofthe oscillators, i.e., the carbon atoms or the vortices. Thebest accordance in the analogy can be achieved when analternating polarization pattern is tuned to the vortex molecule.In this case, all gyrational modes can be identified withvibrational modes in the actual benzene. The symmetryallows to simplify the derivation of the fundamentally dif-ferent dispersion relations of the vortex molecule for thehomogeneous and alternating core polarization patterns. Incontrast to other models, the presented approach includes theeffect of damping and is characterized by only three modelparameters, each of them determined in the experiments. Bothdispersion relations have been confirmed by x-ray transmissionmicroscopy proving that the magnetic vortex molecule featuresa reprogrammable band structure or dispersion relation. ACKNOWLEDGMENTS We thank Ulrich Merkt for fruitful discussions, Benedikt Schulte for support during the evaluation of the measurements,and Michael V olkmann for superb technical assistance. We ac-knowledge the support of the Max-Planck-Institute for Intelli-gent Systems (formerly MPI for Metals Research), DepartmentSch¨utz and the MAXYMUS team, particularly Michael Bech- tel and Eberhard Goering. We thank the Helmholtz-ZentrumBerlin (HZB) for the allocation of synchrotron radiationbeamtime. Financial support of the Deutsche Forschungs-gemeinschaft via the Sonderforschungsbereich 668 and theGraduiertenkolleg 1286 is gratefully acknowledged. This workhas been supported by the excellence cluster “The HamburgCentre for Ultrafast Imaging - Structure, Dynamics, andCentre of Matter at the Atomic Scale” of the DeutscheForschungsgemeinschaft. [1] S.-B. Choe, Y . Acremann, A. Scholl, A. Bauer, A. Doran, J. St¨ohr, and H. A. Padmore, Science 304,420(2004 ).[2] B. Kr ¨uger, A. Drews, M. Bolte, U. Merkt, D. Pfannkuche, and G. Meier, P h y s .R e v .B 76,224426 (2007 ). 024426-4GYRATIONAL MODES OF BENZENELIKE MAGNETIC . . . PHYSICAL REVIEW B 92, 024426 (2015) [3] T. Shinjo, T. 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H ¨anze, C. F. Adolff, M. Weigand, and G. Meier, Phys. Rev. B 91,224417 (2015 ). [13] A. A. Thiele, P h y s .R e v .L e t t . 30,230(1973 ). [14] J. Shibata, K. Shigeto, and Y . Otani, P h y s .R e v .B 67,224404 (2003 ). [15] M. H ¨anze, C. F. Adolff, M. Weigand, and G. Meier, Appl. Phys. Lett.104,182405 (2014 ). [16] C. F. Adolff, M. H ¨anze, A. V ogel, M. Weigand, M. Martens, and G. Meier, Phys. Rev. B 88,224425 (2013 ). [17] E. Wigner, Nachrichten d. Gesell. d. Wissenschaften z. G¨ottingen, Math.-Phys., Klasse 1, 133 (1930). [18] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevB.92.024426 for the supplemental movies and movie descriptions. [19] In order to determine the chiralities, a different setting of the microscope is used that provides in-plane contrast.[20] E. B. Wilson, Phys. Rev. 45,706(1934 ). [21] T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers, J. Appl. Phys. 85,7849 (1999 ). [22] A. V ogel, A. Drews, T. Kamionka, M. Bolte, and G. Meier, Phys. Rev. Lett. 105,037201 (2010 ). [23] S. Jain, V . Novosad, F. Y . Fradin, J. E. Pearson, V . Tiberkevich, A. N. Slavin, and S. D. Bader, Nat. Commun. 3,1330 (2012 ). [24] A. V ogel, T. Kamionka, M. Martens, A. Drews, K. W. Chou, T. Tyliszczak, H. Stoll, B. Van Waeyenberge, and G. Meier, Phys. Rev. Lett. 106,137201 (2011 ). [25] J. Mej ´ıa-L´opez, D. Altbir, A. H. Romero, X. Batlle, I. V . Roshchin, C.-P. Li, and I. K. Schuller, J. Appl. Phys. 100,104319 (2006 ). [26] This is a simplification of the model commonly used to describe the stray-field coupling of magnetic vortices. It corresponds toan analytical expression for the numerical coupling integrals de- s c r i b e di nR e f s .[ 14,31], i.e.,η xx∝(1 r3−3r2y r5),ηyy∝(1 r3−3r2x r5), ηxy=ηyx∝3rxry D5, whereas the vector /vectorr=(rx,ry)Tconnects the centers of two disks. 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PhysRevB.92.064416.pdf

PHYSICAL REVIEW B 92, 064416 (2015) Spin-wave dynamics in permalloy/cobalt magnonic crystals in the presence of a nonmagnetic spacer P. Malag `o,1L. Giovannini,1R. Zivieri,1P. Gruszecki,2and M. Krawczyk2 1Dipartimento di Fisica e Scienze della Terra, Universit `a di Ferrara, Via G. Saragat 1, I-44122 Ferrara, Italy 2Faculty of Physics, Adam Mickiewicz University in Pozna ´n, Umultowska 85, 61-641 Poznan, Poland (Received 16 April 2015; revised manuscript received 24 July 2015; published 12 August 2015) In this paper, we theoretically study the influence of a nonmagnetic spacer between ferromagnetic dots and a ferromagnetic matrix on the frequency dispersion of the spin-wave excitations in two-dimensional bicomponentmagnonic crystals. By means of the dynamical matrix method we investigate structures that are inhomogeneousacross the thickness represented by square arrays of cobalt or permalloy dots in a permalloy matrix. We showthat the introduction of a nonmagnetic spacer significantly modifies the total internal magnetic field, especiallyat the edges of the grooves and dots. This permits the manipulation of the magnonic band structure of spinwaves localized either at the edges of the dots or in matrix material at the edges of the grooves. According to themicromagnetic simulations two types of end modes were found. The corresponding frequencies are significantlyinfluenced by the end modes’ localization region. We also show that, with the use of a single ferromagneticmaterial, it is possible to design a magnonic crystal preserving the properties of bicomponent magnonic crystalsand magnonic antidot lattices. Finally, the influence of the nonmagnetic spacers on the technologically relevantparameters such as group velocity and magnonic bandwidth are discussed. DOI: 10.1103/PhysRevB.92.064416 PACS number(s): 75 .30.Ds,75.78.−n,75.75.−c I. INTRODUCTION Spatial periodicity in a ferromagnetic material modifies the spin-wave (SW) dispersion relation and results in the formationof magnonic bands and band gaps. Magnetic materials withperiodic modulation are called magnonic crystals (MCs) [ 1–3]. Presently, MCs are getting particular interest due to the possi-bility of tailoring frequency spectra of SWs at the nanoscale;as a consequence, it is possible to understand magnetizationdynamics and (a) to design metamaterial devices [ 4,5], (b) to transduce and transmit signals [ 6–8], (c) to realize magnonic transistors [ 9], and (d) to make logic operations [ 10–12]. Among the possible geometries of MCs, the planar MCs are the most often investigated. This is due to the feasibility offabrication of regular patterns and easy access to characterizemagnetic properties, to measure SW dispersion relation anddynamics in time domain, and to visualize SW excita-tions [ 13,14]. In standard SW transmission measurements microwave transducers (microstripes or coplanar waveguides)are used. They allow for effective excitation of SWs with longwavelengths. In this limit the magnetostatic interactions areimportant and propagation of SWs in nanostructures is usuallyinvestigated in the direction perpendicular to the externalmagnetic field, i.e., in the Damon-Eshbach (DE) geometry,where even at zero wave number the relatively high groupvelocity is present. Among planar MCs, the one- and two-dimensional (2D) MCs, i.e., with periodicity along one and two directions, respectively, can be distinguished. The three main groups of2D MCs are arrays of dots, antidot lattices, and bicomponentMCs (BMCs). The first consists of regular arrays of thinferromagnetic dots, the second of negative arrays of theformer, i.e., arrays of holes in thin ferromagnetic film.The last group can be regarded as a superposition of both, i.e., the antidot lattices with holes filled with a different ferromagnetic material. These three groups present distinctfeatures in the SW propagation. The collective magnetizationdynamics in an array of dots is solely due to dynamic dipolecoupling between resonant excitations of the dots; however, its properties are also influenced by static demagnetizingeffects [ 15]. In the case of weak coupling (large separation between dots with respect to their thickness and width), the magnonic spectra consist of flat bands with frequencies related to the eigenmode excitations of the isolated dot [ 16]. By increasing the dynamic dipole coupling, e.g., by decreasingseparation between dots, collective SW excitations with finitebandwidth and preserving properties of the magnetostaticwaves appear. The widening of the bands depends on thedipolar coupling strength and on the stray magnetic field [ 17]. However, the dynamic dipole interaction is effective especially for eigenmodes having the largest total dynamic magneti-zation (averaged over the whole dot), viz., mainly for thefundamental mode [ 18], but also for end modes or low-order backwardlike magnetostatic modes [ 19]. In antidot lattices, the low-frequency part of SW spectra is influenced by theinhomogeneous static demagnetizing field created by the edges of the holes. The presence of holes leads to the formation of wells of the total magnetic field where magnetization dynamicsmainly concentrate [ 20,21]. Indeed, in antidot lattices end modes localized at the edges of the holes and SWs concentratedin channels between holes were found [ 22,23]. These effects disappeared at sufficiently small lattice constants, wherethe exchange interactions start to prevail over the dipole interactions. Recently, also, the effect of magnetization pinning on spin-wave dispersion has been theoretically studied inpermalloy (Py) antidot waveguides by introducing a surfaceanisotropy at the ferromagnetic-air interface [ 24]. Moreover, it has been shown that structural changes in antidot waveguidesbreaking the mirror symmetry of the waveguide can close bandgaps [ 25]. It is also well known from the literature that the Dzyaloshinskii-Moriya interaction induces the tilting of themagnetic moments at the edges and leads to the formation ofa noncollinear structure [ 26] acting as a scattering barrier for spin excitations [ 27] and partly contributes to the formation of 1098-0121/2015/92(6)/064416(10) 064416-1 ©2015 American Physical SocietyP. MALAG `Oet al. PHYSICAL REVIEW B 92, 064416 (2015) end modes along the barrier. The transition from the quantized to the propagative regime of SWs (end modes and fundamentalmode) can be controlled, e.g., by the magnetic field orientationor by the separation between holes [ 28–30]. In addition to the demagnetizing effects, also the shape and size of holesin the antidot lattices influence the SW spectrum. This effectdominates for exchange SWs, i.e., at high frequencies or whenthe lattice constant is small [ 31]. In bicomponent MCs the inhomogeneous demagnetizing field is still present; however,its amplitude depends on the difference between the magneticproperties of the constituent materials. Thus, its influence onSW dynamics is weaker than in antidot lattices and valuablefor the low-frequency modes only. Recently, a bicomponent MC composed of Co circular dots embedded in a Py(Ni 80Fe20) matrix was investigated theoretically and experimentally [ 14,32–34]. The Brillouin light scattering (BLS) measurements showed the existenceof two types of SW excitations concentrated in regionsperpendicular to the external magnetic field containing Codots and in Py matrix between the Co dots [ 32]. Theoretical studies have confirmed that the separation of frequenciesof these SWs is due to a magnetostatic effect [ 33,34] and the splitting of the magnonic band at the boundary of theBrillouin zone (BZ) is connected to the periodicity of themagnetic system [ 14]. However, the full magnonic band gap in bicomponent MCs has not yet been investigated in detail.Changes of dot or antidot shape, their rotation with respect tothe crystallographic axes, and imperfections in their shape orat their edges can further modify SW spectra [ 35–37]. Thus, the large variety of shapes for dots or antidots and of theirarrangements together with magnetic configurations which canbe realized in MCs [ 38–41] makes magnonics an inexhaustible and intriguing topic of research. More specifically, not enoughattention has been given to the study of collective dynamicsin bicomponent MCs where a nonmagnetic spacer separatesthe two magnetic materials. The aim of this paper is thus totheoretically investigate the effect of a nonmagnetic spacer in2D MCs on the dispersions of the relevant SWs accordingto a micromagnetic approach named the dynamical matrixmethod (DMM). This is done to investigate the importantspin dynamics effects due to the significant spatial variationsexperienced by the total inhomogeneous magnetic field be-cause of the nonmagnetic material at the interface betweenthe two ferromagnetic materials. In this respect, the dynamicsin square lattice 2D MCs with square antidots partially filledwith different magnetic materials are studied, but the obtainedresults can be easily generalized to other geometries. Thisis achieved by putting the nonmagnetic spacer around thedots embedded in antidot lattices. In this study two types ofseparation between dots and Py matrix are considered: (a) withthe nonmagnetic spacer located only below the dot, and (b)with the spacer fully around the dot. It is shown that theseseparations create an inhomogeneous static demagnetizingfield which allows for the formation of end modes in thematrix (characteristic for the antidot lattices) and end modesin the dots (characteristic for the array of dots) which werenot yet found in the previous studies [ 14,34]. Moreover, it is shown that similar properties can be achieved using a singleferromagnetic material, i.e., in single component 2D MC. Thisstudy focuses on the important part of magnetism devotedto SW phenomena in composite structures, which is almost unexplored yet in the case of large-scale 2D bicomponentnanopatterned systems. This investigation is also of interest fortechnological applications in the area of magnonics, magneticmemories, and metamaterials. The paper is organized as follows. In the next section the structures and the theoretical method used in the investigationsare described. Then, in Sec. III, the results of calculation of SW spectra showing the influence of nonmagnetic spacers onthe magnonic dispersions are presented. In Sec. IVthe results obtained are discussed and the influence of the inhomogeneityof the total magnetic field is analyzed. Then, in Sec. V,t h e effect of the nonmagnetic spacer on the group velocity andmagnonic bandwidth is investigated. Finally, conclusions aredrawn in Sec. IV. II. STRUCTURE AND METHOD In order to study the dynamical properties of 2D MCs connected with the nonmagnetic spacers betweenferromagnetic materials, the dispersion relations of SWs forfive systems have been calculated. The magnetic systems arecomposed of Py, Co, and nonmagnetic material. All geometriesinvestigated here are based on square lattice and squaremagnetic dots, the lattice constant being a=400 nm. MCs are supposed to be infinite in plane (along xandy). These five systems are depicted in Fig. 1:( a )System 1 (S1): bicomponent MC composed of 30-nm-thick Py film with an array of20-nm-deep square grooves of 200 nm size. In the bottom ofthe grooves there is 10 nm of nonmagnetic material and then Codots (20 nm thick) partially immersed into the grooves. The Codots are in direct contact with Py only at the lateral edges of thedot. (b) System 2-Co (S2 Co): bicomponent MC similar to S1 but with 10-nm-width spacer around the Co dots (200 nm wide).In S2 Co, Co dots and Py matrix are separated by a nonmagnetic spacer. (c) System 2-Py (S2Py): one-component MC with the same geometry of S2Cobut with Py dots. (d) MC composed of square Py dots (10 nm thick and 200 nm wide) surrounded bynonmagnetic spacer and fully immersed in the Py matrix. Thisissystem 3 (S3). (e) An array of squared Co dots (20 nm thick and 200 nm wide) constitutes system 4 (S4). All parameters used in the simulations are typical parameters for Py and Comaterials [ 42,43]: saturation magnetization for Py M S,Py= 750 emu /cm3and for Co, MS,Co=1200 emu /cm3; exchange constants, APy=1.3×10−6erg/cm and ACo=2.0× 10−6erg/cm; gyromagnetic ratios, γPy/2π=2.96 GHz /kOe andγCo/2π=3.02 GHz /kOe. The static and dynamic properties of these magnetic sys- tems have been investigated by means of two micromagneticcodes: Object Oriented MicroMagnetic Framework ( OOMMF ) code [ 42] and the DMM program [ 34,44]. The ground-state magnetization was determined by using OOMMF with 2D periodic boundary conditions; then this magnetic configurationwas used as input to DMM . The DMM with implemented boundary conditions, a finite-difference micromagnetic ap-proach first implemented for isolated ferromagnetic elementsand extended to MCs composed by two ferromagnetic mate-rials [ 34], is applied to study the spin dynamic properties in bicomponent systems where the two ferromagnetic materialsare separated by a nonmagnetic spacer. Since our resultsdo not focus on dissipation properties of collective modes, 064416-2SPIN-WA VE DYNAMICS IN PERMALLOY/COBALT . . . PHYSICAL REVIEW B 92, 064416 (2015) FIG. 1. (Color online) (a) System 1: Top view of the primitive cell and its perpendicular cross section in a bicomponent MC consisting of Co square dots in square array partially immersed in Py. Nonmagnetic spacer (white area) of 10 nm thickness separates the bottom of Co dots from Py. (b) System 2-Co: similar to S1 but with full separation of Co dots (200 nm wide) from Py (10 nm of nonmagnetic spacer from the bottom and lateral sides of Co). (c) System S2-Py: one-component MC with geometry equal to S2Cobut with Py dot. (d) System 3: MC created by square array of square grooves in Py film partially filled with Py dots. Dots are separated from the matrix by 10-nm-thick nonmagnetic spacer. (e) System 4: square array of square Co dots. Red dashed lines in the perpendicular cross sections point at the planes ( z=5 and 25 nm) used in Figs. 2(a)–2(e)to visualize the spatial profiles of SW modes. the dynamics is studied in the purely conservative regime; hence no Gilbert damping energy density contribution isincluded in the equations of motion. For our purposes, inthe DMM two indices are used: (1) an index kto label micromagnetic cells, with k=1,2,..., N , where Nis the total number of micromagnetic cells in the primitive cell;(2) an index j=Py,Co indicating the ferromagnetic material. The number of micromagnetic cells assigned to the jth ferromagnetic material is N jsuch that NPy+NCo=N.F o r each micromagnetic cell the magnetization in reduced unitstakes the form m k=Mk/Ms(k) with Mkthe magnetization in thekth cell and Ms(k) the saturation magnetization depending on the ferromagnetic material through the index k.Hence, in a polar reference frame the magnetization can be written in thefollowing form: m k=(sinθkcosφk,sinθksinφk,cosθk), (1) where φk(θk) is the azimuthal (polar) angle of the magnetiza- tion; for the sake of simplicity the time dependence is omitted. The total energy density E=˜E V—with ˜Ethe energy and V the volume of the system, respectively—depends on the polarand azimuthal angle in each micromagnetic cell, θ kandφk. The total energy ˜Eis the magnetic Hamiltonian and the DMM was developed to study conservative systems corresponding to a purely precessional dynamics. In explicit form, for thesystems under study, the energy density reads E=E ext+Eexch+Edmg, (2) withEextthe Zeeman, Eexchthe exchange, and Edmgthe demagnetizing, respectively. Specifically, E=−MSH·N/summationdisplay k=1mk+N/summationdisplay k=1/summationdisplay n∈{n.n.}Aexch(k,n)1−mk·mn a2 kn +1 2M2 S/summationdisplay klmk·↔ Nml. (3)The first term of Eq. ( 3) corresponds to the Zeeman energy density, where Hindicates the external magnetic field. The second term of Eq. ( 3) is the exchange energy density expressed by means of two sums: the first sum runs overtheNmicromagnetic cells and is indexed by kwhereas the second sum indexed by nranges over the nearest-neighbouring (n.n.) micromagnetic cells of the kth micromagnetic cell that can belong to a different ferromagnetic material. A exchis the exchange stiffness constant and is related to the ferromagneticmaterials through the indices kandn, respectively, while a kndenotes the distance between the centers of two adjacent micromagnetic cells of indices kandn, respectively. When the kth micromagnetic cell is on one of the edges (vertices) of the proper primitive cell, the interaction with the micromagneticcells belonging to the correct nearest supercell (primitive cell)must be taken into account. The last term of Eq. ( 3)i st h e demagnetizing energy density where ↔ Nis the demagnetizing tensor and expresses the interaction among micromagneticcells within the primitive cell and belonging to differentprimitive cells. Note that, unlike the bicomponent systemstudied [ 34], in S2 Cothe intermaterial exchange contribution is set equal to zero, because in the primitive cell the Co dot andthe Py matrix are separated. Instead, in the S1, the exchangecontribution at the interface between the two ferromagnetic materials is set equal to ¯A Py−Co exch=(APy+ACo)/2 because Py matrix and Co dots are in contact. Note that in Eq. ( 2)t h e thermal contribution related to the thermal field is not included.Indeed, the studied dynamics is purely deterministic and notstochastic. Actually, the equations of motion within the DMM correspond to the deterministic Landau-Lifshitz equations andnot to the stochastic Langevin or Fokker-Planck equations [ 45]. The dynamic magnetization δm(r) of each collective mode fulfills the generalized Bloch theorem depending on the Blochwave vector Kand on the two-dimensional lattice vector of the periodic system R.For each micromagnetic cell δm(r) is expressed in polar coordinates depending on the angular 064416-3P. MALAG `Oet al. PHYSICAL REVIEW B 92, 064416 (2015) deviation from the equilibrium position of the azimuthal and polar angles δφk,δθk. In a compact form, the complex generalized Hermitian eigenvalue problem takes the form Av=λBv, (4) where the eigenvalue λ=1 ωwithωthe angular frequency of the given collective mode which is in turn described by theeigenvector v=(δφ k,δθk). The Hermitian matrix Adepends on saturation magnetization of the two ferromagnetic materialsand on the corresponding gyromagnetic ratios. The Hessianmatrix B is expressed in terms of the second derivatives of the energy density with respect to the azimuthal and polar angulardeviations δφ kandδθkcalculated at equilibrium. For further technical details of the DMM applied to several materials see Ref. [ 34]. The use of the DMM for calculating the spectrum of collective spin-wave modes is preferred with respect to theFourier analysis of OOMMF because it has several computa- tional advantages. Among them, just to mention a few, are(a) the system under study does not need to be excited byany magnetic field pulse; (b) the spin-wave modes frequenciesand eigenvectors of any symmetry are determined by meansa single calculation; (c) the spatial profiles of the spin-wavemodes are directly connected to the calculated eigenvectorsallowing to accurately classify each collective excitation;(d) the spectrum is computed directly in the frequency domain;(e) the mode degeneracy is completely taken into account;(f) the differential scattering cross section associated to eachspin-wave mode can be computed accurately starting from thecorresponding eigenvectors. The size of the micromagneticcells used in the static and dynamic simulations is 5 ×5× 10 nm along x,y, andz, respectively. In order to investigate the propagation properties of SWs in MCs, the systems have been studied in the DE geometry,i.e., with the external magnetic field ( H) of magnitude fixed at 2000 Oe parallel to the yaxis and the Bloch wave vector ( k) parallel to the xaxis. III. SPIN-WA VE EXCITATIONS IN MCS In 2D antidot lattices and bicomponent MCs a full magnonic spectrum is very rich with plenty of SW excita-tions [ 33]. As an example, the differential scattering cross section computed at the center of the BZ is displayed inFig. 2for S1. It can be seen that there is a large number of spin-wave modes resulting from the calculation. However, forthe purposes of this study focused on the dispersion behaviorin the first BZ only three modes belonging to the lowest-frequency part of the spectrum, namely, the ones exhibitingan appreciable differential scattering cross section, have beenselected in S1. Nevertheless, note that there are also othercollective modes in the highest-frequency part of the spectrumhaving a non-negligible differential scattering cross section,but in higher BZs. The same conclusions on the differentialscatting cross section can be drawn also for the other systems.The dispersion relations shown in Fig. 3are the ones measured in a typical BLS experiment [ 32,46]. The dispersion relations of SWs in S1 are shown in Fig. 3(a). We classify the collective modes by taking into10 12 14 16 18 200.00.10.20.3DE DEHRIntensity (arb. units) Frequency (GHz)EM FIG. 2. Differential scattering cross section calculated for S1 at the center of the BZ. The arrows label the modes with the highest scattering cross section in the center of the first BZ investigated in this paper. account the region inside the primitive cell where they have the maximum amplitude. In this respect, we named them(1) end mode of the dot (EM d) (where the subscript “d” means dot) mainly localized at the borders of the dots,(2) Damon-Eshbach–like mode in horizontal rows (DE HR) where the superscript “HR” means horizontal rows, and(3) Damon-Eshbach-like (DE) mode; they have frequencies9.94, 12.89, and 14.06 GHz, respectively. The modes (2)and (3) are called Damon-Eshbach–like because they exhibitnodal planes parallel to the local static magnetization in thehigher BZs and no nodal planes in the center of the BZ[see Fig. 4(a)]. This is in accordance to the classification of collective modes given for binary magnonic crystals [ 34]. In the center of the BZ, the DE HR, and DE are the resonance modes called fundamental modes. The DEHRmode is localized in the horizontal rows containing the square dots (withamplitude concentrated mainly in Py), while the DE modehas the maximum amplitude in Co dots and non-negligibleamplitude in the Py film. We point out that the end mode de-tected here has been previously found only in one-componentMCs [ 47,48]. The appearance of the end mode and the different SW amplitude distribution between Py and Co of DE and DE HR modes marks the difference between the S1 and the Co/Pybicomponent MC investigated in Refs. [ 14,34]. We remark that these differences with respect to previously studied systemsare mainly due to (a) the 10-nm-thick nonmagnetic spacerbetween Co dots and Py matrix placed at the bottom of thedots, and (b) the dot shape (these effects will be discussedin the next paragraph). Next, we study the effect of a fullseparation of Co dots from Py matrix on magnonic spectra.In Fig. 3(b), the dispersion curves for S2 Coare presented. By looking at Fig. 3(b) we can see the appearance of two new modes, i.e., the end mode of Py film (EM f) at 11.9 GHz (where the subscript “f” means film) localized at the borderof Py film and the backwardlike mode (BA HR) at 13.86 GHz mainly concentrated in the horizontal rows. The BAHRmode has nodal planes perpendicular to the local static magnetization[see Fig. 4(b) for profiles of the modes]. The frequency of the BA HRin S2Cois higher than the frequency of DEHR.T h i s 064416-4SPIN-WA VE DYNAMICS IN PERMALLOY/COBALT . . . PHYSICAL REVIEW B 92, 064416 (2015) 0.0 0.5 1.0101214 Wave vector, kx(/a)Frequency (GHz)EM dDEHRDE 0.0 0.5 1.068101214DEHR EM f EMdFrequency (GHz) Wave vector, kx(/a)BAHRDE 0.0 0.5 1.010121416Frequency (GHz) Wave vector, kx(/a)BAHR DE EM dEM fDEHR 0.0 0.5 1.010121416 Wave vector, kx(/a)DEBAHRDEHR EM fEMdFrequency (GHz) 0.0 0.5 1.048121620Frequency (GHz) Wave vector, kx(/a)DE EM(a) S1 (b) S2Co (c) S2Py (d) S3 (e) S4 FIG. 3. (Color online) Dispersion relation in the first BZ along the direction perpendicular to the external magnetic field.(a) Dispersion relation of the most relevant modes in S1: end mode of the dot (EM d), Damon-Eshbach–like mode (DE), and DE-like mode in horizontal rows (DEHR) are shown. (b) Dispersion relation in S2Co. The additional dispersion relation of the end mode in Py film (EM f) and backwardlike volume SW (BAHR)a r es h o w n . (c) Dispersion relation of the most relevant modes in S2Py. (d) Dispersion relation in S3. (e) Dispersion relation in the array of Co dots (S4). The black dashed lines in (b)–(d) mark dispersion relation of DE mode in homogeneous Py film of 10 nm thicknesscalculated according to Ref. [ 49]. EMd EMfDEHR BAHR DE EMdDEHRDES1 z= 5 nm z= 25 nmS2Co EMdEMfDEBAHRDEHR S2Py S3 EMfDE EMd BAHR DEHR EMDES4 +1.0 0.0 -1.0Amplitude (arb. units)(a) (b) (c) (d) (e)z= 5 nm z= 25 nm z= 5 nm z= 25 nmz= 5 nm z= 25 nm FIG. 4. (Color online) Spatial profiles (real part of the out-of- plane component of the dynamic magnetization vector) for SWswith large differential scattering cross section in the center of the Brillouin zone. The spatial profiles of SW modes from the bottom part of the Py film (in the plane z=5 nm in left column) and in the plane crossing dots (for z=25 nm in right column) are shown in 3×3 primitive cells, i.e., on the planes marked in Fig. 1with red dashed lines. (a) Spatial profiles of EM, DE HR, and DE modes in S1. (b) Spatial profiles of EM d,E M f,D EHR,B AHR, and DE modes in S2Co. (c) Spatial profiles of EM d,E M f,D E ,B AHR,a n dD EHRmodes in S2Py. (d) Spatial profiles of EM f, DE, EM d,B AHR,a n dD EHR modes in S3. (e) Spatial profiles of EM and DE modes in S4. might be attributed to the strong localization feature of the BAHRin the region filled by Co dots having higher values of the magnetic parameters. By comparing the frequency at the centerof the BZ passing from S1 to S2 Co, we observe a significant decrease of the EM dfrequency from 9.94 to 5.47 GHz and a slight increase of the DE (DEHR) frequencies from 14.06 GHz (12.89 GHz) to 14.67 GHz (13.48 GHz). The presence offive dispersion curves in S2 Cois attributed to the fact that the differential scattering cross section is comparable for the fiveSW excitations at the BZ center. In order to study the effect of the Py matrix on the SW excitation in Co dots, we calculate the dispersion curves of S4[Fig. 1(e)], the array composed of square Co dots. By inspec- tion of Fig. 3(e)we note that the frequency of the EM din S4 (3.5 GHz) is about 6 GHz lower than in S1 and 2.5 GHz lower 064416-5P. MALAG `Oet al. PHYSICAL REVIEW B 92, 064416 (2015) as compared to the corresponding one in S2Co. Instead, the frequency of the DE mode (18.4 GHz) is 4 GHz higher than theone in S1 and 3.5 GHz higher than the one in S2 Co. Therefore, the effect of Py matrix is to lower the frequencies of the DEmode and to raise the frequencies of the EM d. This behavior can be understood by taking into account the variation of themagnitude of the interdot dipolar dynamic coupling and of thestatic demagnetizing field passing from an array of dots (S4) toMCs (S1 and S2 Co) composed of two ferromagnetic materials. To study the effect of dot material and thickness in a Py matrix, we calculate the SW spectra of 2D MCs composed ofPy dots in a Py matrix. It is important to underline that S2 Py and S3 are neither a bicomponent MC nor antidot lattices, butthese structures preserve properties of both with the use ofa single ferromagnetic material. The kind of mode found inS2 Pyand S3 is similar to the one found in S2Co.I nS 2Py,t h e EM d(8.92 GHz) is the lowest-frequency mode as in S2Co.I n S2Pythe EM f(10.46 GHz) has a dispersion curve similar to that of EM d. The DE mode has a frequency of 12.8 GHz at the center of the BZ. The frequencies of BAHRmode (13.8 GHz) are lower than the ones of the DEHRmode (14.12 GHz). We observe that in S2Pythe frequency sequence of DE, DEHR, and BAHRmodes is different with respect to the one in S1 and S2Co[see Figs. 3(a)and3(b)]. In particular, the DE mode frequencies are lower than the DEHRmode ones as in the case of 2D one-component antidot lattices [ 47,48] (for further discussion see Sec. IV). In order to understand the effect of the thickness of Py dots we compute the dispersion curves for S3 shown in Fig. 3(d).I n S3, the EM f(8.92 GHz) is the lowest-frequency mode of the spectrum. The EM dfrequency at the center of the BZ (12.84 GHz) is larger than the one of the DE mode (12.36 GHz);however, the corresponding dispersion curves have a similarbehavior. This frequency inversion as compared to S2 Pyis not surprising because the total magnetic field experienced by the EM dis higher with respect to the field felt by the DE. The DEHRand BAHRhave frequencies 14.68 and 14.12 GHz at the center of the BZ, respectively. Comparing the dispersioncurves in S2 Coand S3, we observe that the order of DE and DEHRfrequency modes in S3 is interchanged with respect to the ones in S2Co. Moreover, also the frequency order of the EM f and the EM dis interchanged with respect to the one in S2Py and S2Co. This interchange can be attributed to the effect of the reduction of the dot thickness that induces a lowering of thetotal magnetic field in the Py film where the EM fis localized. The intensities of the differential scattering cross section ofthe DE, EM d,E M f, and BAHRmodes are comparable but are 40% lower than that of DEHR. In Fig. 4we show the spatial profiles of the real part of the out-of-plane component of the dynamic magnetization for themain modes at the center of the BZ of the systems studied.The spatial profiles are presented at planes z=5 nm and z=25 nm, left and right column of each panels, respectively, along the cross sections indicated in Fig. 1with red dashed lines. Looking at Fig. 4(a)we can see that the EM dis strongly localized at the border of the Co dots and its amplitudedecreases at z=5 nm where only Py is present with respect to z=25 nm. The presence of the Co dots in S1 induces a strong DE HRamplitude decrease inside the region containing the Co dots: indeed, for z<10 nm the amplitude of the DEHRmodeis uniform in the whole rows, while for 20 nm <z< 30 nm its amplitude decreases in the Co dots region. By contrast, forthe EM dthe square Co dots induce an opposite behavior. The amplitude distribution of the DE mode takes a contributionfrom both Co dots and Py matrix through its whole thickness.The DE is also the mode with largest differential scatteringcross section. Its intensity at k x=0 is three times larger than that of the EM dor the DEHRmode (see Fig. 2). Figure 4(b) displays the spatial profiles of the characteristic SW modesof S2 Co. The presence of the nonmagnetic spacer around the Co dots induces the appearance of the EM fthat is strongly localized at the border of the Py matrix close to thenonmagnetic spacer. The amplitude of this mode is almostuniform along the thickness, while that of the EM ddecreases by decreasing z.T h eD EHR,B AHR, and DE modes have uniform amplitude in the region of the Py matrix along thethickness. On the other hand, in the region filled by Co dotstheir amplitude strongly decreases for z>20 nm. In Fig. 4(c) we show the spatial profiles of the collective excitations in S2 Py. The amplitude variation of the EM d,D E , BAHR, and DEHRmodes as a function of zis the same as that in S2Co. Moreover, in S2Pythe amplitude of EM fdecreases by decreasing zfollowing a trend similar to that of the EM d. The amplitude of SW modes of S3 are illustrated in Fig. 4(d). Similarly to what occurs in S2Coand S2Py, the SW amplitude of the DE mode is almost homogeneous across the thicknessof the whole structure and larger in the rows between dots. TheDE HRand BAHRmodes’ amplitude is almost uniform along zin the Py matrix but decreases for z>20 nm in the region filled by Py dots. In Fig. 4(e)are depicted the spatial profiles of collective modes in S4. In this system there is only Co alongzand the amplitudes of EM and DE mode are uniform along the thickness. IV . TOTAL MAGNETIC FIELD ANALYSIS In order to understand the dispersion curves of the in- vestigated structures, we calculate the in-plane componentsof the total (effective) magnetic field at different values ofz. The total static magnetic field, which is the sum of the exchange field, the demagnetizing field, and the Zeeman field,calculated for each micromagnetic cell by the OOMMF code, is averaged along the xdirection for different values of z andy. The behavior of the total magnetic field is strictly related to the orientation of the static magnetization in themagnetic system. In Fig. 5four regions along the thickness are taken into account: (a) 0 nm /lessorequalslantz/lessorequalslant10 nm where only Py is present; (b) 10 nm <z/lessorequalslant20 nm where there are Py and a nonmagnetic spacer; (c) 20 nm <z/lessorequalslant30 nm where in S1 there are Py and Co, in S2 Cothere are Py, a nonmagnetic spacer, and Co, while in S2Pyand S3 there are Py and a nonmagnetic spacer; (d) 30 nm <z/lessorequalslant40 nm where in S1 and S2Cothere is Co, and in S2Pythere is Py. In particular, the presence of a well or a wall in the total magnetic field (see Fig. 5) is due to the saturation magnetization contrast present at interfacesbetween two different materials. Moreover, in MCs showingmagnetization inhomogeneities across the thickness, the totalmagnetic field at interfaces between two materials, present for10 nm <z< 30 nm, influences also collective excitations in the homogeneous part of the structure (for 0 nm <z< 10 nm). 064416-6SPIN-WA VE DYNAMICS IN PERMALLOY/COBALT . . . PHYSICAL REVIEW B 92, 064416 (2015) FIG. 5. (Color online) The ycomponent of the total magnetic field calculated for (a) S1, (b) S2Co,( c )S 2Py, and (d) S3 along the y axis and averaged along x, for four different values of z:z=5n m (in full Py film, black dot-dashed line), z=15 nm (crossing Py and spacer below the dots, red dashed line), z=25 nm (crossing Py matrix and middle of dots, green dotted line), and z=35 nm crossing Co dots (only in S1 and S2Co, blue solid line). The gray vertical rectangles mark the nonmagnetic spacers which separate the dot from the matrix. The insets on the top show a sketch of MCs with lines along whichthe total magnetic field is calculated. The appearance of end modes in MCs is related to the presence of a strong inhomogeneity of the total field resultingin deep wells close to the border of the dots and the matrix.This feature of the total magnetic field in 2D bicomponentMCs depends on two main factors: the shape of the dotand the contrast between the saturation magnetization of thedifferent materials. In particular, the magnetization saturationcontrast enhanced by the presence of the nonmagnetic spacerleads to the formation of an inhomogeneous demagnetizingfield and, as a consequence, to strong inhomogeneities ofthe total magnetic field at the border between two materials(Co/Py, Co/nonmagnetic spacer, and Py/nonmagnetic spacer).Therefore, the presence of a thin nonmagnetic spacer betweentwo ferromagnetic materials not only influences significantlythe SW spectra but can also be an end mode’s creatingfactor. We underline that this important feature, namely, theappearance of end modes, either as EM for EM d, does not depend on the dot shape or on the ferromagnetic materialfor MCs having geometric parameters in the range of theones typical of the recently studied bicomponent systems.Hence, this picture is different from the one occurring inbicomponent systems [ 14,34] where a crucial rule to determine the appearance of end modes was played by a specificcombination of the magnetization saturation contrast and thedot shape. As an example, in a bicomponent MC composed ofcircular Co (Py) dots in direct contact with a Py (Co) matrix,the end mode is present when |/Delta1M S|=|MS,Co−MS,Py|> 250 emu /cm3, but disappears when |/Delta1M S|=200 emu /cm3.Instead, if the bicomponent system is composed of square Co (Py) dots in direct contact with Py (Co) matrix, an endmode is present when |/Delta1M S|>200 emu /cm3. Therefore, a thin nonmagnetic spacer between the two ferromagneticcomponents of MCs not only influences significantly the SWspectra but also is an end mode’s creating factor. We underlinethat this important feature, namely, the appearance of endmodes, either as EM for EM d, does not depend on the dot shape or on the ferromagnetic material. In the following, wediscuss the shape of the total magnetic field in 2D bicomponentMCs introduced by nonmagnetic spacers around dots and itsrelation to the end modes. Figure 5(a)shows the total magnetic field calculated for S1 vs yfor different values of z.Two deep wells are present inside the region of the Co dot above thePy matrix corresponding to z>30 nm. The two wells are still present for 20 nm <z< 30 nm, although with decreasing depth. The two wells disappear for z<20 nm; however, the walls appear in this range. For this reason the EM dis strongly localized in the well of the total magnetic field at the border oft h eC od o tf o r z>20 nm and disappears in the homogeneous part of the system where there is the Py matrix ( z<20 nm) [see Fig. 4(a)]. In Fig. 5(b) is displayed the total magnetic field calculated for S2 Coas a function of yfor different values of z.I tc a n be seen that the positions of the minima of the total magneticfield depend on z. In particular, the total magnetic field has its minimum value in the Py region for z<20 nm, and in the Co region for z>20 nm. These two wells close to the border between Py and the nonmagnetic spacer and the nonmagneticspacer and Co give rise to the two localized modes EM fand EM d, respectively. Thus, the presence of these two end modes is strictly related to the nonmagnetic material that surroundsthe Co dots responsible for the appearance of the two minimain the total magnetic field. Comparing the profiles of the total field at z=15 nm and z=25 nm [red dashed and green dotted line in Figs. 5(a) and5(b)], an increase of the depth of the magnetic wells can be noted in S2 Cowith respect to the one in S1. This explains the decrease of the frequency of the EM din S2Co as compared to the one in S1. Moreover, the wells of the totalfield corresponding to the region filled by the Py matrix close tothe nonmagnetic spacer at z=15 nm, although less deep than the ones in the Co dot, are deep enough to permit localizationof the EM f. By looking at Fig. 5(a)it is also possible to understand that the variation of the total magnetic field due to the nonmagneticspacer induces a change of DE HRand DE mode profiles as a function of z. We observe that the uniform amplitude of DEHR in the horizontal rows [see Fig. 4(a)] is due to the trend of the total magnetic field. Indeed, by looking at Fig. 5(a)(black dot-dashed line), we note that the total magnetic field does notpresent significant inhomogeneities along the ydirection at z=5 nm. Instead, at z=25 nm the DE HRmode is localized only in the Py region [see Fig. 4(a)] and its amplitude vanishes inside the Co dot. On closer inspection of the correspondingtotal magnetic field [Fig. 5(a), green dotted line] we note the presence of a high wall at the border between Py and Cothat prevents the spreading of DE HRinside the Co dot. The DE mode has higher frequency than DEHRand its amplitude spreads also in Co dot for z>20 nm. In S2Co, there is an 064416-7P. MALAG `Oet al. PHYSICAL REVIEW B 92, 064416 (2015) TABLE I. Group velocity vgin the BZ center and bandwidth for EM d,E M f,D E ,a n dD EHRmodes in the MCs investigated in the paper. The two largest group velocities and bandwidths are emphasized in bold. S1 S2CoS2PyS3 S4 vg Bandwidth vg Bandwidth vg Bandwidth vg Bandwidth vg Bandwidth (m/s) (MHz) (m /s) (MHz) (m /s) (MHz) (m /s) (MHz) (m /s) (MHz) EM d 64 162 48 154 0 21 48 446 40 154 DEHR144 750 368 272 160 1378 160 668 – – DE 256 355 522 810 256 1097 152 410 68 203 EM f – – 80 226 48 49 48 173 – – increase of the total magnetic field inhomogeneity as compared to S1 for each value of z, apart from z>30 nm where there is a small reduction [see Figs. 5(a) and5(b)]. This results in an increase of the frequencies of the DE and DEHRmodes. In Fig. 5(c) we plot the total magnetic field for S2Pyin order to investigate the effect of the change of the materialfilling the dots. There are two minima of the total magneticfield: The absolute minimum is located in the Py matrix for10 nm <z< 20 nm and the other minimum is placed in the Py dot for 30 nm <z< 40 nm. In correspondence of the above- mentioned minima, also in S2 Pythere is the appearance of the EM dand of the EM f, respectively. By looking at Figs. 5(b)– 5(d), we can note a qualitative similarity of the behavior of the total magnetic field as a function of yin S2Co,S 2Py, and S3, respectively. In Fig. 5(d), where the Py dot thickness is 10 nm, the magnetic field well in the dot is less deep than the onein S2 Py, while in the Py matrix it has a significant minimum (green dotted line, z=25 nm). This explains the interchange of the frequencies of the EM fand EM dmodes found in S3 with respect to the ones in S2Coand S2Py. Detailed inspection of the total magnetic field profiles shown in Figs. 5(b)–5(d) allows us to notice also the relative change of the magneticfield values among S2 Co,S 2Py, and S3 in the channels parallel to the xaxis containing dots [i.e., area of the DEHRmode, for 100 nm <y< 300 nm in Fig. 5] and lying between the dots [i.e., area of the DE mode for 0 nm <y< 90 nm and 310 nm <y< 400 nm]. In the middle part of these areas the average value of the total magnetic field is almost constantacross the full thickness. In S2 Cothe values of the field are 2.06 and 1.85 kOe in the center of the areas of DE and DEHRmode, respectively, while in S3 the respective values are 1.82 and2.1 kOe. This behavior of the field can explain the frequencyexchange of the DE and DE HRmodes between S2Co,S 2Py, and S3 in Figs. 3(b)–3(d), respectively. V . FEATURES OF THE DISPERSION RELATION In order to fully understand the effect of different position and size of the nonmagnetic spacer on the propagation ofSWs, we compute the group velocity and the bandwidth forthe most relevant modes. The group velocity is important, e.g.,in the transmission measurements with the use of coplanarwaveguide transducers, where SWs with low wave number areusually excited [ 7]. A wide bandwidth is important in order to accommodate incoming and transmitted signal; moreover,it can be used as an indicator of the interaction strength in the MC. The group velocity ( v g) in the DE geometry has been calculated for selected modes close to the center of the BZ, as vg=2π/Delta1ν /Delta1kx, (5) where /Delta1νis the change of the SW frequency due to the change of the wave vector along the xaxis,/Delta1kx(in calculations we set /Delta1kx=0.05π/a). The bandwidth for the selected mode has been calculated as a change of its frequency between the BZcenter and the BZ border, /Delta1ν bw=|ν(kx=π/a)−ν(kx=0)|. The group velocity and the bandwidth of the investigated SWexcitations (EM d,E M f, DE, and DEHR) are calculated and collected in Table I. By looking at Table Iwe can see that for vanishing wave vector, the DE and DEHRmodes in S2Coexhibit the largest group velocities. These larger values of vgcan be attributed to a combination of higher contrast between Co and nonmagneticspacer and Py and nonmagnetic spacer and to a higher Cogyromagnetic ratio. This is an interesting result as S2 Cocan be regarded as the most disruptive structure with respect toa homogeneous thin film. The DE HRmodes in S1, S2Py, and S3 have similar group velocities, while the DE mode of S3has a group velocity smaller than the ones of the DE modesin S1 and S2 Py. The decrease of the group velocity in S3 can be due to the thickness reduction of the Py dots. These groupvelocities can be compared to that of the DE magnetostaticSW in homogeneous Py film of 10 nm thickness calculatedaccording to Eq. ( 5). In this special case the latter turns out to be 880 m/s, a value larger than the ones of the systems studiedas expected. The dispersion relation of the DE magnetostaticSW is superimposed in Figs. 3(b)–3(d) with a black dashed line. We can see that it matches very well with the DE modein S2 Coand the DEHRmodes in S2Pyand S3. This shows that the DE and DEHRmodes, in S2Co,S 2Py, and S3, respectively, propagate in a way similar to that of the DE magnetostatic SWin homogeneous Py film and they travel mainly in the lowerpart of the structure where the dots’ influence on the internalfield is smallest; nevertheless, it changes the group velocityand bandwidth. Comparing the group velocities of DE and DE HRmodes of S1, S2Co,S 2Py, S3, and S4 with the one of the DE magnetostatic SW mode in homogeneous Py film, it can benoted that the presence of two different magnetic materialsand a nonmagnetic spacer reduces the speed of propagation in 064416-8SPIN-WA VE DYNAMICS IN PERMALLOY/COBALT . . . PHYSICAL REVIEW B 92, 064416 (2015) the BZ center. This is probably due to the presence of different magnetic material and nonmagnetic spacer that induce the SWconfinement in particular regions of the primitive cell. The DE and DE HRmode of S2Pyhave the largest bandwidth. It is interesting to note that also the end modes with higherfrequency, EM fand EM din S2Coand S3, have a bandwidth comparable to that of the propagative DEHRand DE modes. This means that also the localized modes can propagate inthese kinds of MCs and their properties can be exploited fortransmitting a signal. VI. CONCLUSIONS Detailed theoretical investigations of the spin-wave spectra in two-dimensional bicomponent MCs with the DMM , in order to identify the influence of a nonmagnetic spacer on the magnonic band structure, have been performed. Square arrays of square grooves in thin Py film filled (or partially filled) withCo or Py square dots have been studied. The conclusions drawnfor these kinds of MCs can be generalized to other kinds of 2Dlattices and of different dot shapes in the nanometric range. Thenonmagnetic spacer breaks exchange interactions between themagnetic materials of the matrix and the dot. However, mostimportantly, this nonmagnetic spacer strongly modifies thetotal magnetic field, especially also at the dot edges. Due tothese changes of the magnetic field, two types of end modesappear in the same structure. These are the end mode localizedin the dot and that localized in the matrix. Their frequenciesstrongly depend on the magnetization of the matrix and of thedot material. Moreover, we have shown that, by employinga single material (Py in our case), it is possible to design aMC preserving the main properties of bicomponent MCs andmagnonic antidot lattices.We have also shown that the introduction of a nonmagnetic spacer and the change of the magnetic dot material allowus to tailor in different ways the SW spectra in MCs. Thisincludes even the interchange of the SW frequency order.This property can be further exploited for modeling themagnonic band structure and magnonic band gaps towardsthe properties desired for practical applications. Moreover, thenonmagnetic spacer breaks the exchange interaction at theborder between the two ferromagnetic materials and allowsthe fabrication of structures where magnetization reversal ofthe dots can take place at magnetic field values differentfrom those causing magnetization reversal in the matrix (dueto different shape or crystalline magnetic anisotropy). Here,there are more possibilities than in one-dimensional (1D)reprogrammable structures [ 50,51], because the anisotropy axis (and the magnetization) of the dots can be in an obliquedirection with respect to the magnetization of the matrix. The results of this study are interesting also for the investigation of the dynamical properties of bicomponent MCscomposed of hard and soft ferromagnetic materials, wherestray magnetic field originating from the dots (made of hardferromagnetic material) influences formation of the domainpattern [ 52] but SW dynamics has not been investigated so far in such structures. ACKNOWLEDGMENTS The research leading to these results has received funding from Polish National Science Centre Project No. DEC-2-12/07/E/ST3/00538, from the European Union’s Horizon2020 research and innovation programme under the MarieSkłodowska-Curie Grant Agreement No. 644348 (MagIC),and from MIUR-PRIN 2010–11 Project No. 2010ECA8P3“DyNanoMag.” [1] M. Krawczyk and H. Puszkarski, Acta Phys. Polon. A 93, 805 (1998). [2] S. A. Nikitov, P. Tailhades, and C. S. Tsai, J. Magn. Magn. Mater. 236,320(2001 ). [3] M. Krawczyk and H. Puszkarski, Phys. Rev. B 77,054437 (2008 ). [4] M. Mruczkiewicz, M. Krawczyk, R. V . Mikhaylovskiy, and V . V . Kruglyak, Phys. Rev. B 86,024425 (2012 ). [5] D. Kumar, J. W. Kłos, M. Krawczyk, and A. Barman, J. Appl. Phys. 115,043917 (2014 ). [6] Y . Au, E. Ahmad, O. Dmytriiev, M. Dvornik, T. Davison, and V . V . Kruglyak, Appl. Phys. Lett. 100,182404 (2012 ). [7] H. Yu, G. Duerr, R. Huber, M. Bahr, T. Schwarze, F. Brandl, and D. Grundler, Nat. 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PhysRevB.94.014412.pdf

PHYSICAL REVIEW B 94, 014412 (2016) Magnon spin transport driven by the magnon chemical potential in a magnetic insulator L. J. Cornelissen,1,*K. J. H. Peters,2G. E. W. Bauer,3,4R. A. Duine,2,5and B. J. van Wees1 1Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands 2Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 3Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai, Japan 4Kavli Institute of NanoScience, Delft University of Technology, Delft, The Netherlands 5Department of Applied Physics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (Received 13 April 2016; published 11 July 2016) We develop a linear-response transport theory of diffusive spin and heat transport by magnons in magnetic insulators with metallic contacts. The magnons are described by a position-dependent temperature and chemicalpotential that are governed by diffusion equations with characteristic relaxation lengths. Proceeding from alinearized Boltzmann equation, we derive expressions for length scales and transport coefficients. For yttriumiron garnet (YIG) at room temperature we find that long-range transport is dominated by the magnon chemicalpotential. We compare the model’s results with recent experiments on YIG with Pt contacts [L. J. Cornelissenet al. ,Nat. Phys. 11,1022 (2015 )] and extract a magnon spin conductivity of σ m=5×105S/m. Our results for the spin Seebeck coefficient in YIG agree with published experiments. We conclude that the magnon chemicalpotential is an essential ingredient for energy and spin transport in magnetic insulators. DOI: 10.1103/PhysRevB.94.014412 I. INTRODUCTION The physics of diffusive magnon transport in magnetic insulators, first investigated by Sanders and Walton [ 1], has been a major topic in spin caloritronics since the discoveryof the spin Seebeck effect (SSE) in YIG |Pt bilayers [ 2–4]. This transverse voltage generated in platinum contacts toinsulating ferromagnets under a temperature gradient can beexplained by thermal spin pumping caused by a temperaturedifference between magnons in the ferromagnet and electrons in the platinum [ 4–7]. The magnons and phonons in the bulk ferromagnet are considered as two weakly interactingsubsystems, each with their own temperature [ 1]. Hoffman et al. explained the spin Seebeck effect in terms of the stochastic Landau-Lifshitz-Gilbert equation with a noise termthat follows the phonon temperature [ 8]. Recently, diffusive magnon spin transport over large dis- tances has been observed in yttrium iron garnet (YIG) that wasdriven either electrically [ 9,10], thermally [ 9], or optically [ 11]. Notably, our observation of electrically driven magnon spintransport was recently confirmed in a Pt|YIG|Pt trilayergeometry [ 12,13]. Here, we argue that previous theories cannot explain these observations, and therefore do not capture the complete physics of magnon transport in magnetic insulators. We present arguments in favor of a nonequilibrium magnonchemical potential and work out the consequences for theinterpretation of experiments. Magnons are the elementary excitations of the magnetic order parameter. Their quantum mechanical creation andannihilation operators fulfill the boson commutation relationsas long as their number is sufficiently small. Just like photonsand phonons, magnons at thermal equilibrium are distributedover energy levels according to Planck’s quantum statistics fora given temperature T. This is a Bose-Einstein distribution *Corresponding author: l.j.cornelissen@rug.nlwith zero chemical potential because the energy and therefore magnon number is not conserved. Nevertheless, it is wellestablished that a magnon chemical potential can parametrizea long-living nonequilibrium magnon state. For instance, parametric excitation of a ferromagnet by microwaves gen-erates high-energy magnons that thermalize much faster bymagnon-conserving exchange interactions than their numberdecays [ 14]. The resulting distribution is very different from a zero-chemical potential quantum or classical distribution func-tion, but is close to an equilibrium distribution with a certaintemperature and nonzero chemical potential. The breakdownof even such a description is then indicative of the creation ofa Bose (or, in the case of pumping at energies much smallerthan the thermal one, Rayleigh-Jeans [ 15]) condensate. This new state of matter has indeed been observed [ 16]. Here, we argue that a magnon chemical potential governs spin and heattransport not only under strong parametric pumping, but also inthe linear response to weak electric or thermal actuation [ 17]. The elementary magnetic electron-hole excitations of nor- mal metals or spin accumulation have been a very fruitful concept in spintronics [ 18]. Since electron thermalization is faster than spin-flip decay, a spin-polarized nonequilibriumstate can be described in terms of two Fermi-Dirac distributionfunctions with different chemical potentials and temperaturesfor the majority and minority spins. We may distinguishthespin (particle) accumulation as the difference between chemical potentials from the spin heat accumulation as the difference between the spin temperatures [ 19]. Both are vectors that are generated by spin injection and governed by diffusionequations with characteristic decay times and lengths. Thespin heat accumulation decays faster than the spin particleaccumulation since both are dissipated by spin-flip scattering,while the latter is inert to energy exchanging electron-electron interactions. Here, we proceed from the premise that nonequilibrium states of the magnetic order can be described by a Bose-Einstein distribution function for magnons thatis parametrized by both temperature and chemical potential, 2469-9950/2016/94(1)/014412(16) 014412-1 ©2016 American Physical SocietyCORNELISSEN, PETERS, BAUER, DUINE, AND V AN WEES PHYSICAL REVIEW B 94, 014412 (2016) where the latter implies magnon number conservation. We therefore define a magnon heat accumulation δTmas the difference between the temperature of the magnons and thatof the lattice. The chemical potential μ mthen represents themagnon spin accumulation , noting that this definition differs from that by Zhang and Zhang [ 20], who define a magnon spin accumulation in terms of the magnon density.The crucial parameters are then the relaxation times governing the equilibration of δT mandμm. When the magnon heat accu- mulation decays faster than the magnon particle accumulation,previous theories for magnonic heat and spin transport shouldbe doubted [ 1,5–7,21]. The relaxation times are governed by the collision integrals that include inelastic (one-, two-, andthree-magnon scatterings involving phonons) and elastic two-and four-magnon scattering processes. At room temperature, two-magnon scattering due to disorder is likely to be negligibly small compared to phonon scattering. Four-magnon scatteringonly redistributes the magnon energies, but does not lead tomomentum or energy loss of the magnon system. Processesthat do not conserve the number of magnons are caused byeither dipole-dipole or spin-orbit interaction with the latticeand should be less important than the magnon-conserving onesfor high-quality magnetic materials such as YIG. At room temperature, the magnon spin accumulation is then essential to describe diffusive spin transport in ferromagnets. Here, we revisit the linear-response transport theory for magnon spin and heat transport, deriving the spin and heatcurrents in the bulk of the magnetic insulator as well asacross the interface with a normal metal contact. The magnontransport is assumed to be diffusive. Formally we are thenlimited to the regime in which the thermal magnon wavelength/Lambda1and the magnon mean-free path /lscript(the path length over which magnon momentum is conserved) are smaller than the systemsizeL. The wavelength of magnons in YIG is (in a simple parabolic band model) a few nanometers at room temperature.Boona et al. [22] find that /lscriptat room temperature is of the same order. As in electron transport in magnetic multilayers,scattering at rough interfaces is likely to render a diffusivepicture valid even when the formal conditions for diffusive bulktransport are not met. Under the assumptions that magnonsthermalize efficiently and that the mean-free path is dominatedby magnon-conserving scattering by phonons or structural andmagnetic disorder, we find that the magnon chemical potentialis required to harmonize theory and experiments on magnonspin transport [ 9]. This paper is organized as follows: We start with a brief review of diffusive charge, spin, and heat transport in metalsin Sec. II A. In Sec. II B, we derive the linear-response expressions for magnon spin and heat currents, starting fromthe Boltzmann equation for the magnon distribution function.We proceed with boundary conditions at the Pt |YIG interface in Sec. II C. In Sec. II D, we provide estimates for relaxation lengths and transport coefficients for YIG. The transportequations are analytically solved for a one-dimensional model(longitudinal configuration) in Sec. III A . In Sec. III B ,w e implement a numerical finite-element model of the experi-mental geometry and we compare results with experiments inSec. III C . We apply our model also to the (longitudinal) spin Seebeck effect in Sec. III D . A summary and conclusions are given in Sec. IV. Generation Absorption Platinum YIG PlatinumMx yz jsjsjcinjcoutjm FIG. 1. Schematic of the 1D geometry [ 13,20]. A charge current jin cis sent through the left platinum strip along +y. This generates a spin current js=jxz=θjin ctowards the YIG |Pt interface and a spin accumulation, injecting magnons into the YIG with spin polarizationparallel to the magnetization M. The magnons diffuse towards the right YIG |Pt interface, where they excite a spin accumulation and spin current into the contact. Due to the inverse spin Hall effect, thisgenerates a charge current j out calong the −ydirection. Note that if M is aligned along −z, magnons are absorbed at the injector and created at the detector. II. THEORY We first review the diffusion theory for electrical magnon spin injection and detection as published by one of usin [17,23]. By introducing the magnon chemical potential, this approach can disentangle spin and heat transport in contrastto earlier treatments based on the magnon density [ 20]o r magnon temperature [ 1,5–7] only. We initially focus on the one-dimensional (1D) geometry in Fig. 1with two normal metal (Pt) contacts to the magnetic insulator YIG. We expressthe spin currents in the bulk of the normal metal contactsand magnetic spacer, and the interface. While Ref. [ 17] focused on the chemical potential, here we include the magnontemperature as well. At low temperatures, the phonon specificheat has been reported to be an order of magnitude larger thanthe magnon one [ 22]. The room-temperature phonon mean- free path (that provides an upper bound for the phonon collisiontime) of a few nm [ 22] corresponds to a subpicosecond transport relaxation time for sound velocities of 10 3–104m/s. From the outset, we therefore take the phonon heat capacityto be so large and the phonon mean-free path and collisiontimes so short that the phonon distribution is not significantlyaffected by the magnons. The phonon temperature T pis assumed to be either a fixed constant or, in the spin Seebeckcase, to have a constant gradient. For simplicity, we alsodisregard the finite thermal (Kapitza) interface heat resistanceof the phonons [ 24]. A. Spin and heat transport in normal metals There is much evidence that spin transport in metals is well described by a spin diffusion approximation. Spin-flipdiffusion lengths of the order of nanometers reported in plat-inum betray the existence of large interface contributions [ 25], but the parametrized theory describes transport well [ 26]. The charge ( j c,α), spin ( jαβ), and heat ( jQ,α) current densities in 014412-2MAGNON SPIN TRANSPORT DRIVEN BY THE MAGNON . . . PHYSICAL REVIEW B 94, 014412 (2016) the normal metals, where the spin polarization is defined in the coordinate system of Fig. 1,a r eg i v e nb y( s e ee . g .[ 27]) jc,α=σe∂αμe−σeS∂αTe−σSH 2/epsilon1αβγ∂βμγ, 2e /planckover2pi1jαβ=−σe 2∂αμβ−σSH/epsilon1αβγ∂γμe−σSHSSN/epsilon1αβγ∂γTe, jQ,α=−κe∂αTe−σeP∂αμe−σSH 2PSN/epsilon1αβγ∂βμγ.(1) Here, μe,Te, and μαdenote the electrochemical potential, electron temperature, and spin accumulation, respectively.The subscripts α,β,γ ∈{x,y,z}are Cartesian components in the coordinate system in Fig. 1,αindicating current direction and βspin polarization. /epsilon1 αβγ is the Levi-Civita tensor and the summation convention is assumed throughout.The charge, spin, and heat current densities are measured inunits of A /m 2,J/m2, and W /m2,respectively, while both the electrochemical potential and the spin accumulation arein volts. The charge and spin Hall conductivities are σ eand σSH, both in units of S /m. Thermoelectric effects in metals are governed by the Seebeck coefficient Sand Peltier coefficient P=STe. Similarly, we allow for a spin Nernst effect via the coefficient SSNand the reciprocal spin Ettingshausen effect governed by PSN=SSNTe. We assume, however, that spin-orbit coupling is weak enough so that we can ignore spinswapping terms, i.e., terms of the form j αβ∼∂βμαand their Onsager reciprocal [ 28]. The spin heat accumulation in the normal metal and therefore spin polarization of the heat currentare disregarded for simplicity [ 19]./planckover2pi1and−eare Planck’s constant and the electron charge. The continuity equation∂ tρe+∇·je=0 expresses conservation of the electric charge density ρe. The electron spin μand heat Qeaccumulations relax to the lattice at rates /Gamma1sμand/Gamma1QT,respectively: ∂tsβ+1 /planckover2pi1∂αjαβ=− 2/Gamma1sμeμβν, (2) ∂tQe+∇·jQ=−/Gamma1QTCe(Te−Tp), (3) where the nonequilibrium spin density sβ=2eμβν,Ceis the electron heat capacity per unit volume, and νthe density of states at the Fermi level. Inserting Eq. ( 1) leads to the length scales /lscripts=/radicalbig σe/(4e2/Gamma1sμν) and/lscriptep=/radicalbig κe/(/Gamma1QTCe) govern- ing the decay of the electron spin and heat accumulations,respectively. At room temperature, these are typically /lscript Pt s= 1.5n m , /lscriptPt ep=4.5 nm for platinum [ 21,29], and /lscriptAu s=35 nm, /lscriptAu ep=80 nm for gold [ 21,30]. B. Spin and heat transport in magnetic insulators Magnonics traditionally focuses on the low-energy, long- wavelength regime of coherent wave dynamics. In contrast,the basic and yet not-well-tested assumption underlying thepresent theory is diffusive magnon transport, which we believeto be appropriate for elevated temperatures in which short-wavelength magnons dominate. Diffusion should be prevalentwhen the system size is larger than the magnon mean-free pathand magnon thermal wavelength (called magnon coherencelength in [ 5]). Magnons carry angular momentum parallel to the magnetization ( zaxis). Oscillating transverse components of the angular momentum can be safely neglected for systemsizes larger than the magnetic exchange length, which is on the order of 10 nm in YIG at low external magnetic fields [ 8]. Not much is known about the scattering mean-free path, but extrapolating the results from Ref. [ 22] to room temperature leads to an estimate of a few nm. Dipolar interactionsaffect mainly the long-wavelength coherent magnons thatdo not contribute significantly at room temperature. Thermalmagnons interact by strong and number-conserving exchangeinteractions. In the Appendix, the magnon-magnon scatteringrate is estimated as ( T/T c)3kBT//planckover2pi1[31,32] or a scattering time of 0.1 ps for YIG with Curie temperature Tc∼500 K at room temperature T=300 K, where T≈Tm≈Tp. According to the Landau-Lifshitz-Gilbert phenomenology [ 33], the magnon decay rate is αGkBT//planckover2pi1[32], with Gilbert damping constant αG≈10−4/lessmuch1 for YIG. Hence, the ratio between the scatter- ing rates for magnon-nonconserving to -conserving processesisα G(Tc/T)3/lessmuch1 at room temperature. These numbers justify the second crucial premise of the present formalism, viz., veryefficient, local equilibration of the magnon system. Since aspin accumulation in general injects angular momentum andheat at different rates, we need at least two parameters for themagnon distribution f, i.e., an effective temperature T mand a nonzero chemical potential (or magnon spin accumulation)μ min the Bose-Einstein distribution function nB: f(x,/epsilon1)=nB(x,/epsilon1)=/parenleftbig e/epsilon1−μm(x) kBTm(x)−1/parenrightbig−1, (4) where kBis Boltzmann’s constant. Both magnon accumu- lations Tm−Tpandμmvanish on, in principle, different length scales during diffusion. Assuming an isotropic (cubic)medium, the magnon spin current ( j m,i nJ/m2) and heat current densities ( jQ,m,i nW/m2) in linear response read as /parenleftBigg2e /planckover2pi1jm jQ,m/parenrightBigg =−/parenleftbigg σm L/T /planckover2pi1L/2eκ m/parenrightbigg/parenleftbigg ∇μm ∇Tm/parenrightbigg , (5) where μmis measured in volts, σmis the magnon spin conductivity (in units of S /m),Lis the (bulk) spin Seebeck coefficient in units of A /m, and κmis the magnonic heat conductivity in units of Wm−1K−1. Magnon-phonon drag contributions jm,jQ,m∝∇Tpare assumed to be absorbed in the transport coefficients since Tm≈Tp. The spin and heat continuity equations for magnon transport read as /parenleftBigg∂ρm ∂t+1 /planckover2pi1∇·jm ∂Qm ∂t+∇·jQ,m/parenrightBigg =−/parenleftbigg /Gamma1ρμ/Gamma1ρT /Gamma1Qμ/Gamma1QT/parenrightbigg/parenleftBigg μm∂ρm ∂μm Cm(Tm−Tp)/parenrightBigg , (6) in which ρmis the nonequilibrium magnon spin density and Qmthe magnonic heat accumulation. Cmis the magnon heat capacity per unit volume. The rates /Gamma1ρμand/Gamma1QTdescribe relaxation of magnon spin and temperature, respectively. Thecross terms (decay or generation of spins by cooling or heatingof the magnons and vice versa) are governed by the coefficients/Gamma1 ρTand/Gamma1Qμ. Equations ( 5) and ( 6) lead to the diffusion equations /parenleftbigg eα μkB eαT/kB 1/parenrightbigg/parenleftbigg ∇2μm ∇2Tm/parenrightbigg =/parenleftBigg e//lscript2 m kB/(/lscriptρTT2) e//parenleftbig kB/lscriptQμμ2 m/parenrightbig 1//lscript2 mp/parenrightBigg/parenleftbigg μm Tm−Tp/parenrightbigg ,(7) 014412-3CORNELISSEN, PETERS, BAUER, DUINE, AND V AN WEES PHYSICAL REVIEW B 94, 014412 (2016) Te TmTp μsμmNM FI lslmleplmp ∆μ∆Tme jxzM FIG. 2. Length scales at normal metal |ferromagnetic insulator (NM|FI) interfaces in Fig. 1. Assuming a constant gradient of the phonon temperature Tpand disregarding Joule heating, the electron temperature Teand magnon temperature Tmr e l a xo nl e n g t hs c a l e s /lscriptep and/lscriptmp. A significant phonon heat (Kapitza) resistance would cause a step in Tpat the interface. The spin Hall effect in the normal metal drives a spin current jxztowards the interface, which will be partially transmitted to the magnon system (causing a nonzeromagnon chemical potential in the FI) and partially reflected back into the NM (causing a nonzero electron spin accumulation in the NM). The electron spin accumulation μ s=μzand the magnon chemical potential μmrelax on length scales /lscriptsand/lscriptm, respectively. with four length scales and two dimensionless ratios. Here, /lscriptm=/radicalBig σm/(2e/Gamma1ρμ)(∂ρm ∂μm)−1is the magnon spin diffusion length (or relaxation length of the magnon chemical potential) and/lscriptmp=/radicalbig κm/(/Gamma1QTCm) is the magnon-phonon relaxation length that governs the relaxation of the magnon tempera-ture. The equilibrium values for magnon chemical potentialand magnon temperature are μ m=0 and Tm=Tp(see Fig. 2). The length scales /lscriptρT=/radicalbig kBσm/(2e2/Gamma1ρTCm) and /lscriptQμ=/radicalBig eκm/(/planckover2pi1kB/Gamma1Qμ)(∂ρm ∂μm)−1arise from the nondiagonal cross terms. The dimensionless ratio αμ=eL/(kBσmTp)i s a measure for the relative ability of chemical-potential andtemperature gradients to drive spin currents. Similarly, α T= /planckover2pi1kBL/(2eκm) characterizes the magnon heat current driven by chemical potential gradients relative to that driven bytemperature gradients. C. Interfacial spin and heat currents The electron and magnon diffusion equations are linked by interface boundary conditions. Spin currents andaccumulations are parallel to the magnetization direction ofthe ferromagnet along the zdirection. We assume that the exchange coupling dominates the coupling between electronsand magnons across the interface. A perturbative treatmentof the exchange coupling at the interface leads to the spincurrent [ 34,35] j int s=−/planckover2pi1g↑↓ 2e2πs/integraldisplay d/epsilon1D (/epsilon1)(/epsilon1−eμz) ×/bracketleftbigg nB/parenleftbigg/epsilon1−eμm kBTm/parenrightbigg −nB/parenleftbigg/epsilon1−eμz kBTe/parenrightbigg/bracketrightbigg , (8) where g↑↓is the real part of the spin-mixing conductance in S/m2,s=S/a3the equilibrium spin density of the magnetic insulator, and S is the total spin in a unit cell with volume a3. The density of states of magnons D(/epsilon1)=√/epsilon1−/Delta1/(4π2J3/2 s) for a dispersion /planckover2pi1ωk=Jsk2+/Delta1. The spin-wave gap /Delta1is governed by the magnetic anisotropy and the applied magneticfield. In soft ferromagnets such as YIG /Delta1∼1 K, which we disregard in the following since we focus on effects at roomtemperature (see e.g. Ref. [ 8]). The heat current is given by inserting /epsilon1//planckover2pi1into the integrand of Eq. ( 8). Linearizing the above equation, we find the spin and heat currents across the interface [ 17] /parenleftBigg j int s jint Q/parenrightBigg =3/planckover2pi1g↑↓ 4e2πs/Lambda13/parenleftBiggeζ(3/2)5 2kBζ(5/2) 5 2ekBT /planckover2pi1ζ(5/2)35 4k2 BT /planckover2pi1ζ(7/2)/parenrightBigg ×/parenleftbigg μz−μm Te−Tm/parenrightbigg . (9) /Lambda1=√4πJs/(kBT) is the magnon thermal (de Broglie) wavelength (the factor 4 πis included for convenience). These expressions agree with those derived from a stochasticmodel [ 5] after correcting numerical factors of the order of unity. In YIG at room temperature /Lambda1∼1n m .T h et e r m proportional to μ zcorresponds to the spin transfer (absorption of spin current by the fluctuating magnet), while thatproportional to μ mis the spin pumping contribution (emission of spin current by the magnet). The prefactor ∼1/(s/Lambda13) can be understood by noting that s/Lambda13is the effective number of spins in the magnetic insulator that has to be agitated andappears in the denominator of Eq. ( 9) as a mass term. In the macrospin approximation, this term would be replaced by thetotal number of spins in the magnet. From Eq. ( 9) we identify the effective spin conductance g sthat governs the transfer of spin across the interface by the chemical potential difference /Delta1μ=μz−μm. In units of S/m2, gs=3ζ/parenleftbig3 2/parenrightbig 2πsg↑↓ /Lambda13. (10) Using the material parameters for YIG from Table IIand the expression for the thermal de Broglie wavelength given above,we find g s=0.06g↑↓at room temperature [ 21,36].gsscales with temperature like ∼(T/T c)3/2, but it should be kept in mind that the theory is not valid in the limits T→TcandT→0. It is nevertheless consistent with the recently reported strongsuppression of g sat low temperatures [ 10,13]. D. Parameters and length scales In this section, we present expressions for the transport parameters derived from the linearized Boltzmann equationfor the magnon distribution function and present numericalestimates based on experimental data. 014412-4MAGNON SPIN TRANSPORT DRIVEN BY THE MAGNON . . . PHYSICAL REVIEW B 94, 014412 (2016) TABLE I. Transport coefficients and length scales [ 17] as derived in the Appendix. Symbol Expression Magnon thermal de Broglie wavelength /Lambda1√4πJs/(kBT) Magnon spin conductivity σm 4ζ(3/2)2e2Jsτ/(/planckover2pi12/Lambda13) Magnon heat conductivity κm35 2ζ(7/2)Jsk2 BTτ/(/planckover2pi12/Lambda13) Bulk spin Seebeck coefficient L 10ζ(5/2)eJskBTτ/(/planckover2pi12/Lambda13) Magnon thermal velocity vth 2√JskBT//planckover2pi1 Magnon spin diffusion length /lscriptm vth/radicalBig 2 3ττmr Magnon-phonon relaxation length /lscriptmp vth/radicalBig 2 3τ(1/τmr+1/τmp)−1 Magnon spin-heat relaxation length /lscriptρT /lscriptm/√αμ Magnon heat-spin relaxation length /lscriptQμ /lscriptm/√αT αμ5 2ζ(5/2)/ζ(3/2) αT2 7ζ(5/7)/ζ(7/2) 1. Boltzmann transport theory Magnon transport as formulated in the previous section is governed by the transport coefficients σm,L,κm, four length scales /lscriptm,/lscriptmp,/lscriptρT, and /lscriptQμ, and two dimension- less numbers αμandαT. In the Appendix, we derive these parameters using the linearized Boltzmann equationin the relaxation time approximation. We consider fourinteraction events: (i) elastic magnon scattering by bulkimpurities or interface disorder, (ii) magnon dissipation bymagnon-phonon interactions that annihilate or create spinwaves and/or inelastic scattering of magnons by magneticdisorder, (iii) magnon-phonon interactions that conserve thenumber of magnons, and (iv) magnon-magnon scattering bymagnon-conserving exchange scattering processes (see alsoSec. II B). The magnon energy and momentum-dependent scattering times for these processes are τ el,τmr,τmp, andτmm.At elevated temperatures they should be computed at magnon energyk BTand momentum /planckover2pi1//Lambda1. Magnon-magnon interactions that conserve momentum do not directly affect transport currentsin our single magnon band model, so the total relaxation rateis 1/τ=1/τ el+1/τmr+1/τmp. The transport coefficients and length scales derived in the Appendix are summarized in Table I. The Einstein relation σm=2eDm∂ρm//planckover2pi1∂μmconnects the magnon diffusion constant Dmdefined by jm=−Dm∇ρmwith the magnon con- ductivity, where ∂ρm/∂μm=eLi1/2(e−/Delta1/k BT)/(4π/Lambda1J s) and Lin(z) is the polylogarithmic function of order n. We observe that the magnon-phonon relaxation length /lscriptmp is smaller than the magnon spin diffusion length /lscriptmsince the latter is proportional to τmr, whereas /lscriptmpis limited by both magnon-conserving and -nonconserving scattering processes.Furthermore, 1 /τ mrcan be estimated by the Landau-Lifshitz- Gilbert equation as ∼αGkBT//planckover2pi1[32], where the Gilbert constant αGat thermal energies is not necessarily the same as for ferromagnetic resonance. 2. Clean systems In the limit of a clean system, 1 /τel→0. At sufficiently low temperatures, the magnon-conserving magnon-phononscattering rate 1 /τ mp∼T3.5[37] (see also the Appendix)loses against 1 /τmr∼αGkBT//planckover2pi1sinceαGis approximately temperature independent. Then, all lengths ∼/Lambda1/α G∼10μm for YIG at room temperature and with αG=10−4from ferromagnetic resonance (FMR) [ 8]. The agreement with the observed signal decay [ 9] is likely to be coincidental, however, since the spin waves at thermal energies have a much shorterlifetime than the Kittel mode for which α Gis measured. σm estimated using the FMR Gilbert damping is larger than the experimental value by several orders of magnitude, which isa strong indication that the clean limit is not appropriate forrealistic devices at room temperature. 3. Estimates for YIG at room temperature The phonon and magnon inelastic mean-free paths derived from the experimental heat conductivity appear to be almostidentical at low temperatures up to 20 K [ 22] but could not be measured at higher temperatures. Both are likelyto be limited by the same scattering mechanism, i.e., themagnon-phonon interaction. We assume here that the magnon-phonon scattering of thermal magnons at room temperatureis dominated by the exchange interaction (which alwaysconserves magnons) rather than the magnetic anisotropy(which may not conserve magnons) [ 38]. Then, τ∼τ mpand extrapolating the low-temperature results to room temperatureleads to an /lscript mpof the order of a nm, in agreement with an analysis of spin Seebeck [ 6] and Peltier [ 21] experiments. The associated time scale τmp∼1–0.1 ps is of the same order asτmmestimated in Sec. II B. On the other hand, τmr∼1n s fromαG∼10−4and therefore /lscriptm∼vth√τmpτmr∼0.1–1μm. The observed magnon spin transport signal decays over asomewhat longer length scale ( ∼10μm). Considering that the estimated τ mris an upper limit, our crude model apparently overestimates the scattering. An important conclusion is,nonetheless, that /lscript m/greatermuch/lscriptmp, which implies that the magnon chemical potential carries much farther than the magnontemperature. Withτ∼τ mp∼0.1–1 ps we can also estimate the magnon spin conductivity σ∼e2Jsτ//planckover2pi12/Lambda13∼105–106S/m, in rea- sonable agreement with the value extracted from our experi-ments (see next section). 014412-5CORNELISSEN, PETERS, BAUER, DUINE, AND V AN WEES PHYSICAL REVIEW B 94, 014412 (2016) III. HETEROSTRUCTURES Here, we apply the model, introduced and parametrized in the previous section, to concrete contact geometries andcompare the results with experiments. We start with ananalytical treatment of the one-dimensional geometry, fol-lowed by numerical results for the transverse configurationof top metal contacts on a YIG film with finite thickness.Throughout, we assume, motivated by the estimates presentedin the previous section, that the magnon-phonon relaxationis so efficient that the magnon temperature closely followsthe phonon temperature, i.e., T m=Tp(only in Sec. III C 3 we study the implications of the opposite case, i.e., Tm/negationslash=Tp andμm=0). This allows us to focus on the spin diffusion equation for the chemical potential μm. This approximation should hold at room temperature, while the opposite regime/lscript mp/greatermuch/lscriptmmight be relevant at low temperatures or high magnon densities: when the magnon chemical potential ispinned to the band edge, transport can be described in terms ofthe effective magnon temperature. The intermediate regime/lscript mp∼/lscriptm, in which both magnon chemical potential and effective temperature have to be taken into account, is leftfor future study. A. One-dimensional model We consider first the one-dimensional geometry shown in Fig. 1. We focus on strictly linear response and therefore disregard Joule heating in the metal contacts as well asthermoelectric voltages by the spin Nernst and Ettingshauseneffects. The spin and charge currents in the metal are thengoverned by /parenleftBigg j c 2e /planckover2pi1js/parenrightBigg =/parenleftbigg σe−σSH −σSH−σe/parenrightbigg/parenleftBigg ∂yμe 1 2∂xμz/parenrightBigg , (11) where the charge transport is in the ydirection, spin transport in thexdirection, and the electron spin accumulation is pointing in the zdirection. The spin and magnon diffusion equations reduce to ∂2μs ∂x2=μz /lscript2s, (12) ∂2μm ∂x2=μm /lscript2m. (13) The interface spin currents ( 8) provide the boundary conditions at the interface to the ferromagnet, while all currents at thevacuum interface vanish. Equations ( 9) and ( 10) lead to the interface spin current density j int s=gs(μint z−μint m), where gs is defined in Eq. ( 10). 1. Current transfer efficiency The nonlocal resistance Rnlis the voltage over the detector divided by current in the injector, also referred to as nonlocalspin Hall magnetoresistance (see below). The magnon spininjection and detection can also be expressed in terms ofthe current transfer efficiency η, i.e., the absolute value of the ratio between the currents in the detector and injectorstrip [ 20] when the detector circuit is shorted. η=R nl/R0 for identical Pt contacts with resistance R0.I nF i g . 3,w elmlm FIG. 3. The current transfer efficiency η(nonlocal resistance normalized by that of the metal contacts) as a function of distance between the contacts in a Pt |YIG|Pt structure calculated in the 1D model. Parameters are taken from Table IIand the Pt thickness t=10 nm. The dashed lines are plots of the functions C1/d(red dashed line) and C2exp (−d//lscriptm) (blue dashed line) to show the different modes of signal decay in different regimes: diffusive 1 /d decay for d</lscript mand exponential decay for d>/lscript m. The constants C1 andC2were chosen to show overlap with ηfor illustrative purposes, but have no physical meaning. plot the calculated ηas a function of distance dbetween the contacts for a Pt thickness t=10 nm and parameters from Table II.ηdecays algebraically ∝1/dwhend/lessmuch/lscriptm,which implies diffusion without relaxation, and exponentially ford/greatermuch/lscript m. The calculated order of magnitude already agrees with experiments [ 9]. The η/primesi nR e f .[ 20] are three orders of magnitude larger than ours due to their much weakerrelaxation. TABLE II. Selected parameters for spin and heat transport in bilayers with magnetic insulators and metals. a, S, and Jsare adopted from [ 39],/lscriptsandθfrom [ 21,29], and σeis extracted from electrical measurements on our devices [ 9]. Note that our values for σeand /lscriptsare consistent with Elliot-Yafet scattering as the dominant spin relaxation mechanism in platinum [ 40]. The mixing conductance, magnon spin diffusion length, and the magnon spin conductivity are estimated in the main text. Symbol Value Unit YIG lattice constant a 12.376 ˚A Spin quantum number per YIG S 10 unit cell Spin-wave stiffness constant in YIG Js 8.458×10−40Jm2 YIG magnon spin diffusion length /lscriptm 9.4 μm YIG spin conductivity σm 5×105S/m Real part of the spin-mixing g↑↓1.6×1014S/m2 conductance Platinum conductivity σe 2.0×106S/m Platinum spin relaxation length /lscripts 1.5n m Platinum spin Hall angle θ 0.11 014412-6MAGNON SPIN TRANSPORT DRIVEN BY THE MAGNON . . . PHYSICAL REVIEW B 94, 014412 (2016) FIG. 4. Experimental spin Hall magnetoresistance (SMR) as a function of platinum strip width. The black squares (left axis) showabsolute resistance changes /Delta1R SMR divided by the device length (18μm) in units of /Omega1/m. The red dots (right axis) show the relative resistivity changes /Delta1ρ/ρ . The origin of the small ηis the inefficiency of the spin Hall mediated spin-charge conversion. The ratio between the spin accumulations in injector and detector ηs=μdet s/μinj sis much larger than ηand discussed in Sec. III C 2 . 2. Spin Hall magnetoresistance The effective spin conductance gsgoverns the amount of spin transferred across the interface between the normal metaland the magnetic insulator. While g scannot be extracted from measurements directly, it is related to the spin-mixing conduc-tanceg ↑↓via Eq. ( 10). In order to determine g↑↓we measured the spin Hall magnetoresistance (SMR) [ 41,42] in devices of Ref. [ 9]. The SMR is defined as the relative resistivity change in the Pt contact between in-plane magnetization parallel andnormal to the current /Delta1ρ/ρ . The expression for the magnitude of the SMR reads as [ 43] /Delta1ρ ρ=θ2/lscripts t2/lscriptsg↑↓tanh2t 2/lscripts σe+2/lscriptsg↑↓cotht /lscripts, (14) where t=13.5 nm is the platinum thickness. Figure 4shows the experimental SMR as a function of platinum strip width.As expected /Delta1ρ/ρ =(2.6±0.09)×10 −4does not depend on the strip width. Using Eq. ( 14) and the values for /lscripts,θ, and σeas indicated in Table II, we find g↑↓=(1.6±0.06)×1014 S/m2,which agrees with previous reports [ 29,42,44]. In Chen et al. ’s zero-temperature theory [ 43]t h es p i n current generated by the spin Hall effect in Pt is perfectlyreflected when spin accumulation and magnetization arecollinear. As discussed above, at finite temperature a fraction ofthe spin current is injected into the ferromagnet in the form ofmagnons. This implies that the SMR should be a monotonouslydecreasing function of temperature. This has been found forhigh temperatures [ 45], but the decrease of the SMR at low temperatures [ 46] hints at a temperature dependence of other parameters such as the spin Hall angle. The current transfer efficiency ηcan be interpreted as a nonlocal version of the SMR [ 10]. The SMR is caused bythe contrast in spin current absorption of the YIG |Pt interface when the spin accumulation vector is normal or parallel tothe magnetization M. In the nonlocal geometry, we measure the voltage in contact 2 that has been induced by a chargecurrent (in the same direction) in contact 1. Since g s<g↑↓,t h e relation |/Delta1ρ/ρ|/greaterorequalslantηmust hold even in the absence of losses in the ferromagnet and detector. This indeed agrees with ourdata. 3. Interface transparency The analytical expression for ηin the one-dimensional geometry is lengthy and omitted here, but it can be simplifiedfor special cases. In the limit of a large bulk magnonspin resistance, the interface resistance can be disregarded.The decay of the spin current is then dominated by thebulk spin resistance and relaxation of both materials. Whenσ m//lscriptm,σe//lscripts/lessmuchgs η=θ2/lscriptmσeσm t/bracketleftbig σ2m+/parenleftbig/lscriptm /lscripts/parenrightbig2σ2e/bracketrightbigsinh−1d /lscriptm, (15) where the Pt thickness is chosen t/greatermuch/lscriptsandθ=σSH/σe is the spin Hall angle. When d/lessmuch/lscriptmwe are in the purely diffusive regime with algebraic decay η∝1/d. Exponential decay with characteristic length /lscriptmtakes over when d/greaterorsimilar/lscriptm. In our experiments (see Table II)σm∼σeand/lscriptm/greatermuch/lscripts,s o η=θ2/lscript2 sσm /lscriptmtσesinh−1d /lscriptm. (16) On the other hand, when σm//lscriptm,σe//lscripts/greatermuchgsthe interfaces dominate and η=θ2g2 s/lscript2s/lscriptm tσeσmsinh−1d /lscriptm, (17) with identical scaling with respect to d, but a different prefactor. According to the parameters in Table IIσm//lscriptm/greatermuch σe//lscripts/greatermuchgs, so spin injection is limited by the interfaces due to the small spin conductance between YIG and platinum. B. Two-dimensional geometry Experiments are carried out for Pt |YIG|Pt with a lateral (transverse) geometry in which the platinum injector anddetector are deposited on a YIG film. The two-dimensionalmodel sketched in Fig. 5captures this configuration but cannot be treated analytically. We therefore developed a finite-elementimplementation of our spin diffusion theory by the COMSOL MULTIPHYSICS (version 4.3a) software package, extending the description of spin transport in metallic systems [ 47]t o magnetic insulators. The finite-element simulations of the spinSeebeck [ 6] and spin Peltier [ 21]e f f e c t si nP t |YIG focused on heat transport and were based on a magnon temperaturediffusion model. Here, we find that neglecting the magnonchemical potential underestimates spin transport by ordersof magnitude because the magnon temperature equilibratesat a length scale /lscript mpof a few nanometers and the magnon heat capacity and heat conductivity are small [ 22]. The magnon chemical potential and the associated nonequilibriummagnons, on the other hand, diffuse on the much longer lengthscale/lscript m. 014412-7CORNELISSEN, PETERS, BAUER, DUINE, AND V AN WEES PHYSICAL REVIEW B 94, 014412 (2016) Detector YIGInjector Interface layerμsDetectordrightdleft tintt tYIGw w w xz y MInterface layerjzxleftjzxright Interface layer FIG. 5. Schematic of the 2D geometry. The relevant dimensions are indicated in the figure. The spin accumulation arising from the charge current through the injector μsis used as a boundary condition on the YIG |Pt interface. The interface layer is used to account for the effect of finite spin-mixing conductance between YIG and platinum. In order to model the experiments in two dimensions, we assume translational invariance in the third direction, whichis justified by the large aspect ratio of relatively small contactdistances compared with their length. With equal magnon andphonon temperatures everywhere, the magnon transport in twodimensions is governed by 2e /planckover2pi1jm=−σm∇μm, ∇2μm=μm /lscript2m, (18) where ∇=x∂x+z∂z. The particle spin current js=(jxx,jzx)i nt h em e t a li s described by 2e /planckover2pi1js=−σe 2∇μx, ∇2μx=μx /lscript2s, (19) where μxis the xcomponent of the electron spin accumu- lation. The spin-charge coupling via the spin Hall effect isimplemented by the boundary conditions in Sec. III B 2 , while the inverse spin Hall effect is accounted for in the calculationof the detector voltage (see Sec. III B 5 ). The estimates at the end of the previous section justify disregarding temperatureeffects. 1. Geometry In order to accurately model the experiments, we define two detectors (left and right) and a central injector, introducing thedistances d leftanddrightas in Fig. 5. We generate a short- (A) and a long-distance (B) geometry. The injector and detectorsare slightly different as summarized in Table III.T h eY I Gfi l m thicknesses are 200 nm for (A) and 210 nm for (B). The YIGfilm is chosen to be long compared to the spin diffusion length(w YIG=150μm) in order to prevent finite-size artifacts. TABLE III. Properties of geometry sets A and B. Pt width Pt thickness Distances w(nm) t(nm) d(μm) Geometry A 140 13 .50 .2–5 Geometry B 300 7 2–42 .52. Boundary conditions Sending a charge current density jcin the +ydirection through the platinum injector strip generates a spin accumula-tionμ sat the YIG |platinum interface by the spin Hall effect (shown in Fig. 5). This is captured by Eq. ( 1) that predicts a spin accumulation at the Pt side of the interface of [ 21] μs≡μx|interface =2θjc/lscripts σetanh/parenleftbiggt 2/lscripts/parenrightbigg , (20) which is used for the interface boundary condition of the magnon diffusion equation. Here, we assume that the contactwith the YIG does not significantly affect the spin accumu-lation [ 43], which is allowed for the collinear configuration since g s<σe//lscripts. The spin orientation of μspoints along −x, parallel to the YIG magnetization. A charge current I=100μA generates spin accumulations in the injector contact of μA s=8.7μV and μB s=7.7μV for geometries A and B, respectively. The uncovered YIG surface is subject to a zero current boundary condition ( ∇·n)μs=0, where nis the surface normal. 3. YIG |Pt interface The interface spin conductance gsis modeled by a thin interface layer, leading to a spin current jint s=−σint s∂μx/∂z, with spin conductivity σint s=gstint. When the interface thick- nesstintis small compared to the platinum thickness tPtwe can accurately model the Pt |YIG interface without having to change the COMSOL code. Varying the auxiliary interface layer thickness between 0 .5<t int<2.5 nm, the spin currents change by only 0 .1%. This is expected because the increased interface layer thickness is compensated by the reducedresistivity of the interface material such that the resistanceremains constant. In the following, we adopt t int=1.0n m . Finally, with Eq. ( 10)gs=0.06g↑↓andg↑↓from Sec. III A 2 we get gs=9.6×1012S/m2. 4. Magnon chemical potential profile A representative computed magnon chemical potential map is shown in Fig. 6(a), while different profiles along the three indicated cuts are plotted in Figs. 6(b)–6(d). The magnon chemical potential along xand at z=− 1 nm (i.e., 1 nm below the surface of the YIG) in Fig. 6(b) is characterized by the spin injection by the center electrode. Globally, μmdecays exponentially with distance from the injector on the scale of /lscriptm. 014412-8MAGNON SPIN TRANSPORT DRIVEN BY THE MAGNON . . . PHYSICAL REVIEW B 94, 014412 (2016) μm (μV) 059 1234678 −15 −10 −5 0 5 10 1502468 x (μm)μm (μV)Linecut along x −200 −100 00510 z (nm)μm (μV)Injector linecut −200 −100 00123 z (nm)μm (μV)Left detector linecut(a) (b) (d) (c)x (nm)00 04 002- 002 004-z (nm) -200015213 FIG. 6. (a) Two-dimensional magnon chemical potential distribu- tion for geometry (A) with dleft=200 nm and dright=300 nm. The lines numbered 1, 2, 3 indicate the locations of the profiles plotted in figures (b), (c), (d), respectively. In (b) we observe a maximumμ mforx=0, i.e., under the injector, followed by a sharp decrease close to the detectors located at x=− 200 and 300 nm because the Pt contacts are efficient (but not ideal) spin sinks. On the outer sides ofthe detectors μ mpartially recovers with distance and finally decays exponentially on the length scale /lscriptm. We also observe that the left and right detector contacts at x= −200 nm and 300 nm, respectively, act as sinks that visibly suppress but do not quench the magnon accumulation. Thefinite mixing conductance and therefore magnon absorptionare also evident from the profiles along zin Figs. 6(c) and6(d):the magnon chemical potential changes abruptly across the YIG|Pt interface by the relatively large interface resistance g −1 s. The magnon chemical potential is much smaller than the magnon gap ( ∼1 K). We are therefore far from the threshold for current-driven instabilities such as magnon condensationand/or self-oscillations of the magnetization [ 32]. 5. Detector contact and nonlocal resistance The spin current density in the detectors is governed by the spin accumulation according to /angbracketleftjzx/angbracketright=−σe 2A/integraldisplay A∂μx ∂zdA/prime, (21) which is an average over the detector area A=wt.T h e observable nonlocal resistance Rnl(normalized to device length) in units of /Omega1/m, Rnl=θ/angbracketleftjzx/angbracketright σeI, (22) is compared with experiments in the next section. C. Comparison with experiments 1. Two-dimensional model Figure 7compares the simulations as described in the previous section with our experiments [ 9]. Figure 7(a) is a linear plot for closely spaced Pt contacts while Fig. 7(b) shows the results for all contact distances on a logarithmic scale. Themagnon spin conductivity σ mand the magnon spin diffusion length /lscriptmare adjustable parameters; all others are listed in Table II. We adopted σm=5×105S/m and /lscriptm=9.4μma s the best fit values that agree with the estimates in Ref. [ 9] and Sec. II D. At large contact separations in geometry (B), the signal is more sensitive to the bulk parameters /lscriptmandσmthan the interface gs. When contacts are close to each other, the interfaces become more important and the results dependsensitively on g sandσmas compared to /lscriptm. For very close contacts ( d< 500 nm) the total spin resistance of YIG is dominated by the interface and our model calculations slightly (b) (a) FIG. 7. (a) Computed nonlocal first harmonic signal as a function of distance on a linear scale. The red open circles show the results for sample (A), while black open squares represent sample (B). The blue triangles are the experimental results [ 9]. The red dashed line is a 1 /dfit of the numerical results for (A). (b) Same as (a) but on a logarithmic scale. 014412-9CORNELISSEN, PETERS, BAUER, DUINE, AND V AN WEES PHYSICAL REVIEW B 94, 014412 (2016) RintsRintsRYIGs μsinjμsdet RPts(a) (b)10−310−210−110010110210−610−510−410−310−210−1100 Distance ( μm)μsdet / μsinjExperimental data Fit from Ref. 9 Circuit model (no relaxation) RPts FIG. 8. (a) Experimental and simulated spin transfer efficiency ηs=μdet s/μinj s. The blue solid line is a fit by the 1D spin diffusion model [ 9]. Since here interfaces are disregarded, μdet s→μinj sfor vanishing contact distances. The red dashed line is obtained fromthe equivalent circuit model in (b) with spin resistances R s Xdefined in the text. This model includes gsbut is valid for d</lscript monly since spin relaxation is disregarded. The interfaces lead to a saturation ofη sat short distances. underestimate the experimental signal and, in contrast to experiments, deviate from the ∼d−1fit that might indicate an underestimated gs.However, a larger gswould lead to deviations at intermediate distances (1 <d< 5μm). 2. Spin transfer efficiency and equivalent circuit model The spin transfer efficiency ηs=μdet s/μinj s, i.e., the ratio between the spin accumulation in the injector and that inthe detector, can be readily derived from the experiments byEq. ( 20). From the voltage generated in the detector by the inverse spin Hall effect V ISHE [48] μdet s=2t θL1+e−2t//lscripts (1−e−t//lscripts)2VISHE, (23) where lis the length of the metal contact. The spin transfer efficiency therefore reads as ηs=t /lscriptsθ2Rnl Rdet(et//lscripts+1)(e2t//lscripts+1) (et//lscripts−1)3, (24) where Rnl=VISHE/Iis the observed nonlocal resistance and Rdetthe detector resistance. Figure 8(a) shows the experimental data converted to the spin transfer efficiency as a function ofdistance dthat is fitted to a 1D magnon spin diffusion model that does not include the interfaces [ 9]. When d→0 and interfaces are disregarded, η sdiverges. This artifact can be repaired by the equivalent spin-resistor circuit in Fig. 8(b)according to which ηs=Rs Pt Rs YIG+2Rs int+2RPts, (25) where Rs Pt=/lscripts/[σeAinttanh(t//lscripts)] is the spin resistance of the platinum strip [ 48],Rs int=1/(gsAint) is interface spin resistance, and Rs YIG=d/(σmAYIG) is the magnonic spin resistance of YIG. AYIG=ltYIGis the cross section of the YIG channel and Aint=wlis the area of the Pt |YIG interfaces. The parameters in Table IIlead to the red dashed line in Fig. 8(a), which agrees well with the experimental data for d</lscript m.N o free parameters were used in this model since we adoptedσ m=5×105S/m as extracted from our 2D model in the previous section. The model predicts that the spin transfer efficiency should saturate for d/lessorsimilar100 nm for gs=9.6×1012S/m2. A pre- dicted onset of saturation at 200 nm is not confirmed by theexperiments, which as pointed out already in the previoussection, could imply a larger g s. Experiments on samples with even closer contacts are difficult but desirable. Basedon the available data, we predict that the efficiency saturatesatη s=4×10−3. The charge transfer efficiency (defined in Sec. III A 1 ) would be maximized at η≈5×10−5, which is still below the SMR /Delta1ρ/ρ =2.6×10−4, as predicted in Sec. III A 2 . 3. Magnon temperature model We can analyze the experiments also in terms of magnon temperature diffusion [ 1] as applied to the spin Seebeck [ 5,6] and spin Peltier [ 21] effects. Communication between the platinum injector and detector is possible via phonon andmagnon heat transport: the spin accumulation at the injectorcan heat or cool the magnon/phonon system by the spin Peltiereffect. The diffusive heat current generates a voltage at thedetector by the spin Seebeck effect. However, pure phononicheat transport does not stroke with the exponential scaling,but decays only logarithmically (see below). The magnontemperature model (which describes the magnons in termsof their temperature only) can give an exponential scaling,but in order to agree with experiments, the magnon-phononrelaxation length must be large such that T m/negationslash=Tpover large distances. This is at odds with the analysis by Schreier et al. and Flipse et al. However, we can test this model by, for the sake of argument, increasing this length scale by four ordersof magnitude to /lscript mp=9.4μm and completely disregard the magnon chemical potential. The spin Peltier heat current Qinj SPE is then [ 21] Qinj SPE=LsTμinj s 2Aint, (26) where Lsis the interface spin Seebeck coefficient, Ls= 2g↑↓γ/planckover2pi1kB/(eMs/Lambda13)[5,6,21], and Ms=μBS/a3is the sat- uration magnetization of YIG. The equivalent circuit is basedon the spin Peltier heat current and the spin thermal resistancesof the YIG |Pt interfaces and the YIG channel. This allows us to find T m−e, the temperature difference between magnons and electrons at the detector interface, which is the driving forcefor the SSE in this model. The equivalent thermal resistancecircuit is shown in Fig. 9(b). Relaxation is disregarded, so 014412-10MAGNON SPIN TRANSPORT DRIVEN BY THE MAGNON . . . PHYSICAL REVIEW B 94, 014412 (2016) RintthRintthRYIGth QSPEinj(a) (b) Tm-e10−310−210−110010110210−710−610−510−410−310−210−1 Distance ( μm)μsdet / μsinj Experimental data κm=1e−2 W/(mK) κm=1e−1 W/(mK) κm=1 W/(mK) FIG. 9. (a) Results of the thermal model for κm=10−2W/(mK) (red curve), κm=10−1W/(mK) (green curve), and κm=1W/(mK) (black curve). Plotted on the yaxis is the spin transfer efficiency resulting from the thermal model ηth=μdet s/μinjs. The blue squares represent the experimental data. (b) The equivalent thermal resistance model. The definitions of the thermal resistances used in the model are given in the main text. At the thermal grounds in the circuit,the temperature difference between magnons and electrons ( T m−e)i s zero. the model is only valid for d</lscript mp. The interface magnetic heat resistance is given by Rth int=1/(κI sAint), with κI sequal to [5,6,21] κI s=h e2kBT /planckover2pi1μBkBg↑↓ πMs/Lambda13, (27) and where μBis the Bohr magneton. The YIG heat resistance Rth YIG=d/(κmAYIG) and from the thermal circuit model we find that Tm−e=Qinj SPE(Rth int)2/(Rth int+Rth YIG), which generates a spin accumulation in the detector by the spin Seebeck effect μdet s=Tm−eg↑↓γ/planckover2pi1kB πMs/Lambda134π e/lscripts σtanh/parenleftbiggt 2/lscripts/parenrightbigg1+e−2t//lscripts (1−e−t//lscripts)2.(28) The thus obtained spin transfer efficiency ηthis plotted in Fig. 9(a) as a function of the magnon spin conductivity κm.F o rκm∼0.1–1 W /(mK) reasonable agreement with the experimental data can be achieved. While Schreier et al. argued that κmshould be in the range 10−2–10−3W/(mK)), κmfrom Table Iis also of the order of 1 W /(mK) at room temperature. Hence, the magnon temperature modelcan describe the nonlocal experiments, provided that themagnon-phonon relaxation length /lscript mpis large. However, from the expression for /lscriptmpthat we gave in Table Iwe find that /lscriptmp∼10μm corresponds to τmp≈τmr∼1 ns and κm∼104 W/(mK), which is at least three orders of magnitude larger than even the total YIG heat conductivity, and is clearlyunrealistic. Thus, requiring /lscriptmp∼10μm while maintaining κm∼1W/(mK) is inconsistent. Also, an /lscriptmpof the order of nanometers as reported by Schreier et al. and Flipse et al. is difficult to reconcile with the observed length scale of the orderof 10μm. Up to now, we disregarded phononic heat transport. As argued, the interaction of phonons with magnons in the spinchannel is weak, but the energy transfer can be efficient.The spin Peltier effect at the contact generates a magnonheat current that decays on the length scale /lscript mp, heating up the phonons that subsequently diffuse to the detector, wherethey cause a spin Seebeck effect. The magnon system is inequilibrium except at distances from injector and detector onthe scale /lscript mpthat we argued to be short. In this scenario, there is no nonlocal magnon transport in the bulk at all, but injectorand detector communicate by pure phonon heat transport.However, this mechanism does not explain the exponentialdecay of the nonlocal signal: the diffusive heat current emittedby a line source, taking into account that the gadoliniumgallium garnet (GGG) substrate has a heat conductivity closeto that of YIG [ 6], decays only logarithmically as a function of distance. D. Longitudinal spin Seebeck effect The spin Seebeck effect is usually measured in the longi- tudinal configuration, i.e., samples with a YIG film grownon GGG and a Pt top contact. Longitudinal spin Seebeckmeasurements are hence local measurements, as opposed tothe nonlocal experiments we have discussed in the precedingsections. However, in the longitudinal configuration our one-dimensional model [ 17] is still applicable. A recent study extracted the length scale of the longitudinal spin Seebeckeffect from experiments on samples with various YIG filmthicknesses [ 49]. A length of the order of 1 μm was found. Similar results were obtained by Kikkawa et al. [50]. We assume a constant gradient ( T L−TR)/d < 0,where TL,TRare the temperatures at the interfaces of YIG to GGG, platinum, respectively, with Tmeverywhere equilibrized to Tp, and disregard the Kapitza heat resistance [cf. Fig. 10(a) ]. At the YIG |GGG interface the spin current vanishes. Figure 10 illustrates the magnon chemical potential profile on the YIGthickness das well as the transparency of the Pt |YIG interface for four limiting cases, i.e., for opaque ( g s<σm//lscriptm) and transparent ( gs>σm//lscriptm) interfaces and a thick ( d>/lscript m) and at h i n( d</lscript m) YIG film, in which analytic results can be derived. We define a spin Seebeck coefficient as the normalized inverse spin Hall voltage VISHE/tyin the platinum film of length tydivided by the temperature gradient /Delta1T/d, with /Delta1T=TL−TRand average temperature T0: σSSE=dVISHE ty/Delta1T. (29) Assuming that the Pt spin diffusion length /lscriptsis much shorter than its film thickness t, we find the analytic expression σSSE=gs/lscripts/lscriptmLθ/bracketleftbig coshd /lscriptm−1/bracketrightbig tσeT0/bracketleftbig gs/lscriptmcoshd /lscriptm+σm/parenleftbig 1+2gs/lscripts σe/parenrightbig sinhd /lscriptm/bracketrightbig.(30) 014412-11CORNELISSEN, PETERS, BAUER, DUINE, AND V AN WEES PHYSICAL REVIEW B 94, 014412 (2016) d > lm YIG PtChemical potential μmd < lm 0 GGGYIG Pt GGG YIG PtChemical potential μm 0GGGYIG PtGGG0 0Opaque Transparent(b) (a) (d) (c)TR TL FIG. 10. Magnon chemical potential μmunder the spin Seebeck effect for a linear temperature gradient in YIG, in the limit of (a)an opaque interface and thick YIG, (b) an opaque interface and thin YIG, (c) a transparent interface and thick YIG, and (d) a transparent interface and thin YIG. In all four cases, μ mchanges sign somewhere in the YIG. For higher interface transparency (larger gs), the zero crossing shifts closer to the Pt |YIG interface. In Fig. 11,σSSE is plotted as a function of the relative thickness d//lscriptmof the magnetic insulator in the transport direction, Pt thickness of t=10 nm and T0=300 K. We adopt Lfrom Table Iand a relaxation time τ∼τmp∼0.1 ps and the parameters from Fig. 11. The normalized spin Seebeck coefficient saturates as a function of don the scale of the magnon spin diffusion length /lscriptm. While experiments at T0/lessorequalslant250 K report somewhat smaller length scales than our /lscriptm,our saturation σSSE∼0.1–1μV/K is of the same order as the experiments [ 51]. In the limit of an opaque interface, σSSEsaturates to σSSE(d/greatermuch/lscriptm)=gs/lscripts/lscriptmLθ tT0σeσm=/parenleftbigggs/lscripts σe/parenrightbigg/parenleftbigg/lscriptm t/parenrightbiggαμθkB e,(31) in terms of the dimensionless ratio αμfrom Eq. ( 7). 02468 1 00.000.050.100.150.200.250.300.35 dlmΣSSEΜVK FIG. 11. Normalized spin Seebeck coefficient as a function of the thickness of the magnetic insulator in the direction of the temperaturegradient. Parameters taken are from Table II, together with a Pt thickness of t=10 nm and temperature of 300 K. The value for the bulk spin Seebeck coefficient Lis taken from the expression in Table Iwithτ=0.1p s .For a transparent interface with /lscriptm/greatermuch/lscriptsandσm∼σe,t h e result is governed by bulk parameters only: σSSE(d→∞ )=/lscriptsLθ tT0σe. (32) This model for the spin Seebeck effect is oversimplified by assuming a vanishing magnon-phonon relaxation lengthand disregarding interface heat resistances. The gradient inthe phonon temperature can give rise to a spin Seebeckvoltage [ 52] even when bulk magnon spin transport is frozen out by a large magnetic field. Nevertheless, it is remarkablethat it gives a reasonable qualitative description for the spinSeebeck effect with input parameters adapted for electricallydriven magnon transport. We conclude that also in thedescription of the spin Seebeck effect the magnon chemicalpotential can play a crucial role. IV . CONCLUSIONS We presented a diffusion theory for magnon spin and heat transport in magnetic insulators actuated by metalliccontacts. In contrast to previous models, we focus on themagnon chemical potential. This is an essential ingredientbecause under ambient conditions /lscript m>/lscript mp, i.e., the magnon chemical potential relaxes over much larger length scalesthan the magnon temperature. We compare theoretical resultsfor electrical magnon injection and detection with nonlocaltransport experiments on YIG |Pt structures [ 9], for both a 1D analytical and a 2D finite-element model. In the 1D model, we study the relevance of interface versus bulk-limited transport and find that, for the materials andconditions considered, the interface spin resistance dominates.For the limiting cases of transparent and opaque interfaces,the spin transfer efficiency ηdecays algebraically ∝1/das a function of injector-detector distance dwhen d</lscript m, and exponentially with a characteristic length /lscriptmford>/lscript m. A 2D finite-element model for the actual sample configura- tions can be fitted well to the experiments for different contactdistances, leading to a magnon conductivity σ m=5×105 S/m and diffusion length /lscriptm=9.4μm. The experiments measure first- and second-order har- monic signals that are attributed to electrical magnon spininjection/detection and thermal generation of magnons byJoule heating with spin Seebeck effect detection, respectively.Here, we focus on the linear response that we argue tobe dominated by the diffusion of a magnon accumulationgoverned by the chemical potential, rather than the magnontemperature. However, we applied our theory also to thestandard longitudinal (local) spin Seebeck geometry. We findthe same length scale /lscript mand a (normalized) spin Seebeck coefficient of σSSE∼0.1–1μV/Kf o r d/greatermuch/lscriptm,which is of the same order of magnitude as the observations [ 49]. ACKNOWLEDGMENTS We would like to acknowledge H. M. de Roosz and J. G. Holstein for technical assistance, and Y . Tserkovnyak,A. Brataas, S. Bender, J. Xiao, and B. Flebus for discussions.This work is part of the research program of the Foundationfor Fundamental Research on Matter (FOM) and supported 014412-12MAGNON SPIN TRANSPORT DRIVEN BY THE MAGNON . . . PHYSICAL REVIEW B 94, 014412 (2016) by NanoLab NL, EU FP7 ICT Grant No. 612759 InSpin, Grant-in-Aid for Scientific Research (Grants No. 25247056,No. 25220910, and No. 26103006) and the Zernike Institutefor Advanced Materials. R.D. is a member of the D-ITPconsortium, a program of the Netherlands Organization forScientific Research (NWO) that is funded by the DutchMinistry of Education, Culture, and Science (OCW). APPENDIX: BOLTZMANN TRANSPORT THEORY Here, we derive our magnon transport theory from the linearized Boltzmann equation in the relaxation time ap-proximation, thereby introducing and estimating the differentcollision times. 1. Boltzmann equation Equations ( 5)–(7) are based on the Boltzmann equation for the magnon distribution function f(x,k,t): ∂f ∂t+∂f ∂x·∂ωk ∂k=/Gamma1in[f]−/Gamma1out[f], (A1) where /Gamma1in=/Gamma1in el+/Gamma1in mr+/Gamma1in mp+/Gamma1in mm and/Gamma1out=/Gamma1out el+ /Gamma1out mr+/Gamma1out mp+/Gamma1out mmare the total rates of scattering into and out of a magnon state with wave vector k, respectively. The subscripts refer to elastic magnon scattering at defects,magnon relaxation by magnon-phonon interactions that do notconserve magnon number, magnon-conserving inelastic andelastic magnon-phonon interactions, and magnon number andenergy-conserving magnon-magnon interactions. We discussthem in the following for an isotropic magnetic insulator andin the limit of small magnon and phonon numbers. The elastic magnon scattering is given by Fermi’s golden rule as /Gamma1 out el=2π /planckover2pi1/summationdisplay k/prime/vextendsingle/vextendsingleVel kk/prime/vextendsingle/vextendsingle2δ(/planckover2pi1ωk−/planckover2pi1ωk/prime)f(k,t), (A2) where Vel kk/primeis the matrix element for scattering by defects and rough boundaries [ 23,37] of a magnon with momentum /planckover2pi1kto one with /planckover2pi1k/primeat the same energy. /Gamma1in elis obtained from this expression by interchanging kandk/prime.I nt h e presence of the in-scattering term (vertex correction) /Gamma1in el, the Boltzmann equation is an integrodifferential rather thana simple differential equation. Gilbert damping parametrizes the magnon dissipation into the phonon bath. According to the linearized Landau-Lifshitz-Gilbert equation [ 32] /Gamma1 out mr=2αGωkf(k,t). (A3) Since the phonons are assumed to be at thermal equilibrium with temperature Tp,/Gamma1in mris obtained by substituting f(k,t)→ nB(/planckover2pi1ωk/kBTp)i n/Gamma1out mr. Magnon-conserving magnon-phonon interactions with ma- trix elements Vmp kk/primeqgenerate the out-scattering rate /Gamma1out mp=2π /planckover2pi1/summationdisplay k/prime,q/vextendsingle/vextendsingleVmp kk/primeq/vextendsingle/vextendsingle2δ(/planckover2pi1ωk−/planckover2pi1ωk/prime−/epsilon1q) ×f(k,t)[(1+f(k/prime,t)]/bracketleftbigg 1+nB/parenleftbigg/epsilon1q kBTp/parenrightbigg/bracketrightbigg , (A4)where /epsilon1q=/planckover2pi1c|q|is the acoustic phonon dispersion with sound velocity cand momentum q.The “in” scattering rate /Gamma1in mp=2π /planckover2pi1/summationdisplay k/prime,q/vextendsingle/vextendsingleVmp kk/primeq/vextendsingle/vextendsingle2δ(/planckover2pi1ωk−/planckover2pi1ωk/prime−/epsilon1q) ×f(k/prime,t)[(1+f(k,t)]nB/parenleftbigg/epsilon1q kBTp/parenrightbigg . (A5) Finally, the four-magnon interactions (two magnons in, two magnons out) generate /Gamma1out mm=2π /planckover2pi1/summationdisplay k/prime,k/prime/prime,k/prime/prime/prime/vextendsingle/vextendsingleVmm k+k/prime,k−k/prime,k/prime/prime−k/prime/prime/prime/vextendsingle/vextendsingle2 ×δ(/planckover2pi1ωk+/planckover2pi1ωk/prime−/planckover2pi1ωk/prime/prime−/planckover2pi1ωk/prime/prime/prime)δ(k+k/prime−k/prime/prime−k/prime/prime/prime) ×f(k,t)f(k/prime,t)[1+f(k/prime/prime,t)][1+f(k/prime/prime/prime,t)], (A6) while /Gamma1in mmfollows by exchanging kk/prime/prime, and k/primeandk/prime/prime/prime. Disregarding umklapp scattering, the magnon-magnon inter-actions conserve linear and angular momentum. V mmtherefore depends only on the center-of-mass momentum and the relativemagnon momenta before and after the collision, which impliesthat/Gamma1 mmdoes not affect transport directly (analogous to the role of electron-electron interactions in electric conduction). The collision rates govern the energy and momentum-dependent collision times τa(k,/planckover2pi1ω) (with a∈{el,mr,mp,mm}). These are defined from the “out” rates via 1 τa(k,/planckover2pi1ω)=/Gamma1out a f(k,t), (A7) replacing f→nB(/planckover2pi1ωk/kBTp) and /planckover2pi1ωkwith /planckover2pi1ωwhere phonons are involved. Here, we are interested mainly in thermal magnons for which the relevant collision times are evaluated at energy /planckover2pi1ω=kBTand momentum k=/Lambda1−1. Then, 1 /τmr∼αGkBT//planckover2pi1. Elastic magnon scattering can be parametrized by a mean-free path /lscriptel=τel(k,/planckover2pi1ω)∂ωk/∂k, and therefore 1 /τel(k,/planckover2pi1ω)=2/lscript−1 el√Jsω//planckover2pi1orτel=/lscriptel/vm, where vm=2√Jsω//planckover2pi1is the magnon group velocity. Estimates for /lscriptelrange from 1 μm[23] under the assumption that /lscriptelis due to Gilbert damping and disorder only, to 500 μm[37]. Therefore, τel∼10–105ps. Since we deduce in the main text that at room temperature τmpis one to two orders of magnitude smaller than thisτel, we completely disregard elastic two-magnon scattering in the comparison with experiments. We adopt the relaxation time approximation in which the scattering terms read as /Gamma1[f]=1 τel/bracketleftbigg f−nB/parenleftbigg/planckover2pi1ωk−μm kBTm/parenrightbigg/bracketrightbigg +1 τmr/bracketleftbigg f−nB/parenleftbigg/planckover2pi1ωk kBTp/parenrightbigg/bracketrightbigg +1 τmp/bracketleftbigg f−nB/parenleftbigg/planckover2pi1ωk−μm kBTp/parenrightbigg/bracketrightbigg +1 τmm/bracketleftbigg f−nB/parenleftbigg/planckover2pi1ωk−μm kBTm/parenrightbigg/bracketrightbigg . (A8) The distribution functions here are chosen such that the elastic scattering processes stop when fapproaches the Bose- Einstein distribution with local chemical potential μm/negationslash=0, 014412-13CORNELISSEN, PETERS, BAUER, DUINE, AND V AN WEES PHYSICAL REVIEW B 94, 014412 (2016) in contrast to the inelastic scattering that causes relaxation to thermal equilibrium with the lattice and μm=0.Similarly, the temperatures TpvsTmare chosen to express that the scattering exchanges energy with the phonons or keeps it in the magnonsystem, respectively. The Boltzmann equation may be linearized in terms of the small perturbations, i.e., the gradients of temperature andchemical potential. The local momentum space shift δfof the magnon distribution function δf(x,k)=τ∂n B/parenleftbig/planckover2pi1ωk kBTp/parenrightbig ∂/planckover2pi1ωk∂ωk ∂k·/parenleftbigg ∇xμm+/planckover2pi1ωk∇xTm Tp/parenrightbigg , (A9) where 1 /τ=1/τmr+1/τmp. The magnon spin and heat currents [Eq. ( 5)] are obtained by substituting δfinto jm=/planckover2pi1/integraldisplaydk (2π)3δf(k)∂ωk ∂k, (A10) jQ,m=/integraldisplaydk (2π)3δf(k)/planckover2pi1ωk∂ωk ∂k. (A11) The magnon spin and heat diffusion [Eq. ( 6)] are obtained by a momentum integral of the Boltzmann equation ( A8)a f t e r multiplying by /planckover2pi1and/planckover2pi1ωk, respectively. The local distribution function in the collision terms consists of the sum of the“drift” term δfand the Bose-Einstein distribution with local temperature and chemical potential f(k,t)=δf+n B([/planckover2pi1ωk−μm(x)]/[kBTm(x)]). (A12)We reiterate that the relatively efficient magnon conserving τm limits the energy, but not (directly) the spin diffusion. 2. Magnon-magnon scattering rate The four-magnon scattering rate is believed to efficiently thermalize the local magnon distribution to the Bose-Einsteinform [ 31,32]. At room temperature, the leading-order correc- tion to the exchange interaction in the presence of magnetiza-tion textures reads as H xc=−Js 2s/integraldisplay dxs(x)·∇2s(x), (A13) where s(x)(s=|s|=S/a3) is the spin density. By the Holstein-Primakoff transformation, the spin-lowering op- erator reads as ˆs−=sx−isy=/radicalbig 2s−ˆψ†ˆψˆψ/similarequal√ 2sˆψ− ˆψ†ˆψˆψ/2√ 2sin terms of the bosonic creation ( ˆψ†) and annihilation ( ˆψ) operators. Hxccan be approximated as a four-particle pointlike interaction term Hmm≈g/integraldisplay dxˆψ†ˆψ†ˆψˆψ, (A14) where g∼kBT/s is the exchange interaction strength at thermal energies. Using Fermi’s golden rule for this interactionyields collision terms as Eq. ( A6) with V mm≈g: 1 τmm(k,/planckover2pi1ω)≈g2 /planckover2pi1/summationdisplay k/prime,k/prime/prime,k/prime/prime/primeδ(/planckover2pi1ωk+/planckover2pi1ωk/prime−/planckover2pi1ωk/prime/prime−/planckover2pi1ωk/prime/prime/prime)δ(k+k/prime−k/prime/prime−k/prime/prime/prime)×nB/parenleftbigg/planckover2pi1ωk/prime kBTp/parenrightbigg ×/bracketleftbigg 1+nB/parenleftbigg/planckover2pi1ωk/prime/prime kBTp/parenrightbigg/bracketrightbigg/bracketleftbigg 1+nB/parenleftbigg/planckover2pi1ωk/prime/prime/prime kBTp/parenrightbigg/bracketrightbigg . (A15) The momentum integrals can be estimated for thermal magnons with k=/Lambda1−1and/planckover2pi1ω=kBTand 1 τmm≈g2 /Lambda16kBT /planckover2pi1≈/parenleftbiggT Tc/parenrightbigg3kBT /planckover2pi1, (A16) with Curie temperature kBTc≈Jss2/3. With parameters for YIGJss2/3/kB≈200 K, which is the correct order of magnitude. The T4scaling of the four-magnon interaction rate results from the combined effects of the magnon density ofstates (magnon scattering phase space) and energy dependenceof the exchange interactions. While the magnon-magnon scattering is efficient at thermal energies, it becomes slow at low energies close to the band edgedue to phase space restrictions and leads to deviations from theBose-Einstein distribution functions that may be disregardedat room temperature. 3. Magnon-conserving magnon-phonon interactions At thermal energies and large wave numbers, the magnon- conserving magnon-phonon scattering [ 37] is dominated bythe dependence of the exchange interaction on lattice distor- tions rather than magnetocrystalline fields. Since we estimateorders of magnitude, we disregard phonon polarization andthe tensor character of the magnetoelastic interaction and startfrom the Hamiltonian H mp=−B s/integraldisplay dxs(x)·∇2s(x)⎛ ⎝/summationdisplay α∈{x,y,z}∂R ∂xα⎞ ⎠, (A17) where Bis a magnetoelastic constant. The scalar lattice displacement field Rcan be expressed in the phonon creation and annihilation operators ˆφ†and ˆφas R=/radicalBigg /planckover2pi12 2ρ/epsilon1[ˆφ+ˆφ†], (A18) where /epsilon1is the phonon energy and ρthe mass density. By the Holstein-Primakoff transformation introduced in the previous 014412-14MAGNON SPIN TRANSPORT DRIVEN BY THE MAGNON . . . PHYSICAL REVIEW B 94, 014412 (2016) section, we find to leading order Hmp≈B/integraldisplay dx(∇ˆψ†)·(∇ˆψ)/parenleftbigg/planckover2pi12 ρ/epsilon1/parenrightbigg⎛ ⎝/summationdisplay α∈{x,y,z}∂ˆφ ∂xα⎞ ⎠+H.c. (A19) This Hamiltonian is the scattering potential in the matrix elements of Eq. ( A5): /vextendsingle/vextendsingleVmp kk/primeq/vextendsingle/vextendsingle2≈B2/planckover2pi12q2 ρ/epsilon1q(k·k/prime)2δ(k−k/prime−q) (A20) which by substitution and in the limit /Lambda1/lessmuch/Lambda1p, where /Lambda1p= /planckover2pi1c/kBTpis the phonon thermal de Broglie wavelength, leads to 1 τmp∼B2 /planckover2pi1ρ/parenleftbigg/planckover2pi1 kBT/parenrightbigg21 /Lambda14/Lambda15p. (A21)In the opposite limit /Lambda1/greatermuch/Lambda1p, 1 τmp∼B2 /planckover2pi1ρ/parenleftbigg/planckover2pi1 kBT/parenrightbigg21 /Lambda17/Lambda12p. (A22) At room temperature /Lambda1≈/Lambda1pand for ρa3=10−24kg both expressions lead to τmp=10(Js/B)2ns [38]. We could not find estimates of Bfor YIG in the literature. In iron, exchange interactions change by a factor of 2 upon small lattice distortion/Delta1a/lessmucha[53]. While the authors of this latter work find that this does not strongly affect the Curie temperature, it leadsto fast magnon-phonon scattering as we show now. 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PhysRevB.87.174417.pdf

Selected for a Viewpoint inPhysics PHYSICAL REVIEW B 87, 174417 (2013) Comparative measurements of inverse spin Hall effects and magnetoresistance in YIG/Pt and YIG/Ta C. Hahn, G. de Loubens,*O. Klein, and M. Viret Service de Physique de l’ ´Etat Condens ´e (CNRS URA 2464), CEA Saclay, 91191 Gif-sur-Yvette, France V. V. N a l e t ov Service de Physique de l’ ´Etat Condens ´e (CNRS URA 2464), CEA Saclay, 91191 Gif-sur-Yvette, France and Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation J. Ben Youssef Universit ´e de Bretagne Occidentale, Laboratoire de Magn ´etisme de Bretagne CNRS, 6 Avenue Le Gorgeu, 29285 Brest, France (Received 18 February 2013; published 13 May 2013) We report on a comparative study of spin Hall related effects and magnetoresistance in YIG |Pt and YIG |Ta bilayers. These combined measurements allow to estimate the characteristic transport parameters of both Pt andTa layers juxtaposed to yttrium iron garnet (YIG): the spin mixing conductance G ↑↓at the YIG |normal metal interface, the spin Hall angle /Theta1SH, and the spin diffusion length λsdin the normal metal. The inverse spin Hall voltages generated in Pt and Ta by the pure spin current pumped from YIG excited at resonance confirm theopposite signs of spin Hall angles in these two materials. Moreover, from the dependence of the inverse spin Hallvoltage on the Ta thickness, we extract the spin diffusion length in Ta, found to be λ Ta sd=1.8±0.7 nm. Both the YIG|Pt and YIG |Ta systems display a similar variation of resistance upon magnetic field orientation, which can be explained in the recently developed framework of spin Hall magnetoresistance. DOI: 10.1103/PhysRevB.87.174417 PACS number(s): 85 .75.−d, 76.50.+g I. INTRODUCTION Spintronics aims at designing devices that capitalize on the interplay between the spin and charge degrees of freedomof the electron. In particular, it is of central interest to study the interconversion from a spin current, the motion of spin angular momentum, to a charge current and the transfer ofspin angular momentum between the conduction electrons of anormal metal (NM) and the magnetization of a ferromagneticmaterial (FM). The separation of oppositely spin-polarizedelectrons of a charge current through spin-orbit coupling iscalled spin Hall effect (SHE). 1,2Its inverse process (ISHE) converts spin currents into charge currents and has recently sparked an intense research activity.3,4as it allows for an electrical detection of the dynamical state of a ferromagnet.5,6 Indeed, a precessing magnetization in a ferromagnet generates a spin current via spin pumping,7which can be converted, at the interface with an adjacent normal layer, to a dc voltageby ISHE. Moreover, electronic transport can also be affected by the static magnetization in the FM as electrons spins separated by SHE can undergo different spin-flip scatteringon the interface with the FM layer. In particular, spin-flippedelectrons are deflected by ISHE in a direction opposite to theinitial current, leading to a reduced total current at constantvoltage. This effect depends on the relative orientation betweenmagnetization and current direction, and has recently been called spin Hall magnetoresistance (SMR). 8 Experimental studies on spin pumping induced inverse spin Hall voltages ( VISH)i nF M |NM bilayers were first carried out with Pt as NM in combination with NiFe as FM5,9–12and more recently with the insulating ferrimagnet yttrium irongarnet (YIG). 6,13–17Although other strong spin-orbit metals have been tried in combination with the metallic ferromagnetsNiFe18,19and CoFeB,20,21inverse spin Hall voltage6,17and magnetoresistance8,22measurements made on YIG |NM have so far been limited to NM =Pt. Still, it would be very interesting to compare VISHand SMR measurements on different YIG |NM systems, including metals having opposite spin Hall angles, such as Pt versus Ta.20,23Ab initio calculations indeed predict the spin Hall angle of the resistive βphase of Ta to be larger and of opposite sign to that of Pt.24The defining parameters for VISHand SMR are the spin diffusion length in the normal metal, λsd, the spin Hall angle /Theta1SH, which quantifies the efficiency of spin to charge current conversion,and the spin mixing conductance ( G ↑↓), which depends on the scattering matrices for electrons at the FM |NM interface7and can be seen as the transparency of the interface for transfer ofspin angular momentum. 25The evaluation of the three above mentioned parameters is a delicate task,26as the measured VISHvoltages and SMR ratio depend on all of them. In this paper, we present a comparative study of YIG |Pt and YIG|Ta bilayers, where we measure both the ISHE and SMR on each sample. We confirm the opposite signs of spin Hallangles in Pt and Ta and the origin of SMR, which has beenexplained in Ref. 8. Thanks to these combined measurements, we can evaluate the spin mixing conductances of the YIG |Pt and YIG |Ta interfaces and the spin Hall angles in Pt and Ta. In order to get more insight on the previously unexplored YIG |Ta system, we study the dependence of ISHE on Ta film thickness,which enables us to extract the spin diffusion length in Ta. The remaining of the manuscript is organized as follows. Section IIgives details on the samples and experimental setup used in this study. In Sec. III, the experimental data of V ISH and SMR obtained on the YIG |Pt and YIG |Ta systems are presented and analyzed. In Sec. IV, we discuss the transport parameters extracted from our measurements. We also 174417-1 1098-0121/2013/87(17)/174417(8) ©2013 American Physical SocietyC. HAHN et al. PHYSICAL REVIEW B 87, 174417 (2013) FIG. 1. (Color online) (a) Standard in-plane FMR spectrum of a bare YIG 200-nm thin film used in this study. (b) Full FMR linewidth vs frequency. comment on the absence of direct effect of a charge current in Pt on the linewidth of our 200-nm-thick YIG samples. Finally,we emphasize the main results of this work in the conclusion. II. EXPERIMENTAL DETAILS A. Samples 1. YIG films Two single-crystal Y 3Fe5O12(YIG) films of 200-nm thick- ness were grown by liquid phase epitaxy on (111) Gd 3Ga5O12 (GGG) substrates,27and labeled YIG1 and YIG2. Epitaxial growth of the YIG was verified by x-ray diffraction and thefilms roughness was determined by atomic force microscopyto be below 5 ˚A. Their magnetic static properties were investigated by vibrating sample magnetometry. The in-planebehavior of the thin YIG films is isotropic with a coercitivitybelow 0.6 Oe. 27The saturation magnetization, found to be 140 emu /cm3, corresponds to the one of bulk YIG. This value was verified by performing ferromagnetic resonance (FMR) atdifferent excitation frequencies. FMR also allows to extract the magnetic dynamic properties of the 200-nm-thick YIG films. A typical FMR spectrum ofthe YIG1 film obtained at 10 GHz and low microwave power(P=− 20 dBm) is presented in Fig. 1(a). The gyromagnetic ratio of our YIG films is found to be γ=1.79×10 7rad/s/Oe. From the dependence of the linewidth on the excitationfrequency, their Gilbert damping α G=(2.0±0.2)×10−4 can be determined, see Fig. 1(b). This value highlights the very small magnetic relaxation of these thin films. Still, thereis an inhomogeneous part to the linewidth [ /Delta1H 0=0.4O ei n Fig. 1(b)]. For one of the two prepared films (YIG2), two to three closely spaced resonance lines could be observed in somecases, which we attribute to distinct sample regions havingslightly different properties. 2. YIG |Pt and YIG |Ta bilayers After these standard magnetic characterizations, the YIG films were cut into slabs with lateral dimensions of1.1 mm ×7 mm in order to perform inverse spin Hall voltage and magnetoresistance measurements. Platinum and tantalumthin films were then grown by rf sputter deposition, at a powerdensity of 4 W /cm 2. The growth of the resistive βphase of Ta was achieved by optimizing the Ar pressure duringthe sputtering process. This study was conducted in parallelonto oxidised Si and GGG(111) substrates. The appearanceof the tetragonal crystalline phase in a narrow window around10 −2mbar was verified by the presence of the characteristic (200)- β-Ta line in the x-ray diffraction spectra.28Theβphase was also confirmed by the resistivity of the films,20which for 10-nm Ta thickness lies at 200 μ/Omega1cm. In order to compare ISHE and SMR on YIG |Pt and YIG |Ta bilayers, a 15-nm-thick Pt and a 3-nm-thick Ta layers weregrown on the YIG1 sample. The conductivities of thesemetallic films are σ Pt=2.45×106/Omega1−1m−1(in agreement with the values reported in Refs. 17and18) andσTa=3.05× 105/Omega1−1m−1, respectively. These two samples have been used to obtain the results presented in Figs. 2and4. The dependence on Pt thickness of both VISH17and magnetoresistance22,29has been studied earlier. In this work, we have used the YIG2sample to study the dependence as a function of the Tathickness, which was varied from 1.5 to 15 nm (1.5, 2, 3,5, 10, and 15 nm). The conductivity of these Ta films increasesfrom 0 .8×10 5to 7.5×105/Omega1−1m−1with the film thickness. This series of samples has been used to obtain the data ofFig. 3. Finally, Pt films with thicknesses 10 and 15 nm were also grown on YIG2, for the sake of comparison with YIG1. B. Measurement setup A 500- μm-wide and 2- μm-thick Au transmission line cell and electronics providing frequencies up to 20 GHz wereused for microwave measurements. The long axis of thesample was aligned perpendicularly to the microwave line,thus parallel to the excitation field h rfas indicated in the inset of Fig. 2.VISHwas measured by a lock-in technique (with the microwave power turned on and off at a frequency of afew kilohertz) with electrical connections through gold leads 174417-2COMPARATIVE MEASUREMENTS OF INVERSE SPIN HALL ... PHYSICAL REVIEW B 87, 174417 (2013) FIG. 2. (Color online) Inverse spin Hall voltage measured at 3.5 GHz for YIG |Ta and YIG |Pt. (Inset) Sketch of the experiment. at equal distance to the area of excitation. Magnetotransport measurements of the YIG |NM slabs were performed using a four-point configuration. The samples were placed at the centerof an electromagnet, which can be rotated around its axis inorder to obtain curves of magnetoresistance versus angle. Themeasurement cell was placed in a cryostat, with the possibilityto cool down to 77 K. All the measurements presented in thispaper were performed at room temperature, except for thosereported in Fig. 5. III. EXPERIMENTAL RESULTS AND ANALYSIS A. Inverse spin Hall voltage: YIG |Pt versus YIG |Ta First, we compare in Fig. 2the inverse spin Hall voltages measured at 3.5 GHz ( P=+ 10 dBm) in the YIG |Pt and YIG|Ta bilayers. It shows that one can electrically detect the FMR of YIG in these hybrid systems.6The spin current Js pumped into the adjacent normal metal by the precessing mag- netization in YIG is converted into a charge current by ISHE, Je=2e ¯h/Theta1SHJs, (1) where eis the electron charge and ¯ hthe reduced Planck constant. This leads to a transverse voltage VISH(across the length of the YIG |NM slab), as sketched in the inset of Fig. 2. Moreover, VISHmust change sign upon reversing the magnetization of YIG because of the concomitant reversalof the spin pumped current J s(hence Je). This is observed in both the YIG |Pt and YIG |Ta systems, where VISHis odd in applied magnetic field, which shows that the voltagegenerated at resonance is not due to a thermoelectric effect. The striking feature to be observed here is the opposite signs of V ISHin these two samples. This remains true at all microwave frequencies (from 2 to 8 GHz) and power levels(from −8t o+10 dBm), which were measured, as well asfor the different YIG |Pt and YIG |Ta bilayers made from YIG1 and YIG2 samples. It thus confirms that the spin Hallangles in Ta and Pt have opposite signs, as predicted by ab initio calculations 24and inferred from measurements where the spin current was generated by a metallic ferromagnet.20,23 Moreover, from the electrical circuit that was used in the measurements (the anode of the voltmeter is on the left inFig. 2, inset), it can be found that /Theta1 Pt SH>0, while /Theta1Ta SH<0. The precise estimation of the spin Hall angles in these twomaterials requires the extra analyses presented in the followingsections. Still, it is interesting to note that the 4 μV amplitude ofV ISHmeasured in Fig. 2on our 15-nm-thick Pt is close to the one reported in Ref. 17(2 to 3 μV) with comparable experimental conditions. B. Dependence of inverse spin Hall voltage on Ta thickness In this work, we have measured the dependence of VISHonly on Ta thickness. The study as a function of Pt thickness wasalready reported in Ref. 17, using a similar 200-nm-thick YIG film (fabricated in the same laboratory). In Fig. 3,w eh a v e plotted using red squares the dependence of V ISHon the Ta thickness measured on the series of samples described above.Here,V ISHis produced by the precession of magnetization in YIG, resonantly excited at 3.8 GHz by the microwave field(P=+ 10 dBm). V ISHincreases from less than 2 μVu pt o 70μV as the Ta layer thickness is reduced from 15 to 2 nm at which the maximal voltage is measured. For the thinnest Talayer ( t Ta=1.5n m ) , VISHdrops to about 10 μV , a value close to the one observed at tTa=10 nm. A similar dependence of VISHon Pt thickness was reported in Ref. 17, where a maximum of voltage was observed between tPt=1.5 nm and tPt=6n m . The resistance measured across the length of the YIG |Ta slab is also plotted with green crosses in Fig. 3as a function oftTa(see right scale). It is interesting to note that both VISH 174417-3C. HAHN et al. PHYSICAL REVIEW B 87, 174417 (2013) FIG. 3. (Color online) Dependence of inverse spin Hall voltage on Ta thickness (red squares, left scale). The microwave frequencyis 3.8 GHz ( P=+ 10 dBm). The lines are theoretical predictions 17 from Eq. (2)for different values of λsd, with the parameters G↑↓= 4.3×1013/Omega1−1m−2and/Theta1SH=− 0.02. The resistance of the samples is also displayed (green crosses, right scale). The resistance of the 1.5-nm-thin Ta sample (95 k /Omega1) is out of range. andRfollow a similar dependence on the Ta thickness, if one excludes the thinnest Ta layer, which might be discontinuous oroxidized, and thus exhibits a very large resistance ( R=95 k/Omega1 is out of range of the graph). To analyze the thickness dependence of the inverse spin Hall voltage, we follow the approach derived in Ref. 17.T h e spin diffusion equation with the appropriate source term andboundary conditions leads to the following expression: V ISH=/Theta1SHG↑↓ G↑↓+σ λsd1−exp (−2tNM/λsd) 1+exp (−2tNM/λsd) ×hLPf sin2(θ) 2etNM[1−exp (−tNM/λsd)]2 1+exp (−2tNM/λsd),(2) where σis the conductivity of the normal metal, tNMis its thickness, Lis the length of the YIG |NM slab excited at frequency fby the microwave field, θis the angle of precession of YIG, and Pis an ellipticity correction factor. The latter depends on the excitation frequency18and, in our case,P/similarequal1.25. From Eq. (2), one can see that the amplitude ofVISHdepends on the transport parameters λsd,G↑↓, and /Theta1SH,a sw e l la s on the resonant precession angle θ. We do not have a direct measurement of θ, but it can be evaluated from the strength of the microwave field hrfand the measured linewidth /Delta1H.30By performing network analyzer measurements and consideringthe geometry of the transmission line, we estimate the strengthof the microwave field h rf/similarequal0.2O ef o ra P=+ 10 dBm output power from the synthesizer. For the series of YIG |Ta samples, it yields a precession angle θ/similarequal3.3◦in YIG at 3.8 GHz.Nevertheless, the measurements presented in Fig. 3are not sufficient to extract independently G↑↓and/Theta1SH. Thethickness dependence ofVISHprimarily depends on λsd, through the argument of the exponential functions in Eq. (2). The spin diffusion length can thus be adjusted to fit the shapeofV ISHvs.tTain Fig. 3. The series of lines in Fig. 3displays the result of calculations based on Eq. (2)for three different values of λsd, using the thickness dependent conductivity σTa measured experimentally. A very good overall agreement to the data is found for a spin diffusion length λTa sd=1.8n m .W e explain the discrepancy observed at tTa=1.5 nm at which the measured voltage is about five times smaller than predicted, bythe fact that the thinnest Ta layer is discontinuous or oxidized,as already pointed out. C. Magnetoresistance: YIG |Pt versus YIG |Ta We now turn to the measurements of dc magnetoresistance in our hybrid YIG |NM bilayers. We have measured the variation of resistance in the exact same samples as the onesstudied by ISHE in Fig. 2,Y I G|Pt (15 nm) and YIG |Ta (3 nm), as a function of the angle of the applied field with respect tothe three main axes of the slabs. In these experiments, theapplied field was fixed to H=3 kOe (sufficient to saturate the YIG), and a dc current of a few mA together with a 6 1/2digits voltmeter were used to probe the resistance of the NM layersin a four-probe configuration. The results obtained by rotatingthe magnetic field in the plane of the sample (angle α), from in-plane perpendicular to the charge current J eto out-of-plane (angle β) and from in-plane parallel to Jeto out-of-plane (angle γ) are presented in Figs. 4(a)–4(c), respectively (see also associated sketches). In both YIG |Pt and YIG |Ta bilayers, we do observe a weak magnetoresistance ( /Delta1R max/R0of 5×10−5and 4 ×10−5, respectively), as it was reported on the YIG |Pt system.22We checked that this weak variation does not depend on the signor strength of the probing current. In contrast to the inversespin Hall voltage measurements presented in Fig. 2,w ea l s o note that the sign (or symmetry) of the effect is identical inYIG|Pt and YIG |Ta. In order to interpret this magnetoresistance, it is important to understand its dependence on all three different angles,α,β, andγ, shown in Fig. 4. If one would just look at the in-plane behavior [see Fig. 4(a)], one could conclude that the NM resistance Rchanges according to some anisotropic magnetoresistance (AMR) effect, as if the NM would bemagnetized at the interface with YIG due to proximity effect. 22 But with AMR, Rdepends on the angle between the charge current Jeand the magnetization (applied field H). Hence no change of Ris expected with the angle β, whereas Rshould vary with the angle γ, which is exactly opposite to what is observed in Figs. 4(b) and4(c), respectively. Therefore usual AMR as the origin of the magnetoresistance in YIG |Pt and YIG|Ta bilayers has to be excluded. Instead, the spin Hall magnetoresistance (SMR) mechanism proposed in Ref. 8is well supported by our magnetoresistance data. In this scenario, the electrons carried by the chargecurrent in the NM layer are deflected by SHE in oppositedirections depending on their spin. Those whose spin is flippedby scattering at the interface with the FM can oppose the 174417-4COMPARATIVE MEASUREMENTS OF INVERSE SPIN HALL ... PHYSICAL REVIEW B 87, 174417 (2013) FIG. 4. (Color online) (a)–(c) Magnetoresistance in YIG |Ta and YIG|Pt as a function of the angle of the applied field ( H=3 kOe) sketched at the top (the samples are the same as the ones measured in Fig. 2). Dashed lines are predictions from Eq. (3)of the SMR theory (see Ref. 8). initial current by ISHE and lead to an increase of resistance. Therefore the spin Hall magnetoresistance depends on therelative angle between the magnetization Mof the FM and the accumulated spins sat the FM |NM interface: R=R 0+/Delta1R maxsin2(M,s). (3) The increase of resistance is maximal when Mand sare perpendicular, because the spin-flip scattering governed byG ↑↓at the interface is the largest. In the geometry depicted in Fig. 4, the charge current is applied along y, hence the spins accumulated at the YIG |NM interface due to SHE are oriented along x. The dashed lines plotted in Figs. 4(a)–4(c) are the prediction of the SMR theory. As can be seen, Eq. (3)explains the presence (absence) of resistance variation upon the appliedfield angles αandβ(γ). Due to demagnetizing effects, the magnetization of YIG is not always aligned with the appliedfield. This is the reason why the measured curves in Figs. 4(a) and4(b) have different shapes, and a simple calculation 30of the equilibrium position of Min combination with Eq. (3) reproduces them quite well. The SMR ratio was also calculated in Ref. 8: SMR=/Delta1R max R0=/Theta12 SH2λ2 sd σtNMG↑↓tanh2/parenleftbigtNM 2λsd/parenrightbig 1+2λsd σG↑↓coth/parenleftbigtNM λsd/parenrightbig.(4) As for the inverse spin Hall voltage VISH[see Eq. (2)], the SMR depends on all the transport parameters G↑↓,/Theta1SH, andλsd, which therefore cannot be extracted individually from a single measurement. In Sec. IV A , we will take advantage of the combined measurements of VISH(see Figs. 2and 3) and SMR (see Fig. 4) to do so. For now, it is interesting to point out that because both SHE and ISHE are at play in spin Hallmagnetoresistance, the SMR depends on the square of the spin Hall angle. This explains the positive SMR for both YIG |Pt and YIG |Ta, even though the spin Hall angles of Pt and Ta are opposite. Finally, it would have been interesting to measure the dependence of SMR on Ta thickness (the dependence on Ptthickness was studied in Refs. 22and29). Unfortunately, it was difficult to realize low noise four-point contacts to investigatethe faint magnetoresistance on the series of Ta samplesprepared to study V ISHversus tTa. From our attempts, we found that the SMR of YIG |Ta (10 nm) is less than 2 ×10−5.T h i s is consistent with the decrease predicted by Eq. (4)(assuming λTa sd=1.8 nm) with respect to the SMR /similarequal4×10−5measured for YIG |Ta (3 nm). IV . DISCUSSION A. Transport parameters As already discussed, both VISHand SMR depend on the set of transport parameters ( G↑↓,/Theta1SH,λsd). By studying VISHas a function of the NM thickness, the spin diffusion length can bedetermined, and we found that in Ta, λ Ta sd=1.8±0.7n m ,s e e Fig. 3. We mention here that from a similar study on YIG |Pt, λPt sd=3.0±0.5 nm could be inferred.17This value lies in the range of spin diffusion lengths reported on Pt, which span overalmost an order of magnitude, 26from slightly more than 1 nm up to 10 nm. We note that the spin diffusion length extracted from the YIG|Ta data of Fig. 3is somewhat shorter than the 2.7 nm inferred from nonlocal spin-valve measurements.23The value found for the spin diffusion length in Ta is short, but reasonableas it represents several times the electronic mean free paths,which are of the order of 0.4 nm. We note here that indeedTa is very resistive but still in the metallic-like regime withsheet resistances below 4 k /Omega1. However, the extraction of physical spin diffusion lengths in these measurements is at theheart of a present controversy. 26It seems indeed that nonlocal measurements give systematically larger values than thoseextracted from ISHE measurements of FMR spin pumping. Inthis respect, we would like to point out that perhaps the modelof Eq. (2)is too simple for the present problem as charge current (for resistivity measurements) and spin currents are intwo different directions: the former is in-plane while the latteris perpendicular to the plane. Hence the spin current has tocross one interface and interacts with the free surface of themetallic layer. It is not clear to us that the relevant quantity inthe problem is really the bulk λ sd. It is not impossible that the relevant spin diffusion length also depends on layer thickness,but this refinement is beyond the reach of the present paper. There is a direct way to get the spin mixing conductance of a FM |NM interface, by determining the increase of damping in the FM layer associated to spin pumping inthe adjacent NM layer. 7Due to its interfacial nature, this effect is inversely proportional to the thickness of the FM 174417-5C. HAHN et al. PHYSICAL REVIEW B 87, 174417 (2013) TABLE I. Transport parameters obtained from the analysis of inverse spin Hall voltage [Figs. 2and 3+Eq.(2)] and spin Hall magnetoresistance [Fig. 4+Eq.(4)] performed on YIG |Ta (1.5– 15 nm) and YIG |Pt (15 nm). YIG|Ta (1.5–15 nm) YIG |Pt (15 nm) σ(106/Omega1−1m−1)0 .08–0.75 2 .45±0.10 λsd(10−9m) 1 .8±0.7n /a [from 1.5 to 10]26 G↑↓(1013/Omega1−1m−2)4 .3±11 2 6.2±14 4 /Theta1SH −0.02±0.008 0.015 0.03±0.04 0.015 and can be measured only on ultrathin films. This was recently achieved in nanometer-thick YIG films grown bypulsed laser deposition, 31,32where spin mixing conductances G↑↓=(0.7−3.5)×1014/Omega1−1m−2have been reported for the YIG|Au interface. Even for 200-nm-thick YIG films as ours, it is possible to obtain the full set of transport parameters thanks to ourcombined measurements of V ISHand SMR on YIG |NM hybrid structures. In fact, from Eqs. (2)and(4), the ratio V2 ISH/SMR does not depend on /Theta1SH, which allows to determine G↑↓. Then, the last unknown /Theta1SHcan be found from the VISHor SMR signal. This is how we proceed to determine the transportparameters, which are collected in Table I. The drawback of this method is that it critically relies on (i) λ sd, which enters in the argument of exponential functions in Eqs. (2) and(4)and (ii) the angle of precession θin the inverse spin Hall experiment, since V2 ISH/SMR∝θ4. Our estimation of θ being within ±25%, the value extracted for G↑↓from the ratio V2 ISH/SMR can vary by a factor up to 8 due to this uncertainty. The spin Hall angle /Theta1SHis less sensitive to other parameters, still it can vary by a factor up to 3. This explains the ratherlarge error bars in Table I. In this study, we did not determine the spin diffusion length in Pt, hence we used the range ofvalues reported in the literature. 26 The spin mixing conductances determined from our com- bined VISHand SMR measurements on YIG |Ta and YIG |Pt bilayers lie in the same window as the ones determined frominterfacial increase of damping in YIG |Au, 31from inverse spin Hall voltage in BiY 2Fe5O12|Au and Pt,33and from first-principles calculations in YIG |Ag.25We would like to point out that despite the large uncertainty, G↑↓for YIG |Ta is likely less than for YIG |Pt. We note that the smaller damping measured in CoFeB |Ta compared to CoFeB |Pt was tentatively attributed to a smaller spin mixing conductance.20 The spin Hall angles that we report for Pt and Ta are both of a few percents. In particular, /Theta1Ta SH/similarequal− 0.02 lies in between the values determined from nonlocal spin-valve measurements(/similarequal−0.004) 23and from spin-torque switching using the SHE (/similarequal−0.12).20 The main conclusion, which arises from the summary pre- sented in Table I, is that the sets of transport parameters deter- mined for the hybrid YIG |Ta and YIG |Pt systems are quite sim- ilar. Apart from the opposite sign of /Theta1SHin Ta and Pt, the main difference concerns the conductivity: σβ−Tais roughly one order of magnitude smaller than σPt. This explains the large inverse spin Hall voltages that can be detected in our YIG |Ta bilayers (up to 70 μVa tP=+ 10 dBm), since from Eq. (2) VISH∝1/σ, which could be a useful feature of the Ta layer.B. Influence of a dc current on FMR linewidth Onsager reciprocal relations imply that if there is an ISHE voltage produced by the precession of YIG, there must also bea transfer of spin angular momentum from the NM conductionelectrons to the magnetization of YIG, through the finite spinmixing conductance at the YIG |NM interface. 25Therefore one would expect to be able to control the relaxation of theinsulating YIG by injecting a dc current in an adjacent strongspin-orbit metal, as it was shown on YIG |Pt in the pioneering work of Kajiwara et al. 6Although this direct effect is well established when the ferromagnetic layer is ultra-thin andmetallic, 34–37only a few works report on conclusive effects on micron-thick YIG6,38,39or provide a theoretical interpretation to the phenomenon.40 The 200-nm-thick YIG films that have been grown for this study are about six times thinner than the one used in Ref. 6, with an intrinsic relaxation close to bulk YIG. Because the spintransfer torque is an interfacial effect and sizable spin mixingconductances have been measured in our YIG |Ta and YIG |Pt bilayers, our samples must be good candidates to observethe direct effect of a dc current on the relaxation of YIG.Due to their large resistance, β-Ta films are not convenient to pass the large current densities required to observe such aneffect (large Joule heating). Therefore we have conducted theseexperiments only on the YIG |Pt films prepared in this work. The inverse spin Hall voltage measurements presented in Fig. 2have therefore been repeated in the presence of a dc current flowing through the Pt layer. This type of experiment,where a ferromagnetic layer is excited by a small amplitudesignal and a spin polarized current can influence the linewidthof the resonance, has already been reported on spin-valve spin-torque oscillators 41,42and NiFe |Pt bilayers.35,43The results obtained on our YIG (200 nm) |Pt (15 nm) at 77 K when the dc current is varied from −40 to +40 mA are displayed in Fig. 5. Let us now comment on these experiments. We first emphasize that the current injected in Pt is truly dc (not pulsed).A sizable Joule heating is thus induced, as reflected by theincrease of Pt resistance. As a consequence, the main effect ofdc current injection at room temperature is the displacementof the resonance towards larger field, due to the decrease ofthe YIG saturation magnetization M s. To avoid this trivial effect, we have performed these experiments directly in liquidnitrogen. In that case, the increase of Pt resistance is verylimited ( +0.2% at ±40 mA). We note that when cooled from 300 K down to 77 K, the peak of the inverse spin Hall voltagemeasured in the YIG |Pt bilayer is displaced towards lower field due to the increase of M sof YIG (from 140 to 200 emu /cm3), and its amplitude slightly decreases. The main conclusion that can be drawn from Fig. 5is that there is basically no effect of the dc current injected in Pt onthe YIG resonance. We stress that the maximal current densityreached in Pt in these experiments is J e=2.4×109Am−2, i.e., twice larger than the one at which YIG magnetizationoscillations were reported in Ref. 6. In our experiments, we are not looking for auto-oscillations of YIG, which requiresthat the damping is fully compensated by spin transfer torque,but only for some variation of the linewidth. The fact that wedo not see any change in the shape of the resonant peak of 174417-6COMPARATIVE MEASUREMENTS OF INVERSE SPIN HALL ... PHYSICAL REVIEW B 87, 174417 (2013) FIG. 5. (Color online) Inverse spin Hall voltage measured at 2.95 GHz ( P=+ 10 dBm) for YIG |Pt as a function of the dc current flowing in the Pt layer. A small current dependent offset ( <0.2μV) has been subtracted to the data. our 200-nm-thin YIG film is thus in contradiction with the observation of bulk auto-oscillations in thicker films.6 We have also performed similar experiments on the other YIG(200 nm) |Pt samples, which were prepared using the two different YIG films grown for this study. Although the currentdensity was increased up to 6 ×10 9Am−2, we were never able to detect any sizable variation of the linewidth of YIG. Instead,we have measured that the dc current can affect the inverse spinHall voltage in different ways. First, when a charge current isinjected into Pt, a non-zero offset of the lock-in signal can bedetected (it was subtracted in Fig. 5). This is due to the increase of Pt resistance induced by the microwave power, as it wasverified by monitoring this offset while varying the modulationfrequency of the microwave. Secondly, the amplitude of theV ISHpeaks can be affected by the dc current (but again, not the linewidth). This effect can at first be confused with someinfluence on the relaxation of YIG, because it displays theappropriate symmetries versus field and current. But instead,we have found that this is a bolometric effect: 44when the YIG is excited at resonance, it heats up, thereby heating theadjacent Pt whose resistance gets slightly larger. Hence anadditional voltage to V ISHis picked up on the lock-in due to the nonzero dc current flowing in Pt. Therefore, one shouldbe very careful in interpreting changes in inverse spin Hallvoltage as the indication of damping variation in YIG. Finally,we observed that at very large current density, the resonancepeak slightly shifts towards larger field due to Joule heating,even at 77 K. V . CONCLUSION In this paper, we have presented and analyzed a com- parative set of data of inverse spin Hall voltage VISHandmagnetoresistance obtained on YIG |Pt and YIG |Ta bilayers. We have detected the voltages generated by spin pumping atthe YIG |Pt interface (already well established) 6and at the YIG|Ta interface. Their opposite signs are assigned to the opposite spin Hall angles in Pt and Ta.24From the thickness dependence of VISH, we have been able to obtain the spin diffusion length in Ta, λTa sd=1.8±0.7 nm, in reasonable agreement with the value extracted from nonlocal spin valvemeasurements. 23From symmetry arguments, we have shown that the weak magnetoresistance measured on our hybridYIG|NM layers cannot be attributed to usual AMR, but is instead well understood in the framework of the recentlyintroduced spin Hall magnetoresistance (SMR). 8By taking advantage of the combined measurements of VISHand SMR performed on the same samples, we have been able to extractthe spin Hall angles in Pt and Ta, as well as the spin mixingconductances at the YIG |Pt and YIG |Ta interfaces. These transport parameters have all been found to be of the same order of magnitude as those already measured 20,31or predicted.25We believe that at least part of the discrepancies between the parameters evaluated in different works26depend on the details of the YIG |NM interface32and on the quality of the NM.18,19,23 Finally, we could not detect any change of linewidth in our YIG|Pt samples by passing large current densities through the Pt layer. One might argue that our high-quality 200-nm YIGthin films are still too thick to observe any appreciable effectof spin transfer torque, which is an interfacial mechanism, orthat the spin-waves which can auto-oscillate under the actionof spin transfer at the interface with Pt are different fromthe uniform mode that we excite with the microwave fieldin our experiments. 6,40If one would estimate the threshold current required to fully compensate the damping of all the 174417-7C. HAHN et al. PHYSICAL REVIEW B 87, 174417 (2013) magnetic moments contained in our YIG films,20,40Jth/similarequal 2eαωM stYIG/(/Theta1SHγ¯h), one would get current densities of about 1011Am−2. This is 20 times larger than the largest current density which we have tried. Thus the lack of avisible effect in our Fig. 5is not a real surprise in itself, but it is inconsistent with the results reported in Ref. 6. Future experiments on ultrathin YIG |NM hybrid films, in which thespin mixing conductance can be directly determined from the interfacial increase of damping, 31might give a definite answer to this point. ACKNOWLEDGMENT This research was supported by the French ANR Grant Trinidad (ASTRID 2012 program). *Corresponding author: gregoire.deloubens@cea.fr 1M. I. Dyakonov and V . I. Perel, JETP Lett. 13, 467 (1971). 2J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 3S. O. Valenzuela and M. Tinkham, Nature (London) 442, 176 (2006). 4T. Kimura, Y . Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. 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Buhrman, Science 336, 555 (2012). 21C.-F. Pai, L. Liu, Y . Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 101, 122404 (2012).22S. Y . Huang, X. Fan, D. Qu, Y . P. Chen, W. G. Wang, J. Wu, T. Y . C h e n ,J .Q .X i a o ,a n dC .L .C h i e n , Phys. Rev. Lett. 109, 107204 (2012). 23M. Morota, Y . Niimi, K. Ohnishi, D. H. Wei, T. Tanaka, H. Kontani,T. Kimura, and Y . Otani, P h y s .R e v .B 83, 174405 (2011). 24T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima, K. Yamada, and J. Inoue, P h y s .R e v .B 77, 165117 (2008). 25X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Europhys. Lett. 96, 17005 (2011). 26L. Liu, R. A. Buhrman, and D. C. Ralph, arXiv: 1111.3702 v3. 27V . Castel, N. Vlietstra, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 86, 134419 (2012). 28R. Hoogeveen, M. Moske, H. Geisler, and K. Samwer, Thin Solid Films 275, 203 (1996). 29N. Vlietstra, J. Shan, V . Castel, J. Ben Youssef, and B. J. van Wees, arXiv: 1301.3266 . 30A. G. Gurevich and G. A. 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PhysRevLett.99.227207.pdf

Microwave Assisted Switching of Single Domain Ni80Fe20Elements Georg Woltersdorf and Christian H. Back Universita ¨t Regensburg, Universita ¨tsstraße 31, 93040 Regensburg, Germany (Received 16 July 2007; published 30 November 2007; publisher error corrected 3 December 2007) We study the switching behavior of thin single domain magnetic elements in the presence of microwave excitation. The application of a microwave field strongly reduces the coercivity of the elements. We showthat this effect is most profound at the ferromagnetic resonance frequency of the elements. Observations using time-resolved magneto-optic Kerr microscopy in combination with pulsed microwave excitation further support that the microwave assisted switching process is indeed based on the coherent motion ofthe magnetization. DOI: 10.1103/PhysRevLett.99.227207 PACS numbers: 75.60.Jk, 75.30.Ds, 75.75.+a, 78.47.+p In magnetic recording the increasing data rates require fast magnetization reversal in small magnetic elements. Thin magnetic elements in the deep submicron range ofsizes favor a single domain state and often show a switch-ing behavior as expected by the Stoner-Wohlfarth theory.However, due to the absence of magnetization reversal processes based on domain wall motion or buckling, this implies that for coherent Stoner-Wohlfarth magneticswitching a field larger than the magnetic anisotropy fieldneeds to be applied along the magnetic easy axis for a duration of several nanoseconds. This long wait time is required since applying a magnetic field opposing themagnetization does not exert a torque on the magnetizationand only thermal fluctuation or a small initial misalignmentof the magnetic field can cause large angle precession of the magnetization. Finally the magnetization precesses into the direction parallel to the applied field due to re-laxation processes. This slow and—if based on thermalfluctuations—to a large extent unpredictable process canbe avoided in two ways. (i) Precessional or ballistic switch- ing; in this case the magnetic field is applied perpendicular to the magnetization and the torque term of the equation ofmotion is used [ 1–3] to drive the magnetization reversal. This type of switching, however, requires a careful timing of pulse length /.0028and magnetic field amplitude Bsince a deviation from the product of /.0028/.0001Bcan lead to multiple switching and a loss of control of the final state [ 4]. (ii) An alternative route is to use an rf field perpendicular to themagnetization in order to assist the magnetic switching. The feasibility of this method was shown in magnetic nanoparticles by superconducting quantum interferencemagnetometry [ 5] and recently by magnetic force micros- copy [ 6]. In this Letter time-resolved Kerr microscopy is employed to show that this mechanism is also present in micrometer-sized thin film elements. The static magneti-zation is probed by a time-resolved method based on themagneto-optic Kerr effect. For a single spin the magnetization dynamics can be described by the Landau-Lifshitz-Gilbert equation of mo- tion [ 7]: d~M dt/.0136/.0255/.0013/.00220/.0137~M/.0002~Heff/.0138/.0135/.0011 M/.0020 ~M/.0002d~M dt/.0021 ; (1) where~Mis the magnetization vector, ~Heffis the effective magnetic field, /.0011is the damping parameter, and /.0013/.0136 g/.0022B=@is the absolute value of the gyromagnetic ratio. The first term on the right-hand side determines the reso- nance frequency and the second term represents dampingand leads to relaxation. For the measurements, we use atime and spatially resolved ferromagnetic resonance(FMR) technique based on the time-resolved magneto-optical Kerr (TRMOKE) effect combined with continuouswave (cw) excitation [ 8]. The spatial resolution of our Kerr setup is roughly 300 nm. Details concerning this technique can be found in [ 9]. The magnetic sample is excited by means of a cw rf field which is phase-locked to the laserpulses. By measuring the TRMOKE signal as a function ofthe time delay between microwave signal and optical probeone obtains amplitude and phase of the magneticprecession. The layer structure of the samples is prepared by sputter deposition in UHV of 5 nm Al=2n mN i 80Fe20=200 nm Au=5n m Ti onto GaAs. A coplanar waveguide with a signal linewidth of 10/.0022mis subsequently defined by optical lithography and dry etching. The magnetic ele-ments are structured in a second e-beam lithography and dry etching step. Two different element sizes were inves-tigated:0:7/.00021:4/.0022m 2hexagons and 1:5/.00023:0/.0022m2hex- agons. The Ni80Fe20structures have a 1:2aspect ratio and a 45/.0014taper. The thickness of the magnetic elements is only 2 nm to ensure that the single domain state is the magnetic ground state [ 10]. A schematic outline of the sample structure with the relative orientations of the dc and rfmagnetic fields is shown in Fig. 1. In addition, an optical microscope image of a typical sample is shown in the upperinset of Fig. 2. An unpatterned region on top of the signal line (not shown) was used to characterize the 2 nm thickNi 80Fe20film. From the Kittel plot shown in Fig. 2, magnetic anisot- ropy,gfactor, and effective magnetization are extracted:PRL 99,227207 (2007)PHYSICAL REVIEW LETTERSweek ending 30 NOVEMBER 2007 0031-9007 =07=99(22)=227207(4) 227207-1 ©2007 The American Physical SocietyHA/.01360:1m T ,g/.01362:2,/.00220Meff/.01360:93 T . The damping parameter is determined from the frequency scan in abias field of 5 mT and amounts to /.0011/.01360:0085 . The reduced M effis expected and a consequence of the perpendicular interface anisotropy present in such thin Ni80Fe20films [11]. The zero-field resonance scan of the two hexagonal structures and the unpatterned films are compared in the lower inset of Fig. 2. The elongated hexagonal shape of the structures leads to a uniaxial shape anisotropy field alongthe long axis of the magnetic elements of about /.0022 0HA/.0136 0:5m T for the larger and /.00220HA/.01361:0m T for the smallerhexagon. This additional internal field in comparison with the extended film is evident in the 250 MHz and 700 MHzoffsets of the zero-field resonance frequency, as can be seen in the lower inset of Fig. 2. As a consequence of the shape anisotropy, the two possible magnetic states of theelements (along the long axis of the elements pointing tothe right or to the left) are thermally stable at room tem- perature. The energy barrier between the two states can be estimated from the uniaxial anisotropy field and leads—for example for the smaller element—to an energy barrier of 1 eV , which is far above k BTat 300 K. In the TRMOKE experiments indeed a stable single domain state is observedin zero field; see Fig. 3. This is important since we want to study the influence of a rf magnetic field on the switching behavior of magnetic elements immersed in an opposing magnetic field. The switching is studied by measuringhysteresis loops of an individual element. Figure 3shows a typical example. The loops are acquired by using the off- resonance dynamic response to the synchronized micro- wave excitation measured by time-resolved Kerr micros-copy. In the particular case shown in Fig. 3we measure at a fixed frequency of 2 GHz. However, the resonance field of the element is at about 5 mT when it is excited at 2 GHz;see the inset of Fig. 2. This means that a field sweep from /.02551m T to/.01351m T is always far away from resonance and the induced precessional motion of the magnetization is small. Nevertheless, the technique applied here allows oneto measure static hysteresis loops on individual magnetic FIG. 2 (color). The main graph shows the Kittel plot for the Ni80Fe20film. The resonance frequencies were determined by measuring the amplitude of the precession as a function offrequency at a low microwave power of 0 dBm. The inset on the top shows an optical microscope image of the microstruc- tures. In the inset on the bottom resonance scans are shown. Thered curve corresponds to the unpatterned Ni 80Fe20film. Blue and olive curves correspond to 1:5/.00023:0/.0022m2and0:7/.00021:4/.0022m2 hexagons, respectively. The microstructures have a built-in shape anisotropy of 0.5 mT and 1 mT leading to zero-field frequencyoffsets 250 and 700 MHz compared to the continuous film, respectively. The black dashed curve shows a resonance scan for the1:5/.00023/.0022m 2hexagon measured in a field of 5 mT.FIG. 1 (color online). Experimental configuration. A coplanar waveguide is used to excite the magnetic microstructures. The magnetic easy axis and the applied magnetic field are parallel to the coplanar waveguide. FIG. 3 (color online). Series of hysteresis loops measured for the1:5/.00023/.0022m2hexagon as a function of microwave power at a fixed frequency of 2 GHz. The microwave frequency is 2 GHzwith a power of (a) 8 dBm, (b) 15 dBm, (c) 16 dBm, and (d) 18 dBm. As the magnetization reverses, the sign of the magnetic response is inverted; see image scans in (a). The slightincrease of the signal with increasing field is due to the fact thatthe amplitude of the magnetic precession increases with field since the resonance of the element at 2 GHz occurs at about 5m T .PRL 99,227207 (2007)PHYSICAL REVIEW LETTERSweek ending 30 NOVEMBER 2007 227207-2elements with an extremely high sensitivity: in the experi- ment the synchronized microwaves are chopped and lock-in detection is used. If the microwave field is applied in- plane and perpendicular to the magnetic easy axis of the elements, as illustrated in Fig. 1the phase of the out-of- plane magnetic response to the microwaves changes by180 /.0014(and hence the signal changes its sign) when the magnetization switches; see Fig. 3. Monitoring the polar Kerr signal as a function of the applied magnetic fieldtherefore allows one to measure the magnetic hysteresisfor individual elements as small as a few hundred nm. When hysteresis loops are measured as a function of microwave power at a fixed frequency one observes agradually decreasing coercive field with increasing micro-wave power while the square shape of the hysteresis loop remains; Figs. 3and4show the coercive field as a function of microwave power measured at a fixed frequency of2 GHz. At a microwave power below 5 dBm the coercivityis independent of the power for both hexagonal element sizes. At a certain threshold power the coercivity is rapidly reduced to zero. The larger threshold power for the smallerhexagon can be expected due to its larger shape anisotropy. In a further experiment this collapse of the hysteresis was measured for applied microwave frequencies between 0.08and 2.0 GHz. The result is shown in Fig. 5in a 2D plot. One can clearly see that for a fixed microwave power the reduction of the coercivity is strongest at the 500 MHz resonance frequency of the 1:5/.00023:0/.0022m 2hexagon; cf. Fig. 2. This can be expected if the reduced coercivity is indeed caused by coherent motion of the magnetization dynamics, allowing the magnetization to spiral out of its local energy minimum when an opposing magnetic field isapplied. Nembach et al. also recently observed a micro- wave induced reduction of the coercivity for much larger multidomain elliptical Ni 80Fe20elements ( 160/.0022m/.000280/.0022m2)[12]. This reduction was explained by an en- hanced domain wall nucleation in a microwave field [ 13] and preferential entropy-driven domain growth in a trans-verse microwave field [ 14]. In this Letter we show that for micron-sized single domain magnetic elements the micro- wave assisted switching is actually based on a coherentmotion of the magnetization. It is important to exclude the possibility that the reduc- tion of the coercive fields is due to thermal heating effects. The large thickness of the Au waveguide underneath the 2 nm thick Ni 80Fe20film provides an efficient heat sink and the applied microwave power is small. The maximumapplied power of 20 dBm corresponds to an in-plane rf- field amplitude of h rf/.00244m T atf/.01362 GHz for the 10/.0022mwide signal line. One can estimate the power absorbed at FMR using PFMR/.0136/.0025f/.003100h2 rfAtFM[15], where /.003100/.002450is the imaginary part of the susceptibility at FMR andAtFMis the volume of the ferromagnetic sample. An upper limit for the expected heating /.0001Tat resonance can be estimated from steady state heating by the thermal conductivity of the substrate using the following relation: /.0001T/.0136tPFMR /.0021, where /.0021/.013655 W=mK for GaAs and t/.0136 200/.0022mis the thickness of the GaAs substrate. These pessimistic considerations only lead to a temperature in- crease of less than 0.2 K at FMR, in line with recent resultsreported in Ref. [ 16]. In addition to the magnetic dissipa- tion there are also electrical losses in the 200 nm thick Au waveguide. Here the temperature increase can be estimatedin the following way: in the frequency range of interest the waveguide has a transmission of more than T/.013690%. The 10/.0022mwide signal line section has a length of 1 mm. If one assumes that the losses are uniformly distributed along this length a dissipation of 10% of P/.013620 dBm would lead to a temperature increase of only /.0001T/.0136 t/.01331/.0255T/.0134P /.0021/.00244K. From these simple estimates one can see that heating FIG. 4 (color online). Coercive field of the of the 0:7/.0002 1:4/.0022m2(diamonds) and 1:5/.00023:0/.0022m2(circles) hexagons as a function of microwave power at a frequency of 2 GHz. FIG. 5 (color). Coercive fields as a function of microwavepower and frequency measured for the 1:5/.00023:0/.0022m 2hexagon. The zero-field resonance occurs at about 500 MHz (see lowerinset in Fig. 2). The coercivity is color-coded and ranges be- tween 0 and 0.4 mT. In addition to the power scale also a scale with the corresponding rf magnetic field is shown.PRL 99,227207 (2007)PHYSICAL REVIEW LETTERSweek ending 30 NOVEMBER 2007 227207-3cannot play a significant role in the reduction of the coercivity. In order to further support the notion that the microwave assisted switching is related to a coherent motion of themagnetization, microwave bursts are used to study theswitching behavior. Using a microwave mixer and a pulse generator short bursts with a carrier frequency of 2 GHz are produced from a 2 GHz cw signal. The rise and fall timesof the pulses are about 200 ps and pulse lengths between0.4 and 2.5 ns are studied. Typical microwave bursts gen-erated by this method are shown in the inset of Fig. 6.I n these measurements again the static magnetization isprobed by the sign of the magnetic response to the micro-wave burst excitation. Figure 6shows the coercive field as a function of the microwave pulse width at a constant microwave power of 18 dBm. The coercivity as a functionof microwave burst duration is reduced in an oscillatingmanner. Dips of the coercive field are evidenced when thepulse period corresponds to a multiple of the period of the2 GHz carrier frequency of the microwave burst (0.5 ns,1.0 ns, and 1.5 ns). As the pulse length grows the preces-sional amplitude grows and it becomes more and more likely that the energy barrier may be overcome. This behavior is consistent with the picture that a coherentspin precession is the leading mechanism of the reducedcoercivity. It is worthwhile to point out that although theduty cycle in the pulsed experiment is reduced by morethan a factor of 6 compared to the experiments with cwexcitation the peak amplitude required to cause a reducedcoercivity remains nearly unchanged. When the peak amplitude of the burst was reduced such that even for longer bursts (several ns) a finite coercivitycan still be observed, a reduction of the coercivity is only observed for pulse lengths below approximately 2 ns. Forlonger pulses the coercivity is not reduced any further. This can be expected by considering the decay time of magnetic excitations. The decay time is determined fromthe damping parameter obtained from the frequency widthof resonance scans at high fields (not shown): /.0028/.0136 2=/.0137/.0011/.0013/.0022 0/.0133Meff/.01352Heff/.0134/.0138 /.01361:4n s [17]. This implies that it is not possible to pump more energy into the magneticsystem after 2 ns. The presence of this behavior also showsthat thermal effects can be excluded since heating would simply lead to a continuous decrease of the coercivity as a function of pulse length. In conclusion, it was shown that the coercive fields in micron-sized uniaxial magnetic elements can be signifi-cantly reduced in the presence of a rf magnetic fieldapplied in-plane and perpendicular to the magnetization.This effect was shown to be a threshold effect. In addition,the reduction of the coercivity is strongest at the zero-field resonance frequency of the magnetic element itself. Measurements with short microwave pulses as a functionof pulse length clearly show that the reduction of thecoercive field is a consequence of coherent motion of themagnetization. The method which was used to measure thehysteresis loops by using the sign of the dynamic responseas introduced in this Letter is new and provides an unpre-cedented signal-to-noise ratio allowing one to study mag- netic switching behavior of individual submicron magnetic elements by magnetic Kerr effect microscopy. Financial support from the DFG Priority Program No. 1133 and Sonderforschungsbereich No. 689 is grate-fully acknowledged. [1] C. H. Back et al. , Phys. Rev. Lett. 81, 3251 (1998). [2] T. Gerrits et al. , Nature (London) 418, 509 (2002). [3] W. Hiebert, L. Lagae, and J. de Boeck, Phys. Rev. B 68, 020402 (2003). [4] H. Schumacher et al. , Phys. Rev. Lett. 90, 017204 (2003). [5] C. Thirion, W. Wernsdorfer, and D. Mailly, Nature Mater. 2, 524 (2003). [6] Y . Nozaki et al. , Appl. Phys. Lett. 91, 082510 (2007). [7] L. D. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935). [8] S. Tamaru et al. , J. Appl. Phys. 91, 8034 (2002). [9] I. Neudecker et al. , J. Magn. Magn. Mater. 307, 148 (2006). [10] R. Cowburn et al. , Phys. Rev. Lett. 83, 1042 (1999). [11] J. O. Rantschler et al. , J. Appl. Phys. 97, 10J113 (2005). [12] H. T. Nembach et al. , Appl. Phys. Lett. 90, 062503 (2007). [13] E. Schloemann, IEEE Trans. Magn. 11, 1051 (1975). [14] A. Krasyuk et al. , Phys. Rev. Lett. 95, 207201 (2005). [15] A. G. Gurevitch, Ferrites at RF Frequencies (Springer, New York, 1960), Chap. 1, p. 1. [16] R. Meckenstock et al. , J. Appl. Phys. 99, 08C706 (2006). [17] G. Woltersdorf et al. , Phys. Rev. Lett. 95, 037401 (2005).FIG. 6 (color online). Pulse length dependence of the coercive field measured for the 0:7/.00021:4/.0022m2hexagon. The pulse length is varied in 30 ps steps between 0.4 and 2.5 ns. The microwave carrier frequency is 2 GHz. The inset shows a 500 ps microwave burst acquired with a fast oscilloscope and the correspondingmagnetic response.PRL 99,227207 (2007)PHYSICAL REVIEW LETTERSweek ending 30 NOVEMBER 2007 227207-4

PhysRevLett.97.077205.pdf

Theoretical Limit of the Minimal Magnetization Switching Field and the Optimal Field Pulse for Stoner Particles Z. Z. Sun and X. R. Wang Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR, China (Received 7 May 2006; published 18 August 2006) The theoretical limit of the minimal magnetization switching field and the optimal field pulse design for uniaxial Stoner particles are investigated. Two results are obtained. One is the existence of a theoreticallimit of the smallest magnetic field out of all possible designs. It is shown that the limit is proportional to the damping constant in the weak damping regime and approaches the Stoner-Wohlfarth (SW) limit at large damping. For a realistic damping constant, this limit is more than 10 times smaller than that of so-called precessional magnetization reversal under a noncollinear static field. The other is on the optimal field pulse design: if the magnitude of a magnetic field does not change, but its direction can vary during a reversal process, there is an optimal design that gives the shortest switching time. The switching timedepends on the field magnitude, damping constant, and magnetic anisotropy. DOI: 10.1103/PhysRevLett.97.077205 PACS numbers: 75.60.Jk, 75.75.+a, 85.70.Ay Fabrication [ 1,2] and manipulation [ 3] of magnetic single-domain nanoparticles (also called the Stoner parti-cles) are of great current interests in nanotechnology and nanosciences because of their importance in spintronics. Magnetization reversal, which is about how to switch amagnetization from one state to another, is an elementaryoperation. One important issue is how to switch a magne-tization fast by using a small switching field. The switchingfield can be a laser light [ 4], or a spin-polarized electric current [ 5,6], or a magnetic field [ 7,8]. Many reversal schemes [ 9,10] have been proposed and examined. However, the issue of theoretical limits of the smallest switching field and the shortest switching time under allpossible schemes are not known yet. Here we report twotheorems on the magnetic-field induced magnetizationreversal for uniaxial Stoner particles. One is about thetheoretical limit of the smallest possible switching field.The other is about the optimal field pulse for the shortestswitching time when the field magnitude is given. Magnetization ~M/.0136~mM of a Stoner particle can be conveniently described by a polar angle /.0018and an azimuthal angle/.0030, shown in Fig. 1(a), because its magnitude Mdoes not change with time. The dynamics of magnetization unit direction ~mis governed by the dimensionless Landau- Lifshitz-Gilbert (LLG) equation [ 3,8], /.01331/.0135/.00112/.0134d~m dt/.0136/.0255~m/.0002~ht/.0255/.0011~m/.0002/.0133~m/.0002~ht/.0134;(1) where/.0011is a phenomenological damping constant whose typical value ranges from 0.01 to 0.22 for Co films [ 11]. The total field ~ht/.0136~h/.0135~hicomes from an applied field ~h and an internal field ~hi/.0136/.0255 r ~mw/.0133~m/.0134due to the magnetic anisotropic energy density w/.0133~m/.0134. Different particle is char- acterized by different w/.0133~m/.0134. In our analysis, we assume it uniaxial with the easy axis along the zdirection, w/.0136 w/.0133cos/.0018/.0134and~hi/.0136/.0255@w/.0133cos/.0018/.0134 @/.0133cos/.0018/.0134^z/.0017f/.0133cos/.0018/.0134^z.According to Eq. ( 1), each field generates two motions, a precession motion around the field and a damping motiontoward the field as shown in Fig. 1(a). In terms of /.0018and/.0030, Eq. ( 1) can be rewritten as [ 3] /.01331/.0135/.00112/.0134_/.0018/.0136h/.0030/.0135/.0011h/.0018/.0255/.0011f/.0133cos/.0018/.0134sin/.0018; /.01331/.0135/.00112/.0134sin/.0018_/.0030/.0136/.0011h/.0030/.0255h/.0018/.0135f/.0133cos/.0018/.0134sin/.0018:(2) Hereh/.0018andh/.0030are the field components along ^e/.0018/.0255and ^e/.0030/.0255directions of ~m, respectively. The switching problem is as follows: in the absence of an external field, the particle has two stable states, ~m0 (pointA) and/.0255~m0(pointB) along its easy axis as shown in Fig. 1(b). Initially, the magnetization is ~m0, and the goal is to reverse it to /.0255~m0by applying an external field. In our analysis, Gilbert damping constant /.0011and the anisotropy f/.0133cos/.0018/.0134are the fixed specifications of the problem, and only applied field variations are investigated. This is incontrast with earlier studies [ 12] where completely differ- ent analysis was performed. There are an infinite number of paths that connect the initial and the target state. L1 and h m−m h−m (m h ) θ(a) y x xyL1 L2Az z Bφ(b) FIG. 1. (a) Two motions of magnetization ~munder field ~h: /.0255~m/.0002~hand/.0255~m/.0002/.0133~m/.0002~h/.0134describe the precession and dis- sipation motions, respectively. (b) Points AandBrepresent the initial and the target states, respectively. The solid curve L1 anddashed curve L2 illustrate two possible reversal routes.PRL 97,077205 (2006)PHYSICAL REVIEW LETTERSweek ending 18 AUGUST 2006 0031-9007 =06=97(7)=077205(4) 077205-1 ©2006 The American Physical SocietyL2 in Fig. 1(b) are two examples. Each of these paths can be used as a magnetization reversal route (path). Let ~hL;s/.0133t/.0134 be the magnetic field pulse of design salong magnetization reversal route L. To proceed, a few quantities must first be introduced. Definition of switching field HL;s.—The switching field HL;sof design salong route Lis defined to be the largest magnitude of ~hL;s/.0133t/.0134for allt,HL;s/.0136maxfj~hL;s/.0133t/.0134j;8tg. Definition of minimal switching field HLon reversal routeL.—The minimal switching field along route Lis defined to be the smallest value of HL;sfor all possible designssthat will force the magnetization to move along L, i.e.,HL/.0136minfHL;s;8sg. Definition of theoretical limit of minimal switching field Hc.—The switching field limit Hcis defined as the smallest value of HLout of all possible routes, i.e., Hc/.0136 minfHL;8Lg. If the applied field is restricted to be static, reversal of a magnetization from AtoBcan only go through so-called ‘‘ringing motion’’ [ 7,8]. The corresponding switching field forms so-called modified Stoner-Wohlfarth (SW) astroid [7]. Strictly speaking, these switching fields are not HLthat exists only for those ballistic reversal paths [ 8]. With the above remark about a static field, we come back to the firstissue about H c. Theorem 1.— For a given uniaxial magnetic anisotropy specified by f/.0133cos/.0018/.0134/.0136/.0255@w/.0133cos/.0018/.0134 @/.0133cos/.0018/.0134, the theoretical limit of the minimal switching field is given by Hc/.0136/.0011/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 1/.0135/.00112pQ, whereQ/.0136maxff/.0133cos/.0018/.0134sin/.0018g,/.00182/.01370;/.0025/.0138. Proof.— To find the lowest possible switching field, it should be noticed that field along the radius direction hrof an external field does not appear in Eq. ( 2). Thus one can lower the switching field by always putting hr/.01360, and the magnitude of the external field is h/.0136/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 h2 /.0018/.0135h2 /.0030q . According to Eq. ( 2),_/.0018and _/.0030are fully determined by h/.0018 andh/.0030and vice versa. It can be shown that h2can be expressed in terms of /.0018,/.0030,_/.0018, and _/.0030 g/.0017h2 /.0136/.01331/.0135/.00112/.0134_/.00182/.01352/.0011f/.0133cos/.0018/.0134sin/.0018_/.0018/.0135/.0133/.0011sin/.0018_/.0030/.01342 /.0135sin2/.0018/.0137_/.0030/.0255f/.0133cos/.0018/.0134/.01382: (3) Hereg/.0133_/.0018;/.0018; _/.0030/.0134does not depend explicitly on /.0030for a uniaxial model. In order to find the minimum of g, it can be shown that /.0030 must obey the following equation: _/.0030/.0136f/.0133cos/.0018/.0134=/.01331/.0135/.00112/.0134; (4) which is from@g @_/.0030j/.0133_/.0018;/.0018/.0134/.01360and@2g @_/.00302j/.0133_/.0018;/.0018/.0134>0. Equation ( 4) is a necessary condition for the smallest minimal switching field. This can be understood as fol- lows. Assume Hcis the minimal switching field along reversal path Ldescribed by /.0018/.0133t/.0134/.0136/.00181/.0133t/.0134and/.0030/.0133t/.0134/.0136 /.00301/.0133t/.0134[i.e.,Hcis the maximum magnitude of the externalfield that generates the motion of /.00181/.0133t/.0134and/.00301/.0133t/.0134]. If/.00301/.0133t/.0134 does not satisfy Eq. ( 4), then one can construct another reversal path L/.0003specified by /.0018/.0133t/.0134/.0136/.00181/.0133t/.0134and/.0030/.0133t/.0134/.0136 /.00302/.0133t/.0134, where/.00302/.0133t/.0134satisfies Eq. ( 4). Because /.0018/.0133t/.0134and _/.0018 are exactly the same on both paths LandL/.0003at an arbitrary timet, the values of g/.0133t/.0134shall be smaller on L/.0003than those onLat anyt. Thus, the maximum g/.0003/.0136/.0133H/.0003c/.01342ofgonL/.0003 will be also smaller than that ( H2c)o nL, i.e.,H/.0003c<Hc. But this is in contradiction with the assumption that Hcis the theoretical limit of the minimal switching field. Hence,/.0030/.0133t/.0134must obey Eq. ( 4) on the optimal path that generates the smallest switching field, H c. Substituting Eq. ( 4) into Eq. ( 3), we have h2/.0136/.0137/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 1/.0135/.00112p_/.0018/.0135/.0011f/.0133cos/.0018/.0134sin/.0018=/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 1/.0135/.00112p /.01382:(5) In order to complete a magnetization reversal, the tra- jectory must pass through all values of 0/.0020/.0018/.0020/.0025.I n particular, it must pass through whatever value of /.0018in that range maximizes f/.0133cos/.0018/.0134sin/.0018on that range. At that maximizing value of /.0018, the trajectory must be such that/.0018is nondecreasing, that is _/.0018/.00210, so that the trajec- tory is proceeding in the correct direction. Substitutingthese constraints into Eq. ( 5), we see that at that point in the trajectory, hmust be at least /.0011Q/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 1/.0135/.00112p , where Q/.0017maxff/.0133cos/.0018/.0134sin/.0018g.—QED. To have a better picture about what this theoretical limit Hcis, we consider a well-studied uniaxial model, w/.0133~m/.0134/.0136 /.0255km2z=2,o rf/.0136kcos/.0018. It is easy to show that the largest h is at/.0018/.0136/.0025=4so thatQ/.0136k=2, and Hc/.0136/.0011/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 1/.0135/.00112pk 2: (6) At small damping, Hcis proportional to the damping constant. The result in the limit of /.0011!0coincides with the switching field in Ref. [ 9] where the time-dependent field always follows the motion of magnetization. At thelarge damping, H capproaches the SW field [ 8] when a noncollinear static switching field is 135/.0014from the easy axis. The solid curve in Fig. 2isHcversus/.0011. For com- parison, the minimal switching fields from other reversalschemes are also plotted. The dotted line is the minimal switching field when the applied field is always parallel to FIG. 2. The switching field hcvs damping constant /.0011under different reversal schemes.PRL 97,077205 (2006)PHYSICAL REVIEW LETTERSweek ending 18 AUGUST 2006 077205-2the motion of the magnetization [ 9]. The curve in square symbols is the minimal switching field when a circularlypolarized microwave at optimal frequencies is applied [ 9]. The dashed line is minimal switching field under a non- collinear static field of 135 /.0014to the easy axis. It saturates to the SW field beyond /.0011c[7,8]. Although the theoretical limit of the switching field is academically important because it provides a low bound tothe switching field so that one can use the theorem toevaluate the quality of one particular strategy, a designusing a field at the theoretical limit would not be interesting from a practical point of view because the switching time would be infinitely long. Thus, it is more important todesign a reversal path and a field pulse such that thereversal time is the shortest when the field magnitude H (H>H c) is given. An exact result is given by the follow- ing theorem. Theorem 2.— Suppose a field magnitude Hdoes not depend on time and H>H c. The optimal reversal path (connects /.0018/.01360and/.0018/.0136/.0025) that gives the shortest switch- ing time is the magnetization trajectory generated by the following field pulse ~h/.0133t/.0134, hr/.0133t/.0134/.01360;h /.0018/.0133t/.0134/.0136/.0011H=/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 1/.0135/.00112p ; h/.0030/.0133t/.0134/.0136H=/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 1/.0135/.00112p /.0136h/.0018=/.0011:(7) Proof.— The reversal time from AtoB[Fig. 1(b)]i sT/.0017R/.0025 0d/.0018= _/.0018. According to Eq. ( 2), one needs /.0133h/.0030/.0135/.0011h/.0018/.0134to be as large as possible in order to make _/.0018maximal at an arbitrary/.0018. SinceH2/.0136h2r/.0135h2 /.0018/.0135h2 /.0030, one has the follow- ing identity: /.01331/.0135/.00112/.0134H2/.0136/.01331/.0135/.00112/.0134h2r/.0135/.0133h/.0030/.0135/.0011h/.0018/.01342/.0135/.0133h/.0018 /.0255/.0011h/.0030/.01342: (8) Thus, (h/.0030/.0135/.0011h/.0018) reaches the maximum of/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 1/.0135/.00112p H whenhr/.01360andh/.0018/.0136/.0011h/.0030, which lead to Eq. ( 7), are satisfied.—QED The optimal shape of field pulse ( 7) appears to depend only on the Gilbert damping constant /.0011and not on f/.0133cos/.0018/.0134. However, those expressions provide the components offield magnitude in a coordinate system relative to the time-varying direction of ~m. The magnetic anisotropy f/.0133cos/.0018/.0134in part determines the trajectory of ~mwhich in turn determines the optimal pulse shape when combinedwith the expressions of Eq. ( 7). It should be pointed out that if they were to change f/.0133cos/.0018/.0134and nothing else, the time-dependent field pulse would be different. Under the optimal design of ( 7),/.0030/.0133t/.0134and/.0018/.0133t/.0134satisfy, respectively, Eq. ( 4) and _/.0018/.0136H=/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 1/.0135/.00112p /.0255/.0011f/.0133cos/.0018/.0134sin/.0018=/.01331/.0135/.00112/.0134:(9) For uniaxial magnetic anisotropy w/.0133~m/.0134/.0136/.0255km2z=2,i ti s straightforward to integrate Eq. ( 9), and to find the reversal timeTfromAtoB[Fig. 1(b)], T/.01362 k/.0133/.00112/.01351/.0134/.0025/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 4/.0133/.00112/.01351/.0134H2=k2/.0255/.00112p : (10) In the weak damping limit /.0011!0,T/.0025/.0025=H while in the large damping limit /.0011!1 ,T/.0025/.0011/.0025/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 H2/.0255k2=4p !1 . For the large field H!1 ,T/.0025/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 /.00112/.01351p /.0025 H, inversely proportional to the field strength. Thus, it is better to make /.0011as small as possible. Then the critical field is low, and the speed is fast(T/.0024/.0025=H ). Figure 3shows the field dependence of the switching time for /.0011/.01360:1, whereTandHare in the units of2=kandk=2, respectively. How much could the so-called ballistic (precessional) reversal strategy [ 7,13] be improved? To answer the ques- tion, let us compare the switching field and time in theballistic reversal with those of theoretical limits for uniax-ial magnetic anisotropy w/.0133~m/.0134/.0136/.0255km 2z=2and/.0011/.01360:1. According to Ref. [ 8], the smallest switching field (in unit ofk=2) for the ballistic connection is H/.01361:02ap- plied in 97.7/.0014to the easy zaxis, and the corresponding ballistic reversal time (in unit of 2=k)i sT/.01365:87. On the other hand, the theoretical limit for the minimal switching field isHc/.00250:1from Eq. ( 6), about one tenth of the minimal switching field in the ballistic reversal [ 8]. For realistic value of /.0011of order of 0.01, the difference between experimentally achieved low switching field and the theo-retical limit is of the order of hundred times. Thus there is avery large room for an improvement. It is also possible toswitch a magnetization faster than that of the conventional ballistic reversal by using a smaller field. For example, to achieve a reversal time of T/.01365:87along the optimal route, the field magnitude can be as lower as H/.01360:547 (instead of H/.01361:02) according to Fig. 3. To illustrate what scale of theoretical reversal is being demonstratedhere, let us consider bulk fcc Co parameters of anisotropyconstantK/.01362:7/.000210 6erg=cm3and saturation magneti- zationMs/.01361445 emu =cm3[14]. Thus the dimensionless reversal time of T/.01365:87and switching field H/.01360:547 correspond to 178psand0:1T, respectively. The field pulse given in Eq. ( 7) requires a constant adjustment of field direction during the magnetization FIG. 3. The field dependence of Tunder the optimal field pulse Eq. ( 7) for/.0011/.01360:1. The field is in the unit of k=2and the unit for time is 2=k.PRL 97,077205 (2006)PHYSICAL REVIEW LETTERSweek ending 18 AUGUST 2006 077205-3reversal. To have a better idea about the type of fields required, we plot in Fig. 4(a) the time dependence of x, y, andzcomponent of the field while its magnitude is kept atH/.01360:547. The time dependence of /.0018and/.0030is also plotted in Fig. 4(b) and4(c). Although the Stoner-Wohlfarth problem of magnetiza- tion reversal for a uniaxial model is of great relevance to the magnetic nanoparticles, it is interesting to generalize the results to the nonuniaxial cases. So far, our results areon the magnetic-field induced magnetization reversal; itwill be extremely important to generalize the results to the spin-torque induced magnetization reversal. It should also be pointed out that it is an experimental challenge to createa time-dependent field pulse given by Eq. ( 7) in order to implement the optimal design reported here. This chal- lenge could be met if a device sensitive to the motion ofa magnetization can be found because a coil can be at-tached to the device to generate the required field. In principle, one may also use three mutually perpendicular coils to generate a given time-dependent field. This can beaccomplished by controlling time-dependent electric cur-rents through the coils. In conclusion, the theoretical limit of the magnetization switching field for uniaxial Stoner particles is obtained. The limit is proportional to the damping constant at weakdamping and approaches the SW field at large damping.When the field magnitude is kept to a constant, and the field direction is allowed to vary, the optimal field pulse and reversal time are obtained. This work is supported by UGC, Hong Kong, through RGC CERG grants (No. 603106). A discussion withProfessor J. Shi is acknowledged.[1] Shouheng Sun, C. B. Murray, D. Weller, L. Folks, and A. Moser, Science 287, 1989 (2000); D. Zitoun, M. Respaud, M.-C. Fromen, M. J. Casanove, P. Lecante,C. Amiens, and B. Chaudret, Phys. Rev. Lett. 89, 037203 (2002). [2] M. H. Pan, H. Liu, J. Z. Wang, J. F. Jia, Q. K. Xue, J. L. Li, S. Qin, U. M. Mirdaidov, X. R. Wang, J. T. Market, Z. Y .Zhang, and C. K. Shih, Nano Lett. 5, 87 (2005). [3] Spin Dynamics in Confined Magnetic Structures I & II , edited by B. Hillebrands and K. Ounadjela (Springer-Verlag, Berlin, 2001); J. Miltat, G. Albuquerque, and A. Thiaville, in Spin Dynamics in Confined Magnetic Structures I , edited by B. Hillebrands and K. Ounadjela (Springer-Verlag, Berlin, 2001). [4] M. V omir, L. H. F. Andrade, L. Guidoni, E. Beaurepaire, and J.-Y . Bigot, Phys. Rev. Lett. 94, 237601 (2005). [5] J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); L. Berger, Phys. Rev. B 54, 9353 (1996). [6] M. Tsoi et al. , Phys. Rev. Lett. 80, 4281 (1998); E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. 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Wetzels, G. E. W. Bauer, and O. N. Jouravlev, Phys. Rev. Lett. 96, 127203 (2006). [11] C. H. Back, D. Weller, J. Heidmann, D. Mauri, D. Guarisco, E. L. Garwin, and H. C. Siegmann, Phys. Rev. Lett. 81, 3251 (1998); C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Weller, E. L. Garwin, and H. C. Siegmann, Science 285, 864 (1999). [12] R. Kikuchi, J. Appl. Phys. 27, 1352 (1956). [13] H. W. Schumacher, C. Chappert, P. Crozat, R. C. Sousa, P. P. Freitas, J. Miltat, J. Fassbender, and B. Hillebrands, Phys. Rev. Lett. 90, 017201 (2003); H. W. Schumacher, C. Chappert, R. C. Sousa, P. P. Freitas, and J. Miltat, Phys.Rev. Lett. 90, 017204 (2003). [14] J. P. Chen, C. M. Sorensen, K. J. Klabunde, and G. C. Hadjipanayis, Phys. Rev. B 51, 11 527 (1995).FIG. 4. Time dependence of different field components, /.0018and /.0030,o f~mfor uniaxial magnetic anisotropy w/.0133~m/.0134/.0136/.0255km2z=2with /.0011/.01360:1andH/.01360:547when the reversal path is optimal. The reversal time is T/.01365:87. (a)x,y, andzcomponents of magnetic field. (b) /.0018/.0133t/.0134. (c)/.0030/.0133t/.0134.PRL 97,077205 (2006)PHYSICAL REVIEW LETTERSweek ending 18 AUGUST 2006 077205-4

PhysRevApplied.7.034004.pdf

Gilbert Damping Parameter in MgO-Based Magnetic Tunnel Junctions from First Principles Hui-Min Tang1and Ke Xia1,2,* 1The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing 100875, China 2Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China (Received 26 November 2016; revised manuscript received 14 February 2017; published 6 March 2017) We perform a first-principles study of the Gilbert damping parameter ( α) in normal-metal/MgO-cap/ ferromagnet/MgO-barrier/ferromagnetic magnetic tunnel junctions. The damping is enhanced by interface spin pumping, which can be parametrized by the spin-mixing conductance ( G↑↓). The calculated dependence of Gilbert damping on the thickness of the MgO capping layer is consistent with experimentand indicates that the decreases in αwith increasing thickness of the MgO capping layer is caused by suppression of spin pumping. Smaller αcan be achieved by using a clean interface and alloys. For a thick MgO capping layer, the imaginary part of the spin-mixing conductance nearly equals the real part, and thelarge imaginary mixing conductance implies that the change in the frequency of ferromagnetic resonance can be observed experimentally. The normal-metal cap significantly affects the Gilbert damping. DOI: 10.1103/PhysRevApplied.7.034004 I. INTRODUCTION Magnetic tunnel junctions (MTJs) with an epitaxial (001)- oriented MgO barrier exhibit a large tunnel magnetoresistance ratio (TMR) at room temperature, which makes them a goodcandidate for applications in magnetic devices [1–6].A p a r t from this large TMR ratio, perpendicular magnetic anisotropy (PMA) and magnetic damping ( α)i nC o F e B =MgO=CoFeB MTJs have been investigated in detailed studies [7–10].T h e MTJ is a promising candidate as memory cells in spin- transfer-torque magnetic random-access memory (STT MRAM) because of their high thermal stability, low STTswitching current, and high magnetoresistance (MR). To produce STT MRAM with long retention time [11]as well as spin-torque oscillators and diodes with stable precession[12–14],al a r g e rP M Ai sn e c e s s a r y .I na d d i t i o n ,t h es w i t c h i n g current in STT-based devices is proportional to α[15–20],s oa small αis desired. Recent experiments show that using a MgO capping layer on CoFeB =MgO=CoFeB (or FeB) MTJs can increase the PMA and decrease α[21–24].T h o u g ht h i c k e n i n gt h eM g O capping layer decreases αand increases the PMA, it also decreases the TMR [25], so a moderately thick MgO capping layer is preferable. Tsunegi et al. [22]found α∼0.0054 and PMA∼3.3erg=cm 2in a MgO-barrier/FeB/MgO-cap struc- ture with a 0.6-nm MgO capping layer using the spin-torquediode effect. The decreases in αappear to come from suppression of spin pumping [26,27] at the ferromagnet – normal-metal (FM/NM) interface [21].H o w e v e r ,M o r i y a m a et al. [28]measured a large voltage signal in MTJs with an Al 2O3tunnel barrier, a behavior which cannot be explainedby the spin-pumping mechanism. It is not clear whether spin- pumping theory is applicable to MgO-based tunnel junctions. Here, we use first-principles calculations to analyze the relation between αand spin pumping in NM/MgO/FM MTJs. Our calculated αestimated from G↑↓agrees well with the experiment [13], and the imaginary part of G↑↓(G↑↓ i) almost equals the real part ( G↑↓ r), especially with a thick MgO capping layer. These results indicate that the decrease in α comes from the suppression of spin pumping, and the large G↑↓ iimplies that the change in frequency of the ferromagnetic resonance can be observed experimentally. A clean interface and an alloy electrode promote small α. Moreover, we find that the capping normal metal greatly affects α. This article is organized as follows. In Sec. II,w e introduce scattering theory in order to estimate αby the parameter G↑↓. In Sec. III, we present our results of αin NM/MgO/FM MTJs. Section IVis our summary. II. SPIN PUMPING AND GILBERT DAMPING PARAMETER For a FM/NM interface, the magnetization in the FM layer can be simplified as a macrospin M¼MsVm, where mis the unit vector, Msis the saturation magnetization, and Vis the total FM volume. When the magnetization precesses, a spin current Ipump sis pumped out of the ferromagnet [26]: Ipump sðtÞ¼ℏ 4π½G↑↓ rmðtÞ×_mþG↑↓ i_mðtÞ/C138; ð1Þ where G↑↓¼ðe2=hÞTrðI−r† ↑r↓Þis the spin-mixing con- ductance, and Iandrσare the unit matrix and the matrix of interface reflection coefficients for spin σspanned by the scattering channels at the Fermi energy [29].*kexia@bnu.edu.cnPHYSICAL REVIEW APPLIED 7,034004 (2017) 2331-7019 =17=7(3)=034004(7) 034004-1 © 2017 American Physical SocietyAn immediate consequence of the spin pump is enhanced Gilbert damping of the magnetization dynamics [26]. When the NM is an ideal spin sink, the spin backflow can be disregarded, and by the conservation of the spinangular momentum, enhanced Gilbert damping can be obtained [30] α¼γℏ 4πMsVZ dE/C18 −df dE/C19 G↑↓ rðEÞ; ð2Þ where γis the gyromagnetic ratio. At finite temperature, the volume Vis replaced by the magnetic coherent volume Va [31,32] . The temperature effects contain two factors: one is the Fermi-Dirac distribution function f¼expf½ðE−EfÞ=KBT/C138þ 1g−1, and the other is the magnetic coherent volume Va¼f2=½3ξð5=2Þ/C138g½ð4πDÞ=ðkBTÞ/C1383=2, where ξis the Riemann ζfunction, Tis the ambient temperature, and D is the spin stiffness constant. At 0 K, Vaequals Vand −∂f=∂Ei sr e d u c e dt ot h e δfunction δðE−EfÞat the Fermi level Ef. In general, the total Gilbert damping contains the bulk part α0and the interface enhancement part α. In Konoto et al.’s[21] experiment, α0accounts for 40% of the total Gilbert damping; however, the decreases in Gilbert dampingwith increasing thickness of the MgO capping layer comes from the interfacial effect, so we focus on the interface- enhancement part. III. GILBERT DAMPING IN Ag=MgO -CAP =FeMTJS We consider an Ag =MgO ðxnmÞ=Fe MTJ, as shown in Fig.1, where the MgO-barrier thickness xranges from 0 to 1.4 nm. In our calculation, we choose bcc Fe as one lead with the lattice constant a Fe¼2.866Å, and the crystal MgO is reduced 4% and rotated 45 to match the bcc Fe. The lattice constant of fcc Ag is aAg¼4.053Å, and Ag is matched with MgO. The interlayer spacing between the Ag monolayer and the MgO surface layer is 2.52 Å for Ag above the O site [33], For the clean junction, we use a 400×400k-point mesh in the full two-dimensional Brillouin zone (BZ) to ensure numerical convergence. For the disordered interface case, we use a 40×40k-point mesh in the full two-dimensional BZ for a 4×4lateral supercell, and 12 configurations are used to ensureconfiguration convergence. The scattering matrix is obtained using a first-principles wave-function methodwith tight-binding linearized muffin-tin orbitals [34]. First, we check how the MgO capping layer and MgO barrier affect G ↑↓. We study Ag =MgO-cap ðnMLÞ= Feð18MLÞ=MgO-barrier ðmMLÞ=Fe MTJs, where the MgO capping layer is nML and the MgO barrier is m ML. Table Ishows the dependences of G↑↓and TMR on the thickness of the MgO capping layer and MgO barrier. When considering the dependence of G↑↓and TMR on n (m), we fix m(n) equal to 3. The TMR is defined in terms of the conductance of the magnetization between thetwo Fe layers parallel ( P) and antiparallel (AP): TMR ¼ ½ðG P−GAPÞ=GAP/C138×100% . Table Ishows that G↑↓decreases with increasing thickness of the MgO capping layer and does not change significantlywith an increasing thickness of the MgO barrier. Gilbert damping is estimated from G ↑↓. As we show in Fig. 2,t h e experimental measurement of G↑↓is dominated by the MgO capping layer. Thus, we can choose the Ag =MgO-cap =Fe structure to study how the MgO capping layer affects theGilbert damping of Ag =MgO-cap =Fe=MgO-barrier = Fe MTJs. The MgO capping layer and MgO barrier have a different effect on the TMR. The TMR decreases with increasingthickness of the MgO capping layer and increases with anincreasing thickness of the MgO barrier, which agrees withprevious experiments [25]. Based on scattering theory, αis proportional to G ↑↓ r, which is related to the in-plane part of the spin current. Weestimate αfrom Eq. (2). Figure 2shows the αdependences of the MgO capping layer thickness of Ag =MgO ðxnmÞ=Fe MTJs with 6.25% OVs at all Fe =MgO and Ag =MgO interfaces, with the thickness of the MgO capping layer ranging from 0 to 1.4 nm. αdecays exponentially (inset of Fig. 2) with the thickness of the MgO capping layer. FIG. 1. Schematic Ag =MgO-cap ð001Þ=Fe MTJs with five MgO monolayers. The red and green atoms in the scatteringregion (MgO) denote O and Mg, the gray atoms on the left leadsdenote Ag, and the blue atoms denote Fe. There are some oxygenvacancies (OVs, yellow) at the two interfaces.TABLE I. Dependences of the spin-mixing conductance ( G↑↓) (in units of 1013Ω−1m−2) and the TMR on the thickness of the MgO capping layer with a 3-ML MgO barrier and on thethickness of the MgO barrier with a 3-ML MgO capping layerin Ag =MgO-cap ðnMLÞ=Feð18MLÞ=MgO-barrier ðmMLÞ=Fe MTJs with a clean interface and an interface with 6.25% OVs(in brackets). nis the number of MgO layers. n(m¼3) G ↑↓ r G↑↓ iTMR (%) 2 1.991(4.252) 0.772(1.174) 1027.5(97.1) 3 0.119(0.315) 0.119(0.289) 449.5(29.1)4 0.009(0.056) 0.016(0.051) 321.0(31.5)5 0.002(0.011) 0.003(0.014) 209.1(2.88)m(n¼3) 2 0.158(0.380) 0.145(0.298) 245.2(1.10)3 0.119(0.315) 0.119(0.289) 449.5(29.1)4 0.099(0.314) 0.105(0.297) 918.9(91.2)5 0.091(0.314) 0.101(0.233) 1288.3(97.4)HUI-MIN TANG and KE XIA PHYS. REV. APPLIED 7,034004 (2017) 034004-2Δα¼0.0029 atT¼0K and Δα¼0.01atT¼300K, the second result of which is comparable to earlier experimental results [21]Δα¼0.007at room temperature in Ta =MgO ð0–1.9nmÞ=Fe80B20=MgO ð1nmÞ=Ta MTJs. As shown in Fig. 2,αdecreases as the MgO capping layer thickens, which agrees well with the experiment. The good agreement between our calculations and experimentalresults indicates that the decreases in αare related to the suppression of the spin-pumping effects through the FM/ NM interface, with inserting MgO layers. To understand why αdecreases as the MgO capping layer thickens, we compare the spin-dependent conduct- ance G ↑,G↓[G↑ð↓Þ¼ðe2=hÞTrðt† ↑ð↓Þt↑ð↓Þ)] and G↑↓ rfor various thicknesses of the MgO capping layer, as shown in Fig.3.G↑,G↓, and G↑↓ rof the sample with the thinnest MgO layer are much larger than those of the thickersample, showing an exponential decay as a function of MgO thickness (inset of Fig. 3).αis estimated from G↑↓ r, and these results indicate that the decrease in αis related to the decrease in spin-dependent conductance. We plot the k∥-resolved transmission and G↑↓ rat the Fermi energy for the epitaxial Ag =MgO ð3MLÞ=Fe MTJs with a clean interface, as shown in Figs. 4(a)–4(c). The logarithm function is applied to the transmission coeffi-cient, and the red (blue) color represents a high- (low-) transmission probability. We find that the majority-spin channel mainly comes from k ∥points near the Γpoint andthat the minority-spin channel comes from bright k∥points near the boundary of the BZ. The G↑↓ rchannel contains features of both the majority and minority spins, indicating thatG↑↓ ris related to spin-dependent conductance. The G↑↓ r channel in some kpoints is larger than 1, such as G↑↓ r¼ 1.99atkxa0¼0.7645 andkya0¼2.8926 , as shown with red circles in Fig. 4(c), where a0¼0.286nm is the lattice constant of Fe. These large G↑↓ rchannels are caused by the resonant states at those kpoints. To understand how the interface roughness affects α,w e study G↑,G↓, and G↑↓ rwith clean 4% OVs and 6.25% OVs at the Fe =MgO and Ag =MgO interfaces, as shown in Fig. 3. G↑,G↓, and G↑↓ rare larger for the Fe =MgO and Ag =MgO interfaces with OVs than for the clean interfaces. Thevalues increase at a higher concentration of OVs. Thisresult agrees with previous studies of FeCo =MgO =FeCo MTJs [35]. As shown in Fig. 5, the origin of the increment of G ↑and G↓for Fe =MgO and Ag =MgO interfaces with OVs that the transmission channels spread much wider into the BZ dueto the diffusive scattering with interface roughness. Thus,FIG. 2. Gilbert damping parameter in Ag =MgO ð0–1.4nmÞ=Fe MTJs as a function of MgO thickness in the presence of 6.25%OVs at the Fe =MgO and Ag =MgO interfaces at T¼0and 300 K. L MgO ¼0nm is calculated from the Ag =Fe interface. For the case at 0 K, we assume 16-ML Fe (about 2 nm). Δα¼αðxÞ-αðx¼∞Þ. αis estimated from Eq. (2), and the temperature effect is considered including both the Fermi-Dirac distribution and themagnetic coherent volume effect. The blue stars are the exper-imental values [21] of the Ta =MgO ð0–1.9nmÞ=Fe 80B20= MgO ð1nmÞ=Ta MTJs. The inset picture is the logarithmic coordinates for the yaxis.FIG. 3. Spin-dependent conductance G↑,G↓(in units of 1013Ω−1m−2) and G↑↓ r(in units of 1013Ω−1m−2)i n Ag=MgO ð0–1.4nmÞ=Fe MTJs as a function of MgO thickness in the presence of a clean interface, 4% OVs, and 6.25% OVs at the Fe =MgO and Ag =MgO interfaces. G↑,G↓, and G↑↓ rdecrease with thickening of the MgO capping layer, and the results at therough interface are larger than at the clean interface. αis estimated from G↑↓ r.GILBERT DAMPING PARAMETER IN MGO-BASED … PHYS. REV. APPLIED 7,034004 (2017) 034004-3theG↑andG↑after integrating over the entire BZ is larger for the rough interface than for the clean interface. G↑↓ ris determined by the coupling between the incoming states(arriving to interface) and magnetizations. The diffusion scattering enhances the conductance electrons, which also enhances the G↑↓ randαfor interfaces with OVs, so the clean interface is better for decreasing smaller α. The ferromagnetic alloys FeCo and FeCoB are mostly used as spin sources in MgO-based multilayer structures because of their perpendicular magnetic anisotropy.Table IIlists the parameters G ↑,G↓,G↑↓ r, and αfor FeCo =MgO MTJs. We consider two kinds of structures: crystalline and alloy FeCo structures. The crystalline FeCostructure consists of alternating Fe and Co atomic layers along bcc (001), which can form a Fe- (Co-) terminated interface with MgO referred to as the “Fe (Co) term. ”For the alloy structure, Fe and Co in every layer (including theinterface layer) are randomly distributed, and we consider three different compositions of Fe xCo1−x, where xis the concentration of Fe. The magnetization of Co is smaller than that of Fe, so the total magnetization MsVis different between Fe and FeCo. The total magnetizations are 2.36, 2.31, 2.18, 1.98, 2.27, and2.22μBin Fe, Fe 0.75Co0.25,F e 0.5Co0.5,F e 0.25Co0.75,F e term, and Co term, respectively, for 1-ML ferromagnetic layers, where μBis the Bohr magneton. Table IIshows that the spin-dependent conductance and G↑↓ rare larger in junctions with Fe as the lead. Both crystalline and alloy FeCo decrease the value of αto lower than that with a Fe lead at T¼300K. When Fe and Co atoms form an alloy, the interfacial resonant states aresuppressed by the random distribution of atoms at the interface, reducing α. The value of αfor Co term is smaller than for all the other structures. For example, for 2LMgO layers, α¼0.346in the Co-term crystalline FeCo structure, which is about 47% smaller than that for the Fe lead. These FIG. 4. k∥-resolved electron transmission probability for (a)–(c) Ag =MgO ð3MLÞ=Fe with a clean interface, (d) –(f) Ir=MgO ð3MLÞ=Fe with a clean interface, and (g) –(i) Pt=MgO ð3MLÞ=Fe with a clean interface at the Fermi energy: (a),(d),(g) for majority-spin ( G↑) channels, (b),(e),(h) for minor- ity-spin channels ( G↓), and (c),(f),(i) for G↑↓ rchannels. Resonant tunnel reflection features are shown with red circles. FIG. 5. k∥-resolved electron transmission probability for Ag=MgO ð3MLÞ=Fe with 6.25% OVs at both the Fe =MgO and Ag =MgO interfaces: (a) majority-spin ( G↑) channels, (b) minority-spin ( G↓) channels. This calculation for electron transmission probability is based on the Keldysh nonequilibriumGreen ’s function [35,36] .TABLE II. Spin-dependent conductances G↑,G↓,G↑↓ r(in units of1013Ω−1m−2) and α(T¼300K) in Ag =MgO ðnMLÞ=FM MTJs with clean interfaces. FM indicates Fe, crystalline, andalloy FeCo structures. Fe term (Co term) indicates that MgO isattached to both Fe (Co) layers of the crystalline FeCo. nis the number of MgO layers, and n¼3is about 0.61 nm. n FM G ↑G↓G↑↓ r αð10−3Þ 2 Fe 2.37 0.74 2.67 0.652 Fe0.25Co0.75 2.32 0.50 1.61 0.455 Fe0.5Co0.5 2.37 0.65 1.64 0.460 Fe0.75Co0.25 2.40 0.96 1.89 0.527 Fe term 2.63 0.31 1.80 0.532 Co term 2.07 0.04 1.22 0.346 3 Fe 0.49 0.07 0.31 0.086 Fe0.25Co0.75 0.47 0.05 0.27 0.077 Fe0.5Co0.5 0.48 0.05 0.27 0.081 Fe0.75Co0.25 0.51 0.06 0.29 0.083 Fe term 0.52 0.03 0.29 0.085 Co term 0.41 0.003 0.22 0.063 TABLE III. G↑↓ r,G↑↓ i(in units of 1013Ω−1m−2) dependence on MgO thickness in Fe =MgO =Ag MTJs with a clean interface and with 6.25% OVs at both the Fe =MgO and Ag =MgO interfaces (in brackets). nis the number of MgO layers and LMgOis the corresponding distance in nanometers. The last row Fe=Ag is n¼0. nL MgO (nm) G↑↓ r G↑↓ i 2 0.41 2.6731(4.9641) 0.6191(0.7751) 3 0.61 0.3144(0.4907) 0.1977(0.2373)4 0.81 0.0474(0.0750) 0.0378(0.0553)5 1.01 0.0096(0.0162) 0.0088(0.0142)6 1.22 0.0021(0.0040) 0.0020(0.0040)7 1.42 0.0005(0.0011) 0.0005(0.0011)Fe=Ag 0 45.81 −0.11HUI-MIN TANG and KE XIA PHYS. REV. APPLIED 7,034004 (2017) 034004-4results indicate that the FeCo lead is better for reducing α, especially for the Co-term crystalline FeCo structure. From Eq. (1),G↑↓ iis related to the out-of-plane part of the spin current. The out-of-plane spin current can be ignored at the NM/FM interface because G↑↓ iis 2 orders of magnitude smaller than G↑↓ r(the value for the Fe =Ag interface in Table III). Our results indicate that the out-of- plane part of the spin current cannot be neglected compared with the in-plane part in the MgO-based MTJs, especially with a thick MgO capping layer (see Table III). G↑↓ imodifies the gyromagnetic ratio and the frequency of ferromagnetic resonance [37]:ð1=γeffÞ¼ð 1=γÞ× f1−½ðℏγG↑↓ iÞ=ð4πMsVÞ/C138g, the frequency ω¼γHeff, the change of the frequency is Δω¼½1=ð1−ℏγG↑↓ i= 4πMsVÞ−1/C138ω. In our calculations, G↑↓ i¼0.78× 1013Ω−1m−2for 2-ML MgO with 6.25% OVs rough- ness at both interfaces, the frequency of the resonances inthe MgO-based MTJ is about ω¼2–14GHz [38], which modifies the frequency by Δω¼0.46–3.22MHz at room temperature and should be able to be observedexperimentally. To understand the effects of capping NMs, we study various NMs on the MgO =Fe interface. Table IVlists the calculated G ↑,G↓, and G↑↓ rin NM =MgO ðnMLÞ=Fe MTJs with a clean interface. We consider Ir, Pt, and Ag as NMsfor 2- and 3-ML MgO layers. We choose fcc Pt (Ir) with the lattice constant a Pt¼3.9242 Å(aIr¼3.839Å). The in- plane lattice constant of the NM is matched with MgO, andthe interlayer spacing between the NM monolayer and theMgO surface layer is 2.364 (2.305) Å for Pt (Ir) above theO site to ensure the space fulfilling. We find that both spin-dependent conductance and G ↑↓ r in Ir=MgO =Fe structures are higher than in Pt =MgO =Fe and Ag =MgO=Fe. To understand the difference, we plot the k∥-resolved transmission and G↑↓ rfor Ir =MgO ð3MLÞ=Fe and Pt =MgO ð3MLÞ=Fe MTJs at the Fermi energy, as shown in Figs. 4(d)–4(i).G↑↓ ris dominated by the area near theΓpoint of the 3-ML MgO barrier, and the transmission probability and area are larger in Ir than in Pt and Ag as aNM. The band structures along the fcc (001) of Ag, Ir, andPt are analyzed in the following to further understand theirdifference, as shown in Fig. 6. The main contribution to the transmission channel is the Δ 1state of MgO-based MTJs [3]. The kinetic energy in the Ir junction is larger than in Pt and Ag at the Fermi energy, so G↑↓ ris larger in Ir=MgO =Fe MTJs. Table IVshows that G↑↓ ris larger than the spin-dependent conductance of Ir =MgO ð2MLÞ=Fe junctions. The relative spin-mixing conductance [39] η¼2G↑↓ r=ðG↑þG↓Þ¼2.5. The large ηimplies a large skewness of the angular- dependent STT of Fe =MgO =Ir structures, which makes this material promising for applications in high-frequency gen- erators [40]. IV. SUMMARY We estimate G↑↓in Ag =MgO ðnMLÞ=Fe MTJs. At room temperature, our calculated αis consistent with the experimental results. The decrease in αis related to the suppression of spin-pumping effects in the Fe =Ag interface caused by inserted MgO layers. Our calculations show that both the interface roughness and site disorder in the ferromagnetic layer affect α, while a clean interface and alloy electrode are better for obtaining smaller α:G↑↓ iis nearly equal to G↑↓ rin the MgO-based MTJs, and the capping NM material affects α. ACKNOWLEDGMENTS We gratefully acknowledge financial support from the National Natural Science Foundation of China (Grants No. 61376105 and No. 21421003). [1] S. S. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant, and S.-H. Yang, Giant tunnelling magneto-resistance at room temperature with MgO (100) tunnelbarriers, Nat. Mater. 3, 862 (2004) .TABLE IV. The calculated G↑,G↓, and G↑↓ r(in units of 1013Ω−1m−2)i nN M =MgO ðnMLÞ=Fe MTJs with a clean interface. NM is Ir, Pt, and Ag; nis the number of MgO layers. n NM G↑G↓G↑↓ r 2 Ir 5.01 2.55 9.33 Pt 3.67 0.78 3.71 Ag 2.37 0.74 2.67 3 Ir 1.24 0.09 0.88 Pt 0.86 0.04 0.52 Ag 0.49 0.07 0.31 FIG. 6. 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PhysRevB.95.174407.pdf

PHYSICAL REVIEW B 95, 174407 (2017) Ultrafast optical excitation of coherent magnons in antiferromagnetic NiO Christian Tzschaschel,1,*Kensuke Otani,2Ryugo Iida,2Tsutomu Shimura,2Hiroaki Ueda,3 Stefan Günther,1Manfred Fiebig,1and Takuya Satoh1,2,4 1Department of Materials, ETH Zurich, 8093 Zurich, Switzerland 2Institute of Industrial Science, The University of Tokyo, Tokyo 153-5805, Japan 3Department of Chemistry, Kyoto University, Kyoto 606-8502, Japan 4Department of Physics, Kyushu University, Fukuoka 819-0395, Japan (Received 30 January 2017; revised manuscript received 3 April 2017; published 5 May 2017) In experiment and theory, we resolve the mechanism of ultrafast optical magnon excitation in antiferromagnetic NiO. We employ time-resolved optical two-color pump-probe measurements to study the coherent nonthermalspin dynamics. Optical pumping and probing with linearly and circularly polarized light along the optic axisof the NiO crystal scrutinizes the mechanism behind the ultrafast magnon excitation. A phenomenologicalsymmetry-based theory links these experimental results to expressions for the optically induced magnetizationvia the inverse Faraday effect and the inverse Cotton-Mouton effect. We obtain striking agreement betweenexperiment and theory that, furthermore, allows us to extract information about the spin domain distribution. Wealso find that in NiO the energy transfer into the magnon mode via the inverse Cotton-Mouton effect is aboutthree orders of magnitude more efficient than via the inverse Faraday effect. DOI: 10.1103/PhysRevB.95.174407 I. INTRODUCTION Antiferromagnetism is rapidly gaining importance as a cru- cial ingredient of spintronics applications [ 1,2]. Because of the absence of a net magnetization in the ground state, it is robustagainst externally applied fields and the formation of domainsis not obstructed by magnetic stray fields. Accordingly, thetechnologies envisaged are mainly based on the application ofspin currents instead of magnetic fields [ 3–9]. In addition, the intimate coupling of the sublattice magnetizations in antifer-romagnets in combination with a strong exchange interactionbetween neighboring spins implies magnetization-dynamicaltimescales, which are typically orders of magnitude faster thanthose of ferro- or ferrimagnetic materials [ 10]. Naturally, ultra- short laser pulses come to mind when accessing the dynamicalproperties of the antiferromagnetic order. In contrast to thermalapproaches, which are based on local heating of the electronicand magnetic systems [ 11], nonthermal excitations would pro- vide a quasi-instantaneous access to the antiferromagnetic spinsystem via spin-orbit coupling. Thus they can fully exploit thefaster timescales inherent to antiferromagnets. The two mostprominent nonthermal magneto-optical effects are the inverseFaraday effect (IFE) [ 12] and the inverse Cotton-Mouton effect (ICME) [ 13]. In a qualitative picture, they represent impulsive stimulated Raman scattering processes, where theIFE is described by an antisymmetric tensor and the ICME by asymmetric tensor [ 14–16]. Consequently, the magneto-optical coupling effectively exerts a torque onto the spin system. The IFE and ICME have been applied to a variety of mate- rial systems [ 17–24], but a clean discrimination in experiment and theory between the two effects for a pure antiferromagnetis still due. A particularly obvious candidate for such ananalysis is antiferromagnetic NiO because of its high orderingtemperature, its simple crystallographic structure, and its well-researched physical properties [ 25–34]. In addition, it may be *christian.tzschaschel@mat.ethz.chan excellent candidate for a clear and insightful experimental and theoretical discrimination between IFE and ICME becauseit has been speculated that in NiO the symmetric part issignificantly larger than the antisymmetric part of the Ramanscattering tensor [ 31]. Consequently, the ICME would be more pronounced than the IFE, even though the ICME is a second-order effect in the magnetic order parameter. Unfortunately,the pronounced magnetic birefringence of NiO [ 28] leads to an inseparable mixture of the polarization-dependent Ramancontributions. Hence the spin oscillations observed in NiOare to date generally induced by such mixture of IFE andICME. Consequently, the mechanism behind the nonthermalexcitation of coherent magnons in NiO has not been identified,let alone quantified [ 35–40]. In this paper, we present a comprehensive experimental and theoretical analysis of IFE and ICME in antiferromagneticNiO. We separate the two effects in a nonthermal polarization-dependent two-color pump-probe measurement. The birefrin-gence resulting from the optical anisotropy is avoided byapplying our measurements to a specific single-domain state.The combination with a symmetry-based phenomenologicaltheory that we develop for quantifying IFE and ICME allowsus to distinguish between the two effects and clarify the drivingforce exciting the magnon oscillations in NiO. Moreover, wecompare the magnon generation efficiencies of the two effects. The paper is organized as follows: the crystallographic and magnetic lattices of NiO are reviewed in Sec. IAwith a special focus on the domain structure. We describe themagneto-optical properties in Sec. II A. Subsequently, based on that description, we develop a theory for the inversemagneto-optical effects in NiO in Sec. II B. In Sec. III A ,t h e optical pump-probe setup is described, and the results of thetheory sections are converted into experimental configurationsthat enable IFE and ICME to be measured and distinguished.Sections III B andIII C present the experimental results ob- tained by linear and circular pump polarizations, respectively.They are discussed in detail in Sec. IV, where we show that magnon excitation via the ICME in NiO is significantly 2469-9950/2017/95(17)/174407(11) 174407-1 ©2017 American Physical SocietyCHRISTIAN TZSCHASCHEL et al. PHYSICAL REVIEW B 95, 174407 (2017) FIG. 1. (a) Crystallographic and magnetic structure of NiO in the defined coordinate system. (b) Graphical representation of the spin dynamics for the in-plane mode and (c) out-of-plane mode. (d)Schematic of the experimental geometry. θdenotes the azimuth angle of the pump polarization relative to the easy axis of the spins, whereas ψparameterizes the setting of the Wollaston prism. The probe pulse is always circularly polarized. more efficient than via the IFE. In Sec. V, conclusions are presented. A. NiO structure NiO is a type-II antiferromagnet with a Néel temperature TNof 523 K [ 27]. In the paramagnetic phase, the crystal has the NaCl-type structure (point group m¯3m). Below TN, spins are coupled ferromagnetically within the {111}planes with neighboring planes being coupled antiferromagnetically[Fig. 1(a)][26]. Furthermore, in the antiferromagnetic phase, there is a rhombohedral distortion along the /angbracketleft111/angbracketrightdirection arising from exchange striction. This distortion correspondsto a reduction of the crystallographic point symmetry to ¯3mand induces a significant uniaxial optical anisotropy of /Delta1n=0.003 [ 28]. The optic axis forms along the direction of the distortion. Because the four independent /angbracketleft111/angbracketrightdirections ([111] ,[11¯1],[1¯11],[¯111]) are energetically degenerate in the paramagnetic phase, the rhombohedral distortion can occuralong any of those directions leading to four twin-domainstates commonly referred to as T-domain states ( T 0−T3). The four T-domain states can be distinguished by their linear birefringence [ 41]. Within each T-domain state, spins point in one of three independent /angbracketleft11¯2/angbracketrightdirections that are perpendicular to the direction of the rhombohedral distortion [ 30]. This creates the formation of three spin domain states, commonly referred to asS-domain states, S 1−S3, leading to a total of twelve possible orientation domain states in NiO [ 34]. The formation of the Sdomains leads to another small magnetostrictive distortion,corresponding to a reduced crystallographic point symmetry 2/m, which is also the point symmetry of the magnetic lattice [32,42]. This distortion, as well as the resulting linear bire- fringence, are approximately two orders of magnitude smallerthan that associated with the Tdomains [ 29] so that they have negligible influence on the polarization of the propagatingpump and probe light. For the symmetry-based polarizationanalysis, however, the full magnetic 2 /msymmetry needs to be considered, as we shall see later. Antiferromagneticordering along the [11 ¯2] direction breaks the threefold rotational symmetry; for the resulting 2 /msymmetry the twofold axis is perpendicular to both the rhombohedraldistortion and the easy axis of the spins, i.e., along [1 ¯10]. With the two sublattice magnetizations M 1(t) and M2(t), we define the ferromagnetic vector M(t)=M1(t)+M2(t) and the antiferromagnetic vector L(t)=M1(t)−M2(t). To study dynamics, it is convenient to split both quantities into a time-independent ground state and describe the excitation by a time-dependent contribution: M(t)=M 0+m(t)=m(t), (1a) L(t)=L0+l(t). (1b) The dynamic contribution may be a superposition of the two eigenmodes of the two sublattice antiferromagnetic system,both of which are optically excitable in NiO [ 35]. For the in-plane mode (IPM), also termed B gmode, the modulation of the antiferromagnetic vector l(t)i sa l o n gt h e[ 1 ¯10] direction, i.e., it occurs within the sheets of ferromagnetically coupledspins. The oscillating magnetization m(t), in contrast, is along the [111] out-of-plane direction. The frequency of thismode is /Omega1 IPM/2π/similarequal0.14 THz at 77 K [ 33,35]. The opposite behavior occurs for the out-of-plane mode (OPM), also termedA gmode. The antiferromagnetic vector is modulated along the [111] direction, whereas the magnetization oscillatesalong [1 ¯10]. The eigenfrequency of the out-of-plane mode is/Omega1 OPM/2π/similarequal1.0 7T H za t7 7K[ 27,31,35–40,43,44]. In contrast to previous publications [ 35], we specifically consider a T0domain on a (111)-cut NiO sample, where the rhombohedral distortion is along the surface normal. There-fore, the optic axis coincides with the propagation direction oflight at normal incidence and optical anisotropy, especially lin-ear birefringence, can be avoided. For this situation, we define areference system: we choose the xaxis to be along the surface normal, i.e., the [111] direction, the zaxis to be along the magnetic easy axis, i.e., the [11 ¯2] direction, and the yaxis per- pendicular to both to form a right-handed coordinate system,i.e., along [1 ¯10]. The orientation is shown in Fig. 1together with a schematic representation of the spin motion for the in-plane mode [Fig. 1(b)] and the out-of-plane mode [Fig. 1(c)]. Using this notation, Eqs. ( 1a) and ( 1b) can be expressed as M(t)=⎛ ⎝m x(t) my(t) 0⎞ ⎠, (2a) L(t)=⎛ ⎝0 0 Lz⎞ ⎠+⎛ ⎝lx(t) ly(t) 0⎞ ⎠. (2b) 174407-2ULTRAFAST OPTICAL EXCITATION OF COHERENT . . . PHYSICAL REVIEW B 95, 174407 (2017) Here, mxandlyare contributions purely from the in-plane mode, whereas myandlxoriginate from the out-of-plane mode. II. PHENOMENOLOGICAL THEORY OF MAGNETO-OPTICAL AND INVERSE MAGNETO-OPTICAL EFFECTS IN NiO We briefly review the phenomenological theory of the Fara- day effect as well as the Cotton-Mouton effect, both of whichare used to detect magnon oscillations in NiO. Furthermore,a phenomenological theory of the inverse magneto-opticaleffects, i.e., the IFE and the ICME, is presented, whichenables the different magnon excitation mechanisms to bedistinguished. These discussions are accompanied by a specialconsideration of the point-group symmetry of NiO. Light-matter interaction is typically described by an inter- action Hamiltonian which, in cgs units, reads [ 45] H int=−/epsilon1ij(M,L) 16πEi(t)E∗ j(t). (3) Here, /epsilon1ij(M,L) is the dielectric tensor, which is in general a complex function of MandL, and Eiis the electric field amplitude with Ei(t)=/Rfractur[Ei(t)eiωt][18,46]. We assume light propagating in xdirection. Thus E=(0,Ey(t),Ez(t)). Expanding the dielectric tensor into a power series in M andL, we obtain with magneto-optical coupling constants kijk andgijkl[47,48]: /epsilon1ij=/epsilon1(0) ij+ikM ijkMk+ikL ijkLk +gMM ijklMkMl+gLL ijklLkLl+gML ijklMkLl. (4) As|M|/lessmuch| L|, the term quadratic in Mcan be neglected. For symmetry reasons, only even orders in Lcan give nonvanishing contributions to the dielectric tensor. This leadsto the simplified equation /epsilon1 ij=/epsilon1(0) ij+ikM ijkMk+gLL ijklLkLl. (5) Note that because of the ac character of Mk(/Omega1) andLk(/Omega1)t h e magneto-optical coupling constants kM ijkandgLL ijklwill depend on the frequency /Omega1. For simplicity, we will henceforth omit the superscripts M and LL as well as the argument /Omega1. Considering the complex dielectric tensor /epsilon1ij(M,L) and the Onsager principle, the absence of absorption leads to /epsilon1ij(M,L)=/epsilon1∗ ji(M,L)=/epsilon1∗ ij(−M,L)=/epsilon1∗ ij(M,−L).(6) Here, /epsilon1∗ ijdenotes the complex conjugate of /epsilon1ij. Equation ( 6) indicates that the diagonal components /epsilon1iiare purely real, whereas the off-diagonal components are in general complex./Rfractur[/epsilon1 ij] is a symmetric tensor, whereas /Ifractur[/epsilon1ij] is antisymmetric. Consequently, the nonzero coefficients in Eq. ( 5) are real- valued and satisfy kijk=−kjikandgijkl=gjikl=gijlk= gjilk. As the birefringence caused by the magnetostriction is neglected in our symmetry analysis [ 29], we set /epsilon1(0) yy=/epsilon1(0) zz≡ /epsilon1(0)and/epsilon1(0) yz=/epsilon1(0) zy≡0. Considering an electromagnetic wave propagating in xdirection, we neglect all xcomponents of the dielectric tensor and assume the following ansatz for theremaining tensor components: /parenleftbigg /epsilon1yy/epsilon1yz /epsilon1zy/epsilon1zz/parenrightbigg =/parenleftbigg /epsilon1(0)+/tildewideαyβ+iξ β−iξ /epsilon1(0)+/tildewideαz/parenrightbigg . (7) With Eq. ( 5) we identify /tildewideαy=gyyzzLzLz+gyyzxLzlx, (8a) /tildewideαz=gzzzzLzLz+gzzzxLzlx, (8b) β=gyzzyLzly, (8c) ξ=kyzxmx. (8d) All other possible contributions to gijklandkijkvanish in compliance with the 2 /msymmetry of the antiferromagnetic order [ 42,48]. As the static magnetic linear birefringence expressed by Eqs. ( 8a) and ( 8b) was not resolved, we assume gyyzz≈gzzzzand redefine: /epsilon1(0)+/tildewideαy=/epsilon1/prime+αy, (9a) /epsilon1(0)+/tildewideαz=/epsilon1/prime+αz, (9b) with αy=gyyzxLzlx, (10a) αz=gzzzxLzlx. (10b) A. Magneto-optical effects We now discuss the eigenvalues and eigenpolarizations of Eq. ( 7) in the simplified case, where only one of the quantities α,β orξis nonzero. The square roots of these eigenvalues are the refractive indices corresponding to the eigenpolarizations.We show that ξleads to the Faraday effect, i.e., circular birefringence, whereas αandβinduce a linear birefringence thus leading to the Cotton-Mouton effect. 1.ξ/negationslash=0andα=β=0 The refractive indices N±and corresponding eigenpolar- izations E±are N±=/radicalbig /epsilon1/prime±ξ (11a) E±=E0√ 2exp/bracketleftbigg iω/parenleftbigg t−N± cx/parenrightbigg/bracketrightbigg (ˆy∓iˆz). (11b) Here, ˆyand ˆzcorrespond to unit vectors along the yand zdirections, ωis the angular frequency of the light, and cis the speed of light. Thus the eigenpolarizations E± describe circularly polarized waves ( σ±), which are subject to different refractive indices N±. Typically, with /epsilon1/prime/greatermuch|kyzxmx|, the circular birefringence /Delta1N=N+−N−≈kyzxmx/√ /epsilon1/primeis linear in mxand results in a rotation of the plane of polarization of linearly polarized light by φF=−ωdkyzxmx c√ /epsilon1/prime∝mx, (12) where dis the sample thickness. Therefore the magnetization component mxcan be studied by analyzing the Faraday rotation of linearly polarized light. 174407-3CHRISTIAN TZSCHASCHEL et al. PHYSICAL REVIEW B 95, 174407 (2017) 2.β/negationslash=0andα=ξ=0 The eigenvalues N±45◦and the corresponding eigenpolar- izations E±45◦are N±45◦=/radicalbig /epsilon1/prime±β, (13a) E±45◦=E0√ 2exp/bracketleftbigg iω/parenleftbigg t−N±45◦ cx/parenrightbigg/bracketrightbigg (ˆy±ˆz).(13b) The eigenpolarizations are linearly polarized with angle ±45◦ relative to the ydirection. Over a propagation distance d,t h i s linear birefringence induces a phase difference of φ45◦≈ωdgyzzyLzly c√ /epsilon1/prime∝ly. (14) Incident circularly polarized light thus becomes elliptically polarized with principal axes along the yandzdirections. 3.α/negationslash=0andβ=ξ=0 The refractive indices Ny,zand eigenpolarizations Ey,zare Ny,z=/radicalbig /epsilon1/prime+αy,z, (15a) Ey=E0exp/bracketleftbigg iω/parenleftbigg t−Ny cx/parenrightbigg/bracketrightbigg ˆy, Ez=E0exp/bracketleftbigg iω/parenleftbigg t−Nz cx/parenrightbigg/bracketrightbigg ˆz. (15b) Hence the eigenpolarizations are linearly polarized along the yandzdirections with different refractive indices Ny,z.O v e r a propagation distance d, this linear birefringence induces a phase difference of φyz≈ωd(gyyzx−gzzzx)Lzlx 2c√ /epsilon1/prime∝lx. (16) Consequently, circularly polarized light becomes ellipti- cally polarized with principal axes aligned at ±45◦.T h e magnetically induced linear birefringence observed in cases2 and 3 are also known collectively as the Cotton-Moutoneffect. To summarize, because each component of the dielectric tensor has a specific dynamical modification, the polarizationof light propagating through the material is altered in a highlyselective way. This selectivity enables the different physicalmechanisms that are responsible for a certain modulationof the magnetization to be distinguished experimentally. Inparticular, only the in-plane magnon mode causes oscillationsinm xandlyand can thus be observed via the Faraday effect (case 1) or the Cotton-Mouton effect (case 2). The out-of-planemagnon mode causes oscillations of l xandmyand is therefore only observable via the Cotton-Mouton effect (case 3). B. Inverse magneto-optical effects The Hamiltonian in Eq. ( 3) with the dielectric tensor defined in Eq. ( 7) can also be used to describe the inverse magneto- optical effects. In accordance with the previous notion, kijkand gijklthen mediate the IFE and the ICME, respectively, with direct and inverse effects parameterized by the same couplingtensors. This assumption should be valid in the region of opticaltransparency [ 49] to which our investigation is restricted. Yet,since the tensor components are dependent on the frequency of both the laser and the magnetization, the actual values for thedirect and inverse effects can still be different, as suggested inRef. [ 50]. However, this has no influence on the results of the ensuing investigation as we focus our discussions exclusivelyon the coupling parameters for the inverse effects. We define the effective magnetic fields H effandheffform andlas the partial derivative of the interaction Hamiltonian with respect to mandl: Heff=−∂Hint ∂m,heff=−∂Hint ∂l. (17) When an ultrashort light pulse irradiates a sample, these effective magnetic fields become the driving force of thenonthermal magnetization dynamics. The Landau-Lifshitz-Gilbert equations for mandlare [45] ∂m ∂t=−γ{M×Heff+L×heff}+Rm, (18a) ∂l ∂t=−γ{M×heff+L×Heff}+Rl, (18b) where γis the gyromagnetic ratio. Anisotropy terms leading to the elliptical precession of mandl, and damping terms are subsumed into Rm,l. Combining these with Eq. ( 17) and the initial conditions M(t=0)=0 and L(t=0)=(0,0,Lz), we obtain ∂m ∂t=γ 16πL2 z[gyzzy{Ey(t)E∗ z(t)+Ez(t)E∗ y(t)}ˆx −{gyyxzEy(t)E∗ y(t)+gzzxzEz(t)E∗ z(t)}ˆy]+Rm, (19a) ∂l ∂t=−iγ 16πLzkyzx{Ey(t)E∗ z(t)−Ez(t)E∗ y(t)}ˆy+Rl. (19b) If the magnetization dynamics are induced by an ultrafast laser pulse of pulse duration τ, which is short compared to the spin oscillation period, i.e., E(t)E∗(t)≈EE∗τδ(t)=I0/cδ(t), the terms RmandRlcan be neglected and Eqs. ( 19a) and ( 19b) can be integrated around t=0: /Delta1m=γτ 16πL2 z[gyzzy{EyE∗ z+EzE∗ y}ˆx −{gyyxzEyE∗ y+gzzxzEzE∗ z}ˆy], (20a) /Delta1l=−iγτ 16πLzkyzx{EyE∗ z−E∗ yEz}ˆy. (20b) These optically induced changes occur instantaneously during the excitation. 1. Excitation by linearly polarized light With linearly polarized light specified by ( Ey(t),Ez(t))= E(t)(sinθ,cosθ), where θdenotes the angle between the direction of polarization and the zaxis (cf. Fig. 1), Eqs. ( 20a) 174407-4ULTRAFAST OPTICAL EXCITATION OF COHERENT . . . PHYSICAL REVIEW B 95, 174407 (2017) and ( 20b) lead to /Delta1mlin=γ 16πcL2 zI0[gyzzysin(2θ)ˆx−(g1−g2cos(2θ))ˆy], (21a) /Delta1llin=0. (21b) Here,g1=(gyyxz+gzzxz)/2 and g2=(gyyxz−gzzxz)/2. Af- ter the quasi-instantaneous generation of mxandmy,t h e spins start to precess around their easy axis orientation witha strong ellipticity that reflects the pronounced magneticanisotropy perpendicular to this axis. The short axis of theellipse is along /Delta1m, whereas the long axis is along /Delta1l[41,51]. The precession can be separated into in-plane and out-of-plane contributions, where for the in-plane mode m xandly oscillate with a π/2 phase difference at frequency /Omega1IPMand for the out-of-plane mode myandlxoscillate at frequency /Omega1OPM. The magnetization dynamics lead to mlin x(t)=γ 16πcL2 zI0gyzzysin 2θcos/Omega1IPMt, (22a) llin y(t)=−γ 16πcAIPML2 zI0gyzzysin 2θsin/Omega1IPMt, (22b) mlin y(t)=−γ 16πcL2 zI0(g1−g2cos 2θ)cos/Omega1OPMt, (22c) llin x(t)=−γ 16πcAOPML2 zI0(g1−g2cos 2θ)sin/Omega1OPMt. (22d) Here, we introduced the anisotropy factors AIPMandAOPM, which account for the magnetic anisotropy and parametrize theellipticity of the spin precession [ 18]. The coupling between the light field and the magnetization is purely described byparameters based on the tensor g ijkl, and therefore based on magnetic linear birefringence. Therefore both modes areexcited by the ICME. 2. Excitation by circularly polarized light With circularly polarized light, σ±=(Ey(t),Ez(t))= E(t)(1,∓i)/√ 2, analogous considerations as for linearly polarized light lead to /Delta1mσ±=−γ 16πcL2 zI0g1ˆy, (23a) /Delta1lσ±=∓γ 16πcLzI0kyzxˆy. (23b) This induces oscillations of mandlaccording to mσ± x(t)=∓γ 16πc1 AIPMLzI0kyzxsin/Omega1IPMt, (24a) lσ± y(t)=∓γ 16πcLzI0kyzxcos/Omega1IPMt, (24b) mσ± y(t)=−γ 16πcL2 zI0g1cos/Omega1OPMt (24c) lσ± x(t)=−γ 16πcAOPML2 zI0g1sin/Omega1OPMt. (24d)Thus, the in-plane mode is linearly dependent on Lz, obtains a 180◦phase change upon changing the pump helicity, and couples via kyzx, which is related to magnetic circular birefringence. Accordingly, it is excited by the IFE, whichcreates an effective magnetic field that exerts a torque on thespin system and contributes the term /Delta1l. Meanwhile, even though induced by circularly polarized light, the out-of-planemode is excited via the ICME. III. EXPERIMENTAL RESULTS A. Optical setup We study the magnon dynamics in NiO by performing impulsive stimulated Raman scattering experiments in thetime domain, which was realized by a pump-probe setup.We optically excite the sample using a 0.98-eV 90-fs laserpulse and probe the transient optical properties of the materialwith a 1.55-eV 50-fs pulse [ 35]. The absorption coefficient of NiO for the pump pulse is approximately 20 cm −1at 77 K [25]. By pumping and probing the sample in the highly transparent regime, we are able to excite and measure theentire volume of our 260- μm-thick NiO slice. Furthermore, we avoid heating effects, which allows us to study the nonthermalmagnetization dynamics. The polarization of both pulsescan be tuned such that any linear or circular polarizationcan be realized for the pump and for the probe pulse. Thetransmitted part of the probe pulse is split into orthogonalcontributions by a Wollaston prism and measured as intensitiesI 1andI2on a balanced pair of photodiodes. The theory presented in the previous section allows to predict the resultingimbalance /Delta1η=/bracketleftbiggI 1−I2 I1+I2/bracketrightbigg pumpon−/bracketleftbiggI1−I2 I1+I2/bracketrightbigg pumpoff(25) between the photodiodes as a function of the orientation of the Wollaston prism, which is parameterized by the angle ψ[cf. Fig. 1(c)], as well as by the pump and probe polarizations. We focus on the Cotton-Mouton effect by probing withcircularly polarized light and measuring the ellipticity of thetransmitted light. This enables both in-plane and out-of-planemodes to be observed. The sample is kept at 77 K for allmeasurements. Eliminating l yby combining Eqs. ( 14) and ( 22b), we find for the in-plane mode excited by linearly polarized light thefollowing dependence of the ellipticity on the pump and probeconditions: /Delta1η lin IPM=CAIPML3 zI0gpu yzzygpr yzzysin 2θcos 2ψsin/Omega1IPMt.(26) Here, we defined C=−γωd/ (16πc2√ /epsilon1/prime). Furthermore, the magneto-optical coupling constants are in general frequencydependent and can therefore be different for the pump and theprobe pulse. This is taken into account by introducing g pu yzzy andgpr yzzy. Analogously, combining Eqs. ( 14) and ( 24b) yields the following dependence for the observation of the in-planemode, when excited by circularly polarized light: /Delta1η σ± IPM=±CL2 zI0kpu yzxgpr yzzycos 2ψcos/Omega1IPMt. (27) 174407-5CHRISTIAN TZSCHASCHEL et al. PHYSICAL REVIEW B 95, 174407 (2017) Similar considerations based on Eqs. ( 16) and ( 22d)a sw e l la s (24d) yield for the out-of-plane mode: /Delta1ηlin OPM=CAOPML3 zI0 ×/parenleftbig gpu 1−gpu 2cos 2θ/parenrightbig gpr 2sin 2ψsin/Omega1OPMt,(28) /Delta1ησ± OPM=CAOPML3 zI0gpu 1gpr 2sin 2ψsin/Omega1OPMt.(29) Thus the present model clearly predicts the measurable signal of the magnon dynamics as a function of pump and probepolarizations. In reverse, it allows the determination of themechanisms leading to magnon excitation. Experimentallyverifying the predictions, which are ultimately summarized inEqs. ( 26)t o( 29), is the core part of the following section. We shall first consider excitations using linearly polarized pump pulses and subsequently circularly polarized light. B. Excitation by linearly polarized light To verify the predictions regarding linearly polarized pump pulses, i.e., Eqs. ( 26) and ( 28), we performed time-resolved measurements for three different settings: (i) the detectionangle ψis fixed at 0 ◦and the pump polarization angle θis varied; (ii) the pump polarization angle θis fixed at 45◦and the detection angle ψis varied; (iii) the detection angle ψis fixed at 45◦and the pump polarization angle θis varied. Figure 2(a) shows time-resolved ellipticity measurements for setting (i). Here and in all following time-dependentmeasurements the pump pulse triggers an instantaneousstep-like change of /Delta1η. This step occurs independent of polarization which suggests a thermal origin [ 22] rather than displacive mechanisms like photoinduced magneticanisotropy effects, which are mostly restricted to systemswith dopants or impurities [ 52,53]. Figure 2(a) reveals a single oscillation with a periodicity of approximately 8 ps.The solid lines are fits according to the equation /Delta1η IPM= η0−Aexp (−t/τ)s i n(/Omega1t+B). [Note that Acorresponds to CAIPML3 zI0gpu yzzygpr yzzysin 2θcos 2ψin Eq. ( 26).] Fitting yields an oscillation frequency /Omega1/2πof 0.13 THz, which is in agreement with the expected value of 0.14 THz for the in-planemode [ 35]. The slight deviation may be temperature related. The initial phase Bturns out to be close to zero, confirming the sinelike time-dependence of Eq. ( 26). The red curve in Fig. 2(b) shows the behavior of the signed amplitude A.I t resembles the predicted sin 2 θfunction, but a fit proportional to sin 2( θ−ζ) reveals a small shift ζ=−6.9 ◦±0.7◦, and thus a deviation from the predicted behavior. As we shall seein Sec. IV A , this phase shift originates from the S-domain substructure of our single Tdomain. Distinct from the red curve, the blue curve in Fig. 2(b) shows the signed amplitude Aof the magnon oscillation for setting (ii). It confirms the expected cos 2 ψdependence of the in-plane mode amplitude in both Eqs. ( 26) and ( 27). To verify the linear dependence on the pump intensity, the pump fluence was reduced from80 mJ cm −2for setting (i) to 40 mJ cm−2for setting (ii). The observed maximum amplitudes of the two curves in Fig. 2(b) differ by a factor of about 2, confirming the predicted behavior. Figure 3(a) shows time-resolved measurements of the magnetically induced linear birefringence for setting (iii).According to our model, this allows for the most efficientFIG. 2. (a) Observation of the in-plane mode from measurements of the Cotton-Mouton effect in setting (i), i.e., ψ=0◦,θvaried. Curves are vertically displaced for clarity. (b) Signed amplitude Aof optically induced magnon oscillation in setting (i) (red) and setting (ii) (blue). A(θ)∝sin 2(θ+ζ).A(ψ)∝cos 2ψ. The difference in the modulation amplitude reflects the difference in pump power as mentioned in the text. observation of the out-of-plane mode. Measurements were performed on the same spot as for Fig. 2(a). A high-frequency modulation of the underlying in-plane mode is clearlyvisible. The solid curves are fits according to /Delta1η=η 0+ A0exp (−t/τ0)−Aexp (−t/τ)s i n(/Omega1t+B)−A/primeexp (−t/τ/prime) sin (/Omega1/primet+B/prime). The exponential terms ( ∼τ0,τ,τ/prime) and the phase shifts ( ∼B,B/prime) are phenomenological additions parameterizing the magnetic damping and the aforementionedS-domain substructure of our single Tdomain, respectively. The fit reveals /Omega1/2π=0.13 THz and /Omega1 /prime/2π=1.07 THz confirming the origin of the observed oscillations as a magnonexcitation. A magnified representation of the region aroundt=0 is given in Fig. 3(b). The sinelike behavior of the out-of-plane mode is in agreement with Eq. ( 28). Figure 3(c) shows the dependence of the signed oscillation amplitudeof the out-of-plane mode A /primeon the pump polarization angle θ. As expected from Eq. ( 28), we observe a cos 2( θ−ζ) dependence with an isotropic, i.e., polarization-independentbackground. This contribution is allowed in materials withmagnetic point groups permitting g yyxz/negationslash=−gzzxzsuch as NiO with its magnetic symmetry 2 /m[48]. This phenomenological description captures all features of the measurements. Thus, 174407-6ULTRAFAST OPTICAL EXCITATION OF COHERENT . . . PHYSICAL REVIEW B 95, 174407 (2017) FIG. 3. (a) Observation of the out-of-plane mode by measurement of the Cotton-Mouton effect in setting (iii), i.e., ψ=45◦,θvaried. Curves are vertically displaced for clarity. (b) Magnification of region between −1a n d+4 ps. The out-of-plane mode is sinelike. Data points around 0 are out of scale. (c) Dependence of the signed amplitude A/primeof the out-of-plane mode on the linear pump polarization angle θ. The solid red line is a fit using A/prime=X1−X2cos 2(θ−ζ). in NiO, with its collinear spin structure, the excitation mechanism relies on a magneto-optical coupling that issymmetry-allowed in the magnetic point group. Note thatin principle, the noncollinearity of the spins in the domainwalls can lead to an additional isotropic contribution viathe inverse magneto-refractive effect [ 54,55]. However, with 150 nm the width of a NiO domain wall is about an order ofmagnitude smaller than the lateral extension of the domain[56], so that any isotropic contribution from the walls would be of the order of a few percent only. This value is expected todiminish further by compensation effects among the varietyof wall types that are possible between the several domainstates within a T 0domain of NiO. We therefore neglect the influence of the inverse magneto-refractive effect and restrictour description to the simplest possible model required toexplain our results. The red line in Fig. 3(c) plots the fitting function X 1− X2cos 2(θ−ζ) with X1=(9±1)×10−5,X2=−(1.6± 0.1)×10−4, and ζ=3.7◦±2.2◦. The phase shift of 3 .7◦ and the presence of the in-plane mode are again caused by the admixture of additional Sdomains to the anticipated single-domain state, which are discussed in detail in Sec. IV A . Summarizing, we are able to observe both magnon modes of NiO by studying the magnetically induced linearFIG. 4. Magnon oscillations induced by circularly polarized light. Curves have been displaced by ±0.1m r a df o rc l a r i t y .( a )O n l yt h e in-plane mode is observed; the signal displays a 180◦phase shift following a change in the pump helicity, indicating excitation via the IFE. (b) The out-of-plane mode has no pump-helicity dependence, indicating excitation via the ICME. birefringence, which can be efficiently probed by circularly polarized light. Furthermore, based on the striking agreementbetween measurement and theory, we can identify the ICMEas the driving mechanism for the optical magnon excitation bylinearly polarized light in NiO. C. Excitation by circularly polarized light After confirming our model theory for the generation of magnons by linearly polarized light, we now consider magnonexcitations driven by circularly polarized optical pulses.Similar to the previous section, two cases can be distinguished,where the detection angle of the Wollaston prism is fixed toeitherψ=0 ◦orψ=45◦. Furthermore, the helicity σ±of the circularly polarized pump pulse can be altered. Four individualmeasurements are obtained (see Fig. 4). Forψ=0 ◦[Fig. 4(a)], only the in-plane mode is observed in agreement with the theory. The cosinelike behavior ofthe probed birefringence accords also with the prediction.Moreover, the in-plane mode obtains a 180 ◦phase shift when the pump helicity is changed. This is a distinct signature of theIFE as the driving mechanism of this oscillation. Qualitatively,the IFE creates an effective magnetic field pulse in xdirection, which acts as a torque on L z, effectively rotating Laround thexaxis. This causes a finite contribution in ly, which can be consequently probed by the induced birefringence via theCotton-Mouton effect. The out-of-plane mode cannot be probed in this geometry because of the sin 2 ψdependence [Eqs. ( 28) and ( 29)]. To clarify its excitation mechanism, we also took measurements atψ=45 ◦, which allows for the observation of the out-of-plane mode. The obtained time-traces [Fig. 4(b)] show the expected sinelike time dependence. Remarkably, the out-of-plane modedoes not obtain a 180 ◦phase shift after a change in the pump 174407-7CHRISTIAN TZSCHASCHEL et al. PHYSICAL REVIEW B 95, 174407 (2017) helicity, just as predicted by Eq. ( 29). Consequently, based on the excellent agreement between theory and all measurementspresented here, we can identify the Cotton-Mouton effect asthe driving mechanism of the out-of-plane mode, even thoughit was excited by circularly polarized light. Note that the weakunderlying signature of the in-plane mode in Fig. 4(b) does not obtain a 180 ◦phase shift, when the pump helicity is changed. Furthermore, it exhibits a sine-like time-dependenceas opposed to the cosinelike time dependence of the in-planemode in Fig. 4(a). Thus it is not excited by the IFE acting on the underlying S-domain substructure, but rather by the ICME [Eq. ( 26)] due to a slight inevitable ellipticity of the circularly polarized pump pulse. IV . DISCUSSION A. Influence of S2and S3domains The coordinate system in Fig. 1was chosen such that the easy axis of the S1-domain state lies along the zaxis. The probed single- T-domain area, however, may also include S2 andS3domains. Their easy axes are rotated around the x axis by 120◦and 240◦, respectively. As we will see in the following, measurements on different spots on the sampleindeed yielded varying compositions of Sdomains. For a more detailed analysis of our data, we therefore have to expand /Delta1η by terms representing the contributions from these domainstates. For pumping with linearly polarized light , probing with circularly polarized light leads to /Delta1η lin IPM=CAIPML3 zI0gpu yzzygpr yzzysin/Omega1IPMt ×[A1cos(2ψ)sin(2θ) +A2cos(2ψ−120◦)sin(2θ−120◦) +A3cos(2ψ−240◦)sin(2θ−240◦)] (30) for the in-plane mode, where A1,2,3represent the area fractions covered by the domain states S1,2,3of the single Tdomain. Thus we impose the boundary condition A1+A2+A3=1. Note that for A1=A2=A3=1/3, the isotropic ¯3msymme- try is recovered as an average across the Tdomain. In this case, Eq. ( 30) simplifies to /Delta1η=1 2CAIPML3 zI0gpu yzzygpr yzzysin(2θ−2ψ)sin/Omega1IPMt,(31) indicating that the observed amplitude depends solely on the difference between pump and detection angle. This behaviorhas been observed for instance in FeBO 3[18]. For parameterizing the degree of S-domain mixing within aN i O Tdomain, it is convenient to consider the setting ψ=0 for which Eq. ( 30) can be rewritten as /Delta1ηlin IPM=CAIPML3 zI0gpu yzzygpr yzzysin/Omega1IPMt ×Alin effsin(2θ−δlin) (32) with δlin=arctan/parenleftbigg√ 3(A2−A3) 4A1+A2+A3/parenrightbigg (33) and Alin eff=4A1+A2+A3 4 cosδlin. (34)FIG. 5. Amplitude of the in-plane mode as a function of pump polarization angle θforψ=−45◦and 0◦. The dependence can be explained by contributions from different S-domain states. Let us now analyze the distribution of the domains probed. For a single S1domain, the amplitude for ψ=±45◦would be zero for all pump angles. However, the amplitude of thein-plane-mode as a function of pump polarization for differentdetection angles (Fig. 5) immediately reveals the presence of a multi-S-domain composition of the sample. A fit of Eqs. ( 30) yields A 1=0.651±0.004, A2=0.064±0.007, (35) A3=0.285±0.012, In combination with Eqs. ( 33) and ( 34), we find δlin=−7.4◦±0.7◦, (36) Alin eff=0.744±0.002. (37) As anticipated, the combination of S1,S2, and S3domains leads to a phase shift δlinin the polarization dependence. A similar analysis of the S-domain composition can be applied for the excitation of the in-plane mode with circularly polarized light , that is, via the IFE. In analogy to Eq. ( 30), we obtain /Delta1ησ± IPM=CL2 zI0kpu yzxgpr yzzycos/Omega1IPMt ×[A1cos(2ψ) +A2cos(2ψ−120◦) +A3cos(2ψ−240◦)], (38) which can be expressed as /Delta1ησ± IPM=CL2 zI0kpu yzxgpr yzzycos/Omega1IPMt ×Aσ± effcos(2ψ−δσ±) (39) with δσ±=arctan/parenleftbigg√ 3(A2−A3) 2A1−A2−A3/parenrightbigg (40) and Aσ± eff=2A1−A2−A3 2 cosδσ±. (41) 174407-8ULTRAFAST OPTICAL EXCITATION OF COHERENT . . . PHYSICAL REVIEW B 95, 174407 (2017) In revisiting Fig. 4, the dominance of the in-plane mode for ψ=0 and its small amplitude of approximately 0.05 mrad for ψ=45◦are striking. They point to the pronounced prevalence of the S1domain state so that, even without an explicit fit of Eq. ( 38), we can conclude that δσ±≈0 andAσ± eff≈1.0 in the probed area. A refinement of our analysis by taking S-domain distributions into account as described in this section enables, in the following, quantitative statementsabout the strength of the magneto-optical coupling constantsin NiO to be made. B. Magneto-optical coupling constants This section focuses on the quantitative analysis of the magneto-optical coupling tensors kijkandgijkl.D u r i n g the analysis of the out-of-plane mode given in Sec. III B,t h e fitting parameters X1andX2were extracted, which are directly related to the magneto-optical coupling constants gyyxz and gzzxz. The extracted values yield gzzxz/gyyxz≈−3.6. This is a significant deviation from the isotropic case with ¯3m symmetry, where the ratio would be −1[42,48]. This is strong confirmation that, although the deviation from the crystallo- graphic point symmetry ¯3mtoward 2 /mby magnetostriction from the Sdomains is small, the magneto-optical properties of NiO have to be discussed in the framework of the magnetic point symmetry 2 /m. Furthermore, by comparing the oscillation amplitude of the in-plane mode in Figs. 2and4, the magnon generation efficiency via IFE and ICME can be compared. The pumpfluences were 80 mJ cm −2in both cases. In Fig. 2,t h e magnon was excited by the ICME with a maximum oscillationamplitude l ICME y of approximately 4.7 mrad. In contrast, for generation via the IFE [Fig. 4(a)], the observed oscillation had an amplitude lIFE yof 0.13 mrad. In both cases, the dynamics were probed via the contribution of lyto the Cotton-Mouton effect. The quantitative evaluation of the two excitation pathsis hindered, however, by the multi- S-domain distribution. With the analysis of the previous section, we can now renormalizethe measured amplitudes for single- S-domain samples. From Eq. ( 32) and ( 39), we see that the ratio of the spin precession amplitudes is determined by l ICME y lIFEy=AIPMAlin eff Aσ± effLzgpu yzzy kpu yzx. (42) The anisotropy factor AIPMcan be derived from the exchange field [ 35]HE=2π×27.4T H z /γand the angular frequency of the mode /Omega1IPM=2π×0.14 THz according to [ 18] AIPM=2γHE /Omega1IPM≈400. (43) The second factor in Eq. ( 42) is geometric and accounts for the distribution of Sdomains within the probed area. With our previously determined values for Alin effandAσ± eff, we conclude that the ratio of the induced magnetizationsisL zgpu yzzy/kpu yzx≈0.1. Consequently, the ICME induces a magnetization, which is about an order of magnitude smallerthan that of the IFE in NiO. Even though NiO is structurallydifferent, this value is in line with the values obtained by Ra-man scattering in rutile structure antiferromagnets [ 57]. Nev- ertheless, in NiO, this is overcompensated by the pronouncedmagnetic anisotropy so that in total the amplitude ratio of the induced magnon oscillation on a single Sdomain equals A IPMLzgyzzy/kyzx≈50. Moreover, we can consider the magnetic anisotropy energy, which applies to the in-plane mode: Haniso=a 2m2 x+b 2l2 y. (44) This anisotropy leads to an elliptical spin motion. Con- sequently, mx=0, when lyis maximized and vice versa. Therefore the ratio of the energies pumped into the magneticsystem by the ICME and the IFE scales with the square of theratio of the l yamplitudes, which is about 502, or 2500. As the IFE and ICME are described by antisymmetric and symmetric tensors kijkandgijkl, respectively, we can now revisit the apparent contradictions in earlier Ramanscattering experiments [ 31]. There, it had been argued that the commonly accepted antisymmetric Raman scattering tensoris not sufficient to explain their results, but a symmetrictensor would. Moreover, they estimated that the symmetriccontribution would be dominant. This is now confirmed,explained, and quantified by our measurements. V . CONCLUSIONS We performed time-resolved pump-probe measurements of two magnon modes in antiferromagnetic NiO. Measurementswere performed on T 0domains on the (111) surface of the sample. Thus, pump and probe pulses were propagating alongthe optic axis of the crystal, which avoids loss of the initial lightpolarization due to birefringence. This allowed us to study thedependence of the amplitude and phase of the induced magnonoscillations on pump polarization in detail. Comparing themeasurements to an analytical model under consideration ofthe full magnetic 2 /mpoint symmetry, we clarified the driving force of the individual magnon modes. Our model predictsclear selection rules for the dependence of the optical responseon the probe conditions, which were verified in experiments. The ICME constitutes the excitation mechanism for both the in-plane and the out-of-plane magnon modes bylinearly polarized light. Its analysis even provides highlysensitive quantitative access to the distribution of the elusiveS-domain substructure of the otherwise dominating T-domain distribution. When circularly polarized pump pulses are used, the general behavior of the in-plane mode is qualitativelydifferent from the out-of-plane mode. Such pulses propagatingalong the xaxis excite the out-of-plane mode via the ICME; the IFE becomes the driving mechanism of the in-plane mode.Comparison of the amplitudes of the magnon oscillationsresulting from ICME and IFE revealed that the energy transferinto the magnetic system via the ICME is about three ordersof magnitude more efficient than via the IFE. Whereas themagneto-optical coefficients parameterizing the ICME areabout an order of magnitude smaller than those of the IFE,the dynamics induced by the ICME are significantly morepronounced due to the strong magnetic anisotropy. Thisresolves the long-standing question about the proclaimeddominance of the second-order ICME over the first-order IFEderived from Raman scattering experiments. 174407-9CHRISTIAN TZSCHASCHEL et al. PHYSICAL REVIEW B 95, 174407 (2017) ACKNOWLEDGMENTS T.S. was supported by KAKENHI (Grants No. 15H05454 and No. 26103004), JST-PRESTO, JSPS Core-to-Core Pro-gram, A. Advanced Research Networks, and thanks ETHZurich for hosting him on a guest Professorship. C.T. and M.F. acknowledge support from the SNSF project 200021/147080and by FAST, a division of the SNSF NCCR MUST. [1] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol. 11,231(2016 ). [2] E. V . Gomonay and V . M. Loktev, Low Temp. Phys. 40,17 (2014 ). [3] J. Železný, H. Gao, K. Výborný, J. 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PhysRevB.87.180403.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 87, 180403(R) (2013) Phase-resolved x-ray ferromagnetic resonance measurements of spin pumping in spin valve structures M. K. Marcham,1L. R. Shelford,2S. A. Cavill,2P. S. Keatley,1W. Yu,1P. Shafer,3A. Neudert,4J. R. Childress,5J. A. Katine,5 E. Arenholz,3N. D. Telling,6G. van der Laan,2and R. J. Hicken1,* 1School of Physics and Astronomy, University of Exeter, Stocker Road, Exeter, Devon, EX4 4QL, United Kingdom 2Diamond Light Source, Harwell Science and Innovation Campus, Didcot, Oxfordshire, OX11 0DE, United Kingdom 3Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 4Helmholtz-Zentrum Dresden-Rossendorf e. V., Institute of Ion Beam Physics and Materials Research, P .O. Box 51 01 09, 01314 Dresden, Germany 5San Jose Research Center, HGST, 3403 Yerba Buena Rd., San Jose, California 95135, USA 6Keele University, Institute for Science and Technology in Medicine, Guy Hilton Research Centre, Thornburrow Drive, Hartshill, Stoke-on-Trent, ST4 7QB, United Kingdom (Received 6 December 2012; revised manuscript received 6 March 2013; published 21 May 2013) Element-specific phase-resolved x-ray ferromagnetic resonance (FMR) was used to study spin pumping within Co50Fe50(3)/Cu(6)/Ni80Fe20(5) (thicknesses in nanometers) spin valve structures with large areas, so that edge effects typical of nanopillars used in standard magnetotransport experiments could be neglected. The phase ofprecession of the Co 50Fe50fixed layer was recorded as FMR was induced in the Ni 80Fe20free layer. The field dependence of the fixed layer phase contains a clear signature of spin transfer torque (STT) coupling due to spinpumping. Fitting the phase delay yields the spin-mixing conductance, the quantity that controls all spin transferphenomena. The STT coupling is destroyed by insertion of Ta into the middle of the Cu layer. DOI: 10.1103/PhysRevB.87.180403 PACS number(s): 76 .50.+g, 72.25.−b, 75.70.Cn, 75 .76.+j The ability of a spin-polarized electric current to exert spin transfer torque (STT) upon a nanoscale ferromagneticelement has led to a revolution in electronics. Electricallyaddressed magnetic random access memory, agile microwavefrequency spin transfer oscillators, and low power spintroniclogic devices are being realized in metal-based structures,fueling research into spin-polarized transport in other classesof material. By also exploiting the spin Hall, spin Seebeck,and precessional spin-pumping effects, there are furtheropportunities to observe new physical effects and constructdevices based upon the flow of pure spin currents. Whilemicroscopic theory for the generation, transfer, and absorptionof spin current has been developed, it now needs to be testedin materials of practical interest. However, the fabrication ofnanostructured devices for spin-polarized current injection andlateral transport of spin current continues to be a formidablechallenge. Although multilayered thin-film stacks can be de-posited with atomic scale precision, additional patterning andion milling processes are required to form nanopillars andlateral spin valves. Processing may modify the structuraland magnetic properties, particularly at edges, in a mannerthat is difficult to characterize and control. Hence there isan urgent need to study spin transfer effects in large-areafilms of the highest structural quality, in which the effectsof nanoscale patterning are absent or negligible. In this waythe intrinsic interfacial and interlayer STT effects can be bettercharacterized. In the spin-pumping effect, magnetization precession within a ferromagnetic (FM) “source” layer pumps pure spincurrent into an adjacent nonmagnetic (NM) layer. 1A nonlocal damping may result from spin scattering in the NM layer.However, if a second FM “sink” layer is added to forma spin valve structure, then the transverse component ofthe spin current may be absorbed by the sink, generatinga STT that acts upon the sink, and further modifying the damping of the source. The STTs generated, by injection ofeither a spin-polarized charge current or a pure spin current,depend upon the spin-mixing conductance g ↑↓. Studies of spin pumping in large-area multilayered films can therefore be usedto predict the performance of nanostructured STT devices. Spin pumping was first observed as an increased damp- ing of the source layer in ferromagnetic resonance (FMR)experiments. 2By varying the thickness of the sink layer, the transverse spin relaxation length within the sink layerhas recently been inferred. 3However, spin current can be destroyed by spin-flip scattering at interfaces and within thespacer layer. Therefore it is essential to also directly observethe response of the sink if the flow of spin current is tobe fully understood. The dynamics of the sink have beendetected in just a few time-resolved magneto-optical Kerreffect (TRMOKE) studies 4–6of epitaxial structures with Ag and Au spacer layers. In this Rapid Communication we presentx-ray ferromagnetic resonance (XFMR) measurements of spinpumping within spin valve structures with polycrystallineCu spacer layers. Element-specific x-ray magnetic circulardichroism (XMCD) allows the magnetization dynamics of thesource and sink layers to be studied independently. It willbe shown that the field-dependent phase of precession of thesink layer provides a clear signature of STT coupling fromwhich the value of g ↑↓may be determined. The present study hence shows how phase-resolved measurements made uponeach oscillator within an ensemble can provide informationabout their mutual interactions. This method has immediateextensions within acoustics, 7plasmonics,8and the interaction of spins in quantum dots coupled by tunneling.9 A spin valve stack consisting of underlayers/ Ta(3)/Ru(2)/Ir80Mn 20(6)/Co50Fe50(3)/Cu(6)/Ni80Fe20(5)/ Ru(7) (thicknesses in nm) was deposited by magnetron 180403-1 1098-0121/2013/87(18)/180403(4) ©2013 American Physical SocietyRAPID COMMUNICATIONS M. K. MARCHAM et al. PHYSICAL REVIEW B 87, 180403(R) (2013) FIG. 1. (Color online) Schematic of the experimental geometry for XFMR measurements. Precession of the magnetization Mabout the bias field His induced by an in-plane rf magnetic field h(t). The x-ray beam is incident at grazing angle θ. sputtering onto an insulating sapphire substrate of 500 μm thickness. Field annealing was used to set the exchange biasfield of the antiferromagnetic IrMn layer. The thickness ofthe Cu spacer layer is small compared to the spin diffusionlength (350 nm). 10However, a second reference stack was deposited with a Cu(2.5) /Ta(1)/Cu(2.5) spacer layer, in which strong spin scattering at the Ta is expected to quenchthe spin accumulation within the NM layer and suppressSTT-induced dynamics of the sink. A combination ofelectron-beam lithography and ion-beam milling was used topattern the magnetic layers of the stack into elements withlateral dimensions of 190 ×400μm 2. Photolithography and further milling were then used to define a 50 /Omega1coplanar waveguide (CPW) within the now exposed nonmagneticTa(5)/[Cu(25) /Ta(3)] ×3/Cu(25) /Ta(5)/Ru(10) underlayers. The elements are sufficiently large that inhomogeneitiesassociated with edges make negligible contribution to thespatially averaged behavior of the element. A 5- μm border was left between the edges of the element and the centraltrack of the CPW to avoid any significant out-of-plane fieldexcitation. Phase-resolved XFMR measurements were made in fluo- rescence yield. 11A continuous wave microwave magnetic field was phase locked to the x-ray pulse train generated by thesynchrotron and used to excite the sample magnetization intoa state of steady precession about an in-plane bias magneticfield. The sample was positioned close to the shorted end ofthe CPW, as shown in Fig. 1, so as to be close to an antinode of the microwave field. The exchange bias field and the appliedfield lay parallel to the length of the CPW in the experiment,of which further details are given elsewhere. 11 Previous TRMOKE studies of the STT-induced dynamics of the sink layer used a spacer of sufficient thickness that theMOKE signal from the source layer was negligible. 5Other studies used a rotatable compensator to suppress the signalfrom the source layer. 4,6In this XFMR study the response of the source and sink layers is distinguished by tuning the x-rayenergy to the Ni L 3edge and Co L3edge, respectively. Theory12predicts that the FMR linewidth of the source will be broadened as it “leaks” spin angular momentum into theadjacent NM layer. The pure spin current pumped into the NMlayer generates a spin accumulation that may be described as aspin splitting of the chemical potential when diffuse scatteringat the interfaces randomizes the electron momentum withinthe NM layer. 13Spin currents driven by diffusion within the FIG. 2. (Color online) Longitudinal MOKE loops acquired from the patterned structures with (a) the Cu spacer and (b) the Cu /Ta/Cu spacer. NM layer flow both in to the sink and back to the source. The back flow partially (fully) compensates the spin currentfrom the source when its magnetization is precessing (inequilibrium). It is assumed that spin current injected into a3dtransition-metal FM layer is completely absorbed near the interface. The absorption of the component of spin angularmomentum transverse to the sink magnetization generatesa STT. The equations of motion take the form of coupledLandau-Lifshitz-Gilbert equations modified to include STTdue to spin pumping, 2 ∂mi ∂t=− |γi|mi×Heff,i+α(0) imi×∂mi ∂t +αSP i/bracketleftbigg mi×∂mi ∂t−mj×∂mj ∂t/bracketrightbigg , (1) where miandmjare unit vectors parallel to the magnetization vectors of layers iandj, respectively. The first term on the right-hand side represents the torque term due to the localeffective field H eff,i, while the second represents the damping within the ith layer due to intrinsic spin-orbit effects and two magnon scattering. The third term describes the enhanceddamping of the ith layer due to spin pumping, while the fourth term represents the STT induced by absorption of spin currentfrom the jth layer. Let us consider the case that the resonance field of the fixed layer ( i=2) lies below that of the free layer ( j=1), and is heavily damped so that the fixed and free layer resonancesoverlap. The direction of the STT acting upon the fixed layerchanges abruptly as the field passes through the free layerresonance value. Above (below) the free layer resonance thedifference in phase between the precession of the fixed layerand the oscillation of the driving field decreases (increases)as the STT partially assists (opposes) the torque term dueto the static applied field. The magnitude of the STT scaleswith the amplitude of the free layer precession, and so, to afirst approximation, the STT generates a bipolar feature in thefield-dependent fixed layer phase that has a width comparableto the FWHM of the free layer resonance. Outside this fieldrange the fixed layer phase returns to the background valueresulting from excitation of the fixed layer by the rf field. The longitudinal MOKE hysteresis loops acquired from the patterned samples are shown in Fig. 2. All dynamic measurements were performed for positive bias field, wherethe free and fixed layer magnetizations are parallel. The freelayer resonance condition was identified by sweeping the biasfield with the delay between the x rays and microwaves set 180403-2RAPID COMMUNICATIONS PHASE-RESOLVED X-RAY FERROMAGNETIC RESONANCE ... PHYSICAL REVIEW B 87, 180403(R) (2013) FIG. 3. (Color online) (a) The imaginary component of the magnetic susceptibility component χyyof the free layer for (a) the Cu spacer at 7 GHz and (b) the Cu /Ta/Cu spacer at 5 GHz. Lorentzian fits to the experimental data (open symbols) are shown as solid red curves. so as to obtain the imaginary component of the magnetic susceptibility component χyyas shown in Fig. 3. The linewidth extracted by Lorentzian fitting was found to be equal to 50 Oe for both samples within experimentalerror. At 7 GHz [Fig. 3(a)] this requires the sum of the damping constants α(0) 1andαSP 1for the free layer to be equal to 0.0105. Due to imperfect impedance matching themicrowave amplitude at the sample had a different frequencydependence for each sample. Excitation frequencies of 7 and5 GHz were used in Figs. 3(a) and3(b), respectively, for which the microwave amplitude was a maximum in each case. InFig. 3(b) a linewidth of 50 Oe at 5 GHz implies that the sum of the damping parameters is equal to 0.0150. Although noattempt is made to separate the contributions to the damping,the damping of the free layer in the absence of spin pumpingis expected to be similar at 5 and 7 GHz. This suggests that thevalue of α SP 1is larger for the Cu /Ta/Cu spacer, as expected if the Ta strongly scatters spins within the spacer layer. Thespin-pumping contribution to the Gilbert damping coefficienthas the form 14 αSP i=gμBRe(g↑↓) 8πMidi, (2) where Miis the saturation magnetization, diis the layer thickness, gis the spectroscopic splitting factor, and Re( g↑↓) is the real part of the spin-mixing conductance, which has notbeen corrected to account for the Sharvin conductance. 12 The fixed layer resonance was not observable in field sweep measurements performed on either sample due to alarge damping resulting from direct contact with the IrMn.However, the precession of the fixed layer could be observedin time delay scans performed at different applied fields asshown in Fig. 4. The phase of the x-ray pulses relative to the microwave field was varied by passing the microwavesthrough an electromechanical delay generator. The delay scansobtained from the Cu and Cu /Ta/Cu samples are shown in Figs. 4(a) and 4(b), respectively. A sine curve with period equal to that of the microwaves was fitted to each scan. Abackground of constant phase and amplitude, arising frominductive pickup, was subtracted from the fitted curves. Thefitted amplitudes are plotted against the applied field inFigs. 4(c) and 4(d). The phase of each fitted curve relative to the microwave field is plotted in Figs. 4(e) and4(f).F r e e layer delay scans (not shown) were also fitted and the phase FIG. 4. (Color online) Fixed (sink) layer delay scans for (a) the Cu spacer at 7 GHz and (b) the Cu /Ta/Cu spacer at 5 GHz. The sine curves (red) are fits to the data (open dots). (c), (d) Fitted amplitudes are plotted as open dots. (e), (f) The phase relative to the driving field of both fixed (open circles) and free (open squares) layers is plotted. The red (fixed layer) and blue (free layer) curves assume (e) αSP 1=0.0050 and αSP 2=0.0034; (f) αSP 1=0.010 and αSP 2=0.0068. values are plotted for comparison. No background subtraction was performed for the much larger free layer signals. Thefree layer phase curve has the sigmoidal shape expected for asimple harmonic oscillator. For the Cu spacer, a clear peak at the free layer resonance (530 Oe) is observed in Fig. 4(c) on top of a broad Lorentzian background due to the FMR of the fixed layer. No clearpeak is seen for the control sample [Fig. 4(d)]. For the Cu spacer, a clear bipolar variation in the phase due toSTT coupling is observed at the free layer resonance field.For the control sample there is perhaps a small dip in thefixed layer phase at the free layer resonance. A unipolarphase variation is characteristic of dipolar coupling, resultingfrom interfacial roughness, or interlayer exchange coupling,although the latter is expected to be negligible for a 6 nm Cuthickness. The amplitude and phase data in Fig. 4were modeled with a linearized macrospin solution of Eq. (1). The saturation mag- netization of the free and fixed layers was assumed to be 815and 2017 emu cm −3, as determined by vibrating sample mag- netometry measurements made on coupon samples, while theexchange bias field was taken from the loops in Fig. 2. Dipolar coupling between fixed and free layers was neglected. The 180403-3RAPID COMMUNICATIONS M. K. MARCHAM et al. PHYSICAL REVIEW B 87, 180403(R) (2013) fixed layer damping constant α(0) 2was varied so as to reproduce the background for the fixed layer response, yielding values of0.45 and 0.35 for Cu and Cu /Ta/Cu, respectively. These values are large but reasonable. A previous study 15showed that the damping constant of a Ni 81Fe19/Fe50Mn 50sample increased from 0.008 to 0.05 as the exchange bias field increased from0 to 120 Oe. Extrapolating to the exchange bias values ofFig. 2yields values for the damping constant comparable to those obtained here. The linewidths observed in Fig. 3constrain the total free layer damping constant ( α (0) 1+αSP 1) to values of 0.0105 and 0.0150 for the Cu and Cu /Ta/Cu samples, respectively. The values of αSP 1andα(0) 1were varied subject to this constraint to give the best simultaneous agreement with the free and fixedlayer responses in Fig. 4. The best agreement for the Cu sample was obtained when α(0) 1=0.0055 and αSP 1=0.0050. From Eq.(2)this implies that αSP 2=0.0034. The intrinsic Gilbert damping of α(0) 1=0.0055 for permalloy is in agreement with the range of values reported in the literature.16We note that the values for αSP 1andαSP 2are also comparable to those found in previous spin-pumping studies (3 – 5 ×10−3).4 The curves in Fig. 4(f) were obtained with α(0) 1=0.005 and αSP 1=0.010. This then implies that αSP 2=0.0068. However, since no evidence of STT was observed in the Cu /Ta/Cu sample, the fourth term on the right-hand side of Eq. (1)was set to zero for both layers. The implication is that strong spinscattering in the Ta layer prevents spin current passing fromone layer to the other. Inserting the fitted α SP 1into Eq. (2)yields Re( g↑↓)=2.64× 1015cm−2. The value of Re( g↑↓) is related to the number of conducting channels per spin and is a measure of the spin-pumping efficiency.17Approximate expressions of Re( g↑↓) ≈1.2n2/3and 0.75 n2/3have been assumed previously14,18 where nis the number of electrons per spin in the spacer layer. Assuming n=4.25×1022cm−3for Cu (Ref. 19) leads to Re( g↑↓)=1.46 and 0.91 ×1015cm−2, respectively. Improved agreement can be expected following correctionfor the Sharvin conductance 14but this requires input from ab initio electronic structure calculations that lie beyond the scope of the present study. Strictly speaking a separate valueofg ↑↓should be introduced to describe each interface at which spin scattering can be expected to occur. Thereforethe values deduced here should be regarded as effective valuesthat describe the two dissimilar interfaces and any internalstructure of the spacer layer. In summary, phase-resolved XFMR measurements of the spin-pumping effect have been demonstrated for spinvalve structures with polycrystalline Cu spacers. The field-dependent phase of precession of the fixed layer at the freelayer resonance provides a clear signature of STT couplingdue to spin pumping. The phase variation is reproduced by amacrospin model that allows the real part of the spin-mixingconductance to be determined. By quantifying the flow of spinangular momentum from the source layer and into the sinklayer, XFMR is a powerful new tool for the study of spintransfer in material systems of practical interest. The presentwork illustrates the more general principle of how measuringthe phase of individual oscillators within an ensemble canprovide unique insight into their mutual interaction. The authors gratefully acknowledge the financial support of EPSRC Grant No. EP/F021755/1. Part of this work wascarried out on beamline I06 at Diamond Light Source. *R.J.Hicken@exeter.ac.uk 1Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 2B. Heinrich, Y . Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. E. W. Bauer, P h y s .R e v .L e t t . 90, 187601 (2003). 3A. Ghosh, S. Auffret, U. Ebels, and W. E. Bailey, Phys. Rev. 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PhysRevLett.99.267201.pdf

Driven Dynamic Mode Splitting of the Magnetic Vortex Translational Resonance K. S. Buchanan,1,*M. Grimsditch,2F. Y . Fradin,2S. D. Bader,1,2and V . Novosad2 1Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, USA 2Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA (Received 10 April 2007; revised manuscript received 16 August 2007; published 27 December 2007) A magnetic vortex in a restricted geometry possesses a nondegenerate translational excitation that corresponds to circular motion of its core at a characteristic frequency. For 40-nm thick, micron-sized permalloy elements, we find that the translational-mode microwave absorption peak splits into two peaks that differ in frequency by up to 25% as the driving field is increased. An analysis of micromagneticequations shows that for large driving fields two stable solutions emerge. DOI: 10.1103/PhysRevLett.99.267201 PACS numbers: 75.40.Gb, 75.30.Ds, 75.75.+a Nonlinear phenomena are ubiquitous in nature, existing in systems ranging from leaky faucets to atmosphericcirculation to optics [ 1–3]. Magnetic systems can be mod- els for improving our understanding of nonlinear phe-nomena since they are experimentally accessible and theequations of motion are generally tractable. In magnetism, nonlinear effects were first observed in high-power ferro- magnetic resonance experiments in the 1950s [ 4,5] where the premature saturation of the main absorption peak andthe emergence of subsidiary peaks were attributed to spin-wave generation [ 6]. These initial nonlinear dynamic stud- ies focused on a saturated magnetic state. More recently,magnetic systems have been shown to exhibit a wealth ofother interesting phenomena, such as spin-wave self- focusing [ 7], symmetry breaking spin-wave Mo ¨bius soli- tons [ 8], and foldover and bistability effects [ 9]. The spin vortex state, an in-plane flux-closure magneti- zation distribution with a small central core, is often ob-served in magnetically soft microstructures [ 10–12]. Spin- polarized scanning tunneling microscopy shows that thecore radius is /.002410 nm , comparable to the material’s ex- change length [ 13]. Because of the Magnus-type force (gyroforce) that acts on the core, magnetic vortices exhibit unique dynamic excitations, including the translational orgyrotropic mode that is characterized by sub-GHz, spiral-like core motion [ 14–20], distinct from the higher fre- quency (GHz range) quantized spin waves observed inrestricted geometries [ 21–23]. The core polarization p/.0136 /.00061determines the handedness of the spiral motion, as demonstrated by time-resolved magneto-optical Kerr [24–26] and x-ray experiments [ 27]; the restoring force is provided mainly by the magnetostatic energy [ 16,28]. Here we explore frequency- and amplitude-dependent dynamics of magnetic vortices in circular and ellipticalmicrodisks excited by a radio frequency (rf) driving field.We find that as the magnitude of the rf field h acis in- creased, the resonance peak corresponding to the vortextranslational mode splits into two well-defined peaks whose separation increases with h ac. The appearance of mode splitting is unusual since for a single dot this is anondegenerate mode. (There is a degeneracy in the sense that the ensemble contains random chiralities and polari-ties with the same eigenfrequency but we find no indicationin simulations that their high-field response should differ.) However, comparing the results with micromagnetic cal- culations and a phenomenological analytical model revealsthat this system is similar to a driven anharmonic oscillatorwhere a nonlinear energy potential can lead to two reso-nance states, thereby resolving the apparent contradictionof a split, nondegenerate mode. Nevertheless, the prevail- ing theory for magnetic vortex dynamics does not fully explain our observations. We use a microwave reflection technique to investigate the excitations of vortices in magnetic microstructures[29]. Elliptical and circular permalloy ( Fe 20Ni80alloy) microdisks were patterned on the central strip of coplanar waveguides (CPW), /.00242000 per waveguide separated by >1/.0022mto minimize dipolar interactions, using e-beam lithography and liftoff. A rf current in the waveguidegenerates an in-plane rf magnetic field that is preferentiallyabsorbed when the frequency coincides with a resonance.The derivatives of the CPW impedance are recorded with respect to a small modulation field applied parallel to the static, in-plane magnetic field H, andh acis calculated from the current in the CPW. We examine three samples: 2:2/.0002 1:1/.0022mellipses, 3:1/.00021:7/.0022mellipses, and circles of diameter 2:2/.0022m, all 40-nm thick, referred to as samples A,B, andC, respectively. All are in the single-vortex ground state with a mixture of chiralities and polarities. Figure 1(a) shows microwave impedance spectra as a function of hacfor sample AwithH/.013660 Oe along the ellipse minor axis, orthogonal to hac. Magneto-optical Kerr effect measurements (not shown) indicate that this is below the vortex annihilation field of 2.5 kOe along the minor (hard) axis. A single, symmetric peak is found at /.0024130 MHz for lowhac.A shacis increased, it broadens and develops a shoulder ( hac/.002411 Oe ) and then splits in two, where one branch increases in frequency and the other decreases, reaching a separation of 40 MHz for hac/.0024 24 Oe . Figure 1(b) shows how the microwave spectraPRL 99,267201 (2007)PHYSICAL REVIEW LETTERSweek ending 31 DECEMBER 2007 0031-9007 =07=99(26)=267201(4) 267201-1 ©2007 The American Physical Societychange as a function of H. Spectra at H/.01360O e , obtained using background subtraction (not shown), are similar tothose obtained at H/.013620 Oe . The increase in resonance frequency with His related to the increasing curvature of the energy profile, as described in [ 28]. Both the critical h ac and the splitting magnitudes, however, are relatively in- sensitive to H. The frequencies below the critical hacvary little for small Hbut take on a negative slope for larger H, which may be related to the asymmetry of the energy profile of the shifted vortex [ 28]. The data for sample B(not shown) are similar except the frequencies are lower ( /.002480 MHz forH/.013620 Oe ) due to the smaller aspect ratio [ 11]. Splitting begins at hac/.00247O e along the major axis, reaches 21 MHz at hac/.002410 Oe , and changes little with H. The critical hacis slightly higher when directed along the ellipse minor axis. For circulardots (sample C), splitting effects were detected at h acof only 3.6 Oe and the splitting reaches 17 MHz at hac/.0024 5:5O e . The vortex translational-mode splitting is thus observed at large hac(nonlinear regime ) for all measured samples and is relatively insensitive to H. Since the vortex translational mode is nondegener- ate, the key question is, Why is mode splitting observed?To explore this further, we use micromagnetic theory and computation, both of which are in excellent agree- ment with experiments for small hac(linear regime ) [18,21,24,26,27,30]. We include additional terms in Thiele’s equation [ 14] to describe the nonlinear demagne- tization and Zeeman energy profiles expected at higher hac and we use micromagnetic simulations to model the effect of increasing hac. In spite of incomplete agreement with experiment, both approaches show evidence for the emer- gence of two allowed modes at high hac, only one of which is accessed at a given time. Thiele’s equation /.0255G/.0002dX dt/.0255DdX dt/.0135@W/.0133X/.0134 @X/.01360de- scribes the motion of the vortex core position X/.0136/.0133X;Y/.0134 in a phenomenological energy well W/.0133X/.0134, wheretis time andDis a damping parameter [ 30]. The gyrovector G/.0136 /.0255Gp^zinduces the spiral motion of the core, where G/.0136 2/.0025LMs=/.0013,/.0013is the gyromagnetic ratio, and Msis the saturation magnetization [ 24–26].W/.0133X/.0134provides the re- storing force. The dipolar portion of W,Wdem, is quadratic for small amplitudes, leading to harmonic-oscillator-like equations of motion. For large displacements, higher orderterms must be included [ 28], leading to equations of mo- tion similar to those describing an anharmonic oscillator. Considering contributions up to 4th order in X,W dem/.0133X/.0134/.0136 /.0133/.0020=2/.0134X2/.0135/.0133/.0012=4/.0134X4for a circular disk. The Zeeman en- ergy isWZ/.0136/.0255M/.0001h, where h/.0136hacsin/.0133!t/.0134^iis the driv- ing magnetic field at frequency !. For small X,Mx/Y, but the general form is Mx/.0136C1Y/.0135C2Y/.0133Y2/.0135X2/.0134, where C1, andC2are adjustable parameters. For /.0012/.0136C2/.01360, the eigenfrequency is !0/.0136G/.02551/.0129/.0129/.0129 /.0129/.0020p[16]. Damping effects may also contribute [ 31], but should primarily affect the amplitude and width of the resonance line. Figure 2shows numerical solutions calculated using the parameters of sample C[32] assuming a solution of the form X/.0136Axcos/.0133!t/.0135/.0030/.0134^i/.0135Aysin/.0133!t/.0135/.0030/.0134^j, whereAx andAyrepresent the amplitudes of the core motion along x andyand/.0030is the phase lag between hacand the vortex response. /.0012was chosen to provide a small narrowing of a) b) 100 150 200 25005001000 A (nm)x5 Oe 1.5 Oe 0.5 Oe 100 150 200 2500200400600 ω/2π(MHz) ω/2π(MHz) FIG. 2 (color online). (a) Calculated vortex core motion am- plitude as a function of !for a circular disk with /.0012> 0shows a foldover effect when hac(values in legend) is increased. Arrows indicate the hysteretic path and the dashed line shows the /.0012/.01360 result for hac/.01361:5O e . (b)Axvs!=2/.0025using a large C2value shows a crossover effect for hac/.01365O e . FIG. 1 (color online). (a) Microwave spectra (real part, scaled) for sample A,H/.013660 Oe , as a function of hac. (b) Spectra showing dynamic splitting as a function of excitation power forH/.013620–140 Oe applied along the ellipse minor axis.PRL 99,267201 (2007)PHYSICAL REVIEW LETTERSweek ending 31 DECEMBER 2007 267201-2Wdem/.0133X/.0134, andC1/.0024MsV=R, whereVis the volume, esti- mated by assuming that the vortex annihilation approxi-mately corresponds to saturation. We also considered 3rdharmonics but found they are suppressed by several orders of magnitude. As the driving field is increased a foldover of the solution is observed, as reported previously for non-linear oscillators [ 33], high-power ferromagnetic resona- tors [ 34], and in nonlinear optics [ 2]. The solution changes little for small C 2(C2/.0136C3/.0136/.0255 0:1C1=R2, considered reasonable based on fits to MvsXfrom micromagnetic simulations), but for larger C2/.0136/.0255 2:5C1=R2crossover behavior is found [Fig. 2(b)]. Both solutions lead to two peaks, as indicated by arrows in Fig. 2. Figure 2(a) shows both frequencies increasing as hacis increased, whereas Fig. 2(b) is qualitatively closer to the experimental trends as it allows for one peak to increase with hacwhile the other decreases. Note, however, that the value of C2used in Fig. 2(b) is larger than can be justified based on simulations. Micromagnetic modeling was conducted based on the Landau-Lifshitz-Gilbert equation [ 35]. To achieve trac- table computational times ( /.00243 days on a workstation), a scaled-down ellipse with dimensions 300/.0002150 nm , 20- nm thick, of permalloy [ 32] was used with 3/.00023n m2cells. A sinusoidal driving magnetic field h/.0133t/.0134/.0136hacsin/.0133!t/.0134was applied parallel to the ellipse major axis ( x) and!was swept slowly to maintain steady-state motion. Micromagnetic modeling results for hac/.01360:1, 10, and 20 Oe are shown in Fig. 3. The amplitude Mxbuilds and then declines as the frequency is swept through resonance[Fig. 3(a)] and/.0030is zero for low !,/.0025at high, and passes through/.0025=2at resonance. Since the absorption measured experimentally is proportional to M xsin/.0133/.0030/.0134, this effec- tively decreases the peak widths in Fig. 3so that they more closely resemble Fig. 1. The forward and reverse sweeps coincide when hacis 0.1 Oe ( <1 MHz difference is due to sweeping). At hac/.013610 Oe ,Mxbuilds more slowly and the response is hysteretic, falling off at 637 MHz on the forward sweep and building at 628 MHz on the reverse. At hac/.013620 Oe the effect is more pro- nounced (builds at 670 and falls at 643 MHz).Simulations at constant !/.0136632 MHz andh ac/.013610 Oe show that varying the initial phase of the driving fieldallows two steady-state resonances to be accessed(M x=Ms/.01360:127,/.0030/.01362:60;Mx=Ms/.01360:216,/.0030/.0136 2:07), confirming that the bistability in Fig. 3is a steady- state phenomenon. There is no indication from theory orsimulations that vortices with different chirality or polarity,other than their direction of core circulation, should behavedifferently. In the experiment, the frequency is increased in discrete steps and the foldover dependence on sweep di- rection cannot be detected. Since we are averaging severalspectra for many dots we expect that the signal shouldreflect both bistable states. The magnitude of h acand the separation of the frequency resonance edges in Fig. 3arequalitatively similar to the experiment and they display a foldover effect similar to Fig. 2(a). Assuming comparable representation of the bistable resonances, the expected line shape resembles a peak with a shoulder, as shown for hac/.0024 11–15 Oe in Fig. 1(a) and not the well-separated peaks observed for larger hac. For higher hac/.013650 Oe , the simulations show that the core becomes unstable and will undergo repeated polariza- tion reversal events. For !above and below resonance, steady-state solutions are obtained; however, for a range of!near resonance the amplitude repeatedly builds, declines after the core reversal, and rebuilds, which would lead to a region of decreased absorption in our experiment.Experimental evidence of core flipping by an in-plane rf field has been shown recently [ 36], which may occur in our samples for the largest h ac; however, micromagnetic mod- eling and theory indicate that the onset of nonlinear be- havior should occur at lower hac. In summary, using a microwave reflection technique we find that driven magnetic vortices exhibit a nonlinear re- a) 0.0000.003 -0.003 0.2 0.0 -0.2 580 600 620 640b)M /Mxs M /MxsM /Mxs (MHz)(MHz)hac= 0.1 Oe hac= 10 Oe600 680 640-0.40.00.4hac = 20 Oe ω/2π ω/2π FIG. 3 (color online). Micromagnetic simulations for a 300/.0002 150 nm ellipse, 20-nm thick, for hacof (a) 0.1 Oe, (b) 10 Oe, and inset of (b) 20 Oe. The frequency was increased [dark (blue)] or decreased [light (green)] at 31:8 kHz=ns, and the xaxis con- verted from time to !=2/.0025.PRL 99,267201 (2007)PHYSICAL REVIEW LETTERSweek ending 31 DECEMBER 2007 267201-3sponse at the vortex translational-mode resonance. For large driving fields we detect a splitting of the translationalmode that is surprising since this mode is nondegenerate. Theory and micromagnetic modeling indicate that a foldover-type effect plays a role in defining this behaviorand suggest that the two peaks in the absorption spectracorrespond to two steady-state solutions that differ in theirphase lag relative to the driving field. We thank K. Yu. Guslienko, A. Slavin, and P. Roy for stimulating discussions. Work at Argonne, including use ofthe Center for Nanoscale Materials, was supported by the U.S. Department of Energy, Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. *Corresponding author. buchanan@anl.gov [1]Nonlinear P henomena and C haos in Magnetic Materials , edited by P. E. Wiggen (World Scientific, River Edge, NJ,1994), p. 248. [2] R. W. Boyd, Nonlinear Optics (Academic, New York, 2002), 2nd ed., p. 576. [3] J. S. Russell, Report on Waves, Report of t he Fourteent h Meeting of t he Britis hAssociation for t he Advancement of Science (John Murray, London, 1844), pp. 311–390. [4] R. W. Damon, Rev. Mod. Phys. 25, 239 (1953). [5] N. Bloomberg and S. Wang, Phys. Rev. 93, 72 (1954). [6] H. Suhl, J. Phys. Chem. Solids 1, 209 (1957). [7] M. Bauer, O. Bu ¨ttner, S. O. Demokritov, B. Hillebrands, V . Grimalsky, Yu. Rapoport, and A. N. Slavin, Phys. Rev. Lett. 81, 3769 (1998). [8] S. O. Demokritov, A. A. Serga, V . E. Demidov, B. Hille- brands, M. P. Kostylev, and B. A. Kalinikos, Nature(London) 426, 159 (2003). [9] Yu. K. Fetisov and C. E. Patton, IEEE Trans. Magn. 40, 473 (2004). [10] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science 289, 930 (2000). [11] K. Yu. Guslienko, V . Novosad, Y . Otani, H. Shima, and K. Fukamichi, Phys. Rev. B 65, 024414 (2001). [12] R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E. Welland, and D. M. Tricker, Phys. Rev. Lett. 83, 1042 (1999). [13] A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Morgenstern, and R. Wiesendanger, Science 298, 577 (2002). [14] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973). [15] D. L. Huber, Phys. Rev. B 26, 3758 (1982).[16] K. Yu. Guslienko, B. A. Ivanov, V . Novosad, H. Shima, Y . Otani, and K. Fukamichi, J. Appl. Phys. 91, 8037 (2002). [17] N. A. Usov and L. G. Kurkina, J. Magn. Magn. Mater. 242, 1005 (2002). [18] B. A. Ivanov and C. E. Zaspel, Phys. Rev. Lett. 94, 027205 (2005). [19] K. S. Buchanan, P. E. Roy, M. Grimsditch, F. Y . Fradin, K. Yu. Guslienko, S. D. Bader, and V . Novosad, Nature Phys. 1, 172 (2005). [20] J. Shibata, Y . Nakatani, G. Tatara, H. Kohno, and Y . Otani, Phys. Rev. B 73, 020403(R) (2006). [21] V . Novosad, M. Grimsditch, K. Yu. Guslienko, P. Vavassori, Y . Otani, and S. D. Bader, Phys. Rev. B 66, 052407 (2002). [22] L. Giovannini, F. Montoncello, F. Nizzoli, G. Gubbiotti, G. Carlotti, T. Okuno, T. Shinjo, and M. Grimsditch, Phys.Rev. B 70, 172404 (2004). [23] B. Hillebrands and K. Ounadjela, Spin Dynamics in Confined Magnetic Structures I , Topics in Applied Physics V ol. 83 (Springer, Berlin, 2002). [24] J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A. Crowell, Phys. Rev. B 67, 020403 (2003). [25] M. Buess et al. , Phys. Rev. Lett. 93, 077207 (2004). [26] C. E. Zaspel, B. A. Ivanov, J. P. Park, and P. A. Crowell, Phys. Rev. B 72, 024427 (2005). [27] S.-B. Choe, Y . Acreman, A. Scholl, A. Bauer, A. Doran, J. Sto¨hr, and H. A. Padmore, Science 304, 420 (2004). [28] K. S. Buchanan, P. E. Roy, M. Grimsditch, F. Y . Fradin, K. Yu. Guslienko, S. D. Bader, and V . Novosad, Phys. Rev.B74, 064404 (2006). [29] V . Novosad, F. Y . Fradin, P. E. Roy, K. Buchanan, K. Yu. Guslienko, and S. D. Bader, Phys. Rev. B 72, 024455 (2005). [30] K. Yu. Guslienko, Appl. Phys. Lett. 89, 022510 (2006). [31] V . S. Tiberkevich and A. N. Slavin, Phys. Rev. B 75, 014440 (2007). [32] 2R/.01361100 nm ,L/.013640 nm ;M s/.0136800 emu=cm3;e x - change constant Aex/.01361:3/.0022erg=cm; no anisotropy; /.0013=/.01332/.0025/.0134/.01362:83 MHz=Oe;/.0011/.01360:01;!0=2/.0025/.0136163 MHz ; /.0020/.0136G! 0=/.0025;/.0012/.0136/.01333/.0002108cm/.02552/.0134/.0020; andDwas calcu- lated as per Ref. [ 30]. [33] L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, New York, 1976), 3rd ed., p. 169. [34] Y . K. Fetisov and A. V . Makovkin, Tech. Phys. 46,8 4 (2001). [35] M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0 (National Institute of Standards and Technology, Gaithersburg, MD, 1999). [36] B. Van Waeyenberge et al. , Nature (London) 444, 461 (2006).PRL 99,267201 (2007)PHYSICAL REVIEW LETTERSweek ending 31 DECEMBER 2007 267201-4

PhysRevB.100.054426.pdf

PHYSICAL REVIEW B 100, 054426 (2019) Hydrodynamics of three-dimensional skyrmions in frustrated magnets Ricardo Zarzuela, Héctor Ochoa,*and Yaroslav Tserkovnyak Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA (Received 7 January 2019; revised manuscript received 31 July 2019; published 20 August 2019) We study the nucleation and collective dynamics of Shankar skyrmions [Shankar, J. Phys. 38,1405 (1977 )] in the class of frustrated magnetic systems described by a SO(3) order parameter, including multilatticeantiferromagnets and amorphous magnets. We infer the expression for the spin-transfer torque that injectsskyrmion charge into the system and the Onsager-reciprocal pumping force that enables its detection by electricalmeans. The thermally assisted flow of topological charge gives rise to an algebraically decaying drag signal innonlocal transport measurements. We contrast our findings to analogous effects mediated by spin supercurrents. DOI: 10.1103/PhysRevB.100.054426 I. INTRODUCTION Recent years have witnessed a growing interest in the transport properties of frustrated (quantum) magnets [ 1–9] since they provide a powerful knob to explore unconventionalspin excitations in phases characterized by a highly degenerateground state. Spin glasses [ 10,11], spin ices [ 12], and spin liquids [ 13], to mention a few examples, belong to this broad family. In the exchange-dominated limit for magnetic inter-actions [ 14], long-wavelength excitations around a local free- energy basin are generically described by the O(4) nonlinearσmodel [ 15,16], whose action reads S=1 4/integraldisplay d3/vectorrd t(χTr[∂tˆRT∂tˆR]−ATr[∂kˆRT∂kˆR]).(1) The order parameter ˆR(/vectorr,t) represents smooth and slowly varying proper rotations of the initial noncoplanar spin config-uration [ 17,18];χandAdenote the spin susceptibility and the order-parameter stiffness of the system, respectively. Phase-coherent precessional states sustain spin supercurrents [ 7], manifested as a long-range spin signal decaying algebraicallywith the propagation distance. This form of spin superfluidity[19] gives rise to a low-dissipation channel for spin transport that could be probed via nonlocal magnetotransport measure-ments [ 20]. The SO(3) order parameter can also host stable three- dimensional solitons akin to skyrmions in chiral models ofmesons [ 21]. In condensed-matter physics, these textures are known as Shankar skyrmions and appear, e.g., in the A phase of superfluid 3He [22,23] and in atomic Bose-Einstein condensates with ferromagnetic order [ 24–26]. These objects are characterized by a different topological number than otherthree-dimensional textures arising in materials characterizedby vectorial order parameters [ 27–29]. Like chiral domain walls in one dimension [ 30] and baby skyrmions in two- dimensional magnets [ 31], suitable spin-transfer torques at the interface bias the injection of Shankar skyrmions into *Present address: Department of Physics, Columbia University, New York, 10027.the frustrated magnet, which diffuse over the bulk as stable magnetic textures carrying quanta of topological charge. Ro-bustness against structural distortions and moderate externalperturbations along with their particlelike behaviors makeskyrmions attractive from the technological standpoint dueto their potential use as building blocks for information andenergy storage [ 32,33]. Frustrated magnets offer a possible realization of these objects, which were originally proposed inlow-energy chiral effective descriptions of QCD [ 21,34] and appear in cosmology [ 35] and string theory [ 36]. In this paper, we construct a hydrodynamic theory for skyrmions in the (electrically insulating) bulk, complementedwith spin-transfer physics at the interfaces with adjacentheavy-metal contacts. Figure 1depicts the device (open) geometry with lateral terminals usually utilized in nonlocaltransport measurements. For suitable reduced symmetries(Rashba-like systems), magnetic torques can pump skyrmioncharge into the frustrated magnet, whose diffusion over thebulk and subsequent flow across the right interface sustains apumping electromotive force in the second terminal. The re-sultant drag of spin current is positive and thermally activated,in sharp contrast to the case of spin superfluid transport. Thestructure of the paper is as follows. In Sec. II, we introduce the winding number describing SO(3) skyrmions and constructa continuity equation for the associated topological current.Appendix Acontains some useful mathematical identities. The expression of the spin-transfer torque favoring skyrmionnucleation and its reciprocal electromotive force are deducedin Sec. III. We obtain the expression for the spin-drag resis- tivity of a proptotypical device in Sec. IV. Some details of the derivation are saved for Appendix B. Finally, we compare the spin-drag signals mediated by skyrmions and spin supercur-rents in Sec. V. II. TOPOLOGICAL CHARGE AND CONTINUITY EQUATION The order-parameter manifold SO(3) is topologically equivalent to the four-dimensional unit hypersphere with an-tipodal points identified. Unit-norm quaternions (so-calledversors) q=(w,v) provide a convenient parametrization of 2469-9950/2019/100(5)/054426(7) 054426-1 ©2019 American Physical SocietyZARZUELA, OCHOA, AND TSERKOVNYAK PHYSICAL REVIEW B 100, 054426 (2019) 1.0 0.8 0.6 0.4 0.2 0 I zy x V FIG. 1. Two-terminal geometry for the electrical injection and detection of skyrmions in frustrated magnets. The inset: Imaginarycomponent of the versor parametrization of the rigid hard cutoff ansatz for skyrmions q={cos[f(˜r)/2],sin[f(˜r)/2]ˆe r}(see the main text for details). Length and color of the arrows correspond to themagnitude of the vector field. rotation matrices: The three-dimensional vector vlies along the rotation axis, whereas the first component wparametrizes the rotation angle, see Appendix A. Skyrmions are topological objects associated with the nontrivial classes of the homo-topy group π 3[SO(3)] =Z, which are labeled by an integer index referred to as the skyrmion charge. The latter is themultidimensional analog of a winding number and admits thefollowing simple expression in terms of versors: Q=/integraldisplay d 3/vectorrj0,j0=/epsilon1klm 12π2det[q,∂kq,∂lq,∂mq],(2) where k,l,m∈{x,y,z}are spatial indices, /epsilon1αβ···μis the Levi- Civita symbol, and det[ ·,·,·,·] denotes the determinant of a 4×4 matrix formed by versors arranged as column vectors. Our choice of prefactor ensures the normalization to unity ofthe skyrmion charge when the mapping q:S 3→S3wraps the target space once. Formulation of a hydrodynamic theory for skyrmions re- quires the stability of these textures, which, in turn, yields thelocal conservation of their charge. In this regard, additionalquartic terms (in the derivatives of the order parameter) inthe effective action given by Eq. ( 1), which may have a dipolar /exchange origin in real systems, preclude the collapse of skyrmions into atomic-size defects [ 21]. We will assume this scenario in what follows and utilize the rigid hard cutoffansatz for stable skyrmions as a simple solution that sufficesto estimate the transport coefficients of our theory [ 37]: ˆR(/vectorr)=exp[−if(˜r)ˆe ˜r·ˆL], (3) where [ ˆLα]βγ=−i/epsilon1αβγrepresent the generators of SO(3), ˜r=|/vectorr−/vectorR|, and ˆe˜r=(/vectorr−/vectorR)/˜ris the unit radial vectorfrom the center /vectorRof the skyrmion. Here, f(˜r)=2π(1− ˜r/R⋆)/Theta1(R⋆−˜r),/Theta1(x) denotes the Heaviside θfunction, and the skyrmion radius reads R⋆=ξ(A/A4)1/2, where ξis a dimensionless prefactor and A4is the strength of the fourth- order term [ 38]. Note that this ansatz corresponds to the rotation around ˆe˜rby the angle fat each point of space. Figure 1also depicts the vector field v(/vectorr)=sin[f(˜r)/2]ˆer associated with the versor parametrization of the rotation matrix ( 3), whose skyrmion charge is Q=−1. Topological invariance (i.e., global conservation) of the skyrmion charge translates into a local conservation law em- bodied in a continuity equation. More specifically, we cancast the skyrmion charge density as the time component ofa topological four-current defined per j μ=1 12π2/epsilon1μμ 1μ2μ3det[q,∂μ1q,∂μ2q,∂μ3q], (4) which satisfies the continuity equation ∂μjμ=0. Here, μ,μ 1–3∈{t,x,y,z}denote spatiotemporal indices. The com- ponents of the associated topological flux read jk=1 32π2/epsilon1klmω·(/Omega1l×/Omega1m), (5) in terms of the angular velocity of the order parameter ω≡ iTr[ˆRTˆL∂tˆR]/2, and the (spin) vectors /Omega1l≡iTr[ˆRTˆL∂lˆR]/2 describing the spatial variations of the collective spin rotationthat defines the instantaneous state of the magnet [ 7]. Note that both scalar and cross products (highlighted in boldcharacters) take place in spin space and that, in the versorparametrization, these quantities can be recast as the Hamiltonproduct 2 ∂ μq∧q∗of the derivatives of the quaternion and its adjoint q∗, see Appendix A. Similarly, the skyrmion charge density takes the form j0=1 16π2/Omega1z·(/Omega1x×/Omega1y). (6) It is worth noting here that, contrary to the case of baby skyrmions, the topological charge is even under time-reversalsymmetry, see Appendix A. Furthermore, the skyrmion flux /vectorj is a pseudovector in real space. III. SPIN-TRANSFER TORQUES AND ELECTROMOTIVE FORCES In the device geometry considered in Fig. 1, the magnet is subject to spin-exchange and spin-orbit coupling with ad-jacent heavy-metal contacts. In what follows, we invoke aminimal symmetry reduction in the bulk, which allows theexistence of magnetic torques τthat couple to the skyrmion flux in Eq. ( 5), and assume that they also operate at the interface. Thus, for our purposes, interfaces serve merely asa medium for the charge current to flow. These torques areonly effective in a volume of width λ(along x) in contact with the metal where this distance characterizes the spatialextension of the proximity effect between the metal contactand the insulating magnet. In order to inject a skyrmion flux /vectorjby a transverse charge current density /vectorJ, we wish to establish the following work (per unit of volume and time) by the magnetic torque: P≡τ·ω=¯h 2e/vectorj·(/vectorJ×/vectorζ), (7) 054426-2HYDRODYNAMICS OF THREE-DIMENSIONAL SKYRMIONS … PHYSICAL REVIEW B 100, 054426 (2019) where /vectorζis a special vector (with units of length) and the scalar and cross products on the right-hand side of the equation (innormal characters) take place in real space. Note that mirror reflection symmetry must be broken along /vectorζforPto be a scalar, i.e., we restrict ourselves hereafter to Rashba-type magnets with /vectorζbeing the corresponding principal axis. The latter should ideally be oriented parallel to the interface. Letus consider the situation depicted in Fig. 1where the principal axis lies along the perpendicular to the basal plane ( /vectorζ=ζˆe z). As can be inferred from Eq. ( 7), we are interested in the magnetic torques that produce work in favor of the skyrmionmotion along the longitudinal direction ( xaxis) when they are induced by a charge current density flowing along thetransverse direction ( yaxis). With account of Eq. ( 5), we obtain from Eq. ( 7) that the spin-transfer torque providing such a work is given by τ=¯h 32eπ2(/vectorJ·/vector/Omega1)×(/vectorζ·/vector/Omega1). (8) This torque involves two spatial derivatives of the order parameter and is dissipative, implying that the injectionof skyrmion charge requires a strong spin-orbit interaction.Heavy-metal contacts, such as platinum contain this basicmicroscopic ingredient, and the effect is likely to be enhanced by the application of a perpendicular electric field ( /vectorE∝/vectorζ) just due to the conventional Bychkov-Rashba effect [ 39]. Spin torques of the form ∝/vectorJ·/vector/Omega1do not couple to the topological flux/vectorjand will, thus, be disregarded along with other torques at the same order of expansion, e.g., τ∝/vectorJ·(/vector∇×/vector/Omega1) that are irrelevant to the skyrmion-injection physics. Skyrmion diffusion over the magnet yields a pumping electromotive force in the second terminal, whose expressioncan be obtained by invoking Onsager reciprocity. Currents andthermodynamic forces are related by the following matrix oflinear-response coefficients: ⎛ ⎝∂ tˆR ∂tm /vectorJ⎞ ⎠=⎛ ⎝·⋆·· ⋆·· ⋆· ·⋆·γχB× ˆLsq ·⋆· ˆLqsˆϑ⎞ ⎠⎛ ⎝ˆfˆR fm /vectorE⎞ ⎠, (9) where ˆfˆR≡−δF/δˆRandfm≡−δF/δm=−m/χ+γB= −ωare the thermodynamic forces conjugate to the order pa- rameter and the nonequilibrium spin density, respectively, and /vectorErepresents the electromotive force. For our construction, we only need to focus on the charge and spin sectors, which arerelated by ˆL sq,ˆLqs(other linear-response coefficients, denoted by·⋆·, are inconsequential for our discussion); Bis an external magnetic field, γis the gyromagnetic ratio, and ˆϑis the conductivity tensor that we assume symmetric (i.e., purelydissipative). Note that it is not obvious whether Onsager re-ciprocal relations can be applied to the order-parameter sectorbecause the SO(3) matrices ˆRare defined with respect to the initial (mutual equilibrium) spin configuration defining a free-energy basin, and microscopic time-reversal symmetry relatesdifferent (and possibly disconnected) basins. However, thenonequilibrium spin-density mdoes not depend on the initial configuration and, therefore, the situation for the spin-chargesectors is analogous to that of bipartite antiferromagnets [ 40].For the torque in Eq. ( 8), we have [ˆL sq]αi=¯h 32π2e/epsilon1αβγϑijζk/Omega1jβ/Omega1kγ, (10) and since the off-diagonal blocks are related by the reciprocal relation ˆLqs=− ˆLT sq, the pumping electromotive force /vectorE= ˆϑ−1ˆLqsfmgenerated in the right terminal becomes /vectorE=¯h 32π2eω·[/vector/Omega1×(/vectorζ·/vector/Omega1)]=¯h 2e/vectorζ×/vectorj. (11) IV . SKYRMION DIFFUSION AND SPIN DRAG Dynamics of the soft modes (center of mass) describing stable skyrmions obey the Thiele equation, M¨/vectorR+/Gamma1˙/vectorR=/vectorf, (12) where M=16π 9(π2+3)χR⋆is the skyrmion inertia and /vectorf= −δF/δ/vectorRrepresents the thermodynamic force conjugate to the skyrmion center, see Appendix B. The friction coeffi- cient/Gamma1=αsM/χis proportional to the Gilbert damping constant αparametrizing losses due to dissipative processes in the bulk [ 41] where s≈¯hS/a3,Sis the length of the microscopic spin operators and adenotes the lattice spacing. Local (quasi)equilibrium within a free-energy basin alongwith translational invariance in the bulk yields Fick’s law forthe topological flux, /vectorj=−D/vector∇j 0, (13) where the diffusion coefficient is related to the friction coef- ficient via the Einstein-Smoluchowski relation D=kBT//Gamma1. Hereafter, we assume that the current is injected into thefrustrated magnet from the left contact in the two-terminalgeometry depicted in Fig. 1. We also assume translational invariance along the transverse directions (i.e., the yzplane). The latter, combined with the continuity equation for thetopological four-current, yields the conservation of the lon-gitudinal bulk skyrmion current in the steady state. It reads j x bulk=D(j0 L−j0 R)/Ltwith j0 L/RandLtbeing the skyrmion charge density at the left /right terminals and the distance between them, respectively. The topological current at the boundaries of the magnet can be cast as jx L=γL(T)¯hζλJL ekBT−¯γL(T)j0 L, (14a) jx R=¯γR(T)j0 R, (14b) where γL(T)=ν(T)e−Esky/kBTis the equilibrium-nucleation rate of skyrmions at the left interface, ν(T) and Eskydenotes the attempt frequency and the skyrmion energy, respectively[42], and ¯ γ L,R(T) represents the skyrmion annihilation rates per unit density [ 43]. The electrical bias in the left terminal favors the nucleation of skyrmions with positive topologicalcharge by lowering the energy barrier in an amount equal tothe work carried out by the magnetic torque in Eq. ( 8); the expression in Eq. ( 14a) corresponds to the leading order in the external bias [ 31]. Continuity of the topological flux sets the steady state, characterized, in linear response, by the drag 054426-3ZARZUELA, OCHOA, AND TSERKOVNYAK PHYSICAL REVIEW B 100, 054426 (2019) resistivity, /rho1drag=λR2 Q Rbulk+RL+RR(15) defined per the ratio of the detected voltage per unit length to the injected charge current density. Here, RQ=h/2e2/similarequal 12.9k/Omega1is the quantum of resistance and Rbulk,RL/Rdenote the drag resistances of the bulk and interfaces of the frustratedmagnet, respectively, R bulk=2π2/Gamma1Lt e2ζ2j0eq,RL/R=2π2kBT e2ζ2γL/R(T), (16) where j0 eq=γL,R(T)/¯γL,R(T)=ρ0e−Esky/kBTis the skyrmion density at equilibrium. V . DISCUSSION The channel for spin transport rooted in the diffusion of skyrmion charge becomes suppressed in the low-temperatureregime as the proliferation of skyrmions in the bulk of themagnet dies out with probability ∝e −Esky/kBT. The frustrated magnet, however, sustains stable spin supercurrents in thepresence of additional easy-plane anisotropies, the latter pre-cluding the relaxation of the phase-coherent precessional stateinto the uniform state. This coherent transport of spin maybe driven by nonequilibrium spin accumulations at the leftinterface, which are induced by the charge current flowingwithin the first terminal via the spin Hall effect [ 7]. Fur- thermore, in the absence of topological singularities in theSO(3) order parameter (namely, Z 2vortices) degradation of the spin superflow only occurs via thermally activated phaseslips in the form of 4 π-vortex lines [ 7]. These events are exponentially suppressed at low temperatures (compared tothe easy-plane anisotropy gap). On the other hand, we canshow through the analog of the Mermin-Ho relation [ 44] derived in Appendix A, /vector∇×/vectorJ α=−(A/2)/epsilon1αβγ/vector/Omega1β×/vector/Omega1γ, (17) that skyrmions crossing streamlines in a planar section of the magnet do not contribute to the generation of phase slips in the superfluid [ 45]. Here, /vectorJα=−A/vector/Omega1αdenotes the αcom- ponent of the spin supercurrent. Therefore, in magneticallyfrustrated systems with weak easy-plane anisotropies, weexpect to observe a smooth crossover from a spin superfluidto a skyrmion conductor driven by temperature as depicted in Fig. 2. For a large separation between terminals, L t/greatermuch 1//Gamma1γ L,R,¯hgL,R/4παs(gL,R’s are the effective interfacial con- ductances), the drag coefficients for both transport channelsreduce to /rho1sky drag=/parenleftbigg¯h e/parenrightbigg2ζ2λj0 eq 2/Gamma1Lt,/rho1SF drag=−/parenleftbigg¯h 2e/parenrightbigg2ϑ2 sH αstdLt,(18) where ϑsHandtddenote the spin Hall angle in the metal con- tacts and the thickness of the detector strip, respectively. Note the algebraical decay /rho1sky,SF drag∝1/Ltand the opposite sign of the drag resistivities in these two spin-transport channels. Thelatter can be intuitively understood as the manifestation ofthe different symmetries under time reversal of the flavorsencoding the information and dragging of the electrical signal: FIG. 2. Sketch of the thermal dependence of the total drag re- sistivity /rho1drag=/rho1SF drag+/rho1sky dragfor frustrated magnets with weak easy- plane anisotropies. Whereas, in the case of the superfluid, this is just the spin flow ascribed to coherent precession, in the case of the skyrmionconductor, the signal is mediated by the flux of the associ-ated topological charge, which is even under time reversal.We note in passing that, remarkably, skyrmions do generatehopfions through the fibration S 3→S2described by a given element of the internal spin frame [ 46]. Finally, experimental platforms well suited to host Shankar skyrmions and observethe aforementioned crossover are amorphous magnets, inparticular, amorphous yttrium iron garnet in which nonlocalspin-transport measurements have been recently reported [ 5]. In conclusion, we have established the hydrodynamic equations governing the diffusion of skyrmion charge withinthe bulk of frustrated magnetic insulators. Interfacial spin-transfer torques inject topological charge into the system,whose steady flow sustains a spin drag signal between themetallic terminals. The algebraic decay of the drag coefficientover long distances manifests the topological robustness ofShankar skyrmions in the SO(3) order parameter. We alsoremark that S 2hopfions could be pumped into the frustrated magnet by suitable spin-transfer torques, therefore, givingrise to a third channel for low-dissipation spin transport.The program developed in this paper can, in principle, beextended to S 2hopfions with the caveat that the Hopf charge density is nonlocal in the order parameter [ 47] and that it is unclear whether these topological excitations are stable withinSkyrme-like models [ 48]. ACKNOWLEDGMENTS This work has been supported by NSF under Grant No. DMR-1742928. R.Z. and H.O. contributed equally to this work. APPENDIX A: VERSOR PARAMETRIZATION In this Appendix, we show that versors (i.e., unit-norm quaternions) provide a convenient parametrization of rotationmatrices. To begin with note that SU(2) is the universal(double) covering of SO(3) and is isomorphic to the unit 054426-4HYDRODYNAMICS OF THREE-DIMENSIONAL SKYRMIONS … PHYSICAL REVIEW B 100, 054426 (2019) hypersphere in R4. The latter means that we can represent a generic SU(2) matrix ˆUby means of a four-component (real) vector q=(w,v), ˆU=wˆ1−iv·σ≡wˆ1−ivxˆσx−ivyˆσy−ivzˆσz, (A1) where σ=(ˆσx,ˆσy,ˆσz) is the vector of Pauli matrices and v= (vx,vy,vz) denotes the vector part of the quaternion q.N o t e that the normalization condition w2+v2=1 arises from the unitary character of SU(2) matrices. The SO(3) matrix ˆR associated with ˆU∈SU(2) reads ˆRαβ=(1−2|v|2)δαβ+2vαvβ−2εαβγwvγ. (A2) Since qand−qparametrize the same rotation ˆR, we conclude that SO(3) ∼=RP3, namely, the group of proper rotations corresponds to the hypersphere S3with antipodal points being identified. In this parametrization of rotations, vlies along the rotation axis, and the first component wparametrizes the rotation angle. The set {ˆ1,−iˆσx,−iˆσy,−iˆσz}defines the basis of quater- nions as a real vector space where addition and multiplicationby scalars is as in R 4. The algebra of Pauli matrices defines a multiplicative group structure, the Hamilton product, q1∧q2≡(w1w2−v1·v2,w1v2+w2v1+v1×v2).(A3) The adjoint of q=(w,v)i sq∗=(w,−v) so that the norm√q∗∧q(=1 in the case of versors) is a real number. Note that the Hamilton product provides a convenient representation ofthe usual matrix product in SO(3) since q ∗corresponds to ˆRT and ˆR1·ˆR2corresponds to q1∧q2. Finally, the O(4) nonlinear σmodel takes the following simple form: L=2/integraldisplay d3/vectorr(χ∂tq∗∧∂tq−A∂iq∗∧∂iq), (A4) in terms of versors. A simple spin-wave analysis of this Lagrangian yields, akin to Néel antiferromagnets, three inde-pendent linear dispersion relations characterized by the soundvelocity c=√ A/χ. 1. Spin currents in versor parametrization We first introduce the fields /Omega1μ=iTr[ˆRTˆL∂μˆR]/2, which describe time ( μ=t) and spatial ( μ=x,y,z) variations of the collective spin rotation defining the instantaneous state ofthe magnet, i∂ μˆU(t,/vectorr)=[/Omega1μ(t,/vectorr)·ˆS]ˆU(t,/vectorr). (A5) Here, [ ˆLα]βγ=−i/epsilon1αβγare the generators of SO(3), and /epsilon1αβγ is the Levi-Civita symbol. In particular, /Omega1t=ωis the angular velocity of the order parameter ˆR. The spin current is given by/vectorJ=−A/vector/Omega1as inferred from the Euler-Lagrange equations, where /vector/Omega1≡/Omega1xˆex+/Omega1yˆey+/Omega1zˆez[7]. With account of the versor parametrization Eq. ( A2), we obtain the identity, /Omega1μ=2w∂μv−2v∂μw+2v×∂μv, (A6) which is just /Omega1μ=2∂μq∧q∗as deduced from the definition of the Hamilton product Eq. ( A3). Note that the scalar part of ∂μq∧q∗is identically zero w∂μw+vα∂μvα=0.The following identity holds in the absence of singularities in the order parameter: ∂μ1/Omega1μ2−∂μ2/Omega1μ1=/Omega1μ1×/Omega1μ2.μ 1,μ2∈{t,x,y,z}. (A7) In terms of the αcomponent of the spin current /vectorJα=−A/vector/Omega1α, the above equation for spatial subindices can be recast as /vector∇×/vectorJα=−A 2/epsilon1αβγ/vector/Omega1β×/vector/Omega1γ, (A8) which is analogous to the Mermin-Ho relation in3He-A[44]. Equation ( A7) can be easily proved in versor notation since ∂μ1/Omega1μ2−∂μ2/Omega1μ1=2∂μ2q∧∂μ1q∗−2∂μ1q∧∂μ2q∗ =/Omega1μ1×/Omega1μ2, (A9) so long as the order parameter is single valued and, therefore, ∂μ1∂μ2q=∂μ2∂μ1q. The internal spin frame of reference is defined locally by the tetrad of vectors ˆeα=ˆR·ˆα,α=x,y,z. By projecting Eq. ( A7) onto these director vectors, we obtain ˆeα·(/Omega1i×/Omega1j)=ˆeα·(∂iˆeα×∂jˆeα). (A10) Furthermore, the projection of the spin current onto the vec- tors{ˆeα}αdefines the components of the internal spin current , namely, the spin current measured in the internal spin frameof the texture, /vectorJ (α)=ˆeα·/vectorJ=[ˆRT·/vectorJ]α. (A11) Equation ( A7) can be recast as [/vector∇×/vectorJ(α)]k=−A 2/epsilon1ijkˆeα·(∂iˆeα×∂jˆeα), (A12) which implies that the circulation of the αcomponent of the internal spin current along a closed loop is proportional tothe solid angle subtended by the surface defined by ˆe αon the planar section enclosed by the loop. Therefore, in the absenceof singularities in the order parameter, the spin current canonly decay in multiples of 4 πAbecause the solid angle is quantized in units of 4 π(provided that ˆe αpoints towards the same direction far away from the phase-slip event). 2. Versors under parity and time-reversal symmetries The order-parameter manifold of magnetic systems with frustrated interactions dominated by exchange is genericallybuilt upon applying SO(3) rotations to a given ground-stateG, which corresponds to a classical solution (a minimum) of the free-energy landscape [ 7,17,18]. These rotations connect physically distinguishable spin configurations with the sameenergy. Nonequilibrium deviations within the free-energybasin are described by smoothly varying (in space and time)elements of SO(3) in this approach. LetˆPand ˆTbe the operators (in spin space) corresponding to the representations of parity and time-reversal symmetryoperations, respectively. Note that the action of these sym-metries on the ground-state |G/angbracketrightleads to isoenergetic states |G /prime/angbracketrightthat belong, in general, to other energy basins. The spin-rotation operator ˆUacting on the whole set of spins is the direct sum of irreducible representations of SU(2) actingon individual spins S i(ilabels here the spatial position). 054426-5ZARZUELA, OCHOA, AND TSERKOVNYAK PHYSICAL REVIEW B 100, 054426 (2019) The identities |G/prime/angbracketright≡ˆTˆU|G/angbracketright= ˆUˆT|G/angbracketrightand|G/prime/prime/angbracketright≡ ˆPˆU|G/angbracketright= ˆUˆP|G/angbracketrightfollow from ˆTSiˆT−1=−Si, (A13a) ˆPSiˆP−1=Si, (A13b) so that ˆPˆUˆP−1=ˆUand ˆTˆUˆT−1=ˆU. With account of Eq. ( A1), we have the identities, •ˆPˆUˆP−1=ˆPwˆP−1−iˆPvˆP−1·ˆPσˆP−1 =w−iv·σ=ˆU/equal1⇒ qˆP/mapsto→q,(A14) •ˆTˆUˆT−1=ˆTwˆT−1+iˆTvˆT−1·ˆTσˆT−1 =w−iv·σ=ˆU/equal1⇒ qˆT/mapsto→q,(A15) Thus, the quaternions that parametrize SO(3) rotations (with respect to the new basin) remain invariant under theinversion operations. APPENDIX B: COLLECTIVE-VARIABLE APPROACH FOR SKYRMIONS Time dependence of the SO(3)-order parameter for the hard cutoff ansatz is encoded in the soft modes of theskyrmion texture, namely, its center of mass: ˆR(t,/vectorr)≡ˆR[/vectorr− /vectorR(t)]. At the same time, the canonical momentum /vector/Pi1conju- gate to /vectorRreads /vector/Pi1=−/integraldisplay d 3/vectorrm·/vector/Omega1. (B1)With account of the equation of motion m=χωand of ∂tˆR≈ −(˙/vectorR·/vector∇/vectorr)ˆRfor rigid skyrmions, we can write the canonical momentum as /Pi1i=Mij˙Rj, where the inertia tensor takes the form Mij=χ/integraldisplay d3/vectorr/Omega1i·/Omega1j =4χ/integraldisplay d3/vectorr(∂iw∂jw+∂iv·∂jv)=Mδij,(B2) withM=16π 9(π2+3)χR⋆. For the final result, we have used the ansatz given in the main text. 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[27] F. N. Rybakov, A. B. Borisov, and A. N. Bogdanov, Phys. Rev. B87,094424 (2013 ). [28] F. N. Rybakov, A. B. Borisov, S. Blügel, and N. S. Kiselev, New J. Phys. 18,045002 (2016 ). [29] X.-X. Zhang, A. S. Mishchenko, G. De Filippis, and N. Nagaosa, P h y s .R e v .B 94,174428 (2016 ). [30] S. K. Kim, S. Takei, and Y . Tserkovnyak, Phys. Rev. B 92, 220409(R) (2015 ). [31] H. Ochoa, S. K. Kim, and Y . Tserkovnyak, Phys. Rev. B 94, 024431 (2016 ). [32] A. Fert, V . Cros, and J. Sampaio, Nat. Nanotechnol. 8,152 (2013 ); Y . Zhou and M. Ezawa, Nat. Commun. 5,4652 (2014 ); X. Zhang, M. Ezawa, and Y . Zhou, Sci. Rep. 5,9400 (2015 ); A. Fert, N. Reyren, and V . Cros, Nat. Rev. Mat. 2,17031 (2017 ). [33] Y . Tserkovnyak and J. Xiao, P h y s .R e v .L e t t . 121,127701 (2018 ). [34] E. Witten, Nucl. Phys. B 223,422(1983 );223,433(1983 ). [35] K. Benson and M. Bucher, Nucl. Phys. B 406,355(1993 ). [36] T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113,843 (2005 ). [37] In addition to its center /vectorR=(X,Y,Z) and radius R,t h ea n s a t z in Eq. ( 3) is implicitly parametrized by two angular offsets θ0(polar) and φ0(azimuthal) describing the orientation of ˆe˜r in the spin frame defined by the three generators of SO(3). This broken symmetry is generically restored by quantumfluctuations, provided that θ 0andφ0are compact variables and, thus, the associated spectrum consists of discrete levelslabeled by integer angular momentum numbers. In the presenceof additional free-energy terms with a quartic dependence onthe derivatives of the order parameter [ 38], fluctuations of theradius around R ∗are energetically penalized. Therefore, only the coordinates /vectorRrepresent true soft modes. [38] In particular, skyrmions are stabilized by the so-called Skyrme term S4=−(A4/8)/integraltext d3/vectorrd tTr[∂μ1ˆRT∂μ2ˆR−∂μ2ˆRT∂μ1ˆR]2, see Ref. [ 21]. [39] Yu. A. Bychkov and E. I. Rashba, Pis’ma Zh. Eksp. Teor. Fiz. 39, 66 (1984) [ Sov. Phys. JETP Lett. 39, 78 (1984)]. [40] K. M. D. Hals, Y . Tserkovnyak, and A. Brataas, P h y s .R e v .L e t t . 106,107206 (2011 ). [41] T. L. Gilbert, IEEE Trans. Magn. 40,3443 (2004 ). [42] A lower bound for the energy of stable skyrmions in the Skyrme model is Esky/greaterorequalslant12π2√AA 4|Q|, see L. D. Faddeev, Lett. Math. Phys. 1,289(1976 ). [43] P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251(1990 ). [44] N. D. Mermin and T.-L. Ho, Phys. Rev. Lett. 36,594(1976 ). [45] By projecting this relation onto the element ˆez≡ˆR·ˆzof the tetrad defining the internal frame of reference, we obtain theexpression −4πAρ 2D skyfor the right-hand side where ρ2D sky= ˆez·(∂xˆez×∂yˆez)/4πis the baby-skyrmion charge density as- sociated with the vector field ˆez:R3/mapsto→S2restricted to planar zsections. As a result, in the absence of disclinations, the spin superflow can decay in multiples of 4 πAonly since the solid angle swept by the zelement of the tetrad is quantized in units of 4 π. The hard cutoff ansatz for skyrmions yields a total solid angle of zero (per planar section). On the other hand, thecoreless 4 πvortex of the SO(3) order parameter on the xyplane correspond to baby skyrmions of the field ˆe z. [46] Hopfions constitute the nontrivial classes of the homotopy group π3(S2)=Z. These topological textures are classified by a linking number, the so-called Hopf charge. In the caseof skyrmion-projected hopfion textures, described by, e.g.,thezelement of the internal frame of reference, q/mapsto→ˆR[q]· ˆz=(2v xvz+2wvy,2vyvz−wvx,1−2v2 x−2v2 y)T, their Hopf charge corresponds to the skyrmion charge given by Eq. ( 2). [47] J. H. C. Whitehead, Proc. Natl. Acad. Sci. USA 33,117 (1947 ). [48] D. Foster, J. Phys. A: Math. Theor. 50,405401 (2017 ). 054426-7

PhysRevB.62.570.pdf

Spin-current interaction with a monodomain magnetic body: A model study J. Z. Sun IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598 ~Received 3 February 2000 ! I examined the consequence of a spin-current-induced angular momentum deposition in a monodomain Stoner-Wohlfarth magnetic body. The magnetic dynamics of the particle are modeled using the Landau-Lifshitz-Gilbert equation with a phenomenological damping coefficient a. Two magnetic potential landscapes are studied in detail: One uniaxial, the other uniaxial in combination with an easy-plane potential term thatcould be used to model a thin-film geometry with demagnetization. Quantitative predictions are obtained forcomparison with experiments. I. INTRODUCTION Recently it has been shown, both theoretically1–5and experimentally6–9that a spin-polarized current, when passing through a small magnetic conductor, will deposit its spin-angular momentum into the magnetic system. It causes themagnetic moment to precess or even switch direction. Thenature of this interaction between the spin current and theferromagnetic moment brings about a new set of precessiondynamics, the details of which remain unexplored. In thispaper, a model system is presented of a monodomain ferro-magnetic body with its dynamics determined by the Landau-Lifshitz-Gilbert ~LLG!equation. The spin-current-induced magnetic precession dynamics are examined, and the resultsobtained compared to controlled thin-film experiments. Thisstudy also brings quantitative insights to the potential use ofspin-current injection as a method for magnetic writing. Centimeter-gram-seconds units are used for this work. Variables are grouped in simple forms where the only rel-evant unit is the energy product. Therefore results should bereadily translatable to meter-kilogram-seconds or any otherengineering units. For numerical simulation a set of dimen-sionless variables are introduced to simplify discussion andto elucidate the basic physics. Table I gives a summary ofthese reduced variables. II. MODEL DEFINITION The ferromagnet is represented by a Stoner-Wohlfarth monodomain magnetic body with magnetization M, situated at the origin, as shown in Fig. 1. For volume calculation only, the body is assumed to have a size of lmalong the ex directions, and ain botheyandezdirections, thus with avolume of a2lm. Assume the shape of the body is close to isotropic, and the energy landscape experienced by Mis de- scribed by three terms ~independent of its geometric param- etersaandlm): an applied field H, a uniaxial anisotropy energyUKwith easy axis along the ezdirection, and an easy-plane anisotropy Upin theey2ezplane, with exbeing its normal direction. The magnetization Mis assumed to be constant in magnitude, its motion represented by a unit di- rection vector nm5M/uMu, which at any instant of time, makes an angle uwith theezaxis, while the plane of Mand ezmakes an angle wwithex. Coordinates ( u,w) completely describe the motion of Min time. A spin-polarized current J enters the magnetic body in the 2exdirection, with spin- polarization factor h, and the spin direction in the ey2ez plane, making an angle fwithezaxis. The current exits in the same direction, but with its average spin direction aligned to that of M. The self-induced magnetic field of the current is ignored here—this is reasonable as long as themagnetic body is small with dimension abelow about 1000 Å , where the spin-current effect is expected to becomedominant over the current-induced magnetic field. The potential energy for MisU5U K1Up1UH, where UK5Ksin2uis the uniaxial anisotropy, with K 5(1/2)MHk, whereHkis the Stoner-Wohlfarth switching field. The easy-plane anisotropy is written as Up 5Kp(sin2ucos2w-1). The magnetic field is applied in the easy plane of ey2ez, making an angle of cwith the easy axisez. Thus UH52M"H52MH(sinusinwsinc 1cosucosc). Define h5H/(2K/M) andhp5Kp/K, U~u,w!5K@sin2u1hpsin2ucos2w22h~sinusinwsinc 1cosucosc!#. ~1! TABLE I. Summary for dimensionless units. Dimensionless variable Conversion relation Normalization quantity Magnetization m5M/Ms Saturation magetization Ms Magnetic field h5H/Hk Uniaxial-anisotropy field Hk Easy-plane anisotropy field hp5Kp/K54pMs/Hk Uniaxial-anisotropy field Hk. Effective spin current hs5(\/2e)hJ/lmMsHk Uniaxial-anisotropy energy MsHk/2 Natural time unit t5Vkt/(11a2) Ferromagnetic resonance frequency Vk5gHkPHYSICAL REVIEW B 1 JULY 2000-I VOLUME 62, NUMBER 1 PRB 62 0163-1829/2000/62 ~1!/570~9!/$15.00 570 ©2000 The American Physical SocietyIf one takes the usual thin-film situation of shape anisotropy and lets the easy-plane anisotropy energy be Kp52pMs2 thenhp54pMs/Hk. The torque Mexperiences within unit volumelm3(unit area) under potential well Eq. ~1!can be written as GU lm52nm3U~u,w! ~2! with „U(u,w)5(]U/]u)eu1(1/sin u)(]U/]w)ew, whereeu andeware unit vectors for uandwrotation, respectively. The three terms in potential energy Ulead to three terms in torque GU. First the uniaxial anisotropy term: G1 lmK5~2 sin ucosu!@~sinw!ex2~cosw!ey#. ~3! Second the easy-plane anisotropy term: G2 lmK522hp@~cosusinucosw!ey2~coswsinwsin2u!ez#. ~4! Third, the applied field term: G3 lmK52h@~sinwcoscsinu2cosusinc!ex 2~coswcoscsinu!ey1~sinucoswsinc!ez#. ~5! Spin current also brings a torque to M. We assume that the magnetic body absorbs the angular-momentum from the spin current only in the direction perpendicular to M.10This causes a net torque on M, which can be expressed in vector form as: G45snm3~ns3nm!52lmKhsnm3~ns3nm!, ~6! wheres5(\/2e)hJis the spin-angular momentum deposi- tion per unit time. h5(J"2J#)/(J"1J#) is the spin- polarization factor of the incident current J. The spin direc- tion of the incident current is in the ey2ezplane, and makesan angle fwith theezaxis.nsis a unit vector whose direc- tion is that of the initial spin direction of the current. Also we define hs5s 2lmK5S\ 2eDhJ 2lmK5S\ 2eDhI 2lma2K~7! as the spin-current amplitude in dimensionless units. In com- ponent form Eq. ~6!becomes G4 lmK52hs$2~sinucosw!~sinusinwsinf1cosucosf!ex 1@~cosu!~sinfcosu2cosfsinusinw! 1sin2ucos2wsinf#ey1@~sinu!~sinucosf 2sinwsinfcosu!#ez%. ~8! The dynamics of Munder the influence of torque G5( i514 Gi can be described using the Landau-Lifshitz-Gilbert equation as dnm dt1aSnm3dnm dtD51 2VK( i514SGi lmKD, ~9! where ais the LLG damping coefficient and g5gmB/\is the gyromagnetic ratio. In our case, g52. Here we intro- duced a characteristic frequency unit VK5gHk. Equation ~9!can be written in component form using a natural time unitt5VKt/(11a2): Fu8 w8G5( i514Fui8 wi8G ~10! with Fu18 w18G52Fasinucosu cosuG, Fu28 w28G52hpF~sinw1acosucosw!sinucosw ~coswcosu2asinw!coswG, Fu38 w38G52hFcoswsinc 1a~sinucosc2cosusinwsinc! [~sinucosc2cosusinwsinc! 2acoswsinc]/sin uG, Fu48 w48G5hsFacoswsinf 1sinwsinfcosu2cosfsinu (coswsinf 2asinwsinfcosu)/sin u 1acosfG, FIG. 1. Model geometry definition and related mathematical symbols.PRB 62 571 SPIN-CURRENT INTERACTION WITH A MONODOMAI N...where () 85d/dt(). Equation ~10!can be numerically evalu- ated. It is the basis for all numerical studies discussed below. III. ANALYTICAL RESULTS In this section we discuss the analytical solutions to Eq. ~10!. For simplicity we only consider the on-axis geometry where both applied field hand the spin direction nsare along the easy-axis ez. Further, we assume a small cone-angle limit uuu!1. In this case Eq. ~10!becomes du dt52u@a~11h!1hp~sinw1acosw!cosw1hs# dw dt5212hp~cosw2asinw!cosw2h1hsa. ~11! A coordinate transformation of u15ucoswandu25usinw could further simplify this equation set for small cone-angle motion. However, we decide to keep using the polar coordi-nate system here, as it allows easy comparison with numeri-cal results that involve large cone-angle motions. A. Unperturbed equation of motion In this case, c50,hs50, and a50. For finite hp, Eq. ~11!can be solved to give w~t!5arctanFS«11 «D1/2 cotvptG u~t!5u0F~2«11!1cos2vpt 2~«11!G1/2 ~12! and in implicit form simply from energy considerations: u25«u02 ~«1cos2w!, ~13! where «5(11h)/hpandvp5hpA«(11«). An initial con- dition of u5u0!1 and w5p/2 is assumed. This derivation is valid only for «.0, that is for h. 21. Similar small uorbits can be obtained for «!21 with «(11«).0. We will restrict our discussion to these two regions. When «(11«),0, the trajectory changes shape to include large oscillations in u, violating the small uassump- tion. This corresponds to an unperturbed orbit crossing the equator with periodic oscillations of Mfromezto2ezdi- rection. B. Average system energy Use the constant-energy motion trajectory @Eqs.~12!and ~13!#as a starting point, and treating the damping and spin current as a perturbation, the average rate of energy change ^dU/dt&is obtained using Eq. ~1!, with Eq. ~11!foru8and w8. The average energy variation rate thus obtained is1 KKdU dtL52~2hp«u02!hs2~2hp«u02!ahpF2«~11«!A 1~112«!1hs hp2BG ~14! with A[K1 «1cos2wL52«11 2«~11«! B[Kcoswsinw «1cos2wL50 ~15! as long as «(«11).0. Therefore, 1 KKdU dtL522~11h!FS11h11 2hpDa1hsGu02.~16! C. Low-field switching threshold Foruhu,1, Eq. ~16!gives the on-axis stability threshold for spin-current-driven motion at the small cone-angle limit. For spin current, instability occurs when the magnitude of hs exceeds a critical value. In this case, hs,hsc52~11h11 2hp!a. ~17! Placing real-life units back in, we have for the magnitude of the critical spin-injection current: Ic51 hS2e \Da ucosfu~a2lmHkMs!S112pMs Hk1H HkD, ~18! which was the same as shown in Ref. 7 but now also in- cludes an easy-plane anisotropy 2 pMs/Hk. This relation is also consistent with the results obtained by Katine et al.9 It is curious to notice that the easy-plane anisotropy hp does not affect the magnetic switching threshold of uhu51, yet it does affect the threshold for spin-current-induced switch. For large easy-plane anisotropy uhscu’ahp/2. This is because a magnetic-field-driven switch can occur with M practically rotating only in the easy plane, whereas a spin-current-induced switch has to involve significant amount ofout-of-plane precession. It is also important to mention that Eq. ~17!only gives a threshold for an instability towards an increasing cone angle insmall ulimit.Itdoesnotguaranteethattheconeanglewill increase indefinitely and a switching event will follow. For largehpsystems the actual switching requires a spin current with larger magnitude than dictated by Eq. ~17!, as will be discussed later using a numerical example. D. High-field switching threshold Foruhu@hp11, and with a large spin-current hspushing the moment in the opposite direction as hdoes, one obtains another threshold, either for current or for applied field, for the high-field forced alignment of Mwith respect to applied field. This relates to the stable small cone-angle ( u0!1) solution for the unperturbed ( a50,hs50) orbit in the limit572 PRB 62 J. Z. SUNof«(«11).0 but «,21. In this case, because Mandhare in opposite directions along the easy axis, a small u0stability corresponds to an energy maximum . Using Eq. ~16!and keeping track of the signs with regard to the relative align- ment of Mandh, one gets the threshold fields for field- induced switching under a large spin current hs: uhac(6)u5uhsu a6S111 2hpD, ~19! wherehac(1)is the threshold field for Mto switch from anti- parallel to parallel to an hincreasing in magnitude, while hac(2)is the threshold field for switching of Mback from a parallel to antiparallel state with respect to has the value of his reduced. In real-life units, if one assumes a zero-field threshold current Ic}2pMs, then Eq. ~19!can be rewritten as Hac(6)52pMsSIbias Ic61D, ~20! whereIbiasis the bias current of the junction. Equations ~19! and~20!are related to the intermediate magnetoresistance states observed in Fig. 3 ~d!of Ref. 9, as will be discussed below using a numerical example. IV. NUMERICAL STUDIES Numerical studies of Eq. ~10!are organized as follows. First we discuss the time evolution of the magnetization M. This is followed by a study on the effect of spin current on the magnetic switching, both in terms of sweeping field H and sweeping current hs. We then discuss the speed of switching under spin-current drive as it compares to the morefamiliar field-driven reversal process. In the end we brieflydiscuss the device and material implications of this mecha-nism. We use the reduced units introduced in previous sections for our simulation. A summary of the units and their refer-ence values are given in Table I. In most simulation results discussed below we set the LLG damping coefficient a 50.01, unless differently specified for individual cases. A. Time evolution of M under the influence of a spin-current First consider the simple uniaxial anisotropy case with hp50. The time evolution of Munder the influence of a uniaxial anisotropy field is one of a spiral motion traced by the tip of M. The damping action causes a decrease of the cone angle, and the moment eventually comes to rest in the direction parallel to the easy-axis ez. This is well known. Under the influence of a spin-current hs,Mwill pick up an additional precession corresponding to the spin-angular mo-mentum deposition. The balance between the damping term and that of h sdetermines the final resting direction of M,a s described by du dt52u@a~11h!1hs# dw dt52~11h!1hsa ~21!that has a solution of u~t!5u0exp~2t/t1! t151/@a~11h!1hs# ~22! with a threshold spin current of hsc52a~11h!. ~23! Given an initial state such that Mis stationary and slightly tilted away from the uniaxial direction ezatt50, the time- dependent evolution of M(t) is illustrated in Fig. 2 for dif- ferent values of spin current hs. A characteristic of a spin- current-induced switch of M(t) is the reversal of its precession direction when it crosses the equatorial position.This comes from the sign change in the spin-current-inducedtorque term in Eq. ~6!. A purely magnetic-field-driven switch ofM( t) does not have this precession reversal. For finite values of hp, as one may expect, the precession in general follows an elliptically distorted trajectory, with thecone angle more spread out in the easy plane, while becom-ing confined normal to the easy plane. An example of thissituation is shown in Fig. 3. Later we will show that a large h p(@1) does not only compress the precession cone angle into the easy plane, it can also introduces a steady-state pre-cession for spin currents with a magnitude slightly above the low-cone-angle stability threshold h scfrom Eq. ~17!. B. Spin-current induced switching As shown above, in the case of pure uniaxial anisotropy with the spin polarization aligned to that of the easy axis, whenhsexceedshsc,Mswitches its orientation to become aligned with the spin-polarization. This can be traced out as an hysteresis loop in M(hs), as shown in Fig. 4. A system- atic dependence of the switching field hscon applied field h is found, following Eq. ~23!. Forh50,M(hs) is always symmetric against origin. That does not necessarily mean M(I) is symmetric. This is because the amount of net torque deposition depends sensi- FIG. 2. The precession of magnetization under the influence of a spin current. Uniaxial anisotropy alone. ~a!Time dependence of Mz.~b!Time dependence of Mx.~c!A 3D portrait of the spiral motion of the tip of M. North pole is ezdirection.PRB 62 573 SPIN-CURRENT INTERACTION WITH A MONODOMAI N...tively on the condition of the interface responsible for spin- current injection. This situation can be phenomenologicallyhandled by introducing an effective spin polarization con- taining a sign dependence on I, i.e., h!h6in Eqs. ~7!and ~18!. C. The effect of a strong easy-plane anisotropy The hysteresis loop M(hs) changes its shape upon the introduction of a large easy-plane anisotropy. This is illus- trated in Fig. 5. For hp.5, before a complete reversal of M, a sloped M(hs) region is seen to develop when uhsufirst exceeds uhscuas defined by Eq. ~17!. This region corresponds to a steady-state precession with an oblong-shaped trajec- tory. This can be seen in the time dependence of Mz(t), as shown in Fig. 6. This large angle steady-state precession is a result of an increase in effective damping for large cone-angle dynamics.As one increases the easy-plane anisotropy, the precessionbecomes increasingly nonlinear and complex, which chan-nels more energy into the higher frequency modes that give more dissipation to M( t) per unit time. A balance can al- ways be established between increased energy injection from increasing hsand the increased damping from increasing cone angle, as long as the maximum cone angle does notcross the equator. This is the region where a steady-stateprecession is formed. Once the precession crosses the equa-tor, however, due to the sign change of the torque term @Eq. ~6!#, the precession accelerates, and a switching of M( t) results. Figure 5 also shows the dependence of M(hs) hysteresis on applied field h. While Eq. ~17!does dictate the onset of M reversal ~see bottom inset, Fig. 5 !, the threshold current po- sition corresponding to the completion of Mreversal ~de- fined ashsc1,2shown in Fig. 5 !does not follow from that of Eq.~17!, but rather has a stronger dependence in h, as shown in Fig. 5. Furthermore, the dependence of hsc1,2on appliedfieldhis asymmetric. When the direction of his to decrease hsc2, the magnitude of hscdecreases asymptotically towards hsc52a(11h11 2hp) from Eq. ~17!. It does not decrease belowhschowever until h,21. Then a sudden switch oc- curs and hsc2drops to zero. This is reasonable, since h, 21 is the condition for a magnetic-field induced moment reversal without the assistance of spin current, naturally hsc2 becomes zero. On the other hand, if his to increase the magnitude of hsc1, as occurred on the left-side transition shown in Fig. 5, hsc1’s change is not bounded by hsc, hence larger field dependence in the ~broadened !switching field is observed there. D. High-field switching threshold An example of the simulated high-field switching thresh- old behavior is shown in Fig. 7. Here a large spin current hs56.0 is applied with its polarization along the 2ezdirec- tion. Also included is an easy-plane anisotropy of hp5190. The applied field is swept from 2800 to 800 along the ez axis. This is a situation very similar in quantitative terms to the experiment shown in Fig. 3 ~d!of Katine et al.’s paper.9 TheM(h) behavior consists of four regions. ~1!Forh,0 in Fig. 7, the effect of both applied field and the spin current is to force Mto point to 21, henceMpoints to2ez. ~2!Between h50 andh5hp: This region corresponds to an unperturbed orbit involving large cone angles where « ,0 and «(11«),0. In the present situation, the competi- tion between applied field hwhich now favors a 11 direc- tion forM, and that of the spin current ~still pointing towards 2ez) causes a strong steady-state precession when 1 ,h !hp.A sh!hpa stable resting position develops for Mthat points out of the easy plane and making an angle with the 2ezdirection, results in an Mzvalue between 0 and 21.As hincreases in value, Mincreasingly tilts back towards 2ez, away from h, causing Mzto approach 21. It is at first counterintuitive that Mzin this region should become closer to21 as the applied field is being increased. But this is actually not surprising once one realizes that in this region FIG. 3. The precession of magnetization under the influence of a spin current. Uniaxial anisotropy plus an easy-plane anisotropy ofh p55. The uniaxial-anisotropy-alone trace of a50.01 is included for comparison. The elliptical precession is apparent here, with thecone-angle being compressed in the direction normal to the easyplane. Panels have the same definition as in Fig. 2. FIG. 4. Spin-current-driven reversal of magnetization. Uniaxial anisotropy only, no easy-plane anisotropy is added. The switchingcurrenth scshows linear dependence on applied easy-axis field has predicted by Eq. ~23!.574 PRB 62 J. Z. SUNthe behavior of Munder the influence of hsis to seek a resting position with energy maximum . ~3!Between h5hpandh5hac(1)5uhsu/a1(11hp/2), fol- lowing Eq. ~19!. In this region Mzis completely forced to 21.~4!Whenh.hac(1), where finally the effect of applied field takes over, and Mswitches direction to rest along the direction of applied field, and Mz511. OnceMswitches direction to align with h, the stability criteria for small cone angle, Eq. ~16!changes sign, hence on its way back, hac(2)5uhsu/a2(11hp/2). The monodomain threshold hac(6)can only give a rough estimate to the high-field threshold observed in Katine et al.’s experiment. For a real thin-film sample such as the one used by Katine et al.,the magnetic dynamics between h50 andh5hac(6)is not even approximately monodomain in nature. This is because in this region large cone-angle mo- tion as well as resting positions with significant out-of-the- plane component of Mis involved, which would favor spin- wave excitation or domain formation. The fact that thesystem in this parameter region seeks out an energy maxi-mum rather than a minimum, further increases the likelihoodfor the film to break into complex domains or to excite spinwaves. This may account for the wide plateau observed inRef. 9. A proper treatment of these is however beyond thescope of this paper. E. Effect of spin current on the M Hswitching characteristics: Spin-current-induced distortion to astroids Here we study the switching behavior of M(H) as a func- tion of spin current hs. We focus on on-axis geometries, where the relative angle fbetweennsandezis either zero or p. Without the presence of a spin current, the M(H) switch- ing characteristic for simultaneous easy- and hard-axis fieldpresence is an analytically solvable energy minimumproblem. 11The resulting switching boundary forms an as- troid shape, with the boundary curves defined by FIG. 5. Spin-current induced magnetic switching hysteresis loop M(hs), with a strong easy-plane anisotropy hp5190~chosen to emulate a cobalt thin-film’s demagnetization field 4 pMs). The on- set position in hsforMswitching follows the estimate given in Eq. ~17!. It is not very sensitive to the change of hf r o m0t o1 ,a s expected ~sinceh!hpin this range of h). However, the beginning portion of the switching curve is much more gently sloped. This isdue to the presence of a steady-state precession as discussed in thetext and in Fig. 6. FIG. 6. The evolution of steady-state precession and the completion of M(hs) switch as the precessing moment crosses equator upon increasing hs. Initial u5p20.01. Small deviation from pis added to shorten the initial build-up time for precession amplitudes. Curves ~b!–~e!are progressively offset in vertical di- rection. A crossover from steady-state precession and completeswitching occurs within 1.2127497 ,h s,1.2127498. FIG. 7. Numerical result for the high-current, high-field behav- ior. Field his applied along ez. Spin-current polarization is along 2ez. From origin to h5hp, the competition between applied field and the spin-current causes a deflected final resting angle for the moment. Between h5hpand point A with hA5hac15(hs/a)1(1 1hp/2) according to Eq. ~19!, spin-current effect causes the mo- ment to seek out the energy maximum for its resting direction,hencem z521. At point A, the energy term from applied field finally takes over, and a switching of magnetic moment from 21t o 1 occurs. This switching is hysteretic—upon reversing the sweepdirection of applied field, the moment does not switch back to -1 until point Cwhereh C5hac25(hs/a)2(111 2hp). The net hyster- esis opening between points AandCisdh5hp12;hp.PRB 62 575 SPIN-CURRENT INTERACTION WITH A MONODOMAI N...heasy2/31hhard2/351. ~24! The effect of a spin current on the shape of the astroid is shown in Fig. 8, for a monodomain magnetic moment withonly a uniaxial anisotropy term. Notice that the amount ofspin current required to significantly change the shape of theastroid is within a factor of 2 of the zero-field critical current h sc. The increase in magnitude of the switching field on the left side of the astroid is interesting to observe, as this is aregion where the spin-momentum deposition completelychanged the magnetic system’s trajectory of motion, distort-ing significantly the astroid boundary. While without thepresence of the spin current a large cone-angle precessionwill develop, the spin current stabilizes the small cone-angleprecession, and hence this region is now treatable as a per-turbation to the constant-energy trajectory. The introduction of an easy-plane anisotropy h pdoes not affect the zero-spin-current switching astroid @Eq.~24!#. However, it does alter the effect the spin current has on theshape of the astroid. The evolution of the switching charac- teristics for h p5190 is shown in Fig. 9. The amount of spin current required to affect the shape of the astroid again is around the zero-field critical current hsc, as determined by Eq.~17!. The presence of a strong easy-plane anisotropy completely suppresses the increase of switching field magni-tude on the left side of the astroid. V. SWITCHING SPEED The reversal of Munder a spin-current-driven situation is different from that of a magnetic-field-driven case. For field-driven reversal, in small damping limit ( a!1), the reversal time for magnetization Malong its easy axis depends primarily on the initial dynamics of the moment. For a system with «5(11h)/hp&0, the unperturbed orbit for small ucan be used to estimate the amount of time for the cone angle to evolve from its initial value u0tou.I nt h e limit ofhp@1, to the leading order of «,i ti s t~u!51 hpA2«lnu1Au22u02 u0. ~25! Therefore the asymptotic behavior of the initial reversal- related switching time is ~setting u;1):t0’1 hpA2«ln11A12u02 u0}Hu12hu21/2~h!11! 2lnu0 ~u0!01!. ~26! This relation is verified by numerical simulation for a spe- cific set of conditions: hp5190 and a50.01 and 0.001, re- spectively, as shown in Fig. 10. For current-driven reversal the process is somewhat dif- ferent. Since current-driven reversal is determined by thebalance of damping-related dissipation and the spin-currentinduced energy gain, damping plays a much more critical role—it determines the value of threshold h sc. For the uniaxial anisotropy-only situation, Eq. ~22!gives t~u!’uhs2hscu21ln~u/u0!. ~27! A similar scaling behavior is found numerically in the large hplimit, as shown in Fig. 11. To compare the situation between a field-driven reversal and a spin-driven switch, one examines the behavior of t(u) for the same amount of relative overdrive amplitude infield and in spin current. Following Eq. ~17!, for a given amount of overdrive amplitude uhsu5(11d)uhscu,t }(adhp)21ln(u/u0), whereas for the same amount of over- drive in field uhu511d,t}(dhp)21/2ln(2u/u0). Thus for a spin-current switch with a fixed percentage of overdrive, the speed is directly proportional to its threshold current ahp, and hence to a, whereas for magnetic-field-driven switch, a doesn’t matter as long as a!1. Another limit for a spin-current switch is when the current is well-above the threshold. In this case, t0}uhsu21ln(u/u0). Thus in a large current limit, the switching time for a spin- injection process is independent of a, and is determined by the amount of spin current injected. Thus, in a large spin-current limit, the total amount of spins needed for a reversalevent is independent of the magnitude of the spin current. To get some feelings for real materials, consider a pat- terned cobalt film. Assume a uniaxial anisotropy field of H k5100 Oe from the film’s in-plane shape. In the direction perpendicular to the film, a demagnetization field of 4 pMs ’1.83104Oe’180Hkis present, thus hp’180, similar to FIG. 8. Uniaxial anisotropy only: effect of spin-current injection on the shape of M(H). The zero-current switching characteristics reproduces the well-known ‘‘astroid’’ shape. For this simulation a50.01, thus hsc50.01. FIG. 9. Uniaxial anisotropy plus a strong in-plane anisotropy of hp5190. The effect of spin-current injection on the shape of M(H) is quite different from those shown in Fig. 8 with only uniaxialanisotropy. In this case, h sc50.96 as calculated from Eq. ~17!.576 PRB 62 J. Z. SUNthehp5190 used in the simulation for Figs. 10 and 11. The time conversion is t’Vk21t50.568(ns) twith Vk 5(2mB/\)Hk’1.763109s21. When driven at twice the threshold value, for on-axis-only a magnetic-field-driven switch, and at an initial angle of u051023, the initial- reversal part of the switching time is about t0’0.34 ns, ac- cording to data shown in Fig. 10. With the same amount of overdrive ( hs52hsc), and the same u051023, the spin- current driven process will involve a t0’3.98 ns. VI. MATERIALS-RELATED DEVICE CONSIDERATIONS Equation ~18!has important implications for device appli- cations. First of all, there is a fundamental limit on howsmall the critical current can be if it were to be used forswitching a memory element. The limit is set by the memorybit size required for thermal stability. This was briefly dis-cussed in Ref. 7 where the numerical estimates were basedon a definition of super-paramagnetic transition temperatures that allows a magnetic lifetime of 1 s. Here we discuss thiswith a more realistic magnetic stability requirement of 100 yr. The thermal transition lifetime tLof the magnetic body can be expressed as tL5tAexp(E0/kBT) with the thermal activation barrier E0for moment reversal set by the uniaxial anisotropy barrier height: E051 2a2lmMsHk.tAis associated with the basic magnetic attempt frequency with 1/ tA ’109Hz. If one sets the magnetic lifetime tL.100 yr 53.153109s as criterion for super-paramagnetic transition, this gives a super-paramagnetic transition temperature TS such that E0/kBTS;ln(tL/t0);42.60. Thus E0542.60kBTS anda2lmMsHk585.19kBTS. From Eq. ~18!, this means the threshold current for a 100 yr stability against thermal rever-sal has to be larger than I c>1 hS2e \Da~85.19kBTS!. ~28! Taking h50.1 and a50.01 as a conservative estimate of a typical magnetic metal such as cobalt, and setting operating FIG. 10. Initial reversal-related switching time t0for magnetic field-driven reversal. hp5190. Here we define t0as the amount of time it takes for Mto evolve from u0tou’p/2.~a!t0is largely determined by the unperturbed motion of M. Adding damping changes the ringing characteristic after the initial reversal, but itdoes not significantly alter the initial switching time. As shown inthe text, an asymptotic relation t;2logu0is held for zero ~or a small !a.~b!t0scales essentially as uh21u21/2. Again the result is fairly robust against adding a small a. The top inset shows time dependencies of Mzand the definition of t0in our numerical procedure. FIG. 11. The reversal process for a spin-current-driven process. ~a!Initial reversal-time dependence on starting angle u0. A scaling oft0}2t1lnu0is demonstrated, as discussed in the text. ~b!The scaling of t0ashsapproaches hsc. It is described by t0}uhs 2hscu21, withhscas described by Eq. ~17!. Upper inset shows the time-dependent evolution of Mz(t) and the definition of the initial reversal time t0.PRB 62 577 SPIN-CURRENT INTERACTION WITH A MONODOMAI N...temperature TS5130°C 5400 K, we have the minimum threshold current for technologically interesting applications ofIc;140mA. For thin-film devices in current-perpendicular geometry, we can estimate the amount of current density required formagnetic switching. Assume that there are ways to neutralizethe demagnetizing field of the film, the threshold currentdensity can then be expressed as J c5a hS2e \D~lmHkMs!S11H HkD, ~29! wherelmcan be viewed as the thickness of the magnetic switching layer. The critical current density is then directly proportional to the film thickness lm. Again, taking cobalt as an example where we assume a uniaxial anisotropy term of Hk5100 Oe, and a saturation magnetization Ms.1.5 3103e m u/c m3, in zero-applied field, one has Jc54.6 3104lm(A/c m2), where lmis in angstroms. An all-metal current-perpendicular pillar can probably take around 107to 108A/c m2of current density without short-term damage. This gives a reasonable working film thickness of at least100 Å . For magnetic tunneling junctions, however, the prac- ticalJ cfrom materials and electrical integrity point of view is limited to about 106A/c m2. This means to directly inject spin current across a tunneling barrier into the magneticbody, the magnetic body would prefer to have softer anisot- ropy energy product H kMsto give a reasonable working film thickness of well-above 1 0 Å . This can perhaps be done by a careful selection of electrode material and its shape—alow-aspect ratio Permalloy magnetic dot perhaps will work. Combining the requirements of thermal stability @Eq. ~28!#and current-density limit @Eq.~29!#, the lateral dimen- sion of the magnetic body can be determined as well. To have aT S5400 K, again use the parameters for cobalt as we did before, and set lm515 Å , one has a;1500 Å . Thesenumbers give a rough estimate to the relevant device dimen- sions, although they should not be taken literally. For onething at such high aspect ratios it is questionable whether thefilm will remain single domained for its dynamic processes. VII. SUMMARY A preliminary study is presented here for the basic dy- namic properties of a magnetic moment under the influenceof a spin current. The magnetic moment is found to precessunder the torque associated with spin-current-induced angu-lar momentum deposition. The competition between thespin-current-related energy gain and the LLG damping-related energy dissipation determines the precession process.Under appropriate conditions, the precession will lead to areversal of the resting direction of the magnetic moment,causing a magnetic switch. Quantitative predictions are madefor the threshold spin current for such a switch, as well as thegeneral dependence of the switching process on the magneticenvironment experienced by the moment. The switchingspeed under spin-current injection is predicted to be compa-rable to present-day field-driven switching processes, al-though the two processes are intrinsically different and theyfollow different asymptotic scaling behaviors with regard tothe initial and drive conditions. The spin current is also pre-dicted to affect the magnetic switching characteristics of themoment, causing a distortion to the astroid-shaped switchingcharacteristics. ACKNOWLEDGMENTS I wish to thank John Slonczewski and Roger Koch at IBM Research and Professor Dan Ralph at Cornell University forfruitful discussions. I would also like to thank Roger Kochfor help setting up the computing environment for part ofthis simulation work. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1~1996!. 2J. C. Slonczewski, J. Magn. Magn. Mater. 195, L261 ~1999!. 3L. Berger, Phys. Rev. B 54, 9353 ~1996!. 4L. Berger, J. Appl. Phys. 49, 2156 ~1978!. 5Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B 57, R3213 ~1998!. 6M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 ~1998!. 7J. Z. Sun, J. Magn. Magn. Mater. 202, 157 ~1999!. 8E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867 ~1999!. 9J. A. Katine, J. F. Albert, R. A. Buhrman, E. B. Meyers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 ~2000!. 10This is a phenomenological assumption that simplifies the mathwhile still keeping the essence of the physics. Details of the spin angular momentum transfer process is significantly more com-plex, and is certainly interface dependent. Experimental quanti-fication remains to be obtained. A theoretical derivation basedon microscopic quantum mechanics for a specific five-layer ge-ometry was given by Slonczewski in Ref. 1, which gave Eq. ~6! an additional factor of (1/ h)g(nsnm)5(1/h)@241(1 1h)3(31nsnm)/4h3/2#21. The angular dependence of this fac- tor is not significant for small h. 11J. C. Slonczewski, IBM Research Memorandum No. 003.111.224 ~1956!. Also see L. D. Landau and E. M. Lifshitz, Electrody- namics of Continuous Media ~Pergamon, New York, 1960 !, Sec. 37, pp. 150–151.578 PRB 62 J. Z. SUN

PhysRevB.77.134424.pdf

Electron magnetic resonance study of transition-metal magnetic nanoclusters embedded in metal oxides Vincent Castel and Christian Brosseau * Laboratoire d’Electronique et Systèmes de Télécommunications, Université de Bretagne Occidentale, CS 93837, 6 Avenue Le Gorgeu, 29238 Brest Cedex 3, France /H20849Received 25 October 2007; revised manuscript received 28 February 2008; published 10 April 2008 /H20850 Here, we report on the results of an electron magnetic resonance /H20849EMR /H20850study of a series of Ni /ZnO and Ni //H9253-Fe2O3nanocomposites /H20849NCs /H20850to probe the resonance features of ferromagnetic /H20849FM/H20850Ni nanoclusters embedded in metal oxides. Interest in these NCs stems from the fact that they are promising for implementingthe nonreciprocal functionality employed in many microwave devices, e.g., circulators. We observe that theEMR spectrum is strongly affected by the metallic FM content and its environment in the NC sample. Wereport the existence of broad and asymmetric features in the EMR spectra of these NCs. Our temperaturedependent EMR data revealed larger linewidth and effective gfactor, in the range of 2.1–3.7 /H20849larger than the free electron value of /H110152/H20850, for all samples as temperature is decreased from room temperature to 150 K. The line broadening and asymmetry of the EMR features are not intrinsic properties of the metallic nanophase butreflect the local /H20849nonmagnetic or magnetic /H20850environment in which they are embedded. Furthermore, the results of a systematic dependence of the room temperature EMR linewidth and resonant field on the Ni content andthe corresponding effective microwave losses measured in previous works show a remarkable correlation. Thiscorrelation has been attributed to the dipolar coupling between magnetic nanoparticles in the NCs. DOI: 10.1103/PhysRevB.77.134424 PACS number /H20849s/H20850: 75.50.Tt, 75.75. /H11001a, 76.30.Fc, 76.50. /H11001g I. INTRODUCTION Interest in the polarization and magnetization mechanisms in nanocomposites /H20849NCs /H20850is on the rise. From the fundamen- tal point of view, the main scientific drivers are the multi-plicity of new and interesting effects, e.g., product propertysuch as magnetoelectricity, 1–4and the observation that many of the ideas central to the understanding of two-dimensionalelectromagnetism and magnetism indicate the need for newtheoretical and experimental approaches. Respective systemsmight be realized in granular mixtures, layered compounds,thin films, or artificial multilayers. Another reason to givesome attention to magnetic NCs is the current increase ininterest in biological physics and in the development of newpharmaceutical products. 5–9The particular interest among the biophysical and chemical sciences is triggered by the factthat plasmonic NCs can provide promising platforms for thedevelopment of multimodal imaging and therapyapproaches. 10,11From the practical side, such studies are im- portant for developing techniques that are able to producenanostructures in a controlled manner. However, expecta-tions for integrated nanoelectronic devices exhibiting newfunctionalities will become more realistic when there is com-plete understanding of the microscopic mechanisms that con-trol changes in electronic structure that scale with the clusterdimensions. While much attention has focused on exploring the micro- structure of materials by using conventional methods of char-acterization, recent work has recognized the value of conven-tional electron magnetic resonance /H20849EMR /H20850techniques /H20849see, e.g., Refs. 12–14/H20850which can reveal both the average mag- netic behavior and its microscopic inhomogeneity. Thesetechniques have proven to provide reliable approaches to thestability of nanoparticle dispersions which can be affected byaggregation and agglomeration due to their high surface en-ergy, secondary crystallization, and Ostwald ripening, just to name a few indicative ones. 12,15–19At this length scale, con- duction and magnetic properties considerably deviate frombulk, e.g., by showing a significant enhanced magnetic mo-ment for sizes up to a few hundred atoms. 4,9Within this context, a powerful attribute of the EMR is its ability toexperimentally probe local scale. As yet, few experimental approaches for the EMR analy- sis of magnetic NCs have been discussed. Only a few se-lected magnetic nanoparticle systems, including /H9253-Fe 2O3 /H20849Refs. 12,15, and 16/H20850and ferrites,20–23were investigated by the EMR methods. So far, hardly any data exist on the EMRcharacterization of magnetic metal and/or metal oxide NCs.It is also worth emphasizing that several analytical 24and numerical25,26approaches have been put forward to analyze the resonance spectra of magnetic submicron particles. De-spite this motivation to develop a full understanding of fer-romagnetic /H20849FM/H20850metallic nanoclusters dispersed into a vari- ety of /H20849magnetic or not /H20850hosts from EMR, several basic features in the gigahertz EMR modes remain unclear. In par-ticular, there is no simple theory which allows the measuredspectrum to be related to the underlying microstructure. An-other basic issue is to determine whether the linewidth is anintrinsic feature of metallic clusters or arises as a conse-quence of surface interactions or other perturbations. Our original intent for the present work was to investigate Ni /ZnO and Ni / /H9253-Fe 2O3NCs. A series of recent microwave frequency-domain spectroscopy /H20849gigahertz /H20850studies of the quasistatic effective permittivity and magnetic permeabilityposed fundamental questions concerning polarization andmagnetization mechanisms in these NCs. 27–32Here, these materials were chosen because our group has previously pub-lished detailed accounts of experimental and modeling ap-proaches on the static magnetic and microwave response ofthe samples under study. 33–38This analysis allowed us toPHYSICAL REVIEW B 77, 134424 /H208492008 /H20850 1098-0121/2008/77 /H2084913/H20850/134424 /H208499/H20850 ©2008 The American Physical Society 134424-1simulate magnetization dynamics on solving the Landau– Lifshitz–Gilbert equation coupled to the Bruggeman effec-tive medium approach. Traditionally, particle and aggregatesize information are considered irrelevant to the wave trans-port since rapid oscillations of the electromagnetic wave onlength scales larger than any scale /H9264of the medium inhomo- geneities are integrated out in the conventional effective me-dium analysis, where it is specifically assumed that /H9264/H11270/H9261 with/H9261the wavelength of the probing wave, and thereby the NC can be treated as a structureless continuummedium. 30,31,34–36,39,40Some very recent work has demon- strated that ferromagnetic resonance /H20849FMR /H20850measurements of these NCs,37,38or of hot isostatic pressed /H20849hipped /H20850polycrys- talline yttrium iron garnet41are very sensitive to details of the spatial magnetic inhomogeneities. Although these studieshave advanced the understanding of wave transport phenom-ena in granular NCs, striking discrepancies remain betweenexperiments and corresponding results obtained from exist- ing phenomenological models and numerical simulations. Animportant question of broad fundamental interest is why theeffective permittivity and magnetic permeability appear to bevery sensitive to the details of the nanoparticle cluster struc-ture, thus suggesting a breakdown of the continuum-levelmodeling and bringing NC physics concepts to light. Theanswer to this question constitutes the key for understandingthe response of magnetic clusters to electromagnetic probes. In the present study, we report on a detailed EMR study of magnetic metal and/or metal oxide NCs. This paper providestwo main results. By varying the Ni volume fraction in thesamples, we observed that the EMR signal reflects the prop-erties of the metallic FM nanophases as well as the localnonmagnetic or magnetic environment in which they are em-bedded. Our data also indicated an empirical correlation be-tween the systematic dependence of EMR linewidth andTABLE I. Selected physical properties of the powders investigated in this study. Powder ZnO /H9253-Fe2O3 Ni Average particle sizea–d49 nm 23 nm 35 nm Powder color White Brown BlackSpecific surface area BET a/H20849m2g−1/H20850 22 51 15.6 MorphologycElongated Nearly spherical, facetedSpherical Crystal phasea,dWurtzite Maghemite /H20849Cubic spinel /H20850Fm3m/H20849225/H20850 ccp Densitya/H20849gc m−3/H20850 5.6 5.2 8.9 aFrom manufacturer product literature. bDetermined from specific surface area. cChecked by TEM images. dDetermined by XRD. In the XRD analysis, the possible influence of strain on the online broadening was neglected. This may result in an underestimate of the particle size. TABLE II. Overview of NCs compositions: ƒ xdenotes the volume fraction of the X species, ƒ pis the porosity of the samples and ƒ resinis the volume fraction of resin. The uncertainty on ƒ xis typically of the order of 5%. Material designation ƒ Ni ƒZnO ƒp ƒresin ƒ/H9253-Fe2O3 nNiZ1 0.49 0.08 0.28 0.15 nNiZ2 0.42 0.17 0.27 0.14nNiZ3 0.38 0.21 0.26 0.15nNiZ4 0.33 0.26 0.25 0.15nNiZ5 0.29 0.30 0.26 0.15nNiZ6 0.25 0.35 0.25 0.15nNiZ7 0.18 0.44 0.23 0.14nNiZ8 0.09 0.54 0.22 0.15nNiZ9 0 0.63 0.21 0.16 1–nNiF 0.08 0.26 0.13 0.532–nNiF 0.17 0.25 0.14 0.443–nNiF 0.29 0.27 0.12 0.32 4–nNiF 0.50 0.26 0.15 0.09 5–nNiF 0.04 0.26 0.25 0.55VINCENT CASTEL AND CHRISTIAN BROSSEAU PHYSICAL REVIEW B 77, 134424 /H208492008 /H20850 134424-2resonant field on the Ni content and the corresponding mi- crowave loss features. The rest of the paper is organized as follows. Section II gives some background on the samples we have investigatedand discusses technical details on the EMR characterization.This is followed in Sec. III by a presentation of an experi-mental analysis of the effects of varying the Ni content in theNCs on the EMR spectra. A comparison with previous FMRlinewidth data and effective microwave losses is given inSec. IV . Next, Sec. V presents the main conclusions of thisstudy and we make some comments regarding possible im-plications of these results. II. EXPERIMENT The same samples were used as in a previous study.33 Room temperature-pressed NC compacts were made underthe application of a uniaxial pressure of 64 MPa for 2 min.The morphology and size of the starting powders /H20849Table I/H20850 were determined by transmission electron microscopy/H20849TEM /H20850by using a 200 kV Phillips apparatus. Bright field TEM images indicate that /H9253-Fe 2O3and Ni particles are ho- mogeneous with nearly spherical shape, whereas thewurtzite-type ZnO particles are rod shaped with an aspectratio of about 3:1. The phase purity was checked by x-raypowder diffraction /H20849XRD /H20850by using Cu K /H92511radiation, and the crystallite sizes were determined from the full width athalf maximum of the strongest reflection by using theWilliamson-Hall method after applying the standard correc-tion for instrumental broadening. The average crystallinesizes of the samples were also determined from analysis ofbright field that cross-sectional TEM images were found tobe consistent with the ones obtained from XRDmeasurements. 30–34The XRD analysis of spherelike clusters suggests that Ni existed in the form of metal. These clusters,composed of Ni nanoparticles, are randomly and uniformlydistributed in the matrix and are isolated from each other bythe metal oxide and epoxy phases. The Ni cluster size in-creases with the Ni content. The metallic nanoparticle tendsto be oxidized, arising from the large surface-area-to-volumeratio and the high electronegativity of metallic nickel. Al-though not studied in detail, the presence of secondaryphases, e.g., NiO, was estimated to be less than 0.3 vol %from XRD. In addition, Ni nanoparticles can be protectedfrom oxidation by encapsulation in the epoxy phase. The EMR experiments were performed in two sets of samples /H20849see Table II/H20850. The first series /H20849Ni /ZnO /H20850comprising nominal Ni volume fractions in the range of 0%–49% con-sisted of mixtures of Ni and ZnO nanoparticles and epoxyresin /H2084915 vol % /H20850. The second series /H20849Ni / /H9253-Fe 2O3/H20850had nomi- nal Ni volume fractions in the range of 4%–50% and /H9253-Fe 2O fractions were chosen to maintain the total content of themagnetic phase constant /H1101560 vol %. For the two sets of NCs, the residual porosities were estimated to be between 21and 28 vol %. For the present EMR measurements, samples were cut to pieces of about 1 /H110031/H110031m m 3. EMR signals were recorded on a heterodyne spectrometer in a continuous wave modeX-mode microwave /H20849F=9.4 GHz /H20850with a 500 mW Varianklystron, a Bruker resonance TE 102cavity, a Varian electro- magnet with maximum field amplitudes of 800 mT, and anitrogen flux cryosystem /H20849Oxford Instruments /H20850for low- temperature measurements in the range from 150 K to roomtemperature /H20849RT/H20850. For g-factor calibration, the 1,1-diphenyl- 2-picrylhydrazyl standard has been used /H20849g=2.0037 /H20850. The EMR signals recorded were the first derivatives of the powerabsorption, dP /dH, as a function of the applied magnetic field Hby using 100 kHz modulation amplitude and lock-in technique /H20849EG&G Princeton /H20850for calculations of gfactors, peak-to-peak linewidths /H20849/H9004H pp/H20850, and resonance field /H20849Hres/H20850. For the g-factor measurements, the cavity frequency /H9263was measured at each temperature and Hresis determined by the location of the zero of the absorption derivative. Then, g =0.714 48 /H1100310−6/H9263/Hres, where /H9263is in gigahertz and Hresin kilo-oersted. The EMR measurement was performed on cool-ing the sample. EMR signals were also obtained for neat nanoparticles by using either powders placed in an EMR tube, and pumped to/H1102110 −3Pa in order to eliminate the moisture and oxygen ef- fect, or in loose packed form with 15 vol % epoxy. For bothsamples, the EMR line cannot be accurately fitted to a singleLorentzian line shape /H20849not shown /H20850. Figures 1and2show that H res/H20849/H9004Hpp/H20850, for both neat and compacted powders of Ni and /H9253-Fe 2O3, appreciably increases /H20849decreases /H20850with TforTsin the range of 150–300 K. The broadening of the EMR featureis sharply decreased by upshifting the resonance field. Sys-tematic EMR characteristics follow the same monotonictrends; however, the data reveal that the dilution of nanopar-ticle powder in the epoxy host matrix and compaction havefor effect to downshift the EMR line and to decrease its peakwidth compared to the neat powder EMR features. The EMRlinewidth is affected by inhomogeneities and is quite large,i.e., 0.85–2.6 kOe. The difference between powder andloose packed form with epoxy can be attributed to someaspects of the microstructure, such as porosity, grain bound-aries and other extended defects, the presence of localstrains, or anisotropy in the randomly oriented magneticclusters. Figure 3illustrates that grapidly decreases with increasing Tfor the neat Ni powder, whereas for /H9253-Fe 2O3 powder, gshows little Tdependence and does not signifi- cantly deviate from g/H110152.1–2.2. A similar behavior has been reported in ferrite nanoparticles.12For ease of comparison, data are plotted as g/H20849T/H20850/g/H20849RT/H20850. Part of the gshift with low- ering Tcan be attributed to the increase in the demagnetizing field. We would like also to emphasize that no measurableEMR signal for ZnO nanoparticle powder was detected. For our low conductivity NCs and considering the mea- sured values of the effective electromagnetic parameters ofthese NCs, 31,37,38we find that the skin depth is in the 102–103mm size range in the gigahertz frequency range, i.e., much larger than the sample thickness. Thus, one cansafely assume a full and homogeneous penetration of themicrowaves into our samples. In composites with uniformdispersions of magnetic nanoparticles, the conductivity ofthe NC is mainly determined by the interparticle distance andeddy currents, produced within the particle is extremelysmall at high frequency, which are limited to individual par-ticles or aggregates. We note that Ramprasad et al. 27have shown, in their phenomenological modeling of the propertiesELECTRON MAGNETIC RESONANCE STUDY OF … PHYSICAL REVIEW B 77, 134424 /H208492008 /H20850 134424-3of magnetic nanoparticle composites, that in the 0.1–10 GHz frequency range, particles with radii smaller than 100 nm areexpected to encounter negligible eddy current losses. Thiswas found true even at high particle volume fraction, whenclustering of particles could result in aggregates much largerthan the actual particles. Furthermore, no signature for per-colation threshold is apparent for the data collected; thus, weinfer from the Ni content dependence of /H9268dcthat the Ni nano- aggregates should be separated in the ZnO matrix. It hasbeen recognized by now that in ferromagnetic NCs, theshielding and dissipation due to eddy currents rapidly dimin-ish with decreasing the particle size. This has for effect toreduce the dielectric losses in metallic nanoparticles. III. ELECTRON MAGNETIC RESONANCE LINEWIDTH PARAMETERS We now present the data underlying our conclusions sum- marized above. Full sets of absorption versus field derivativeprofiles, shown in arbitrary units, which correspond to rep-resentative EMR spectra of Ni /ZnO and Ni / /H9253-Fe 2O3NCs,were obtained in the restricted temperature range from 300 to 150 K. Typical spectra are displayed in Figs. 4/H20849a/H20850and 4/H20849b/H20850at six different temperatures for Ni /ZnO /H20849Ni content of 17.5 vol % /H20850and Ni //H9253-Fe 2O3/H20849Ni content of 50.3 vol % /H20850 NCs, respectively. Note first that the EMR spectra display aseverely distorted line shape suggestive of inhomogeneousbroadening, which is typical of superparamagnetic resonancespectra. 25Unfortunately, the spectra cannot be meaningfully deconvoluted. Possible explanations range from size andshape distributions to magnetic anisotropy /H20849see, for example, Refs. 25and26/H20850. A consensus on the cause of these has yet to be established partly because it is difficult to make quan-titative evaluations of the EMR line in composites of finemagnetic particles and also because there may not be a singlecause. To analyze the temperature dependence of the EMR spec- tra,/H9004H ppandHresare shown in Fig. 5/H20849Fig. 6/H20850as a function of temperature for the Ni /ZnO NCs /H20849Ni //H9253-Fe 2O3NCs /H20850. The inset of Fig. 1indicates how /H9004Hppand Hreswere actually measured. The data of the lower panels of Figs. 5and6serve to make four important points. First, although different indetail, the Tdependence of /H9004H ppin Figs. 5and6has impor- tant features in common, namely, as Tis increased, /H9004Hpp monotonically decreases. Also, the line shape becomes quite asymmetrical at the lower temperatures. For the three lowestNi volume fractions in Ni / /H9253-Fe 2O3NCs, a very weak de-150 180 210 240 270 3001.01.21.41.61.82.02.22.42.62.02.22.42.62.83.03.2 Ni powder compacted with epoxy (15 vol% ) Ni neat powder T(K)/CID39Hpp(kOe) (b)H(kOe)dP/dH (a.u)/CID39Hpp Hres (a)Ni powder compacted with epoxy (15 vol% ) Ni neat powderHres(kOe) FIG. 1. /H20849a/H20850The line position, Hres, as a function of temperature for neat Ni powder and a Ni /H2084956 vol % /H20850powder compact with 15 vol % epoxy /H20849see text for details /H20850. The inset shows that the reso- nant field Hresis determined by the location of the zero of the absorption derivative. The dotted and dashed lines serve as guidesfor the eye. /H20849b/H20850S a m ea si n /H20849a/H20850for the peak-to-peak linewidth /H9004H pp corresponding to the peak-to-peak separation in the absorption de- rivative /H20851inset of /H20849a/H20850/H20852.120 150 180 210 240 270 3000.81.01.21.43.123.163.20 (b)/CID74Fe2O3powder compacted with epoxy (15 vol% )) /CID74Fe2O3neat powder /CID39Hpp(kOe) T(K)(a)/CID74Fe2O3powder compacted with epoxy (15 vol% ) /CID74Fe2O3neat powderHres(kOe) FIG. 2. /H20849a/H20850The line position, Hres, as a function of temperature for neat /H9253-Fe2O3powder and a /H9253-Fe2O3/H2084956 vol % /H20850compact with 15 vol % epoxy. The dotted and dashed lines serve as guides for theeyes. /H20849b/H20850S a m ea si n /H20849a/H20850for the peak-to-peak linewidth /H9004H pp.VINCENT CASTEL AND CHRISTIAN BROSSEAU PHYSICAL REVIEW B 77, 134424 /H208492008 /H20850 134424-4cline around 1.1 kOe is observed. At low temperatures, the spin-spin interactions cause spin dynamics to freeze out, andspins essentially behave as static spins. At higher tempera-tures, when the time scale of the dynamics of the spins isfast, it might be expected that the conduction electron spin-lattice relaxation time in metals, T 1, is the characteristic time for the return to thermal equilibrium of a spin system drivenout of equilibrium by the microwave field at resonance. It iswell established that in pure metals, T 1is limited by the scattering of conduction electrons by the random spin-orbitpotential of nonmagnetic impurities or phonons. 14These changes in /H9004Hpp/H20849T/H20850and g/H20849T/H20850are reflected in the motional narrowing of the EMR line.15,25Second, when these values are compared to the experimental data of neat and compactedpowder with epoxy, i.e., Figs. 1and 2, it is seen that the dependence of /H9004H pponTmatches rather well that for neat powder, except for the Ni //H9253-Fe 2O3NCs containing the two largest Ni volume fractions. We also notice that for theNi / /H9253-Fe 2O3sample with 50 vol % Ni, one recovers very similar values of /H9004Hppthan for those displayed in Fig. 5for Ni /ZnO samples. Thus, the line broadening is not an intrin- sic feature of Ni but arises as a consequence of surface in-teractions, reflecting the fact that interaggregate interactionsinduce collective behavior between the magneticnanophases. It seems likely that the interfaces play a key rolein determining the ultimate performance of magnetic nano-structures: the surface anisotropy and magnetization signifi-cantly modify the static states and dynamic properties. 42Fur- ther, the broadening of the EMR features is sharply increasedfor a large Ni content in Ni / /H9253-Fe 2O3NCs. That the Hres versus Tdependences do not map the Hresof powderedsamples is also a noticeable fact. A natural question is why we observe a substantial downshift of the EMR line of themagnetically diluted samples in comparison with neat pow-ders. It is rather likely that this downshift is a manifestationof powder compaction during sample fabrication. Indeed, weobserved /H20849not shown /H20850an apparent correlation between the residual porosity and the effective density of the NCsamples, e.g., we simultaneously measured a 30% drop ofthe porosity with a 10% increase in the density as the appliedpressure during the compact fabrication process is changedfrom 33 to 230 MPa. Third, it is worth noting that /H9004H ppfor Ni /ZnO NCs is larger than the corresponding value of /H9004Hpp for Ni //H9253-Fe 2O3NCs for a given Ni concentration. Interest- ingly, there is also a clear trend toward lower Hresfor smaller temperature, as displayed in Fig. 5. Fourth, the EMR lines continuously shift to high fields as the temperature is in-creased. For the Ni /ZnO NCs, the temperature variation H res/H20849T/H20850resembles /H20851see Fig. 5/H20849a/H20850/H20852that observed for neat or compacted powder, in stark contrast to what is observed forNi / /H9253-Fe 2O3NCs, i.e., Fig. 6/H20849a/H20850. Figure 3summarizes the dependence of the lowering of the effective gfactor on temperature for the two kinds of NCs. The inset shows the actual values of g/H20849RT/H20850. For Ni /ZnO NCs, these values rapidly fall to the gfactor corre-150 180 210 240 270 3001.01.11.21.3 0 1 02 03 04 05 06 07 08 09 0 1 0 02.02.22.42.62.83.03.23.43.63.8Ni/ZnO Ni//CID74-Fe2O3 NP: PCE: Ni content (vol%)g( R T )powder: Ni /CID74Fe2O3g(T)/g(RT) T(K)Ni/ZnO: 9 vol% 18 29 33 42 Ni//CID74Fe2O3: 4v o l % 8 17 29 50 FIG. 3. Effective gfactor, normalized to the RT value g/H20849RT/H20850,o f Ni /ZnO /H20849open symbols /H20850, and Ni //H9253-Fe2O3/H20849filled symbols /H20850NC samples as a function of temperature. The corresponding values forthe neat powders of Ni /H20849open stars /H20850and /H9253-Fe2O3/H20849filled stars /H20850have been shown for comparison. The inset shows g/H20849RT/H20850versus the Ni content in the Ni /ZnO /H20849open circles /H20850and Ni //H9253-Fe2O3/H20849filled tri- angles /H20850NC samples, Ni powder compact /H20849open square /H20850with 15 vol % epoxy and /H9253-Fe2O3powder compact /H20849filled square /H20850with 15 vol % epoxy /H20851phosphorus-containing /H20849PCE /H20850/H20852, and neat Ni /H20849open star/H20850and/H9253-Fe2O3/H20849filled star /H20850powder /H20849NP/H20850samples. The solid lines serve as guides for the eyes. 02468 1 0 1 2(a) H(kOe)151 K180 K209 K240 K270 KdP/dH (a.u.)Ni/ZnO 17 vol% 294 K (b) 148 K180 K210 K240 K270 KNi//CID74Fe2O350 vol%dP/dH (a.u.)294 K FIG. 4. /H20849a/H20850X-band EMR spectra /H20849absorption derivative /H20850for a representative Ni /ZnO NC sample /H20849Ni content of 17.5 vol % /H20850and different temperature values. /H20849b/H20850X-band EMR spectra /H20849absorption derivative /H20850for a representative Ni //H9253-Fe2O3sample /H20849Ni content of 50.3 vol % /H20850and different temperature values.ELECTRON MAGNETIC RESONANCE STUDY OF … PHYSICAL REVIEW B 77, 134424 /H208492008 /H20850 134424-5sponding to neat powder /H20851g/H20849RT/H20850/H110152.2/H20852. By contrast, g/H20849RT/H20850is practically constant for Ni //H9253-Fe 2O3NCs. The clear and sys- tematic decrease in the gfactor can be found up to the high- est accessible temperature of 300 K. The experimental be-havior could arise from several sources. First, there ismotional narrowing. Another possible mechanism is the in-terplay of demagnetizing field effects and of the presence ofshort-range magnetic ferrimagnetic ordering due to the /H9253-Fe 2O3nanoparticles. Figure 7/H20849Fig. 8/H20850illustrates how /H9004Hppand Hresat RT change as a function of the metallic FM nanophase contentfor the Ni /ZnO NCs /H20849Ni / /H9253-Fe 2O3NCs /H20850. For Ni /ZnO NCs, /H9004Hppsignificantly increases as Ni volume fraction is in- creased followed by a sharp drop at Ni content of/H1101530 vol %, while in the same Ni fraction range, /H9004H pp monotonically increases for the Ni //H9253-Fe 2O3specimens. These data serve to make two further points. First, the broad-ening of the EMR line for Ni / /H9253-Fe 2O3samples is sharply increased by increasing the Ni volume fraction. We interpretthe broadening of the EMR features in these samples as aris-ing from the strong collective magnetostatic intergranular in-teractions of the nanosized FM clusters with the surroundingferrimagnetic matrix. Second, it is interesting to relate themaximum of /H9004H ppto the electrical transport properties of the Ni /ZnO NCs characterized by the four-point probe technique.37In Ref. 37, we discussed a set of electrical trans- port data of these materials and observed that the RT dcconductivity /H9268dcversus Ni content data for Ni /ZnO NCs collected at low field exhibit an S-shaped curve /H20849notperco- lativelike process /H20850with an exponential increase between 10 and 30 vol % Ni and a change of slope at about 30 vol % Ni. Generally, in the solid state, we classify EMR lines into those that are homogenously broadened and those that areinhomogeneously broadened. 13,14The main contributions to homogeneous broadening are the magnetic dipolar coupling,spin-lattice interaction, interaction with radiation field, andmotionally narrowing fluctuations of local fields. 13,14For the inhomogeneous case, the line broadening mechanism distrib-utes the resonance frequencies over an unresolved band, e.g.,inhomogeneous external magnetic field, anisotropic interac-tions in the randomly oriented set of spins, unresolved hy-perfine structure, and strain distribution. Thus, the distribu-tion in local fields will make the spins in various parts of thesample feel different field strengths. Here, in the analysis ofour FMR spectra of the NCs under study here, 37we found that inhomogeneity based line broadening mechanisms, dueto the damping of surface and/or interface effects and inter-particle interaction, affect the FMR effective linewidth.These remarks suggest that the /H9004H ppversus Tand Ni content is most likely associated with two contributions to the ob-served EMR linewidth: on the one hand, there is the homo-2.02.22.42.62.83.0 150 200 250 3001.41.61.82.02.22.42.62.842% 18 9 (a)Ni/ZnOHres(kOe)neat compacted neat (b)42% 18 9 /CID39Hpp(kOe) T(K) FIG. 5. /H20849a/H20850The line position, Hres, as a function of temperature for Ni /ZnO NC samples. The number indicates the Ni volume frac- tion. For the purpose of comparison, we have represented the valuesofH resfor neat Ni powder and a Ni /H2084956 vol % /H20850powder compact with 15 vol % epoxy as dashed and dotted lines, respectively. /H20849b/H20850 S a m ea si n /H20849a/H20850for the peak-to-peak linewidth /H9004Hpp. For the purpose of comparison, we have represented the values of Hresfor neat Ni powder and a Ni /H2084956 vol % /H20850powder compact with 15 vol % epoxy as dashed and dotted lines, respectively.2.82.93.03.13.2 150 200 250 300 3500.81.21.62.02.42.8neat (a)50% 29 17 8 4Hres(kOe) neat (b)Ni//CID74-Fe2O3 50% 29 17 8 4 T(K)/CID39Hpp(kOe) FIG. 6. Same as in Fig. 5for Ni //H9253-Fe2O3samples. For the purpose of comparison, we have represented the values of Hresfor neat Ni powder and a Ni /H2084956 vol % /H20850powder compact with 15 vol % epoxy as dashed and dotted lines, respectively.VINCENT CASTEL AND CHRISTIAN BROSSEAU PHYSICAL REVIEW B 77, 134424 /H208492008 /H20850 134424-6geneous line broadening mechanism involving dipolar inter- action and spin-lattice relaxation, which is stronglytemperature dependent /H20849motional narrowing /H20850. On the other hand, the unsymmetrical line shape, especially at low tem-peratures, is an indication that the EMR is broadened in aninhomogeneous manner and the EMR linewidth is expectedto increase as a function of magnetic field. 37 IV. COMPARISON WITH FERROMAGNETIC RESONANCE SPECTRA AND MICROWAVE LOSSES It may be meaningful to compare our EMR parameters to values of the linewidths measured over the 9–10 GHz fre-quency range recently reported in FMR experiments, inves-tigating the surface anisotropy contribution to the anisotropyof Ni and /H9253-Fe 2O3nanoparticles,37and to effective micro- wave losses measured in order to probe the evolution oflarge-wave-vector spin wave modes in these NCs. 38The FMR response of magnetic nanostructures is a rich area in itsown right, and several models, 43,44e.g., two-magnon scatter- ing theory, have been applied to the problem of magneticheterogeneities in coarse-grained heterostructures. Fromthese FMR measurements, 37it was pointed out that the char- acteristic intrinsic damping dependent on the selected mate-rial and the damping due to surface and/or interface effectsand interparticle interaction were estimated. Inhomogeneousdamping due to surface and/or interface effects increases with diminishing particle size, whereas damping due to in-teractions increases with increasing volume fraction of mag-netic particles /H20849i.e., reducing the separation between neigh- boring magnetic phases /H20850in the composite. Figures 7and8summarize the main findings of this work. The upper and lower panels in Figs. 7and8provide a direct comparison of /H9004H ppandHresas a function of Ni content and at RT between, on the one hand, the EMR /H208499.40 GHz /H20850fea- tures and the nominal /H208499 and 10 GHz /H20850uniform FMR mode, and on the other hand, the nominal /H208499.40 GHz /H20850microwave losses. A number of interesting features are worth remarking.First, all of the curves show nonmonotonic variations. Sec-ond, as can be realized from these graphs, the striking mainfeature is the qualitative similarity in the three types of mea-surements. Third, as seen in Figs. 7and 8, substituting a nonmagnetic by a magnetic host matrix not only shifts theposition of the resonance but also sensitively affects its line-width. This is caused by locally changing the interactionsbetween magnetic nanoparticles. It has been recognized thata shortening of T 1can result from weak dipole interactions, whereas strong interactions may result in slowing down ofthe relaxation. 45The analysis of the FMR spectra was inter- preted in Ref. 37as arising from aggregates of magnetic nanoparticles, each of which resonates in an effective mag-netic field composed of the applied field, the average /H20849mag- netostatic /H20850dipolar field, and the randomly oriented magnetic anisotropy field. The importance of the Ni concentration has2.42.62.83.03.23.4 10 20 30 40 501.41.61.82.02.22.42.6 123456(a)Ni/ZnOHres(kOe)FMR: 9G H z 10 GHz from [39] EMR 9.40 GHz FMR: 9G H z 10 GHz from [39] EMR 9.40 GHz (b)/CID39Hpp(kOe) Ni content (vol.%)Losses 9.40 GHz from [40]Losses L (dB) FIG. 7. EMR line peak-to-peak width /H9004Hp.p.and position, Hres, /H208499.40 GHz /H20850plotted as a function of the Ni content in the Ni /ZnO NC samples at RT. Comparison with FMR data /H208499 and 10 GHz /H20850 from Ref. 39and microwave loss data /H208499.40 GHz /H20850from Ref. 40.2.93.03.13.23.33.43.53.63.7 0 1 02 03 04 05 01.21.62.02.42.8 04812FMR: 9G H z 10 GHz from [39] EMR 9.40 GHzNi//CID74-Fe2O3 (a)Hres(kOe) FMR: 9G H z 10 GHz from [39] EMR 9.40 GHz (b)/CID39Hpp(kOe)Losses L (dB) Ni content (vol.% )Losses 9.40 GHz from [40] FIG. 8. Same as in Fig. 7for Ni //H9253-Fe2O3samples.ELECTRON MAGNETIC RESONANCE STUDY OF … PHYSICAL REVIEW B 77, 134424 /H208492008 /H20850 134424-7also been discussed in relation to measurements of the spin wave group velocity induced by the samples.38The presence of nonmagnetic phases of specific type and volume fractionoffers the possibility of controlling the magnetic and micro-wave properties of NCs. V. CONCLUDING REMARKS In summary, a systematic EMR study in Ni /ZnO and Ni //H9253-Fe 2O3NCs, for temperatures ranging from 150 up to RT, has been presented. Two compounds with different prop-erties /H20849diamagnetic ZnO and ferrimagnetic /H9253-Fe 2O3/H20850and structural disorder were chosen in order to vary the magneticinteractions between the nanoparticles in these heterostruc-tures. There are general reasons to expect that the EMR linebroadening and position are not intrinsic features of the me-tallic FM content, but arise as a consequence of the interac-tion between aggregates and other interface perturbations.The motional narrowing offers an explanation for the widevariation in the degree of broadening of the EMR line as afunction of temperature. The strength of the coupling, asmanifested by the EMR linewidth, can be significantly modi-fied by the metallic FM content. One very interesting findingis that there is a clear correlation between EMR linewidth,the corresponding FMR features, and the effective micro-wave losses measured in these heterostructures. The correla-tion found here is far from trivial and we regard it as moti-vation for the development of models underlying the processof resonance in granular heterostructures in which all thedetails take place. More specifically, although a large numberof theoretical calculations have been performed to under-stand the phenomenon of resonance in nanostructures, nofirst-principles theoretical calculation has been reported tounderstand the role of the nonmagnetic phase in tuning theFM of nanoclusters which can be a useful reference for ex-perimentalists. This study is part of a larger effort to identify essential factors governing magnetoelectricity in dense nanostructuredcompacts and to explore control possibilities due to their rich behaviors under magnetic and electric perturbations. 46There are/H20849at least /H20850three directions in which the present work could be extended. First, one wishes to know if our experimentalfindings do extend to functional multiferroic NCs motivatedby the desire to be able to simultaneously manipulate differ-ent combinations of microwave properties by the applicationof external fields. Second, the question begs to be asked:what is the impact of having extremely dense nanocompacts,e.g., by using hot isostatic pressing in order to have a nearlycomplete elimination of porosity, on EMR linewidths. Third,numerous experimental challenges exist when consideringmagnetic NCs because they offer a promising avenue towardnanoelectronics and spintronics. EMR can give important in-sights complementing information from direct studies of themorphological structure of magnetic nanoclusters. Investiga-tions at these length scales are in their infancy, and muchroom exists for improvement. There remains significant workahead in continuing to understand the growth modes of nano-particle aggregates, e.g., for Ni clusters containing up to 800atoms, regularly spaced peaks in the mass spectra of a certainmagic cluster sizes have been interpreted as icosahedralgrowth patterns. 47The applicability of metals in nanoelec- tronic and spintronic devices in which information is pro-cessed by using electron spins will depend on a sufficientlylong spin lifetime, i.e., long T 1or narrow EMR linewidth.48 We hope to discuss the magnetism of Ni clusters and the resulting microwave frequency-domain spectroscopy in fu-ture work. ACKNOWLEDGMENTS One of the authors /H20849V .C. /H20850gratefully acknowledges finan- cial support from the Conseil Régional de Bretagne. We wishto thank J. Ben Youssef for helpful conversations. We alsowish to thank N. Kervarec for her assistance in EMR experi-ments. *Also at Département de Physique, Université de Bretagne Occi- dentale. brosseau@univ-brest.fr 1J. 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PhysRevB.92.024403.pdf

PHYSICAL REVIEW B 92, 024403 (2015) Propagating spin waves excited by spin-transfer torque: A combined electrical and optical study M. Madami,1E. Iacocca,2,3S. Sani,4G. Gubbiotti,5S. Tacchi,5R. K. Dumas,2J.˚Akerman,2,4and G. Carlotti1 1Physics Department, University of Perugia, Perugia, Italy 2Physics Department, University of Gothenburg, Gothenburg, Sweden 3Department of Applied Physics, Division for Condensed Matter Theory, Chalmers University of Technology, Gothenburg, Sweden 4Materials Physics, School of Information and Communications Technology, Royal Institute of Technology (KTH), Kista, Sweden 5Istituto Officina dei Materiali del Consiglio Nazionale delle Ricerche (IOM-CNR), Unit `a di Perugia c/o Dipartimento di Fisica, Perugia, Italy (Received 13 April 2015; revised manuscript received 9 June 2015; published 2 July 2015) Nanocontact spin-torque oscillators are devices in which the generation of propagating spin waves can be sustained by spin transfer torque. In the present paper, we perform combined electrical and optical measurementsin a single experimental setup to systematically investigate the excitation of spin waves by a nanocontactspin-torque oscillator and their propagation in a Ni 80Fe20extended layer. By using microfocused Brillouin light scattering we observe an anisotropic emission of spin waves, due to the broken symmetry imposed bythe inhomogeneous Oersted field generated by the injected current. In particular, spin waves propagate on theside of the nanocontact where the Oersted field and the in-plane component of the applied magnetic field areantiparallel, while propagation is inhibited on the opposite side. Moreover, propagating spin waves are efficientlyexcited only in a limited frequency range corresponding to wavevectors inversely proportional to the size of thenanocontact. This frequency range obeys the dispersion relation for exchange-dominated spin waves in the farfield, as confirmed by micromagnetic simulations of similar devices. The present results have direct consequencesfor spin wave based applications, such as synchronization, computation, and magnonics. DOI: 10.1103/PhysRevB.92.024403 PACS number(s): 85 .75.−d,75.30.Ds,81.07.Lk,78.35.+c I. INTRODUCTION Nanocontact (NC)-based spin-torque oscillators (STOs) [ 1] have been extensively studied in recent years due to theirpotential applications in a variety of fields, including spin-tronics, magnonics [ 2], and data storage [ 3]. Nanocontact- based spin-torque oscillators typically consist of an extendedpseudo-spin-valve where two magnetic layers are separatedby a nonmagnetic spacer. One of the magnetic layers is a hard magnet that acts as a spin polarizer and a reference layer, and it is referred to as the “fixed” layer. The second “free,” magneticlayer is softer and thinner, making it susceptible to spin-transfer torque (STT) induced dynamics [ 4,5]. Depending on the material used for the free layer, unique magnetodynamicalmodes can be generated in NC-STOs. On the one hand, byusing Co/Ni multilayers exhibiting perpendicular magneticanisotropy, NC-STOs have been shown to support steadymagnetization dynamics at low external fields [ 6] and even the excitation of localized modes known as magnetic dissipativedroplets [ 7–10]. On the other hand, using soft magnets such as NiFe, the existence of propagating [ 11,12], localized [ 12–15], and vortex [ 16,17] modes has been demonstrated, depending on both the strength and the direction of the external magnetic field. The propagating mode was predicted theoretically by Slonczewski [ 18] for a perpendicularly magnetized NC-STO, i.e., by applying an external magnetic field perpendicular tothe device plane strong enough to saturate the free layer. Thismode has been shown to consist of exchange-dominated spinwaves (SWs) propagating radially away from the NC regionat frequencies above the ferromagnetic resonance (FMR)frequency with a wavenumber inversely proportional to theNC diameter. The propagating nature of this mode lends itselfto coupling NC-STOs to achieve better spectral features bysynchronization [ 19–23], to perform computation [ 24,25], or to propagate information in magnonic devices [ 26,27]. The existence of the propagating mode was demonstrated only recently by employing either electrical measurements [28,29], to study SW-mediated synchronization, or micro- focused Brillouin light scattering ( μ-BLS) [ 11], to directly detect the emitted SWs. Although these experiments pro-vided unambiguous proof of the propagating character ofthis mode, micromagnetic simulations including the Oerstedfield generated by the bias current [ 13,30] predicted that the propagation is generally anisotropic, especially if the applied field is tilted away from the device normal. This has tremendous consequences for the design and applicationof devices utilizing STT-driven SWs. However, to the bestof our knowledge, no direct experimental evidence of such asymmetric emission has been provided to date for out-of-plane magnetized NC-STOs with well-defined propagatingspin waves. Evidence of asymmetric emission has been experimentally demonstrated only for in-plane magnetized NC-STOs by the studies of Demidov et al. [15,31] to date. In the present paper, we combine electrical characterization andμ-BLS in a single experimental setup to systematically investigate both the electrical properties of the STT-induceddynamics and the characteristics of the propagating SWsexcited in an out-of-plane magnetized NC-STO. Evidence isprovided for anisotropic SW propagation in these devices, asreproduced by micromagnetic simulations. More importantly,the STT-generated SWs are found to propagate within awavevector range dictated by both the SW dispersion in the farfield and the diameter of the NC. In fact, micromagnetic sim-ulations reveal that nonlinear dynamics are strongly dampedclose to the NC (near-field region), while only wavevectorsobeying the SW dispersion relation are able to propagatein the far-field region, i.e., at distances larger than the NC 1098-0121/2015/92(2)/024403(7) 024403-1 ©2015 American Physical SocietyM. MADAMI et al. PHYSICAL REVIEW B 92, 024403 (2015) dimensions. These results are relevant for the development of SW-based devices, including the synchronization of NC-STOarrays, computation, and magnonics. A. Experimental details Samples were fabricated on a SiO 2(1μm) covered silicon wafer through the following processing steps: apseudo-spin-valve stack of Pd (8 nm)/Cu (15 nm)/Co (8 nm)/Cu (8 nm)/Ni 80Fe20(4.5n m ) / C u( 3n m ) / P d( 3n m ) w a s d e - posited by magnetron sputtering; then, making use ofphotolithography, rectangular 8 ×16-μm 2mesas were defined. A 30 nm thick SiO 2layer was deposited on top of the mesas to realize electric insulation of the devices. Defined inthis insulation layer above each mesa were 2 ×4μm 2ground electrodes and circular NCs with 100-nm diameter. Finally,contact pads with ground-signal-ground (GSG) geometry, asdepicted in Fig. 1(c), were fabricated with sputter deposition of 1μmC u/20 nm Au in a photolithography prepared lift-off pattern. The final device allows for μ-BLS optical access on one side of the NC only, as shown in Fig. 1(c). Measurements in the frequency domain were performed using a broadband spectrum analyzer (SA). A dc was injectedinto the NC while the device impedance was continuouslymonitored in order to avoid damage due to Joule heating.The direction of the injected dc is negative (i.e., electronsdrift from the free to the fixed layer) as required to obtain anSTT-induced enhancement of the magnetization dynamics inthe free layer [ 11]. Notably, this approach is only sensitive to the magnetodynamics generated very near the NC, along thepath of the dc. Microfocused Brillouin light scattering measurements were performed applying an external magnetic field Hat an angle of about 15 ◦from the sample normal by means of a customized projected field electromagnet. As has alreadybeen demonstrated in a previous paper [ 11], the out-of-plane magnetized NC geometry favors the emission of propagatingSWs in the extended portion of the NiFe free layer. At the sametime, the slight tilt of the external field allows us to define thedirection of the in-plane component of the field, H IP,a ss h o w n in Fig. 1. This choice is crucial in defining the direction of the propagating SWs, as we will demonstrate below. The emittedSWs were experimentally detected at room temperature bymeans of a μ-BLS setup described in detail elsewhere [ 32]. We measured the Stokes side of each spectrum only, in orderto halve the total acquisition time. This approach is justifiedby the symmetry of the Stokes and anti-Stokes sides of thespectrum in a measurement performed at normal incidence ina thin magnetic film [ 33]. To summarize, our setup allows us to electrically investigate the generated dynamics of the NC-STO while simultaneouslycharacterizing the emitted SWs by μ-BLS. B. Micromagnetic simulations Micromagnetic simulations were performed using the graphic processing unit (GPU) accelerated software Mumax3[34]. The extended free layer was modeled by a disk with a di- ameter of 2 .2μm and absorbing boundary conditions (ABCs) to prevent spurious generation and reflection of SWs. TheABCs were implemented by linearly increasing the Gilbert damping coefficient over 200 nm towards the disk edge, sothat the active area of the model was a 1 .8μm diameter disk. We considered the device nominal free layer thickness to be4.5 nm and the magnetic parameters as follows: saturation magnetization μ oMS=0.69 T, Gilbert damping α=0.01, exchange stiffness A=11 pJ/m, and no magnetocrystalline anisotropy. These parameters define the cell discretization to4.3×4.3×4.5n m 3, below the exchange length λex≈6n m , resulting in a 512 ×512 mesh. Spin-transfer torque was con- sidered only in a cylindrical region defined by the NC ofdiameter d=100 nm positioned in the geometrical center of the disk. We assumed a polarization consistent with a Co layer,P=0.3, and a symmetric torque λ=1, as has previously been shown to model similar devices with high accuracy [ 13,14]. The direction of the fixed layer was calculated numericallyby solving the magnetostatic boundary conditions for a Cothin film with saturation magnetization μ oMP=1.5T . B o t h the external applied field and the current-generated Oerstedfield were included in the simulations. Unless specified, thesimulations were performed with a fixed time step of 10 fs atroom temperature. The spectral characteristics of the generated dynamics can be estimated from 10 ns long simulations sampled at 10 ps,returning a frequency resolution of ≈97 MHz. Field-dependent simulations indicate that high field magnitudes are required toexcite well-defined SWs. This can be understood from thefact that demagnetizing fields arise at the boundaries of thesimulated disk, favoring a vortex ground state. Applied fields ofmagnitude μ oH> 650 mT are numerically found to preclude the formation of a vortex and to allow the excitation of STT-induced propagating SWs. II. RESULTS AND DISCUSSION A. Anisotropic spin-wave emission and decay length The first step of our experiment was to investigate the effect of varying the direction of the in-plane component of theexternal magnetic field ( H IP) on the spatial distribution of SWs emitted below the NC area. To perform such an experimentthe external field intensity was set to μ oH=+ 700 mT and the field direction was tilted so that its in-plane componentwas antiparallel to the Oersted field on the side of the NCwhich is accessible to μ-BLS, as shown in Fig. 1(a).B y injecting a dc of I=30 mA into the NC, we were able to obtain steady-state oscillations at a frequency of 15 .5 GHz with an intensity of about +9 dB over noise, as measured by the SA. In order to map the intensity of the emitted SWs,we then performed a two-dimensional μ-BLS scan over an area of about 2 ×1μm 2with a step size of 250 nm on the optically accessible NC side at a distance of about 1 μmf r o m the NC itself. The measured two-dimensional map, shownin Fig. 1(a), was obtained by integrating the measured SW intensity in a frequency range of 0 .5 GHz around the central value of 15 .5 GHz. This μ-BLS map clearly shows emission from the NC area, confined in a relatively narrow beam whichpropagates in the direction perpendicular to both the directionof the Oersted field and H IP. If the external field polarity is reversed ( μoH=− 700 mT), the Oersted field and HIPbecome 024403-2PROPAGATING SPIN W A VES EXCITED BY SPIN- . . . PHYSICAL REVIEW B 92, 024403 (2015) (a) (b) (c) FIG. 1. (Color online) (a, b) SW intensity maps measured by μ-BLS in the green area of panel (c) for two different directions of the in-plane component ( HIP) of the external field, H=700 mT and I=30 mA. (c) Scanning electron microscope (SEM) image of the device with the GSG pads, NC position and the direction of the Oersted (Oe) field generated by the current ( I) flowing into the NC. parallel on the optically accessible NC side [ 29] but no SWs are detected in this case, as shown by the μ-BLS map in Fig. 1(b) [35]. This clearly demonstrates anisotropic SW emission, confirming previous numerical predictions indicating that thelargest SW intensity is emitted on the side of the NC where 1 . 01 . 21 . 41 . 61 . 82 . 02 . 2200300400500600 µ-BLS intensity best-fit curve intensity (arb. units) distance from the NC ( µm) FIG. 2. (Color online) SW intensity (circles) measured by μ-BLS as a function of the distance ( X−X0) from the NC position. Best-fit curve (line) obtained using Eq. ( 1).the Oersted and the HIPfields are antiparallel [ 13,30]. As the second step of this investigation, we measured the decay of theemitted SW intensity ( i) as a function of the distance from the NC position ( X−X 0), by scanning the laser probe along theXdirection. The results of these measurements are shown in Fig. 2(black dots) together with the result of a best fit analysis obtained using the following analytical expression: i(X)=i0+A X−X0·e−X−X0 l (1) where i0is the baseline, X0is the NC position (fixed at X0=0), and lis the decay length of SW intensity; Aand lare the free parameters in the fit routine, which returns a value of l=500±50 nm. This is in very good agree- ment with the expected value obtained from the expressionl=v g/2αω=500 nm, where vg=1.0μm/ns is the SW group velocity extracted from the simulated dispersion curveof the free layer (see Fig. 5),α=0.01 is the conventional value of damping in NiFe, and ω=2πfis the SW angular frequency ( f=15.5 GHz). B. Dependence of the spin-wave emission on the injected current and applied field intensity Next, we turn our attention to the characteristics of the emitted SWs by simultaneously measuring the spectra 024403-3M. MADAMI et al. PHYSICAL REVIEW B 92, 024403 (2015) -20 -18 -16 -14 -12 -10 -8 -6 frequency (GHz)(a) (b) (d) -20 -18 -16 -14 -12 -10 frequency (GHz)µ0H=550 mT µ0H=800 mTµ0H=550 mT µ0H=800 mT (e)(c) FIG. 3. (Color online) (a) μ-BLS and (b) SA spectra measured as a function of the intensity of the external field Hf o rafi x e dv a l u eo ft h e injected dc I=30 mA. (c, b) Color plots of the sequence of spectra in panels (a) and (b). (e) Results of micromagnetic simulations obtained under the same conditions. obtained electrically and optically when varying either the intensity of the external magnetic field Hor the direct current I. The results reported in Fig. 3were obtained by injecting a constant current I=30 mA and applying an external magnetic field of variable intensity (in the range H=550−800 mT) with its in-plane component ( HIP) directed as in Fig. 1(a),t o favor the emission of SWs on the optically accessible side ofthe device. The first two panels show the μ-BLS spectra of SW intensity measured at a fixed position of about 1 .0μm away from the NC [Fig. 3(a)] and the sequence of spectra acquired with the SA [Fig. 3(b)]a saf u n c t i o no f H.T h e same data are reported in Figs. 3(c) and 3(d) on color scale bidimensional plots. The data obtained from μ-BLS features two well-defined peaks. The lowest frequency peak, which isnot observed on the SA spectra [Fig. 3(d)], exhibits a constant intensity and can be measured even for I=0 [cf. red spectrum in Fig. 4(a)]. It corresponds to the thermally activated FMR mode of the NiFe layer, consistent with Kittel’s equation asaf u n c t i o no f H. The highest frequency peak of Fig. 3(c) is instead observed in the SA spectra at the same frequency forall the measured fields and corresponds to the SWs emittedby the STT-driven precession of the magnetization under theNC. The blue-shift of this peak with respect to the FMRfrequency is a signature of the propagating character of theSWs emitted away from the NC, which is further confirmedby the fact that its signal is measurable up to about 2 .2μm away from the NC itself, as was shown in Fig. 2. The positive field tunability of this mode is estimated to be 24 MHz /mT. A comparison of the above experimental results with thoseobtained from micromagnetic simulations [Fig. 3(e)] accounts for a very good quantitative agreement over the entire rangeof fields we investigated. Careful inspection of the measuredspectra reveals the presence of a mode transition for a fieldvalue of H=675 mT, denoted by two resonant frequencies in both the μ-BLS and SA spectra. Corresponding to this mode transition, the SA spectrum is strongly modified, exhibiting asharper and more intense peak. The occurrence of such modetransitions is very common in this kind of NC-STO devices[28,29] and reflects the complicated nonlinear dynamics that can also be affected by unique and local features of the realNC. As a matter of fact, nominally identical devices presentone or more mode transitions at different field/current values,or even no transitions at all. A direct comparison of the measured peaks corresponding to the FMR and STT modes in the μ-BLS spectra [Fig. 3(a)] suggests that the two modes have comparable intensities. Thisresult may seem surprising at first, since a much larger intensityshould be expected for STT-driven SWs than for thermallyexcited SWs. The reason for this apparent inconsistencylies in the wavevector content of the two modes and howit is efficiently collected by the optics of our experimentalsetup. By combining the effect of the finite collection angle[θ=48.6 ◦,NA=sin(θ)=0.75] of our microscope objective with the effect of the uncertainty in the in-plane componentof the wavevector of the scattered photons, resulting fromthe limited spatial extent of the laser spot on the sample(≈300 nm), it is possible to demonstrate that the efficiency of our apparatus in detecting SWs with small wavelengths 024403-4PROPAGATING SPIN W A VES EXCITED BY SPIN- . . . PHYSICAL REVIEW B 92, 024403 (2015) -20 -18 -16 -14 -12 -10 frequency (GHz)-20 -18 -16 -14 -12 -10 -8 -6 frequenc y shift (GHz)FMR I=23 mA I=38 mAI=0 mA FMRI=23 mA I=38 mA (a) (b) (d)(c) FIG. 4. (Color online) (a) μ-BLS and (b) SA spectra measured as a function of the intensity of the injected dc ( I) for a fixed value of the external field H=680 mT. (c, b) Color plots of the sequence of spectra in panels (a) and (b). decreases very rapidly as the wavelength is reduced below 300 nm [ 11]. Since the diameter of the NC under investigation is only 100 nm, we expect the wavelength of propagating SWsto be smaller than 200 nm [ 18], which means we can detect them with relatively low efficiency in our apparatus. Figure 4shows the results of a second set of simultaneous μ-BLS and SA characterizations obtained by setting the external field intensity to a fixed value of H=680 mT and varying the intensity of the injected dc ( I) over the range between 23.0 and 38 .0m A . T h e fi r s t μ-BLS spectrum (red line) in Fig. 4(a) was measured at I=0m A , s h o w i n g t h e thermal FMR mode. In the following sequence of μ-BLS spectra [Fig. 4(a)], the FMR signal was no longer acquired, in order to reduce the acquisition time and because its frequencydoes not vary with current, so the only visible peak inthe spectra is the one corresponding to the STT-driven SWexcitation. Figure 4(b) shows the corresponding sequence of SA spectra, while Figs. 4(c) and4(d) report the same μ-BLS and SA measurements in bidimensional color scale plots.Similarly, as in the case of the field characterization (Fig. 3), we observed a blue-shift of the STT-driven SW frequencieswith respect to the FMR, as well as a positive currenttunability that we estimate to be 250 MHz /mA. On increasing I, a mode transition is observed for the value I=29.0m A ;this is characterized by a relatively large frequency jump of about 2 GHz, clearly visible in the SA sequence of spectra,and accompanied by a dramatic reduction in linewidth andan increase in maximum intensity. It is interesting to notehow the signal coming from propagating SWs, as measuredbyμ-BLS, is easily detectable only over a finite range of currents, I=29.0−35.0 mA, after the mode transition. Before the mode transition ( I< 29.0 mA) the SW signal is hardly detectable, probably because of the very low STT efficiency, asis confirmed by the electric measurements. More interesting isthe existence of an upper bound ( I=35.0 mA), above which the intensity of the emitted SWs rapidly decreases. This isdiscussed in greater detail in the following paragraphs. The existence of a well-defined dc range (and a corresponding frequency range), in which propagatingSWs are experimentally detected, is a reproducible featurethat we observed in several NC-STO devices with slightlydifferent NC diameters, in the range 80 −120 nm, obtaining results which are in good qualitative agreement with thosepresented in this paper. In order to shed more light onthis effect, we calculated the dispersion curve for SWs inthe out-of-plane magnetized NiFe layer ( H=680 mT). The resulting frequency vs 1 /λplot is shown in Fig. 5(b), where it is compared with the frequency vs I plot of 024403-5M. MADAMI et al. PHYSICAL REVIEW B 92, 024403 (2015) I=29 mA I=36 mA FMR = 150 115 nm(b) (a) FIG. 5. (Color online) (a) μ-BLS spectra measured as a function of the intensity of the injected dc ( I) for a fixed value of the external fieldH=680 mT. (b) Simulated SW dispersion curve of the NiFe free layer. Fig. 5(a). We can see good quantitative agreement between the frequency of the FMR mode, as measured by μ-BLS, and the simulated frequency at 1 /λ=0(k=2π/λ=0). From a direct comparison between the two panels in Fig. 5,i t is clear that the finite frequency range detected experimentallycorresponds to a finite wavelength range of λ=115−150 nm. This is the central result of the present paper, as it suggeststhat propagating SWs emitted by an out-of-plane magnetizedNC-STO have a limited range of accessible wavelengths,which is ultimately associated with the NC diameter andthe exchange-dominated dispersion relation of SWs [ 36]. In other words, a NC-STO cannot efficiently excite exchange-dominated SWs with a wavelength much shorter (or longer)than its own NC diameter. This has tremendous implicationsfor the synchronization of NC-STOs, since the wavelengthunambiguously determines the phase difference between thedevices and ultimately the phase-locking condition. The above observations can be further tested using micro- magnetic simulations. Due to the finite size of the simulateddisk, much stronger fields are required to obtain a coherentSW emission. Consequently, we set a field of magnitudeH=800 mT tilted 15 ◦from the plane normal in the following simulations. To characterize the allowed propagating SWs,we performed simulations at T=0 K and determined their wavelength as a function of distance via smoothed pseudo-Wigner-Ville (SPWV) transform. By this method, it is possibleto tune the real and reciprocal space resolution by makinguse of smoothing windows. The wavelength content of thegenerated SW is determined by performing a SPWV transformon a simulation snapshot along the Xdirection, where we used a Gaussian window of 4 .3 nm for the real space and a Hann window of 1 .9μm −1for the reciprocal space. We then averaged 100 SPWV transforms, each one calculated fromdifferent snapshots of the same time-dependent simulation,to further improve the accuracy of the method. The resultsfor NC-STO driven at 34 and 25 mA are shown as colorplots in Figs. 6(a) and 6(b), respectively, where the vertical black lines indicate the boundaries of the NC and the colorrepresents the normalized SW magnitude in logarithmic scale.The choice of the values for Iwas made in order to have one value below the SW emission threshold (25 mA) and the otherone right in the middle of that range (34 mA), as confirmedby Fig. 5(a). We remind the reader that this figure must be understood as the wavelength content ( yaxis) as a function of distance for the SW propagating along X. Whether the I = 34 mA I = 25 mA FIG. 6. (Color online) SPWV transform of simulated SWs along Xfor (a) I=34 mA and (b) I=25 mA and an external field of H=800 mT. The vertical black lines indicate the boundaries of the NC while the white dashed lines indicate wavelength obtained fromthe SW dispersion relation at each frequency, namely (a) 19 .2 GHz and (b) 17 .1 GHz. The color contrast indicates the normalized magnitude in logarithmic scale of the SW associated with eachwavelength ( yaxis) as a function of X. NC-STO is driven at 34 mA [Fig. 6(a)]o r2 5 m A[ F i g . 6(b)], most of the wavelength content is rapidly damped within about0.2μm from the NC center (near-field region). This strong wavelength content describes the nonlinear forced dynamicsof the system due to the presence of strong/local effects inthe NC area such as the STT effect and the Oersted field.However, when the NC is driven at 34 mA [Fig. 6(a)], a band of wavelengths is observed to prevail up to the simulationboundary, and only in the +Xdirection. This wavelength band is in good quantitative agreement with the wavelengthanalytically obtained from the SW dispersion relation for NiFeat the driven NC-STO frequency (white dashed line) andsimilar to the NC diameter. In contrast, when the NC-STOis driven at 25 mA [Fig. 6(b)], this wavelength band, although present, is rapidly damped within about 0 .6μm, corresponding to a scenario in which the Oersted field strongly distortsthe magnetic landscape. It is worth mentioning that theseresults indicate that SW do propagate in the −Xdirection as well, as expected from exchange coupling, but their energyis negligible for distances greater than 0 .5μm from the NC. An additional interesting feature of Fig. 6is that the wavelengths close to the NC boundary are lower (higher) in the +X(−X) direction corresponding to the decrease (increase) of the localfield magnitude induced by the Oersted field. III. CONCLUSIONS The properties of propagating SWs emitted by an out-of- plane magnetized NC-STO were investigated by means ofa combined radio frequency (RF) and μ-BLS experimen- tal setup. We experimentally demonstrated the anisotropicemission of SWs, which is concentrated on the side of theNC where the Oersted field and the in-plane component ofthe external field are antiparallel. The analysis of both thefield and the current tunability of the device showed a clearblue-shift of the STT-excited SW frequency with respect tothat of the FMR, as well as the presence of abrupt modetransitions. More importantly, it was found that propagatingSWs are only efficiently excited over a limited interval ofwavelengths comparable with the NC diameter. Given thedispersion relation in the far field, this corresponds to a range 024403-6PROPAGATING SPIN W A VES EXCITED BY SPIN- . . . PHYSICAL REVIEW B 92, 024403 (2015) of frequencies of a few gigahertz. We believe these results will be of the utmost importance for further progress in NC-STOsynchronization, computation, and magnonic applications viaemitted SWs. ACKNOWLEDGMENTS Support from the European Community’s Seventh Frame- work Programme (FP7/2007-2013) under Grant No. 318287“LANDAUER” and by the Ministero Italiano dell’Universit `a e della Ricerca (MIUR) under the PRIN2010 project (No.2010ECA8P3) is gratefully acknowledged. Support from theSwedish Research council (VR), the Swedish Foundation forStrategic Research (SSF), and the Knut and Alice WallenbergFoundation is gratefully acknowledged. E. Iacocca acknowl-edges support from the Swedish Research Council, Reg. No.637-2014-6863. [1] T. Silva and W. Rippard, J. Magn. Magn. Mater. 320,1260 (2008 ). [ 2 ]R .L .S t a m p s ,S .B r e i t k r e u t z ,J . ˚Akerman, A. V . Chumak, Y . Otani, G. E. W. Bauer, J.-U. Thiele, M. Bowen, S. A. Majetich,M. Kl ¨aui, I. L. Prejbeanu, B. Dieny, N. M. Dempsey, and B. Hillebrands, J. Phys. D: Appl. Phys. 47,333001 (2014 ). [3] J. ˚Akerman, Science 308,508 (2005 ). [4] J. Slonczewski, J. Magn. Magn. Mater. 159,L1(1996 ). [5] L. Berger, Phys. Rev. B 54,9353 (1996 ). [6] W. Rippard, A. Deac, M. Pufall, J. Shaw, M. Keller, S. Russek, G. Bauer, and C. Serpico, Phys. Rev. B 81,014426 (2010 ). [7] M. Hoefer, T. Silva, and M. Keller, P h y s .R e v .B 82,054432 (2010 ). [8] S. Mohseni, S. Sani, J. Persson, T. Nguyen, S. 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PhysRevApplied.15.024017.pdf

PHYSICAL REVIEW APPLIED 15,024017 (2021) Giant Perpendicular Magnetic Anisotropy Enhancement in MgO-Based Magnetic Tunnel Junction by Using Co /Fe Composite Layer Libor Vojá ˇcek ,1,2,*Fatima Ibrahim ,1Ali Hallal ,1Bernard Dieny ,1and Mairbek Chshiev1,3,† 1Univ. Grenoble Alpes, CNRS, CEA, Spintec, 38000 Grenoble, France 2CEITEC BUT, Brno University of Technology, Purkynova 123, 612 00 Brno, Czech Republic 3Institut Universitaire de France (IUF), 75231 Paris Cedex 5, France (Received 17 August 2020; revised 8 December 2020; accepted 8 December 2020; published 8 February 2021) Magnetic tunnel junctions with perpendicular anisotropy form the basis of the spin-transfer torque magnetic random-access memory (STT MRAM), which is nonvolatile, fast, dense, and has quasi-infinite write endurance and low power consumption. Based on density-functional-theory (DFT) calculations, wepropose an alternative design of magnetic tunnel junctions comprising Fe (n)Co(m)Fe(n)|MgO storage layers [ nand mdenote the number of monolayers (ML)] with greatly enhanced perpendicular magnetic anisotropy (PMA) up to several mJ/m 2, leveraging the interfacial perpendicular anisotropy of Fe |MgO along with a strain-induced bulk PMA discovered within bcc Co. This giant enhancement dominates the demagnetizing energy when increasing the film thickness. The tunneling magnetoresistance (TMR) estimated from the Julliere model is comparable with that of the pure Fe |MgO case. We discuss the advan- tages and pitfalls of a real-life fabrication of the structure and propose the Fe (3ML)Co(4ML)Fe(3ML)as a storage layer for MgO-based STT MRAM cells. The large PMA in strained bcc Co is explained in the framework of second-order perturbation theory by the MgO-imposed strain and consequent changes in theenergies of d yzand dz2minority-spin bands. DOI: 10.1103/PhysRevApplied.15.024017 I. INTRODUCTION MgO-based magnetic tunnel junctions (MTJs) are used in today’s hard-disk-drive read heads and a variety of magnetic field sensors for their supremely high tun- neling magnetoresistance (TMR) effect [ 1]. Hard-disk drives, however, are approaching their scaling limits [2]. Besides, spin-transfer torque magnetic random-access memory (STT MRAM), also based on MTJs comprising an MgO tunnel barrier, is entering into volume production for eFLASH replacement and SRAM type of applications. They are nonvolatile, fast (5–50 ns cycle time), can be made relatively dense (Gbit), with low power consumption (100 fJ/write event), and exhibit very good write endurance [3–5]. The building block of STT MRAM is a cell with (1) high TMR for good readability, (2) high spin-transfer torque efficiency for good writability, and (3) high mag- netic anisotropy for good thermal stability and therefore memory retention [ 6,7]. All of these requirements must be satisfied together, which is the case in perpendicu- larly magnetized Co-Fe-B |MgO MTJs as long as the cell diameter remains larger than approximately 30 nm [ 8]. *libor.vojacek@vutbr.cz †mair.chshiev@cea.frBelow this diameter, the perpendicular anisotropy pro- vided by the Co-Fe-B |MgO interface becomes too weak in regards to thermal fluctuations so that the memory reten- tion reduces excessively. In this work, we focus on improv- ing the third requirement—the perpendicular magnetic anisotropy (PMA)—of the MgO-based MTJ, therefore allowing improved downsize scalability of out-of-plane magnetized MRAM. Although heavy metals like Pt or Pd can enhance PMA [9,10], they do so by increasing the spin-orbit coupling (SOC) parameter ξ. This is, however, associated with the undesirable side effect of increasing the Gilbert damp- ing [ 11], thus increasing the spin-transfer torque switch- ing current [ 12,13]. To avoid this problem, recipes based on purely 3 dmetallic elements were developed [ 14,15]. However, these recipes are based on Fe /Ni or Co /Ni alter- nating atomic layers, yielding structures that are intrin- sically complex to fabricate or may require excessively high deposition or annealing temperature. They can also get deteriorated upon the annealing necessary to obtain good crystallization of the MgO barrier and surrounding magnetic electrodes. In this work, we propose a different approach based on introducing a bulk Co interlayer into a simple Fe |MgO MTJ. The latter exhibits comparable or stronger PMA than the aforementioned Fe /Ni or Co /Ni alternating atomic 2331-7019/21/15(2)/024017(8) 024017-1 © 2021 American Physical SocietyLIBOR VOJÁ ˇCEK et al. PHYS. REV. APPLIED 15,024017 (2021) layers. In addition, the PMA characterized by the mag- netocrystalline anisotropy energy EMCA increases with the film thickness at a comparable or higher rate. Lastly, the Co Curie temperature (1404 K) is significantly higher than that of Fe (1043 K) and twice higher than that of Ni (631 K), which provides higher temperature stability [ 16]. The paper is organized as follows. In Sec. II, based on DFT calculations, we propose a MTJ with enhanced PMA and high TMR and discuss the aspects useful for a real-life fabrication of the structure. In Sec. III, the sys- tematic calculations supporting our proposal are presented. In Sec. IV, the large bulk Co |MgO PMA, crucial to the enhancement, is explained by the electronic structure and the second-order perturbation theory. II. MTJ WITH GREATLY ENHANCED PMA The DFT calculations are performed using the Vienna ab initio simulation package (VASP) [17,18]. Besides the electronic structure, the primary output of the calcula- tions is the magnetocrystalline anisotropy energy EMCA and its atomic site-resolved contributions. Positive (neg- ative) value of EMCA indicates PMA (in-plane anisotropy), respectively. See the Supplemental Material [ 19] for calcu- lation details [ 20–22]. A. PMA enhancement In this study, we find that a significant contribution to PMA originates from the bulk of epitaxial bcc Co on top of MgO. Its origin is presented in detail in Secs. IIIandIV. We exploit this finding and propose to enhance the PMA of conventional Fe |MgO MTJ by replacing the bulk Fe layers with Co. The effect can be further enhanced by sand- wiching the magnetic layer between two MgO barriers. The proposed improved MTJ storage layer thus takes the form MgO |Fe(n)Co(m)Fe(n)|MgO with n≥2a n d m≥3, as shown in Fig. 1(c). It is required to have at least two Fe atoms at the MgO interface and three successive Co atoms in the bulk to obtain the PMA enhancement (see Fig. S1 within the Supplemental Material [ 19] for details). Structures with different nand mare systematically investigated. The thickness of MgO in all the calcula- tions is chosen to be 5 (6) monolayers for an odd (even) number of metal layers, respectively. Figure 1(a) shows the effective PMA, which is a sum of the magnetocrys- talline anisotropy energy EMCAand dipole-dipole–induced demagnetization energy EDDas a function of n,m. One can see that the effective PMA does not vanish with increasing thickness, but interestingly, it grows steadily. This con- trasts with the pure Fe |MgO case (gray line), where the demagnetizing energy EDDdrives the magnetization in plane for thicknesses above 11 monolayers. The variation ofEMCA and EDDas a function of mfor n=2 is shown in Fig.1(b). (a) (c) (b) FIG. 1. (a) Effective PMA ( EMCA+EDD)i n MgO|Fe(n)Co(m)Fe(n)|MgO as a function of number of monolayers n,m. There is no perpendicular to in-plane magnetic anisotropy switching compared to pure MgO |Fe|MgO (gray diamonds; its thickness is m+4 ML, the same as the overall thickness for n=2). (b) EMCA,EDD, and the effective PMA (EMCA+EDD)f o r n=2. The effective PMA increases with the Co thickness. (c) Supercell of the MgO |Fe(2)Co(3)Fe(2)|MgO with periodic boundary conditions applied in all directions.Produced by VESTA [23]. The dipole-dipole energy EDDis obtained by summing over all the dipole-dipole interactions [ 24,25] (see the Supplemental Material [ 19] for calculation details). EDD acts mainly as demagnetizing energy [ 24], favoring an in-plane magnetization orientation. EDDis proportional to the product of the magnetic moments of the inter- acting atoms ( μ1·μ2). Thus, replacing the bulk Fe with Co decreases its magnitude knowing that in the bulk of the layer μFe≈2.5μBwhile μCo≈1.8μB, where μBis the Bohr magneton. The effect of the bulk Co is there- fore twofold: it increases the positive EMCA (discussed in detail in Secs. IIIand IV) and diminishes the negative demagnetizing dipole-dipole contribution. Replacing the MgO on one side of the metal film with vacuum, as tested on the n=4,m=6 structure, decreases the EMCA by 16% (see the Supplemental Material [ 19]). This is predominantly due to the drop in EMCA of the two interfacial Fe atoms [ 26]. (The effect of replacing the MgO by vacuum in pure bcc Fe (Co, Ni )|MgO can also be seen in Fig. S3 within the Supplemental Material [ 19].) However, at reduced thicknesses ( n=2,m=3), this drop is much more dramatic, namely 36%, or even 86%, when both interfaces are in contact with vacuum. This explains why 024017-2GIANT PMA ENHANCEMENT IN MgO-BASED MTJ. . . PHYS. REV. APPLIED 15,024017 (2021) Hotta et al.[14] observed no enhancement of anisotropy in Fe(2)Co(3)Fe(2)|MgO. They mimicked the presence of MgO by setting the lattice parameter equal to that of bulk MgO while, in reality, having vacuum at the interfaces. Note that changing the MgO thickness from 3 ML to 7 ML has a negligible effect on the EMCA in the order of 0.01 mJ/m2, as tested on the Fe (2)Co(3)Fe(2). B. Tunneling magnetoresistance Since we are interested in implementing this proposed storage layer in a full MTJ stack, we investigate its expected TMR amplitude. A large TMR of 410% at room temperature has been observed previously in pure bcc Co|MgO|Co MTJs [ 27]. In addition, Co in combination with Fe is often used for its record-holding TMR values. Therefore, we expect the high TMR to be present also in the proposed Fe |MgO MTJs with the inserted Co bulk layer. We estimate the TMR of the structures from the Julliere formula [ 28] with the spin polarization calculated from the local density of states with /Delta11symmetry of the interfacial oxygen at the Fermi level. The values are very high, around 300%. Note that the PMA does not originate solely from the interface in the proposed structure, but also from the bulk, in contrast to pure Fe |MgO [ 26]. This sug- gests that we might be able to separately tune the PMA (by the bulk Co) and the TMR (by the Fe |MgO interface). As mentioned in the introduction, the proposed MTJ is conceptually much simpler and more robust against annealing than the alternating layer-based MTJ [ 14,15] with similar properties. However, there are two main issues that we address regarding the fabrication of our structure, namely the stability of the bcc Co phase and the robustness against the Fe-Co interface not being atomically sharp. C. Fabrication of the metastable bcc Co Although the natural form of Co is hcp, the metastable bcc Co phase can be grown at room temperature [ 29–31]. It has been successfully grown on top of Fe with thick- ness up to 15 ML [ 32], with well-defined interfaces and no visible interdiffusion. The observed strain of 10% in bcc Co|MgO is considerable but still within the limit of what is experimentally realizable [ 33]. Indeed, Yuasa et al.[27] fabricated bcc Co (4ML)|MgO(10ML )|Co(4ML)MTJ and measured a record-holding TMR of 410% at room temper- ature. From our structural relaxation simulations, it follows that the bcc Co is preserved on top of MgO while it trans- forms into the fcc phase when surrounded by vacuum.Therefore, the bcc phase will probably be most stable if the device is used as a double-barrier MTJ. This also pro- vides higher PMA from the interfacial Fe, compared to single-barrier MTJ. FIG. 2. The effect of interdiffusion on the effective PMA in two selected structures. For the (minimal) Fe (2)Co(3)Fe(2)struc- ture, there is a significant PMA decrease of 73% at 0.5-ML atomic intermixing. For the thicker Fe (3)Co(4)Fe(3),t h eP M A is reduced only by 22%, demonstrating the robustness against interfacial roughness. We expect this robustness in the thicker structures in general. D. Decrease of PMA with interfacial roughness The sharpness of the Fe-Co interface is another rele- vant factor to consider. From the simulations it follows that any interdiffusion is fatal for the PMA when the Fe or Co thickness is less than 2 ML or 3 ML, respec- tively ( n<2o r m<3; see Fig. S1 within the Supplemental Material [ 19]). Robustness can be achieved at larger Fe and Co thicknesses. In Fig. 2, one may see the effec- tive PMA in the Fe (2)Co(3)Fe(2)and Fe (3)Co(4)Fe(3) structures with 0.5-ML (50%) interdiffusion and when the interface layers are swapped (1-ML interdiffusion). The drop in the effective PMA of Fe (3)Co(4)Fe(3)is only 22% at 0.5-ML interdiffusion, compared to a drop of 73% for Fe (2)Co(3)Fe(2). This robustness against surface roughness is to be expected in the thicker structures in general. Larger Co thickness is favorable as it increases the PMA (Fig. 1), but on the other hand, thicker bcc Co will probably be harder to fabricate [ 27]. Larger Fe thickness provides robustness against interdiffusion and might stabilize the bcc Co, as it is generally easier to grow bcc Co on Fe than on MgO (Co does not wet well on oxides due to its high surface tension while Fe does [ 34,35]). On the other hand, the PMA is decreased with thicker Fe, as shown in Fig. 1(a) (for layer-resolved behavior, see Fig. S4 within the Supplemental Material [ 19]). Looking at Fig. 1(a) and considering all the mentioned aspects, the MgO |Fe(3ML)Co(4ML)Fe(3ML)|MgO seems like a promising candidate as a storage layer for STT MRAM cells with highly improved thermal stability 024017-3LIBOR VOJÁ ˇCEK et al. PHYS. REV. APPLIED 15,024017 (2021) (a) (b) (c) FIG. 3. (a) Thickness dependence of EMCA in bcc Fe (Co, Ni )|MgO. Surprisingly, the PMA in Co increases steadily. (b) Layer- resolved EMCA in the structure with 15 ML of metal. Layer 1 is the interfacial layer. The layer number increases towards the bulk of the material. The largest contribution for Fe comes from the interface; for Co, it comes from the bulk. (c) EMCA in purely bulk bcc Fe, Co, and Ni as a function of c/aratio. Dashed lines indicate the typical value of c/ain the bulk of the given metal |MgO (see text for details). At its typical strain, the EMCA for Co is the same as in the bulk layers in Fig. 3(b). compared to conventional STT MRAM. Indeed, when the storage layer is sandwiched between two MgO layers, the anisotropy per unit area is of the order of 2 mJ/m2from the interfacial contribution minus approximately 1.2 mJ/m2 from demagnetizing energy (dependent on the chosen stor-age layer thickness) yielding a net effective PMA per unit area approximately 0.8 mJ/m 2[4]. In comparison, the net anisotropy per unit area in the proposed structure is approximately 2.2 mJ/m2being almost 3 times larger. This means that for the same thermal stability factor, the cell area could be reduced by a factor of 3 compared to conventional MRAM [ 7,36]. III. PMA IN BCC Fe(Co, Ni) |MgO THIN FILMS The idea of the improved MTJ proposed above is driven by our systematic investigation of the thickness depen- dence of EMCA in pure bcc (001) Fe, Co, and Ni |MgO ultrathin films. The EMCA as a function of metallic layer thickness is presented in Fig. 3(a). While for Fe the EMCA converges to a constant value [ 26], we observe a steady increase for Co. The behavior for Ni is more subtle. To elu- cidate why the trend varies among the three metals, in Fig. 3(b), we show the layer-resolved contributions to the EMCA (the contributions from each atomic layer separately). For Fe, the main contribution to EMCA comes from the first two interfacial layers [ 26,37]. Increasing the thickness does not affect the electronic properties of the interfacial layers in a significant way [ 26] (see also Fig. S5 within the Supplemental Material [ 19]). The bulk layers almost do not contribute to the PMA. Hence the EMCA does not change.In contrast, all the bulk layers of Co seem to contribute with a significant positive EMCA value, as evident from Fig. 3(b). Hence, the EMCA grows almost linearly with the number of added bulk Co layers [see Fig. 3(a)]. This observation is the cornerstone of this paper. For Ni, the influence of the interface manifests itself as deep as 6 ML, with the two interfacial monolayers contributing a negative EMCA. This is the reason for the in-plane anisotropy in the 5-ML structure, as shown in Fig.3(a). Although the deeper bulk layers contribute posi- tively, the EMCA does not grow monotonically as expected because the interfacial contributions in Ni do change upon thickness increase (see Fig. S5 in the Supplemental Mate- rial [ 19]). The bcc Ni |MgO is problematic also because of the large strain of approximately 15%. The large positive bulk EMCA in Co|MgO is caused by the strain that is induced within the Co by the MgO. To confirm this hypothesis, we have calculated EMCA as a function of the c/aratio in the primitive bcc unit cell for each of the metals shown in Fig. 3(c).T h e aand blat- tice parameters were set to the value of the relaxed bulk bcc Fe, Co, or Ni unit cell, while the clattice parameter was varied. The typical relaxed c/aratios we found within the bulk metal layers in Fe, Co, and Ni interfaced with MgO are 0.94, 0.89, and 0.85, respectively [dashed lines in Fig. 3(c)]. It clearly follows from Fig. 3(c) that for Fe and Ni, this bulk strain-induced EMCA is very small at their typ- ical bulk strains (dashed lines). However, we observe a large positive contribution of ≈0.5 mJ/m2for Co, which is the same value expected from Fig. 3(b). Indeed, artificially setting c/a= 1 in a previously relaxed Co|MgO 024017-4GIANT PMA ENHANCEMENT IN MgO-BASED MTJ. . . PHYS. REV. APPLIED 15,024017 (2021) (a) (b) (c) (d) (e) (f) FIG. 4. (a) Based on the orbital-resolved local density of states, the four contributions to the bcc Co EMCA(c/a) calculated from the second-order perturbation theory [ 41] are shown separately and in total. The values extracted from the DFT calculation with spin-orbit coupling included [“DFT”; see Fig. 3(c)] are shown for comparison. The classical Bruno term /Delta1E↓⇒↓ on its own reproduces the DFT curve for c/a<1 to a certain extent. (b) The Bruno /Delta1E↓⇒↓ term from Fig. 4(a) divided into contributions from individual virtual excitations. The excitation from dyztodz2states and vice versa ( dyz↔ dz2; red circles) is the one that causes the overall increase for c/a<1 .( c )S a m ea sF i g . 4(a)but calculated from the orbital-projected band structure, where additional aspects are taken into account (see text for details). The correspondence with the DFT curve is hence much better. (d) The orbital-resolved DOS for bcc Co with c/a=1. There is a peak in the dz2and dx2−y2minority states right above the Fermi level. (e) The orbital-resolved DOS for bcc Co with c/a=0.90. The strain causes the overall spreading of the DOS. Both the dz2and the dx2−y2peaks are pushed further above EF. (f) The most bulklike Co from the Fe (3)Co(12)Fe(3)structure. Its features are very similar to the bcc Co with c/a=0.90 in Fig. 4(e), supporting the applicability of the results of Sec. IVto the proposed structures of Sec. II. eliminates the bulk contribution to EMCA, thus confirming that strain plays the central role. We note that the calcu- lated EMCA (c/a) dependence in Fig. 3(c)corresponds well to previous findings [ 38,39], where the focus was limited toc/a>1 . IV . PERTURBATIVE TREATMENT OF STRAIN-INDUCED ANISOTROPY IN BCC COBALT The magnetocrystalline anisotropy is due to the spin- orbit coupling [ 40]. The magnitude of spin-orbit coupling constant is several 10 meV, which is much less than thewidth of 3 dbands. Therefore, the E MCA can be calculated within the second-order perturbation theory framework directly from the orbital-resolved (A) density of states or (B) band structure. Both of these are obtained from a DFTcalculation with the spin-orbit coupling not included. This treatment allows us to link changes in EMCA directly to changes in the electronic structure. A. Density of states The magnetic anisotropy energy can be calculated from the local density of states (LDOS) at a particular atom as [41] EMCA=/Delta1E↓⇒↓+/Delta1E↑⇒↑−/Delta1E↑⇒↓−/Delta1E↓⇒↑,( 1 ) where [ 42] /Delta1Eσ⇒σ/prime=ξ2 4/summationdisplay μμ/primePμμ/prime/integraldisplayEF −∞dε/integraldisplay∞ EFdε/primeρσ μ(ε)ρσ/prime μ/prime(ε/prime) ε/prime−ε. (2) 024017-5LIBOR VOJÁ ˇCEK et al. PHYS. REV. APPLIED 15,024017 (2021) Hereξis the spin-orbit coupling constant, μ( μ/prime)is an occupied (unoccupied) orbital with spin σ( σ/prime)and density of states ρσ μ(ρσ/prime μ/prime)at energy ε( ε/prime). The constant Pμμ/prime= |/angbracketleftμ|Lz|μ/prime/angbracketright|2−| /angbracketleftμ|Lx|μ/prime/angbracketright|2, where Lz(Lx)is the orbital momentum operator in the out-of-plane (in-plane) direc- tion, respectively. The Pμμ/primematrix for dorbitals is given in the Supplemental Material [ 19]. We use ξCo=84 meV [43]. The physical picture encompassed in Eqs. (1)and(2) is that virtual excitations of electrons from occupied to unoccupied orbitals give rise to positive, negative, or zero contribution to EMCA. The sign depends on the Pμμ/primecon- stant, and hence on the two interacting orbitals μandμ/prime. The closer the two orbitals are to each other, and hence to the Fermi level EF, and the larger their density of states ρ, the larger this contribution. The excitations are divided into four terms [Eq. (2)] based on the orbitals’ spins. The two “spin-conservation terms” have a plus sign, while the two “spin-flip terms” have a minus sign. In Fig. 4(a), we aim to reproduce the DFT-calculated results from Fig. 3(c) using the model from Eqs. (1) and (2). We show the four terms from Eq. (1), their sum (“ EMCA”), and the Co DFT curve from Fig. 3(c) for comparison (“DFT”). The EMCA curve does not reproduce the DFT curve well, so the model from Eq. (2)must be an oversimplification. Despite this, we can use it to draw several qualitative conclusions. (1) The minority-to-minority excitation term /Delta1E↓⇒↓ (the only term that is taken into account in the origi- nal Bruno approach [ 40]) on its own can reproduce the increase of EMCA for c/a<1 observed in the DFT curve. To link this increase directly to changes in the LDOS in Fig. 4(d), in Fig. 4(b), we show all the excitations con- tributing to /Delta1E↓⇒↓ separately. We see that the increase is caused by excitations from dyztodz2minority orbitals and vice versa ( dyz↔ dz2). This is linked to the strain- induced changes in the LDOS: for c/a<1 [Fig. 4(e)], the dz2peak in the unoccupied minority states is shifted further above the Fermi level, diminishing the negative dyz→ dz2contribution. This increases the overall EMCA. The increase is counteracted by a decrease in the posi- tive dxy↔ dx2−y2excitation contribution. This decrease is due to the strain-induced shift of the minority unoccupied dx2−y2peak, located immediately above EFfor c/a=1. (2) The two contributions that come from excitations to majority-spin states, /Delta1E↑⇒↑ and−/Delta1E↓⇒↑, are small. The reason is that there are almost no unoccupied majority- spin states, especially near the Fermi level, as we see in Figs. 4(d)–4(f). Moreover, these two contributions tend to cancel each other. Hence they can often be neglected [ 42]. Note that including the porbitals gives only a minor correction of approximately 1%.B. Band structure Next, we calculate the EMCA(c/a) directly from the (orbital-resolved) band structure [Eqs. (4)–(6) in [ 41]]. This approach intrinsically includes many aspects that are neglected in the calculation from LDOS in Sec. IV A , namely that (1) the projection coefficient of a Bloch state onto a particular dorbital is a complex number, (2) virtual excitations also happen in between atoms at different sites, not only on site, and that (3) a virtual excitation generally includes four orbitals, not only two (see Fig. 1 in Ref. [ 44]). All of these have proven to be essential for the model to be more accurate. The whole calculation is nicely described by Miura et al.[41]. In short, setting LORBIT =12 in the VASP calcula- tion provides the real and imaginary parts of the projection coefficients c, which we use to calculate the joint local den- sity of states G(see Ref. [ 41]). Taking its real part and performing summation over several variables, one may obtain the four contributions to EMCA from Eq. (1).W e useξCo=84 meV [ 43]. The calculation results are plotted in Fig. 4(c).T h e model is much better than the one in Fig. 4(a), while the main features are retained, namely that the /Delta1E↓⇒↓ term governs the overall trend. The −/Delta1E↑⇒↓ term serves to refine the shape, but in addition, causes an excessive overall decrease. The EMCA(c/a=1) is not zero in the EMCA curve, as it should be by symmetry arguments and as it is in the DFT curve. Despite that, the difference EMCA(c/a=0.90)- EMCA(c/a=1) in the EMCA and DFT curves correspond well to each other. Analyzing the contributions to /Delta1E↓⇒↓ from individual excitations, we confirm the results of Sec. IV A , namely that the main positive change in EMCA for c/a<1 is due to the dyz→ dz2virtual excitation, and the main negative change is due to the dx2−y2→ dxyexcitation. V . CONCLUSIONS We propose an alternative concept of MTJ with strongly enhanced perpendicular magnetic anisotropy based on introducing a Co interlayer into the bulk of conven- tional Fe |MgO MTJ. DFT calculations confirm that the PMA enhancement overcomes the negative demagnetiz- ing energy in these Fe (n)Co(m)Fe(n)|MgO structures. The TMR shows values similar to the pure Fe |MgO case. There is a trade-off between the enhancement magnitude and its robustness against the Fe-Co interfacial diffu- sion in a prospective real-life fabrication process. The Fe(3ML)Co(4ML)Fe(3ML)seems of strong potential as a storage layer for MgO-based STT MRAM cells. The design is based on the presented systematic study of PMAin bcc Fe (Co, Ni )|MgO, showing clearly that the MgO- imposed compressive strain induces a significant bulk PMA in bcc Co. We explain the PMA enhancement in bcc Co via the second-order perturbation theory approach and 024017-6GIANT PMA ENHANCEMENT IN MgO-BASED MTJ. . . PHYS. REV. APPLIED 15,024017 (2021) attribute it mainly to the strain-induced changes in ener- gies of the minority-spin filled dyzand unfilled dz2orbitals around the Fermi level. ACKNOWLEDGMENTS Computational resources are partially supplied by the project “e-Infrastruktura CZ” (e-INFRA LM2018140) pro- vided within the program “Projects of Large Research, Development and Innovations Infrastructures”. B.D. acknowledges ERC funding via ERC Adv grant No. MAG- ICAL 669204. Co-funded by the Erasmus +Programme of the European Union. [1] C. H. Bajorek, Magnetoresistive (MR) Heads and the Ear- liest MR Head-Based Disk Drives: Sawmill and Corsair, Computer History Museum, Mountain View, CA, Tech.Rep. (2014). [2] H. J. Richter, Density limits imposed by the microstructure of magnetic recording media, J. Magn. Magn. Mater. 321, 467 (2009). [3] J. Meena, S. Sze, U. Chand, and T.-Y. Tseng, Overview of emerging nonvolatile memory technologies, Nanoscale Res. Lett. 9, 526 (2014). [4] A. V. Khvalkovskiy, D. 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PhysRevLett.121.097204.pdf

Theory of the Topological Spin Hall Effect in Antiferromagnetic Skyrmions: Impact on Current-Induced Motion C. A. Akosa,1,2,*O. A. Tretiakov,3G. Tatara,1,4and A. Manchon2 1RIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan 2King Abdullah University of Science and Technology (KAUST), Physical Science and Engineering (PSE) Division, Thuwal 23955-6900, Saudi Arabia 3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4RIKEN Cluster for Pioneering Research (CPR), 2-1 Hirosawa, Wako, Saitama, 351-0198 Japan (Received 5 September 2017; revised manuscript received 11 May 2018; published 30 August 2018) We demonstrate that the nontrivial magnetic texture of antiferromagnetic Skyrmions (AFM Sks) promotes a nonvanishing topological spin Hall effect (TSHE) on the flowing electrons. This effect results ina substantial enhancement of the nonadiabatic torque and, hence, improves the Skyrmion mobility. This nonadiabatic torque increases when decreasing the Skyrmion size, and, therefore, scaling down results in a much higher torque efficiency. In clean AFM Sks, we find a significant boost of the TSHE close to the vanHove singularity. Interestingly, this effect is enhanced away from the band gap in the presence of nonmagnetic interstitial defects. Furthermore, unlike their ferromagnetic counterpart, the TSHE in AFM Sks increases with an increase in the disorder strength, thus opening promising avenues for materialsengineering of this effect. DOI: 10.1103/PhysRevLett.121.097204 Introduction. —As the spintronics community advances the search for high-efficiency, high-density, and low-power- consuming spintronic devices, alternative materials otherthan conventional ferromagnets (FMs) are being continu- ously introduced and explored. Besides FMs, antiferromag- nets (AFMs) have recently drawn significant attention [1,2]. The experimental observation of bulk spin-orbit torques(SOTs) in locally inversion asymmetric CuMnAs [3],t h e demonstration of AFM-assisted zero-field SOT switching [4,5], and the achievement of large anomalous and spin Hall effects in noncollinear AFMs [6–8]open promising per- spectives for the implementation of AFMs into efficient spindevices. The latter effect is particularly intriguing, since itemerges from the coexistence of spin-orbit coupling (SOC)- driven Berry curvature and noncollinear magnetism. In addition, it has also been predicted that AFM textures suchas domain walls driven by SOTs can move much faster thantheir FM counterparts due to the absence of Walker break- down [9,10] . Therefore, the interplay between topological spin transport and the dynamics of AFM textures is apromising route to explore towards the realization of efficientcurrent-driven control of the AFM order parameter. Recently, ferromagnetic Skyrmions (FM Sks) have been proposed as good candidates for technological applicationsdue to their weak sensitivity to defects [11–13], ultralow critical current density [13–19], enhanced nonadiabatic torque [20,21] , and substantial TSHE [22,23] . In spite of these remarkable properties, FM Sks suffer from the so- called Skyrmion Hall effect [19,24,25] , a motion transverse to the current flow. This parasitic effect hinders the robustelectrical manipulation of FM Sks. In contrast, both the analytical theory and micromagnetic simulations recently showed that, in AFM Sks, the Skyrmion Halleffect vanishes by symmetry [26–30]. In this Letter, we demonstrate that the nontrivial magnetic texture of AFM Sks promotes a nonvanishing TSHE on the flowing electrons. This effect results in a substantial enhance- ment of the nonadiabatic torque and, hence, improves theSkyrmion mobility. This nonadiabatic torque increases as the Skyrmion size decreases, and, as a result, scaling down results in a much higher torque efficiency. In clean systems, we find asignificant enhancement of the TSHE close to the van Hove singularity. Most importantly, unlike FM Sks [23],t h eT S H E in AFM Sks increases in the presence of nonmagneticinterstitial defects. Moreover, the TSHE is enhanced awayfrom the band gap in the presence of these defects. Phenomenological model. —Motivated by the prediction of a metastable single AFM Sk on a square lattice [29–32], our analysis begins with an isolated G-type (i.e., checker- board) AFM Sk with equivalent sublattices aandb. The conduction electrons are coupled to the N´ eel order nðr;tÞ via an exchange energy J. For a smooth and slowly varying N´eel order parameter, the emergent electromagnetic fields of the ηsublattice acting on electrons with spin σare derived as [33,38 – 40] Eη;σ em¼ðσℏ=2eÞPη σNt;iðrÞei; ð1aÞ Bη;σ em¼−ðσℏ=2eÞPη σNx;yðrÞz; ð1bÞPHYSICAL REVIEW LETTERS 121, 097204 (2018) 0031-9007 =18=121(9) =097204(5) 097204-1 © 2018 American Physical Societywhere Nμ;νðrÞ¼ð∂μn×∂νnÞ·n, with μ;ν∈ðt; x; y Þ, σ¼þ ð−Þ1for↑ð↓Þspin, Pη σ¼ð1þσηPkÞ=2, where Pk¼J=εkis the polarization of the density of state per sublattice, and εkis the energy dispersion [40,41] . Notice thatPktakes a value of 1 close to the van Hove singularity or as the exchange Jgoes to infinity. In these limits, the two sublattices behave like two independent parallel ferromag- nets ( Pa ↑¼1,Pa ↓¼0) and ( Pb ↑¼0,Pb ↓¼1). Therefore, unlike FM Sks, in which electrons feel an emergent electromagnetic field of opposite sign for different spins, in real AFM Sks (finite J), the magnitude of this field is both spin- and sublattice-dependent and strongly dependson the dispersion [42]. Our analysis is based on an AFM Sk with radius r 0, embedded in a large system of radius R≫r0moving rigidly with velocity v[i.e.,∂tn¼−ðv·∇Þn]. We chosewithout the loss of generality the profile given as n¼ðcosΦsinθ; sinΦsinθ;cosθÞ, where cos θ¼pðr2 0−r2Þ=ðr2 0þr2Þ andΦ¼qArgðxþiyÞþcπ=2define the polar and azimu- thal angles, respectively. The constants p,q,a n d c, which take values of /C61, define the polarization, vorticity, and chirality, respectively [33]. Under the action of an external electric field along the xaxis (i.e., E¼Ex), the local charge and spin current densities per sublattice read [33] jη e¼ð1=2Þ½σ0xþησxyðrÞy/C138E þηðℏ=4Þ¯P0σ0Nx;yðrÞðv×zÞ; ð2aÞ jη s¼n⊗ð1=2Þ½ηx−pqβTðrÞy/C138bJ þðpq=2ÞαTðrÞn⊗ðv×zÞ; ð2bÞ where σ0ðHÞ=2is the longitudinal (ordinary Hall) conduc- tivity of sublattice ηand σxyðrÞ¼ðℏ=2eÞ¯PHσHNx;yðrÞ is the nonlocal steady state transverse conductivity. bJ¼γℏP0σ0E=2eMsquantifies the adiabatic torque, while αTðrÞ¼pqλ2 ENx;yðrÞand βTðrÞ¼pqλ2 HNx;yðrÞ are dimensionless nonlocal contributions to the Gilbert damping and nonadiabatic torque, respectively. Here Ms is the saturation magnetization, and the constants λ2 H¼ℏ˜PHσH=ð2eP0σ0Þand λ2 E¼γℏ2˜P0σ0=ð4e2MsÞare length scales associated with the emergent magnetic and electric fields, respectively [21]. In the above expressions, P0ðHÞ≡Pa 0ðHÞ¼−Pb 0ðHÞis the longitudinal (ordinary Hall) current polarization, where Pη 0ðHÞ¼ ðση;↑ 0ðHÞ−ση;↓ 0ðHÞÞ=ðση;↑ 0ðHÞþση;↓ 0ðHÞÞ.F i n a l l y ,w ea l s on e e dt o define the effective polarizations ¯P0;H¼ðP0ðHÞþPkÞ=2 and ˜P0ðHÞ¼ð1þP0ðHÞPkÞ=2[33]. Interesting physics of charge and spin transport in AFM Sks can be inferred from Eq. (2). Indeed, since ηchanges sign on different sublattices, there is (i) no macroscopictransverse (along y) charge current, i.e., no topological Hall effect (THE) [29,30] , (ii) no macroscopic longitudinal(along x)spin current, and (iii) a nonzero transverse spin current, i.e., a finite TSHE [26,42] . The physical origin of the TSHE as illustrated in Fig. 1stems from the interplay between the emergent magnetic field, Eq. (1b), and the dispersion of the underlying system. The emergent mag-netic field deflects flowing electrons with opposite spins to opposite directions, and the inherent twofold degeneracy ensures that a continuous transverse pure spin current flows in the system. To elucidate the effect of this transverse spin current on the mobility of Skyrmions, the impact of the topologicalspin current derived in Eq. (2b)on the dynamics of an AFM Sk is investigated. To achieve this, we calculate thecorresponding total spin torque as τ T¼−∇·js, where js¼jasþjbsand, for the sake of completeness, we include nonadiabatic effects [43] via a constant nonadiabaticity β and the Gilbert damping torque with damping constant α such that the total spin torque is given as τ¼αn×∂tn− βbJn×∂xn−τTto obtain τ¼αn×∂tn−βbJn×∂xn þαTðrÞn×∂tn−βTðrÞbJn×∂xn: ð3Þ It appears clear from Eq. (3)that, just as in FM Sks [20,21] , the transverse spin current flowing in AFM Sks directlyenhances the nonadiabatic torque and the damping.Moreover, this nonadiabatic topological torque increaseswhen decreasing the Skyrmion size. As a result, theefficiency of the current-driven motion increases whenthe Skyrmion becomes smaller. We follow the standard theoretical scheme employed to study the dynamics of antiferromagnetic textures [44–49] supplemented by the derived topological torque to obtainthe equation of motion of the N´ eel order parameter as [33] 1 ¯a˜γ∂2tnþαeffðrÞ∂tn¼γfnþβeffðrÞbJ∂xn; ð4Þ where ˜γ¼γ=ð1þα2Þ,αeffðrÞ¼αþαTðrÞ,βeffðrÞ¼ βþβTðrÞ, and fnis the effective field derived from the FIG. 1. Schematic illustration of the physical origin of the TSHE in an AFM Sk. (a) For J≫tand close to the van Hove singularity, two bands are essentially decoupled ( Pη k¼η), and the emergent magnetic field (EMF) results to a substantial TSHE. (b) For J∼t (away from the van Hove singularity), a strong transition betweendegenerate bands results to a substantial reduction of the TSHE.PHYSICAL REVIEW LETTERS 121, 097204 (2018) 097204-2magnetic energy E¼Rdr½ð¯a=2Þm2þA 2ð∇nÞ2/C138asfn¼ −δnE, where ¯aandAare the homogeneous and inhomo- geneous exchange constants, respectively [46]. The termi- nal velocity calculated from Eq. (4)is given as vy¼0and vx¼ðβeff=αeffÞbJ; ð5Þ where the effective parameters are given as [33] αeff¼αþ4 3λ2 E r2 0and βeff¼βþ4 3λ2 H r2 0: ð6Þ To provide a qualitative estimate of our predicted effect, using realistic material parameters Ms¼800KA=m, α¼0.01,β¼0.02,P0¼0.7,Pk¼0.4,σ0¼14.75=ðμΩmÞ, σH=σ0¼0.045=T, and je¼5×1011A=m2, we obtain λ2 E¼ 0.225nm2andλ2 H¼13.54nm2. These values translate to a longitudinal velocity of up to 391m=s for a Skyrmion size of 10 nm, showing that, for small Skyrmions sizes, thetopological torque produces a sizable effect. Before we proceed, we note that, even though the topological torque discussed above does not rely onSOC, the latter is expected to be ubiquitous in systemspromoting noncollinear magnetic textures such as AFMSks [38–40]. Indeed, SOC has several effects on spin transport, depending on its symmetry. In bulk materials, it contributes to spin relaxation, which results in nonadiabatictorque that gives rise to a Skyrmion mobility that isindependent of r 0[43]. In magnets lacking inversion symmetry, such as in magnetic multilayers, interfacial(Rashba-like) SOC produces (mostly) a fieldlike torque,while the spin Hall effect arising from an adjacent heavymetal induces a dampinglike torque. The former does notcontribute to AFM Sks mobility, while Velkov et al. [28] showed that the latter induces a mobility that is proportionaltor 0. Finally, one also needs to consider the spin Hall effect inside the AFM itself. In the case of ferromagnetic vortices,Manchon and Lee [50]showed that the spin Hall effect acts in the same way as the nonadiabatic torque, thus inducinga mobility that does not depend on the vortex radius.Therefore, we expect the Skyrmion mobility to be domi-nated by the topological torque discussed above in the limitof small Skyrmions. Tight-binding model. —Our theoretical predictions in Eqs. (2)and(6)are verified by means of a tight-binding model of an isolated AFM Sk on a square lattice describedby the Hamiltonian H¼X iϵiˆc† iˆci−tX hijiˆc† iˆcj−JX iˆc† imi·ˆσˆci;ð7Þ where Jis the exchange energy that couples the spin of electrons ˆσto the local magnetization mi,tis the nearest- neighbor hopping, and ϵiand ˆc† i(ˆci) are the on-site energy and the spinor creation (annihilation) operator of site i,respectively. We consider an AFM Sk of radius 12a0to ensure that the texture is smooth and slowly varying [21], for both the strong ( J¼5t) and the intermediate ( J¼2t=3) exchange limits [22]using the KWANT code [51]. The Hall transport is investigated via a four-terminal system [23] with a scattering region of size 102×102a2 0(i.e., 51×51 AFM unit cells) and compared with an equivalent FM Sk. In a clean system, we find a substantial TSHE in AFM Sks in both the strong exchange limit and close to thevan Hove singularity [blue line in Figs. 2(a)and2(b)]. In the intermediate exchange limit, however, the transition between degenerate bands is strong [40]; this results in an overall reduction of the TSHE. Furthermore, since theTSHE increases with the Skyrmion density [23], a sub- stantial spin current capable of inducing magnetizationdynamics and/or switching on an adjacent attached FM layer can be expected. Moreover, unlike FM Sks, AFM Sks exhibit no THE due to the cancellation of the charge currentcontributions from both sublattices [29] [green line in Figs. 2(a) and2(b)]. Our numerical results are consistent with our analytical predictions in Eq. (2). To model real materials, we investigated the impact of nonmagnetic impurities which are omnipresent in experi- ments. This is done via randomized on-site energies ϵ i¼Vi∈½−ðW=2Þ;ðW=2Þ/C138, where Wdefines the strength of the disorder, and average over 104configurations to ensure convergence. Two classes of defects are considered: (i)interstitial defects, which preserve the coherence between the sublattices within the antiferromagnetic unitcell, referred to as symmetric scattering (SS), and (ii) dis-order that induces decoherence within the unit cell, referred to as asymmetric scattering (AS) [33]. We find that, in both the strong and intermediate exchange limits, as shown inFigs. 3(a)and3(b), respectively, the presence of disorder-4 -2 0 2 4 Transport energy (t)-0.10 0.1 Hall angles FMTH FMTSH AFM TH AFM TSH-6 -3 0 3 6 Transport energy (t)0 0.1 Hall angles -1.1 -0.8 Transport energy (t)0 0.040.08 Hall angles -5.6 -5.2 Transport energy (t)0 0.020.04Hall angles(b) (a) (c) (d)3/t2 = J t5 = J J = 5t J = 2t/3 FIG. 2. Computed THE and TSHE for a FM Sk and AFM Sk as a function of the Fermi energy in the (a) strong and (b) inter-mediate exchange limits. Insets (c) and (d) represent an enlarge-ment around the purple region in (a) and (b), respectively.PHYSICAL REVIEW LETTERS 121, 097204 (2018) 097204-3progressively quenches the TSHE for the FM Sk (black curve); in contrast, in the case of the AFM Sk, only ASquenches the TSHE (red curve). In fact, SS disorder enhances the TSHE (blue curve) as long as the coherence between the two sublattices is preserved (region I) [33,40] . A further increase in the disorder strength eventually leads to the onset of decoherence (region II), resulting in the reduction of the TSHE. Finally, we numerically verify the scaling law of the topological torques with respect of the Skyrmion size given by Eq. (3). To achieve this, we follow the scheme outlined in Ref.[21], consider a large system size of 302×302a 2 0, and calculate the local spin transfer torque from the nonequili- brium spin density induced by a voltage bias of 0.2t.T h e calculated torque is then projected on ∂xn(adiabatic) or n×∂xn(nonadiabatic), integrated over space, and normal- ized accordingly to obtain the scaling law with respect to the Skyrmion size. From this, we computed the nonadiabatic torque as βeffbJ¼½Rτ·ðn×∂xnÞd2r/C138=RNx;yðrÞd2r[33]. Our numerical calculations as depicted in Figs. 4(a)and4(b) show good correspondence with our analytical predictions inEq.(6)in both the strong and intermediate exchange limits.Conclusion. —Micromagnetic simulations originally predicted that Skyrmions have, in principle, limitedsensitivity to local and edge defects owing to very weak interactions [11,12] and their finite spatial extension [15,17,52] . Indeed, the ability of a defect to pin a Skyrmion increases when the size of the Skyrmion becomes compa-rable to the size of the defect [11]. Hence, scaling down the Skyrmion towards sub-100-nm size results in low Skyrmion mobility and large critical depinning currentsin polycrystalline systems [53]. What makes AFM Sks remarkable in this respect is the fact that the torque efficiency itself increases when reducing the Skyrmion size, as discussed above. While this topological torquecontributes only to the transverse motion of FM Sks, it drives the longitudinal motion of AFM Sks and, therefore, directly competes with the enhanced pinning potential.This unique property could be a substantial advantage to compensate the increasing pinning upon size reduction. Furthermore, our calculations show that the TSHE isenhanced in the presence of moderate disorder that is omnipresent in real materials, demonstrating the robustness of the proposed approach for device applications. This work was supported by Grant-in-Aid for Scientific Research(B) No. 17H02929, from the Japan Society for the Promotion of Science and Grant-in-Aid for Scientific Research on Innovative Areas No. 26103006 from theMinistry of Education, Culture, Sports, Science andTechnology (MEXT) of Japan. O. A. T. acknowledges support by the Grants-in-Aid for Scientific Research (No. 25247056, No. 17K05511, and No. 17H05173) fromthe MEXT of Japan, MaHoJeRo (DAAD Spintronics network, Project No. 57334897), and by JSPS and RFBR under the Japan-Russia Research CooperativeProgram. A. M. acknowledges support from the King Abdullah University of Science and Technology (KAUST). C. A. A. thanks R. Cheng for useful discussions. *collins.akosa@riken.jp [1] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol. 11, 231 (2016) . [2] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and T. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018) . [3] P. Wadley et al. ,Science 351, 587 (2016) . [4] S. Fukami, C. Zhang, S. DuttaGupta, A. Kurenkov, and H. Ohno, Nat. Mater. 15, 535 (2016) . [5] A. van den Brink, G. Vermijs, A. Solignac, J. Koo, J. T. Kohlhepp, H. J. M. Swagten, and B. Koopmans, Nat. Commun. 7, 10854 (2016) . [6] C. Sürgers, G. Fischer, P. Winkel, and H. v. Löhneysen, Nat. Commun. 5, 3400 (2014) . [7] S. Nakatsuji, N. Kiyohara, and T. Higo, Nature (London) 527, 212 (2015) . [8] A. K. Nayak et al. ,Sci. Adv. 2, e1501870 (2016) . [9] O. Gomonay, T. Jungwirth, and J. Sinova, Phys. Rev. Lett. 117, 017202 (2016) . FIG. 3. Dependence of the TSHE on impurity for the (a) strong and (b) intermediate exchange limits. In region I (II), the mean free path of the system is longer (shorter) than the system size. Notice the enhancement of the TSHE in the presence ofan impurity for symmetric (blue curve) compared to both asymmetric (red curve) defects and a FM Sk (black curve). 10 15 20 Skyrmion radius (a0)00.511.52Normalized torqueAdiabatic torque Nonadiabatic torque 10 15 20 Skyrmion radius (a0)012345Normalized torqueAdiabatic torque Nonadiabatic torque51 0 1 5 1/r02 (a0-2)012Normalized torque 51 0 1 5 1/r02 (a0-2)024Normalized torque 10-5 10-8 10-5 10-8 10-3 10-3 F = -3.2tF = -6.0t(a) (b) (d) (c) FIG. 4. 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PhysRevB.81.054418.pdf

Stochastic properties and Brillouin light scattering response of thermally driven collective magnonic modes on the arrays of dipole coupled nanostripes M. P. Kostylev1and A. A. Stashkevich2 1School of Physics–M013, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia 2LPMTM CNRS (UPR 9001), Université Paris 13, 93430 Villetaneuse, France /H20849Received 15 October 2009; revised manuscript received 10 January 2010; published 16 February 2010 /H20850 In the present paper, the problem of thermal excitation of collective /H20849magnonic /H20850Bloch magnetostatic modes on a one-dimensional array of magnetic stripes has been addressed. It has been shown that partially phase-correlated oscillations localized on individual stripes can be regarded as an ensemble of individual harmonicoscillators interpretable in terms of independent degrees of freedom of the magnetic system subject to thelow-energy Rayleigh-Jeans statistics. Numerical simulations of the Brillouin light scattering spectra, based onthis approach, have shown that the nth Bloch mode in strongly coupled stripes contributes mainly to the scattering in the nth Brillouin zone. Our calculations have also confirmed numerically the noncoherent wide- angle character of the BLS, demonstrated experimentally earlier. DOI: 10.1103/PhysRevB.81.054418 PACS number /H20849s/H20850: 75.30.Ds, 78.35. /H11001c I. INTRODUCTION From both fundamental and application viewpoints, mag- netization dynamics in ferromagnetic media has become ofutmost significance today. Rapid advances in spintronics dur-ing the last decade have contributed massively to theprogress in this field. Suffice it to mention such discoveriesas the giant magnetoresistance 1,2and precessional switching applied to read-write processes in magnetic data storage.3,4 In the case of relatively low-angle precession, magnetic dy- namics manifests itself through magnetic excitations propa-gating in the bulk of such materials known as spin waves/H20849SW /H20850, which are an object of great interest themselves. His- torically, there are two different philosophies regarding thisphenomenon from different points of view. It was with theincoherent SW driven by thermal agitation, better known asmagnons, that the research in this domain began, as early asin the 1930s. 5Since then such magnetic excitations, with a very short wavelength on the scale of the lattice parameterand hence entirely dominated by short-range exchange inter-actions, have become one of the fundamental notions in thesolid-state physics of ferromagnetic media. Another view ofthe problem, seeing it from a completely different angle, isdue to extensive application of ferrite materials to micro-wave components, beginning from the 1950s. In this case,the excitations, better known, as magnetostatic waves/H20849MSW /H20850, are of long-wavelength nature, which is dictated by the macroscopic characteristic size of the microwave ele-ments themselves. As a result, their behavior is completelydescribed solely by the long-range dipole-dipole interactions/H20849DDI /H20850. Further development of the latter concept in the 1960s was based on the breakthrough in the technology ofsingle-crystal ferrite films, especially those based on yttriumiron garnet, with extremely low losses at microwave frequen-cies, suitable for effective signal processing, typically in de-lay lines in the frequency range from 2 to 20 GHz. 6The coherent MSWs were excited by an external microwavesource, i.e., by a special MSW antenna. The characteristicsize of the ferrite films employed /H20849their thickness /H20850as well as that of the microstrip MSW antennas /H20849their width /H20850being in the micrometric range, the magnetic waves excited were ofcombined dipole-exchange nature. 7Extensive research in SW dynamics over the last few decades has lead to further merging of the two ideasthus leading to a generic concept of the dipole-exchangeSW, either coherent or incoherent. One of the stepscontributing to the reconciliation of the two points ofview, especially important in the context of this paper, wasdue to the major improvements of the Brillouin lightscattering /H20849BLS /H20850techniques, that took place in the late 1970s and early 1980s. While earlier setups, lacking in sen-sitivity, had to resort to generation of coherent SWs by ex-ternal sources, 8,9in the updated versions10preference was given to thermal magnons,11i.e., incoherent SWs. In spite of their low intensity, they ensured, owing to their thermal na-ture, excitation of modes with all possible wave numbers andtemporal frequencies, within the range permitted by the SWspectra. The latter development spawned a series of papersproviding adequate theoretical support for a newly emergedtechnique. 12,13It relied on the modification of the fluctuation- dissipation theorem /H20849FDT /H20850, developed somewhat earlier, for a magnetic system /H20849see, for example, Refs. 14and15/H20850. The latter allows to relate the Fourier transform of the correlationfunction /H20855m i/H20849x/H11032,r/H6023,t/H20850mj/H20849x,r/H6023=0,t=0/H20850/H20856, determining the inten- sity of the light scattered at a given angle to the dynamic susceptibilities gij/H20849x,x/H11032,K/H6023,/H9275/H11006i/H9255/H20850, /H20885dtd2r/H20855mi/H20849x/H11032,r/H6023,t/H20850mj/H20849x,r/H6023=0 ,t=0 /H20850/H20856exp /H20851i/H20849−K/H6023·r/H6023+/H9275t/H20850/H20852 /H11008/H20851gij/H20849x,x/H11032,K/H6023,/H9275+i/H9255/H20850 −gij/H20849x,x/H11032,K/H6023,/H9275−i/H9255/H20850/H20852, in the limit /H9255→0.gij/H20849x,x/H11032,K/H6023,/H9275/H11006i/H9255/H20850, being responses to delta-type excitations hi/H20849x/H20850=/H9254/H20849x−x/H11032/H20850, can be as well re- garded as magnetic Green’s functions. Optical properties ofthe multilayered ferromagnetic structure are taken into ac-count via the optical Green’s-functions formalism. Typicallyapplied to magneto-optical /H20849MO /H20850interactions in thin metal ferromagnetic structures, it has proved to be a very powerfultheoretical tool, allowing to extract from the structure of theBLS spectra valuable information on the intrinsic magneticPHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 1098-0121/2010/81 /H208495/H20850/054418 /H2084914/H20850 ©2010 The American Physical Society 054418-1parameters of the investigated sample, unattainable by any other technique. In more recent papers16some specialized versions of this general theoretical approach have been re-ported. In spite of a rapid progress, within a span of the last 5 years, of an innovative micro-Brillouin modification, typi-cally employing externally excited coherent SW localized onindividual nanoelements, 17,18the classic “thermal magnon BLS” is still indispensable,16,19–21especially for the investi- gation of collective SW modes existing on the arrays of fer- romagnetic elements, forming a one-dimensional /H208491D /H20850or two-dimensional /H208492D /H20850structure.22–24Not surprisingly, it is largely and successfully used to this end until now. Although powerful and effective, the FDT implies an analysis of the magnetic system on the microscopic level,which accentuates the quantum-mechanical aspect of theproblem. An alternative approach, more consistent with apurely microscopic nature of the investigated phenomena, can be developed. Moreover, it is possible to adapt, withouttoo much difficulty, this formalism to the case of utmostimportance nowadays, namely, that of nanostructured ferro-magnetic films. In its main features, this technique can bereduced to the classical problem of the Brownian motiondriven by the Langevin force. The goal of this paper is theoretical description of the collective SW modes traveling in a periodic one-dimensionalarray of ferromagnetic stripes /H20849magnonic modes /H20850, typically referred to as Bloch waves, and driven by a thermal “Lange-vin” source of magnetic field. Periodic structures of this typeare also known as 1D magnonic crystals. As their closestanalogs /H20849photonic and phononic crystals /H20850they manifest all major features characteristic of wave propagation in periodicmedia, such as band gaps and Brillouin zone /H20849BZ /H20850. The latter has been confirmed experimentally in Refs. 25and26 /H208492D case /H20850. What makes their wave behavior especially interest- ing, from the physical point of view, is a strongly pro-nounced magnetically adjustable dispersion. Theoreticalanalysis of three-dimensional magnonic crystals reveals, notsurprisingly, an even richer spectrum of wave phenomena. 27 Our paper, focusing on the stochastic properties of ther- mally driven collective magnonic modes on 1D arrays ofdipole coupled nanostripes is organized in the following way.At a first stage, a theoretical formalism is developed, allow-ing representation of such modes as an ensemble of indi-vidual harmonic oscillators interpretable in terms of indepen-dent degrees of freedom of the magnetic system subject tothe low-energy Rayleigh-Jeans statistics. 28To this end, the mathematical approach described in our earlier paper, Ref.29, will be further generalized. The second part of this paper will be dedicated to the spatial correlation function playing akey role in various phenomena, such as Brillouin light scat-tering. More specifically, the correlation /H20849coherence /H20850length l cfor collective modes on an array of dipole coupled stripes, coupled through DDIs, will be estimated. The latter describesthe spatial interval within which the phases of local pointMO scatterers are sufficiently correlated which ensures thespatial coherence of the inelastically scattered light. Finally,in the last part the results obtained in the previous paragraphwill be applied to the numerical estimation of the BLS an-gular spectrum. Special attention will be paid to the influenceof the correlation /H20849coherence /H20850length l cand the size of the incident optical beam don the form of such spectra. More specifically, numerical simulations theoretically modeling thetransition from incoherent inelastic scattering to coherent in-elastic diffraction, experimentally studied in Ref. 30, will be performed. II. THEORY A. Basic thermodynamics The geometry of the problem is illustrated in Fig. 1.W e study collective magnetic modes existing on a periodic arrayof parallel infinite ferromagnetic stripes with a width wand a thickness Lseparated by a distance /H9004. This corresponds to a spatial period equal to T=w+/H9004. The stripes are magnetized by an external field He /H6023zalong their axis, i.e., along z, which means that the induced static magnetization within them Me/H6023z is homogeneous. We limit our investigation to the case of the lowest purely dipole magnetic mode. We also assume thatthe aspect ratio of the stripes is small L/w/H112701. In other words, we study quasi-Damon-Eshbach /H20849DE /H20850collective /H20849magnonic /H20850modes formed via dipole interactions between the oscillations localized on individual stripes. They propa-gate along the “ x” axis and are characterized by a homoge- neous distribution of the magnetization along the vertical “ y” axis. The latter will make the dependence of the dipole fieldand the dynamic magnetization on yirrelevant which allows one to reduce the initial 2D problem to a 1D problem byaveraging across the film thickness. Thus application of thehighly effective approximate approach proposed in Ref. 31 and generalized for the geometry studied in Ref. 29is fully justified. The linearized Landau-Lifschitz equation describing ther- mally excited collective magnetic modes existing on an arrayof infinite ferromagnetic stripes can be written as /H11509 /H11509tmx/H20849x,t/H20850+/H9275Hmy/H20849x,t/H20850−/H9275M/H20851G/H20849x,x/H11032/H20850/H20002my/H20849x/H11032,t/H20850/H20852 =/H9275M 4/H9266hy/H20849th/H20850/H20849x,t/H20850 /H11509 /H11509tmy/H20849x,t/H20850−/H9275Hmx/H20849x,t/H20850−/H9275Mmx/H20849x,t/H20850 −/H9275M/H20851G/H20849x,x/H11032/H20850/H20002mx/H20849x/H11032,t/H20850/H20852=−/H9275M 4/H9266hx/H20849th/H20850/H20849x,t/H20850/H20849 1/H20850 FIG. 1. Geometry of the considered array of dipole-interacting stripes.M. P. KOSTYLEV AND A. A. STASHKEVICH PHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 054418-2where /H9275H=/H9253H,/H9275M=/H92534/H9266M, G/H20849x,x/H11032/H20850/H20002m/H20849x/H11032,t/H20850 =/H20848−/H11009/H11009G/H20849x,x/H11032/H20850m/H20849x/H11032,t/H20850dx/H11032is the convolution integral and G/H20849x,x/H11032/H20850is a magnetic Green’s function describing DDI within each stripe, as well as between different stripes, G/H20849x,x/H11032/H20850=1 2/H9266Lln/H20875/H20849x−x/H11032/H208502 L2+/H20849x−x/H11032/H208502/H20876, when xand x/H11032are within any stripes, G/H20849x,x/H11032/H20850=0 , when xor/and x/H11032is/are outside any stripe. h/H20849th/H20850/H20849x,t/H20850is a delta-correlated thermal Langevin force, de- scribing thermal excitation of the magnetic modes studied,/H20855h/H9251/H20849th/H20850/H20849x,t/H20850h/H9252/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856=C/H9254/H20849x−x/H11032/H20850/H9254/H20849t−t/H11032/H20850/H9254/H9251/H9252, /H208492a/H20850 where Cis a constant, /H9254/H20849t−t/H11032/H20850is the Dirac delta function, and/H9254/H9251/H9252is the Kronecker delta with /H9251,/H9252=x,y,z. We also assume that its mean value is zero /H20855h/H9251/H20849th/H20850/H20849x,t/H20850/H20856=0 . /H208492b/H20850 Note that we consider h/H9251/H20849th/H20850/H20849x,t/H20850to be completely real. Introducing circular variables, corresponding to circular polarizations m/H11006=mx/H11006imy,h/H11006/H20849th/H20850=hx/H20849th/H20850/H11006ihy/H20849th/H20850/H208493/H20850 allows us to simplify the system /H208491/H20850in such a way that the second equation is reduced to a complex conjugate of thefirst one /H20902/H11509 /H11509tm/H20849x,t/H20850−i/H20873/H9275H+/H9275M 2/H20874m/H20849x,t/H20850−i/H9275M 2/H20851m/H11569/H20849x,t/H20850+G/H20849x,x/H11032/H20850/H20002m/H11569/H20849x/H11032,t/H20850/H20852 =−i/H9275M 4/H9266h/H20849th/H20850/H20849x,t/H20850; c.c. /H20903/H208494/H20850 Here we used the identities m−=m+/H11569,h−/H20849th/H20850=h+/H20849th/H20850/H11569, and de- noted m/H11013m+,h/H20849th/H20850=h+/H20849th/H20850.I nE q . /H208494/H20850the circular polarizations of opposite directions, denoted by mand m/H11569, are coupled through the presence of DDIs which is unavoidable in theDE geometry and which is described by the third term on theleft-hand side of Eq. /H208494/H20850. Physically this means, that the po- larization eigenvectors are not circular, like in the case of theclassic ferromagnetic resonance, but elliptic. We will dealwith this later. The solution to Eq. /H208494/H20850, which describes a traveling wave on a periodic structure, just as in the case considered in Ref.29, can be represented in the form of Bloch waves, 32 mkn/H20849x/H20850=m˜kn/H20849x/H20850exp /H20849ikx /H20850, /H208495/H20850 where kis a Bloch wave vector taking on continuous values, andm˜kn/H20849x/H20850is a spatially periodical function with the period T,m˜kn/H20849x+jT/H20850=m˜kn/H20849x/H20850. Here nis a mode number and jis an integer. It should be noted that in classic wave science thesolution /H208495/H20850is known as Floquet’s theorem. 33Being eigenfunctions of the Hermitian integral operator /H9261/H20849k,n/H20850mkn/H20849x/H20850=G/H20849x,x/H11032/H20850/H20002mkn/H20849x/H11032/H20850/H20849 6/H20850 the individual Bloch waves are mutually orthogonal /H20885 −/H11009/H11009 mkn/H20849x/H11032/H20850mk/H11032n/H11032/H11569/H20849x/H11032/H20850dx/H11032=/H9254nn/H11032/H9254/H20849k−k/H11032/H20850. /H208497/H20850 The general solution to Eq. /H208494/H20850is sought in the form of a complete set of individual Bloch waves within the first Bril-louin zone m/H20849x,t/H20850=/H20858 n=1/H11009/H20885 −/H9266/T/H9266/T akn/H20849t/H20850m˜kn/H20849x/H20850exp /H20849ikx /H20850dk. /H208498/H20850 To obtain equations in amplitudes akn/H20849t/H20850let us insert Eq. /H208498/H20850 into Eq. /H208494/H20850and project both sides of the Eq. /H208494/H20850on the eigenvectors, Eq. /H208495/H20850, taking advantage of the mutual or- thogonality. Thus the problem is reduced to the solution of a standard nonhomogeneous system of spin-wave equations ofmotion 28 /H20902/H11509 /H11509takn/H20849t/H20850+iAa kn/H20849t/H20850+iB /H20841k/H20841na−kn/H20849t/H20850/H11569=−i/H9275M/H20885 −/H11009/H11009 h/H20849th/H20850/H20849t,x/H20850m˜kn/H20849x/H20850/H11569exp /H20849−ikx /H20850dx c.c., /H20903/H208499/H20850STOCHASTIC PROPERTIES AND BRILLOUIN LIGHT … PHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 054418-3where A=− /H20849/H9275H+/H9275M 2/H20850,B/H20841k/H20841n=−/H9275M 2−/H9275M/H9261/H20849/H20841k/H20841,n/H20850 Since the kernel in the integral operator in Eq. /H208496/H20850is sym- metric its eigenvalues are real /H9261/H20849k,n/H20850=/H9261/H20849k,n/H20850/H11569. Moreover, from Eq. /H208496/H20850it follows /H9261/H20849k,n/H20850/H11569=/H9261/H20849−k,n/H20850, which means that /H9261/H20849k,n/H20850=/H9261/H20849−k,n/H20850=/H9261/H20849/H20841k/H20841,n/H20850. To obtain two separate equations in bkn/H20849t/H20850andb−kn/H20849t/H20850/H11569, i.e., to diagonalize the system /H208499/H20850we apply the Bogoliubov trans- formation akn/H20849t/H20850=u/H20841k/H20841nbkn/H20849t/H20850+v/H20841k/H20841nb−kn/H20849t/H20850/H11569. This results in an equation for normal elliptic spin-wave am- plitudes, /H11509 /H11509tbkn/H20849t/H20850+/H20849/H9253/H20841k/H20841n+i/H9275/H20841k/H20841n/H20850bkn/H20849t/H20850=fkn/H20849t/H20850. /H2084910a /H20850 Here /H9275/H20841k/H20841n= sgn /H20849A/H20850/H20881A2−B/H20841k/H20841n2=− /H20853/H9275H/H20849/H9275H+/H9275M/H20850 +/H9275M2/H20851/H20841/H9261/H20849/H20841k/H20841,n/H20850/H20841−/H92612/H20849/H20841k/H20841,n/H20850/H20852/H208541/2, /H2084910b /H20850 u/H20841k/H20841n=/H20881A+/H9275/H20841k/H20841n 2/H9275/H20841k/H20841n,v/H20841k/H20841n= − sgn /H20849A/H20850B/H20841k/H20841n /H20841B/H20841k/H20841n/H20841/H20881A−/H9275/H20841k/H20841n 2/H9275/H20841k/H20841n, /H2084910c /H20850 fkn/H20849t/H20850=−i/H9275M/H20853u/H20841k/H20841nRkn/H20849t/H20850+v/H20841k/H20841n/H20851Rkn/H20849t/H20850/H20852/H11569/H20854, /H2084910d /H20850 where Rkn/H20849t/H20850=/H20885 −/H11009/H11009 /H20851h/H20849th/H20850/H20849t,x/H20850m˜kn/H20849x/H20850/H11569exp /H20849−ikx /H20850/H20852dx, /H2084910e /H20850 and following Ref. 28, have phenomenologically introduced magnetic damping /H9253/H20841k/H20841nthrough replacing /H9275/H20841k/H20841nwith/H9275/H20841k/H20841n +i/H9253/H20841k/H20841n.I nE q . /H2084910a /H20850we have omitted the second complex conjugate equation in the system: it returns formally the fre-quency of the same magnon but a negative sign. The solution of Eq. /H2084910a /H20850can be written as follows: b kn/H20849t/H20850=/H20885 0t dt/H11032fkn/H20849t/H11032/H20850exp /H20851−/H20849/H9253/H20841k/H20841n+i/H9275/H20841k/H20841n/H20850/H20849t−t/H11032/H20850/H20852 +bkn/H20849t=0 /H20850exp /H20851−/H20849/H9253/H20841k/H20841n+i/H9275/H20841k/H20841n/H20850t/H20852. /H2084911/H20850 In the state of thermodynamic equilibrium, i.e., for t /H112711//H9253/H20841k/H20841n, the second term disappears. Each collective mode is fully characterized by its integer index “ n” and a continu- ous Bloch wave number k. To estimate the energy carried by each mode in the state of thermodynamic equilibrium weneed to calculate the correlation function of the wave ampli- tudes /H20855b kn/H20849t/H20850bkn/H11569/H20849t/H20850/H20856fort/H112711//H9253/H20841k/H20841n, Sknn/H20849t/H20850=/H20855bkn/H20849t/H20850bkn/H11569/H20849t/H20850/H20856 =/H20885 0t dt/H11032/H20885 0t dt/H11033/H20855fkn/H20849t/H11032/H20850fkn/H20849t/H11033/H20850/H11569/H20856exp /H20851i/H9275/H20849t/H11032−t/H11033/H20850/H20852 /H11003exp /H20851−/H9253/H20841k/H20841n/H208492t−t/H11032−t/H11033/H20850/H20852. /H2084912/H20850According to Eq. /H20849A6 /H20850the autocorrelation function reads /H20855fkn/H20849t/H20850fkn/H20849t/H11032/H20850/H11569/H20856=2C/H9275M2/H9254/H20849t−t/H11032/H20850/H20849u/H20841k/H20841n2+v/H20841k/H20841n2/H20850 =2C/H9275M2/H9254/H20849t−t/H11032/H20850A /H9275/H20841k/H20841n. Here we have made use of Eqs. /H208499/H20850and /H2084910c /H20850. Thus Sknn/H20849t/H20850=2C/H9275M2A /H9275/H20841k/H20841n/H20885 0t dt/H11032exp /H20851−2/H9253/H20841k/H20841nt/H11032/H20852 =C/H9275M2A /H9253/H20841k/H20841n/H9275/H20841k/H20841n/H208511 − exp /H20849−2/H9253/H20841k/H20841nt/H20850/H20852→ t/H112711//H9253C/H9275M2 A /H9253/H20841k/H20841n/H9275/H20841k/H20841n. In thermodynamics it is the occupation number Nkn/H20849t/H20850that describes the thermal energy of the mode identified by itsnumbers kandnin the state of equilibrium. In other words N kn/H11013Sknnand Nkn/H11013Skn/H20849t=/H11009/H20850=C/H9275M2A 2/H9253/H20841k/H20841n/H9275/H20841k/H20841n. /H2084913/H20850 Therefore for the constant Cwe have C=2/H9253/H20841k/H20841n/H9275/H20841k/H20841nNkn /H9275M2A. /H2084914/H20850 On the other hand in the thermodynamic equilibrium the spin-wave amplitude, Eq. /H2084913/H20850, should obey the Rayleigh- Jeans distribution,28thus it should have the form Nkn=kBT /H20841/H9275kn/H20841, /H2084915/H20850 therefore C=2 /H9275M2kBT /H20841A/H20841/H9253/H20841k/H20841n /H2084916a /H20850 and /H20855fkn/H20849t/H20850fkn/H20849t/H11032/H20850/H11569/H20856=4/H9253/H20841k/H20841nkBT /H20841/H9275kn/H20841/H9254/H20849t−t/H11032/H20850. /H2084916b /H20850 Expression /H2084916b /H20850represents a modification of the FDT for the collective DE mode, in the form of a Bloch wave, propa-gating on an array of dipole coupled ferromagnetic stripes.Similar formulas are known since long ago as fluctuation-dissipation relation in classical Brownian motion 15or Nyquist’s noise theorem, relating the mean-square open-circuit thermal noise voltage to its resistance, in electricalengineering. 34As one sees from this expression all peculiari- ties which are characteristic to the Bloch wave origin of col-lective excitations are hidden in the form of wave dispersion /H9275kn. The general form of Eq. /H2084916b /H20850coincides with one known for spin waves in a continuous magnetic medium28 which facilitates understanding. B. Coherence length of thermally excited collective modes Now we investigate the spatial correlation functions char- acterizing distributions of magnetization in magnetostaticM. P. KOSTYLEV AND A. A. STASHKEVICH PHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 054418-4modes at a given temporal frequency . Knowledge of such functions is instrumental for understanding a number of sig-nificant phenomena in spin-wave physics. For example, theyplay a major role in mechanisms of the Brillouin light scat-tering by thermal magnons. To streamline our analytical cal-culations we employed a highly effective mathematical for-malism developed in the field of Fourier optics in the1970s. 35,36 First, we pass from the direct temporal space “ t”t ot h e frequency space “ /H9275” via the direct Fourier transformation. The solution of Eq. /H2084910a /H20850can be rewritten in the frequency domain as follows: bkn/H20849/H9275/H20850=fkn/H20849/H9275/H20850 /H20851/H9253/H20841k/H20841n+i/H20849/H9275/H20841k/H20841n+/H9275/H20850/H20852/H2084917a /H20850 with bkn/H20849/H9275/H20850=Fˆt→/H9275/H20851bkn/H20849t/H20850/H20852= lim /H9008→/H110091 2/H9008/H20885 −/H9008/H9008 dtb kn/H20849t/H20850exp /H20849−i/H9275t/H20850, fkn/H20849/H9275/H20850=Fˆt→/H9275/H20851fkn/H20849t/H20850/H20852. /H2084917b /H20850 Similarly, one can rewrite Eq. /H208492a/H20850in the frequency domain as /H20855h/H9251/H20849th/H20850/H20849x,/H9275/H20850h/H9252/H20849th/H20850/H20849x/H11032,/H9275/H20850/H20856=C/H9254/H20849x−x/H11032/H20850/H9254/H9251/H92521/H20849/H9275/H20850, /H2084918/H20850 where 1 /H20849/H9275/H20850is the Fourier transform of the Dirac delta func- tion in Eq. /H208492a/H20850, as defined in the space of generalized functions.35,36It is equals to the constant 1 on the whole /H9275axis. This allows us to obtain an explicit formula for fkn/H20849/H9275/H20850, fkn/H20849/H9275/H20850=−i/H9275M/H20851u/H20841k/H20841nRkn/H20849/H9275/H20850+v/H20841k/H20841nRkn/H20849/H9275/H20850/H11569/H20852/H20849 19a /H20850 with Rkn/H20849/H9275/H20850=/H20885 −/H11009/H11009 /H20851h/H20849th/H20850/H20849x,/H9275/H20850m˜kn/H20849x/H20850/H11569exp /H20849−ikx /H20850/H20852dx, h/H20849th/H20850/H20849x,/H9275/H20850=Fˆt→/H9275/H20851h/H20849th/H20850/H20849x,t/H20850/H20852. /H2084919b /H20850 Thus the spatial distribution of the magnetization at a given frequency can be written explicitly as b/H20849x,/H9275/H20850=/H20858 n=1/H11009/H20885 −/H9266/T/H9266/T dkb kn/H20849/H9275/H20850m˜kn/H20849x/H20850exp /H20849ikx /H20850, /H2084920/H20850 which allows us to define the corresponding spatial correla- tion function, averaged over the ensemble, in the followingway, /H20855/H20855b/H20849x, /H9275/H20850b/H11569/H20849x+/H9264,/H9275/H20850/H20856x/H20856=/H20883/H20885 −/H11009/H11009 b/H20849x,/H9275/H20850b/H11569/H20849x+/H9264,/H9275/H20850dx/H20884 =/H20885 −/H11009/H11009 /H20855b/H20849x,/H9275/H20850b/H11569/H20849x+/H9264,/H9275/H20850/H20856dx. /H2084921/H20850 Substituting Eq. /H2084920/H20850into Eq. /H2084921/H20850and taking into account the orthogonality of the eigenfunctions /H208497/H20850we obtain /H20855/H20855b/H20849x,/H9275/H20850b/H11569/H20849x+/H9264,/H9275/H20850/H20856x/H20856=/H20858 n/H20858 n/H11032/H20885 −/H9266/T/H9266/T dkexp /H20849−ik/H9264/H20850/H20885 −/H9266/T/H9266/T dk/H11032/H20855bkn/H20849/H9275/H20850bk/H11032n/H11032/H20849/H9275/H20850/H11569/H20856/H20885 −/H11009/H11009 dxm˜kn/H20849x/H20850m˜k/H11032n/H11032/H11569/H20849x+/H9264/H20850exp /H20849i/H20849k−k/H11032/H20850x/H20850. /H2084922/H20850 To begin with, let us find the expression for /H20855bkn/H20849/H9275/H20850bk/H11032n/H11032/H20849/H9275/H20850/H11569/H20856, making use of Eqs. /H2084917a /H20850and /H20849A6 /H20850, /H20855bkn/H20849/H9275/H20850bk/H11032n/H11032/H20849/H9275/H20850/H11569/H20856=/H20855fkn/H20849/H9275/H20850fk/H11032n/H11032/H11569/H20849/H9275/H20850/H20856 /H20851/H9253/H20841k/H20841n+i/H20849/H9275/H20841k/H20841n+/H9275/H20850/H20852/H20851/H9253/H20841k/H11032/H20841n/H11032−i/H20849/H9275/H20841k/H11032/H20841n/H11032+/H9275/H20850/H20852=2C/H9275M2A/H9254/H20849k−k/H11032/H20850/H9254nn/H11032 /H9275/H20841k/H20841n/H20851/H9253/H20841k/H20841n2+/H20849/H9275/H20841k/H20841n+/H9275/H208502/H20852. Thus finally one obtains /H20855/H20855b/H20849x,/H9275/H20850b/H11569/H20849x+/H9264,/H9275/H20850/H20856x/H20856=/H20858 n/H20885 −/H9266/T/H9266/T dkF nk/H20849/H9264/H20850exp /H20849−ik/H9264/H208502CA/H9275M2 /H9275/H20841k/H20841n/H20851/H9253/H20841k/H20841n2+/H20849/H9275/H20841k/H20841n+/H9275/H208502/H20852/H2084923a /H20850 with Fnk/H20849/H9264/H20850=/H20885 −/H11009/H11009 m˜kn/H20849x/H20850m˜kn/H11569/H20849x+/H9264/H20850dx. /H2084923b /H20850 In other words, Fnk/H20849/H9264/H20850is the autocorrelation of the periodic distribution of the dynamic magnetization m˜kn/H20849x/H20850. For small magnetic damping /H20841Vk0n0/H20849g/H20850/H20841//H20849/H9253/H20841k/H20841nT/H20850/H112711/H20849the parameter Vk0n0/H20849g/H20850is explained below /H20850two important approximations, simplifying further calculations, are justified. First, only the kvectors in the vicinity of k0values, that satisfies /H9275/H20841k0/H20841n0+/H9275=0, contribute to the integral /H2084923a /H20850. Therefore we may expand the eigenfrequency /H9275/H20841k/H20841nin a Taylor series, in theSTOCHASTIC PROPERTIES AND BRILLOUIN LIGHT … PHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 054418-5vicinity of k0, keeping only the linear term /H9275/H20841k/H20841n+/H9275=/H11006/H20841Vk0n0/H20849g/H20850/H20841/H20849k−k0/H20850. This approximation is valid not very close to the center and the edges of the first Brillouin zone. /H20849Near these special points Vk0n0/H20849g/H20850→0 and the second-order term of Taylor-series expansion should be taken into account. /H20850Second, the limits of integration can be pushed up to /H20849−/H11009,/H11009/H20850which will allow us to use the method of residues. Thus we obtain /H20855b/H20849x,/H9275/H20850b/H11569/H20849x+/H9264,/H9275/H20850/H20856=2/H9266 /H20841Vk0n0/H20849g/H20850/H20849/H9275/H20850/H20841CA/H9275M2 /H9253/H20841k0/H20841n0/H9275/H20841k0/H20841n0Fn0k0/H20849/H9264/H20850exp/H20875−/H9253k0n0 /H20841Vk0n0/H20849g/H20850/H20841/H20841/H9264/H20841/H20876exp /H20849−ik0/H9264/H20850, /H2084924/H20850 and hence /H20841/H20855b/H20849x,/H9275/H20850b/H11569/H20849x+/H9264,/H9275/H20850/H20856/H20841=2/H9266 /H20841Vk0n0/H20849g/H20850/H20849/H9275/H20850/H20841CA/H9275M2 /H9253/H20841k0/H20841n0/H9275/H20841k0/H20841n0Fn0k0/H20849/H9264/H20850exp/H20875−/H9253k0n0 /H20841Vk0n0/H20849g/H20850/H20841/H20841/H9264/H20841/H20876. /H2084925/H20850 It should be noted that the indices k0andn0indicate that the corresponding values are calculated for k0andn0that satisfy /H9275/H20841k0/H20841n0+/H9275=0. Note that Eqs. /H2084925/H20850and /H2084926/H20850contain only the term of the total solution which originates from the traveling-wave partof the spin-wave excitation Green’s function. 37The term which corresponds to the source reactance was neglected asit is localized at the source and does not contribute to coher-ence of magnetization precession at a distance from thesource. The obtained formula allows one estimating the spin- wave coherence length l c. We define it as the distance at which the value of correlation function is exp /H208491/H20850times smaller than its original value /H20849/H20855b/H20849x,/H9275/H20850b/H11569/H20849x+lc,/H9275/H20850/H20856=exp /H20849 −1/H20850/H20855b/H20849x,/H9275/H20850b/H11569/H20849x,/H9275/H20850/H20856/H20850. Therefore the correlation length /H20849or coherence /H20850length for the collective mode is defined as fol- lows: lc/H20849k,n/H20850=/H20841Vkn/H20849g/H20850/H20841 /H9253/H20841k/H20841n. /H2084926/H20850 If the magnetic damping is small we may measure the coher- ence length in the number of coupled stripes j. Therefore we assume that lc/H20849k,n/H20850=jT, we obtain j=/H20841Vkn/H20849g/H20850/H20841 /H9253/H20841k/H20841nT. /H2084927/H20850 The coherence length is proportional to the group velocity of the collective mode /H20841Vkn/H20849g/H20850/H20841=/H11509/H9275/H20841k/H20841n /H11509k. It may be calculated from Eq. /H2084910b /H20850by taking its derivative over kwhich gives Vkn/H20849g/H20850=−/H9275M/H20875/H9275M 2+/H9275M/H9261/H20849/H20841k/H20841,n/H20850/H20876 /H9275/H20841k/H20841n/H11509/H9261/H20849k,n/H20850 /H11509k. /H2084928/H20850 Moreover, the derivative/H11509/H9261/H20849k,n/H20850 /H11509kin Eq. /H2084928/H20850can be expressed as the following integral:/H11509/H9261/H20849k,n/H20850 /H11509k=/H20885 −/H11009/H11009 dx/H20885 −/H11009/H11009 dx/H11032i/H20849x/H11032−x/H20850m˜kn/H20849x/H20850G/H20849x,x/H11032/H20850m˜kn/H20849x/H11032/H20850 /H11003exp /H20851ik/H20849x/H11032−x/H20850/H20852dx/H11032. /H2084929/H20850 This formula will be used below in numerical calculations. Figure 2shows variation in the coherence length with k X Data9.0e+9 1.0e+10 1.1e+10 1.2e+10 1.3e+10 1.4e+10 1.5e+10 1.6e +Bloch wavenumber k(105rad/cm) 0.00.20.40.60.81.01.21.4 Frequenc y(GHz)9 1 01 11 21 31 41 51 6Coherence length lcin number of periods 024681012141618n=1n=1n=2 n=2n=3 n=4 n=3n=5 n=4 FIG. 2. Dispersion /H20849upper panel /H20850and coherence length of col- lective excitations. Parameters of calculation: stripe width: 350 nm,stripe thickness: 40 nm, stripe separation: 70 nm, saturation mag-netization 4 /H9266M=10 000 G, applied field: 500 Oe, Gilbert magnetic damping constant: /H9251=0.008, gyromagnetic constant /H9253/2/H9266 =2.82 MHz /Oe. nin the figure denotes the mode number. The fundamental mode is n=1.M. P. KOSTYLEV AND A. A. STASHKEVICH PHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 054418-6and/H9275. From this figure one sees that only the lowest /H20849fun- damental /H20850n=1 mode of collective excitations can propagate considerable distances across the periodic structure. As onecan see from the upper panel, this mode is the only modewith considerable dispersion and thus with a considerablegroup velocity. The latter allows this mode to cross a numberof stripes during its relaxation time 1 / /H9253/H20841k/H20841n. The coherence length drops near the middles /H20849k=2l/H9266/T,l=0,1,... /H20850and the edges of Brillouin zones /H20851k=/H208492l+1/H20850/H9266/T/H20852, where Bloch waves represent standing-wave oscillations. It is worth noticing that the decrease in the group velocity till zero toward the middle of the first BZ is very sharp andhappens in a very narrow krange 0–100 rad/cm. That is why the change in the curvature near k=0 is not seen it the upper panel of Fig. 2. Such a narrow range of kvalues where the group velocity changes from the maximum to zero is a sig-nature of strong dipole coupling of stripes in this example.Furthermore, this reflects the fact that across the BZ the di-pole coupling is strongest for the zone middle where themotion of magnetization is quasihomogeneous across thewhole array, as the Bloch wavelength is much larger than thestructure period. With increase in the distance betweenstripes this krange increases and flattening of dispersion near k=0 becomes visible for /H9004/H11022w. Figure 3shows variation in the coherence length with the distance between adjacent stripes /H9004. This graph was calcu- lated for the fundamental collective mode and for a Blochnumber near the middle of the first BZ /H20849k=0.17 /H9266/T/H20850. Withthe increase in the distance, dipole coupling of stripes de- creases. This results in a decrease in the bandwidth of thefirst magnonic band /H20849 /H9275k=/H9266/T,n=1−/H9275k=0,n=1/H20850and, consequently, in the slope of the dispersion /H9275k,n=1/H20849k/H20850. The former is evi- denced by the upper panel of this figure which shows in-crease in frequency for the fundamental mode with decreasein the dipole coupling. At large separations the frequencytends to one for uncoupled stripes for which the width of themagnonic band is zero and the dispersion slope is zero too.The collective mode coherence length is zero. With the in-crease in the dipole coupling the collective dipole field ofstripes pushes the frequency for the middle of BZ down andfor the edge of the first BZ up with respect to the uncoupledstripes /H20849see Fig. 6 in Ref. 29/H20850. The dispersion slope increases and the coherence length grows, respectively. C. BLS intensities Making use of Eq. /H2084925/H20850one can also estimate BLS inten- sities seen at particular angle of incidence of the laser light inthe conventional /H20849reciprocal space-resolved BLS /H20850. The BLS spectroscopy, in its conventional nonmicro-BLS version, is based on the analysis of the intensity of the lightinelastically scattered by a magnon in the inverse k /H20849s/H20850space as a function of the magnon frequency /H9275. The latter is related to the polarization induced through the interaction of theincident light wave and a magnetostatic mode in the follow-ing way: I/H20851k /H20849s/H20850,/H9275/H20852=/H20855E/H20851k/H20849s/H20850,/H9275/H20852E/H11569/H20851k/H20849s/H20850,/H9275/H20852/H20856=/H20855Fˆx→k/H20849s/H20850/H20851P/H20849x/H20850/H11569P/H11569/H20849x/H20850/H20852/H20856. /H2084930/H20850 P/H20849x/H20850/H11569P/H11569/H20849x/H20850=/H20848−/H11009/H11009P/H20849/H9264/H20850P/H11569/H20849x+/H9264/H20850d/H9264corresponds to the autocor- relation of the polarization with its complex conjugate,which is denoted by the symbol /H11569. Here we do not consider the complex tensor nature of the MO interactions thus focus-ing on the stochastic aspect of the problem. One can findcalculations of the effective cross section elsewhere /H20849see, for example, Refs. 38and39/H20850. Typically, in the conventional BLS technique the divergence of the incident optical beam issmall to ensure resolution in the inverse kspace and the dependence of the MO cross section on the angle of inci-dence can be neglected. That is why we define the inducedpolarization, in the scalar approximation, as P/H20849x/H20850 =b/H20849x, /H9275/H20850E/H20849i/H20850/H20849x,/H9275/H20850. In the particular case of a plane incident light wave char- acterized by an in-plane wave vector k/H20849i/H20850, the Bloch wave number kin Eq. /H2084923a /H20850is to be replaced by k/H20849i/H20850+k. Thus one obtains the classic formula38 I/H20851k/H20849s/H20850,/H9275/H20852=/H20885 −/H11009/H11009 d/H9264/H20855b/H20849x,/H9275/H20850b/H11569/H20849x+/H9264,/H9275/H20850/H20856exp /H20853i/H20851k/H20849s/H20850−k/H20849i/H20850/H20852/H9264/H20854, /H2084931/H20850 where the ensemble average is estimated through the FDT. Actually, however the incident optical beam is of finite size both in the direct and inverse space E/H20849i/H20850/H20849x,/H9275/H20850 =D/H20849x/H20850exp /H20849ik/H20849i/H20850x/H20850. Here function D/H20849x/H20850describes the distribu- tion of the optical field in space. ThusXD a t a0 200 400 600 800 1000Frequency (GHz) 10.010.210.410.610.811.0 Stripes eparation (nm)0 200 400 600 800 100 0Coherence length lcin number of periods 0246810121416 FIG. 3. Coherence length of the fundamental collective mode n=1 as a function of strip separation /H9004for a Bloch wave number near the middle of the first Brillouin zone k=0.17/H9266/T. Parameters of calculation are the same as for Fig. 2.STOCHASTIC PROPERTIES AND BRILLOUIN LIGHT … PHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 054418-7I/H20851k/H20849s/H20850,/H9275/H20852=/H20885 −/H11009/H11009 d/H9264/H20885 −/H11009/H11009 dx/H20855D/H20849x/H20850b/H20849x,/H9275/H20850D/H20849x+/H9264/H20850b/H11569/H20849x+/H9264,/H9275/H20850/H20856exp /H20853i/H20851k/H20849s/H20850−k/H20849i/H20850/H20852/H9264/H20854 =Fˆ/H9264→k/H20849s/H20850−k/H20849i/H20850/H20855/H20853D/H20849x/H20850b/H20849x,/H9275/H20850/H11569/H20851D/H20849x/H20850b/H20849x,/H9275/H20850/H20852/H11569/H20854/H20856. /H2084932/H20850 The latter means that in the inverse q=k/H20849s/H20850−k/H20849i/H20850space the Fourier transform of D/H20849x/H20850b/H20849x,/H9275/H20850and its complex conjugate are multiplied. To be specific let us suppose that the light intensity within the optical spot of width dis homogeneous, i.e., D/H20849x/H20850= Rect /H208492x/d/H20850, where Rect /H208732x d/H20874=/H209041 if−d 2/H11021x/H11021d 2 0if x/H11350d 2and x /H11349−d 2./H20841 To begin with, we consider a particular magnetostatic mode ncharacterized by a particular value of the Bloch wave number k. Thus, making use of Eq. /H2084924/H20850, the autocorrelation function in Eq. /H2084932/H20850can be rewritten /H20855D/H20849x/H20850b/H20849x,/H9275/H20850/H11569/H20851D/H20849x/H20850b/H20849x,/H9275/H20850/H20852/H11569/H20856=/H20885 −/H11009/H11009 dxRect/H208732x d/H20874Rect/H208752/H20849x+/H9264/H20850 d/H20876/H20855b/H20849x,/H9275/H20850b/H11569/H20849x+/H9264,/H9275/H20850/H20856 =H/H20849/H9275/H20841k/H20841n/H20850Fnk/H20849/H9264/H20850exp/H20875−/H9253kn /H20841Vkn/H20849g/H20850/H20841/H20841/H9264/H20841/H20876exp /H20849−ik/H9264/H20850/H20875Rect/H208732/H9264 d/H20874/H20002Rect/H208732/H9264 d/H20874/H20876. To simplify notations we have put2/H9266 /H20841Vkn/H20849g/H20850/H20841CA/H9275M2 /H9253/H20841k/H20841n/H9275/H20841k/H20841n=H/H20849/H9275/H20841k/H20841n/H20850. In other words, in Eq. /H2084932/H20850the Fourier operator is applied to a product of three functions of /H9264, which means that this operator returns a double convolution of the correspondingFourier transforms, Fˆ /H9264→q/H20875Rect/H208732/H9264 d/H20874/H20002Rect/H208732/H9264 d/H20874/H20876=d2Sinc2/H208492q/d/H20850, Fˆ/H9264→q/H20877exp/H20875−/H9253kn /H20841Vkn/H20849g/H20850/H20841/H20841/H9264/H20841/H20876exp /H20849−ik/H9264/H20850/H20878=2/H9253kn /H20841Vkn/H20849g/H20850/H20841 /H20849q−k/H208502+/H20875/H9253kn /H20841Vkn/H20849g/H20850/H20841/H208762 and /H20851see Eq. /H2084923b /H20850/H20852 Fˆ/H9264→q/H20851Fnk/H20849/H9264/H20850/H20852=Fˆ/H9264→q/H20851m˜kn/H20849/H9264/H20850/H11569m˜kn/H20849/H9264/H20850/H20852=/H20841M˜kn/H20849q/H20850/H208412 with M˜kn/H20849q/H20850=Fˆ/H9264→q/H20851m˜kn/H20849/H9264/H20850/H20852andq=k/H20849s/H20850−k/H20849i/H20850. The periodic function m˜kn/H20849x/H20850can be regarded as a convo- lution of the distribution of the dynamic magnetization on a single element skn/H20849/H9264/H20850=Rect /H20849/H9264 T/2/H20850m˜kn/H20849/H9264/H20850with a periodic comb of delta functions with Tspacing, known as the Dirac comb and conventionally denoted comb /H20849/H9264 T/H20850, m˜kn/H20849/H9264/H20850=skn/H20849/H9264/H20850/H20002comb/H20873/H9264 T/H20874 with comb /H20873/H9264 T/H20874=/H20858 l=−/H11009/H11009 /H9254/H20849/H9264−lT/H20850. Consequently, its Fourier transform is a product of the re- spective Fourier transformsFˆ/H9264→q/H20851m˜kn/H20849/H9264/H20850/H20852=Skn/H20849q/H20850T 2/H9266comb/H20873q /H9004q/H20874 and consequently Fˆ/H9264→q/H20851m˜kn/H20849/H9264/H20850/H11569m˜kn/H20849/H9264/H20850/H20852=T 2/H9266/H20858 l=−/H11009/H11009 /H20841Skn/H20849q−l/H9004q/H20850/H208412/H9254/H20873q−l/H9004q /H9004q/H20874. Here Skn/H20849q/H20850=Fˆ/H9264→k/H20851skn/H20849/H9264/H20850/H20852and/H9004q=2/H9266/T. We have also used the well-known relation Fˆ/H9264→q/H20851comb /H20849/H9264 T/H20850/H20852=1 /H9004qcomb /H20849q /H9004q/H20850. Thus the first convolution yields /H20841M˜kn/H20849q/H20850/H208412/H20002/H20851d2Sinc2/H208492q/d/H20850/H20852 =T 2/H9266/H20858 l=−/H11009/H11009 /H20841Skn/H20849q−l/H9004q/H20850/H208412/H9254/H20873q−l/H9004q /H9004q/H20874/H20002d2Sinc2/H208492q/d/H20850 =Td2 2/H9266/H20858 l=−/H11009/H11009 /H20841Skn/H20849q−l/H9004q/H20850/H208412Sinc2/H208752/H20849q−l/H9004q/H20850 d/H20876. A second convolution leads to the final result I/H20849q,/H9275/H20850=Td2 /H9266/H9253kn /H20841Vkn/H20849g/H20850/H20841/H20858 l=−/H11009/H11009/H20885 −/H11009/H11009 dq/H11032/H20841Skn/H20849q/H11032−l/H9004q/H20850/H208412 /H11003Sinc2/H208752/H20849q/H11032−l/H9004q/H20850 d/H208761 /H20849q−q/H11032−k/H208502+/H20875/H9253kn /H20841Vkn/H20849g/H20850/H20841/H208762. /H2084933/H20850 The square of the Sinc function, known as Fejer kernel, is a delta sequence. Thus if the width of the beam in Eq. /H2084933/H20850 tends to the infinity d→/H11009, it must be replaced by a DiracM. P. KOSTYLEV AND A. A. STASHKEVICH PHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 054418-8delta function. Due to the filtering properties of the latter the integration in Eq. /H2084933/H20850disappears and one obtains for an infinitely wide beam I/H20849q,/H9275/H20850=I0/H20849k,n/H20850/H20858 l=−/H11009/H11009/H20841Skn/H20849l/H9004q/H20850/H208412 /H20849q−l/H9004q−k/H208502+/H20873/H9253kn /H20841Vkn/H20849g/H20850/H20841/H208742, I0/H20849k,n/H20850=Td2 /H9266/H9253kn /H20841Vkn/H20849g/H20850/H20841. /H2084934a /H20850 In the inverse qspace, each collective mode, due to its peri- odic character, will give rise to multiple responses of Lorent-zian type. The weight of each peak is determined by theFourier coefficients /H20841S kn/H20849l/H9004q/H20850/H208412while the width of each Lorentzian line depends entirely on the coherence length /H20851see Eqs. /H2084926/H20850and /H2084934a /H20850/H20852. This not surprising, since the coherence length lc/H20849k,n/H20850describes the spatial localization of coherent scattering sources. In the general case, however each Lorentzian line is smeared by the finite angular spectrum of the incident lightway, which is mathematically taken into account through aconvolution of the Lorentzian function and the Fejer kernel/H20851see Eq. /H2084933/H20850/H20852. Without unnecessary loss of accuracy, the main lobe of the latter can be replaced by an equivalentrectangular function /H20849see below /H20850, which makes the calcula- tion straightforward, extremely rapid and reliable. The expression /H2084934a /H20850can be rewritten in the domain of temporal frequencies /H9275, I/H20849q,/H9275/H20850=I0/H20849k,n/H20850/H20841Vkn/H20849g/H20850/H208412 /H9253/H20841k/H20841n2+/H20849/H9275/H20841k/H20841n+/H9275/H208502/H20858 l/H20841Skn/H20849l/H9004q/H20850/H208412. /H2084934b /H20850 The BLS lines for all the responses will be centered at the frequency of resonance excitation of the mode by a thermalsource /H20841 /H9275/H20841k/H20841n/H20841=/H20841/H9275/H20841. The responses will be seen at incidence angles which correspond to the transferred wave numbers k+l/H9266/T,l=0 ,/H110061,/H110062,... /H2084935/H20850 /H20849Recall, − /H9266/T/H11021k/H11021/H9266/T/H20850. In which Brillouin zone /H20849l+1 −/H9266/T+2/H9266l/T/H11021q/H11021/H9266/T+2/H9266l/T/H20850a mode gives a maximum response depends on the mode eigenprofile m˜kn/H20849x/H20850/H20849through Skn/H20850. The fundamental mode is characterized by a quasiho- mogeneous distribution of dynamic magnetization across thestripes and thus gives the maximum response in the first BZ.The next mode is antisymmetric and has one node across thestripe width. Its spectrum is obviously composed from oddharmonics of the structure period 2 /H9266l/T,l=/H110061,/H110063,... with the maximum response in the second BZ /H20849l+1=2 /H20850, etc. A real BLS setup has a finite qresolution as it collects light from a finite range of incidence angles /H9004/H9258. Then the corresponding range of uncertainty /H9004kinq/H20849and thus in k/H20850 results in broadening of the BLS line. For simplicity wemay assume that within /H9004 /H9258intensity of all spectral compo- nents of light incident on the sample is the same. Thus in thescattered light resonance lines for eigenexcitations with fre-quencies ranging from /H9275k−/H9004k/2nto/H9275k+/H9004k/2nwill be present with equal amplitude. Then in order to account for this effect one has to substitute the term1 /H9253/H20841k/H20841n2+/H20849/H9275/H20841k/H20841n+/H9275/H208502in Eq. /H2084934b /H20850byits integral over the range of uncertainty in /H9004k:U/H20849/H9004k/H20850 =/H20848k−/H9004k/2k+/H9004k/2 dk/H11032 /H9253/H20841k/H11032/H20841n2+/H20849/H9275/H20841k/H11032/H20841n+/H9275/H208502. Approximating /H9275k/H11032n=/H9275kn+Vkn/H20849g/H20850/H20849k/H11032−k/H20850 one obtains U/H20849/H9004k/H20850=/H20875arctan/H20873/H9275kn+/H9275+Vkn/H20849g/H20850/H9004k /H9253kn/H20874 + arctan /H20873/H9275+/H9275kn+Vkn/H20849g/H20850/H9004k /H9253kn/H20874/H20876//H20849/H9253knVkn/H20849g/H20850/H20850./H2084936/H20850 Figure 4shows plots of I/H20849q,/H9275/H20850in which gray scale is for the mode intensity I. It was calculated using Eq. /H2084934a /H20850with the term1 /H9253/H20841k/H20841n2+/H20849/H9275/H20841k/H20841n+/H9275/H208502substituted by U/H20849k/H20850as defined by Eq. /H2084936/H20850. The uncertainty in the transferred wave number was taken 5% of the width of a Brillouin zone 2 /H9266/T. The lower panel of this figure is for stripes placed far apart from one another.Dipole coupling of stripes in this geometry is small for allmodes which results in a small group velocity and, conse-quently in a small coherence length. As a result, all themodes are practically dispersionless, as previously seen innumerous experiments. 16,19,21,40 The middle panel is for strongly dipole coupled stripes. From this panel one clearly sees the opposite tendency: thelowest /H20849fundamental /H20850mode gives rise to a BLS response in the first BZ, the second one in the second BZ, and so on.Cross sections of this figure along the lines q=0.2/H1100310 5and 0.6/H11003105rad /cm are given in Fig. 5. Positions of the respec- tive cross sections are shown in Fig. 4by the respective vertical lines. The middle panel is for an array of strongly dipole coupled stripes but consisting of wider stripes than for theupper panel. The width of the stripe is chosen such as it iscomparable with the free propagation path in an unstructured film /H9253Vunstruct/H20849g/H20850. From this figure one sees the BLS responses for the highest-order modes practically collapse into a con-tinuous dispersion law for an unstructured film. This phe-nomenon was previously observed on uncoupled stripes. 20 Recall that the highest-order modes have a negligible widthof the magnonic band and thus are not practically dipolecoupled. BLS intensity for stripes at a large distance fromeach other is given for comparison in the lower panel. In the last series of calculations we have estimated BLS intensity for different collection solid angles which introducedifferent uncertainties in the transferred wave numbers. Thiswork was partially inspired by Ref. 30in which BLS from a periodical array of elongated nanodots was experimentallystudied. To apply our theory we treat rows of nanodots as500-nm-wide stripes of infinite length with 250 nm inter-stripe separation. Our calculation shows that these effectivestripes are efficiently dipole coupled. The collective modedispersion as seen by BLS with a small collection solid angleis shown in Fig. 6, upper right panel. The upper left panel demonstrate the same calculation for a collection angle which corresponds to 1 2of the width of the BZ for this qua- sicrystal lattice /H20849/H9004q=/H9266/T/H20850. One sees a broad intensity peak for the main mode and much narrower peaks for the higher-order modes. Cross sections of the 2D plots in the upperpanels along the edge of the first BZ q= /H9266/Tare shown in the lower panels.STOCHASTIC PROPERTIES AND BRILLOUIN LIGHT … PHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 054418-9The lowest collective mode gives rise to the broadest in- tensity peak for /H9004q=/H9266/T. This mode is characterized by the largest correlation length due to the largest group velocity/H20849dispersion slope /H20850. Furthermore, it is characterized by the largest frequency band /H20849magnonic band /H20850. The closest higher- order mode forms a much smaller magnonic band. Further-more, it is almost dispersionless which results in a muchsmaller correlation length for this mode. Thus, in our calcu-lation decreasing the collection angle for the case of themode with a large correlation length results in a considerablenarrowing of the BLS peak and decrease in its intensity. Onthe contrary, the first higher-order mode does not exhibit aconsiderable change in the peak width with decrease in thecollection angle. Importantly, the peak intensity decreasesconsiderably similar to the lowest-order mode. FIG. 4. “Intensities” of collective modes as seen by Brillouin light scattering technique for different stripe widths wand separa- tions d. Upper panel: w=350 nm and /H9004=70 nm. Middle panel: w=1050 nm and /H9004=70 nm. Lower panel: w=/H9004=350 nm. Laser beam width d=/H11009. Brighter gradations of gray correspond to larger intensity. The other parameters of calculation are the same as in Fig.2. The Gilbert damping constant entering the expression for mag- netic damping /H9253nk=/H9251/H9275nkis as for permalloy /H9251=0.008. /H20849The spot structure seen in the data is an artefact of presentation of calculateddata by the plotting software used. It is connected with discreetnessof input data for the software. /H20850Frequenc y(GHz)7 8 9 1 01 11 21 3Intens ity(arb.u nit.) 024681012 FIG. 5. Cross sections of the 2D plot in Fig. 4, middle panel, along the lines q=0.2/H11003105rad /cm /H20849solid line /H20850and 0.6 /H11003105rad /cm /H20849dashed line /H20850. Positions of the respective cross sec- tions are shown in Fig. 4by the respective vertical lines. FIG. 6. Intensities for different collecting lens apertures. Left panels: the lens collects the light from the solid angle corresponding to1 2of the Brillouin zone. Right panel: 1/10 of the Brillouin zone. Upper panels: intensities. Lower panels: cross sections of the upperpanels along the edge of the first Brillouin zone /H9266/T. Brightness scale is the same for both upper panels.M. P. KOSTYLEV AND A. A. STASHKEVICH PHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 054418-10Thus the behavior of the peak amplitude is in good agree- ment with Ref. 30, which is not the case for the evolution of the peak width. We suggest the following explanation for thisdisagreement. First, the array in Ref. 30is a set of nanodots. The nanodots as 2D objects are characterized by a muchricher spectrum of eigenoscillations with broader frequencybands than the quasi-1D nanostripes /H20849see, e.g., Ref. 41/H20850. Therefore, increasing the collection angle results in collect-ing a BLS response from a larger number of eigenexcitationsthus from a larger frequency band. If the dynamics is drivenby a spectrally narrow source, like a microwave generator,the increase in the collection angle does not result in achange in the peak width, as the peak width is given by thelinewidth of the microwave generator, which is negligible,and the instrumental linewidth of the Sandercock interferom-eter. One has to note that a periodic stripe array represents a diffraction lattice for the incident laser light. In particular, theauthors of Ref. 30note that the diffracted beams originating from the patterned sample are clearly visible to the nakedeye. Thus multiple maxima of diffraction of the laser light inreflection can be formed. This scattering is “elastic” as thefrequency of light is conserved. In the backscattering geom-etry of BLS experiment the light which has elastically scat-tered into all orders of diffraction n dcan then scatter from all harmonics m˜knlof the collective modes. This modifies the resonance scattering condition /H2084935/H20850. The more precise condi- tion for resonance scattering of the light reads, q=k+2/H9266/H20849l−nd/H20850/T. /H2084937/H20850 Following this condition inelastically scattered light col- lected at an incidence angle /H9258=arcsin /H20853/H20851k+2/H9266/H20849l −nd/H20850/T/H20852/H20849/H9261las/4/H9266/H20850/H20854will represent a combination of responses from all orders of elastic diffraction ndand of inelastic scat- tering from all harmonics lof the collective modes which satisfy the above condition for /H9258. This reflects the double- scattering nature of this contribution: at the first stage, thelight is diffracted elastically by the relief lattice and after thisthe second MO scattering occurs. In the present calculationfor simplicity reasons we do not account for this effect, as forahighly pronounced collective behavior the stripes should be closely spaced /H20849/H11021200 nm apart /H20850. This means that the elastic scattering is dominated by near-field mechanisms, whichmakes it relatively inefficient. As a final note for this section we now discuss validity of our theory of the magneto-optical interaction. Its obviouslimitation is its scalar character. An important consequenceof this is a loss of the so-called Stokes-anti-Stokes asymme-try of BLS peaks for the Damon-Eschbach wave. 42,43This effect represents a difference in the BLS amplitudes for thepositive and negative frequencies. This difference is seen,e.g., in Fig. 5 of Ref. 22. One can separate two contributions to this effect. One is connected to the fact that the magneto-optical interaction isdescribed by a tensor magnitude: the magneto-optical tensor. Its action on the circularly polarized vector amplitude of dy- namic magnetization depends on the direction of the trans-ferred wave number. An interested reader can find an exten-sive discussion concerning this point in Ref. 44.A sw ed onot include the magneto-optical tensor into the derivation of Eq. /H2084934a /H20850, and treat the magneto-optical interaction as a sca- lar, we loose this contribution to the Stokes/anti-Stokesasymmetry. The second contribution to the Stokes-anti-Stokes asym- metry appears in experiments for larger wave numbers suchaskLis on order of 1. This contribution is related to the surface character of the Damon-Eshbach wave and to theskin depth law valid for the optical field in the sample. Insimple words, the thickness profile of the Damon-Eshbachwave propagating along the film surface facing the incidentoptical beam has a larger overlap integral with the opticalfield than the mode profile for the wave localized at the sec-ond film surface. This contribution is not included in ourtheory either, since to include it one should have treated themagnetization dynamics in the stripes thickness resolved, asit was done in a recent paper. 45In the present paper, to keep the results simple we use the simple quasi-one-dimensionaldescription of the magnetization dynamics /H20849see discussion in the beginning of Sec. II A /H20850. This description is valid for kL /H110210.5. 46For these wave numbers the Damon-Eshbach wave localization at the film surfaces is not important and is ne-glected from very beginning of the derivation. III. CONCLUSION Formation of collective magnetostatic modes via dipolar coupling between individual elements is the main physicalmechanism underlying magnonic wave phenomena in peri-odic ferromagnetic structures. In the present paper, this fun-damental problem has been addressed for the case of ther-mally driven MSW on a one-dimensional array of magneticstripes. It has been shown that partially phase-correlated oscilla- tions localized on individual stripes can be regarded as anensemble of individual harmonic oscillators interpretable interms of independent degrees of freedom of the magneticsystem subject to the low-energy Rayleigh-Jeans statistics. Further theoretical analysis, based on the spatial correla- tion approach, has revealed the importance of the parameterl cknown as correlation or coherence length of a Bloch mode, driven by a thermal “Langevin magnetic source.” The latterdescribes the number of dipole coupled individual oscilla-tions localized on individual stripes, whose phases are effec-tively correlated according to the Bloch wave number q, im- posed by the collective mode. Numerical simulations of the BLS spectra, based on this approach, have shown that the nth Bloch mode in strongly coupled stripes contributes mainly to the scattering in the nth Brillouin zone. This is not the case for weakly coupledstripes with higher values of interwire spacing. In such ge-ometries l ccan decrease drastically and, as a result, for ex- ample, the fundamental Bloch mode will contribute signifi-cantly to the scattering in several lowest Brillouin zones. Ourcalculations have also confirmed numerically the noncoher-ent wide-angle character of the BLS, demonstrated experi-mentally in Ref. 30. ACKNOWLEDGMENTS Support by the Australian Research Council and LPMTP,STOCHASTIC PROPERTIES AND BRILLOUIN LIGHT … PHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 054418-11Université Paris-13 is gratefully acknowledged. APPENDIX Let us try to estimate the correlation function, describing the Langevin force exciting elliptically polarized Blochmodes, which is given below. This expression will be ex-tremely useful in the context of the main text of the paper /H20855f kn/H20849t/H20850fk/H11032n/H11032/H20849t/H11032/H20850/H11569/H20856=/H9275M2/H20851u/H20841k/H20841nu/H20841k/H11032/H20841n/H11032/H20855Rkn/H20849t/H20850Rk/H11032n/H11032/H20849t/H11032/H20850/H11569/H20856 +v/H20841k/H20841nv/H20841k/H11032/H20841n/H11032/H20855Rkn/H20849t/H20850/H11569Rk/H11032n/H11032/H20849t/H11032/H20850/H20856 +u/H20841k/H20841nv/H20841k/H11032/H20841n/H11032/H20855Rkn/H20849t/H20850Rk/H11032n/H11032/H20849t/H11032/H20850/H20856 +v/H20841k/H20841nu/H20841k/H11032/H20841n/H11032/H20855Rkn/H20849t/H20850/H11569Rk/H11032n/H11032/H20849t/H11032/H20850/H11569/H20856/H20852./H20849A1 /H20850Each particular correlation function, out of four, can be evaluated independently. We will begin with the first and thethird. /H20855R kn/H20849t/H20850Rk/H11032n/H11032/H20849t/H11032/H20850/H11569/H20856=/H20885 −/H11009/H11009/H20885 −/H11009/H11009 dxdx /H11032mkn/H11569/H20849x/H20850mk/H11032n/H11032/H20849x/H11032/H20850 /H11003/H20855h/H20849th/H20850/H20849x,t/H20850h/H20849th/H20850/H11569/H20849x/H11032,t/H11032/H20850/H20856, /H20849A2a /H20850 /H20855Rkn/H20849t/H20850Rk/H11032n/H11032/H20849t/H11032/H20850/H20856=/H20885 −/H11009/H11009/H20885 −/H11009/H11009 dxdx /H11032mkn/H11569/H20849x/H20850mk/H11032n/H11032/H11569/H20849x/H11032/H20850 /H11003/H20855h/H20849th/H20850/H20849x,t/H20850h/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856. /H20849A2b /H20850 In Eq. /H20849A2 /H20850a key role is played by the averaged expressions within the brackets, /H20855h/H20849th/H20850/H20849x,t/H20850h/H20849th/H20850/H11569/H20849x/H11032,t/H11032/H20850/H20856=/H20855/H20851hx/H20849th/H20850/H20849x,t/H20850+ihy/H20849th/H20850/H20849x,t/H20850/H20852/H20851hx/H20849th/H20850/H20849x/H11032,t/H11032/H20850−ihy/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20852/H20856 =/H20855hx/H20849th/H20850/H20849x,t/H20850hx/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856+/H20855hy/H20849th/H20850/H20849x,t/H20850hy/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856+i/H20855hx/H20849th/H20850/H20849x,t/H20850hy/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856+/H20855hy/H20849th/H20850/H20849x,t/H20850hx/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856, /H20849A3a /H20850 /H20855h/H20849th/H20850/H20849x,t/H20850h/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856=/H20855/H20851hx/H20849th/H20850/H20849x,t/H20850+ihy/H20849th/H20850/H20849x,t/H20850/H20852/H20851hx/H20849th/H20850/H20849x/H11032,t/H11032/H20850+ihy/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20852/H20856 =/H20855hx/H20849th/H20850/H20849x,t/H20850hx/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856−/H20855hy/H20849th/H20850/H20849x,t/H20850hy/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856+i/H20855hx/H20849th/H20850/H20849x,t/H20850hy/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856+/H20855hy/H20849th/H20850/H20849x,t/H20850hx/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856. /H20849A3b /H20850 Here we used the expression of the Langevin force in circular polarizations h/H20849x,t/H20850=hx/H20849x,t/H20850+ihy/H20849x,t/H20850/H20851see Eq. /H208493/H20850/H20852. It should be reminded that the Cartesian components of the thermal magnetic field are purely real. Making use of Eq. /H208492/H20850one can rewrite Eq. /H20849A3 /H20850as /H20855h/H20849th/H20850/H20849x,t/H20850h/H20849th/H20850/H11569/H20849x/H11032,t/H11032/H20850/H20856=2C/H9254/H20849x−x/H11032/H20850/H9254/H20849t−t/H11032/H20850, /H20849A4a /H20850 /H20855h/H20849th/H20850/H20849x,t/H20850h/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856=0 . /H20849A4b /H20850 Moreover /H20855h/H20849th/H20850/H11569/H20849x,t/H20850h/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856=/H20855h/H20849th/H20850/H20849x,t/H20850h/H20849th/H20850/H11569/H20849x/H11032,t/H11032/H20850/H20856/H11569=2C/H9254/H20849x−x/H11032/H20850/H9254/H20849t−t/H11032/H20850, /H20849A4c /H20850 /H20855h/H20849th/H20850/H11569/H20849x,t/H20850h/H20849th/H20850/H11569/H20849x/H11032,t/H11032/H20850/H20856=/H20855h/H20849th/H20850/H20849x,t/H20850h/H20849th/H20850/H20849x/H11032,t/H11032/H20850/H20856/H11569=0 . /H20849A4d /H20850 Inserting Eqs. /H20849A4a /H20850and /H20849A4b /H20850in Eqs. /H20849A2a /H20850and /H20849A2b /H20850, respectively, one obtains /H20855Rkn/H20849t/H20850Rk/H11032n/H11032/H20849t/H11032/H20850/H11569/H20856=2C/H9275M2/H20885 −/H11009/H11009/H20885 −/H11009/H11009 dxdx /H11032mkn/H11569/H20849x/H20850mk/H11032n/H11032/H20849x/H11032/H20850/H9254/H20849x−x/H11032/H20850/H9254/H20849t−t/H11032/H20850 =2C/H9275M2/H9254/H20849t−t/H11032/H20850/H20885 −/H11009/H11009 dxmkn/H11569/H20849x/H20850mk/H11032n/H11032/H20849x/H20850=2C/H9275M2/H9254/H20849t−t/H11032/H20850/H9254/H20849k−k/H11032/H20850/H9254nn/H11032, /H20849A5a /H20850 /H20855Rkn/H20849t/H20850Rk/H11032n/H11032/H20849t/H11032/H20850/H20856=0 . /H20849A5b /H20850 Similarly /H20855Rkn/H20849t/H20850/H11569Rk/H11032n/H11032/H20849t/H11032/H20850/H20856=/H20855Rkn/H20849t/H20850Rk/H11032n/H11032/H20849t/H11032/H20850/H11569/H20856/H11569=2C/H9275M2/H9254/H20849t−t/H11032/H20850/H9254/H20849k−k/H11032/H20850/H9254nn/H11032, /H20849A5c /H20850 /H20855Rkn/H11569/H20849t/H20850Rk/H11032n/H11032/H11569/H20849t/H11032/H20850/H20856=/H20855Rkn/H20849t/H20850Rk/H11032n/H11032/H20849t/H11032/H20850/H20856/H11569=0 . /H20849A5d /H20850 Inserting Eqs. /H20849A5a /H20850–/H20849A5d /H20850into Eq. /H20849A1 /H20850and taking into account the orthonormality of the eigenfunctions Eq. /H208497/H20850one finally obtainsM. P. KOSTYLEV AND A. A. STASHKEVICH PHYSICAL REVIEW B 81, 054418 /H208492010 /H20850 054418-12/H20855fkn/H20849t/H20850fk/H11032n/H11032/H20849t/H11032/H20850/H11569/H20856=/H20851u/H20841k/H20841nu/H20841k/H11032/H20841n/H110322C/H9275M2/H9254/H20849t−t/H11032/H20850/H9254/H20849k−k/H11032/H20850/H9254nn/H11032+v/H20841k/H20841nv/H20841k/H11032/H20841n/H110322C/H9275M2/H9254/H20849t−t/H11032/H20850/H9254/H20849k−k/H11032/H20850/H9254nn/H11032/H20852 =2C/H9275M2/H9254/H20849t−t/H11032/H20850/H9254/H20849k−k/H11032/H20850/H9254nn/H11032/H20849u/H20841k/H20841n2+v/H20841k/H20841n2/H20850=2AC/H9275M2 /H9275/H20841k/H20841n/H9254/H20849t−t/H11032/H20850/H9254/H20849k−k/H11032/H20850/H9254nn/H11032. /H20849A6 /H20850 Similar calculations are applicable for the correlation function in the frequency domain /H20855fkn/H20849/H9275/H20850fk/H11032n/H11032/H11569/H20849/H9275/H20850/H20856=/H9275M2/H20877u/H20841k/H20841nu/H20841k/H11032/H20841n/H11032/H20885 −/H11009/H11009/H20885 −/H11009/H11009 mkn/H20849x/H20850/H11569mk/H11032n/H11032/H20849x/H11032/H20850/H20855/H20851hx/H20849th/H20850/H20849x,/H9275/H20850+ihy/H20849th/H20850/H20849x,/H9275/H20850/H20852/H20851hx/H20849th/H20850/H20849x/H11032,/H9275/H20850−ihy/H20849th/H20850/H20849x/H11032,/H9275/H20850/H20852/H20856dxdx /H11032 +v/H20841k/H20841nv/H20841k/H11032/H20841n/H11032/H20885 −/H11009/H11009/H20885 −/H11009/H11009 mkn/H20849x/H20850mk/H11032n/H11032/H20849x/H11032/H20850/H11569/H20855/H20851hx/H20849th/H20850/H20849x,/H9275/H20850−ihy/H20849th/H20850/H20849x,/H9275/H20850/H20852/H20851hx/H20849th/H20850/H20849x/H11032,/H9275/H20850+ihy/H20849th/H20850/H20849x/H11032,/H9275/H20850/H20852/H20856dxdx /H11032 +u/H20841k/H20841nv/H20841k/H11032/H20841n/H11032/H20885 −/H11009/H11009/H20885 −/H11009/H11009 mkn/H20849x/H20850/H11569mk/H11032n/H11032/H20849x/H11032/H20850/H11569/H20855/H20851hx/H20849th/H20850/H20849x,/H9275/H20850+ihy/H20849th/H20850/H20849x,/H9275/H20850/H20852/H20851hx/H20849th/H20850/H20849x/H11032,/H9275/H20850+ihy/H20849th/H20850/H20849x/H11032,/H9275/H20850/H20852/H20856dxdx /H11032 +v/H20841k/H20841nu/H20841k/H11032/H20841n/H11032/H20885 −/H11009/H11009/H20885 −/H11009/H11009 mkn/H20849x/H20850mk/H11032n/H11032/H20849x/H11032/H20850/H20855/H20851hx/H20849th/H20850/H20849x,/H9275/H20850−ihy/H20849th/H20850/H20849x,/H9275/H20850/H20852/H20851hx/H20849th/H20850/H20849x/H11032,/H9275/H20850−ihy/H20849th/H20850/H20849x/H11032,/H9275/H20850/H20852/H20856dxdx /H11032/H20878 which leads finally to /H20855fkn/H20849/H9275/H20850fk/H11032n/H11032/H11569/H20849/H9275/H20850/H20856=2AC/H9275M2 /H9275/H20841k/H20841n1/H20849/H9275/H20850/H9254/H20849k−k/H11032/H20850/H9254nn/H11032. /H20849A7 /H20850 1G. 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PhysRevLett.124.117202.pdf

Coherent Spin Pumping in a Strongly Coupled Magnon-Magnon Hybrid System Yi Li,1,2Wei Cao ,3Vivek P. Amin,4,5Zhizhi Zhang ,2,6Jonathan Gibbons,2Joseph Sklenar,7John Pearson ,2 Paul M. Haney,5Mark D. Stiles,5William E. Bailey,3,*Valentine Novosad,2Axel Hoffmann,2,‡and Wei Zhang1,2,† 1Department of Physics, Oakland University, Rochester, Michigan 48309, USA 2Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 3Materials Science and Engineering, Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA 4Maryland Nanocenter, University of Maryland, College Park, Maryland 20742, USA 5Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 6School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan 430074, China 7Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48202, USA (Received 1 September 2019; accepted 23 January 2020; published 17 March 2020) We experimentally identify coherent spin pumping in the magnon-magnon hybrid modes of yttrium iron garnet/permalloy (YIG/Py) bilayers. By reducing the YIG and Py thicknesses, the strong interfacial exchange coupling leads to large avoided crossings between the uniform mode of Py and the spin wavemodes of YIG enabling accurate determination of modification of the linewidths due to the dampinglike torque. We identify additional linewidth suppression and enhancement for the in-phase and out-of-phase hybrid modes, respectively, which can be interpreted as concerted dampinglike torque from spin pumping.Furthermore, varying the Py thickness shows that both the fieldlike and dampinglike couplings vary like 1=ffiffiffiffiffiffit Pyp, verifying the prediction by the coupled Landau-Lifshitz equations. DOI: 10.1103/PhysRevLett.124.117202 Coherent phenomena have recently become an emerging topic for information processing with their success inquantum computing [1,2]. In spintronics, exchange- induced magnetic excitations, called spin waves, or mag-nons [3,4], are good candidates for coherent information processing where information can be encoded by both the amplitude and the phase of spin waves. For example, theinterference of coherent spin waves can be engineered forspin wave logic operations [5–7], the coherent interaction of spin-torque oscillators leads to mutual synchronization[8–13], which can be applied in artificial neural networks [14,15] . and the coherent coupling between magnons and microwave cavities [16–23]opens up new opportunities for magnon-based quantum information science [24,25] . Recently, strong coupling between two magnonic systems has enabled excitations of forbidden spin wavemodes [26–28]and high group velocity of propagating spin waves [29,30] . The coupling is dominated by the exchange interaction at the interface of the magnetic bilayers, providing a new pathway to coherently transfer magnon excitations between two magnetic systems possessingdistinctive properties: from conductor to insulator, fromuniform to nonuniform mode, and from high-damping tolow-damping systems. However, the underlying physicalmechanisms of the coupling are still not fully understood. First, what are the key parameters that dictate the coupling efficiency and enable one to reach the strong-couplingregime? Second, with the interfacial exchange couplingacting as a fieldlike torque, is there a dampinglike torque associated with spin pumping [31–34]? To resolve both questions, large separations of the two hybrid modes arerequired in order to quantitatively analyze the coupling mechanism. The second question is also important for optimizing the coherence of spin wave transfer in hybridsystems. Furthermore, the parasitic effect on the incoherentspin current from the conduction band is absent [35–37] when using magnetic insulators such as yttrium iron garnet(Y 3Fe5O12, YIG) [30,38,39] , which facilitates the study of spin pumping coherency. In this work, we study YIG/permalloy (Ni 80Fe20, Py) bilayers. By using much thinner YIG and Py films than studied in previous works [26,28] , we achieve an exchange- induced separation of the two hybrid modes much larger than their linewidths, allowing us to study the evolution of their linewidths in the strong-coupling regime. We find apronounced suppression of the total linewidth for the in- phase hybrid modes and a linewidth enhancement for the out-of-phase hybrid modes. The linewidths can be under- stood from the Landau-Lifshitz-Gilbert (LLG) equation with interfacial exchange coupling and mutual spin pump- ing, which provide the fieldlike and dampinglike interlayer coupling torques, respectively. Furthermore, the thicknessdependence of the two coupling strengths agrees with the modeling of coupled LLG equations with mutual spin pumping. The sign of the fieldlike torque also reconfirms that the YIG and Py are coupled antiferromagnetically [26]. Our results provide important insights for improving thePHYSICAL REVIEW LETTERS 124, 117202 (2020) 0031-9007 =20=124(11) =117202(6) 117202-1 © 2020 American Physical Societycoupling strength and coherence in magnon-magnon hybrid systems and pave the way for coherent information process- ing with exchange-coupled magnetic heterostructures. The samples consist of YIG ð100nmÞ=PyðtPyÞbilayers where tPyvaries from 5 to 60 nm. YIG(100 nm) films were deposited by magnetron sputtering from a YIG target onto Gd3Ga5O12ð111Þsubstrates and annealed in air at 850°C for 3 h to reach low-damping characteristics [40]. Before the deposition of Py films on top of YIG, the YIG surfaces were ion milled in situ for 1 min in order to enable good exchange coupling between Py and YIG [41]. For each Py thickness, one additional reference Py film was deposited on a Si =SiO 2substrate during the same deposition. The hybrid magnon dynamics were characterized by broadband ferromagnetic resonance with field modulationon a coplanar waveguide [Fig. 1(a)]. An in-plane magnetic field H Bsaturates both the YIG and Py magnetizations. Their Kittel modes, which describe spatially uniform magnetization precession, are formulated as ω2=γ2¼μ2 0HrðHrþMsÞ, where ωis the mode frequency, γ=2π¼ðgeff=2Þ× 27.99GHz=T is the gyromagnetic ratio, Hris the resonancefield, and Msis the magnetization [42]. For YIG, the spatially nonuniformperpendicularstandingspinwave(PSSW)modes can also be measured. An effective exchange field Hexwill lower the resonance field by μ0HexðkÞ¼ð 2Aex=M sÞk2, where Aexis the exchange stiffness, k¼nπ=t,nlabels the index of PSSW modes, and tis the film thickness [43]. Figure 1(b)shows the line shapes of the resonance fields for the first three resonance modes of YIG ( n¼0, 1, 2) and the Py uniform mode ( n¼0) measured for tPy¼9nm. For illustration, the YIG ( n¼0) resonance is shifted to zero field. An avoided crossing is clearly observed when the Py uniform mode is degenerate with the YIG ( n¼2) mode. This is due to the exchange coupling at the YIG/Py interface [26–28]providing a fieldlike coupling torque. Both in-phase and out-of-phase YIG/Py hybrid modes are strongly excitedbecause the energy of the Py uniform mode is coherentlytransferred to the YIG PSSW modes through the interface [26]. The full-range frequency dependencies of the extracted resonance fields are plotted in Fig. 1(c). To analyze the two hybrid modes, we analyze our results with two independent Lorentzians because it facilitates a transparent physical picture and the fit line shapes agree well with our measure-ments. The mode crossing happens at ω c=2π¼9.4GHz (black dashed line), which corresponds to the minimal resonance separation of the two hybrid modes. Fitting to the Kittel equation, we extract μ0MYIGs¼0.21T, μ0MPy s¼0.86T. From the exchange field offset as shown in Fig. 1(b), an exchange stiffness Aex¼2.6pJ=mi s calculated for YIG, which is similar to previous reports [44]. The avoided crossing can be fitted to a phenomenologi- cal model of two coupled harmonic oscillators, as pre-viously shown in magnon polaritons [16–18,20] : μ 0H/C6c¼μ0HYIGrþHPy r 2/C6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi/C18 μ0HYIGr−HPy r 2/C192 þg2cs ;ð1Þ where HYIGðPyÞ r is the resonance field of YIG (Py) and gcis the interfacial exchange coupling strength. HYIGrandHPy r are both functions of frequency and are equal at ωc. Note that for in-plane biasing field, the resonance field isnonlinear to the excitation frequency. This nonlinearity will be accounted for in the analytical reproduction of Eq.(1). The fitting yields g c¼8.4mT for tPy¼9nm. Next, we focus on the linewidths of the YIG-Py hybrid modes. Figure 2(a) shows the line shape of the two hybrid modes for tPy¼7.5nm at ωc=2π¼9.4GHz (same value as for 9-nm Py). These two eigenmodescorrespond to the in-phase and out-of-phase magnetization precession of Py and YIG with the same weight, so they should yield the same total intrinsic damping. Nevertheless,a significant linewidth difference is observed, with the extracted full width at half maximum linewidth μ 0ΔH1=2 varying from 3.5 mT for the lower field resonance to 8.0 mT for the higher field resonance. Figure 2(b) shows(a) (b) (c) FIG. 1. (a) Illustration of the magnetization excitations in the YIG/Py bilayer with a coplanar waveguide. (b) Line shapes of theYIGð100nmÞ=Pyð9nmÞsample for the first three resonance modes of YIG and the uniform mode of Py. The field axis isshifted so that the resonance field of the YIG ðn¼0Þmode is zero. (c) Unshifted evolution of the four modes in (b). Curvesshow the fits as uncoupled modes. The vertical dashed line denotes where the YIG ðn¼2Þand Py ðn¼0Þmodes cross on the frequency axis at ω c=2π¼9.4GHz.PHYSICAL REVIEW LETTERS 124, 117202 (2020) 117202-2the full-range evolution of the linewidth. Compared with the dotted lines which are the linear extrapolations of the YIG ( n¼2) and Py linewidths, the linewidth of the higher- field hybrid mode (blue circles) exceeds the Py linewidth and the linewidth of the lower-field hybrid mode (green circles) reduces below the YIG linewidth when the fre-quency is near ω c. This is the central result of this Letter. It suggests a coherent dampinglike torque which acts along or against the intrinsic damping torque depending on the phase difference of the coupled dynamics of YIG and Py,the same as the fieldlike torque acting along or against the Larmor precession. The dominant mechanism for the dampinglike torque is the spin pumping from the concerteddynamics of YIG and Py [31,32] . Because spin pumping is dissipative, we determine the mode with a broader (narrower) linewidth as the out-of- phase (in-phase) precession mode. In Fig. 2(a)the broader- linewidth mode exhibits a higher resonance field than thenarrower-linewidth mode. This is a signature of antiferro- magnetic exchange coupling at the YIG/Py interface [26]. From the resonance analysis we also find that all theSiO 2=Py samples show lower resonance fields than the Py samples grown on YIG [45], which agrees with the antiferromagnetic nature of the YIG/Py interfacial coupling. To reproduce the data in Fig. 2(b), we introduce the linewidths as the imaginary parts of the resonance fields in Eq.(1):μ0ðH/C6cþiΔH/C6 1=2Þ ¼μ0HYIGrþHPy r 2þiμ0κYIGþκPy 2 /C6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/C18 μ0HYIGr−HPy r 2þiμ0κYIG−κPy 2/C192 þ˜g2cs ; ð2Þ where κYIGðPyÞis the uncoupled linewidth of YIG (Py) from the linear extraction (dotted lines) in Fig. 2(b), and ˜gc¼ gcþiκcis the complex interfacial coupling strength with an additional dampinglike component κcfrom spin pumping. In order to show the relationship between the spin pumpingfrom the coherent YIG-Py dynamics and the incoherent spin pumping from the individual Py dynamics, we identify the latter as the linewidth enhancement of Py(7.5 nm), ΔHPy sp, between the linearly extrapolated YIG/Py [red dots in Fig.2(b)] and Si =SiO 2=Py [red stars in Fig. 2(b)]. Then, we quantify the coherent dampinglike coupling strength κc asκcðωÞ¼βμ0ΔHPy spðωÞ, where βis a unitless and fre- quency-independent value measuring the ratio between the coherent and incoherent spin pumping. For the best fit value, β¼0.82, Eq. (2)nicely reproduces the data in Fig. 2(b).F o r comparison, if we set κcðωÞ¼0in Eq. (2), we obtain the blue and green dashed curves, which result in identical linewidth at ωcas opposed to the data in Fig. 2(a). In order to understand the physical meaning of ˜gc,w e consider the coupled LLG equations of the YIG/Py bilayer[26,32,34] in the macrospin limit: dmi dt¼−μ0γimi×Heffþαimi×dmi dt−γimi×J Mitimj þΔαi/C18 mi×dmi dt−mj×dmj dt/C19 ; ð3Þ wheremi;jis the unit magnetization vector, Heffis the effective field including HB,Hex, and the demagnetizing field, and αiis the intrinsic Gilbert damping. The index is defined as ði; jÞ¼ð 1;2Þor (2,1). In the last two coupling terms, Jis the interfacial exchange energy and Δαi¼ γiℏg↑↓=ð4πMitiÞis the spin pumping damping enhance- ment with g↑↓the spin mixing conductance. The two terms provide the fieldlike and dampinglike coupling torques,respectively, between m iandmj. To view the dampinglike coupling on a similar footing, we define its coupling energy J0as J0ðωÞ¼g↑↓ 4πℏω: ð4Þ Here J0describes the number of quantum channels per unit area ( g↑↓) for magnons ( ℏω) to pass through [31,34] ; similarly, Jdescribes the number and strength of exchange bonds between YIG and Py per unit area. From the definition, we can express the spin pumping linewidth(a) (b) FIG. 2. (a) The line shape of the YIG ð100nmÞ=Pyð7.5nmÞ sample at ωc=2π¼9.4GHz, showing different linewidths be- tween the two hybrid modes of YIG ðn¼2Þand Py ðn¼0Þ resonances. (b) Linewidths of the two hybrid modes as a functionof frequency. Dotted lines show the linear fit of the linewidthsfor the two uncoupled modes. Dashed curves show the theo-retical values with κ c¼0. Solid curves show the fits with finite κc.PHYSICAL REVIEW LETTERS 124, 117202 (2020) 117202-3enhancement as μ0ΔHispðωÞ¼J0ðωÞ=M iti, in pair with the exchange field term in Eq. (3). By solving Eq. (3)we find κiðωÞ¼αiω γiþJ0ðωÞ Miti; ð5aÞ gc¼fðωcÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J M1t1J M2t2s ; ð5bÞ κcðωcÞ¼fðωcÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J0ðωcÞ M1t1J0ðωcÞ M2t2s ; ð5cÞ with the dimensionless factor fðωÞaccounting for the precession elliptical asymmetry. fðωÞ¼1for identical ellipticity ( M1¼M2) and fðωcÞ¼0.9in the case of YIG and Py; see the Supplemental Material for details [45]. Equation (5)shows that both gcandκcðωcÞare propor- tional to 1=ffiffiffitip, which comes from the geometric averaging of the coupled magnetization dynamics. This is in contrast to the1=tidependence of the uncoupled exchange field and spin pumping damping enhancement for a single layer, ass h o w ni nE q . (5a).I nF i g . 3(a), a good fitting of g cto1=ffiffiffiffiffiffitPyp rather than 1=tPyvalidates the model. In the limit of zero Py thickness, the model breaks down due to the significance of boundary pinning and the assumption of macrospin dynam- ics, as reflected in the reduction of gcattPy¼5nm. For the dampinglike coupling, we plot βinstead as a function of tPyin order to minimize the variation in the quality of interfacial coupling and the frequency depend- ence of κcðωcÞ. By taking the ratio between κcðωcÞand μ0ΔHPy spðωcÞfrom the analytical model, we obtain the macrospin expression β¼fðωcÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiMPytPy=M YIGtYIGpwith fðωcÞ¼0.9. Figure 3(b)shows that the extracted β2varies linearly with tPy, rather than being independent of it aswould be expected for incoherent spin pumping. The fit is not perfect, which may be caused by (i) the variation of inhomogeneous broadening of Py in YIG/Py bilayers or(ii) the multipeak line shapes in YIG [see YIG n¼0line shapes in Fig. 1(b)] due to possible damage during the ion milling process. If we calculate βfrom the macrospin approximation, the prediction, shown in the red dashed arrow in Fig. 3(b), differs significantly from the experimental data. To accountfor the difference, we consider a spin wave model for the YIG/Py bilayer, where finite wave numbers exist in both layers and are determined from the boundary condition[46]. For simplicity, we consider free pinning at the two exterior surfaces of YIG and Py and Hoffmann exchange boundary conditions for the interior interface of YIG/Py[47]. From the spin wave model, we find an additional factor offfiffiffi 2p in Eqs. (5b) and(5c); see the Supplemental Material for details [45]. This factor arises because the nonuniform profile of the PSSW mode in YIG reduces the effective mode volume by a factor of 2 compared with the uniform mode. A similar effect has been previouslydiscussed in spin pumping from PSSW modes [48,49] .I n Fig. 3(b) the theoretical calculation from the spin wave model (cyan dashed arrow) is close to the experimentalvalues. This is an additional evidence of the coherent spin pumping in YIG/Py bilayers. Figure 4compares the values of Jand J 0obtained from the hybrid dynamics. For convenience we esti- mate the value of J0from Eq. (5c),a s J0ðωcÞ¼ κcðωcÞ=fðωcÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMYIGtYIGMPytPy=2p. Noting the frequency (a) (b) FIG. 3. (a) Extracted gcas a function of tPy. (b) Extracted β2as a function of tPy. In both panels, the solid and dashed curves are the fits of data to the coherent and incoherent models, respec- tively. In (b), the red and cyan dotted arrows show the theoreticalpredictions for the coherent models based on the macrospin andspin wave approximations, respectively. Error bars indicate singlestandard deviations found from the fits to the line shape.FIG. 4. Thickness dependence of J(circles) and J0ðωcÞ(tri- angles), which are calculated from gcandκcðωcÞ, respectively. Blue points denote the results for YIG ð100nmÞ=PyðtPyÞand red points for YIG ð50nmÞ=PyðtPyÞ. The blue stars denote J0spðωcÞ,i n which ΔHPy spðωcÞis calculated from the Py linewidth enhance- ment from Py ðtPyÞto YIG ð100nmÞ=PyðtPyÞ. Error bars indicate single standard deviations found from the fits to the line shape.PHYSICAL REVIEW LETTERS 124, 117202 (2020) 117202-4dependence of J0ðωÞ, all the values of J0ðωÞin this work are obtained around ωc=2π¼9GHz. We can also calculate J0ðωcÞfrom the uncoupled spin pumping effect, as J0spðωcÞ¼μ0ΔHPy spðωcÞMPytPy. For the YIG/Py interface, the value of Jstays at the same level; the value of J0ðωcÞ fluctuates with samples but is well aligned with J0spðωcÞ, which again supports that the dampinglike interfacial coupling comes from spin pumping. Furthermore, we have also repeated the experiments for a thinnerYIGð50nmÞ=PyðtÞsample series and obtained similar values of JandJ 0ðωcÞ, as shown in Fig. 4. Table Isummarizes the values of J,J0, and g↑↓for the YIG/Py interface, where J0is taken from the vicinity of ωc=2π¼9GHz and g↑↓is calculated from J0ðωcÞby Eq.(4). The value of Jis much smaller than a perfect exchange-coupled interface, which is not surprising giventhe complicated and uncharacterized nature of the YIG/Pyinterface. For Py, the interfacial exchange energy can be estimated [46] by2A ex=a, where for Py Aex¼12pJ=m [49] and the lattice parameter a¼0.36nm. We find 2Aex=a¼68mJ=m2, 3 orders of magnitude larger than J. Comparing with similar interfaces, our reported Jis similar to YIG/Ni ( 0.03mJ=m2[27]) and smaller than YIG/Co ( 0.4mJ=m2[26]). A different interlayer exchange coupling from Ruderman-Kittel-Kasuya-Yosida interaction may generate a larger J[50–52]but a smaller g↑↓[53]. There could also be a fieldlike contribution of Jfrom g↑↓ [23,26,54 –56]. But since the exchange Jdominates in the coupled dynamics, it is difficult to distinguish the spinmixing conductance contribution in our experiments. In conclusion, we have characterized the dampinglike coupling torque between two exchange-coupled ferromag- netic thin films. By exciting the hybrid dynamics in the strong-coupling regime, this dampinglike torque can eitherincrease or suppress the total damping in the out-of-phase or in-phase mode, respectively. The origin of the damp- inglike torque is the coherent spin pumping from thecoupling magnetization dynamics. Our results reveal newinsight for tuning the coherence in magnon-magnon hybrid dynamics and are important for magnon-based coherent information processing. Work at Argonne on sample preparation was supported by the U.S. DOE, Office of Science, Office of Basic EnergySciences, Materials Science and Engineering Division under Contract No. DE-AC02-06CH11357, while work at Argonne and National Institute of Standards andTechnology (NIST) on data analysis and theoreticalmodeling was supported as part of Quantum Materials for Energy Efficient Neuromorphic Computing, an Energy Frontier Research Center funded by the U.S. DOE,Office of Science. Work on experimental design atOakland University was supported by AFOSR under Grant No. FA9550-19-1-0254 and the NIST Center for Nanoscale Science and Technology, AwardNo. 70NANB14H209, through the University ofMaryland. 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PhysRevB.95.054422.pdf

PHYSICAL REVIEW B 95, 054422 (2017) Spin transfer and spin-orbit torques in in-plane magnetized (Ga,Mn)As tracks L. Thevenard,1,*B. Boutigny,1N. G ¨usken,1L. Becerra,1C. Ulysse,2S. Shihab,1A. Lema ˆıtre,2 J.-V . Kim,2V . Jeudy,3and C. Gourdon1 1Sorbonne Universit ´es, UPMC Univ Paris 06, CNRS, Institut des Nanosciences de Paris, 4 place Jussieu, 75252 Paris, France 2Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Universit ´e Paris-Saclay, 91460 Marcoussis, France 3Laboratoire de Physique des Solides, CNRS - Universit ´e Paris-Sud, b ˆat. 510, 91405 Orsay, France (Received 15 October 2016; revised manuscript received 10 January 2017; published 16 February 2017) Current-driven domain wall motion is investigated experimentally in in-plane magnetized (Ga,Mn)As tracks. The wall dynamics is found to differ in two important ways with respect to perpendicularly magnetized(Ga,Mn)As: the wall mobilities are up to ten times higher and the walls move in the same direction as thehole current. We demonstrate that these observations cannot be explained by spin-orbit field torques (Rashba andDresselhaus types) but are consistent with nonadiabatic spin transfer torque enhanced by the strong spin-orbitcoupling of (Ga,Mn)As. This mechanism opens the way to domain wall motion driven by bulk rather thaninterfacial spin-orbit coupling as in ultrathin ferromagnet/heavy metal multilayers. DOI: 10.1103/PhysRevB.95.054422 I. INTRODUCTION Nonvolatile memory and logic devices based on magnetic domain-wall (DW) manipulation [ 1,2] remain technologically challenging as they require narrow DWs, large velocities,and low voltage/current density operation, features that aredifficult to combine in a given material. This has led toconstant endeavors to optimize the torques experienced byDWs under an applied current. Initial works focused on spintransfer torques (STT [ 3–7]). Recently, torques originating from the spin-orbit interaction (SOI) have been evidencedin ferromagnetic/nonmagnetic multilayers [ 8–14]. The mechanisms involved in DW propagation are still partiallyunder debate, but signatures of the Rashba effective field,the spin Hall effect, and the chiral Dzyaloshinskii-Moriyainteraction (DMI) have been suggested [ 8,11,15,16]. These observations have been limited to out-of-planemagnetized heavy metal/metallic ferromagnet combinations(Pt/Co/AlOx,GdOx or Gd [ 13,17], Pt/Co/Ni/Co [ 12], Pt/ or Ta/CoFe/MgO [ 11,14]), in which the main source of SOI was interfacial inversion asymmetry along the sample normal z. The purpose of this work is to investigate experimentally thecase of a ferromagnet that is its own source of (bulk) spin-orbitinteraction, without relying on adjacent layers. For this, westudied the dilute magnetic semiconductor (Ga,Mn)As, inwhich the complex anisotropy and rich spin-orbit couplingphysics enable numerous configurations to be tested. In (Ga,Mn)As, spin-orbit coupling gives rise to two effects very different in magnitude. The main one is described by theKohn-Luttinger (KL) Hamiltonian. It splits the manifold ofvalence states with L=1 orbital quantum number and S= 1/2 spin into J=3/2 and J=1/2 states with J=L+S [18]. For nonzero kwave vectors, the J=3/2 states are further split into heavy hole ( J=±3/2) and light hole ( J=±1/2) states, each with twofold degeneracy. The second and muchweaker effect is a lifting of this degeneracy analogous to ak-dependent magnetic field. This small spin-orbit effect arises from the lack of centrosymmetry of the zinc-blende lattice *thevenard@insp.jussieu.fr(k3Dresselhaus term). A further lowering of the symmetry induced by epitaxial strain ( ε) yields a Dresselhaus term linear ink. An even weaker Rashba term, also linear in k, exists due to the nonequivalence of [110] and [1 ¯10] directions induced during the growth [ 19], formally equivalent to an in-plane shear strain or an electric field perpendicular to the interface [ 18]. This is reminiscent of the one encountered in the z-asymmetric metallic stacks mentioned earlier. These spin-orbit effects have two consequences. First, the heavy-light hole splitting modifies the usual spin transfertorques in the presence of a domain wall since valence statesare not pure spin states. A significant hole reflection should oc-cur at the domain wall, resulting in spin accumulation [ 20–22]. The resulting spin transfer torque is expected to be up to tenfoldmore efficient than the standard torque but has up to now notbeen evidenced experimentally. Secondly, Rashba or Dressel-haus spin-orbit splittings of the valence bands result, under cur-rent, in an out-of-equilibrium hole spin polarization. The anti-ferromagnetic exchange interaction between the carrier spinsand the Mn spins then yields corresponding effective spin-orbitfields on the magnetization [ 23–26]. These fields have been evidenced experimentally [ 27–30] to be in-plane and perpen- dicular to the current density, but their effect on DWP remainsto be shown. Note that these terms are about 100 times weakerthan the Rashba term in metals which were claimed to beresponsible for fast DWP against the electron direction [ 9,13]. To explore these spin-orbit interaction effects, we have worked on in-plane magnetized (Ga,Mn)As tracks, a configu-ration that has been rarely studied up to now [ 31,32]. We show that DWs propagate at high mobilities under current and oppo-site to the direction given by the spin-transfer torque observedin out-of-plane magnetized (Ga,Mn)As [ 33–37], a radically different phenomenology. We demonstrate our observations tobe only partially reconcilable with SO field torques. We sug-gest instead that they may be a signature of the Kohn-LuttingerHamiltonian-induced spin accumulation at the domain wall. II. SAMPLES AND EXPERIMENTAL SETUP DW propagation was studied on a 50 nm thick epilayer of (Ga,Mn)As grown on (001) GaAs. After an 8 h /200◦C 2469-9950/2017/95(5)/054422(10) 054422-1 ©2017 American Physical SocietyL. THEVENARD et al. PHYSICAL REVIEW B 95, 054422 (2017) FIG. 1. (a),(b) Two track configurations and device schematics: the hole current (red arrows) flows either parallel (C //) or perpendicular (C⊥) to the magnetization (dark arrows) whose easy direction is along the crystallographic axis [1 ¯10]. With the reservoir grounded, the hole current direction shown in this schematics therefore results from a negative potential applied to the opposite tip of the track. The hydrogenated (Ga,Mn)As:H has turned paramagnetic, thus defining the stripe. (c),(g),(j) Phenomenology of current-induced DW motion. The effective spin-orbit field ( /vectorHSO) is the sum of the Rashba and the Dresselhaus contributions (see text). (d)–(f) and (h),(i) Longitudinal Kerr microscopy images (divided by a reference image taken after saturation) showing domain wall displacements under the application of successive current pulses (1 ,2, etc.). The white dotted lines are guides for the eyes and the black dashed ones materialize the edges of the track and reservoir. (d)–(f) 2 μmw i d eC //track, 70 ns long current pulses of J=24.5G Am−2under μ0H=1.1m T( Teff=40 K). (h) 10 μmw i d eC ⊥track, 120 ns long current pulses of J=14.9G Am−2,n oa p p l i e dfi e l d( Teff=80 K). A domain is easily nucleated in the middle of the track under current by /vectorHSO(pulses 1 and 5). (i) 2 μmw i d eC ⊥track, 100 ns long current pulses of J=23.5G Am−2under μ0H=1.3m T( Teff=50 K). anneal, the Curie temperature reached T C=116 K and the magnetically active Mn concentration x=3.7%. The layer was grown under compressive strain, which led to themagnetization lying in the plane. SQUID magnetometry andcavity ferromagnetic resonance (FMR) evidenced a uniaxialmagnetic anisotropy with the easy axis along the [1 ¯10] direction. The tracks were 2, 4, and 10 μm wide and 95 μm long, oriented either parallel to the easy axis—“C //” configuration— or perpendicular to it—“C ⊥” configuration, with one end leading into a large grounded reservoir [Figs. 1(a) and1(b)]. The reservoir and tracks were patterned by locally passivatingthe layer at 130 ◦C using a hydrogen plasma over a Ti mask [ 38]. This selectively turns the (Ga,Mn)As paramagnetic, whilst maintaining the bidimensionality of the layer, thuslimiting diffraction-related imaging issues. It also preventsthe lattice from relaxing perpendicular to the track, whichwould pull the easy axis towards the track direction [ 32], and therefore render C //and C ⊥configurations nonequivalent anisotropywise. Finally, Cr(10 nm) /Au(200 nm) contacts were thermally evaporated on the sample. Domains were observedwith an in-house built longitudinal Kerr microscope, using aλ=635 nm LED source [ 39]. A generator was used to apply a continuous or pulsed voltage. In (Ga,Mn)As, Rashba and Dresselhaus spin-orbit fields are positively (negatively) collinear for J//[110] ( J//[1¯10]//x), with the total SO effective field /vectorH SOlying transverse to the tracks [Figs. 1(c) and1(g)]. The field intensity for J//[110] is about three times larger than for J//[1¯10] [ 27–30]. This geometry is distinct from those explored in metallic structuresin two respects: (i) the spin-orbit effect is generated in themagnetic layer itself, and does not require a distinct high-SOImaterial next to it; (ii) the spin-orbit field can be eitherperpendicular or collinear to the magnetization, a configurationthat has not been explored in DWP yet due to the lackof strongly uniaxial in-plane metals subject to spin-orbitfields [ 40]. III. PHENOMENOLOGY We first describe the DW propagation phenomenology under field and current [Figs. 1(c)–1(j)]. After saturation by an external field Hsat, a reversed domain was nucleated by a high current pulse. In the narrowest, 2 and 4 μm wide tracks, 054422-2SPIN TRANSFER AND SPIN-ORBIT TORQUES IN IN- . . . PHYSICAL REVIEW B 95, 054422 (2017) DWs required a small field opposite to Hsatto depin. Under this small field only, note that the DWs did not move. Inthe 10 μm track, the pinning of DWs was sufficiently low to enable current-induced motion without external magneticfield. In the C //track [see Figs. 1(d)–1(f)], we observed that DWs only depin in the hole current direction, regardless ofthe DW charge (tail-to-tail or head-to-head), as summarized inFig.1(c). Holes flow in the same direction as the conventional current, i.e., opposite the electrons. We will for now call thiseffect “STT-like”, and suppose the DW feels a local field /vectorH STT oriented parallel to the easy axis. In the C ⊥tracks, the DW behavior is more complex [see Figs. 1(h) and1(i)], and seems to consist of two competing effects. It depends onthe DW polarity which we label [Fig. 1(g)]p=+1(p=−1) when the magnetization of the first domain crossed by thecurrent is +π/2(−π/2) rotated with respect to the current direction. We observed that regardless of the current andH satsigns, p=−1 DWs alway propagate, whereas p=+1 DWs are always pinned, as summarized schematically inFig. 1(g). This suggests the current creates an effective field /vectorH effpointing in the magnetization direction of the expanding domain, competing with the STT-like contribution. When afield is needed to depin, as in the 2 μm wide tracks, an identical phenomenology was observed [images in Fig. 1(i)]. Note that in one particular configuration, DWs occasionally depinned inthe direction opposite to the current flow [Fig. 1(j)]. Finally, the reproducibility of these observations was verified in detailbetween 4 K and 80 K, as well as the robustness of thephenomenology [ 41]. These results are in stark contrast with those obtained by transport-only measurements on planar(Ga,Mn)As with biaxial anisotropy. They had shown either no dependence on current polarity [ 31], or deduced indirectly a DW propagation direction against the hole current [ 32]. To get an insight on the nature of the torques at play in the DW motion, we then acquired hysteresis cycles on the2μmw i d eC ⊥track for different values of a dc current [see Fig.2(a)]. In order to maintain a constant effective temperature (Teff=83 K), the temperature rise during sample excitation was carefully characterized and adjusted for by tuning thecryostat temperature (see Appendix A). Figure 2(a) shows that the hysteresis cycles are shifted in opposite directionsfor positive or negative currents, indicating that the effectof current is equivalent to an effective magnetic field alongthe easy axis. In order to study this effect, we monitoredthe depinning field out of a particular defect along thestripe, H dep, as a function of current, and compared it to its value without any current: δHdep(J)=Hdep(J)−Hdep(J= 0). Positive and negative dc currents were used, so thatthe four different relative orientations of current/field wereexplored [see side schematics in Fig. 2(b)], allowing us to disentangle the two competing effects. They reproducedthe blocking/passing configurations observed under pulsed current in Figs. 1(i)and1(j). In the resulting δH dep(J)p l o t of Fig. 2(b),δHdep<0(δHdep>0) implies that the current made it easier (more difficult) to depin the DW. At lowcurrent density and in the south-east quadrant of Fig. 2(b), it becomes very difficult to pinpoint precisely how differentthe depinning field is from the J=0 one. As the current density becomes higher, however, we notice that the δH dep(J) data essentially consists of two intersecting lines of positive (negative) slopeδHdep+ J(δHdep− J). We can therefore extract an ef- fective fieldlike contributionμ0Heff J=μ0 2(|δHdep+ J|+|δHdep− J|)= (3.6±0.2)×10−2mT/GA m−2and a domain-independent, STT-like contributionμ0HSTT J=μ0 2(|δHdep+ J|−|δHdep− J|)=(6± 1)×10−3mT/GA m−2. We therefore conclude thatHeff Jis six times larger than the STT-like contributionHSTT J, and is of the same order of magnitude as the total spin-orbit field efficiencyin Refs. [ 27–30]. Note that H effexhibits the same symmetry as the Oersted field accompanying the passage of the current.Micromagnetic simulations under J=20 GA m −2show that the transverse ( y) component of the Oersted field is around ±0.2 mT (in the center of the top/bottom interfaces), and its out-of-plane ( z) component is ±0.8 mT (at midheight of the track edges). However, the substrate being quite insulating, thecurrent density is mostly confined to the magnetic layer, andwe expect the average Oersted field in the track to be close tozero. FIG. 2. Estimation of the current-induced STT-like and FL-like contributions in the 2 μmw i d eC ⊥track, at Teff=83 K. (a) Normalized hysteresis cycles under continuous current J=±8.3G Am−2, and without current. They are obtained by taking longitudinal Kerr images as the field is cycled, and averaging the signal over the entire surface of the track. Hysteresis cycles averaged on the reservoir under current coincide with those averaged on the track without current. (b) The DW depinning field out of a given trap Hdep(J) is monitored as a function of current density and compared to the one without applied current Hdep(J=0). The difference is indicated in the yaxis. The four schematics represent the physical situations encountered in the four corresponding quadrants of the plot, with the same arrow legend as in Fig. 1. 054422-3L. THEVENARD et al. PHYSICAL REVIEW B 95, 054422 (2017) FIG. 3. Domain-wall velocity: (a) velocity versus field curves (C //and C ⊥2μm wide tracks) at constant temperature and current density. (b) Velocity versus current density for tracks C //(2μmw i d e , μ0H=1.1 and 1.2 mT) and C ⊥(2μmw i d e , μ0H=1.3m T ,a n d1 0 μmw i d e , no field). Measurements taken at Teff=49±1Kf o rt h e2 μm wide tracks, and at Teff=77±2 K for the 10 μm wide one. We can now summarize the current-induced DW mo- tion phenomenology in (Ga,Mn)As with uniaxial in-planeanisotropy as follows. When the current flows colinear tothe magnetization (C //tracks), DW motion occurs in the hole current direction regardless of domain charge. This isopposite to the direction observed for spin-transfer torquein perpendicularly magnetized (Ga,Mn)As films. When thecurrent flows perpendicular to the magnetization (C ⊥tracks), this STT-like contribution competes with an effect six timeslarger, an effective transverse field H effproportional to the current. Finally, we performed DW velocity measurements on the C //and C ⊥tracks (see Fig. 3). Displacements under increasing current pulse lengths were obtained as describedin Appendix B. Figure 3(a) shows v J(H) curves displaying the DW velocity as a function of the applied magnetic field,in the presence of current pulses of constant amplitude. Themaximum measurable velocity is determined by the tracklength. After a depinning regime at low fields, the DW velocityincreases linearly with field, reaching up to 300 m s −1on the 2μmw i d eC //track. These velocities are typical of those measured under field only on a very similar nonpatternedlayer [ 39], and result from the large DW width of in-plane (Ga,Mn)As. In the C ⊥tracks, velocities are overall smaller [up to 150 m s−1,F i g . 3(a)]. Comparing C //and C ⊥vJ(H) curves taken at the same current density shows that the∂v ∂Hmobilities lie in a ratio of 4:1 (measured after the depinning regime).In the stationary regime, we expect field mobilities to varyin a first approximation like the static DW width, estimatedby micromagnetic simulations [Fig. 4(a)] to be almost three times larger in the charged DWs of C //tracks ( /Delta10=40 nm), than in the uncharged DWs of C ⊥(/Delta10=15 nm). The measured mobility ratio is therefore a signature of thestationary regime. Also, the precessional regime is expectedto be preceded by a large velocity plateau, seen under field onlyon unpatterned samples [ 39] and in micromagnetic simulations for C ⊥and C //tracks [Fig. 4(b)]. This plateau was not observed experimentally under applied field/current. The field was then kept constant (1.1, 1.2, 1.3, or 0 mT), and current pulses ( J=7–25 GA m−2) were applied at constant effective temperature. This generated vH(J) curves [Fig. 3(b)]. Once more, velocities of up to 300 m s−1were observed on the 2 μmw i d eC //track, and up to 150 m s−1in C ⊥tracks, again pointing to the stationary regime. The resulting current mobility∂v ∂Jis 11±1m m3C−1for the 2 μmw i d eC // tracks ( Teff=49 K) and 10 μmw i d eC ⊥track ( Teff=77 K), over ten times larger than the mobilities measured on out-of-plane magnetized (Ga,Mn)As [ 34–37]. No field assistance was required for the wider 10 μmC ⊥track [Fig. 3(b)], on which creep motion was also observed at low current densities(J=7G Am −2, velocities too small to appear on the curve). (b)(a) mxmy 0 1 23 4 5050100150200 FIG. 4. Micromagnetic simulations ( T=60 K micromagnetic parameters of the sample, and T=0 in mumax code [ 42]) for both track configurations. (a) Static domain wall width. Smaller DW widths translate into smaller mobilities under field. (b) Field-driven DW propagation, subtracting surface magnetic charges at wire endsto simulate infinite wire. The best agreement with measurements on unpatterned samples [ 39] was obtained with α=0.025. The precessional regime is reached at a few mT, in the plateau of thecurve. 054422-4SPIN TRANSFER AND SPIN-ORBIT TORQUES IN IN- . . . PHYSICAL REVIEW B 95, 054422 (2017) In the 2 μmC⊥track, a lower mobility of 6 ±1m m3C−1was measured. IV . POSSIBLE TORQUES AT PLAY To make sense of these unexpected results, we begin by considering two types of current-induced torques: torquesthat push DWs unidirectionally regardless of their polarityor charge, which will be named STT-like, or torques driven bythe Rashba and Dresselhaus spin-orbit effective fields. A. Spin-orbit effective field torques Dresselhaus and Rashba terms in strained (Ga,Mn)As induce a total effective field /vectorHsoproportional to the current density and transverse to the track direction [Figs. 1(a) and 1(b)], expected to be slightly larger in C ⊥than in C//tracks [ 27–30,43]. Similar to what has been calculated and observed in metals [ 7,44–47], this field can act on the magnetization via two torques: a fieldlike torque [ 48] (FL-SOT) /vectorHso×/vectorMor a Slonczweski-like torque (SL-SOT) /vectorM×/vectorHSLwith /vectorHSL∝/vectorM×/vectorHso. The FL-SOT is sensitive to the charge or polarity of the DW, but not to its chirality(magnetization orientation inthe DW), while the SL-SOT is sensitive to the magnetization configuration within theDW (Bloch/N ´eel, chirality). Both torques have been cal- culated [ 26,43] and measured [ 27–30,43] in monodomain (Ga,Mn)As and (Ga,Mn)(As,P) and found to be of the sameorder of magnitude [ 43]. We have summarized schematically their expected effect on DWs of C //and C ⊥tracks in Fig. 5.I nC //tracks, the FL-SOT simply stabilizes a N ´eel DW structure against Walker breakdown, and possibly imposes a DW chirality during its creation [Fig. 5(a)]. In C ⊥tracks however, /vectorHsois collinear to the domain magnetization, so will act like the effective field /vectorHeffevidenced earlier. Hence we suggest μ0/vectorHeff=μ0/vectorHSO. We established above that at 83 K it varies with current as3.6×10 −2mT/GA m−2[Fig. 2(b)], close to the (2 .0–10.6)× 10−2mT/GA m−2SO field efficiencies found by other authors[27–30]f o r J//[110]. This yields for our typical current densities (e.g., J=20 GA m−2)μ0HSO=0.4–2.7m T ,v e r y much of the order of the applied static fields. Its directionis represented in Figs. 1(g) and5(b),5(d) by a green hollow arrow. Note that the total spin-orbit field we find for J//[110] is of opposite sign to the one found in previous studies ofin-plane magnetized (Ga,Mn)As devices [ 27–30,43]. These measurements were done on samples quite similar to ours inMn content, Curie temperature, of varying anisotropy (uniaxialor biaxial), and monodomain or not. A notable differencewould be our sample thickness (50 nm), e.g., twice that ofthe thickest of these studies (25 nm). One could tentativelysay that a thicker layer would reduce the Rashba (interfacial) contribution to the total /vectorH socompared to thinner layers. However, since it is unclear what governs the sign of theRashba field, and how exactly it varies from layer to layer,it is very difficult to infer anything from this observation. Amore flagrant discrepancy is that the studies of Refs. [ 27–30] are all based on magnetotransport measurements, whereas weproceed via a direct visualization of domains. However, witha correct characterization of both hole current and appliedfield directions, this should not affect the sign of μ 0HSO.T h e reason for this sign difference therefore remains elusive for themoment. The SL-SOT was proposed to explain similar intriguing DW propagation direction and velocities in metals [ 49]. We label it “efficient” (propagation in the stationary regime)when it tilts the magnetization out of the plane of rotationof the DW (materialized by dotted lines in Fig. 5), and “inefficient” (propagation in the precessional regime only)when it merely rotates the DW magnetization. As representedschematically in Figs. 5(a) and 5(b), the SL-SOT will be inefficient for C //and C ⊥. We also consider DW structures other than the N ´eel or transverse ones [ 50]. The SL-SOT can then induce efficient propagation for C //tracks provided a significant Bloch component is present [Fig. 5(c)]. For the resulting DW propagation direction to be independent of theDW charge, as observed experimentally [Figs. 1(c)–1(f)], FIG. 5. Effective fields acting on the magnetization (blue arrows) involved in the fieldlike torque (green hollowed arrows) and the Slonczweski-like torque (black hollowed arrows), /vectorHSL∝/vectorHso×/vectorM. Their effect on C //(a),(c) and C ⊥(b),(d) configurations, and supposing N´eel (a),(b) or Bloch (c),(d) domain walls are represented using dashed contours if the field is only efficient in the precessional regime, or continuous contours if it is efficient in the stationary regime. The direction of the hole current is indicated by a red arrow and the dotted contoursmaterialize the plane of rotation of the magnetization in the domain wall. The direction of /vectorH sohas only been inferred experimentally in (b). 054422-5L. THEVENARD et al. PHYSICAL REVIEW B 95, 054422 (2017) DWs would however need to be chiral, meaning that the magnetization would for instance need to point “up” in head-to-head DWs, and “down” in tail-to-tail DWs. This propertyusually accompanies the DMI [ 16], a point that would require further theoretical development for the case of (Ga,Mn)As.This torque would however be inefficient in the C ⊥tracks [Fig. 5(d)]. To summarize, Rashba and Dresselhaus spin-orbit effective fields could only explain the fieldlike torque observed in C ⊥ tracks. To make sense of the STT-like effects with a torqueinvolving an effective field proportional to the current, onewould need it to be along z, instead of in-plane and transverse to the track. Under the form of a Slonzweski-like torque /vectorM× /vectorH eff,z×/vectorM, it could indeed push DWs efficiently along the hole current on both types of tracks, provided it had the correctsign and DWs were chiral. Such a field has recently beenevidenced in monodomain Ta/CoFeB/TaOx trilayers [ 51]. It is equivalent to an in-plane electric field perpendicular to thetrack, and proportional to the current density. In Yu et al. [51] it originated from the lateral structural asymmetry inducedby the wedged cross section of their sample, but its originin our case would be unclear. It would also be problematicto reconcile it with current-induced domain wall propagationin out-of-plane magnetized (Ga,Mn)As and (Ga,Mn)(As,P): a/vectorH eff,zfield would prevent consecutive DWs from shifting synchronously under current, as has been observed in thesesamples [ 35–37]. To conclude on this part on spin-orbit effective fields, we wish to comment on an implicit hypothesis made in our approach. Here the spin-orbit field /vectorH sois assumed to be only weakly affected by the presence of a domain wall [as repre-sented schematically in Figs. 1(c)and1(g)]. However, domain walls could be the locus of significant Mn (and therefore hole)depolarization [ 20,52], which could in turn modify the local spin-orbit field amplitude. /vectorH sohas so far mainly been measured in monodomain samples, and for current/magnetization onhigh symmetry axes [ 27,29]. Let us mention however that Li et al. [30] have measured /vectorH soon 10μm wide devices probably accommodating DWs, and found it to be similar to those ofthe 80 nm wide monodomain devices of Fang et al. [29] along [1 ±10],[100], which supports our initial hypothesis. Kurebayashi et al. [43] moreover studied, on a monodomain sample, the dependence of the SL field /vectorH z∝/vectorHso×/vectorMon the angle ( /vectorM,/vectorJ), and evidenced weak anisotropic effects. B. Spin-transfer-like torques We now consider the effect of STT-like torques, and focus on the generic “nonadiabatic” term β/vectorM×[(/vectorJ·/vector∇)/vectorM], where Jis the current density and βis a phenomenological factor related to spin accumulation at the domain wall. In thestationary regime, a velocity proportional to the current densityis expected, with v∝ β αJ,αbeing the Gilbert damping term. Previous work [ 35–37] done on 25–50 nm thick out-of- plane magnetized (Ga,Mn)As or (Ga,Mn)(As,P) has evidenceda negative mobility of DWs driven by current. Assuming aspin-relaxation transfer torque [ 3], ∂v ∂J=βsr α|Pc|μB eMs<0w a s then justified by the negative carrier polarization Pcarising from the antiferromagnetic Mn spin/hole spin interaction,and a value of βsr/α≈−1.0±0.5 was found [ 53]. The positive mobilities measured on both C//and C ⊥tracks, however, suggest that in planar (Ga,Mn)As this effect is in factdominated by a counterpropagating one. From the velocitycurves [Fig. 3(b)], we estimate β/α≈12 (see numerical details in Appendix C)f o rt h eC //tracks. Assuming βsr/α≈ −1 implies that an STT-like mechanism of opposite sign needs to account for the remaining 13. Note that the spin-relaxationtransfer torque is very probably present though, since themeasured M s(T) curve exhibits a very standard shape, which is consistent with the efficient mutual polarization of the Mn andhole spin populations. This contrasts with ultrathin metallicfilms sandwiched between other layers, which end up beingpoorly spin polarized due to their weak relative conductivity inthe stack [ 11,13]. In C ⊥tracks a lower ratio of β/α < +0.7–5 was estimated from both the velocity curves and the hysteresiscycles taken under dc current (Fig. 2and Appendix C). The term βphenomenologically accounts for many differ- ent microscopic phenomena leading to spin relaxation suchas spin-flip scattering or DW-induced relaxation [ 4,54]. In metals it also covers the appearance of a DW resistance atabrupt interfaces [ 55–58], leading to a momentum transfer force [ 7,59–62] never clearly identified experimentally [ 63]. Two contributions have been identified in the spin-relaxationnonadiabatic torque. The first one is “interband” and resultsfrom the modification of the electron wave functions under anapplied electric field [ 21,22,43,54]. It is weakly affected by the (Ga,Mn)As KL SOI. The second one is “intraband” and reflectsthe modification of the band populations by the electric field,via the Fermi-Dirac coefficient. In (Ga,Mn)As, the “interband”component dominates as a result of the strong SOI, withpredicted [ 20,21]β/α≈10. In particular, it can overcome the intrinsic limited efficiency of a total momentum-conservingtorque transfering exactly ¯ hbetween conduction carriers and local magnetic moments. Interestingly, Garate et al. [21] predict a sign opposite to the traditional adiabatic STT incertain cases for the intraband component. The large effectiveβ/α observed in our in-plane tracks makes this torque the most likely mechanism at work. The influence of the anisotropy andof the domain-wall width having not been addressed yet inthese calculations, the reason why this contribution wouldbe absent in out-of-plane magnetized (Ga,Mn)As remainselusive. We have calculated k-space maps of the spin and orbital components S zandLzby/vectork·/vectorptheory for in-plane and out-of-plane magnetized (Ga,Mn)As, and have not foundany noteworthy differences. In particular, the calculated holepolarization at the Fermi energy is identical. Instead, apossibility to be explored is the influence of the domain-wallwidth. Indeed we do observe that the high positive mobilityunder current seems to decrease with domain-wall width:β/α≈+12 for /Delta1≈40 nm [C //tracks, Fig. 4(a)],β/α < +0.7–5 for /Delta1≈15 nm (C ⊥tracks), and finally β/α≈−1f o r /Delta1≈5 nm [out-of-plane magnetized (Ga,Mn)As]. Although this agrees with the tendency of the SOI-induced torques toincrease with domain-wall width calculated in ballistic nickeldomain walls [ 58], it disagrees with the prediction of Nguyen et al. [20]. At this stage, based on this phenomenological study and to reconcile all current-induced observations in(Ga,Mn)As/(Ga,Mn)(As,P), one can infer that the KL SOI 054422-6SPIN TRANSFER AND SPIN-ORBIT TORQUES IN IN- . . . PHYSICAL REVIEW B 95, 054422 (2017) spin transfer torque is responsible for DW propagation along the hole current in in-plane layers with wide domain walls,but is absent in out-of-plane layers. The domain-wall widthis however clearly not the only relevant parameter, sinceDe Ranieri et al. [37] have observed current-induced DWP opposite the hole current on perpendicularly magnetized(Ga,Mn)(As,P) tracks with N ´eel DWs. Further theoretical work on the domain-wall width and anisotropy depen-dence of this torque would therefore greatly enrich thisdiscussion. C. Spin Hall and anomalous Hall effects torques Finally, we mention some of the other effects likely to affect current-driven DWP. Among them, the spin Halleffect [ 64] torque that appears when a ferromagnet is adjacent to a nonmagnetic high spin-orbit coupling metal has proveddecisive to explain propagation against the electron flow inultrathin metallic layers [ 10–14]. It is however very unlikely to exist in the bulk of a ferromagnet. One could argue thatif current leaked into the GaAs substrate, the presence of thespin-asymmetric vacuum/(Ga,Mn)As/GaAs interfaces couldin theory pump a spin current perpendicular to the layer. Thescaling of the tracks’ resistance with their width howeverpoints to a sufficiently insulating substrate to neglect thiscontribution. Closely related to it, the anomalous Hall effect islarge in (Ga,Mn)As. It was recently shown both theoreticallyand experimentally to be a possible source of large β/α ratio in Permalloy [ 65,66]. This effect however seems limited to vortex domain walls. V . CONCLUSIONS We have observed current-induced domain-wall propa- gation in uniaxial in-plane (Ga,Mn)As tracks. The current-dependent DW mobility is up to ten times higher and ofopposite sign than in out-of-plane magnetized (Ga,Mn)Asand cannot be explained by the arguments put forward inmetallic structures, where similar effects were evidenced. Inparticular, the fieldlike and Slonczweski-like torques asso-ciated with the Rashba/Dresselhaus spin-orbit fields alonecannot account for these observations. The existence ofan efficient Kohn-Luttinger SOI spin-transfer mechanism,overshadowing the usual spin-relaxation channel seems sofar the most likely candidate, with the constraint that itwould need to be much stronger in in-plane than in out-of-plane (Ga,Mn)As/(Ga,Mn)(As,P) layers. This work, however,provides a strong motivation to further study in-plane mag-netized (Ga,Mn)As tracks, and more generally to engineeruniaxial in-plane materials showing strong intrinsic spin-orbit interactions. This should allow one to explore thepossibility of obtaining these high mobilities without therequirement of a pure-spin current source like in metallicheterostructures. ACKNOWLEDGMENTS We thank A. Thiaville for his continuous interest in this work, as well as V . Cros, J. Sampaio, and A. Manchonfor stimulating discussions. We acknowledge M. Bernard,S. Majrab, and M. Rostiche for their technical assistance, andJ. von Bardeleben for the cavity-FMR measurements. This work has been supported by the French RENATECH network. APPENDIX A: ESTIMATION OF THE EFFECTIVE TEMPERATURE As routinely done on (Ga,Mn)(As,P) and (Ga,Mn)As samples [ 35–37], the strong temperature dependence of the resistance Rwas used to evaluate the effective temperature of the tracks. After taking a calibration R(T) curve under very low current, the cryostat was set at a given temperature. Aconstant voltage Uwas then applied, and the track resistance Rmeasured. This yielded a R(P) curve where P=U 2/R. Using the low current R(T) calibration curve, this was turned into a Tstat(P) curve where Tstatis the track temperature in the stationary regime. In a 1D heat diffusion model, thislinear relationship depends solely on the track dimensions,the substrate thickness, and the thermal conductivity K[67]. We could therefore extract experimentally an effective valueofKfor the different tracks, and from different starting temperatures T 0. Typical values of K=100–150 W m−1K−1 were obtained. This in turn was used to estimate the ef- fective temperature ( Teff) after short current pulses, using the specific heat from Ref. [ 68] (e.g., C=77 J kg−1K−1 around 50 K): /Delta1T(τ)=P 2πKlln/parenleftbigg16Dτ w2/parenrightbigg , (A1) /Delta1T(τ→∞ )=P πKl/bracketleftbigg3 2+ln/parenleftbigg2L w/parenrightbigg/bracketrightbigg , (A2) Teff stat=T0+/Delta1T, (A3) where wandlare respectively the track width and length, L=350μm is the substrate thickness, D=K/ρC is the heat diffusion coefficient, and ρis the mass density. APPENDIX B: METHODOLOGY OF DW VELOCITY MEASUREMENT The procedure for image acquisition was identical to the one used in Ref. [ 39]. An image was first taken in zero field, after saturating the sample ( μ0Hsat=±8 mT). Consecutive images (after field/current application) were divided by thisreference image in order to enhance the domain contrast.Different DW propagation behaviors—pinned, depinning, anddepinned regimes—were observed depending on the value ofcurrent/field [Fig. 6(a)]. Given the high velocity of the DWs, short pulses were required. The velocity was obtained as theslope of the averaged displacements versus pulse length τ [Fig. 6(b)]. For each τ, several acquisitions were made, giving a distribution of displacements. APPENDIX C: EXPERIMENTAL ESTIMATION OF β This was done using either the velocity curves taken under pulsed current [Fig. 3(b)], or the hysteresis cycles taken under dc current (Fig. 2). The magnetization at saturation was determined by SQUID: Ms=33 kA /m at 49 K and Ms= 16 kA /m at 77 K. The domain wall widths were taken as 15 nm (40 nm) for C ⊥(C//) tracks (Fig. 4). 054422-7L. THEVENARD et al. PHYSICAL REVIEW B 95, 054422 (2017) FIG. 6. 2 μmw i d eC //tracks: (a) position of the DW versus current pulse number at fixed applied field μ0H=0.7m Ta n d Teff≈62 K, pulse duration τ=100 ns. Pinned (magenta), depinning (blue), and depinned (black) regimes are clearly identified for increasing current densities. No pr1opagation or depinning are observed for negative values of the current. (b) Domain wall velocity determination from the averaged domain-wall displacement, at fixed applied field μ0H=1.3m T , J=21.75 GA m−2,a n dTeff≈49 K. From the velocity curves : in the stationary regime, the velocity is given by∂v ∂J=β αPcμB eMsfor the C //tracks, and ∂v ∂J=βsr αPcμB eMs+γ/Delta1 α∂μ0HSO ∂Jfor the C ⊥tracks.∂μ0HSO ∂J=3.6× 10−2mT/GA m−2was determined experimentally from the hysteresis cycles at 83 K (Fig. 2). For lack of lower temperature measurement, we will take this as a lower boundary of the 49and 77 K ∂μ0HSO ∂Jvalues, giving an upper boundary of β/α.T h e polarization of holes at the Fermi energy Pcwas calculated byk.ptheory with a hole density of p=3×1020cm−3: |Pc|=0.53 at 49 K and |Pc|=0.4a t7 7K .From the hysteresis cycles : converting the STT-like contri- bution into a value of βwithμ0HSTT J=β γ/Delta1, following Ref. [ 69]. For C //tracks :a tTeff=49 K, the mobility of the 2 μm wide track∂v ∂J=+11.1±0.5m m3C−1leads to β/α≈12. 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RevModPhys.84.119.pdf

Domain wall nanoelectronics G. Catalan Institut Catala de Recerca i Estudis Avanc ¸ats (ICREA), 08193, Barcelona, Spain Centre d’Investigacions en Nanociencia i Nanotecnologia (CIN2), CSIC-ICN, Bellaterra 08193, Barcelona, Spain J. Seidel Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Department of Physics, University of California at Berkeley, Berkeley, California 94720, USASchool of Materials Science and Engineering, University of New South Wales, Sydney NSW 2052, Australia R. Ramesh Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Department of Physics, University of California at Berkeley, Berkeley, California 94720, USADepartment of Materials Science and Engineering, University of California at Berkeley, Berkeley, California 94720, USA J. F . Scott Department of Physics, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom (published 3 February 2012) Domains in ferroelectrics were considered to be well understood by the middle of the last century: They were generally rectilinear, and their walls were Ising-like. Their simplicity stood in stark contrast to the more complex Bloch walls or Ne ´el walls in magnets. Only within the past decade and with the introduction of atomic-resolution studies via transmission electron microscopy, electronholography, and atomic force microscopy with polarization sensitivity has their real complexity been revealed. Additional phenomena appear in recent studies, especially of magnetoelectric materials, where functional properties inside domain walls are being directly measured. In thispaper these studies are reviewed, focusing attention on ferroelectrics and multiferroics but making comparisons where possible with magnetic domains and domain walls. An important part of this review will concern device applications, with the spotlight on a new paradigm of ferroic deviceswhere the domain walls, rather than the domains, are the active element. Here magnetic wall microelectronics is already in full swing, owing largely to the work of Cowburn and of Parkin and their colleagues. These devices exploit the high domain wall mobilities in magnets and theirresulting high velocities, which can be supersonic, as shown by Kreines’ and co-workers 30 years ago. By comparison, nanoelectronic devices employing ferroelectric domain walls often have slower domain wall speeds, but may exploit their smaller size as well as their different functionalproperties. These include domain wall conductivity (metallic or even superconducting in bulkinsulating or semiconducting oxides) and the fact that domain walls can be ferromagnetic while the surrounding domains are not. DOI: 10.1103/RevModPhys.84.119 PACS numbers: 77.80.Fm, 68.37.Ps, 77.80.Dj, 73.61.Le CONTENTS I. Introduction 120 II. Domains 121 A. Boundary conditions and the formation of domains 121B. Kittel’s law 121C. Wall thickness and universality of Kittel’s law 122D. Domains in nonplanar structures 123 E. The limits of the square root law: Surface effects, critical thickness, and domains in superlattices 124F. Beyond stripes: Vertices, vortices, quadrupoles, and other topological defects 125 G. Nanodomains in bulk 128H. Why does domain size matter? 130 III. Domain Walls 130 A. Permissible domain walls: Symmetry and compatibility conditions 130 B. Domain wall thickness and domain wall profile 131 C. Domain wall chirality 133D. Domain wall roughness and fractal dimensions 134REVIEWS OF MODERN PHYSICS, VOLUME 84, JANUARY–MARCH 2012 0034-6861 =2012=84(1)=119(38) 119 /C2112012 American Physical SocietyE. Multiferroic walls and phase transitions inside domain walls 136 F. Domain wall conductivity 138 IV. Experimental Methods for the Investigation of Domain Walls 138 A. High-resolution electron microscopy and spectroscopy 138 B. Scanning probe microscopy 139 C. X-ray diffraction and imaging 141 D. Optical characterization 141 V. Applications of Domains and Domain Walls 142 A. Periodically poled ferroelectrics 142 1. Application of Kittel’s law to electro-optic domain engineering 143 2. Manipulation of wall thickness 143 B. Domains and electro-optic response of LiNbO 3 144 C. Photovoltaic effects at domain walls 144D. Switching of domains 145E. Domain wall motion: The advantage of magnetic domain wall devices 145 F. Emergent aspects of domain wall research 147 1. Conduction properties, charge, and electronic structure 147 2. Domain wall interaction with defects 1493. Magnetism and magnetoelectric properties of multiferroic domain walls 149 VI. Future Directions 150 I. INTRODUCTION Ferroic materials (ferroelectrics, ferromagnets, ferroelas- tics) are defined by having an order parameter that can pointin two or more directions (polarities), and be switched be- tween them by application of an external field. The different polarities are energetically equivalent, so in principle they all have the same probability of appearing as the sample is cooled down from the paraphase. Thus, zero-field-cooledferroics can, and often do, spontaneously divide into small regions of different polarity. Such regions are called ‘‘domains,’’ and the boundaries between adjacent domains are called ‘‘domain walls’’ or ‘‘domain boundaries.’’ The ordered phase has a lower symmetry compared to the parentphase, but the domains (and consequently domain walls) capture the symmetry of both the ferroic phase and the para- phase. For example, a cubic phase undergoing a phase tran- sition into a rhombohedral ferroelectric phase will exhibit polar order along the eight equivalent 111-type crystallo-graphic directions, and domain walls in such a system sepa- rate regions with diagonal long axes that are 71 /C14, 109/C14, and 180/C14apart. We begin our description with a general discus- sion of the causes of domain formation, approaches to under- standing the energetics of domain size, factors that influencethe domain wall energy and thickness, and a taxonomy of the different domain topologies (stripes, vertices, vortices, etc.). As the article unfolds, we endeavor to highlight the common-alities and critical differences between various types of fer- roic systems. Although metastable domain configurations or defect- induced domains can and often do occur in bulk samples,an ideal (defect-free) infinite crystal of the ferroic phase is expected to be most stable in a single-domain state (Landauand Lifshitz). Domain formation can thus be regarded insome respect as a finite size effect, driven by the need to minimize surface energy. Self-induced demagnetization or depolarization fields cannot be perfectly screened and alwaysexist when the magnetization or polarization has a componentperpendicular to the surface. Likewise, residual stresses dueto epitaxy, surface tension, shape anisotropy, or structuraldefects induce twinning in all ferroelastics and most ferro- electrics. In general, then, the need to minimize the energy associated with the surface fields overcomes the barrier forthe formation of domain walls and hence domains appear.Against this background, there are two observations and acorollary that constitutes the core of this review: (1) The surface-to-volume ratio grows with decreasing size; consequently, small devices such as thin films, which are the basis of modern electronics, can have small domains and a high volume concentration ofdomain walls. (2) Domain walls have different symmetry, and hence different properties, from those of the domains theyseparate. The corollary is that the overall behavior of the films may be influenced, or even dominated, by the properties of the FIG. 1. Schematic of logic circuits where the active element is not charge, as in current complementary metal oxide semiconductor(CMOS) technology, but domain wall magnetism. From Allwood et al. , 2005 .120 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012walls, which are different from those of the bulk material. Moreover, not only do domain walls have their own proper- ties but, in contrast to other types of interface, they are mobile. One can therefore envisage new technologies wheremobile domain walls are the ‘‘active ingredient’’ of the device, as highlighted by Salje (2010) . A prominent example of this idea is the magnetic ‘‘racetrack memory’’ where thedomain walls are pushed by a current and read by a magnetic head ( Parkin, Hayashi, and Thomas, 2008 ); in fact, the entire logic of an electronic circuit can be reproduced using mag- netic domain walls ( Allwood et al. ,2 0 0 5 ) (see Fig. 1). Herbert Kroemer, Physics Nobel Laureate in 2000 for his work on semiconductor heterostructures, is often quoted for his dictum ‘‘the interface is the device.’’ He was, of course,referring to the interfaces between different semiconductor layers. His ideas were later extrapolated, successfully, to oxide materials, where the variety of new interface propertiesseems to be virtually inexhaustible ( Mannhart and Schlom, 2010 ;Zubko et al. , 2011 ). However, this review is about a different type of interface: not between different materials, but between different domains in the same material. Paraphrasing Kroemer, then, our aim is to show that ‘‘thewall is the device.’’ II. DOMAINS A. Boundary conditions and the formation of domains The presence and size of domains (and therefore the concentration of domain walls) in any ferroic depends on its boundary conditions. Consider, for example, ferroelec-trics. The surfaces of a ferroelectric material perpendicular to its polar direction have a charge density equal to the dipolar moment per unit volume. This charge generates anelectric field of sign opposite to the polarization and magni- tudeE¼P=" (where "is the dielectric constant). For a typical ferroelectric ( P¼10/C22C=cm 2,"r¼100–1000 ), this depolarization field is c.a. 10–100 kV =cm, which is about an order of magnitude larger than typical coercive fields. So, if nothing compensates the surface charge, the depolarizationwill in fact cancel the ferroelectricity. Charge supplied by electrodes can partly screen this depolarization field and, although the screening is never perfect ( Batra and Silverman, 1972 ;Dawber, Jung, and Scott, 2003 ;Dawber et al. , 2003 ;Stengel and Spaldin, 2006 ), good electrodes can stabilize ferroelectricity down to films just a few unit cellsthick ( Junquera and Ghosez, 2003 ). But a material can also reduce the self-field by dividing the polar ground state into smaller regions (domains) with alternating polarity, so that theaverage polarization (or spin, or stress, depending on the type of ferroic material considered) is zero. Although thisdoes not completely get rid of the depolarization (locally, each individual domain still has a small stray field), the mechanism is effective enough to allow ferroelectricity tosurvive down to films of only a few unit cells thick ( Streiffer et al. , 2002 ;Fong et al. , 2004 ). The same samples (e.g., epitaxial PbTiO 3onSrTiO 3substrates) can in fact show either extremely small (a few angstroms) domains or an infinitely large monodomain configuration just by changing the boundary condition ( Fong et al. , 2006 ), i.e., by allowingfree charges to screen the electric field so that the formation of domains is no longer necessary (and it is noteworthy that such effective charge screening can be achieved just by adsorbates from the atmosphere). An important boundary condition is the presence or other- wise of interfacial ‘‘dead layers’’ that do not undergo theferroic transition. Dead layers have been discussed in thecontext of ferroelectrics, where they are often proposed as explanations for the worsening of the dielectric constant of thin films, although the exact nature, thickness, and evenlocation of the dead layer, which might be inside the elec-trode, is still a subject of debate ( Sinnamon, Bowman, and Gregg, 2001 ;Stengel and Spaldin, 2006 ;Chang et al. , 2009 ). In ferroelectrics, dead layers prevent screening causing do- mains to appear ( Bjorkstam and Oettel, 1967 ;Kopal et al. , 1999 ;Bratkovsky and Levanyuk, 2000 ). More recently, Luk’yanchuk et al. (2009) proposed that an analogous phenomenon may take place in ferroelastics, so that ‘‘ferroe-lastic dead layers’’ can cause the formation of twins (Fig. 2). Surfaces have broken symmetries and are thus intrinsicallyuncompensated, so interfacial layers are likely to be a general property of all ferroics, including, of course, multiferroics (Marti et al. , 2011 ). B. Kittel’s law For the sake of simplicity, most of this discussion will assume ideal open boundary conditions and no screening ofsurface fields. The geometry of the simplest domain morphol-ogy, namely, stripe domains, is depicted in Fig. 3. Although a FIG. 2. Surface ‘‘dead’’ layers that do not undergo the ferroic transition can cause the appearance of ferroelastic twins in other-wise stress-free films. Dead layers also exist in other ferroics such as ferroelectrics and ferromagnets. From Luk’yanchuk et al. , 2009 . δd yxzY wdY d yxzY wdY FIG. 3 (color online). Schematic of the geometry of 180/C14stripe domains in a ferroelectric or a ferromagnet with out-of-plane polarity.G. Catalan et al. : Domain wall nanoelectronics 121 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012stripe domain is by no means the only possible domain structure, it is the most common ( Edlund and Jacobi, 2010 ) and conceptually the simplest. It also captures the physics ofdomains that is common to all types of ferroic materials. For more specialized analyses, the reader is referred to mono- graphs about domains in different ferroics: ferromagnets(Hubert and Schafer, 1998 ), ferroelectrics ( Tagantsev, Cross, and Fousek, 2010 ), and ferroelastics or martensites (Khachaturyan, 1983 ). Domain size is determined by the competition between the energy of the domains (itself dependent on the boundary conditions, as emphasized above) and the energy of thedomain walls. The energy density of the domains is propor-tional to the domain size: E¼Uw, where Uis the volume energy density of the domain and wis the domain width. Smaller domains therefore have smaller depolarization, de-magnetization, and elastic energies. But the energy gained by reducing domain size is balanced by the fact that this requires increasing the number of domain walls, which are themselvesenergetically costly. The energy cost of the domain walls increases linearly with the number of domain walls in the sample, and there-fore it is inversely proportional to the domain size (n¼1=w). Meanwhile, the energy of each domain wall is proportional to its area and, thus, to its vertical dimension. Ifan individual domain wall stopped halfway through thesample, the polarity beyond the end point of the wall wouldbe undefined, so, topologically, a domain wall cannot dothis; it must either end in another wall (as it does for needledomains) or else cross the entire thickness of the sample. For walls that cross the sample, the energy is proportional to the sample thickness. Thus, the walls’ energy density per unitarea of thin film is E¼/C27d=w , where /C27is the energy density per unit area of the wall. Adding up the energy costs ofdomains and domain walls, and minimizing the total withrespect to the domain size, leads to the famous square root dependence: w¼ffiffiffiffiffiffiffiffi /C27 Udr : (1) Landau and Lifshitz (1935) and Kittel (1946) proposed this pleasingly simple model within the context offerromagnetism, where the domain energy was provided by the demagnetization field (assuming spins pointing out of plane). It is nevertheless interesting to notice that Kittel’sclassic article predicted that pure stripes were in fact ener-getically unfavorable compared to other magnetic domainconfigurations (see Fig. 4); this is because his calculations were performed for magnets with relatively small magnetic anisotropy. Where the anisotropy is large, as in cobalt, stripes are favored, and this is also the case for uniaxialferroelectrics or for perovskite ferroelectrics under in-planecompressive strain (which strongly favors out-of-plane po-larization). Closure domains are common in ferromagnets(where anisotropy is intrinsically smaller than in ferroelec-trics), but the width of the ‘‘closure stripes’’ also scales as the square root of the thickness ( Kittel, 1946 ). We return again to the subject of closure domains toward the end ofthis section, as it has become a hot topic in the area of ferroelectrics and multiferroics. Kittel’s law was extended by Mitsui and Furuichi (1953) for ferroelectrics with 180 /C14domain walls, by Roitburd (1976) for ferroelastic thin films under epitaxial strain, byPompe et al. (1993) and Pertsev and Zembilgotov (1995) for epitaxial films that are simultaneously ferroelec- tric and ferroelastic, and, more recently, by Daraktchiev, Catalan, and Scott (2008) for magnetoelectric multifer- roics. The square root dependence of stripe domain widthon film thickness is therefore a general property of allferroics, and it also holds for other periodic domainpatterns ( Kinase and Takahashi, 1957 ;Craik and Cooper, 1970 ;Thiele, 1970 ). C. Wall thickness and universality of Kittel’s law The exact mathematical treatment of the ‘‘perfect stripes’’ model assumes that the domain walls have zero or at least negligible thickness compared to the width of the domains. Inreality, however, domain walls do have a finite thickness /C14, which depends on material constants ( Zhirnov, 1959 ).Scott (2006) observed that for each given material one could rewrite the square root dependence as w2 /C14d¼G; (2) where Gis an adimensional parameter. This equation is also useful in that it can be used in reverse in order to estimate thedomain wall thickness of any ferroic with well-definedboundary conditions ( Catalan et al. , 2007a ). Indirect versions of it have been calculated for the specific case of ferroelec-trics ( Lines and Glass, 2004 ;De Guerville et al. , 2005 ), but in fact Eq. ( 2) is independent of the type of ferroic and allows comparisons between different material classes. Schilling et al. (2006a) did such a comparison and showed explicitly that, while all ferroics scaled with a square root law, ferro-magnetic domains were wider than ferroelectric domains.Meanwhile, the walls of ferromagnets are also much thickerthan those of ferroelectrics ( Zhirnov, 1959 ), so that when the square of the domain size is divided by the wall thickness as per Eq. ( 2), all ferroics look the same (see Fig. 5), meaning FIG. 4. Kittel’s classic study of the minimum energy of different domain configurations: I are ‘‘closure stripes’’ with no demagneti- zation; II are conventional stripes; and III is a monodomain with the polar direction in plane. Note that in the early calculations formagnetic domains, the conventional stripes were not stable at any finite thickness, due to the small anisotropy assumed. From Kittel, 1946 .122 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012thatGis the same for all the different ferroics. The value of G has been calculated ( De Guerville et al. , 2005 ;Catalan et al. , 2007a ,2009 )1as G¼1:765ffiffiffiffiffiffi/C31x /C31zs ; (3) where Gdepends on the anisotropy between in-plane ( /C31x) and out-of-plane ( /C31z) susceptibilities, but in practice the dependence on material properties is weak because theyare inside a square root. Equation ( 2) is useful in several ways. First, it allows one to estimate domain wall thicknesses just by measuring do-main sizes, and it is easier to measure wide domains than it isto measure narrow domain walls. As we discuss in thefollowing sections, domain wall thicknesses have tradition-ally been difficult to determine precisely due to their narrow- ness (see Sec. III.B ). Second, Eq. ( 2) is also a useful guide asto what the optimum crystal thickness should be in order to stabilize a given domain period, and this may be useful, forexample, in the fabrication of periodically poled ferroelec-trics for enhancement of the second-harmonic generation. Specific examples of this are discussed in detail in Sec. V of this review. Although Eq. ( 2) may appear slightly ‘‘miraculous’’ in that it links in a simple and useful way some quantities that are notat first sight related, closer inspection removes the mystery. Adirect comparison between Eqs. ( 1) and ( 2) shows that at heart, the domain wall renormalization of Kittel’s law is a consequence of the fact that the domain wall surface energydensity /C27is of the order of the volume energy density U integrated over the thickness of the domain wall /C14, i.e., /C27/C24U/C14, which one could have guessed just from a dimensional analysis. We emphasize also that these equa-tions are derived assuming open boundary conditions and are not valid when the surface fields are screened. D. Domains in nonplanar structures Kittel’s simple arguments can be adapted to describe more complex geometries. For instance, one can extend them to calculate domain size in nonplanar structures such as nano-wires and nanocrystals or nanodots. The interest in thesethree-dimensional structures stems originally from the factthat they allow the reduction of the on-chip footprint ofmemory devices. The size of the domains in simple three- dimensional shapes such as, say, a parallelepiped (cuboid) can be readily rationalized by adding up the energy of thedomain walls plus the surface energy of the six faces of theparallelepiped with lateral dimensions d x,dy, and dz. Minimizing this with respect to domain width wleads to (Catalan et al. , 2007b ) w2¼ffiffiffi 2p 2/C27 ðUx=dxÞþðUy=dyÞþðUz=dzÞ; (4) where /C27is the energy per unit area of the domain walls, and Ux,Uy, and Uzare the contributions to the volume energy density coming from the x,y, and zfacets of the domains. Equation ( 4) becomes the standard Kittel law when two of the dimensions are infinite (thin-film approximation). It can alsobe seen that domains become progressively smaller as the sample goes from thin film (one finite dimension) to column (two finite dimensions) to nanocrystal (three finite dimen-sions) ( Schilling et al. , 2009 ). These arguments also work for the grains of a polycrystal- line sample (ceramic or nonepitaxial film), which are gener-ally found to have small domains that scale as the square root of the grain size rather than the overall size dimensions ( Arlt, 1990 ). Arlt also observed and rationalized the appearance of bands of correlated stripe domains, called ‘‘herringbone’’domains (see Fig. 6)(Arlt and Sasko, 1980 ;Arlt, 1990 ). The concept of correlated clusters of domains was latergeneralized for more complex structures as ‘‘metadomains’’or ‘‘bundle domains’’ ( Ivry, Chu, and Durkan, 2010 ), and their local functional response was studied using piezores- ponse force microscopy (PFM) ( Anbusathaiah et al. , 2009 ;100101102103104105106107108 100101102103104105106100101102103104105106107w2(nm2) Rochelle salt (ferroelectric) Rochelle salt (ferroelectric) Co (ferromagnetic) PbTiO3 (ferroelectric) PbTiO3 (ferroelectric) BaTiO3 (ferroelastic)w2/( n m ) film thickness(nm) FIG. 5 (color online). Comparisons between stripe domains of different ferroic materials show (i) that all of them scale with the same square root dependence of domain width on film thickness; (ii) that Kittel’s law holds true for ferroelectrics down to smallthickness; (iii) that when the square of the domain size is normal- ized by the domain wall thickness, the different ferroics fall on pretty much the same master curve. Adapted from Catalan et al. , 2009 . 1We note that different values have been given for the exact numerical coefficient. The discrepancies are typically factors of 2 and are due to the different conventions regarding whether /C14is the domain wall thickness or the correlation length, and whether wis the domain width or the domain period. It is therefore important to carefully define the parameters: Here /C14is twice the correlation length (which is a good approximation to the wall thickness),whereas wis the domain size (half the domain period).G. Catalan et al. : Domain wall nanoelectronics 123 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012Ivry, Chu, and Durkan, 2010 ). Herringbone domains appear only above a certain critical diameter, above which thedomain size dependence gets modified: The stripes scale asthe square root of the herringbone width, while the herring-bone width scales as r 2=3(where ris the grain radius), so that the stripe width ends up scaling as r1=3(Arlt, 1990 ). Randall and co-workers also studied in close detail the domain size dependence within ceramic grains ( Cao and Randall, 1996 ;Randall et al. , 1998 ) and concluded that the square root dependence is valid only within a certain range ofgrain sizes, with the scaling exponent being smaller than 1 2for grains larger than 10/C22m, and bigger than1 2for grains smaller than1/C22m. The same authors observed cooperative switching of domains across grain boundaries, as did Gruverman et al. (1995a ,1995b ,1996) , evidence that the elastic fields associ- ated with ferroelastic twinning are not easily screened and can therefore couple across boundaries. Similar ideas underpin the description of domains in nano- columns and nanowires, where domain size is found to bewell described by Eq. ( 4) with one dimension set to infinity (Schilling et al. , 2006b ). An interesting twist is that the competition between domain energy and domain wall energycan be used not just to rationalize domain size, but to actuallymodulate the orientation of the domains just by changing therelative sample dimensions ( Schilling et al. , 2007 ) (see Fig. 7). These are a few examples, but there is still work to be done. The geometry of domains in noncompact nanoshapes such asnanorings or nanotubes, for example, remains to be rational-ized. The interest in such structures goes beyond purelyacademic curiosity, as ferroelectric nanotubes may have real life applications in nanoscopic fluid-delivery devicessuch as ink-jet printers and medical drug delivery implants. Another important question that is only beginning to be studied concerns the switching of the ferroelectric domainsin such nonplanar structures: Spanier et al. (2006) showed that it was possible to switch the transverse polarization even in ultrathin nanowires (3 nm diameter), while Gregg and co- workers have shown that the longitudinal coercive field canbe modified by introducing notches or antinotches along thewires ( McMillen et al. , 2010 ;McQuaid, Chang, and Gregg, 2010 ). The same group of authors are also pioneering re- search on the static and dynamic response of correlatedbundles of nanodomains, showing that such metadomainscan, to all intents and purposes, be treated as if they were domains in their own right ( McQuaid et al. , 2011 ). E. The limits of the square root law: Surface effects, critical thickness, and domains in superlattices In spite of its simplicity, the square root law holds over a remarkable range of sizes and shapes. It is natural to ask when or whether this law breaks down. For large film thicknessthere is no theoretical threshold beyond which the law shouldbreak down, and, experimentally, Mitsui and Furuichi (1953) observed conformance to Kittel’s law in crystals of millimeter thickness. In epitaxial thin films, however, screening effectsand/or defects have been reported to induce randomness andeven stabilize monodomain configurations in PbTiO 3films thicker than 100 unit cells ( Takahashi et al. , 2008 ). As for the existence of a lower thickness limit, Kittel’s derivation makesa number of assumptions that are size dependent. One of themis that the domain wall thickness is negligible in comparison with the domain size. Domain walls are sharp in ferroelastics and even more so in ferroelectrics ( Merz, 1954 ;Kinase and Takahashi, 1957 ;Zhirnov, 1959 ;Padilla, Zhong, and Vanderbilt, 1996 ;Meyer and Vanderbilt, 2002 ), so that this assumption is robust all the way down to an almost atomic scale ( Fong et al. , 2004 ), but this is not the case for ferromagnets, where domain walls are thicker (10–100 nm).For ferromagnets, Kittel’s law breaks down at film thick- nesses of several tens of nanometers ( Hehn et al. , 1996 ). A second assumption of Kittel’s law is that the two sur- faces of the ferroic material do not ‘‘see’’ each other. That isto say, the stray field lines connecting one domain to itsneighbors are much denser than the field lines connecting one face of the domain to the opposite one. However, if and/ or when the size of the domains becomes comparable to thethickness of the film, the electrostatic interaction with theopposite surface starts to take over ( Kopal, Bahnik, and Fousek, 1997 ).Takahashi et al. (2008) recently suggested that the square root law breaks down at a precise thresholdvalue of the depolarization field. Below that critical thickness,the domain size no longer decreases but it increases again, and diverges as the film thickness approaches zero. Neglecting numerical factors of order unity and also neglect-ing dielectric anisotropy, the critical thickness for a ferro-electric is ( Kopal, Bahnik, and Fousek, 1997 ;Streiffer et al. , 2002 )d C/C25/C27ð"=P2Þ(where "¼"0"ris the average dielec- tric constant), while for ferroelastic twins in an epitaxial FIG. 6. (Left) Classic herringbone twin domain structure in large grains of ferroelastic ceramics, and (right) bundles of correlatedstripes in smaller grains. From Arlt, 1990 . x 300nm 300nmyz dy>dxdy<dx .. .. .. .. ..z xy FIG. 7. Ferroelastic and ferroelectric 90/C14domains in single- crystal nanocolumns of BaTiO 3. The domains arrange themselves so as to have the depolarizing fields only on the narrowest dimen-sion of the column, thus minimizing the overall surface energy. Adapted from Schilling et al. , 2007 .124 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012structure it is ( Pertsev and Zembilgotov, 1995 )dC¼ ½/C27=Gðsa/C0scÞ2/C138, where Gis the shear modulus and saand scare the spontaneous tetragonal strains (along aandcaxes, respectively). The theoretical divergence from the square rootlaw for ferroelastic twins in epitaxial films is shown in Fig. 8. Notice that, as a rule of thumb, these critical thicknesses for domain formation are reached when the size of thedomains becomes comparable to the size of the interfacial dead layers ( Luk’yanchuk et al. , 2009 ). They are typically in the 1–10 nm range, and therefore ferroelectric andferroelastic domains persist even for extremely thin layers,as shown by Fong et al. (2004) for single films and by Zubko et al. (2010) in fine-period superlattices. In the particular case of epitaxial ferroelastics there are further geometrical constraints on the domain size that are not readily captured by continuum theories. Ferroelastic twinningintroduces a canting angle between the atomic planes ofadjacent domains. The canting angle /C11is, for the particular case of 90 /C14twins in tetragonal materials (e.g., BaTiO 3orPbTiO 3),/C11¼90/C14/C02tan/C01ða=cÞ. The existence of this cant- ing angle, combined with the tendency of the bigger domainsto be coplanar with the substrate, introduces a geometrical lower limit to domain size ( Vlooswijk et al. , 2007 ): in order to ensure coplanarity between Bragg planes across the small-est domain, the minimum domain size must be w mina¼ c=sinð/C11Þ(see Fig. 9). For the particular case of PbTiO 3, wmina¼7n m . This geometrical minimum domain size ap- plies only to films that are epitaxial ( Ivry, Chu, and Durkan, 2009 ;Vlooswijk, Catalan, and Noheda, 2010 ). F. Beyond stripes: Vertices, vortices, quadrupoles, and other topological defects A final question regarding the domain scaling issue con- cerns what happens to domains beyond the square root range?Other domain morphologies are possible that can be reached in extreme cases of confinement, or when the polarization is coupled to other order parameters. In the ultrathin-film re-gime, for example, atomistic simulations predict that theperfect 180 /C14domains of ferroelectrics should become akin to the closure configuration of ferromagnets ( Kornev, Fu, and Bellaiche, 2004 ;Aguado-Fuente and Junquera, 2008 ) (see Fig. 10). It may seem preposterous to care about a domain structure that takes place only in films that are barely a few unit cells thick, but with the advent of ferroelectric super- lattices these domains become accessible, as the thickness ofeach individual layer in the superlattice can be as thin as onesingle unit cell ( Dawber et al. , 2005 ;Zubko et al. , 2010 ). In the weak-coupling regime, the ferroelectric slabs within the superlattice act as almost separate ultrathin entities(Stephanovich, Luk’yanchuk, and Karkut, 2005 ), so that it is quite possible that these closure stripes are achieved. It is worth noticing that the orientation of the in-plane component of the polarization is such that, if the domain walls werepushed toward each other, there would be a head-to-headcollision of polarizations; the electrostatic repulsion between these in-plane components might explain why it seems to be almost impossible to eliminate the domain walls in ferroelec-tric superlattices ( Zubko et al. , 2010 ). On a related note, while the 180 /C14domain walls of ferro- electrics have traditionally been considered nonchiral (i.e., the polarization just decreases, goes through zero, and in- creases again, but does not change orientation through thewall), recent calculations challenge this view and show thatthey do have some chirality, i.e., the polarization rotates within them as in a magnetic Bloch wall ( Lee et al. , 2009 ). Therefore, when the domains are sufficiently small to becomparable to the thickness of the walls, the end result willbe indeed something resembling the closure stripe configu- ration of Fig. 10. The existence of this domain wall chirality might seem surprising, but it was explained two decadesago by Houchmandzadeh, Lajzerowicz, and Salje (1991) :I f there is more than one order parameter involved in a ferroic (and perovskite ferroelectrics are always ferroelastic as well as ferroelectric), then the coupling introduces chi-rality. This, of course, is also true of magnetoelectric multi-ferroics ( Seidel et al. , 2009 ;Daraktchiev, Catalan, and Scott, 2010 ). The theoretical prediction of ferroelectric closurelike structures where domain walls meet an interface has been FIG. 9 (color online). Schematic of the geometrical minimum domain size in a tetragonal twin structure such that widercdomains are coplanar with the substrate while the narrow adomains are tilted with the inherent twinning angle /C11. From Vlooswijk, Catalan, and Noheda, 2010 . FIG. 8. Calculated domain size for 90/C14ferroelastic domains in an epitaxial film as a function of film thickness. Below a certain critical thickness the domain size stops following the square root depen-dence and begins to diverge. This critical thickness is of the order of the domain wall thickness. From Pertsev and Zembilgotov, 1995 .G. Catalan et al. : Domain wall nanoelectronics 125 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012experimentally confirmed by two different groups ( Jiaet al. , 2011 ;Nelson et al. , 2011 ) (see Fig. 11). It is worth noticing here that the arrangements in Fig. 11 are not a classic fourfold closure structure, with four walls at90 /C14converging in a central vertex. Instead, these domain structures should be seen as half of a closure quadrant. Thebifurcation of a quadrant into two threefold vertices, withwalls converging at angles of 90 /C14and 135/C14, was predicted by Srolovitz and Scott (1986) ; their schematic depiction of the bifurcation process is reproduced in Fig. 12. The reverse process of coalescence of two threefold vertices to formone fourfold vertex, also predicted by Srolovitz and Scott, was recently observed in BaTiO 3by Gregg et al. (private communication). Vertices are a topological singularity closely related to vortices, the main difference being that avortex implies flux closure, whereas a vertex is just a con-fluence of domain walls; some vertices are also vortices (e.g.,the vertices of 90 /C14closure quadrants in ferromagnets and ferroelectrics), but others are not. Vortices are frequently observed in ferromagnetic nanodots (Shinjo et al. , 2000 ). At the vortex core, the spin must necessarily point out of the plane of the nanodot: This out-of-plane magnetic singularity is extremely small, yet stable,and could therefore be useful for memories. Ferroelectricvortices are also theoretically possible ( Naumov, Bellaiche, and Fu, 2004 ), and Naumov and co-workers predicted that such structures are switchable and should yield an unusuallyhigh density of ‘‘bits’’ for memory applications ( Naumov et al. , 2008 ). So far, there is tantalizing experimental evidence for vor- tices in ferroelectrics ( Gruverman et al. , 2008 ;Rodriguez et al. , 2009 ;Schilling et al. , 2009 ). However, although vortices almost certainly appear as transients during switch- ing ( Naumov and Fu, 2007 ;Gruverman et al. , 2008 ;Sene et al. , 2009 ), it is difficult to observe static ferroelectric vortices, or even just closure structures, in conventionaltetragonal ferroelectrics. This is because a simple quadrantarrangement generates enormous disclination strain ( Arlt and Sasko, 1980 ) (see Fig. 13); for dots above a certain critical FIG. 11 (color online). Observation of closurelike polar arrange- ments at the junction between ferroelectric domain walls and aninterface, for thin films of BiFeO 3(left) and PbTiO 3(right). Note that the wall angles are 135/C14,90/C14,135/C14, as in the Srolovitz-Scott model, not 120/C14. Adapted from Nelson et al. , 2011 (left) and Jia et al. , 2011 (right). FIG. 12. A fourfold vertex in a 90/C14quadrant is predicted by a Pott’s model to bifurcate into two threefold vertices. From Srolovitz and Scott, 1986 .Above Tc Below Tc FIG. 13. (Left) Schematic illustration of the disclination stresses that are generated in the center of a closure structure of a tetragonal ferroelectric or ferroelastic; (right) experimental observation that ferroelastic stripes appear within the quadrants, probably in order toalleviate the stress. Adapted from Schilling et al. , 2009 . FIG. 10 (color online). Ferroelectric ‘‘closure stripes’’ predicted by atomistic simulations of ultrathin films. From Kornev, Fu, and Bellaiche, 2004 andAguado-Fuente and Junquera, 2008 .126 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012size, alleviation of associated stresses will be provided by the formation of ferroelastic stripe domains within each quadrant.A back-of-the-envelope calculation allows us to estimate thesize at which the stripes will break the quadrant configuration. We do so by comparing the elastic energy stored within a single quadrant domain with the energy cost of a domainwall. The dimensions of the nanodot are L/C2L/C2d,Gis the elastic shear modulus, and "is the disclination strain, which is of the same order of magnitude as the spontaneous strain.The elastic energy density stored in a quadrant of volume L 2d=4is given by Eelastic¼1 2Gs2L2d 4: (5) The energy cost of the first wall to divide the quadrant is the surface energy density of the wall ( /C27) times the area of the new domain wall: Ewall¼/C27ffiffiffi 2p 2Ld: (6) When these two quantities are equal, the quadrant configura- tion stops being energetically favorable. By making Eelastic¼ Ewallwe therefore obtain an approximate critical size L¼4ffiffiffi 2p/C27 Gs2; (7) which, for the case of BaTiO 3(G¼55 GPa ,/C27¼ 3/C210/C03J=m2, and s¼0:01), gives a critical size of only 3 nm. That small size explains why in larger ferroelectricnanocubes one observes a quadrantlike structure split bymultiple ferroelastic stripes ( Schilling et al. , 2009 ) (see Fig. 13). More recently, ferroelectric flux closure has been confirmed in metadomain formations consisting of finely twinned quadrants ( McQuaid et al. , 2011 ). Equation ( 7) shows that, in order to find a ‘‘pure’’ (non- twinned) ferroelectric quadrant structure, one will have tolook for ferroelectrics with small spontaneous strain and highdomain wall energy. BiFeO 3has a large domain wall energy (Catalan et al. , 2008 ;Lubk, Gemming, and Spaldin, 2009 ) due to the coupling of polarization to antiferrodistortive andmagnetic order parameters ( BiFeO 3is simultaneously ferro- electric, ferroelastic, ferrodistortive, and antiferromagnetic),while at the same time its piezoelectric deformation is small.That helps stabilize closure structures in this material ( Balke et al. , 2009 ;Nelson et al. , 2011 ). In purely magnetic materials, of course, vortex domains are well known and even their switching dynamics are nowbeing studied, as illustrated in Fig. 14: Note that this figure shows that one can create magnetic vortex domains by re-petitive application of demagnetizing fields to single-domainsoft magnets. Similarly, Ivry et al. (2010) observed that application of depolarizing electric fields has a similar effect in ferroelectrics. As mentioned earlier, a close relative of vortices and closure domains is what we call ‘‘vertex’’ domains. A vertexis the intersection between two or more domain walls in aferroic. In the classic quadrant structure, the vertex is a four-fold intersection between 90 /C14domains, while in a needle domain the vertex is a twofold intersection. It is important to note that each of the domain walls intersecting the vertex isequivalent through symmetry; that is, they cannot be different walls, such as (011) and (031), a point to which we return below. Using topological arguments, Janovec (1983) showed that the number Nof domain walls intersecting at the vertex is equal to the dimensionality of the order parameter. Janovec and Dvorak further developed the theory in a longer review in 1986. However, complicating the general theory of Janovec is the fact that several order parameters might coexist (as inmultiferroic materials), and that the domains do not neces- sarily have the same energy. The energetics and stability of vertex domains were ana- lyzed by Srolovitz and Scott (1986) using Potts and clock models. They showed that fourfold vertices, such as are found inBa 2NaNb 5O15(Pan et al. , 1985 ) can, in some materials, spontaneously separate into pairs of adjacent threefold verti- ces. There is an apparent paradox regarding closure domainsbetween the group theoretic predictions of Janovec (1983) andJanovec and Dvorak (1986) , and the clock-model calcu- lations of Srolovitz and Scott (1986) . In particular, Janovec states that threefold closure vertices are forbidden, whereas Srolovitz and Scott show that they may be energetically favored over fourfold vertices. The paradox is reconciled as follows: What Janovec specifically forbids are isolated three- fold vertices with three 120 /C14angles between the domain walls. What Scott and Srolovitz predict is a separation of energetically metastable fourfold vertices into closely spaced pairs of threefold vertices; but these pairs each consist of one original 90/C14angle between domain walls, and two 135/C14 angles along the line between the vertex pairs. Hence this FIG. 14. Dynamic response of magnetic vortices, from the work of Cowburn’s and co-workers: (a)–(e) Hysteresis curves showing the decay of a single-domain state into a vortex state via a series of minor hysteresis cycles. The entire decay process is shown in (a).The arrowed solid line indicates the direction of the transition fromsingle domain to vortex state. The dashed line outlines the Kerr signal corresponding to the positive and negative applied saturation fields. The first three and the last demagnetizing cycles are dis-played in separate panels; (b) first cycle, (c) second cycle, (d) third cycle, and (e) 18th cycle. From Ana-Vanessa, Xiong, and Cowburn, 2006 .G. Catalan et al. : Domain wall nanoelectronics 127 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012should properly be regarded as not a threefold domain vertex but rather a fourfold vertex that has separated slightly at itscenter. This phenomenon is analogous to the separation of thefourfold closure domains in BaTiO 3. Another example of vertex structures that does not satisfy the basic model of Janovec is that in thiourea inclusion compounds ( Brown and Hollingsworth, 1995 ). In this case inclusions in a thiourea matrix result in large strains (straincoupling is not directly included in the Janovec model). Theresult, illustrated in Fig. 15, is a beautiful 12-fold vertex structure. Note that this is despite the fact that the orderparameter is of N¼2dimensions in thiourea ( Toledano and Toledano, 1987 ). The reason is strain. The domain cluster shown in thiourea is of domain walls of different symmetry,notably f130gandf110g. Yet another example of domain wall vertices is provided by the charge density wave domainsobserved by Chen, Gibson, and Fleming (1982) in 2H-TaSe 2(see Fig. 15); this system, with three spatial in- plane orientations and þand/C0out-of-plane distortions, is equivalent to ferroelectric YMnO 3. Both violate the simpler requirement described by Janovec (1983) that the number of domains Nat a vertex must equal the dimensionality nof the order parameter and require incorporation of coupling termsplus energy considerations to determine the equilibriumstructure, as done by Janovec et al. (1985 ,1986) .O na more general level, Saint-Gregoire et al. (1992) showed that domain wall vertex structure classifications consist of36 twofold vertices with five equivalence classes, 96 fourfoldvertices of ten classes, and 63 sixfold vertices of nine classes. It is notable that, even where walls carrying opposite þP zand /C0Pzpolarizations meet, the vertex can still have a polar point group (rod) symmetry, which is not intuitively obvious, butcan be useful as these rods are analogous in this respect to thepolar singularity at the core of a vortex. Note also that the so-called layer groups, such as 2 z0, keep the central plane of a wall invariant, whereas the other groups do not. Rod groups can be chiral; for example, a regular sixfold vertex withsymmetry 6 z0has a helical structure with polarization along z. There are two equivalent sixfold vertices with the same helicity by opposite polarization; the chirality does not dictatethe polarization. This situation is also encountered in multiferroic YMnO 3. Although the sixfold vertices of YMnO 3were observed long ago by Safrankova, Fousek, and Kizhaev (1967) , interest has been rekindled by more recent studies studying these forma-tions in detail ( Choi et al. , 2010 ;Jungk et al. , 2010 ) (see Fig.16). The correct domain analysis requires the tripled unit cell of Fennie and Rabe (2005) for proper description, and not the simpler primitive cell proposed by Van Aken et al. (2004) . The coupling of ferroelectricity to the other order parameters (antiferromagnetic and antiferrodistortive) yieldsthe required dimensionality for the sixfold vertices toform. YMnO 3is also interesting because its domain walls are less conducting than the domains ( Choi et al. ,2 0 1 0 ), which is the exact opposite of what happens in the otherpopular multiferroic, BiFeO 3(Seidel et al. ,2 0 0 9 ). The issue of domain wall conductivity is extensively discussed in latersections. Recently, the functional properties of vertices and vortices are also starting to be studied. In the case of BiFeO 3, for example, it has been found that the conductivity of ferroelec-tric vortices is considerably higher than that of the domainwalls, which are in turn more conductive than the domains(Balke et al. , 2011 ). G. Nanodomains in bulk Kittel’s law implies that small domains can appear in small or thin samples, but nanodomains occur in some bulk com-positions. Trivially, any material with a first-order phasetransition will experience the nucleation of small nonperco-lating domains above the nominal T c. In the case of BaTiO 3, these can occur more than 100/C14above Tc(Burns and Dacol, 1982 ). This, however, has little implication for the functional FIG. 15 (color online). (Left) Twelvefold ferroelectric domain vertex in thiourea. From Brown and Hollingsworth, 1995 . (Right) Sixfold vertex intersection between charge density wave domains in2H-TaSe 2(Chen, Gibson, and Fleming, 1982 ). Schematic in (a) and actual microscopy image in (b). These formations are topologically equivalent to the vertex domains YMnO 3. FIG. 16 (color online). Observation of sixfold vertices in domain ensembles of multiferroic YMnO 3: (left) from Safrankova, Fousek, and Kizhaev, 1967 ; (middle) from Choi et al. , 2010 , and (right) from Jungk et al. , 2010 .128 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012properties because the volume fraction occupied by such nanodomains is small. But there are other material familieswhere nanodomains are inherent. These are nearly alwayslinked to systems with competing phases and frustration, andthe functional properties of nanoscopically disordered mate-rials are often striking: colossal magnetoresistance in man- ganites, superelasticity in tweedlike martensites, and giant electrostriction in relaxors, to name a few. Relaxors combine chemical segregation at the nanoscale and nanoscopic polar domains ( Cross, 1987 ;Bokov and Ye, 2006 ). The key technological impact of these materials lies in their large extension under applied fields for piezoelectricactuators and transducers ( Park and Shrout, 1997 ). Despite many papers on the basic physics of relaxor domains, aholistic theory is still missing. The presence of polar domainsin the cubic phases of relaxors, where they are nominallyforbidden, may be caused by flexoelectricity and internalstrains due to local nonstoichiometry ( Ahn et al. , 2003 ). When mixed with ordinary ferroelectrics such as PbTiO 3,o r subjected to applied fields E, these nanodomains increase in size to become macroscopic ( Mulvihill, Cross, and Uchino, 1995 ;Xu et al. , 2006 ). As for the shape of the domains, in pure PbZn 1=3Nb2=3O3(PZN), the domain walls may be spindlelike ( Mulvihill, Cross, and Uchino, 1995 ) or dendritic (Liu, 2004 ) but become increasing lamellar with increasing additions of PbTiO 3. The condensed h110idomain structure is stable in perovskites and rather unresponsive to fields E along [111] ( Xuet al. , 2006 ), and the polar nanoregions arise from a condensation of a dynamic soft mode along [110], asshown via neutron spin-echo techniques ( Matsuura et al. , 2010 ). Multiferroic (magnetoelectric) relaxors also exist (Levstik et al. , 2007 ;Kumar et al. , 2009 ), but little is yet known about their domains. From the perspective of this review, the key point about relaxors is that, since they are formed by nanodomains, theymust have a large concentration of domain walls. It is there-fore reasonable to expect that the domain walls contribute to the extraordinary electromechanical properties of these ma- terials. Rao and Yu (2007) show that indeed there is an inverse correlation between domain size and piezoelectricbehavior, and suggest that the linking mechanism is a field-induced broadening of the domain walls. On the other hand,domain walls may contribute not only by their static proper-ties or broadening, but also by their dynamic response(motion) under applied electric fields, as suggested by the Rayleigh-type analyses of Davis, Damjanovic, and Setter (2006) andZhang et al. (2010) . Polar nanodomains also exist in another nonpolar material, SrTiO 3, which is important as it is the most common substrate for growing epitaxial films of other perovskites. SrTiO 3is cubic at room temperature, but tetragonal and ferroelastic below 105–110 K ( Fleury, Scott, and Worlock, 1968 ). It is also an incipient ferroelectric whose transition to a macro- scopic ferroelectric state is frustrated by quantum fluctuationsof the soft phonon at low temperature; hence, the material is also called a ‘‘quantum paraelectric’’ ( Muller and Burkard, 1979 ). By substituting the oxygen in the lattice for a heavier isotope, 18O, the lattice becomes heavier, and the phonon slows down and freezes at a higher temperature, causing aferroelectric transition ( Itoh et al. , 1999 ). However, polar nanodomains have been detected even in the normal 16O composition of SrTiO 3(Uesu et al. ,2 0 0 4 ;Blinc et al. , 2005 ), and their local symmetry is triclinic and not tetragonal (Blinc et al. , 2005 ). The ferroelectric phase of the heavy- isotope composition is also poorly understood, but it has finely structured nanodomains reminiscent of those observed in relaxors ( Uesu et al. , 2004 ;Shigenari et al. , 2006 ), while relaxorlike behavior has also been observed in SrTiO 3thin films ( Jang et al. , 2010 ). Again, the high concentration of domain walls concomitant with this fine domain structure shows important implications for functionality, since the domain walls of SrTiO 3are thought to be polar ( Tagantsev, Courtens, and Arzel, 2001 ;Zubko et al. , 2007 ). We also note thatSrTiO 3at low temperatures has giant electrostriction comparable to that observed in relaxor ferroelectrics ( Grupp and Goldman, 1997 ). The above are examples of nanodomains that appear spon- taneously in some special materials. But nanodomains canalso be made to appear in conventional ferroelectrics by clever use of poling. Fouskova, Fousek, and Janous ˇek estab- lished that domain wall motion enhanced the electric-field response of ferroelectric material ( Fouskova, 1965 ;Fousek and Janous ˇek, 1966 ), and domain engineering of crystals yields a piezoelectric performance far superior to that of normal ferroelectrics ( Zhang et al. , 1994 ;Eng, 1999 ; Bassiri-Gharb et al. , 2007 ). However, a newer and more relevant twist is that even static domain walls may signifi- cantly enhance the properties of a crystal, due to the superior FIG. 17 (color online). Measurements and calculations relating decreased domain size (and thus increased domain wall concentration) to enhancement of piezoelectricity in BaTiO 3single crystals. The results suggest that the increased piezoelectric coefficient is due to the internal piezoelectricity of the domain walls. From Hlinka, Ondrejkovic, and Marton, 2009 .G. Catalan et al. : Domain wall nanoelectronics 129 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012piezoelectric properties of the domain wall itself (see Fig. 17). The concept of ‘‘domain wall engineering’’ was introduced by Wada and co-workers as a way to enhance thepiezoelectric performance of ferroelectric crystals ( Wada et al. , 2006 ). At present, however, the size or even the exact mechanism whereby domain walls contribute to the piezo-electric enhancement is still a subject of debate ( Hlinka, Ondrejkovic, and Marton, 2009 ;Jin, He, and Damjanovic, 2009 ). H. Why does domain size matter? The above is a fairly comprehensive discussion of the scaling of domains with device size and morphology. Themain take-home message is that, as device size is reduced,domain size decreases in a way that can often be described by Kittel’s law in any of its guises. Thus, the concentration of domain walls will increase. We can quantify this domain wallconcentration fairly easily: Let us just rearrange the terms ofthe ‘‘universal’’ Kittel’s law, [Eq. ( 2)]: /C14 w¼ffiffiffiffiffiffiffi /C14 Gds : (8)This equation shows that, as the film thickness ddecreases, the fraction /C14=w (i.e., the fraction of the material that is made of domain walls) increases. Taking standard values for the domain wall thickness /C14(typically 1–10 nm), we can see that, for 100-nm-thick films, between 6% and 20% of the film’s volume will be domain walls. Of course, as mentioned before,this percentage assumes that the surface energy is unscreened, so a correction factor must be applied when there is partial screening (the most general case). However, Eq. ( 8) is not completely unrealistic: Strain, for example, cannot be screened at all, and therefore ferroelastic domains (which inperovskite multiferroics tend to be ferroelectric and/or mag- netic as well) can indeed be small. By way of illustration, consider the extremely dense ferroelastic domain structure in Fig. 18. The high concentration of domain walls is important be- cause domain walls not only have different properties from domains but, for specific applications, they can in fact be better ( Wada et al. , 2006 ). A sufficiently large number density of walls can therefore lead to useful emergent behav- ior in samples with nanodomains. This idea is barely in itsinfancy, but already there are hints that it could work. Daumont and co-workers, for example, report a strong corre- lation beween the macroscopic magnetization of a nominally antiferromagnetic thin film, and its concentration of domain walls (see Fig. 18). The rest of this review will discuss the properties of domain walls, the experimental tools used to characterize them, and their possible technological applications. III. DOMAIN WALLS A. Permissible domain walls: Symmetry and compatibility conditions Polar ferroics are those for which an inversion symmetry is broken: space inversion for ferrroelectrics or time inversion for ferromagnets. In these cases, domain walls separating regions of opposite polarity are possible, and they are called 180/C14walls (in reference to the angle between the polar vectors on either side of the wall). 180/C14walls tend to be parallel to the polar axis, so as to avoid head-to-head con- vergence of the spins or dipoles at the wall, as these are energetically costly due to the magnetic or electrostatic re- pulsion of the spins or dipoles. It is nevertheless worth mentioning that, although energetically costly, head-to-head180 /C14walls are by no means impossible. 180/C14head-to-head domains have been studied for decades in ferroelectrics. When they annihilate each other, large voltage pulses are emitted, called ‘‘Barkhausen pulses’’ ( Newton, Ahearn, and McKay, 1949 ;Little, 1955 ); these voltage spikes are orders of magnitude larger than thermal noise. Most recently, head-to- head (charged) 180/C14walls have been directly visualized using high-resolution transmission electron microscopy and found to be about 10 times thicker than neutral walls ( Jia et al. , 2008 ) (see Fig. 19). The difference in thickness be- tween neutral and charged walls was historically first ob- served by Bursill, who noted the bigger thickness of thelatter ( Lin and Bursill, 1982 ;Bursill, Peng, and Feng, 1983 ; Bursill and Peng, 1986 ). According to Tagantsev (2010) , this FIG. 18 (color online). (Top) The ferroelastic domains of ortho- rhombic TbMnO 3film grown on cubic SrTiO 3are so small (/C255n m ) as to be comparable to the domain wall thickness, so that approximately 50% of the material is domain wall. (Bottom) The same authors report a strong correlation between inverse domain size (and thus domain wall concentration) and remnantmagnetization in the films. From Daumont et al. , 2009 ,2010 .130 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012increased thickness is due to the aggregation of charge car- riers at the wall in order to screen the strong depolarizing fieldof the head-to-head dipoles. An interesting corollary to thisobservation is that the thickness of charged domain walls insemiconducting ferroelectrics will be different depending on whether they are head to head or tail to tail, due to the different availability of majority carriers; for example, in ann-type semiconductor, there is an abundance of electrons, and so head-to-head domain walls can be efficiently screened,while tail-to-tail cannot, meaning that the latter will bebroader ( Eliseev et al. , 2011 ). Domain wall thickness is further discussed in Sec III.B . The order parameter in ferroelastic materials is the sponta- neous strain, which is not a vector but a second-rank tensor.Since the spontaneous strain tensor does not break inversion symmetry, purely ferroelastic materials do not have 180 /C14 domains. Instead, a typical example of ferroelastic domains (also called twins) is the 90/C14twins in tetragonal materials, where the spontaneous lattice strains in adjacent domains areperpendicular. In the case of 90 /C14domains, the locus of the wall is the bisector plane at 45/C14with respect to the f001g planes, because along these planes the difference between the spontaneous strains of the adjacent domains is zero, and thus the elastic energy cost of the wall is minimized (also knownas the invariant plane). In the case of multiferroics that aresimultaneously ferroelectric and ferroelastic, the polar com-patibility conditions (e.g., no head-to-head polarization) mustbe added to the elastic ones. Fousek and Janovec did preciselythat and compiled a table of permissible domain walls in ferroelectric and ferroelastic materials ( Fousek and Janovec, 1969 ;Fousek, 1971 ). Whenever the domains are in an epi- taxial thin film, there are further elastic constraints imposed by the substrate, as analyzed in the paper by Speck and Pompe (1994) . A case study of permissible walls in epitaxial thin films of rhombohedral ferroelectricis was done by Streiffer et al. (1998) , and this is relevant for BiFeO 3(space group R3c). In this case, the polar axis is the pseudocubic diagonal h111i, and domain walls separating inversions of one, two, or all three of the Cartesian components of thepolarization are possible (these are called, respectively, 71 /C14, 109/C14, and 180/C14walls). More generally, Aizu (1970) explained that the number of ferroic domain states, and thus of possible domain walls, is given by the ratio of the point group orders of the high- and low-symmetry phases, although Shuvalov, Dudnik, and Wagin (1985) argued that a higher number of domains (‘‘superorientational states’’) may be permissible than given by the Aizu rule, as indeed observed in ferroelastic YBa 2Cu3O7/C0/C14(Schmid et al. , 1988 ). Another important rule is given by Toledano (1974) : It is necessary and sufficient for ferroelastic phase transitions that the crystal undergoes a change in crystal class (trigonal and hexagonal is regarded as a single superclass in this argument). The converse of that rule is that if there is no change in crystal class, then thematerial is not ferroelastic, and thus naturally there will not be any ferroelastic twin walls. Further restrictions apply to the type of domain walls that can exist in magnetoelectric mate- rials ( Litvin, Janovec, and Litvin, 1994 ). Because these rules place strict conditions on what types of walls can exist in a ferroic, domain wall taxonomy can help clarify not only the true symmetry of the ferroic phase in a material, but also its relationship with the paraphase. Anillustrative example is yet again BiFeO 3: The classification of its domain walls allowed the determination that the high- temperature /C12phase (above 825/C14C) was orthorhombic (Palai et al. ,2 0 0 8 ). The existence of orthorhombic twins was also used by Arnold et al. (2010) to argue that the highest-symmetry phase of BiFeO 3should be cubic, even though this cubic phase may be ‘‘virtual,’’ as it probably occurs above the (also orthorhombic) /C13phase and beyond the melting temperature in most samples; however, Palai et al. (2010) found Raman evidence that a reversible orthorhombic-cubic transition exists in some specimens. The determination of this ‘‘virtual paraphase’’ is not trivial,since previously other authors had argued that the ultimate paraphase of BiFeO 3should be hexagonal R3c,a si n LiNbO 3(Ederer and Fennie, 2008 ), which is likely incorrect. The correct determination of the paraphase symmetry is of utmost importance because polar displacements are mea-sured with respect to it. B. Domain wall thickness and domain wall profile Experimentally, domain wall thicknesses can be measured accurately only by using atomic-resolution electron micros- copy techniques. Theoretical estimates can be obtained usinga variety of methods, ranging from ab initio calculations to phenomenological treatments or pseudospin models. We FIG. 19 (color online). High-resolution transmission electron mi- croscopy image of a head-to-head charged domain wall in ferro-electric PbðZr;TiÞO 3. The domain wall is found to be approximately 10 unit cells thick, which is about 10 times thicker than for normal (noncharged) ferroelectric domain walls. From Jia et al. , 2008 , Nature Publishing Group.G. Catalan et al. : Domain wall nanoelectronics 131 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012begin this section by offering a simple physical model that captures the essential physics of domain wall thickness. The volume energy density of any ferroic material has at least two components: one from the ordering of the ferroic order parameter, and one from its gradient. Inside the do- mains, there is no gradient, and so only the homogeneous partof the energy has to be considered. The leading term in thisenergy is quadratic: U¼ 1 2/C31/C01P02for ferroelectrics, 1 2/C22/C01M02for ferromagnets, and1 2Ks02, where Kis the elastic constant and s0is the spontaneous strain. Meanwhile, inside the domain walls there is a strong gradient whose energy contribution is also quadratic, since it obviously cannot de-pend on whether you cross the wall from left to right orvice versa. Although the exact shape of the gradient is bestdescribed as a hyperbolic tangent, as a first approximationone can linearize the polarization profile across the wall asPðxÞ¼P 0½x=ð/C14=2Þ/C138(/C0/C14=2<x< /C0/C14=2). In this linearized approximation, the gradient is simply the switched polariza- tion (or magnetization, or strain) divided by the wall thick-ness, and hence the gradient energy can be approximated as 1 2kð2P0=/C14Þ2(where kis the gradient coefficient or ‘‘ex- change’’ constant, since it measures the energy cost of locally changing the order parameter with respect to its nearest neighbors). Although in this discussion we use polarization as an example, all the equations and conclusions are valid forany other type of ferroic material. The wall also has a contribution from the ferroic ordering, which changes across the wall: It is zero exactly at the centerof the wall and it grows to reach the saturation value at the beginning and end of the wall. The energy density per unit area of the wall is obtained by integrating the two energyterms across its thickness. Hence, /C27¼Z /C14=2 /C0/C14=2/C201 2k/C182P0 /C14/C192 þ1 2/C31/C01PðxÞ2/C21 dx ¼2kP02 /C14þ1 6/C31/C01P02/C14: (9) The actual domain wall thickness will be that which mini- mizes this domain wall energy density; hence @/C27 @/C14¼0¼/C02kP02 /C142þ1 6/C31/C01P02; (10) which leads to /C14¼2ffiffiffi 3pffiffiffiffiffiffi k/C31p : (11) More elaborate phenomenological treatments are based on Landau theory ( Zhirnov, 1959 ), the simplest potential being G¼a 2P2þb 2P4þk 2/C18@P @x/C192 : (12) Variational minimization of the order parameter across a domain wall at x¼0yields tanhðx=/C21Þ, with the correlation length /C21¼2P/C01 0ffiffiffiffiffiffiffiffiffiffiffi 2k=bp . Using P0¼ffiffiffiffiffiffiffiffiffiffiffiffiffi /C0a=bp and/C31¼ /C01=2aand defining the domain wall thickness as /C14¼2/C21, we get /C14¼2ffiffiffiffiffiffiffiffiffiffi ffi /C02k as ¼4ffiffiffiffiffiffi /C31kp : (13)Note the remarkable similarity between Eqs. ( 11) and ( 13), despite the linear simplification assumed in the former. An issue that appears to have been neglected by most [but not all, see Tagantsev, Courtens, and Arzel (2001) ] phenome- nological analyses of ferroelectric and ferroelastic domainwalls is that the existence of large strain gradients at the walls must necessarily lead to considerable flexoelectricity inside them. Zubko (2008) performed some preliminary calculations for the gradients inside the ferroelastic domain walls ofSrTiO 3, using the strain profile calculated by Cao and Barsch (1990) . The results are shown in Fig. 20. Assuming a ferroelastic correlation length /C24/C244/C23A(one unit cell) and a flexoelectric coefficient of 10/C08(Zubko et al. , 2007 ) (mea- sured at room temperature and therefore smaller than the low- temperature value), the flexoelectric polarization in the middle of the domain wall is of the order of 5m C =m2 (0:5/C22C=cm2), which is not negligible. This is only an approximate result, however, because the flexoelectric inter- action must be incorporated into the domain wall structurecalculations in a self-consistent manner, rather thana posteriori . We parenthetically note that strain gradients are significant at the nanoscale, and therefore flexoelectric effects are expected to be important. The large flexoelectriceffects associated with nanodomains in ferroelectric thin (a) (b) FIG. 20 (color online). (a) Ferroelastic strain components in the low-temperature domain walls of SrTiO 3and (b) flexoelectric polar- izations caused by the strain gradients in the walls. From Zubko, 2008 .132 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012films are currently being studied by several groups ( Catalan et al. , 2011 ;Lee et al. , 2011 ;Luet al. , 2011 ). At any rate, putting typical values into Eq. ( 13), it is found that ferroelectrics and ferroelastics have typical domain wall thicknesses in the range of 1–10 nm, whereas ferromagnetshave typically thicker walls in the range 10–100 nm. Thisdifference in thickness is not entirely surprising: Wall thick-ness is given by the competition between exchange andanisotropy (in ferromagnets) with the corresponding terms being dipolar energy and elastic anisotropy energy (in ferro- electrics). The exchange constant measures the energy cost oflocally changing a spin, a dipole, or an atomic displacement(depending on the type of ferroic) with respect to its neigh-bors; in phenomenological treatments this was introduced as the energy cost of creating a gradient in the ferroic order parameter, exchange ¼ðk=2ÞðrPÞ 2. If this energy is big, the ferroic will try to reduce the size of the gradient by increasingthe thickness of the domain wall. Likewise, the softness of the order parameter (its suscep- tibility) will also tend to broaden the walls: A material that has high susceptibility, dielectric, magnetic, or elastic, allows its order parameter to fluctuate more easily, meaning thatbroad domain walls, with a large number of unit cells de-parted from the equilibrium value, are still relatively cheap.Zhirnov (1959) offered a similar argument: The anisotropy measures the energy cost of misaligning the order parameter with respect to the crystallographic polar axes; if this energyis big, the ferroic will try to minimize the number of mis-aligned spins, dipoles, and strains by making the wall as thinas possible. Because both ferroelectricity and ferroelasticityare, at heart, structural properties, their anisotropy (arising from structural anisotropy such as, e.g., the tetragonality of a perovskite ferroelectric) will normally be larger than that offerromagnets, and thus their wall thickness will be smaller.Hlinka (2008) andHlinka and Marton (2008) have recently discussed the role of anisotropy on the domain wall thickness of the different phases of ferroelectric BaTiO 3. The anisot- ropy argument is completely analogous to the susceptibilityone, just by realizing that susceptibility is inversely propor-tional to anisotropy. It follows from the above that materialsthat are uniaxial and have small susceptibility should have far narrower domain walls than ferroics with several easy axes (so that they are more isotropic) and large permittivity; inparticular, one may expect morphotropic phase boundaryferroelectrics to have anomalously thick domain walls, sothat a significant volume fraction of the material may be madeof domain walls. This is also the case for ultrasoft magnetic materials such as permalloys, or structurally soft materials such as some martensites and shape-memory alloys ( Ren et al. , 2009 ). Domain wall thickness has traditionally been a contentious issue for ferroelectrics, where it has been hard to measure experimentally. The earliest electron microscopy measure- ments were reported by Blank and Amelinckx (1963) , and they placed an upper bound of 10 nm on the 90 /C14wall thickness of barium titanate. Bursill and co-workers ( Lin and Bursill, 1982 ;Bursill, Peng, and Feng, 1983 ,Bursill and Lin, 1986 ) used high-resolution electron microscopy to confirm that the domain walls of LiTaO 3andKNbO 3are indeed thin and atomically sharp in the case of 180/C14walls.Meanwhile, Floquet et al. (1997) combined high-resolution transmission electron microscopy with x-ray diffraction tomeasure a width of 5 nm for the 90 /C14walls of BaTiO 3.Shilo, Ravichandran, and Bhattacharya (2004) used atomic force microscopy (AFM) to measure the same type of walls in PbTiO 3; although the tip radius of scanning probe micro- scopes (AFM, PFM) is typically 10 nm, a careful statisticalanalysis allowed the intrinsic domain widths of ferroelectricand ferroelastic 90 /C14walls to be extracted; a wide range of thicknesses between 1 and 5 nm were recorded. They sug- gested that the intrinsic width is less than 1 nm, and that the broadening observed in some measurements is due to theaccumulation of point defects at the wall. The thickness offerroelectric 180 /C14walls is harder to measure experimentally and is discussed in more detail in Sec. IV, but reliable theoretical predictions ( Merz, 1954 ;Kinase and Takahashi, 1957 ;Padilla, Zhong, and Vanderbilt, 1996 ;Meyer and Vanderbilt, 2002 ) and recent measurements by Jia et al. (2008) indicated that they are atomically sharp, confirming the measurements of Bursill. An interesting and still not fully resolved problem is that of the domain wall thickness in multiferroics. In materials withweak coupling, it is assumed that the two ferroic parameters have essentially independent correlation lengths and thus different thicknesses for the two ferroic parameters, even ifthe middle of the wall is shared ( Fiebig et al. , 2004 ). In the converse situation of one order parameter being completelysubordinated to the other, e.g., a proper ferroelectric and animproper ferroelastic such as BaTiO 3, or a proper magnet and an improper ferroelectric such as TbMnO 3, it seems that the principal order parameter dictates a unique thickness of the shared domain wall, so that the ferroelectric domain walls ofTbMnO 3are predicted to be as thick as those of ferromagnets (Cano and Levanyuk, 2010 ). In the intermediate case of two proper order parameters with moderate coupling, it seemsthat there will still be two correlation lengths for each order parameter, but each will be affected by the coupling; Daraktchiev, Catalan, and Scott (2010) have shown that the ferroelectric wall thickness in a magnetoelectric material withbiquadratic coupling is /C14 MP/C1721=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/C18/C12b/C20 2/C13/C12a/C0/C11/C132/C0/C11/C12b/C19s ffi/C14P/C18 1þ/C13a b/C11þOð/C132Þ/C19 ; (14) where /C14MPis the ferroelectric wall thickness in the magneto- electric material, and /C14Pis the ferroelectric wall thickness in the absence of magnetoelectricity. This is thicker than thewalls of normal ferroelectrics and thus more magnetlike,which also agrees with the bigger width of the ferroelectricdomains of BFO compared to those of normal ferroelectrics (Catalan et al. , 2008 ). C. Domain wall chirality In magnetism, the spin is quantized, so it cannot change its magnitude across the wall. Instead, then, the magnetization reverses through rigid rotation of the spins. The rotation plane may be contained within the plane of the domain wall (Ne ´elG. Catalan et al. : Domain wall nanoelectronics 133 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012walls), or it may be perpendicular to it (Bloch walls). Ne ´el and Bloch walls are generically termed as Heisenberg-like, orchiral. Ferroelectric polarization, on the other hand, is notquantized, so it is allowed to vary in magnitude. This canproduce domain walls where the polarization axis does notchange orientation but simply decreases in size, changes sign,and increases again. Such nonchiral domain walls are calledIsing-like (see schematic of different types of walls inFig.21). In ferroelectrics, 180 /C14Ising walls should be favored against chiral walls for two reasons: First, the piezoelectriccoupling between polarization and spontaneous strain meansthat rotating the polarization away from the easy axis has abig elastic cost. Second, there is also a large electrostatic cost,as any change in the polarization perpendicular to the domainwall will cause, via Poisson’s equation, an accumulation ofcharge at the walls: /C1D¼"/C1P¼/C26. It is worth mentioning that the above assumes that the dielectric constant is constant;when it is not, then the correct form of Poisson’s equation is/C1D¼"/C1PþE/C1r"¼/C26. The permittivity gradient can be important in thin films ( Scott, 2000 ), and must be important also for the domain wall, where large structural changes takeplace within a narrow region. For the above reasons, 180 /C14domain walls in ferroelectrics have been traditionally viewed as Ising-like. This commonassumption, however, has recently been challenged by Lee et al. (2009) andMarton, Rychetsky, and Hlinka (2010) , who show that ferroelectric 180 /C14domain walls of perovskite ferroelectrics can be at least partially chiral. The fact thatchirality can appear in a system where none would be ex-pected was examined by Houchmandzadeh, Lajzerowicz, and Salje (1991) . They showed that, whenever there are two order parameters involved (as in any multiferroic system), thecoupling between them can induce chirality at the domainwalls. Perovskite ferroelectrics are multiferroic, because theyare both ferroelectric and ferroelastic. While their 180 /C14walls tend to be seen as purely ferroelectric, they nevertheless havean elastic component, because the suppression of the polar-ization inside the wall affects its internal strain ( Zhirnov, 1959 ). Domain walls in BiFeO 3are also multiferroic, and in a big way, ferroelectricity, ferroelasticity, antiferromagnetism, andantiferrodistortive octahedral rotations all occur in this mate- rial. It is therefore not surprising that the domain walls of this material are found to be chiral ( Seidel et al. , 2009 ). Unlike in normal ferroelectrics, the rotation of the polar vector is quite rigid, meaning that the component of the polarization per- pendicular to the domain wall is not constant. This polar discontinuity means that there is charge density at the walls (see Poisson’s equation above). In order to screen this chargedensity, charge carriers aggregate to the wall, and this carrier increase has been hypothesized to be a cause for the increased conductivity at the domain walls of BiFeO 3(Seidel et al. , 2009 ;Lubk, Gemming, and Spaldin, 2009 ). The issue of domain wall conductivity is discussed in greater detail in Secs. III.F andV.F.1 . Chirality has important consequences for magnetoelectric materials. Magnetic spin spirals can by themselves cause ferroelectricity: indeed, a spin spiral arrangement is knownto cause weak ferroelectricity in some multiferroics (Newnham et al. , 1978 ;Mostovoy, 2006 ;Cheong and Mostovoy, 2007 ). The relationship between spin helicity and polarization is valid not just for bulk but also for the local spin arrangement inside a domain wall; thus, ferromag-netic Ne ´el walls are expected to be electrically polarized. Experimental evidence for this was provided by Logginov et al. (2008) , who applied a voltage to an AFM tip placed near the ferromagnetic domain wall of a garnet, and observed the domain wall to shift its position in response to the voltage (see Fig. 22). Since the garnet is itself centrosymmetric, the piezoelectric response of the domain wall was attributed to its spin spiral. D. Domain wall roughness and fractal dimensions Irregular domain walls have been studied in thin films of ferromagnets ( Lemerle et al. ,1 9 9 8 ), ferroelectrics ( Tybell et al. , 2002 ;Paruch, Giamarchi, and Triscone, 2005 ), and multiferroics ( Catalan et al. , 2008 ). Quantitatively, the ir- regular morphology can be characterized by a roughness coefficient (see Fig. 23), which describes the deviations ( u) from a straight line (the ideal domain wall) as the length of FIG. 22 (color online). (a) Logginov et al. applied voltage pulses to a sharp tip in the vicinity of a ferromagnetic Ne ´el wall in a magnetic garnet. (b) The wall was observed to move toward or away from the tip depending on the polarity of the voltage, suggesting that the domain wall is electrically polarized even though the garnetitself is a nonpolar material. The domain wall polarization is caused by the spin spiral inherent to the Ne ´el wall. From Logginov et al. , 2008 . FIG. 21 (color online). (a) Ising wall, (b) Bloch wall, (c) Ne ´el wall, and (d) mixed Ising-Ne ´el wall. Recent calculations show that domain walls in perovskite ferroelectrics tend to be of mixedcharacter. From Lee et al. , 2009 .134 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012wall is increased ( Lemerle et al. , 1998 ;Paruch, Giamarchi, and Triscone, 2005 ). The wandering deviation uincreases with the distance traveled along the wall, resulting in a power law dependence of the correlation coefficient, BðLÞ/C17 h½uðxþLÞ/C0uðxÞ/C1382i/L2/C16, where /C16is the roughness exponent. If the domain wall closes in on itself, forming a ‘‘bubble’’ domain, the roughness coefficient of the wall becomes adirect proxy for the Hausdorff dimension, which relates thearea contained within the domain ( A) to the domain wall perimeter (see Fig. 23); thus, films with rough walls have fractal domains in the sense that the perimeter does not scaleas the square root of the area, but as the square root of thearea to the power of the Hausdorff dimension, P/A H=2 (Rodriguez et al. , 2007a ;Catalan et al. , 2008 ). Since domain size is dictated by the competition between the domain energy (proportional to the area of the domain)and wall energy (proportional to the domain perimeter) it is quite natural that the scaling of the domain size should reflect the Hausdorff dimension of the domains, or, equivalently, theroughness coefficient of the walls. Catalan et al. (2008) showed that, when the domains are fractal, the Kittel lawmust be modified as w¼A 0tH?=ð3/C0HllÞ, where A0is a constant, tis the film’s thickness, and H?andHkare the perpendicular and parallel (out-of-plane and in-plane with respect to the film’s surface) Hausdorff dimensions of the walls. When both these dimensions are 1 (i.e., smooth walls), the classic Kittelexponent ( 1 2) is recovered. In the particular case of BFO films, the dimension was found to be 1.5 in the in-planedirection, and 2.5 in total, consistent with domain walls that meander in the horizontal direction but are completely straight vertically, much like hanging curtains. This is fully expected in ferroelectric 180/C14walls, as any vertical bend would incur in a strong electrostatic cost due to Poisson’s equation. The fractional dimensionality has also been ob-served in studies of switching dynamics ( Scott, 2007 ), as discussed later in this section. In a perfect system, the domain wall energy cost is mini- mized by minimizing the wall area, i.e., by making the wall assmooth as possible. Whenever domain walls are rough, then, it is because they are being pinned by defects in the lattice. The upshot of this is that the roughness of a domain wall contains information about the type of defects present in the sample ( Natterman, 1983 ). Specifically, the theoretical rough- ness coefficient in a random bond system is /C16¼2=3for a linelike domain wall ( Huse, Henley, and Fisher, 1985 ;Kardar and Nelson, 1985 ), and this has been experimentally verified for ultrathin ferromagnetic films ( Lemerle et al. , 1998 ). In the more general case, it is /C16¼4/C0D=5(Lemerle et al. , 1998 ). Random bond systems can be viewed as systems with a variable depth of the double well. If the asymmetry of the double well changes, then one speaks of random field sys- tems, for which the roughness coefficient is /C16¼4/C0D=3 (Fisher, 1986 ;Tybell et al. , 2002 ), where Dis the dimen- sionality of the wall, which can be fractional. For ferro- electric thin films the roughness coefficient was found to be 0.26, consistent with the random bond system of di- mensionality D¼2:7(Paruch, Giamarchi, and Triscone, Lu A P(a) (b) (c) (d) FIG. 23 (color online). (a) The probability of having a deviation ( u) from the straight line increases with the distance dbetween two points of the wall, resulting in a power law relationship between the size of the wall and its horizontal length. By the same token, if the domain wall closes in on itself (b), the perimeter will not increase as the square root of the area (as would be the case for a smooth circular domain), but asP/A H=2, where His the Hausdorff dimension. (c) Measurement of domain wall roughness in PFM-written ferroelectric domain walls of BiFeO 3thin films and (d) measurement of the Hausdorff dimension in spontaneous domains of the same BiFeO 3films. Panels (c) and (d) partially adapted from Catalan et al. , 2008 .G. Catalan et al. : Domain wall nanoelectronics 135 Rev. Mod. Phys., V ol. 84, No. 1, January–March 20122005 ). In multiferroic BiFeO 3, the roughness was larger, /C16¼0:56(Catalan et al. , 2008 ). Since the roughness of the walls arises directly from the local pinning by defects, and pinning slows down the motionof the domain walls, it is natural to relate the roughness of thedomain walls to their dynamics. This has been done both forferromagnetic films ( Lemerle et al. , 1998 ) and for ferroelec- tric films ( Tybell et al. , 2002 ;Paruch, Giamarchi, and Triscone, 2005 ). The domain wall velocity is characterized by an exponent that, similar to the roughness exponent, is alsodirectly related to the type of pinning defects in the samples.Specifically, the velocity of the wall is v¼v 0exp/C20 /C0U kT/C18Fcrit F/C19/C22/C21 ; where FandFcritare the applied and critical fields (magnetic or ferroelectric) of the sample, Uis an activation energy, and /C22is the critical exponent, which is related to the roughness exponent by /C22¼ðDþ2/C16/C02Þ=ð2/C0&Þ(Lemerle et al. , 1998 ). The value of /C22depends on whether the domain wall motion proceeds by creep or by viscous flow; in ferroelectric thin films /C22¼1was measured, consistent with a creep process (see Fig. 24). The study of the switching dynamics in ferroelectric thin films generally yields an effective dimensionality that is notinteger but fractional. In early switching studies Scott et al. (1988) found from fits to the Ishibashi theory that dimension- ality of the domain kinetics was often D¼2:5(approxi- mately). At the time it was not clear whether this was aphysical result or an artifact of the Ishibashi approximations(especially the simplifying assumption that wall velocities v were independent of domain radius r—actually vvaries as 1=r). However, more recent studies ( Scott, 2007 ) indicate that D¼2:5is physically correct; Scott also calculated by inter- polation the critical exponents in mean field for D¼2:5and found, for example, that the order parameter exponent /C12¼ 1 4for a second-order transition, compared with /C12¼1 2for bulk D¼3. Since this is the same1 4exponent as in a bulk tricritical transition, second-order transitions for D¼2:5may be mis- taken as tricritical. E. Multiferroic walls and phase transitions inside domain walls The idea that domain walls have their own symmetry and properties is not new. Shortly after Ne ´el hypothesized the existence of antiferromagnetic domains ( Ne´el, 1954 ),Li (1956) showed that such walls would have uncompensated spins that could account for the weak ferromagnetism mea- sured in /C11-Fe2O3. An important and often overlooked aspect of Li’s classic model is that the size of the uncompensatedmoment at the wall is inversely proportional to the wallthickness. This, to some extent, is trivial: An atomically sharpantiphase boundary should have a fully uncompensated pair of moments (see Fig. 25), whereas in a broad domain wall the gradual change means that only the fractional differencebetween nearest neighbors is uncompensated. Although ithas not been explicitly stated anywhere, a natural corollaryis that the domain walls of antiferroelectrics should be ferro- electric, or at least pyroelectric. One must bear in mind that the walls of ferroelectrics are atomically sharp, as discussedabove, so antiferroelectric domain walls are expected to beclose to perfect antiphase boundaries, although we know ofno studies of domain wall thickness in antiferroelectrics. In the case of multiferroics, the interplay between the symmetries of all the phases involved is more complex and can lead to rich behavior. Privratska, Janovec, and othersmade a theoretical survey of which properties are allowedinside the domain walls as a function of the space group of theferroic material. Based on this, they predicted that the domain walls of ferroelastics can be polar ( Janovec, Richterova ´, and Privratska, 1999 ), as confirmed by atomistic calculations for CaTiO 3(Gonc¸alves-Ferreira et al. , 2008 ) and experimentally inferred for SrTiO 3(Zubko et al. , 2007 ).Privratska and Janovec (1997 ,1999) , and Privratska (2007) also predicted that there can be net magnetization inside the ferroelectric FIG. 24. Ferroelectric domain wall speed as a function of applied electric field for films of various thicknesses. The critical exponent was found to be /C22¼1, characteristic of creep. From Tybell et al. , 2002 . FIG. 25 (color online). The relative heights of the boxes illustrate how a sharp antiphase boundary must have a net magnetization and polarization in its center that is bigger than that of a broad domain wall such as a chiral wall. Adapted from Li, 1956 .136 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012domain walls of multiferroics. Among the symmetries where this domain wall magnetization is allowed is the space groupR3c(Privratska and Janovec, 1999 ) (i.e., that of BiFeO 3). Symmetry analyses are not quantitative. The knowledge that a property is symmetry allowed is essential, but one stillneeds to know how big that property is. This quantitativeanalysis can be achieved using phenomenological approachessuch as those pioneered by Lajzerowicz and Niez (1979) , who were the first to realize that it was possible for domain walls to undergo their own internal phase transitions. This is pos- sible because the free energy inside the wall is different fromthat inside the domains: Inside the domains the order parame-ter is homogeneous and there are no gradients, whereas insidethe domain walls the order parameter is suppressed and there are strong gradients. Since the free energy of the wall is different, its thermodynamic properties must also be differ-ent, and hence its thermal evolution and phase transitions mayalso be different. This idea becomes, of course, more interesting when sev- eral order parameters are involved. These order parameters may be present at finite temperature (as in multiferroic materials), or they may be suppressed or ‘‘latent’’ (havelow or ‘‘negative’’ critical temperatures) but still may beable to manifest inside the domain walls. The phenomenonof emerging order parameters inside a wall was first explained byHouchmandzadeh, Lajzerowicz, and Salje (1991) in the context of ferroelastic materials. They realized that if asecondary ferroic order is latent (suppressed) due to a positivecoupling energy to the primary order parameter, it will beable to emerge wherever the primary order parameter is zero,i.e., in the middle of the domain wall. For example, it is known that, in perovskites, rotations of the oxygen octahedra are normally opposed to ferroelectric polarization. Hence,where such rotations are suppressed, polarization may beable to emerge, as was theoretically calculated for the anti-ferrodistortive antiphase boundaries of SrTiO 3(Tagantsev, Courtens, and Arzel, 2001 ). Given that BiFeO 3is known to also have strong octahedral rotations ( Megaw and Darlington, 1975 ) that oppose the polarization ( Dupe´et al. , 2010 ), it seems eminently plausible that the ferrodistortive antiphaseboundaries of this material also have a polar enhancement. Daraktchiev, Catalan, and Scott (2010) studied in some detail the analytical solutions for domain walls in multifer- roics with biquadratic coupling between the order parameters,/C13P 2M2; the biquadratic coupling was chosen because (i) it is the smallest power that is symmetry allowed for all materials(the coupling term places no constraint on inversion of either order parameter), and (ii) an effective biquadratic interaction will always be present when strain mediates the coupling,since electrostriction couples strain to the square of polariza-tion while magnetostriction couples strain to the square ofmagnetization. Strain coupling terms are of course large in ferroelectrics, and therefore they will always be important for multiferroics. Another important aspect of this biquadraticcoupling that perhaps has not been emphasized enough isthat, because it is even, the solutions must also be even,meaning that any emerging parameter inside the wall willhave at least two equally stable polarities and may be switch- able between them. This can be seen, for example, in Fig. 26, which shows that there are two equivalent least-energytrajectories connecting the ferroelectric double well through two saddle points at þMand/C0M, and thus there are two possible magnetic polarities for the wall. Experimentally, Pyatakov et al. (2011) demonstrated the converse situation by showing switching of the ferroelectric polarity inside amagnetic domain wall. The exact phenomenology, of course, depends on the sign and symmetry of the coupling elements, and the possibilities are far too numerous to be describedhere, but the basic principle is always the same: Start with the Landau expansion of the free energy and examine the con- sequence of forcing one of the order parameters to be zero, asin the middle of the domain wall. In this context, it is useful to also look at the recent work of Marton, Rychetsky, and Hlinka (2010) , who found new phases with enhanced electrome- chanical properties inside the domain walls of a typical perovskite ferroelectric such as BaTiO 3. So far group-symmetry arguments and phenomenological (thermodynamic) models have been mentioned, but there arealso microscopic calculations for some systems. For example, Goltsev and others calculated the profile of the magnetization across the domain walls of YMnO 3(Goltsev et al. , 2003 ; Fiebig et al. , 2004 ). Meanwhile, Lubk, Gemming, and Spaldin (2009) calculated the octahedral rotations, polariza- tion profile, and band gap across the domain walls of BiFeO 3, and Gonc¸alves-Ferreira et al. (2008) calculated the polar displacements inside the ferroelastic walls of CaTiO 3. Generally, the reduced thickness of ferroelectric and ferroe-lastic walls means that they are computationally affordable for first-principles calculations, whereas the broader magnetic walls tend to require analytical approaches or finite-element FIG. 26 (color online). (a) Ferroelectric polarization profile across the wall, (b) magnetization profile across the wall, (c) relationshipbetween the order parameters across the wall, showing that the magnetization can be either positive or negative, and (d) free-energy landscape, showing one of the two least-energy trajectories con-necting the two ferroelectric polarities through one of the saddle points at M/C2220. Adapted from Daraktchiev, Catalan, and Scott, 2010 .G. Catalan et al. : Domain wall nanoelectronics 137 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012and molecular dynamics calculations. Multiferroic walls fall somewhere in between in terms of thickness, although we suspect that the magnetic part of their behavior will always be more difficult to compute using first-principles approaches. F. Domain wall conductivity The changes in structure (and as a consequence electronic structure) that occur at ferroelectric (multiferroic) domain walls can thus lead to changes in transport behavior. Indeed, domain wall (super)conductivity was studied by Aird and Salje (1998) . Reducing WO 3with sodium vapor, they observed preferential doping along the ferroelastic do- main walls. Transport measurements showed superconduc- tivity, while magnetic measurements did not; this suggested that the superconductivity was located only at the domain walls, which provided a percolating superconductive path while occupying a small volume fraction of the crystal. Later, Bartels et al. (2003) used a conducting-tip scanning probe microscope to show the converse behavior: The domain walls of a calcium-doped lead orthophosphate crystal were found to be more resistive than the domains. The above effects relied on preferential doping along domain walls, but the differential domain wall conductivity was reproduced also in undoped multiferrroics, although with different transport behavior: The domain walls of BiFeO 3 were found to be more conductive than the domains ( Seidel et al. , 2009 ), while those of YMnO 3were found to be more insulating ( Choi et al. , 2010a ). Multiferroic YMnO 3, a so- called improper ferroelectric multiferroic, in which ferroelec- tricity is induced by structural trimerization coexisting with magnetism, domain walls are found to be insulating ( Choi et al. , 2010 ). The increase of the Y-O bond distance at domain walls may be responsible for the reduction of local conduc- tion. The observed conduction suppression at domain walls at high voltages (still much less than the electric coercivity) is in striking contrast with what was reported on BiFeO 3. A useful clue for interpreting these results is perhaps the analysis of the paraphase. Whereas the high-temperature, high-symmetry phase of BiFeO 3is more conducting than the ferroelectric phase ( Palai et al. , 2008 ), the converse is true for YMnO 3(Choi et al. , 2010 ). This illustrates an important point: In some respects the internal structure of the walls can be considered to be in the paraelectric state; by way of trivial example, the 180/C14domain walls of a ferroic are nonpolar, just like its paraphase. The examples of BiFeO 3and YMnO 3suggest that the paraphaselike behavior can be ap- plied to domain wall properties other than just the polariza- tion: The insulating paraelectric state of YMnO 3is consistent with the insulating nature of its domain walls, and conversely the conducting state of the paraphase of BiFeO 3is consistent with its domain wall conductivity. Nevertheless, in the con-ductivity of the BiFeO 3walls at least there are several other considerations: octahedral rotations, electrostatic steps aris- ing from rigid rotation of the polar vector, and increased carrier density at the wall are all thought to play a role in the domain wall conductivity of BiFeO 3and potentially also of other perovskites. A more detailed discussion of these factors is provided in Sec. V.F.1 .The resistive behavior of purely magnetic domain walls has also been studied. The domain walls of metallicferromagnets were found to be more resistive than the do- mains due to spin scattering ( Viret et al. , 2000 ;Danneau et al. ,2 0 0 2 ). On the other hand, the domain walls of man- ganites (which are ferromagnetic and ferroelastic) have beenpredicted to be more conducting than the domains, due to strain coupling: The Jahn Teller distortion is smaller and theoctahedral rotation angle is straighter inside the domain wallthan outside ( Salafranca, Yu, and Dagotto, 2010 ), leading to increased orbital overlap and thus bigger bandwidth. The same interplay between octahedral rotation straighteningand increased conductivity has been postulated for the do-main walls of bismuth ferrite ( Catalan and Scott, 2009 ), and the straightening of the octahedral rotation angle inside thewalls of this material has been confirmed by electron micros-copy ( Borisevich et al. , 2010 ). To complete the picture, it should be mentioned that enhanced domain wall conductivity has also been observed in standard ferroelectrics such asPbðZr;TiÞO 3(Guyonnet et al. , 2011 ), suggesting that this may be a more general property than previously thought. The challenge is now to make a resistive switching device based on domain walls. Two approaches may be pursued here. One was suggested by Lee and Salje (2005) , who observed that the percolation of a zigzag configuration offerroelastic walls between the two surfaces of a crystal couldbe controlled by bending. The other approach pursued isselective doping. The experimental study of domain wallconductivity and the electronic devices that can be madeusing wall properties will be the subject of the following sections. IV. EXPERIMENTAL METHODS FOR THE INVESTIGATION OF DOMAIN WALLS A variety of structural and near-field probes are available to probe both the macroscopic and microscopic details of do-main walls. Atomic-scale imaging of the domain wall struc- ture is now possible with transmission electron microscopy, but of particular emphasis in this review is scanning probe(scanning tunneling microscopy, atomic force microscopy,conducting AFM (c-AFM), and the related piezoforce micro-socopy) techniques, which allow the probing of actual func-tional properties of the domain walls. Readers interested inthe details of all of these structural probe techniques are referred to several reviews on this subject; here we give an overview of the information pertinent to domain walls. A. High-resolution electron microscopy and spectroscopy Among the methods available for the investigation of domain walls is high-resolution electron microscopy ( Goo et al. , 1981 ;Bursill, Peng, and Feng, 1983 ;Bursill and Lin, 1986 ;Stemmer et al. , 1995 ;Hytch, 1998 ;Lichte, 2002 ;Jia, 2003 ). This method allows direct visualization of the lattice distortion across the domain wall by measuring the continu-ous deviation of a set of planes with respect to the undistortedlattice (exit-wave reconstruction) ( Foeth et al. , 1999 ). Current, state-of-the-art techniques permit atomic-scale resolution at 0:5/C23Athrough aberration-corrected imaging138 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012(Figs. 27and 28). The exit-wave reconstruction approach eliminates the effects of objective-lens spherical aberrations, and images can be directly interpreted in terms of the pro-jection of the atomic columns ( Allen et al. , 2004 ). Weak- beam transmission electron microscopy has been used for aquantitative analysis of the thickness fringes that appear onweak-beam images of inclined domain walls. By fittingsimulated fringe profiles to experimental ones, it is possible to extract the thickness of the domain walls in a quantitative way. Regarding high-resolution transmission electron micros-copy (HRTEM) images of domain walls, it has to be takeninto account that the samples in these kinds of experimentsare thin (typically a few nanometers) so that surface pinningof the domain walls could play an important role ( Meyer and Vanderbilt, 2002 ). The atomic displacements across a typical wall are on the order of 0.02 nm, which still makes directimaging and interpretation a challenge ( Gopalan, Dierolf, and Scrymgeour, 2007 ). HRTEM also offers the possibility of imaging the local polarization dipoles at atomic resolution,thus quantitatively measuring the local polarization and in- vestigating the domain structure ( Jiaet al. , 2008 ). Using the negative spherical-aberration imaging technique in an aberration-corrected transmission electron microscope,large differences in atomic details between charged and un-charged domain walls have been reported, and cation-oxygendipoles near 180 /C14domain walls in epitaxial PbZr 0:2Ti0:8O3 thin films have been resolved on the atomic scale ( Jiaet al. , 2008 ).Elemental and electronic structure analysis by electron- energy-loss spectroscopy has also been applied to the study ofdomain walls ( Jia and Urban, 2004 ;Urban et al. ,2 0 0 8 ). Using high-resolution imaging in an aberration-corrected TEM, the concentration of oxygen in BaTiO 3twin bounda- ries was measured at atomic resolution. These measurementsprovide quantitative evidence for a substantial reduction ofthe oxygen occupancy, i.e., the presence of oxygen vacanciesat the boundaries. It was also found that the modified Ti 2O9 group unit formed reduces the grain boundary energy and provides a way of accommodating oxygen vacancies occur- ring in oxygen-deficient materials. This type of atomically resolved measurement technique offers the potential to studyoxide materials in which the electronic properties sensitivelydepend on the local oxygen content (important in view ofcurrent work on LaAlO 3=SrTiO 3superconducting interfa- ces). The attraction between domain walls and vacancies isfurther discussed in Sec. V. B. Scanning probe microscopy Atomic force microscopy and its variations (e.g., c-AFM, PFM) are well suited for direct writing (‘‘ferroelectric lithog-raphy’’) and characterization of prototype ferroelectric struc- tures (Fig. 29), including domain walls ( Eng, 1999 ). These methods provide the tools to get information about localmechanisms of twin-wall broadening that cannot be obtainedby existing experimental methods ( Shilo, Ravichandran, and Bhattacharya, 2004 ). With conductive AFM (c-AFM) one can artificially modify the domain structure as a function of pulsewidth and amplitude ( Tybell et al. ,2 0 0 2 ). PFM is also under continuous development and is currently undergoing a shift of focus from imaging static domains to (i) dynamic character-ization of the switching process (with developments such asstroboscopic PFM and PFM spectroscopy) and (ii) the struc-ture of domain walls ( Gruverman et al. , 2005 ;Jungk, Hoffmann, and Soergel, 2006 ;Rodriguez et al. , 2007a , 2007b ;Morozovska et al. , 2008 ;Kalinin et al. , 2010 ) (see also Fig. 30). Let us take a closer look at the relationship between domain walls and the effect of electric fields on ferroelectrics.When the applied field is higher than the coercive field, thewalls will move; however, the threshold field at which pre-existing domain walls begin to move can be much lower than FIG. 29 (color online). (a) PFM amplitude and (b) PFM phase images of a BFO sample with 109/C14stripe domains; (c) simultaneously acquired c-AFM image of the same area showing that each 109/C14domain wall is electrically conductive. From Seidel et al. , 2010 . FIG. 28 (color online). (a) 71/C14and (b) 109/C14domain walls in bismuth ferrite. FIG. 27 (color online). Atomic-scale TEM image of the electricdipoles formed by the relative displacements of the Zr/Ti cationcolumns and the O anion columns in PbZr 0:2Ti0:8O3. Adapted from Jiaet al. , 2008 .G. Catalan et al. : Domain wall nanoelectronics 139 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012the coercive field required for nucleation ( Gopalan, Dierolf, and Scrymgeour, 2007 ;Choudhury et al. , 2008 ). From preliminary theory and experiments, Gopalan, Dierolf, and Scrymgeour (2007) argued that the thickness of the domain wall is different when an external electric field is applied because of changes in its Landau energy potential. This prediction is also supported by the calculations of Rao and Yu (2007) andHlinka, Ondrejkovic, and Marton (2009) .F o r PbTiO 3, the effect of an applied electric field leads to an increase in the wall thickness of 2 to 4 times and for LiNbO 3 the thickness increases up to 10%–30%, but this phenomenon is expected to be general for all compositions. At the same time, a thicker (or diffuse) domain wall has a lower threshold field ( Choudhury et al. , 2008 ) for lateral movement, as it is not narrow enough to drag against the washboardlike Peierls potential. This effect was used to explain low thresholdfields for domain reversal in ferroelectrics. However, the same authors predict that an increase in wall thickness at surfaces will be of influence only when the crystal thickness is 1–10 nm and that, in general, the region with the lowest wall thickness will dominate the threshold field for the mo- tion of the entire wall. Analysis of both complete area images and individual line- scan profiles provides essential information about local mechanisms of twin-wall broadening, which cannot be ob- tained by other experimental methods ( Shilo, Ravichandran, and Bhattacharya, 2004 ). Surface topography measured usingatomic force microscopy is compared with candidate dis- placement fields, and this allows for the determination ofthe twin-wall thickness and other structural features. Closed-form analytical expressions for vertical and lateral PFM profiles of a single ferroelectric domain wall for the conical and disk models of the tip, beyond point charge and sphereapproximations have been investigated ( Morozovska et al. , 2008 ). Here the analysis takes into account the finite intrinsic width of the domain wall and dielectric anisotropy of thematerial. The analytical expressions provide insight into themechanisms of PFM image formation and can be used for aquantitative analysis of the PFM domain wall profiles. Ferroelectric thin films typically contain various structural defects such as cationic and/or anionic point defects, dislo-cations, and grain boundaries. Since the electric and stressfields around such defects in a ferroelectric thin film arelikely to be inhomogeneous, it is expected that the switchingbehavior near a structural defect will be different fromthe one found in a single-domain state. The role of a single ferroelastic twin boundary has been studied in tetragonal PbZr 0:2Ti0:8O3ferroelectric thin film ( Choudhury et al. , 2008 ). It was shown that the potential required to nucleate a 180/C14domain is lower near ferroelastic twin walls (/C241:6V) compared with /C242:6Vaway from the twin walls. A recently increased interest in combined PFM and con- ductivity measurements arises from both nonvolatile memory application perspective and a potential for electroresitive memory devices ( Yang et al. , 2009 ). The work of Gruverman, Isobe, and Tanaka (2001) explored the interplay of domain dynamics and conductivity at interfaces in thinferroelectric films. The combination of local electromechani-cal and conductivity measurements revealed a connectionbetween local current and pinning at bicrystal grain bounda-ries in bismuth ferrite ( Rodriguez et al. , 2008 ). Electroresistance in ferroelectric structures was recently re- viewed by Watanabe (2007) . The presence of extended de- fects and oxygen vacancy accumulation has been shown toinfluence transport mechanisms at domain walls ( Seidel et al. , 2010 ). Recently, direct probing of polarization- controlled tunneling into a ferroelectric surface was shown(Garcia et al. , 2009 ;Maksymovych et al. , 2009 ). Scanning- near-field optical microscopy has been used to observe pin- ning and bowing of a single 180 /C14ferroelectric domain wall under a uniform applied electric field ( Yang, 1999 ;Kim et al. , 2005 ). Typically the imaging resolution in PFM is about 5–30 nm . The achievable resolution is ultimately limited by the tip-sample contact area, which is nominally determined by the radius of the tip apex. There are additional mechanisms for resolution broadening such as electrostatic interactions andthe formation of a liquid neck under ambient conditions in thetip-surface junction. The PFM amplitude typically providesinformation on the magnitude of the local electromechanicalcoupling under the tip, and the PFM phase gives informationabout the ferroelectric domain orientation. Scanning tunneling spectroscopy can be used to directly probe the superconducting order parameter at nanometer length scales. Scanning tunneling microscopy (STM) andspectroscopy (STS) have been used to investigate the elec-tronic structure of ferroelastic twin walls in YBa 2Cu3O7/C0/C14 FIG. 30 (color online). Schematic piezoresponse across a single 180/C14domain wall in lithium niobate crystal. (a) The surface displacement (solid line) due to the electric field across the domainwall displayed in (e). The dotted line is the original surface plane. (b) The piezoresponse, both XandYsignals, across the domain wall. Xis the product of amplitude ( R) and the sine of the phase, q, andYis the product of amplitude and cosine of the phase. (c) The piezoresponse, both Xsignal and Ysignals, on both þcand/C0c surfaces plotted in a vectorial XYplane. (d) The amplitude and phase of the piezoresponse across the domain wall. (e) Schematicdomain structure and electrical field. From Tian, 2006 .140 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012(YBCO) ( Maggio-Aprile et al. , 1997 ). Twin boundaries play an important role in pinning the vortices and therebyenhancing the currents that YBCO can support while remain-ing superconducting. An unexpectedly large pinning strengthfor perpendicular vortex flux across such boundaries wasfound, which implies that the critical current at the boundaryapproaches the theoretical ‘‘depairing’’ limit. In the case of insulators, STM and STS are by definition a lot more difficult to implement, primarily because a reliabletunneling current cannot be used to establish proximal con-tact. The emergence of ferroelectrics with smaller band gapsand the possibility of conduction at domain walls (see later)has stimulated renewed interest in exploring STM as a probeof the local electronic structure. The emergence of combinedAFM and STM or SEM (scanning electron microscopy) andSTM systems should be a boon in terms of exploring theelectronic properties of domain walls in such insulatingmaterials. Research using such combined tools is in itsinfancy ( Wiessner et al. , 1997 ;Yang et al. , 2005 ;Garcı ´a, Huey, and Blendell, 2006 ;Chiu et al. , 2011 ). C. X-ray diffraction and imaging Diffuse x-ray scattering can also be used to investigate the structure of domains and domain walls in densely twinnedferroelastic crystals ( Bruce, 1981 ;Andrews and Cowley, 1986 ;Locherer, Chrosch, and Salje, 1998 ). The scattering is characterized by strong, well-defined Bragg peaks, with adiffuse streak between these arising from the domain walls(see Fig. 31). The streak typically is several orders of magni- tude lower in intensity. Comparison with an analytical modelfor the scattering allows one to extract the effective domainwall width on the unit-cell level ( Locherer, Chrosch, and Salje, 1998 ). Critical fluctuations and domain walls of, for example, KH 2PO4(KDP) and KD2PO4(DKDP) were inves- tigated ( Andrews and Cowley, 1986 ). The intensity of thecritical scattering near two different reciprocal lattice points was determined and used to find the atomic displacements inthe ferroelectric fluctuations. The x-ray scattering from the domain walls was observed below T cand enabled measure- ments to be made of the width of the domain walls and theatomic displacements in the walls. The width of the domainwalls was shown to increase with temperature toward T c. Synchrotron x-ray sources have also been used for direct imaging of strain near ferroelectric 180/C14domain walls in congruent LiNbO 3andLiTaO 3crystals and in BaTiO 3crys- tals ( Kim, 2000 ;Rogan, 2003 ;Jach, 2004 ). Direct evidence for wide regions of strain on length scales of many micro- meters associated with 180/C14domain walls in congruent LiNbO 3andLiTaO 3crystals was found. The observed strain contrast in symmetric high-resolution diffraction images inBragg geometry arises in part from curvature in the basalplanes across a domain wall as well as from lateral variation in the lattice spacing of the basal planes extending across a wall. In BaTiO 3local triaxial strain fields around 90/C14do- mains were found. Specifically, residual strain maps in aregion surrounding an isolated, approximately 40-/C22m-wide, 90 /C14domain were obtained, revealing significant residual strains. D. Optical characterization Ferroelectrics offer the possibility of engineering their domain structure down to the nanometer regime and thereforeallow for interesting optical functionality such as mode shap-ing and frequency conversion, as well as the integration intocompact optical devices ( Chen et al. , 2001 ;Kurz, Xie, and Fejer, 2002 ;Scrymgeour et al. , 2002 ). Ferroelectric crystals in general are anisotropic and show birefringence. Regions with different orientations of the polar axis are, for example,easily differentiated by polarization microscopy ( Tarrach et al. , 2001 ). Because of the symmetry of the optical indica- trix, regions of opposite polarization cannot be distinguished by linear optics. Nonlinear effects, however, such as second- harmonic generation ( Dolino, 1973 ) or the electro-optic ef- fect ( Hubert and Levy, 1997 ;Otto et al. , 2004 ) revealed the eccentricity of the crystal. Regarding materials, lithium nio-bate ( LiNbO 3) and lithium tantalate ( LiTaO 3) emerged as key technological materials for photonic applications (se Fig. 32). High quality of crystal growth, optical transparency over a FIG. 32. Piezoelectric force microscopy phase contrast images of domain shapes in LiNbO 3andLiTaO 3. From Scrymgeour et al. , 2005 . FIG. 31 (color online). X-ray intensity profile of the ð400Þ=ð040Þ peak along (110) in a WO 3crystal. The dashed line shows a Gaussian fit to the contribution from the domain walls. The bold solid line is the overall fit. Figure courtesy of Ekhard Salje, adapted from Locherer, Chrosch, and Salje, 1998 .G. Catalan et al. : Domain wall nanoelectronics 141 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012wide frequency range (240 nm to 4:5/C22mwavelength), and their large electro-optic and nonlinear optical coefficients are the main advantages of these materials. Emerging fields of optical communication, optical data storage, optical displays,biomedical device applications, and sensors all rely heavily on such ferroelectric materials as a versatile solid-state pho- tonic platform. The process of domain control is difficult andhas received tremendous attention over the past years. The central focus of work has been set on developing a funda- mental understanding of shaping and controlling domainwalls in ferroelectrics, specifically in lithium niobate and lithium tantalate, for photonic applications. An understanding of the domain wall phenomena has been approached at the macroscale and the nanoscale.Different electric-field poling techniques have been devel- oped and used to create domain shapes of arbitrary and controlled orientation. A theoretical framework based onthe Landau-Devonshire theory is typically used to determine the preferred domain wall shapes in these materials ( Lines and Glass, 2004 ). Differences in the poling characteristics and domain wall shapes between the materials as well as differences in material composition have been identified to be related to nonstoichiometric defects in these crystals.(Shur, 2006 ). V. APPLICATIONS OF DOMAINS AND DOMAIN WALLS When applying an external field to a material with do- mains, the walls will move so as to expand those domains that are energetically favored by the field and contract those thatare not. Having a large density of mobile domain walls facilitates this change in domain populations and can there- fore dramatically enhance the susceptibility of any ferroic, beit magnetic susceptibility, elastic compliance, dielectric con- stant or piezoelectric coefficient. The contribution of domain walls to the susceptibility of ferroelectrics was first studied inPrague more than four decades ago ( Fouskova, 1965 ;Fousek and Janous ˇek, 1966 ), and presently there exist good articles and reviews about the dynamic domain wall contribution tosusceptibility and piezoelectricity ( Zhang et al. , 1994 ; Bassiri-Gharb et al. , 2007 ) so we do not dwell on this topic here. Instead, we focus on the applications of staticferroelectric domain configurations (chiefly, electro-optical devices) on one hand, and on the newer concept of devicesexploiting domain wall shift (the so-called racetrackmemories). A. Periodically poled ferroelectrics Prior to the recent flurry of activity on domain engineering, the primary device application requiring control and manipu-lation of ferroelectric domains involved periodic poling offerroelectrics. This application is for nonlinear optics, such as second-harmonic generation: The efficiency of the wave- length conversion is increased by having periodic antiparalleldomains, with the maximum theoretical efficiency beingachieved when the wavelength of the pump laser matchesthe full repeat length of a pair of þPand/C0Pdomains, as first pointed out by Bloembergen in his Nobel prize-winning work(Armstrong et al. ,1 9 6 2 ). The production of highly efficient nonlinear electro-optic devices via the technique of periodically poling ferroelectric crystals (quasiphase matching) emphasized devices madefrom lithium niobate ( LiNbO 3) and lithium tantalate (LiTaO 3), both congruent and stoichiometric, KTP (KTiOPO 4), and tungsten bronzes of the barium sodium niobate family. Generally speaking, these have been success-ful commercial devices, but a few problems remain thatprevent optimization of real products. First, the domainwidths are sometimes not stable with time; second, there is a particular problem in achieving narrow (submicrometer) widths. In this section we examine some real device parame-ters and suggest that the crystal (or film) thicknesses have notbeen optimized in a way that is compatible with the domainwidths, connected through the Landau-Lifshitz-Kittel law(see Table I). Since the original report of efficient nonlinear optics from phase-matched periodically poled ferroelectrics by Armstrong et al. (1962) there have been numerous develop- ments and commercial production of such devices, startingabout two decades ago, first in Japan ( Yamada et al. , 1993 ), and then in the USA ( Myers et al. , 1995 ). From the early 1990s interest was perhaps evenly divided between KTP(KTiOPO 4)(Chen and Risk, 1994 ;Karlsson, 1997 ;Reid, TABLE I. Some device parameters for periodically poled ferroelectrics. Wall thicknesses tmeasured using PFM or optical methods overestimate the true crystallographic wall thicknesses, as pointed out by Jungk, Hoffmann, and Soergel (2007) . The quoted wall thickness for KTP is therefore likely to be too high. Sample thickness d Domain width w Wall thickness t Domain depth d0 (mm) (nm) (nm) (nm) Reference KTiOPO 4 0.5 mm 283–360 nm 20–80 ( aface) /C25100 nm Wittborn et al. (2002) ;Canalias, Pasiskevicius, and Laurell (2005) ; Laurell and Canalias (2009) 0.5 mm 65 ( bface) Canalias, Pasiskevicius, and Laurell (2006) ;Canalias et al. (2006) LiNbO 3 1m m 150 nm –6/C22m <3n m <100 nm Shur et al. (2000) ;Rosenman et al. (2003a ,2003b) ;Grilli et al. (2005) ; Jungk, Hoffmann, and Soergel (2007) LiTaO 3 0.15 875 nm Mizuuchi, Yamamoto, and Kato (1997)142 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 20121997 ;Wang, 1998 ;Rotermund, 1999 ) and lithium niobate (Hu, Thomas, and Webjo ¨rn, 1996 ;Galvanauskas, 1997 ; Penman, 1998 ). This mix of materials continued in more recent studies ( Rosenman et al. , 2003a ,2003b ;Tiihonen, Pasiskevicius, and Laurell, 2006 ;Canalias, Pasiskevicius, and Laurell, 2006 ;Canalias et al. , 2006 ;Lagatsky et al. , 2007 ; Henriksson et al. , 2006 ;Hirohashi et al. , 2007 ), augmented by results on the tungsten bronzes of the Ba2NaNb5O 15 family ( Jaque et al. , 2006 ). Note that, as pulled from the congruent melt, LiNbO 3is not stoichiometric; the spectro- scopic differences between congruent and stoichiometric LiNbO 3were first illuminated by Okamoto, Wang, and Scott (1985) . We also note parenthetically that LiNbO 3=LiTaO 3andKTiOPO 4are both nonferroelastic fer- roelectrics since their crystal classes do not change at their ferroelectric phase transitions (rhombohedral-rhombohedraland orthorhombic-orthorhombic, respectively). This meansthat only 180 /C14domains are present, which simplifies switch- ing dynamics, and it implies that there is no hysteretic stress during switching, which minimizes energy cost in poling. 1. Application of Kittel’s law to electro-optic domain engineering As discussed in Sec. II, the domain width wis proportional to the square root of the crystal (or film) thickness d(see Fig. 1). The proportionality constant is material dependent and not easily evaluated, and this was simplified by the group of Luk’yanchuk ( De Guerville et al. , 2005 ) and by some of us (Catalan et al. , 2007a ): By dividing the Landau-Lifshitz- Kittel formula by the domain wall thickness /C14, a dimension- less constant results [Eq. ( 2)]. Our basic hypothesis, already advanced by Catalan et al. (2007a) , is that periodically poled electro-optic devices often have domain periodicities anddomain walls that are unstable with time because they are fabricated at thicknesses and widths that do not satisfy the Kittel equation [Eq. ( 2)]. We consider how to improve and optimize this situation. Note three things in Eq. ( 2): First, the domain wall thick- ness/C14is assumed to be an intrinsic constant, whereas, in fact, /C14can be manipulated experimentally to optimize stability. Secondly, dis not necessarily the thickness of the crystal, but the depth of the domains, which can be much less thanthe total film thickness. And, finally, the material-specificparameter in Eq. ( 2) is the square root of the ratio of in-plane and out-of-plane electric susceptibilities. As shown, this ratio is different for the three electro-optic materials recently ex-plored for periodically poled devices. The devices are typi-cally fabricated on specimens that are 0.15 to 1.0 mm in thickness. The typical domain widths that one strives for (in order to match visible or near-visible wavelengths) are100–900 nm, and the typical wall thicknesses are 10 nm orless. These numbers do not satisfy the Kittel law; in particu- lar, for a domain wall thickness of 10 nm, a 500-nm domain width would be thermodynamically stable in LiNbO 3or KTiOPO 4only for a much thinner specimen of /C2510/C22m, significantly less than the actual 1 mm. It is not easy to circumvent thermodynamics or to fool Mother Nature. If one constrains the domain widths to be smaller than the equilibrium Kittel value via spatially abruptapplied fields, thermodynamics ‘‘retaliates’’ by making thestable domains not penetrate through the sample from anode to cathode of the applied poling voltage, but instead only partially to a few micrometers in depth ( Batchko et al. , 1999 ). ForBa 2NaNb 5O15the out-of-plane (polar-axis) dielectric constant is 32. The in-plane is biaxial but nearly isotropic at 222 ( xaxis) and 227 ( yaxis) ( Warner, Coquin, and Frank, 1969 ). So the square root of the ratio is 2.65. Hence for excitation with 1:064-/C22mNd:YAG (yttrium aluminum gar- net) (doubling to the green at 532 nm), and remembering that two ferroelectric stripes make up one full wavelength, we calculate the optimum thickness dfrom Eq. ( 2):w2=ðd/C14Þ¼ 2:455/C22:65¼5:5.F o r/C14of 1.0 nm (more about this choice below), this yields an optimum d¼180/C22m.F o r LiNbO 3 (congruent) the out-of-plane dielectric constant is 27.9 and the in-plane one is 85.2 ( Smith and Welsh, 1971 ). Therefore the ratio is 3.05 and w2=ðd/C14Þ¼2:455/C21:75¼4:3.F o r KTP, the dielectric constant is unusually low (average 13.0); the in-plane values are 11.3 and 11.9 (average 11.6), and the out-of-plane one is frequency dependent but /C2517:5 at low frequencies and 15.4 at high frequencies ( Bierlein and van Herzeele, 1989 ;Noda et al. , 2000 ). Using the high-frequency value, this gives a susceptibility ratioof 1.2, for which the square root is 1.1. Hence, w 2=ðd/C14Þ¼ 2:455/C21:1¼2:7. For the 1-nm domain wall thickness given above as an example, this requires a crystal thickness two and one-half times as great as that for LiNbO 3, approximately 0.45 mm. These three materials thus require different thick- nesses for optimum phase matching at 1/C22m, as shown in Fig.33. As can be seen, the different dielectric anisotropy of the three materials has a relatively small impact on the optimum domain size, due to the fact that the crystal anisot- ropy is inside a square root of a square root. A much bigger variation of optimum domain size can be obtained by tuning instead the domain wall thickness, as discussed in the next section. 2. Manipulation of wall thickness The wall thickness parameter /C14is not an intrinsic constant. It can be increased by an order of magnitude by impurity10-610-510-410-310-210-810-710-610-510-4 =10nmw (m) d (m) BNN LNO KTP =1nm FIG. 33 (color online). Domain size as a function of thickness for three uniaxial ferroelectrics commonly used in electro-optical ap-plications. The continuous lines are calculated assuming domain wall thickness of 1 nm, while the dashed lines are for a wall thickness of 10 nm.G. Catalan et al. : Domain wall nanoelectronics 143 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012doping and it can also be increased via application of an electric poling field orthogonal to the polar axis. a. Doping We see in Reznik et al. (1985) that impurity doping can greatly increase the domain wall thickness in ferroelectrics.This will decrease the stability thickness dfor a given wavelength, or alternatively permit longer-wavelengthelectro-optic devices for a fixed film thickness. Similarly,wall thicknesses in congruent lithium niobate are about10 times thicker than in stoichiometric specimens. For many years only LiNbO 3grown from a congruent melt was available for study. These crystals have 1% (or 1021cm/C03) defects. The spectroscopic difference between congruent andstoichiometric LiNbO 3was first shown by Okamoto, Wang, and Scott (1985) andChowdhury (1978) . Recent periodically poled LiNbO 3devices favor stoichiometric samples because their domain walls are more stable. See, for example, Chu et al. (2008) . b. Photovoltaic tensor and off-axis poling The theoretical model of Rao and Wang (2007) implies that off-axis poling can significantly widen wall thicknesses.This brings us into a more general discussion of photovoltaic tensors. Over the years, perhaps misled by the standard text byLines and Glass (2004) , which implies that photovoltaic response in LiNbO 3is along the polar zaxis, many scientists failed to recognize that the photovoltaic tensor is neitherdiagonal nor second rank, despite the correct theory ofChen (1968 ,1969) . As a result, large voltages and fields can arise perpendicular to the polar axis when illuminated. In a 1-W beam at 514.5 nm wavelength, focused to /C2550/C22m diameter, lithium niobate exhibits a field of approximately40 kV =cmin the xyplane, due to the /C12 15photovoltaic tensor component ( Odulov, 1982 ,Anikiev et al. , 1985 ;Reznik et al. , 1985 ;Chaib, Otto, and Eng, 2003 ) (note that we used the reduced notation, the photovoltaic tensor is third rank).These off-axis photovoltages can be mitigated via application of a thermal gradient, which causes charge diffusion tomitigate the photovoltaic effect via the Seebeck effect (Kostritskii et al. , 2007 ,2008 ). Hence it would be useful to more carefully examine the effects of off-axis poling and ofphotovoltage normal to the polar axis. In particular, in thepresence of high-intensity laser light the symmetry of LiNbO 3is actually lowered; the threefold symmetry axis is lost. B. Domains and electro-optic response of LiNbO 3 In assessing the microscopic dynamics of domains in poled lithium niobate, it is useful to point out that the local electricfield is not necessarily along the polar axis, and that for electro-optic devices, in the presence of light there is a strong electric polarization induced perpendicular to this threefold c axis. This was first established by Chen (1969) , and later evaluated quantitatively by Anikiev et al. (1985) who found an orthogonal electric field of /C2540 kV =cmin the presence of moderately focused 0.5 W power at 514.5 nm from an argonlaser. Most recently this result has been confirmed by Chaib, Otto, and Eng (2003) . We emphasized this point here because it is contrary to the claims in the textbook by Lines and Glass (2004) (its first edition was written two years before Chen’s work). It can significantly influence domain widths. All ofthese effects differ in congruent and stoichiometric LiNbO 3, as do the phonon spectra ( Scott, 2002 ) and especially the quasielastic scattering ( Chowdhury, 1978 ;Okamoto, Wang, and Scott, 1985 ). C. Photovoltaic effects at domain walls Recently it was reported that an anomalous photovoltaic effect in BFO thin films arises from a unique, new mecha-nism, namely, structurally driven steps of the electrostatic potential at nanometer-scale domain walls ( Yang et al. , 2010 ; -4-4-40-4-4-4 -15 -10 -5 0 5 10 15-3x10-4-2x10-4-1x10-401x10-42x10-43x10(a) (b) (c) -4 Dark currentPhoto-currentCurrent density (A/cm2) Voltage (V)05 0 1 0 0 1 5 0 2 0 005101520 100 nmVoc (V) Electrode distance ( m)(d) FIG. 34 (color online). Light and dark I-Vmeasurements on ordered arrays of 71/C14domain walls in bismuth ferrite showing large open circuit voltages above the band gap of the material. (a), (b) Schematics of the electrode geometry (c) corresponding I-Vmeasurement perpendicular to the domain walls over 200/C22mdistance; (d) linear scaling of open circuit voltage with the number of domain walls.144 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012Seidel et al. , 2011 ). In conventional solid-state photovoltaics, electron-hole pairs are created by light absorption in a semi- conductor and separated by the electric field spanning a micrometer-thick depletion region. The maximum voltage these devices can produce is equal to the semiconductor electronic band gap, although in noncentric systems such asferroelectrics the photovoltage can be bigger than the band gap ( Sturman and Fridkin, 1992 ). Interestingly, domain walls can give rise to a fundamentally different mechanism for photovoltaic charge separation, which operates over a dis- tance of 1–2 nm and produces voltages that are significantlyhigher than the band gap (see Fig. 34). The separation happens at previously unobserved nanoscale steps of the electrostatic potential that naturally occur at ferroelectric domain walls in the complex oxide BiFeO 3. Electric-field control over domain structure allows the photovoltaic effect to be reversed in polarity or turned off. Currently, the overall efficiency of those photovoltaic devices is limited by the conductivity of the bulk bismuth ferrite material. Methods to increase the carrier mobility as well as inducing the spatially periodic potential in an adja-cent material with a lower gap than BFO are possible routes to achieve larger current densities under white light illumi- nation, and more generally, they demonstrate what the source of periodic potential and the PðVÞcurrent flow can be in different materials. Low-band-gap semiconductors withasymmetric electron and hole mobilities are possible candi- dates to show such an effect. In addition, photoelectrochemic effects at domain walls are a possible further interesting route, e.g., for applications in water splitting ( Kudo and Miseki, 2009 ). D. Switching of domains Rather comprehensive reviews of ferroelectric domain switching have been published elsewhere ( Shur, Gruverman, and Rumentsev, 1990 ;Scott, 2000 ), and so, after a few brief remarks, we concentrate on what is new andparticularly pertinent to thin ferroelectric films with high densities (volume fractions) of domain walls or twin boundaries. At present the best way to monitor domain wall switching is probably via measurement of displacement current IðtÞ versus time t. This gives a rapid rise followed by a roughly Gaussian peak. Assuming that the rise is not current limited from the drive voltage source and that the decay is notlimited by the RCtime constant, such data are popularly fitted to a model due to Ishibashi and Takagi (1971) and based upon earlier work by Avrami (1939) for the analogous problem of crystal growth. The fitting parameters used involve a characteristic switching time tð0Þand, importantly, a dimensionality D. One of the important aspects of this theory is that for a given dimensionality there is a preciseprediction of the dimensionless ratio iðmÞtðmÞ=P, where iðmÞ is the maximum displacement current density during switch- ing and occurs at time tðmÞ, and Pis the spontaneous polarization. In principle the dimension Dis an integer, but because of other approximations made in the model,particularly that the domain wall speed vis independent of domain radius r(it actually varies as 1=r), noninteger valuesusually result from least-squares fitting to the data. Other approximations are not so important, but Dalton, Jacobs, and Silverman (1971) pointed out that the model fails mathe- matically for finite dimensions. This model has been used extensively to fit data as functions of field E, thickness d, temperature T, and fatigue cycles n. An interesting obser- vation is that domain dimensionality Doften decreases from /C253to 2 or 1 with fatigue ( Araujo et al. , 1986 ). Of particular interest is what happens in thin films of highly twinned ferroelectrics and ferroelastics. This was first described by Bornarel, Lajzerowicz, and Legrand (1974) , who found that in such cases ferroelectric polar domain wallscould strongly interact with nearby ferroelastic nonpolarwalls, tilting both walls and making the nonpolar wallsslightly polar. This was recently demonstrated rather spec- tacularly in ferroelectric tris-sarcosine calcium chloride by Jones et al. (2011) . Of relevance here is also the fact that ferroelectric domain walls are easily pinned by defects andvacancies, so the switching properties and dielectric contri-bution of the walls can be modified by manipulating thedopant chemistry. Fujii et al. (2010) discussed this in some detail and explicitly showed how defect dipoles are more effective at domain wall pinning than are oxygen vacancies. E. Domain wall motion: The advantage of magnetic domain wall devices The development of prototype devices based upon mag- netic domain devices has been pioneered by Cowburn and co- workers ( Allwood et al. , 2005 ;Allwood, Xiong, and Cowburn, 2006a ,2006b ;Atkinson et al. ,2 0 0 6 ) as listed and shown schematically in Fig. 1. His devices are suitably small for commercial production (typically 15F2, where F2is the square of the feature size F), and extremely fast. His devices include NOT gates, AND gates, ( Zeng et al. ,2 0 1 0 ) shift registers ( O’Brien et al. , 2009 ), read/write memory devices ( Allwood, Xiong, and Cowburn, 2006a ,2006b ), signal fan-out devices, and data input and crossover structures(Allwood et al. , 2005 ). On the fundamental physics side, this group also investigated vortex domain wall transitions, which have a close relationship with the ferroelectric vortex do- mains discussed in Sec. III. They also examined magnetic comb structures (shown in Fig. 35) in detail ( Lewis et al. , 2010 ). It is not an exaggeration to say that microelectronic devices based upon magnetic domain dynamics are a full decade ahead of those based upon ferroelectric domains. In some important respects, it is difficult for ferroelectrics to catch up, literally. This is because magnetic domain wallsinvolve only flipping of spins (no mass) and can easily bedriven at near the speed of sound ( km=s). In fact, they can even be driven supersonically, with acoustic phonons being produced in high magnetic fields by supersonic magnetic domain walls at a phase angle related to that in the analogousproblems of bow waves in water or in Cerenkov radiation(Demokritov et al. , 1988 ,1991 ). By comparison, ferroelec- tric walls have real momentum and they cannot travel faster than the theoretical limit set by the transverse acoustic pho- non, the speed of sound, for otherwise the sonic boom wouldshatter the crystal. Moreover, domain walls satisfy a ballisticG. Catalan et al. : Domain wall nanoelectronics 145 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012equation of motion with viscous damping ( Dawber et al. , 2005 ) which causes a saturation or ‘‘terminal velocity’’ that is often (but not always) below the speed of sound. On the other hand, the terminal velocity, however, is dependent on the sample and the technique used to deliverthe voltage pulse, and it can be significantly raised. Millermeasured ferroelectric domain wall velocities in LiNbO 3over 9 orders of magnitude from 10/C09mm=supward and found that they saturate near 1:0m=sat high fields ( Miller, 1998 ). But more recent studies by Gruverman set a higher limit,between 10 and 100 m =s(Gruverman, Wu, and Scott, 2008 ). And much faster switching, with velocities approaching thespeed of sound, was achieved by Li and co-workers in directmeasurements of the switching dynamics using an ultrafastphotoconducting switch enabled electric pulse with a rise time of tens of picoseconds ( Liet al. , 2004 ). They showed intrinsic switching time scales of 50–70 psec in fullyintegrated capacitors of 2/C23/C22m 2in lateral dimensions,fabricated with films of 150 nm thickness, suggesting veloc- ities of 2000 –3000 m =s. It should be noted that such mea- surements are not routine: Indeed, it is quite likely that asignificant number of measurements in the literature are compromised by either the rise time of the pulse generator, theRCtime constant of the measurement system, or the rise time of the oscilloscope system. It is also worth mentioningthat ultrahigh fields can be achieved in sufficiently thinsingle-crystal samples: Morrison et al. (2005) reported fields of1:3G V =minBaTiO 3lamellae, and nobody knows what the domain wall speed is under such high fields. As well as maximizing domain wall speed, the future development of ferroelectric domain wall devices will proba- bly require denser domain arrays, so that the walls travel ashorter distance, or rely on device designs that do not requirehigh wall speeds, such as domain conduction devices. Thefirst criterion can be readily met, since ferroelectric domainsare known to be generally narrower than their magneticcounterparts, as seen in previous sections. The second (con- ductivity of domain walls) has also been discussed and will be examined further in the next section. Another important question, which is only now begin- ning to be explored, is that of domain wall dynamics inmagnetoelectric multiferroics. The different response of themagnetic and ferroelectric components of multiferroicwalls to external fields has been proposed by Fontcuberta and co-workers as a new mechanism for eliciting switchable control of exchange bias in hexagonalmultiferroics such as YMnO 3(Skumryev et al. , 2011 ). On the theoretical front, little is yet known about the dynam-ics of coupled domain walls, so this line of work certainlymerits further attention. Parkin and co-workers have made considerable progress in memory devices based upon magnetic domain wall mo- tion, introducing the concept of the racetrack memory (Fig. 36). This design concept in principle offers storage densities that are larger than conventional solid-state mem-ory devices such as flash memory with a better read andwrite performance. Key in these devices is the fact thatmagnetic domain walls can exhibit considerable momentum,moving about a micron after a current pulse is applied (Thomas, Moriya, and Rettner, 2010 ). This is about an order of magnitude less than the inertial travel distance of ferro-electric domain walls subjected to large pulsed fields, but itis not negligible. This is important for magnetic domainmemories because it implies that the spatial positioning ofwalls can be precisely controlled by the current pulse length.(This is somewhat surprising, since the magnetic domain motion follows the Landau-Lifshitz-Gilbert equation, which is first order in time, whereas ferroelectric domain wallssatisfy Newton’s equations, which are second order in timeand hence explicitly display momentum.) The basic mechanism to ‘‘push’’ domain walls along the racetrack using a current is the ‘‘spin torque.’’ The underlyingprinciple is that when spin-polarized electrons in a ferromag-netic material pass through a magnetic domain wall, there is a torque on the electrons that acts to reorient their spin mag- netic moments along the magnetization direction (Fig. 37). Angular momentum in this system is conserved by a reactiontorque, termed the spin transfer torque, which acts from the FIG. 35. (a) Electron microscope images of a magnetic strip and magnetic comb structures; the scale bar is 1/C22min all cases. Magnetic comb structures are designed to speed up domain walls,as experimentally demonstrated in (b). Note the high velocity of the magnetic domain walls (in excess of 1500 m =s) that can be achieved in the combed structures. From Lewis et al. , 2010 .146 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012electrons onto the material magnetization in a way such that it displaces the domain wall in the direction of the electron flow(Stiles and Miltat, 2006 ;Ralph and Stiles, 2008 ). Domain wall motion can be achieved when the current density throughthe device is sufficiently high. Current designs of racetrack memories use a spin-coherent electric current to move magnetic domains along a nano-scopic permalloy wire with a cross section of 200/C2100 nm 2. As a current is passed through the wire, the domain walls passby read and write heads. A memory device is made frommany such elements. Improvements in domain wall detection capabilities, based on the development of new magnetoresis-tive materials, allow the use of increasingly smaller magneticdomains to reach higher storage densities. The basic opera- tion of the racetrack magnetic domain wall memory system is described by Parkin, Hayashi, and Thomas (2008) , and recent details concerning wall pinning discussed by Jiang et al. (2010) , although the basic idea that magnetic domain walls could be moved into precise positions was developed adecade earlier by Ono et al. (1999a ,1999b) . A recent review of this aspect of magnetic electronics (‘‘spintronics’’) has been given by Bader and Parkin (2010) , which follows an earlier review on fundamentals and applications of nanomag-netism by Bader (2006) . F. Emergent aspects of domain wall research We focus on some emergent behavior at domain walls, particularly in materials such as multiferroics, that exhibitcoupled order parameters, i.e., the charge and spin degree offreedom are coupled. In order to focus the discussion on whatis understood and what remains to be explored, we use the multiferroic BiFeO 3as our model system. The richness of phase evolution and electronic properties in this system isnow well established, and we are beginning to understand themanipulation of its electronic structure, correlation effects,and order parameter evolution on the unit-cell level. What arethe consequences and the opportunities? This we discuss next. 1. Conduction properties, charge, and electronic structure By far one of the most fascinating aspects of research on a bismuth ferrite as a multiferroic has to do with the changes in electronic structure as a function of crystal chemistry andparticularly at domain walls. Rhombohedral BiFeO 3has been shown to possess ferroelectric domains in thin films that areinsulatorlike, whereas conduction in its domain walls issignificant ( Seidel et al. , 2009 ) (Fig. 38). The observed conductivity correlates with structurally driven changes in both the electrostatic potential and the local electronic struc-ture, which shows a decrease in the band gap at the domainwall. In light of the intriguing electrical conductivity, detailed electronic properties of the domain walls have been inves-tigated by Lubk, Gemming, and Spaldin (2009) . The layer- by-layer densities of states was calculated to see if the structural deformations in the wall region lead to a closingof the electronic band gap. In particular, the ideal cubicstructure, in which the 180 /C14Fe-O-Fe bond angles maximize the Fe 3d–O2phybridization and hence the bandwidth, has a significantly reduced band gap compared to the R3cstructure. Figure 39shows the local band gap extracted from the layer- by-layer densities of states across the three wall types. In allcases a reduction in the band gap in the wall can be seen, withthe 180 /C14wall showing the largest effect. In no case, however, does the band gap approach zero in the wall region. The samefirst-principles calculations supporting the experimentalwork of Seidel et al. also give insight into the changes in the Fe-O-Fe bond angle in BiFeO 3, in addition to the fact that walls in which the rotations of the oxygen octahedra do not FIG. 37 (color online). Schematic of spin scattering from an interface with a ferromagnet in a simple limit of ideal spin- dependent transmission and reflection. From http://www.nist.gov/ cnst/epg/spin_transfer_torque.cfm. FIG. 36 (color online). The racetrack: a ferromagnetic nanowire.Pulses of highly spin-polarized current move domain walls coher- ently in either direction via spin torque. (a) A vertical-configuration racetrack. Magnetic patterns in the racetrack before and after thedomain walls have moved down one branch of the U, past the read and write elements, and then up the other branch. (b) A horizontal configuration. (c) Reading data from the stored pattern by measur-ing the tunnel magnetoresistance of a magnetic tunnel junctionelement connected to the racetrack. (d) Writing data by the fringing fields of a domain wall moved in a second ferromagnetic nanowire. (e) Arrays of racetracks on a chip for high-density storage. FromParkin, Hayashi, and Thomas, 2008 .G. Catalan et al. : Domain wall nanoelectronics 147 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012change their phase when the polarization reorients are sig- nificantly more favorable than those with rotation disconti- nuities, i.e., antiphase octahedral rotations are energetically costly. The analysis of the local polarization and electronic prop- erties also revealed steps in the electrostatic potential for all wall types, and these must also contribute to the conductivity. Steps in the electrostatic potential at domain walls are corre-lated with (and caused by) small changes in the component ofthe polarization normal to the wall ( Seidel et al. , 2010 ). These changes in normal polarization are a consequence of the fair rotation of the polar vector across the domain wall and are not exclusive of BiFeO 3. Tetragonal PbTiO 3, for example, shows a similar effect for a 90/C14wall ( Meyer and Vanderbilt, 2002 ) (Fig. 40). Extended phase-field calculations for tetrago- nalBaTiO 3also allow calculating the intrinsic electrostatic potential drop across the 90/C14domain wall, regardless of the consideration of the ferroelectric as an n-type semiconductoror dielectric ( Hong et al. , 2008 ). This potential change creates a large electric field that promotes an asymmetriccharge distribution around the walls, where electrons and oxygen vacancies concentrate on the opposite sides. The increased charge density presumably promotes increasedconductivity. As mentioned, the semirigid rotation of the polar vector across a ferroelectric-ferroelastic wall leads to an electrostaticpotential that is screened by free charges which enhance the local charge density and thus, presumably, the conductivity. Since this polar rotation is not exclusive of BiFeO 3, other perovskite ferroelectrics should also be expected to displayenhanced conductivity, and perhaps this mechanism is behindthe enhanced conductivity recently reported also for the domain walls of PbðZr;TiÞO 3(Guyonnet et al. , 2011 ). In BiFeO 3, several other factors might be further helping the conductivity enhancement: First, the magnetoelectric cou-pling between polarization and spin lattice is such that themagnetic sublattice rotates with the polarization ( Zhao et al. , 2006 ;Lebeugle et al. , 2008 ). Since spins rotate rigidly (see Sec. III.B ), they might favor a more rigid rotation of the polarization and hence a bigger electrostatic step at the wall(and, of course, the polarization of BiFeO 3is itself bigger than that of other known perovskite ferroelectrics, whichmeans that all other things being equal a rigid polar rotation inBiFeO 3will cause a bigger electrostatic step). Also, the increased spin alignment at the wall should lower the mag-netic contribution to the band gap ( Dieguez and In ˜iguez, 2011 ). But perhaps the most obvious consideration is the fact that BiFeO 3has an intrinsically smaller band gap than other prototypical perovskite ferroelectrics ( /C242:7e V instead of 3.5–4 eV). This means that the screening charges accumu- lated at the wall will be closer to the bottom of the conductionband and hence will more easily contribute to the conductiv-ity. It would be interesting to see if highly insulating FIG. 40. 90/C14domain walls in lead titanate: (a) potential steps at domain walls; (b) theoretical conduction and valence band alignment; (c) potential in equilibrium. From Meyer and Vanderbilt, 2002 . FIG. 39 (color online). Local band gap at domain walls in bismuth ferrite extracted from the layer-by-layer densities of states. From Lubk, Gemming, and Spaldin, 2009 . FIG. 38 (color online). (a) The three different types of domain walls in rhombohedral bismuth ferrite. Arrows indicate polarization directions in adjacent domains. (b) In-plane PFM image of a written domain pattern in a monodomain BFO (110) film showing all threetypes of domain wall. (c) Corresponding c-AFM image showingconduction at both 109 /C14and 180/C14domain walls; note the absence of conduction at the 71/C14domain walls. This stands in contrast with recent results reporting enhanced conductivity in the 71/C14walls (Farokhipoor and Noheda, 2011 ). Adapted from Seidel et al. , 2009 .148 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012single-crystal samples such as those studied by Chishima et al. (2010) display the same domain wall conductivity as do the thin-film samples studied so far. The current density of these single-crystal samples can be as low as 10/C09A=cm2 even at electric fields in excess of 50 kV =cm, while typical resistivities of thin films are in the region of 106–108/C10c m , comparable to the resistivity of good quality BiMnO 3 (Eerenstein et al. , 2005 ). The role of defect accumulation at the walls also deserves close scrutiny, because defects control the transport behavior, as recently emphasized by Farokhipoor and Noheda (2011) . Localized states are found in the spectrum of ferroelectricsemiconductors, and states localized at the walls and insidethe domain but close to the wall split off from the bulk continuum. These nondegenerate states have a high disper- sion, in contrast with the ‘‘heavy-fermion’’ states at an iso-lated domain wall ( Idlis and Usmanov, 1992 ). Charged double layers can be formed due to coupling between polar-ization and space charges at ferroelectric-ferroelastic domainwalls ( Xiao et al. , 2005 ). Charged domain wall energies are about 1 order larger than the uncharged domain wall energies (Gureev, Tagantsev, and Setter, 2009 ), and phenomenological calculations show decoration of walls by defects such asoxygen vacancies. The presence of charge and defect layersat the walls means that such walls promote electrical failure by providing a high conductivity pathway from electrode to electrode ( Xiao et al. , 2005 ).Eliseev et al. (2011) have also shown how the relative sign between the wall charge and thetype of majority carriers also matters: positively chargedwalls in an n-type ferroelectric are more easily screened (and thus have smaller thickness and lower energy) than negatively charged walls, due to the bigger abundance of screening charges. Vice versa for negatively charged wallsin ap-type ferroelectric semiconductor. The control of the electronic structure at walls by doping and strain in ferroelectric and ferroeleastic oxides opens a way to effectively engineer nanoscale functionality in such mate- rials. For the case of BiFeO 3A-site doping with Ca, and magnetic B-site substitution such as Co or Ni, might prove to be a viable way to achieve new domain wall properties bymanipulating the electronic structure, spin structure, and di- polar moment in this material ( Yang et al. , 2009 ). Of obvious future interest is the question of what sets the limits to thecurrent transport behavior at walls: Can one ‘‘design’’ thetopological structure of the domain wall to controllably induceelectronic phase transitions within the wall arising from thecorrelated electron nature? Is it possible to trigger an Anderson transition by doping of domain walls or straining them? Recently, some of us reported the observation of tunable electronic conductivity at domain walls in La-doped BFOlinked to oxygen vacancy concentration ( Seidel et al. , 2010 ). The conductivity at 109 /C14walls is thermally activated with activation energies of 0.24 to 0.5 eV. From a broader perspec- tive, these results are the first step toward realizing thetantalizing possibility of inducing an insulator-metal transi-tion ( Imada, Fujimori, and Tokura, 1998 ) locally within the confines of the domain wall through careful design of theelectronic structure, the state of strain, and chemical effects at the domain wall. For actual device applications the magni- tude of the wall current needs to be increased. The choice ofthe right shallow-level dopant and host material might prove to be key factors in this respect. Further study of correlations between local polarization and conductivity is an exciting approach to understanding the conduction dynamics andassociated ferroelectric properties in the presence of strongcoupling between electronic conduction and polarization incomplex oxides. 2. Domain wall interaction with defects Defect–domain wall interaction is an important area of research that deserves increased attention ( Robels and Arlt, 1993 ;Gopalan, Dierolf, and Scrymgeour, 2007 ). Point de- fects can broaden the wall ( Shilo, Ravichandran, and Bhattacharya, 2004 ;Lee, Salje, and Bismayer, 2005 ). The width of twin walls in PbTiO 3, for example, can be strongly modified by the presence of point defects within the wall. Theintrinsic wall width of PbTiO 3is about 0.5 nm, but clusters of point defects can increase the size of the twin wall up to 15 nm ( Salje and Zhang, 2009 ). Trapped defects at the domain boundary play a significant role in the spatial varia-tion of the antiparallel polarization width in the BaMgF 4 single crystal as seen by PFM ( Zeng et al. , 2008 ), and asymmetric charge distribution around 90/C14domain walls in BaTiO 3have also been reported, where electrons and oxygen vacancies concentrate on the opposite sides ( Hong et al. , 2008 ). Interaction between the order parameter and the point defect concentration causes point defects to accumulate within twin walls ( Salje and Zhang, 2009 ); conversely such defects contribute to the twin-wall kinetics and hysteresis, asthey tend to clamp the walls. Oxygen vacancies, in particular, have been shown to have a smaller formation energy in the domain wall than in the bulk, thereby confirming the ten-dency of these defects to migrate to, and pin, the domainwalls ( He and Vanderbilt, 2003 ). This leads to a mechanism for the domain wall to have a memory of its location during annealing ( Xiao et al. , 2005 ). 3. Magnetism and magnetoelectric properties of multiferroic domain walls An important question to ask at this point is what is the true state of magnetism at a multiferroic domain wall.Temperature-dependent transport measurements are a pos- sible route to follow to understand the actual spin structure and whether it exhibits a glasslike or ordered ferromagneticstate. Of interest is the effect of extra carriers introduced intothe system, e.g., by doping or electric gating, on magnetism.Is there a way to change the magnetic interaction from super- exchange to double exchange? The strength of the coupling between the ferroelectric and antiferromagnetic walls inBiFeO 3is an issue that still needs to be resolved from both a theoretical and an experimental perspective. The role of the dimensionality on electrical and magnetoelectrical transport needs to be elucidated and compared to known systems, suchas manganites ( Dagotto, 2003 ;Salafranca, Yu, and Dagotto, 2010 ). We note that the interaction between ferroelectric and antiferromagnetic domain walls has been studied in model multiferroics such as YMnO 3(Goltsev et al. , 2003 ) and BiFeO 3(Gareeva and Zvezdin, 2011 ). In both cases it hasG. Catalan et al. : Domain wall nanoelectronics 149 Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012been shown that the antiferromagnetic domain walls are significantly wider (by /C241–2orders of magnitude) compared to the ferroelectric walls. This is also in agreement with the phenomenological predictions of Daraktchiev, Catalan, and Scott (2010) for coupling-mediated wall broadening. VI. FUTURE DIRECTIONS It is safe to say that the phenomena and physics of domain walls in ferroelectrics form an exciting and growing field of interest. As device size is reduced, the number density of domain walls grows, with consequences for functionalbehavior. Based on the differences between domain wall types, the inclusion of a ‘‘wrong’’ domain type could give the system entirely different properties. With the current developments surrounding the conductive properties there are many remaining questions and some new ones. Electronic conduction was predicted for ferroelectric domain walls based on the fact that charged double layers may formon either side of these walls ( Hong et al. , 2008 ). Since the ferroelectric in which the conductive walls were found is a multiferroic, the obvious next step is to verify that multi- ferroics also have this double layer and determine whether it is responsible for the conduction. If this double layer isindeed present and responsible for the conduction, it should be interesting to combine this with the idea that multiferroics have broader walls compared to pure ferroelectrics. Is there a maximum on the thickness of a domain wall to still have this double layer and be conductive? In a more general sense one could ask oneself whether the physics and assumptions based on the findings in pure ferroelectrics are valid for the multi-ferroic materials as well. Conversely, we need to also identify the aspects of domain wall behavior that are exclusive to multiferroics: Order parameter coupling and chirality are two features of multiferroic walls that have unique roles. Another front of research is the investigation of dynamic conductivity at domain walls ( Maksymovych et al. , 2011 ). This addresses important factors: a possible electric-field induced distortion of the polarization structure at the domainwall, the dependence of conductivity on the degree of dis- tortion, and weak-pinning scenarios of the distorted wall. The domain wall is likely not a rigid electronic conductor, instead offering a quasicontinuous spectrum of voltage-tunable elec- tronic states ( Maksymovych et al. , 2011 ). This is different from ferroelectric domains, where switching may give rise to discrete (often only two) conductance levels ( Garcia et al. , 2009 ;Maksymovych et al. , 2009 ). The intrinsic dynamics of domain walls and other topological defects are expected not only to influence future theoretical and experimental inter- pretations of the electronic phenomena, but also to pose the possibility of finding unique properties of multiferroic do- main walls, e.g., magnetization and magnetoresistance within an insulating antiferromagnetic matrix ( Heet al. , 2011 ), also due to order parameter coupling and localized secondary order parameters ( Salje and Zhang, 2009 ;Daraktchiev, Catalan, and Scott, 2010 ). Of future interest is the question of what sets the limits to the current transport behavior at walls: Can one design the topological structure of the domainwall to controllably induce electronic phase transitions within the wall arising from the correlated electron nature? Is itpossible to trigger an Anderson transition by doping of domain walls ( Yang et al. , 2009 ) or straining them? The observation of superconductivity in ferroelastic walls of WO 3 certainly points to various exciting and unexplored areas of domain boundary physics ( Aird and Salje, 1998 ). Another interesting direction for domain wall engineering in ferroelectrics is by the size and design of the system, andthis includes not only the domains themselves but also theirhierarchical self-organization into bigger metastructures.Recently, Schilling et al. (2009) presented work on nano- ferroelectrics, which shows considerably more domain wallsper unit volume, thanks to the size constraints in two and three dimensions, as opposed to the single finite dimension of thin films. Yet another unique design feature of the samples ofSchilling et al. is that they are free-standing ferroelectrics, unlike those that are grown on a substrate, and for whichintrinsic surface tension can play a bigger role ( Luk’yanchuk et al. , 2009 ). Such nanostructures are also prone to new types of topological defects beyond the classic domain walls; for example, recent work by Hong et al. (2010) shows that arrays of ferroelectric nanowires have switchable quadrupoles andthus potential as nanodevices. Exotic topological defects innanostructures (vertices, vortices, quadrupoles, etc.) are cur-rently an active area of research. Another interesting feature that is being studied intensively is the fact that the domains in nanocrystals clearly show organization on several length scales, with correlation not just between narrow stripe domains but also between packetsof stripes. Ivry et al. (2011) found a variety of mesoscopic- scale domain packets or bundles with considerable cross-talkacross PbðZr;TiÞO 3(PZT) grain boundaries. More strikingly, McQuaid et al. (2011) showed metadomains or superdo- mains that are composed of thin stripes but reproduce on amesoscopic scale the exact shape and functional behavior of closure domains such as those of Figs. 10and11. The physics and functional engineering of mesoscopic metadomains orbundles is a rapidly growing topic that will no doubt see moreactivity in the near future. Several applications have been suggested to make use of domain walls in ferroelectric materials based on their addi-tional functionalities as well as their affects on existing devices. Uses that have been mentioned are as a local strain sensor incorporated on an AFM probe or a multilevelresistance-state device that is written by an electrical current(Be´a and Paruch, 2009 ). Other possibilities include nonvola- tile memories, piezoelectric actuators, ultrasound trans-ducers, surface acoustic wave devices, and opticalapplications ( Gopalan, Dierolf, and Scrymgeour, 2007 ). For existing devices, the discovery of conducting domain walls stimulates engineers to prevent their products from having thewrong domain walls that could cause leakage and prevent itsuse in ferroelectric memories. This use of ferroelectrics inmemory has recently been reviewed, and it has been arguedthat conductivity may not be a detriment but an opportunityfor new memory reading mechanisms ( Be´a and Paruch, 2009 ; Garcia et al. , 2009 ;Maksymovych et al. , 2009 ;Zubko and Triscone, 2009 ;Jiang et al. , 2011 ). Experimental results and theoretical investigations in recent years have convincingly demonstrated that certaintransition metal oxides and some other materials have150 G. Catalan et al. : Domain wall nanoelectronics Rev. Mod. Phys., V ol. 84, No. 1, January–March 2012dominant properties driven by spatial inhomogeneity. Strongly correlated materials incorporate physical interac-tions (spin, charge, lattice, and/or orbital hybridization), al- lowing complex interactions between electric and magnetic properties, resulting in ferromagnetic, or antiferromagneticphase transitions. Of even higher interest are the heterointer-faces formed between correlated materials showing new stateproperties. Domain walls are only one example of ‘‘natu-rally’’ occurring interfaces in such materials. The challenge is to determine whether such complex interactions can be controlled in those materials or heterointerfaces at suffi-ciently high speeds and densities to enable new logic devicefunctionality at the nanometer scale. Parameters such asinterface energy, switching speed and threshold, tunability, dynamics of the states, and size dependencies need to be quantified to determine if domain boundary materials couldbe employed as a building block for information processingsystems. In addition, there are some new phenomena associated with ferroelectric domain walls that merit fundamental study: As shown in Sec. III, 2D arrays of vertex domains on ferro- electric surfaces often come in pairs of threefold vertices(Srolovitz and Scott, 1986 ). Fourfold vertices of domains exist in barium sodium niobate, and sixfold domains arewell known since 1967 in YMnO 3(Safrankova, Fousek, and Kizhaev, 1967 ). And so one might ask whether these arrays of vertex domains ‘‘melt’’ at temperatures below theCurie temperature at which stripe domains disappear, i.e., doferroelectric vertex arrays undergo Kosterlitz-Thouless melt-ing ( Kosterlitz and Thouless, 1973 ) involving defect pair production and annihilation? A second and rather deep phe- nomenon has been recently discovered by Schilling et al. (2011) : Vertex domains in rectangular ferroelectrics have off- centered vertices, whose position can be calculated accordingto a Landau theory with aspect ratio replacing temperature;the resulting novel shape-generated phase transition can therefore occur at0 K (quantum criticality). The ferromagnetic properties of ferroelectric walls in para- magnetic and antiferromagnetic materials ( Goltsev et al. (2003) ;Daraktchiev, Catalan, and Scott (2008 ;2010 ) suggest that much more research and development should be done on domain walls in multiferroics and also on the dynamics of domain walls in these materials ( Skumryev et al. , 2011 ): We note in this respect that BiFeO 3is no longer the only room- temperature multiferroic, nor Cr2O3the only good room- temperature magnetoelectric, with the lead-iron-tantalate,lead-iron-niobate, lead-iron-tungstate family now being studied in various laboratories (Josef Stefan Institute, University of Puerto Rico, University of Cambridge), andnew chromates being reported at Florida State, all of whichfunction at room temperature. Many of these are relaxorlikesystems and therefore have intrinsic nanodomains. In summary, we have provided an overview on ferroelectric and multiferroic nanodomain and domain wall electronics.The state of understanding and especially of application lagsthat for magnetic domains, with which comparisons are made;the work of Cowburn et al. and Parkin et al. makes it hard for ferroelectric domain electronics to compete with magnetic devices based on spatial manipulation; this is simply because the magnetic domains have greater mobilities. Thus it isunlikely that ferroelectrics will provide the equivalent of race- track memories, or fast AND orNOT gates, as developed by those groups. Instead it is likely that they will provide com- plementary devices that exploit the electrical conductivity and/or ferromagnetisn of ferroelectric domain walls. Hence these may involve fewer memory devices but more intercon- nects, switches, and sensors and actuators. 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PhysRevB.103.024524.pdf

PHYSICAL REVIEW B 103, 024524 (2021) Spin pumping between noncollinear ferromagnetic insulators through thin superconductors Haakon T. Simensen , Lina G. Johnsen , Jacob Linder, and Arne Brataas Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (Received 25 November 2020; revised 11 January 2021; accepted 12 January 2021; published 22 January 2021) Dynamical magnets can pump spin currents into superconductors. To understand such a phenomenon, we develop a method utilizing the generalized Usadel equation to describe time-dependent situations in supercon-ductors in contact with dynamical ferromagnets. Our proof-of-concept theory is valid when there is sufficientdephasing at finite temperatures, and when the ferromagnetic insulators are weakly polarized. We derive theeffective equation of motion for the Keldysh Green’s function focusing on a thin film superconductor sandwichedbetween two noncollinear ferromagnetic insulators, one of which is dynamical. In turn, we compute the spincurrents in the system as a function of the temperature and the magnetizations’ relative orientations. When theinduced Zeeman splitting is weak, we find that the spin accumulation in the superconducting state is smaller thanin the normal states due to the lack of quasiparticle states inside the gap. This feature gives a lower backflow spincurrent from the superconductor as compared to a normal metal. Furthermore, in superconductors, we find thatthe ratio between the backflow spin current in the parallel and antiparallel magnetization configuration dependsstrongly on temperature, in contrast to the constant ratio in normal metals. DOI: 10.1103/PhysRevB.103.024524 I. INTRODUCTION Superconductivity and ferromagnetism are conventionally considered antagonistic phenomena. Superconductors (SCs)in contact with ferromagnets (FMs) lead to mutual suppres-sion of both superconductivity and ferromagnetism [ 1,2]. Despite this apparent lack of compatibility, several intriguingeffects also emerge from the interplay between superconduc-tivity and ferromagnetism [ 3,4]. A singlet s-wave SC either in proximity with an inhomogeneous exchange field [ 5], or experiencing a homogeneous exchange field and spin-orbit coupling [ 6,7], induces spin-polarized triplet Cooper pairs. The generation of spin-polarized Cooper pairs is of particu-lar interest, paving the way for realizing dissipationless spintransport [ 4]. In recent developments, the combination of magnetization dynamics and superconductivity has gainedattention. This is motivated by spin-pumping experimentsreporting observations of pure spin supercurrents [ 8,9]. Ex- hibiting a wide range of interesting effects and phenomena,SC-FM hybrids are promising material combinations in theemerging field of spintronics [ 10]. It is well known that the precessing magnetization in FMs generates spin currents into neighboring materials via spinpumping [ 11–13]. The injection of a spin current into a neigh- boring material generates a spin accumulation, which in turngives rise to a backflow spin current into the FM. Spin pump-ing has a reactive and a dissipative component, characterizedby how it affects the FM’s dynamics. Reactive spin currentsare polarized along the precession direction of the magnetiza-tion, ˙m, and they cause a shift in the ferromagnetic resonance (FMR) frequency. Dissipative spin currents resemble Gilbertdamping and are polarized along m×˙m, relaxing the magne- tization toward its principal axis. The dissipative spin currentenhances the effective Gilbert damping coefficient [ 14], and broadens the FMR linewidth [ 12,15]. In SCs, both quasiparticles and spin-polarized triplet Cooper pairs can carry spin currents. In the absence of spin-polarized triplet pairs, spin pumping is typically much weakerthrough a superconducting contact than a normal metal (NM)[16,17]. The reduced efficiency is because the supercon- ducting gap /Delta1prevents the excitation of quasiparticles by precession frequencies ω< 2/Delta1. When spin-polarized triplet pairs are present, spins can flow even for low FMR fre-quencies as pure spin supercurrents. Reference [ 8] reported evidence for such pure spin supercurrents. An enhanced FMRlinewidth was measured in a FM–SC–heavy-metal hybridsystem as it entered the superconducting state, which is a sig-nature of an enlarged dissipative spin current [ 18]. The authors attributed this observation to spin transport by spin-polarizedtriplet pairs. These findings and the rapid development ofspintronics have lately sparked a renewed interest in spintransport through FM|SC interfaces [ 9,19–27]. Several earlier works have also considered spin transport resulting from mag-netization dynamics in SC-FM hybrids [ 28–35]. Progress has been made in developing a theoretical under- standing of the spin pumping through SCs [ 17,19,21–23,25 ]. For instance, assuming suppression of the gap at the interface,Ref. [ 17] computed the reduced spin-pumping efficiency in the superconducting state using quasiclassical theory. How-ever, to the best of our knowledge, a full understanding ofthe boundary conditions’ complicated time dependence be-tween dynamical ferromagnets and superconductors is notyet in place. This development is required to give improvedspin-pumping predictions in multilayers of FMs, SCs, andNMs. Furthermore, spin-pumping in superconducting sys-tems with a noncollinear magnetization configuration remains 2469-9950/2021/103(2)/024524(12) 024524-1 ©2021 American Physical SocietySIMENSEN, JOHNSEN, LINDER, AND BRATAAS PHYSICAL REVIEW B 103, 024524 (2021) theoretically underexplored, but it can provide additional in- sight into spin-transport properties. We present a self-consistent method designed to solve the explicit time dependence arising from magnetization dynam-ics by using the generalized Usadel equation. The explicittime dependence complicates the treatment and understandingof spin-transport properties. We aim to describe a consistentproof-of-concept approach that is as simple as possible tounderstand. We will therefore use simplifying assumptionsthat are justified in weak insulating ferromagnets. Hopefully,the main message is then less hindered by subtleties. (i) Weexplore trilayers with a thin film SC between two noncollinearferromagnetic insulators (FMIs). (ii) We exclusively considerthe imaginary part of the spin-mixing conductance in thecontacts between the FMIs and the SC film. (iii) We considerinsulating ferromagnets. The first assumption requires that theinterface resistance is larger than the superconductor’s bulkresistance in the normal state, and that the superconductor isthinner than the coherence length. The second assumption isvalid in weak ferromagnets. Our first main result is the equation of motion for the Green’s function in the SC film when the magnetizationprecesses. Based on these results, we present quantitativepredictions for the spin current as a function of temperatureand the relative magnetization orientation between the FMIs. II. THE GENERALIZED USADEL EQUATION AND ITS SOLUTION In this section, we will first present the generalized Usadel equation, taking into account the magnetization precession.We will demonstrate that it is possible to find an approxi-mate solution to the time dependence when the precessionfrequency is sufficiently slow. In superconductors, we willdiscuss how this approach requires sufficient dephasing, sinceotherwise the peaks in the density of states invalidate theadiabatic assumption. Finally, we will solve the generalizedUsadel equation and compute the resulting spin-current drivenby the magnetization precession. Our analytical approach issupplemented by a numerical solution demonstrating the con-sistency of our assumptions. A. The Generalized Usadel equation in a FMI|SC|FMI trilayer The generalized Usadel equation determines the time evo- lution of the electron Green’s function ˇGin the dirty limit. In a SC the generalized Usadel equation reads [ 36] −iD∇ˇG◦∇ˇG+i∂t1ˆτ3ˇG(t1,t2)+iˇG(t1,t2)∂t2ˆτ3 +[ˆ/Delta1(t1)δ(t1−t/prime)◦,ˇG(t/prime,t2)]=0, (1) where Dis the diffusion coefficient and δ(t) is the Dirac delta function. The symbol ◦denotes time convolution, (a◦b)(t1,t2)=/integraldisplay∞ −∞dt/primea(t1,t/prime)b(t/prime,t2), (2) IMFC S FMI e e FIG. 1. FMI|SC|FMI trilayer. The superconductor is a thin film. The large red arrows depict the magnetic moments of localized d electrons in the FMIs. The green cloud illustrates a gas of selectrons with spin up (red) and down (blue). An attractive interaction between theselectrons (red sawtooth-like line) gives rise to superconductiv- ity. The s-dexchange interaction at the interfaces gives rise to the indirect exchange interaction between the left and right FMI (wiggly gray lines). The precessing magnetization in the left FMI gives riseto spin currents j s Landjs Rfrom the FMIs into the SC. and [ a◦,b]=a◦b−b◦a.ˇGand ˆ/Delta1are matrices, ˇG=/parenleftbiggˆGR ˆGK 0 ˆGA/parenrightbigg ,ˆ/Delta1=⎛ ⎜⎝00 0 /Delta1 00 −/Delta1 0 0/Delta1∗00 −/Delta1∗00 0⎞ ⎟⎠,(3) where R,A, and Kdenote the retarded, advanced, and Keldysh components, respectively. /Delta1is the superconducting gap. We choose to work in the gauge where /Delta1=/Delta1∗is real. In our notation, the hat (e.g., ˆG) denotes 4 ×4 matrices in the sub- space of particle-hole ⊗spin space. The inverted hat (e.g., ˇG) denotes matrices spanning Keldysh space as well. σiare Pauli matrices spanning spin space, where i∈{0,x,y,z}, andσ0is the identity matrix. τiare Pauli matrices span- ning particle-hole space, where i∈{0,1,2,3}, andτ0is the identity matrix. To simplify the notation, we will omit outerproduct notation between matrices in spin and particle-holespace. Consequently, τ iσjshould be interpreted as the outer product of the matrices τiandσj. Moreover, we use the following notation for matrices that are identity matrices inspin space: ˆ τ i≡τiσ0. We consider thin film SCs sandwiched between two iden- tical, homogeneous, weakly magnetized FMIs, illustrated inFig. 1. Because of the insulating nature of the FMIs, we disre- gard any tunneling through the FMIs. The interaction betweenelectrons in the SC region and the FMIs is therefore localizedat the interfaces. This s-dexchange interaction couples the localized delectrons in the FMIs to the selectrons in the SC at the interface. In thin film SCs, where the thickness ofthe superconductor is much shorter than the coherence length,L S/lessmuchξS, we can approximate the effect of the s-dexchange interaction as an induced, homogeneous magnetic field in theSC [ 37–40]. Furthermore, in computing the transport prop- erties, this assumption requires that the interface resistances(inverse ”mixing” conductances) are larger than the SC’s bulkresistance in the normal state. When L S/lessmuchξS, the Green’s function changes little throughout the SC, and we thereforeneglect the gradient term in the generalized Usadel equation 024524-2SPIN PUMPING BETWEEN NONCOLLINEAR … PHYSICAL REVIEW B 103, 024524 (2021) within the SC. The resulting effective generalized Usadel equation for the FMI|SC|FMI trilayer then reads i∂t1ˆτ3ˇG(t1,t2)+iˇG(t1,t2)∂t2ˆτ3+[ˆ/Delta1(t1)δ(t1−t/prime)◦,ˇG(t/prime,t2)] +meff[m(t1)·ˆσδ(t1−t/prime)◦,ˇG(t/prime,t2)]=0, (4) where m(t)=mL(t)+mR(t), and where mL/Ris the magne- tization unit vector for the left /right FMI. meffis the effective magnetic field that each of the two identical FMIs wouldseparately induce in the SC (in units of energy), and ˆσ= diag( σ,σ ∗), where σis the vector of Pauli matrices in spin space. Note that when mL=−mR, the effective magnetic field in the superconductor vanishes, in agreement with theconclusions of Ref. [ 41]. The effective generalized Usadel Eq. ( 4) was phenomeno- logically derived. We find the same equation by includingboundary conditions to the FMIs [ 42,43], and then averaging the Green’s function over the thickness of the superconductor.In principle, one could also have included other terms that arehigher order in both the Green’s functions and magnetizations.However, we consider weak ferromagnets, where the phasedifference /Delta1ϕ=ϕ ↑−ϕ↓in the spin-dependent reflection co- efficients r↑/↓is small. Then it is sufficient to include the imaginary part of the spin mixing conductance, which resultsin Eq. ( 4). In other words, we disregard the real part of the mixing conductance, which is central in strong ferromagnets[17]. B. Gradient expansion in time and energy The Green’s function ˇG(t1,t2) correlates wave functions at times t1and t2. By shifting variables to relative time τ≡t1−t2and absolute time t≡(t1+t2)/2, and performing a Fourier transformation in the relative time coordinate, thefollowing identity holds [ 44,45]: F{(a◦b)(t 1,t2)}=exp/braceleftbiggi 2/parenleftbig ∂a E∂b t−∂a t∂b E/parenrightbig/bracerightbigg a(E,t)b(E,t), (5) where Fdenotes Fourier transform in τ,a(E,t) and b(E,t) are the Fourier transforms of a(τ,t) and b(τ,t) in the relative time coordinate, and ∂a(b) E(t)denotes partial differentiation of the function a(b) with respect to the variable E(t). We will now Fourier transform and rewrite the generalized Usadel Eq. ( 4) into ( E,t) coordinates. The first two terms of Eq. ( 4) contain time differential oper- ators. After rewriting these terms as the relative and absolutetime coordinates, and Fourier-transforming the relative timecoordinate, we find [ 46] F/braceleftbig i∂ t1ˆτ3ˇG(t1,t2)+iˇG(t1,t2)∂t2ˆτ3/bracerightbig =E[ˆτ3,ˇG(E,t)]+i 2{ˆτ3,∂tˇG(E,t)}. (6) The remaining two terms in Eq. ( 4) contain commutators of time convolutions of one-point functions /Delta1(t1) and m(t1) and the Green’s function ˇG(t1,t2). These two terms transform equally. We will therefore consider only the term containingthe magnetization in detail. By straightforward substitutioninto the term containing the magnetization of Eq. ( 4)i n t oEq. ( 5), we find that F{[m(t 1)·ˆσδ(t1−t/prime)◦,ˇG(t/prime,t2)]} =exp/braceleftbigg −i 2∂m t∂ˇG E/bracerightbigg m(t)·σˇG(E,t) −exp/braceleftbiggi 2∂m t∂ˇG E/bracerightbigg ˇG(E,t)m(t)·σ. (7) In the following, we drop the arguments Eandtto ease the notation. We proceed by expanding the exponential function with differential operators, exp/braceleftbigg −i 2∂m t∂ˇG E/bracerightbigg (m·σ)ˇG−exp/braceleftbiggi 2∂m t∂ˇG E/bracerightbigg ˇG(m·σ) =[(m·ˆσ),ˇG]−/parenleftbiggi 2/parenrightbigg {∂t(m·ˆσ),∂EˇG} +1 2!/parenleftbiggi 2/parenrightbigg2/bracketleftbig ∂2 t(m·ˆσ),∂2 EˇG/bracketrightbig −1 3!/parenleftbiggi 2/parenrightbigg3/braceleftbig ∂3 t(m·ˆσ),∂3 EˇG/bracerightbig +(···), (8) where {... ,... }denotes an anticommutator. Here and later on, for ease of notation, we drop the superscript of the differ-ential operators. Instead, we let the differential operators onlyact on the factor directly to the right of it. We keep terms onlyup to linear order in the gradients. This is justified when /vextendsingle/vextendsingle/vextendsingle/vextendsingle1 23/bracketleftbig ∂n t(m·ˆσ),∂n EˇG/bracketrightbig ij/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessmuch/vextendsingle/vextendsingle/bracketleftbig ∂ n−2 t(m·ˆσ),∂n−2 EˇG/bracketrightbig ij/vextendsingle/vextendsingle,(9) /vextendsingle/vextendsingle/vextendsingle/vextendsingle1 23/braceleftbig ∂n t(m·ˆσ),∂n EˇG/bracerightbig ij/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessmuch/vextendsingle/vextendsingle/braceleftbig ∂ n−2 t(m·ˆσ),∂n−2 EˇG/bracerightbig ij/vextendsingle/vextendsingle,(10) where ∂n tdenotes the nth partial derivative with respect to t. The magnetization precesses at a frequency ω. Therefore, ω must be much smaller than the energy gradient of the Green’sfunction. First, to avoid a diverging energy gradient of theGreen’s function, we assume finite temperatures. Second, weadd a phenomenological dephasing parameter δ=1/τ depto the Green’s function, E→E+iδ, where τdepis a character- istic dephasing time. We then find that the requirements ( 9) and ( 10) are satisfied when ( ωτdep)2/8/lessmuch1 and ( ωβ)2/8/lessmuch1, where β=1/kBTis the inverse temperature. To linear order, the effective generalized Usadel equation in the FMI|SC|FMI trilayer reads E[ˆτ3,ˇG]+i 2{ˆτ3,∂tˇG}+[ˆ/Delta1,ˇG]−i 2{∂tˆ/Delta1,∂ EˇG} +meff[m·ˆσ,ˇG]−imeff 2{∂tm·ˆσ,∂EˇG}=0. (11) In the next section, we will supplement this equation with terms arising from spin-memory loss. C. Spin relaxation To obtain a realistic model, we additionally need to include some sort of spin relaxation mechanism in the generalizedUsadel Eq. ( 11). As a simple model, we model the relaxation as a coupling to a NM reservoir, parametrized by the coupling 024524-3SIMENSEN, JOHNSEN, LINDER, AND BRATAAS PHYSICAL REVIEW B 103, 024524 (2021) coefficient V. This coupling relaxes the Green’s function in the SC toward the equilibrium solution around the Fermilevel in the NM reservoir. The effective generalized Usadelequation including this relaxation reads E[ˆτ 3,ˇG]+i 2{ˆτ3,∂tˇG}+[ˆ/Delta1,ˇG]−i 2{∂tˆ/Delta1,∂ EˇG} +meff[m·ˆσ,ˇG]−imeff 2{∂tm·ˆσ,∂EˇG} +iV[ˇN,ˇG]−V 2{∂EˇN,∂tˇG}=0, (12) where ˇNis the equilibrium Green’s function in the NM reservoir. Additionally, this coupling gives a dephasing E→ E−iVin the Green’s function in the SC. This relaxation is therefore a possible source of the dephasing which we havealready introduced in Sec. II B. D. Parametrization We now aim to express the generalized Usadel Eq. ( 12)i n a form that is easier to treat both analytically and numerically.We use a parametrization [ 47] that maps the eight nonzero components of ˆG R,ˆGA, and ˆGKonto two scalars (chargesector) and two vectors (spin sector), one of each reflecting the normal and anomalous parts of the Green’s function. Weexpand the Green’s function as (this applies to the R,A, and Kcomponents) ˆG=/summationdisplay i∈{0,1,2,3}/summationdisplay j∈{0,x,y,z}Gijτiσj, (13) where Gij=1 4TrˆGτiσj. We gather the nonzero components into the following functions: G0≡G30, G≡[G0x,G3y,G0z], F0≡G1y, F≡[−G2z,G10,G2x].(14) The scalar G0and the vector Gdescribe the diagonal elements in particle-hole space of ˆG. The scalar F0and the vector F characterize the corresponding anomalous off-diagonal ele-ments of ˆG. By inserting the definitions ( 13) and ( 14)i n t o the effective generalized Usadel Eq. ( 12), we arrive at the following parametrized differential equations for the normalcomponents: ∂GR/A 0 ∂t=meff/parenleftbigg∂GR/A ∂E/parenrightbigg ·/parenleftbigg∂m ∂t/parenrightbigg −i/parenleftBigg ∂FR/A 0 ∂E/parenrightBigg/parenleftbigg∂/Delta1 ∂t/parenrightbigg , (15) ∂GR/A ∂t=2meff(GR/A×m)+meff/parenleftBigg ∂GR/A 0 ∂E/parenrightBigg/parenleftbigg∂m ∂t/parenrightbigg −i/parenleftbigg∂/Delta1 ∂t/parenrightbigg/parenleftbigg∂FR/A ∂E/parenrightbigg , (16) ∂GK 0 ∂t=meff/parenleftbigg∂GK ∂E/parenrightbigg ·/parenleftbigg∂m ∂t/parenrightbigg −i/parenleftbigg∂FK 0 ∂E/parenrightbigg/parenleftbigg∂/Delta1 ∂t/parenrightbigg −2V/bracketleftbigg GK 0−/parenleftbig GR 0−GA 0/parenrightbig tanh/parenleftbiggβE 2/parenrightbigg/bracketrightbigg −iVβ 2/parenleftbigg∂GR 0 ∂t+∂GA 0 ∂t/parenrightbigg sech2/parenleftbiggβE 2/parenrightbigg , (17) ∂GK ∂t=2meff(GK×m)+meff/parenleftbigg∂GK 0 ∂E/parenrightbigg/parenleftbigg∂m ∂t/parenrightbigg −i/parenleftbigg∂/Delta1 ∂t/parenrightbigg/parenleftbigg∂FK ∂E/parenrightbigg −2V/bracketleftbigg GK−(GR−GA) tanh/parenleftbiggβE 2/parenrightbigg/bracketrightbigg −iVβ 2/parenleftbigg∂GR ∂t+∂GA ∂t/parenrightbigg sech2/parenleftbiggβE 2/parenrightbigg . (18) We also obtain additional equations given in Appendix A for the anomalous components F0andFfor the R,A, and Kcomponents. These equations ( A1)–(A4) are large and less transparent algebraic expressions. Lastly, we need the gapequation, /Delta1=−iN 0λ 4/integraldisplayωD −ωDdE FK 0, (19) where ωDis the Debye cutoff energy, N0is the Fermi-level electron density of states, and λis the BCS electron-phonon coupling constant. We will hereafter refer to /Delta10as the gap at zero temperature, and /Delta1as the gap at the temperature and effective magnetic field that is being considered. For a self-consistent solution, all of the equations ( 15)– (18), (A1)–(A4), and ( 19) are needed. If we assume a static gap, however, only Eqs. ( 15)–(18) are needed to determine the time evolution of the Green’s functions once we know theirsolution at a given time t.E. Spin currents and effects on FMR The magnetization dynamics in FMs generates spin cur- rents into neighboring materials. In the trilayer FMI|SC|FMIunder consideration, these spin currents read j s X,x=−iN0meff 8/integraldisplay∞ −∞dETr{σxτ3[mX·ˆσ◦,ˇG]K}, (20) js X,y=−iN0meff 8/integraldisplay∞ −∞dETr{σyτ0[mX·ˆσ◦,ˇG]K}, (21) js X,z=−iN0meff 8/integraldisplay∞ −∞dETr{σzτ3[mX·ˆσ◦,ˇG]K}, (22) where mXis the magnetization at interface X∈{L,R}, and where positive signs indicate spin-currents going from theFMIs into the SC. After expanding the convolution productsin Eqs. ( 20)–(22) to first order in time and energy gradients, we find j s X=N0meff 4/integraldisplay∞ −∞dE/bracketleftbigg1 2/parenleftbigg∂GK 0 ∂E/parenrightbigg/parenleftbigg∂mX ∂t/parenrightbigg +(GK×mX)/bracketrightbigg . (23) 024524-4SPIN PUMPING BETWEEN NONCOLLINEAR … PHYSICAL REVIEW B 103, 024524 (2021) The first term in this expression is the so-called spin- pumping current arising from the imaginary part of themixing conductance. The spin-pumping current equals j p=(N0meff/2)∂mX/∂tboth in SCs and NMs. The second term in Eq. ( 23) is the backflow spin current jbdue to spin- accumulation in the SC [ 48]. The spin-pumping current is independent of temperature, relative magnetization angles,and of whether the system is superconducting or not. Thebackflow spin current depends on these system parameters,and it will therefore be our main focus henceforth. If we assume that the magnetizations of the FMIs are uniform, the Landau-Lifshitz-Gilbert equation for the left FMIcan be written ∂m L ∂t=−γ0mL×Beff+α0/parenleftbigg mL×∂mL ∂t/parenrightbigg −γ0 Msdjs L,(24) where γ0is the gyromagnetic ratio of the ferromagnetic spins, Beffis the effective field in the FMI, α0is the Gilbert damping parameter, Msis the saturation magnetization in the FMI, and dis the thickness of the FMI. If we express js Lin reactive and dissipative components, js L=Cr∂mL ∂t+Cd(mL×∂mL ∂t), we find the following renormalized properties in the FM: γ0→γ=γ0 1+Crγ0 Msd, (25) α0→α=γ γ0/parenleftbigg α0+Cdγ0 Msd/parenrightbigg . (26) For later convenience, we define the reactive and dissipative spin currents, js r≡Cr∂mL ∂tandjs d≡Cd(mL×∂mL ∂t). III. RESULTS AND DISCUSSIONS We will now use the equations of motion of ( 15)–(18), (A1)–(A4), and the gap Eq. ( 19), to find the spin current generated by FMR in a FMI|SC|FMI trilayer. We considerhomogeneous magnetizations m LandmRin the left and right FMIs, respectively. The angle between the principal axes ofthe magnetizations is θ. The left magnetization is precessing circularly around its principal axis at a precession angle ϕwith angular frequency ω. The right magnetization is static. The system is illustrated in Fig. 1. We will initially search for an analytical solution by treat- ing the dynamic magnetization component as a perturbationfrom an equilibrium solution. Due to the complexity of theequations, we first assume that the gap is static. This approxi-mation enables us to solve the problem for arbitrary relaxation V. Section III A presents this analytical approach. In prin- ciple, it is also possible to find a self-consistent analyticalsolution. However, the solution becomes extremely complexin the presence of relaxation due to the coupling betweenthe retarded /advanced and Keldysh Green’s functions. Hence, the full self-consistent problem is better suited for numer-ical treatments. In Sec. III B , we compare the results of a self-consistent numerical solution to the analytical solution inSec. III A . We additionally outline a self-consistent analytical solution in Appendix Bin the absence of relaxation. This latter solution has restricted physical relevance, but is supplied forthe convenience of further work in this framework.A. Analytical solution with static gap approximation We first separate the magnetization vector minto a static and a dynamic component, m=m(0)+m(1). The static component m(0)=m(0) L+m(0) Ris the sum of the static magne- tizations of the left and right FMIs. The dynamic componentm (1)is the dynamic part of mL. It has magnitude δmand precesses around the zaxis with angular frequency ω,m(1)= δm[cos(ωt),sin(ωt),0]. This decomposition of the magne- tization vectors is illustrated in Fig. 1. We now assume the following: (i) The dynamic magnetization component is muchsmaller than the gap, m effδm/lessmuch/Delta1. (ii) The fluctuations in the gap are much smaller than the dynamic magnetizationamplitude, δ/Delta1/lessmuchm effδm. Assumption (i) enables us to expand the Keldysh Green’s function components in the perturbation δm, GK 0=GK(0) 0+GK(1) 0+(···), GK=GK(0)+GK(1)+(···),(27) where the nth-order terms are assumed to be ∝δmn. We con- sider the first-order expansion in δmonly, and we choose therefore to disregard second- and higher-order terms. As-sumption (ii) implies that the generalized Usadel equationsfor the advanced and retarded Green’s functions [Eqs. ( 15) and ( 16)] decouple from the Keldysh component. In what fol- lows, we will derive the solution for the Keldysh component.The retarded /advanced Green’s functions can then be found simply by substituting K→R/Aand by setting V=0i nt h e Keldysh component solution. To first order in δm, the effective generalized Usadel equa- tions for the Keldysh component read ∂G K(1) 0 ∂t=meff/parenleftbigg∂GK(0) ∂E/parenrightbigg ·/parenleftbigg∂m(1) ∂t/parenrightbigg −2V/bracketleftbigg GK(1) 0−/parenleftbig GR(1) 0−GA(1) 0/parenrightbig tanh/parenleftbiggβE 2/parenrightbigg/bracketrightbigg , (28) ∂GK(1) ∂t=2meff(GK(0)×m(1)+GK(1)×m(0)) +meff/parenleftbigg∂GK(0) 0 ∂E/parenrightbigg/parenleftbigg∂m(1) ∂t/parenrightbigg −2V/bracketleftbigg GK(1)−(GR(1)−GA(1)) tanh/parenleftbiggβE 2/parenrightbigg/bracketrightbigg . (29) We propose the Ansätze GK(1) 0=GK(1) 0+eiωt+GK(1) 0−e−iωt, GK(1)=GK(1) +eiωt+GK(1) −e−iωt. (30) After inserting the Ansätze in Eq. ( 30) into Eqs. ( 28) and ( 29), we note that the differential equations separate into decoupledequations for the +/−components. By solving for G K(1) 0±and 024524-5SIMENSEN, JOHNSEN, LINDER, AND BRATAAS PHYSICAL REVIEW B 103, 024524 (2021) GK(1) ±, we obtain GK(1) 0±=meff±ω ±ω+2V/parenleftbigg∂G(0) ∂E/parenrightbigg ·m(1) ± +2V ±ω+2V/parenleftbig GR(1) 0−GA(1) 0/parenrightbig tanh/parenleftbiggβE 2/parenrightbigg , (31) GK(1) ±=meffA−1 ±ωB±ωm(1) ± +2VA−1 ±ω(GR(1)−GA(1)) tanh/parenleftbiggβE 2/parenrightbigg , (32) where the matrices A±ωandB±ωare defined as A±ω=⎛ ⎜⎝±iω+2V−2meffm(0) z 2meffm(0) y 2meffm(0) z ±iω+2V−2meffm(0) x −2meffm(0) y 2meffm(0) x ±iω+2V⎞ ⎟⎠,(33) B±ω=⎛ ⎜⎝±iωC(E)−2GK(0) z 2GK(0) y 2GK(0) z ±iωC(E)−2GK(0) x −2G(0) y 2GK(0) x ±iωC(E)⎞ ⎟⎠, (34) and where C(E)=tanh/parenleftbiggβE 2/parenrightbigg/parenleftbigg∂GR(0) 0 ∂E−∂GA(0) 0 ∂E/parenrightbigg +/parenleftbigg/bracketleftbig GR(0) 0−GA(0) 0/bracketrightbig −iV/bracketleftbigg∂GR(0) 0 ∂E+∂GA(0) 0 ∂E/bracketrightbigg/parenrightbigg ×β 2sech2/parenleftbiggβE 2/parenrightbigg . (35) The solution to GK(1)is particularly simple when θ=0o r θ=π.F o rθ=0, we obtain GK(1) θ=0=meff/parenleftbig 2GK(0) z+ωC(E)/parenrightbig(4meff+ω)m(1)+2V ω∂m(1) ∂t (2V)2+(4meff+ω)2 +2V2m(1)−V ω(4meff+ω)∂m(1) ∂t (2V)2+(4meff+ω)2GK(0) z, (36) where we have inserted m(0) z=2. We observe that a finite V introduces a component of GKparallel to ∂mL/∂t. When we insert this component into the spin current in Eq. ( 23), we see that it generates both a reactive and a dissipative backflowcurrent, j b randjb d. Hence, even though the spin pumping current is purely reactive, the backflow spin current can indeedcarry a dissipative part due to relaxation in the SC. Moreover,we note that the effective magnetic field 2 m effsuppresses the amplitude of GK(1). This feature is due to Hanle precession of GK(1)around the effective magnetic field, which reduces the effect of the excitation. When θ=π, the Hanle precession is more or less absent due to a very small effective magnetic field ∝meffsinϕ. Under the assumption that the precession angle is sufficiently small,sinϕ/lessmuchω/m eff, we obtain GK(1) θ=π=meffωC(E)ωm(1)+2V ω∂m(1) ∂t (2V)2+ω2. (37) As a control check, we can verify that we obtain the instanta- neous equilibrium solution ( GR(1)−GA(1)) tanh( βE/2) when V/greatermuchω.In the second line of C(E)i nE q .( 35), we have iso- lated the source of nonequilibrium behavior of GK.T h i s nonequilibrium part arises from the energy gradient ofthe distribution function, and is therefore proportional tosech 2(βE 2). In the normal metal limit, we have ∂GR 0/∂E=0, and/integraltext∞ −∞dE C (E)=4 is therefore constant and independent of temperature. The spin current is therefore independent oftemperature in the NM limit. The coefficient C(E)i nE q .( 35) predicts that the nonequi- librium effects mostly arise within a thermal energy interval±β −1from the Fermi level. There are two tunable parame- ters that affect the number of quasiparticle states within thisenergy interval in a SC: First, at higher temperatures, the en-ergy interval in which quasiparticles can be excited broadens.The more overlap there is between this energy window andthe gap edge, the larger we expect the spin accumulation tobe. Another thermal effect is that the gap /Delta1decreases with increasing temperature, which enhances the above-mentionedeffect. Second, the effective magnetic field introduces a spin-split density of states, which pushes half of the quasiparticlestates closer to the Fermi level. An additional effect is that thegap decreases with an increasing effective magnetic field, aneffect that moreover is temperature-dependent. Therefore, theeffective magnetic field also affects the number of quasipar-ticle states within a thermal energy interval from the Fermilevel. Both the temperature and effective magnetic field canhence be tuned to increase the spin accumulation. The spinaccumulation in turn generates a backflow spin current intothe FMIs. We therefore expect a larger backflow spin currentfrom a SC at higher temperatures and for stronger effectivemagnetic fields. We will now evaluate the angular and temperature de- pendence of the backflow spin current for a particular FMI|SC|FMI trilayer. We choose the parameters in the SC so that they match those of Nb. That is, we choose1/V=τ sf/2π∼10−10s[49] and a critical temperature Tc=9.26 K [ 50]. Moreover, we use an effective magnetic field strength meff=0.1/Delta10, and a magnetization precession angleϕ=arcsin(0 .01). Last, we use a precession frequency ω=0.005/Delta10≈10 GHz, which is an appropriate frequency for, e.g., yttrium iron garnet (YIG). The relaxation introduces a dephasing V=0.05/Delta10, which is sufficient to justify the gradient expansion. The gap /Delta1=/Delta1(T,θ,meff) is found by solving the gap equation self-consistently [ 51] to zeroth order in the dynamic magnetization, as well as checking that the freeenergy of the superconducting state is lower than in the normalmetal state. The assumptions (i) and (ii) underlying the staticgap approximation can be satisfied for any effective field m eff providing we choose an appropriate precession amplitude, δm, which can be tuned with the ac magnetic field used to exciteFMR in the FMI. In the FMI|SC|FMI trilayer, the expression for the backflow spin current in Eq. ( 23) implies that there is a static RKKY contribution to the spin current. This RKKY contribution isdue to the finite G Kclose to the Fermi level. However, other terms also contribute to the RKKY interaction beyond thequasiclassical theory. Therefore, we subtract the instantaneousRKKY-like static contribution to the spin current. Figure 2plots the backflow spin current as a function of θ for two different temperatures, T=0.1T candT=0.9Tc.T h e 024524-6SPIN PUMPING BETWEEN NONCOLLINEAR … PHYSICAL REVIEW B 103, 024524 (2021) FIG. 2. The reactive (red) and dissipative (blue) backflow spin current, normalized to the density of states N0, as a function of θ through the left interface of a FMI|SC|FMI trilayer for two different temperatures, T=0.1Tc(upper plot) and T=0.9Tc(lower plot). We have used the parameters given in the main text, with meff=0.1/Delta10. In the lower plot, we have also plotted the spin current through an analogous FMI|NM|FMI trilayer (dotted lines). spin-pumping currents in both cases are purely reactive and equal to jp/N0=10−5J2/m. The first striking observation is that the spin current is much lower in the SC system atT=0.1T cthan at T=0.9Tc. Singlet pair formation hinders injection of spin currents into the superconductor. Next, weobserve that the total spin current grows as θapproaches π, which is the case for both the SC and NM systems, and at bothtemperatures. This is due to the decreased impact of Hanleprecession on the spin accumulation as the effective magneticfield decreases. Moreover, we note that the reactive spin cur-rent is favored close to θ=0, whereas the dissipative spin current is favored close to θ=π. This is because the Hanle precession affects the reactive and dissipative spin currentdifferently. Inspecting Eq. ( 36), we see that the reactive and dissipative spin current are suppressed by a factor ∝(m eff)−1 and∝(meff)−2close to θ=0, respectively. For large effective magnetic fields, that is, close to θ=0, the dissipative spin current is therefore strongly suppressed compared to the re-active spin current. Close to θ=π, where Hanle precession is negligible, the reactive and dissipative spin currents aresuppressed ∝V −2and∝V−1, respectively, as can be seen inFIG. 3. (a) The temperature dependence of the total spin current in the superconducting system for two relative magnetization angles,θ=0a n dθ=π. The spin current is normalized to the normal metal limit, where the spin current is independent of temperature. (b) The ratio j s,θ=0 d/js,θ=π d plotted as a function of temperature for both the SC and NM systems. We have used the parameters given in the main text. The lowest temperature included is T=0.03Tcin order to ensure that the gradient expansion is justified. Eq. ( 37). Hence, the dissipative spin current dominates close toθ=π. Let us now explore the temperature dependence in detail. In Fig. 3(a) we plot the total spin current as a function of temperature for two angles, θ=0 and θ=π, and for dif- ferent effective field strengths meff. We have normalized the spin currents with the respect to the analogous NM limitspin currents. The latter are independent of temperature. Dueto the gradient expansion, the parameters must satisfy thecondition β −1/greatermuchω/√ 8≈0.003kBTc. We therefore restrict the temperature analysis to T/greaterorequalslant0.03Tc. First, we observe that the spin currents approach the NM limit at the criticalfields for the respective effective magnetic fields. We havealready discussed this behavior, which is due to the amount ofquasiparticle states within a thermal energy interval from theFermi energy. This entails an overall decrease in the total spincurrent for the θ=0 configuration, and an increase for the θ=πconfiguration. This is due to the nature of the backflow spin current. In the θ=0 configuration, the backflow spin current is dominated by a reactive component that counteractsthe spin-pumping current. In the θ=πconfiguration, the backflow spin current is dominated by a dissipative compo-nent. This spin current is oriented almost 90 ◦relative to the spin-pumping current, and therefore increases the total spincurrent. 024524-7SIMENSEN, JOHNSEN, LINDER, AND BRATAAS PHYSICAL REVIEW B 103, 024524 (2021) Next, Fig. 3(a) demonstrates that the temperature depen- dence of the normalized spin current for the θ=0 andθ=π states differ. To investigate this further, we plot the ratiobetween the dissipative spin currents in the parallel and an-tiparallel configurations, j s,θ=0 d/js,θ=π d, both in the NM and SC state, in Fig. 3(b). Here, we observe that this ratio is a constant function of temperature in the NM limit, whereas itdepends strongly on temperature in the superconducting state.The ratio peaks at slightly different temperatures for differenteffective fields m effin the superconducting state. The height of the peak increases with an increasing effective field meff. As the temperature approaches Tc, the ratio in the SC state converges toward the NM limit result. This behavior is due to the aforementioned effect of temperature and effective magnetic field. In the parallel con-figuration, the effective magnetic fields of the two FMIs addconstructively and cause a strong spin-splitting in the densityof states. In the antiparallel configuration, the effective fieldsadd destructively and cause only a weakly spin-split densityof states. At very low temperatures, the difference betweenthe parallel and antiparallel configurations is small for thechosen values of m eff. This is because neither state has a large density of states close to the almost δ-function-like thermal energy interval around the Fermi level. At slightly higher tem-peratures, the states that are pushed closer to the Fermi levelstart overlapping with the thermal energy interval E F±β−1. The difference between the two states is maximized for someintermediate temperature, k BT/lessorapproxeql/Delta1(T), where we observe the peaks in Fig. 3(b). At even higher temperatures, the thermal energy interval broadens further. The difference between theparallel and antiparallel states then starts decreasing for highertemperatures, and eventually approaches the NM limit. B. Numerical analysis We aim here to briefly present a numerical solution to the problem that was solved analytically in Sec. III A .O u rm a i n goal is to evaluate whether the assumption of a static gap canbe justified to a good approximation. A subsidiary goal is toshow the time evolution of the gap, and the usefulness of anumerical method in this framework also when the static gapapproximation is not valid. We see from Eqs. ( 16) and ( 18) that the vectors G A,GR, andGKprecess around the effective magnetic field. For such a class of equations, employing a fourth-order Runge-Kuttamethod is suitable for obtaining a numerical solution. Totest the validity of the static gap approximation, we want toperform a simulation of the system where the oscillationsin the gap are maximized. This is expected to occur wherethe magnitude of the effective field oscillates with the largestamplitude. From Eq. ( B9) one can show that this occurs at θ=π/2 in the absence of relaxation, and we hence expect it to occur at θ=π/2 also with the inclusion of relaxation. Figure 4(a) forθ=π/2 shows the fluctuation of the gap δ/Delta1(t) normalized to /Delta1 0over one period 2 π/ω and for sev- eral temperatures T, with meff=0.1/Delta10. The gap oscillates harmonically with frequency ωfor all temperatures up to T=0.85Tc. At temperatures close to the critical temperature for the given effective magnetic field, the gap shows a non-linear response to the dynamical magnetization. This effectFIG. 4. (a) The fluctuations of the gap δ/Delta1(t) plotted over one period 2 π/ω at different temperatures for θ=π/2. (b) The detailed temperature dependence of the gap fluctuation amplitude max |δ/Delta1| for different magnetization angles θ. The gap fluctuations are nor- malized to the gap at zero temperature, /Delta10, and we have used meff=0.1/Delta10. is visible for T=0.95Tc, and is due to the increased sensi- tivity to fluctuations in the magnetic field as the temperatureapproaches the critical temperature. In Fig. 4(b), we further explore θand the temperature dependence of the gap fluc- tuation amplitude, max |δ/Delta1|, in the linear response regime. We observe that the fluctuations are largest at θ=π/2, and that they are maximized at about T≈0.8T c. Moreover, we observe that the fluctuations are not larger than about5.5×10 −5/Delta10. Let us now briefly remind the reader that the formal requirement for the static gap approximation wasδ/Delta1/lessmuchm effδm, where δmis the dynamic magnetization am- plitude. We have meffδm≈0.001/Delta10/greatermuchδ/Delta1/lessorequalslant5.5×10−5/Delta10, which implies that the static gap assumption is an excellentapproximation in this instance. IV . CONCLUSION We have derived an effective, time-dependent generalized Usadel equation in noncollinear FMI|SC|FMI trilayers witha thin superconducting layer and weakly magnetized FMIs.We have provided analytical solutions to these equations interms of perturbations in the dynamic magnetization, firstunder the assumption of a static gap, and then a self-consistentsolution in the absence of relaxation. Lastly, we have provided 024524-8SPIN PUMPING BETWEEN NONCOLLINEAR … PHYSICAL REVIEW B 103, 024524 (2021) numerical procedures to obtain self-consistent solutions of the full equations without any further simplifications. From the solutions to the generalized Usadel equation, we computed the spin currents generated by ferromagnetic reso-nance in one of the FMIs. We have explored this spin currentas a function of both temperature and relative magnetizationangle between the FMIs. The spin current has been decom-posed into a reactive and a dissipative part, which changethe effective gyromagnetic ratio and Gilbert damping coef-ficient of the FMI. We found that the backflow spin currentis generally largest when the magnetization orientations ofthe FMIs are antiparallel. The ratio between the spin currentin the parallel and antiparallel configuration strongly dependson temperature in the SC. The origin is the Zeeman splitting of the quasiparticles at the gap edge. Lastly, we performed anumerical simulation to verify that the static gap assumptionis a good approximation in our regime, also showing theusefulness of a numerical solution in this framework. ACKNOWLEDGMENTS This work was supported by the Research Council of Norway through its Centres of Excellence funding scheme,Project No. 262633 “QuSpin,” as well as by the EuropeanResearch Council via Advanced Grant No. 669442 “Insula-tronics.” APPENDIX A: ADDITIONAL PARAMETRIZED USADEL EQUATIONS In the main text, we provided four of the generalized Usadel equations, Eqs. ( 15)–(18), that were equations of motion for the normal components of the Green’s functions. The remaining four equations that are needed to solve a system with nonzeroanomalous Green’s functions self-consistently are given as follows: FR/A 0=i/parenleftBig /Delta1/parenleftbig meffm·GR/A−(E−iV)GR/A 0/parenrightbig −im2 eff 2m·/parenleftbig∂m ∂t×∂FR/A ∂E/parenrightbig/parenrightBig m2 effm2−(E−iV)2, (A1) FR/A=i/Delta1GR/A (E−iV) −i/parenleftbig meff/Delta1/parenleftbig meffm·GR/A−(E−iV)GR/A 0/parenrightbig m−im3 eff 2/bracketleftbig m·/parenleftbig∂m ∂t×∂FR/A ∂E/parenrightbig/bracketrightbig m+imeff 2/bracketleftbig m2 effm2−(E−iV)2/bracketrightbig/parenleftbig∂m ∂t×∂FR/A ∂E/parenrightbig/parenrightbig (E−iV)/bracketleftbig m2 effm2−(E−iV)2/bracketrightbig , (A2) FK 0=i/parenleftbig /Delta1/parenleftbig meffm·GK−EGK 0/parenrightbig −im2 eff 2m·/parenleftbig∂m ∂t×∂FK ∂E/parenrightbig/parenrightbig m2 effm2−E2−iVtanh/parenleftbigβE 2/parenrightbig/bracketleftbig m·(FR+FA)−E/parenleftbig FR 0+FA 0/parenrightbig/bracketrightbig /parenleftbig m2 effm2−E2/parenrightbig −βVsech2/parenleftbigβE 2/parenrightbig/bracketleftbig m·/parenleftbig∂FR ∂t−∂FA ∂t/parenrightbig −E/parenleftbig∂FR 0 ∂t−∂FA 0 ∂t/parenrightbig/bracketrightbig 4/parenleftbig m2 effm2−E2/parenrightbig , (A3) FK=i/Delta1GK E−i/parenleftbig meff/Delta1/parenleftbig meffm·GK−EGK 0/parenrightbig m−im3 eff 2/bracketleftbig m·/parenleftbig∂m ∂t×∂FK ∂E/parenrightbig/bracketrightbig m+imeff 2(m2 effm2−E2)/parenleftbig∂m ∂t×∂FK ∂E/parenrightbig/parenrightbig E/parenleftbig m2 effm2−E2/parenrightbig +iVtanh/parenleftbigβE 2/parenrightbig/bracketleftbig/braceleftbig m·(FR+FA)−E/parenleftbig FR 0+FA 0/parenrightbig/bracerightbig m−(m2−E2)(FR+FA/parenrightbig/bracketrightbig E/parenleftbig m2 effm2−E2/parenrightbig +βVsech2/parenleftbigβE 2/parenrightbig/bracketleftbig/braceleftbig m·(∂FR ∂t−∂FA ∂t)−E/parenleftbig∂FR 0 ∂t−∂FA 0 ∂t/parenrightbig/bracerightbig m−(m2−E2)/parenleftbig∂FR ∂t−∂FA ∂t/parenrightbig/bracketrightbig 4E/parenleftbig m2 effm2−E2/parenrightbig , (A4) where the notation is defined in the main text. APPENDIX B: SELF-CONSISTENT SOLUTION IN THE ABSENCE OF SPIN RELAXATION We will derive here a self-consistent solution to the generalized Usadel equations, Eqs. ( 15)–(18), Eqs. ( A1)–(A4), and the gap Eq. ( 19), in the absence of spin relaxation ( V=0). This solution has restricted physical relevance, and it only applies in the limit where the precession frequency is much larger than the relaxation rate. However, it is included as a proof of concept that aself-consistent solution is in principle possible. The derivation follows the lines of what was presented in Sec. III A , with a few exceptions. In addition to the perturbation expansion in Eqs. ( 27), we also expand F 0=F(0) 0+F(1) 0+F(2) 0+(···), F=F(0)+F(1)+F(2)+(···), /Delta1=/Delta1(0)+/Delta1(1)+/Delta1(2)+(···).(B1) 024524-9SIMENSEN, JOHNSEN, LINDER, AND BRATAAS PHYSICAL REVIEW B 103, 024524 (2021) We have dropped the retarded /advanced and Keldysh superscript in order to keep the derivation as general as possible. This derivation hence applies to all Green’s-function components. We also propose one additional Ansatz , /Delta1(1)=/Delta1(1) +eiωt+/Delta1(1) −e−iωt. (B2) If we insert this into the generalized Usadel equations to first order in δm, and with V=0, we obtain the solutions G(1) 0±=meff/parenleftbigg∂G(0) ∂E/parenrightbigg ·m(1) ±−i/parenleftbigg∂F(0) 0 ∂E/parenrightbigg /Delta1(1) ±, (B3) G(1) ±=meff˜A−1 ±ω˜B±ωm(1) ±±ω/Delta1(1) ±˜A−1 ±ω∂F(0) ∂E, (B4) where ˜A±ω=⎛ ⎝±iω 2meff−m(0) z 2meffm(0) y 2meffm(0) z ±iω −2meffm(0) x −2meffm(0) y 2meffm(0) x ±iω⎞ ⎠ (B5) and ˜B±ω=⎛ ⎜⎜⎝±iω∂G(0) 0 ∂E−2G(0) z 2G(0) y 2G(0) z ±iω∂G(0) 0 ∂E−2G(0) x −2G(0) y 2G(0) x ±iω∂G(0) 0 ∂E⎞ ⎟⎟⎠. (B6) To solve for /Delta1(1)(t), we look closer at the gap equation given in Eq. ( 19). If we insert the generalized Usadel equation for FK 0[Eq. ( A3)] into the gap equation while using V=0, divide both sides by /Delta1, and assume that |meffm(1)|/lessmuchδ, the first- and second-order gap equations read 1=N0λ 4/integraldisplayωD −ωDdE/parenleftbig meffm(0)·GK(0)−EGK(0) 0/parenrightbig m2 eff(m(0))2−E2, (B7) 0=/integraldisplayωD −ωDdE1 m2 eff(m(0))2−E2/braceleftBigg meff(m(1)·GK(0)+m(0)·GK(1))−EGK(1) 0−2m2 eff(m(0)·m(1))/parenleftbig meffm(0)·GK(0)−EGK(0) 0/parenrightbig m2 eff(m(0))2−E2/bracerightBigg . (B8) Here, we used m(0)·(∂m(1) ∂t×∂FK(0) ∂E)=0, since FK(0)/bardblm(0). We have moreover used E→E+iδand meff|m(0)·m(1)|/lessmuch|m(0)|δ, ensuring that the expansion is also valid when Re {E}→ meff|m(0)|.E q .( B7)i ss i m p l yt h e zeroth-order gap equation, while Eq. ( B8) must be used to find self-consistent solution to the first-order Green’s-function components. All that remains now is to insert the Ansätze Eqs. ( 30) and ( B2) into Eq. ( B8). 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PhysRevB.95.140404.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 95, 140404(R) (2017) Self-focusing skyrmion racetracks in ferrimagnets Se Kwon Kim,1Kyung-Jin Lee,2,3and Yaroslav Tserkovnyak1 1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA 2Department of Materials Science and Engineering, Korea University, Seoul 02841, South Korea 3KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, South Korea (Received 10 February 2017; published 14 April 2017) We theoretically study the dynamics of ferrimagnetic skyrmions in inhomogeneous metallic films close to the angular momentum compensation point. In particular, it is shown that the line of the vanishing angular momentumcan be utilized as a self-focusing racetrack for skyrmions. To that end, we begin by deriving the equations ofmotion for the dynamics of collinear ferrimagnets in the presence of a charge current. The obtained equations ofmotion reduce to those of ferromagnets and antiferromagnets at two special limits. In the collective coordinateapproach, a skyrmion behaves as a massive charged particle moving in a viscous medium subjected to a magneticfield. Analogous to the snake orbits of electrons in a nonuniform magnetic field, we show that a ferrimagnet withnonuniform angular momentum density can exhibit the snake trajectories of skyrmions, which can be utilized asracetracks for skyrmions. DOI: 10.1103/PhysRevB.95.140404 Introduction. A free particle with a magnetic moment precesses at a frequency proportional to the applied magneticfield and its gyromagnetic ratio, which is the ratio of itsmagnetic moment to its angular momentum. When a magnetis composed of equivalent atoms, its net magnetizationand net angular momentum density are collinear with theproportionality given by the gyromagnetic ratio of constituentatoms. The magnetic and spin variables then represent the samedegrees of freedom, and thus are interchangeable in describingthe magnetization dynamics. One-sublattice ferromagnets andtwo-sublattice antiferromagnets are examples of such magnets. When a magnet consists of inequivalent atoms, however, its magnetization and spin density can constitute independentdegrees of freedom [ 1]. One such class of magnets is rare-earth transition-metal (RE-TM) ferrimagnetic alloys [ 2], in which the moments of TM elements and RE elements tend to beantiparallel due to superexchange. Because of the differentgyromagnetic ratios between RE and TM elements, one canreach the angular momentum compensation point and themagnetization compensation point by varying the relativeconcentrations of the two species or changing the temperature.These compensation points (CPs) are absent in ferromagnetsand antiferromagnets, which have been mainstream materialsin spintronics [ 3], and thereby have been bringing a novel phenomenon to the field such as the ultrafast optical magne-tization reversal [ 2]. In particular, the angular momentum CP may provide a tunable crossover between the ferromagnetic(away from the CP) [ 4] and antiferromagnetic (at the CP) [ 2,5] regimes of collective dynamics, with a promise for unusualbehavior in the vicinity of the CP. Here, we are exploring thisquestion in regard to the topological spin-texture dynamics. Topological solitons in magnets [ 6] have been serving as active units in spintronics. For example, a domain wall, whichis a topological soliton in quasi-one-dimensional magnetswith easy-axis anisotropy, can function as a memory unit, asdemonstrated in magnetic domain-wall racetrack memory [ 7]. Two-dimensional magnets with certain spin-orbit coupling canalso stabilize another particlelike topological soliton, whichis referred to as a skyrmion. Skyrmions have been gainingattention in spintronics as information carriers, an alternative todomain walls, because of fundamental interest as well as their practical advantages such as a low depinning electric current[8]. Several RE-TM thin films such as GdFeCo and CoTb have been reported to possess perpendicular magnetic anisotropyand a bulk Dzyaloshinskii-Moriya interaction [ 9,10], and thus are expected to be able to host skyrmions under appropriateconditions. In this Rapid Communication, we study the dynamics of skyrmions in metallic collinear ferrimagnets, in the vicinity ofthe angular momentum CP in RE-TM alloys. To that end,we first derive the equations of motion for the dynamicsof general collinear magnets in the presence of an electriccurrent. The resultant equations of motion reduce to those offerromagnets and antiferromagnets at two limiting cases. Thedynamics of a skyrmion is then derived within the collectivecoordinate approach [ 11]. Generally, it behaves as a massive charged particle in a magnetic field moving in a viscousmedium. When there is a line in the sample across whichthe net angular momentum density reverses its direction, theemergent magnetic field acting on skyrmions also changes itssign across it. Motivated by the existence of a narrow channelin two-dimensional electron gas localized on the line acrosswhich the perpendicular magnetic field changes its direction[12], we show that, under suitable conditions, the line of the vanishing angular momentum in RE-TM alloys can serve asa self-focusing racetrack for skyrmions [ 13]a sar e s u l to ft h e combined effects of the effective Lorentz force and the viscousforce. We envision that ferrimagnets with a tunable spin densitycan serve as a natural platform to engineer an inhomogeneousemergent magnetic field for skyrmions, which would provideus a useful knob to control them. Main results. The system of interest to us is a two- dimensional collinear ferrimagnet. Although the angular mo-mentum can be rooted in either the spin or the orbital degrees offreedom, we will use the term, spin, as a synonym for angularmomentum throughout for the sake of brevity. For temper-atures much below than the magnetic ordering temperature,T/lessmuchT c, the low-energy dynamics of the collinear ferrimagnet can be described by the dynamics of a single three-dimensionalunit vector n, which determines the collinear structure of 2469-9950/2017/95(14)/140404(5) 140404-1 ©2017 American Physical SocietyRAPID COMMUNICATIONS SE KWON KIM, KYUNG-JIN LEE, AND YAROSLA V TSERKOVNYAK PHYSICAL REVIEW B 95, 140404(R) (2017) the magnet [ 4]. Our first main result, which will be derived later within the Lagrangian formalism taken by Andreev andMarchenko [ 4] for the magnetic dynamics in conjunction with the phenomenological treatment of the charge-induced torques[14], is the equations of motion for the dynamics of nin the presence of a charge current density Jand an external field h to the linear order in the out-of-equilibrium deviations ˙n,J, andh, s˙n+s αn×˙n+ρnרn =n×fn+ξ(J·∇)n+ζn×(J·∇)n, (1) where sis the net spin density along the direction of n,sα andρparametrize the dissipation power density P=sα˙n2 and the inertia associated with the dynamics of n, respectively, andfn≡−δU/δ nis the effective field conjugate to nwith U[n] the potential energy [ 15]. Here, ξandζare the phenomenological parameters for the reactive and dissipativetorques due to the current, respectively. When the inertia vanishes, ρ=0, the obtained equations of motion are reduced to the Landau-Lifshitz-Gilbert equation forferromagnets augmented by the spin-transfer torques [ 16,17], in which s α/sandξcan be identified as the Gilbert damping constant and the spin polarization of conducting electrons,respectively. When the net spin density vanishes, s=0, it corresponds to the equations of motion for antiferromagnets[14]. The equations of motion for the dynamics of a two- sublattice ferrimagnet in the absence of an electric current anddissipation, s α=0 and J=0, have been obtained by lvanov and Sukstanskii [ 18]. The low-energy dynamics of rigid magnetic solitons in two-dimensional collinear magnets can be derived from Eq. ( 1) within the collective coordinate approach [ 11], where the dy- namics of the order parameter is encoded in the time evolutionof the soliton position, n(r,t)=n 0[r−R(t)]. The resultant equations of motion for the position of a circularly symmetricsoliton, which are obtained by integrating Eq. ( 1) multiplied byn 0×∂Rn0over the space, are our second main result, M¨R=Q˙R×B−D˙R+FU+FJ, (2) where M≡ρ/integraltext dxdy (∂xn0)2is the soliton mass [ 19],D≡ sα/integraltext dxdy (∂xn0)2is the viscous coefficient, FU≡−dU/d R is the internal force, and ( FJ)i≡/integraltext dxdy [ξn0·(J·∇)n0× ∂in0−ζ∂in0·(J·∇)n0] is the force due to the charge current. The first term on the right-hand side is the effective Lorentzforce on the soliton, which is proportional to its topologicalcharge Q=1 4π/integraldisplay dxdy n0·(∂xn0×∂yn0), (3) which measures how many times the unit vector n0(r) wraps the unit sphere as rspatially varies [ 20], and the fictitious magnetic field B≡Bˆz=−4πsˆz. (4) According to the equations of motion, a skyrmion in chiral ferrimagnets, which is characterized by its topological chargeQ=±1, behaves as a massive charged particle in a magnetic field moving in a viscous medium. The fictitious magnetic fieldis proportional to the net spin density salong the direction of(a) (b) xy zxy z s>0s<0 Q=−1 Q=1F Q=−1 Q=1F FIG. 1. Schematic illustrations of a steady-state skyrmion motion [Eq. ( 5)] in the presence of a current-induced force F=Fˆx. Four possible types are classified by its skyrmion charge Qand the sign of the net spin density s. See the main text for discussions. the order parameter n, which leads us to consider collinear magnets with tunable sto look for a possibly interesting dynamics of a skyrmion. The RE-TM ferrimagnetic alloys [ 2] are such materials. For example, Co 1−xTbxhas been shown to exhibit the vanishing angular momentum s≈0a tx≈17% at room temperature [ 10] by varying the chemical composition. As another example, the angular momentum compensationtemperature of Gd 22%Fe75%Co3%has been reported as T≈ 220 K [ 21]. A skyrmion can be driven by an electric current, as can be seen in Eq. ( 2). In the presence of the corresponding current- induced force FJ≡Fˆx, the direction of which is defined as thexaxis, the steady state of a skyrmion is given by ˙R→V=F B2+D2(Dˆx−QBˆy). (5) See Fig. 1for illustrations of a steady-state skyrmion motion forF> 0. The skyrmion with a topological charge Q=1 moves down for s<0 and up for s>0, while moving to the right regardless of the sign of s. If the ferrimagnet is prepared in such a way that s<0f o r y>0 and s>0 fory<0, the skyrmion with Q=1 will move along the horizontal line y=0 after a certain relaxation time because it is constantly pushed back to the line via the effective Lorentzforce. Note that the skyrmion experiences no Lorentz forceon the angular momentum compensation line, and thus willmove as an antiferromagnetic skyrmion strictly along the lineat a potentially higher speed compared to a ferromagneticskyrmion [ 22]. A similar phenomenon has been predicted for a magnetic vortex moving around the interface between twoferromagnetic materials having opposite signs of the differencebetween the Gilbert damping constant αand the nonadiabatic spin-transfer torque coefficient η[23]. To corroborate the qualitative prediction, we numerically solve the equations of motion [Eq. ( 2)] in their dimensionless form, Id 2˜R d˜t2+4πsQ sαd˜R d˜t׈z+Id˜R d˜t=˜Fˆx, (6) in which time, length, and energy are measured in units of the relaxation time τ≡ρ/sα, the characteristic length scale for the skyrmion size l[24], and /epsilon1≡s2 αl2/ρ, respectively, where I=/integraltext dxdy (∂xn0)2is a dimensionless number determined by the skyrmion structure. The symbols with the tilde willdenote the dimensionless quantities throughout. Figure 2(a) shows the two trajectories of skyrmions of charge Q=1 140404-2RAPID COMMUNICATIONS SELF-FOCUSING SKYRMION RACETRACKS IN FERRIMAGNETS PHYSICAL REVIEW B 95, 140404(R) (2017) 05 −5(a) (b)˜y024 ˜y 0 0.1−0.1s/sα 0 0.1−0.1s/sα 0˜x20 40 60 800˜x20 40 60 800˜Vx8 10 20˜t −2 −4˜Y(0) = 2 ˜Y(0) =−3 FIG. 2. Trajectories of skyrmions with the topological charge Q=1 in the presence of a current-induced force F=Fˆx,w h i c h are obtained by numerically solving the dimensionless equations of motion for the dynamics of skyrmions in Eq. ( 6). (a) Two trajectories for the monotonic net angular momentum density s. The inset shows the convergence of the skyrmion velocities. (b) Multiple trajectories for the periodic net angular momentum density s. See the main text for detailed discussions. departing from ( ˜X,˜Y)=(0,2) and ( ˜X,˜Y)=(0,−3) with a zero initial velocity under the following configurations: I= π/2,˜F=4π, ands/sα=−0.1 tanh( ˜y). We refer to the paths as skyrmion snake trajectories due to their shapes, analogousto the electronic snake orbits in an inhomogeneous magneticfield [ 12]. The inset shows that the skyrmion speed converges as˜V x→˜F/I after a sufficiently long time, ˜t/greatermuch1. Figure 2(b) depicts the multiple trajectories of skyrmions when the netspin density is spatially periodic, s/s α=−0.1s i n ( 2 π˜y/5). Skyrmions are attracted to angular momentum compensationlines and their velocities converge to a finite value. This leadsus to state our third main result: Self-focusing narrow guidesfor skyrmions can be realized in certain ferrimagnets such asthe RE-TM alloys along the lines of the angular momentumcompensation points, which can be useful in using skyrmionsfor information processing by, e.g., providing multiple parallelskyrmion racetracks in one sample [ 25]. The dynamics of collinear magnets. The derivation of the equations of motion for the dynamics of collinear magnetsin [Eq. ( 1)] is given below, which follows the phenomeno- logical approach taken for antiferromagnets by Andreev andMarchenko [ 4]. Within the exchange approximation that the Lagrangian is assumed invariant under the global spinrotations, we can write the Lagrangian density for the dynamicsof the directional order parameter nin the absence of an external field as L=−sa[n]·˙n+ρ˙n 2 2−U[n], (7) to the quadratic order in the time derivative, where a[n]i st h e vector potential for the magnetic monopole, ∇n×a=n[26].The first term accounts for the spin Berry phase associated with the net spin density along n; the second term accounts for the inertia for the dynamics of n, which can arise due to, e.g., the relative canting of sublattice spins [ 15]. Next, the effects of an external field can be taken into account as follows. The conserved Noether charge associatedwith the symmetry of the Lagrangian under global spinrotations is the net spin density, and it is given by s=sn+ ρn×˙n. The magnetization in the presence of an external field Hcan be then written as M=g lsn+gtρn×˙n+χH, where glandgtare the gyromagnetic ratios for the longitudinal and transverse components of the spin density with respect to thedirection n, respectively, and χis the magnetic susceptibility tensor. The relation M=∂L/∂H[4] requires the susceptibility to be χ ij=ρg2 t(1−ninj), with which the Lagrangian is extended to L=−sa[n]·˙n+ρ(˙n−gtn×H)2 2−U[n], (8) whereU[n] includes the Zeeman term, −glsn·H. Finally, the dissipation can be accounted for by the Rayleigh dissipationfunction R=s α˙n2/2, which is half of the dissipation rate of the energy density P=2R. The equations of motion obtained from the Lagrangian and the Rayleigh dissipation function aregiven by Eq. ( 1) without the current-induced torques. Current-induced torques. To derive the torque terms due to an electric current, it is convenient to begin by phenomenologi-cally constructing the expression for the charge current densityJ pumpinduced by the magnetic dynamics, and subsequently to invoke the Onsager reciprocity to obtain the torque terms asdone for antiferromagnets in Ref. [ 14]. To the lowest order of the space-time gradients and to the first order in the deviationsfrom the equilibrium, we can write two pumping terms thatsatisfy the appropriate spatial and spin-rotational symmetries, ˙n·∂ inandn·(˙n×∂in). The resultant expression for the induced current density is given by Jpump i/σ=ζ˙n·∂in+ξn·(∂in×˙n), (9) where σis the conductivity. To invoke the Onsager reciprocity that is formulated in the linear order in the time derivative of the dynamicvariables, we turn to the Hamiltonian formalism instead ofthe Lagrangian formalism. We shall restrict ourselves hereto the case of a vanishing external field for simplicity, butit can be easily generalized to the case of a finite externalfield. The canonical conjugate momenta of nis given by p≡∂L/∂˙n=ρ(˙n−g tn×h)−sa. The Hamiltonian density is then given by H[n,p]=p·˙n−L=(p+sa)2 2ρ+U, (10) which resembles the Hamiltonian for a charged particle subjected to an external magnetic field [ 27]. The Hamilton equations are given by ˙n=∂H ∂p≡−hp, (11) ˙p=−∂H ∂n−∂R ∂˙n≡hn−sα˙n=hn+sαhp, (12) 140404-3RAPID COMMUNICATIONS SE KWON KIM, KYUNG-JIN LEE, AND YAROSLA V TSERKOVNYAK PHYSICAL REVIEW B 95, 140404(R) (2017) where hpandhnare conjugate fields to pandn, respectively. In terms of the conjugate fields, the pumped charge current isgiven by J pump=−ζ∂in·hp−ξ(n×∂in)·hp.B yu s i n gt h e Onsager reciprocity and Ohm’s law for the current J=σE, we can obtain the torque terms in Eq. ( 1). Discussion. Let us discuss approximations that have been used in this Rapid Communication. First, we have developedthe theory for the dynamics of collinear magnets within theexchange approximation [ 4], in which the total energy is invariant under the simultaneous rotation of the constituentspins. The relativistic interactions including the magneticanisotropy, which weakly break the exchange symmetry ofthe magnet, are added phenomenologically to the potentialenergy. Second, when studying the dynamics of skyrmions ininhomogeneous ferrimagnetic films, we have considered thenonuniform spin density s, while neglecting possible spatial variations of the other parameters such as inertia ρor damping s α. As long as skyrmions are attracted to the line of vanishing angular momentum due to the combined effects of the effectiveLorentz force, the viscous force, and the current-induced force,smooth variations of various parameters away from it shouldnot significantly affect the dynamics.Ferrimagnetic RE-TM alloys not only have an angular momentum CP, which we have focused on in this RapidCommunication, but also a magnetic moment CP. Motivatedby the attraction of skyrmions toward the angular momentumcompensation lines that we have discussed, it would be worthlooking for an interesting phenomenon that can occur on themagnetic moment compensation line. For example, since themagnetic moment governs the magnetostatic energy, there maybe unusual magnetostatic spin-wave modes [ 28] localized at the line. In addition, we have considered the dynamics of asoliton in two-dimensional ferrimagnets driven by an electriccurrent. 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PhysRevLett.126.196601.pdf

Spin-Flip Diffusion Length in 5 dTransition Metal Elements: A First-Principles Benchmark Rohit S. Nair ,1Ehsan Barati ,1,‡Kriti Gupta,1Zhe Yuan ,2,*and Paul J. Kelly1,2,† 1Faculty of Science and Technology and MESA+Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands 2The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, 100875 Beijing, China (Received 13 October 2020; accepted 16 April 2021; published 12 May 2021) Little is known about the spin-flip diffusion length lsf, one of the most important material parameters in the field of spintronics. We use a density-functional-theory based scattering approach to determine valuesofl sfthat result from electron-phonon scattering as a function of temperature for all 5dtransition metal elements. lsfdoes not decrease monotonically with the atomic number Zbut is found to be inversely proportional to the density of states at the Fermi level. By using the same local current methodology tocalculate the spin Hall angle Θ sHthat characterizes the efficiency of the spin Hall effect, we show that the products ρðTÞlsfðTÞandΘsHðTÞlsfðTÞare constant. DOI: 10.1103/PhysRevLett.126.196601 Spin-orbit coupling (SOC) leads to the loss of spin angular momentum. A current of electrons injected from aferromagnet into a nonmagnetic material loses its spinpolarization over a length scale of l sf, the spin-flip diffusion length (SDL) [1–4], making the observation of spin currents difficult. The giant magnetoresistance (GMR)effect was discovered in magnetic multilayers [5,6] only when the thickness of the spacer layers separating themagnetic films was made to be of order l sf. A review of the SDL in metals and alloys some twenty years after thediscovery of GMR concerned mainly the free-electron-like metals Cu, Ag, Au, and Al that have large values of l sf; for just a few of the transition metal elements, there was asingle, low temperature entry [7]. With the “rediscovery ” [8,9] of the spin Hall effect (SHE) [10]and its observation in semiconductors [11,12] and metals [13,14] , this situation has changed radically [15,16] . However, even for well- studied materials like Pt, values of l sfreported over the last decade span an order of magnitude [16,17] . Discernible trends in lsfhave not been reported for different transition metal elements. The SHE is another consequence of SOC whereby the passage of a charge current through a metal gives rise to a transverse spin current that can enter an adjacent magnetic material and exert a torque on its magnetization causing itto switch its orientation. The efficiency of the SHE is givenby the spin Hall angle (SHA) Θ sHthat is the ratio of the transverse spin current (measured in units of ℏ=2) to the charge current (measured in units of the electron charge−e). From being a curiosity, the SHE has rapidly become a leading contender to form the basis for a new magneticmemory technology [18] bringing with it the need to find materials with optimal values of Θ sHwith a primary focuson heavy metals like Pt [14,19] ,T a [20], and W [21]. Striking discrepancies between different room temperature(RT) measurements, with reported values of, e.g., Θ Pt sH ranging between 1% and 11% [15], led to the realization that the bulk parameters lsfandΘsH, as well as the resistivity ρwere very sample dependent and needed to be determined simultaneously. Doing this did not, however,lead to a consensus about the values of these parameters[16]. Whether the SHA is determined using spin pumping and the inverse SHE [14,22 –24], the SHE and spin-transfer torque [25], or nonlocal spin injection [19,26] , interfaces are always involved leading to the suggestion that interfaceprocesses like interface spin flipping (spin memory loss)[27–30]or an interface SHE [31,32] should be taken into consideration in interpreting experiment. Attempts to do sohave, if anything, made matters worse with recentlydetermined values of l Pt sfranging from 1.4 to 11 nm and ΘPt sHranging between 3% and 39%; see Table V in Ref. [17]. The only attempt we are aware of to study the intrinsic SDL theoretically is the phonon-induced spin relaxation work by Fabian and Das Sarma on aluminium [33]that is not readily generalized to transition metals. The purpose ofthis Letter is to present benchmark calculations of l sfand ΘsHfor all bulk 5dmetals at temperatures where the resistivity is dominated by electron-phonon scattering andthe extrinsic scattering that dominates low temperaturemeasurements, but whose microscopic origin is seldomknown, can be disregarded. These first quantitative theo-retical predictions give us insight into how l sfmay be modified and confirm long-standing speculations aboutrelationships between ρ,l sf,ΘsH, and the temperature T. Method.—We recently demonstrated that it was possible to distinguish bulk transport properties from the interfacePHYSICAL REVIEW LETTERS 126, 196601 (2021) 0031-9007 =21=126(19) =196601(7) 196601-1 © 2021 American Physical Societyeffects that are inherent to scattering formulations of transport [34]by evaluating local charge and spin currents from the solutions of quantum mechanical scattering calculations [17,35] as sketched in the inset to Fig. 1. Our density-functional-theory scattering calculations include temperature-induced lattice and spin disorder in the adiabatic approximation [36,37] as well as SOC [38]. An example of the results of such a calculation is shown inFig. 1, where we plot the natural logarithm of the spin current that results from injecting a fully polarized current into a length Lof room temperature (RT, T¼300K) disordered Ta where a Gaussian distribution of random atomic displacements was chosen to reproduce the exper- imentally observed resistivity of bulk Ta [39]. From the near-perfect exponential decay over 5 orders of magnitude, we extract a value of l Ta sfðT¼300KÞ∼6.2nm that is independent of the lead material [17]. Spin memory loss at the left interface manifests itself in the deviation from exponential behavior close to z∼0[35]. Results.—The parameter ξcontained in the SOC term ξl:sof the Pauli Hamiltonian scales as the square of the atomic number Z[40,41] as shown in the inset to Fig. 2.For the 5delectrons, ξdincreases monotonically from ∼0.22eV for Hf to ∼0.63eV for Au. This might lead us to expect a decreasing trend in lsfwith Z. The calculated RT values of lsfshown in Fig. 2for all 5delemental metals exhibit no such decrease. For example, Ta and W are both bcc and as neighbors in the periodic table have very similar values of ξdyetlW sfis some 5 times larger than lTa sf. We find instead that the dominant trend is given by the inverse of the Fermi level density of states (DOS), gðεFÞ, shown in orange in Fig. 2averaged over an energy window of /C6kBTabout the Fermi energy. A low DOS means fewer possibilities to scatter with a spin flip. The same correlation with the inverse DOS is found on calculating lsfas a function of band filling for bcc and fcc structures (not shown). Reported experimental values of lsfat RT are 1.8 and 1.9 nm for bcc Ta, 2.1 nm for bcc W, between 1 and 11 nmfor Pt, and from 27 to 86 nm for Au [16]. The lack of any correlation with the “intrinsic ”values we calculate suggests that the measurements are dominated by other, as yet unidentified, factors. Temperature dependence. —By varying the mean square displacement of the atoms in the scattering region toreproduce experimental resistivities, we can study the temperature dependence of l sfdue to electron-phonon coupling. For temperatures in the range 100 –500 K, the product ρðTÞlsfðTÞis plotted for all 5delements in Fig. 3, where it is seen to be independent of temperature within the error bars of the calculations. (The large value of lW sf∼ 30nm at 300 K requires calculations with an excessively long geometry putting values for 100 and 200 K out of reach.) This is in agreement with predictions made byElliott and Yafet for doped semiconductors and alkali FIG. 1. Natural logarithm of a spin current injected into RT bcc Ta as a function of the coordinate zin the transport direction, L→R. The upper inset sketches the transport geometry with a scattering region Swith temperature dependent lattice disorder sandwiched between ideal semi-infinite ballistic leads, LandR.A fully polarized spin current injected from the left lead Lundergoes spin flipping leading to spin equilibration on a length scale given by the spin-flip diffusion (SFD) length lsfas suggested by the red arrows. An unpolarized charge current injected from Lundergoes spin dependent scattering leading to a transverse spin Hall currentdepicted by the purple arrow. The lower inset shows the spincurrent on a linear scale. The current was extracted from the resultsof a scattering calculation for a two-terminal Ta ↑jTajTa configu- ration using a 7×7lateral supercell where Ta ↑indicates an artificial, fully polarized Ta lead. The red line is a weighted linearleast squares fit; the error bar in the value 6.20/C60.06results from different “reasonable ”weightings and cutoff values [17].FIG. 2. Black (left): spin-flip diffusion length lsffor5d transition metals calculated at room temperature (300 K), theerror bars correspond to the spread of values for ten differentconfigurations. For hcp metals, the c-axis values are shown. Orange (right): inverse of ¯gðε FÞ, the density of states averaged over an energy window of /C6kBTabout the Fermi energy εF. Inset: spin-orbit coupling parameter ξdas a function of the square of the atomic number Zfor the 5delements.PHYSICAL REVIEW LETTERS 126, 196601 (2021) 196601-2elements [42,43] but now for Fermi surfaces that are far more complex than those they considered, for which their approximations are not applicable. As such, this result is nontrivial. While lsfvaries from 4 to 50 nm at room temperature, the product ρlsfspans a smaller range, varying between 0.5 and 2fΩm2. Although ξdattains its maximum value for Au, the d bands are then completely filled and well below the Fermi energy so Au is expected to have a long SDL. The low resistivity and weak effective SOC of Au make a directcalculation computationally very challenging. To determine l Au sfat room temperature, we considered an elevated temperature of T¼1000 K and then assumed that ρðTÞlsfðTÞwas a constant in order to estimate the RTvalue of lsf∼50nm shown in Fig. 2(as a black asterisk) and given in Table I. Relationship of τsftoτ.—The product ρðTÞlsfðTÞis expected to be a constant when momentum scattering is dominated by phonons and τsf∝τ[42,43] . We can estimate τand τsfas follows. In the relaxation time approximation, the conductivity is given in terms of the kdependent velocities υnðkÞ¼ð 1=ℏÞ∇kεnðkÞfor band nas σij¼e2X nZZZd3k 8π3τnðkÞυniðkÞυnjðkÞ/C18 −∂f ∂ε/C19 ε¼εnðkÞ; which becomes an integral over the Fermi surface SFwhen −ð∂f=∂εÞ→δðε−εFÞin the low temperature limit and assuming τðkÞ¼τ½εnðkÞ/C138soσ¼e2gðεFÞτðεFÞhv2 Fi. Both gðεFÞand hv2 Fican be evaluated from standard bulk LMTO electronic structure calculations [44]. Since σ≡1=ρis known [39,45] ,τcan be evaluated as τ¼σ e2gðεFÞhv2 Fi: The diffusion coefficient D[3]can be determined from the Einstein relation σ¼De2gðεFÞ. Using the spin-flip dif- fusion length lsfevaluated from the exponential decay of an injected spin current and the relationship l2 sf¼Dτsfallows us to determine τsf. The ratio of the spin relaxation time, τsf, toτis finally τsf τ¼½e2ρlsfgðεFÞ/C1382hv2 Fi: In spite of the apparent complexity of this relationship, Fig.4shows that the factor dominating the Zdependence of both relaxation times is the inverse Fermi-level density of states. In this sense τsf∝τ, consistent with ρlsfbeing independent of temperature. We note that the electron-phonon coupling and phonon-modulated SOC effects enterFIG. 3. Product of the spin-flip diffusion length and resistivity for5dmetals calculated at different temperatures. For hcp metals, thec-axis values are shown. The error bars are estimated from the average spreads over 10 configurations for both lsfand ρ. TABLE I. Room temperature transport parameters of 5dmetallic elements (El.): resistivity ρ(μΩcm); Fermi level density of states ¯gðεFÞ(states/eV .atom); diffusion constant D(cm2=s); Fermi velocity υF≡hυ2 Fi1=2(108cm=s); relaxation time τ(fs); spin-flip diffusion length lsf(nm); spin relaxation time τsf(fs); spin Hall angleΘsH(%). kand⊥refer to parallel to and perpendicular to the hcp hexagonal axis, respectively. El. Latt. ρ ¯gðεFÞD υF τ lsf τsfΘsH Hf hcp k35.6 0.79 4.94 0.40 9.07 4.97 50 1.35 hcp⊥59.0 2.98 5.47 4.2 60 0.98 Ta bcc 12.1 1.25 7.37 0.89 2.79 6.16 50 −0.50 W bcc 5.49 0.35 51.2 1.13 12.0 29.6 170 −0.40 Re hcp k19.7 1.04 4.52 0.71 2.69 4.46 44 −1.28 hcp⊥25.8 3.45 2.05 3.22 30 −1.90 Os hcp k9.48 0.63 14.77 0.88 5.73 6.06 25 0.71 hcp⊥10.0 3.41 5.44 5.90 25 −0.66 Ir fcc 5.31 0.90 18.4 0.76 9.68 14.1 110 0.22Pt fcc 10.8 1.62 5.37 0.43 8.71 5.21 50 4.02Au fcc 2.27 0.29 160 1.39 24.8 50.9 160 0.25 FIG. 4. Momentum relaxation time τand spin-flip relaxation time τsfin fs for all 5delements at room temperature.PHYSICAL REVIEW LETTERS 126, 196601 (2021) 196601-3our calculation implicitly via the resistivity ρwhile the density of states gðεFÞis calculated with SOC explicitly included in the Hamiltonian. Room temperature values ofall the transport parameters calculated as described above are given in Table I. Spin Hall angle. —Because of the correlation between the SHA and SDL observed in measurements [27],i ti s desirable to determine Θ sHusing the same approximations as were used to calculate lsf. Most quantitative theoretical studies of the SHE [47–50]are based upon the Kubo formalism and have focused on the so-called intrinsic contribution that does not consider the role of the elec-tron-phonon scattering mechanism that dominates the resistivity of elemental metals at room temperature where the vast majority of Θ sHdeterminations have been made [16]. In the linear response regime, the scattering theory we use is equivalent to the Kubo formalism [51]and therefore includes the intrinsic contribution as well as that fromelectron-phonon coupling. The advantage of the scattering formalism is that extrinsic mechanisms can be included on an equal footing. By calculating the transverse spin currentresulting from a longitudinal charge current using the localcurrent method [17] introduced previously to study Θ Pt sH [31], we determined ΘsHas a function of temperature for all 5delements. The5dm e t a ls p i nH a l lc o n d u c t i v i t y( S H C ) σsH¼ΘsH×σ is shown for a number of different temperatures in Fig. 5and is seen to be weakly dependent on temperature, except for Re. The element ( Z) dependence is in qualitative agreement with the linear response calculations by Tanaka et al. who used anempirical “quasiparticle damping parameter ”to represent disorder [48]. In these latter calculations, the correct bcc, fcc, and hcp structures were used for each element but Hf andAu were not considered. Based upon a rigid band study for the fcc structure, Guo et al. identified a switch from positive values of σ sHfor band filling corresponding to Pt to negative values for band fillings corresponding to Ta and W [47]and this has been interpreted in terms of Hund ’s third rule [52].W e reproduced the peaks observed by Guo et al. as a function of band filling for the fcc structure in [17]and note that they originate in regions of the Brillouin zone with degeneracies that are strongly affected by SOC. The energies and numbers of these degeneracies depend on the lattice structure andexplicit calculation for the bcc and hcp structures shows that the correspondence between Guo ’s rigid band picture for the fcc structure and observations for materials with otherstructures is accidental [53]. For example, it predicts a small, negative SHC for Hf. However, for hcp Hf we find a small, positive value of σ sHin agreement with recent experi- ments [54,55] . Our finding of a weak temperature dependence of σsHis in qualitative agreement with the temperature independencefound for Pt by Isasa et al. [57] who, however, reported a substantially lower value of σ sH∼1200 ðΩcmÞ−1(note that we use the ℏ=2econvention [58]). A subsequent meas- urement by the same group yielded a room temperature spin Hall conductivity of ∼3200 ðΩcmÞ−1[30]in excellent agreement with our value. At low temperatures in cleansamples, they observed an enhancement of σ sHwith decreasing temperature. Such an enhancement was recently found [59] in calculations in which phonon modes of Pt were explicitly populated as a function of temperature [37] and it was pointed out that the temperature independence of σsHthat we find is characteristic of the classical equiparti- tion approximation [59]. When the resistivity is no longer linear in temperature, our description of thermal disorder in terms of a Gaussian distribution of uncorrelated atomic displacements is not suitable for studying the weak scatter-ing limit. This is typically well below 100 K [39]. In view of the proportionality of Θ Pt sHto the resistivity ρ [31]and of lPt sfto the conductivity [28,37] , their product is expected to be independent of temperature. ΘsH×lsfis shown in the inset to Fig. 5for all 5delements for a number of temperatures between 100 and 500 K. As a function of Z, it is seen to follow the trend of σsHand as a function of temperature it is indeed approximately constant for allmetals except rhenium for which additional theoretical and experimental studies are desirable. It remains to be seen howΘ sH×lsfbehaves when different scattering mecha- nisms are present simultaneously, in particular interfacescattering and bulk thermal disorder. Conclusions. —We have presented a comprehensive ab initio study of two important spin-orbit coupling related transport parameters in the 5dmetals as a function of temperature verifying the generality of the Elliot-YafetFIG. 5. The spin Hall conductivity σsH(¼ΘsH×σ) calculated at different temperatures compared to the room temperature spinHall conductivity for all 5delements (except Hf and Au) reported by Tanaka et al. [48]using a Green function method (GFM) and a phenomenological quasiparticle damping rate to account fordisorder. The inset shows the product Θ sH×lsfas a function of temperature. Only one value is plotted for Au based on anextrapolation of l sfto RT. For hcp metals, the c-axis values are shown.PHYSICAL REVIEW LETTERS 126, 196601 (2021) 196601-4mechanism and establishing numerical benchmarks for experiment. The values of ρlsf,σsH, andΘsHlsfcalculated in this work can be directly used to predict the values of lsf andΘsHfor the most experimentally relevant temperatures. In particular, the direct correspondence between the spin- flip diffusion length and density of states at the Fermi level implies that spin-flip scattering can be effectively con- trolled by alloying, whereas a high spin Hall conductivity may be achieved by tuning the Fermi level with respect to degeneracies at high symmetry points. Our results indicate that the magnitude of the SOC is not the sole determinant of lsfandΘsHbut that crystal structure and associated details of the electronic structure are just as important. This work was financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek ”(NWO) through the research programme of the former “Stichting voor Fundamenteel Onderzoek der Materie, ”(NWO-I, formerly FOM) and through the use of supercomputer facilities of NWO “Exacte Wetenschappen ”(Physical Sciences). R. S. N (Project No. 15CSER12) and K. G. 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PhysRevB.79.184423.pdf

Microscopic model for current-induced switching of magnetization for half-metallic leads N. Sandschneider *and W. Nolting Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany /H20849Received 22 April 2009; published 20 May 2009 /H20850 We study the behavior of the magnetization in a half-metallic ferromagnet/nonmagnetic insulator/ ferromagnetic metal/paramagnetic metal tunnel junction. It is calculated self-consistently within the nonequi-librium Keldysh formalism. The magnetic regions are treated as band ferromagnets and are described by thesingle-band Hubbard model. We developed a nonequilibrium spectral density approach to solve the Hubbardmodel approximately in the switching magnet. By applying a voltage to the junction it is possible to switchbetween antiparallel /H20849AP/H20850and parallel /H20849P/H20850alignments of the magnetizations of the two ferromagnets. The transition from AP to P occurs for positive voltages while the inverse transition from P to AP can be inducedby negative voltages only. This behavior is in agreement with the Slonczewski model of current-inducedswitching and appears self-consistently within the model, i.e., without using half-classical methods such as theLandau-Lifshitz-Gilbert equation. DOI: 10.1103/PhysRevB.79.184423 PACS number /H20849s/H20850: 85.75. /H11002d, 72.25. /H11002b, 73.23. /H11002b, 73.40.Rw There has been considerable interest in the phenomenon of current-induced switching of magnetization since it was first proposed over 10 years ago.1,2The basic idea behind this effect is as follows. The spin direction of electrons moving ina ferromagnet /H20849FM1 /H20850will be mostly aligned parallel to the magnetization axis. When these spin-polarized electrons aretransported to a second ferromagnet, e.g., by applying a volt-age, then the spin angular momentum of the itinerant elec-trons will exert a torque on the local magnetic moment. Thistorque is known as the spin-transfer torque. It will have aninfluence on the direction of magnetization. If the parametersof the materials are chosen in the right way and if the currentthrough the junction is high enough it is even able to switchthe magnetization of one ferromagnet from parallel to anti-parallel or vice versa relative to the other one. This effectwas seen both in all-metallic junctions 3–5such as Co/Cu/Co and in magnetic tunnel junctions /H20849MTJs /H20850consisting of two ferromagnets divided by a thin nonmagnetic insulator.6–8In this paper we focus on a special case of the latter, where theferromagnetic lead is half-metallic, i.e., there are only elec-trons of one spin direction present at the Fermi energy. Some of the possible technological applications of spin- transfer torques in MTJs have been discussed by Diao et al. 9 Most of the theoretical work in this area of research have been focused on the Landau-Lifshitz-Gilbert /H20849LLG /H20850 equation,10–14which is a macroscopic, half-classical equa- tion. The torques entering this equation were usually calcu-lated in a microscopic picture while treating the interactionson a mean-field level. In this paper we propose a modelwhich takes interactions beyond mean field into account. Wemake no use of the LLG equation or other macroscopic ap-proaches and thus we stay on the quantum-mechanical levelthroughout this paper. We will start the presentation of the theory by introducing a model Hamiltonian which describes the magnetic tunneljunction shown schematically in Fig. 1. There are two ferro- magnetic metals /H20849LandR/H20850divided by a nonmagnetic insula- tor /H20849I/H20850and additionally a paramagnetic metal /H20849P/H20850which is necessary to have a well-defined chemical potential on theright side of the second ferromagnet. Each region consists ofa single s-like band. The two outer leads LandPare treated as semi-infinite. The total Hamiltonian consists of several parts H=H L+HLI+HI+HRI+HR+HRP+HP, /H208491/H20850 where HL/H20849R/H20850describes the left /H20849right /H20850ferromagnet, HIthe insulator, and HPthe paramagnet. Both insulator and para- magnet are assumed to be noninteracting so their Hamilto-nians consist of the kinetic energy only H X=/H20858 kX/H9268/H20849/H9280kX−VX/H20850dkX/H9268+dkX/H9268 /H20849X=I,P/H20850, /H208492/H20850 where dkX/H9268/H20849dkX/H9268+/H20850is the annihilation /H20849creation /H20850operator of an electron with wave vector kXand spin /H9268./H9280kXis the disper- sion of the lattice which throughout this paper is chosen as a tight-binding bcc lattice. The applied voltage Vwill shift the center of gravity of the paramagnet by VP=Vand half of that amount for the insulator, VI=V/2. Positive voltage, V/H110220, will shift the bands to lower energies while negative appliedvoltages result in a shift toward higher energies. The Hamiltonians of the left /H20849L/H20850and right /H20849R/H20850ferromag- nets are formally almost identical. Besides the kinetic energythey also include on-site Coulomb interaction. They aregiven in a mixed Bloch-Wannier representation /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /0 /0 /0 /0 /0 /0/0 /0 /0 /0 /0 /0 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 Lµ eVInsulator FM 1 FM 2 PM µP FIG. 1. Schematic picture of the magnetic tunnel junction with applied voltage V. The conduction bands are shown as rectangles. Occupied states in the metals are hatched and the directions ofmagnetization in the ferromagnets are symbolized by thick arrows.PHYSICAL REVIEW B 79, 184423 /H208492009 /H20850 1098-0121/2009/79 /H2084918/H20850/184423 /H208496/H20850 ©2009 The American Physical Society 184423-1HM=/H20858 kM/H9268/H20849/H9280kM−VM/H20850ckM/H9268+ckM/H9268+UM 2/H20858 iM/H9268nˆiM/H9268nˆiM−/H9268,/H208493/H20850 where Mstands for either LorR. The Hubbard Udetermines the interaction strength. nˆiM/H9268=ciM/H9268+ciM/H9268is the occupation number operator. The voltage Vshifts only the band center of the right ferromagnet, VR=Vwhile the left ferromagnet is not directly influenced by V, i.e., VL=0. The remaining three terms of the Hamiltonian are respon- sible for the coupling between the different regions. Thesecouplings act as a hybridization 15between the bands and therefore the Hamiltonians are /H20849M=L,R;X=I,P/H20850, HMX=/H20858 kMkX/H9268/H20849/H9280kMkXckM/H9268+dkX/H9268+ H.c. /H20850. /H208494/H20850 They are characterized by the coupling constants /H9280kMkXwhich determine the strength of the hybridization between the different bands. In general the couplings are wave vectordependent but for the sake of simplicity we neglect this de-pendence, /H9280kMkX/H11013/H9280MX/H11013/H9280XM. Furthermore we assume the coupling between the ferromagnets and the insulator to be equal so that /H9280LI=/H9280RI/H11013/H9280MI. Altogether there remain two cou- plings /H9280MIand/H9280RPwhich cannot be calculated within this model so they will be treated as parameters. The main topic of this work will be the calculation of the nonequilibrium magnetization mof the right ferromagnet within the Keldysh formalism.16It can be calculated with the help of the Fourier transform of the so-called lesser Green’s function defined as GkR/H9268/H11021/H20849t,t/H11032/H20850=i/H20855ckR/H9268+/H20849t/H11032/H20850ckR/H9268/H20849t/H20850/H20856, m=n↑−n↓=1 2/H9266iN/H20885 −/H11009+/H11009 dE/H20858 kR/H20851GkR↑/H11021/H20849E/H20850−GkR↓/H11021/H20849E/H20850/H20852,/H208495/H20850 where n/H9268=/H20855nˆ/H9268/H20856is the occupation number of particles with spin/H9268in the right ferromagnet. In order to derive the lesser Green’s function one first has to calculate the retarded one, GkR/H9268r/H20849E/H20850=/H20855/H20855ckR/H9268;ckR/H9268+/H20856/H20856E. By using the equation of motion method one finds GkR/H9268r/H20849E/H20850=1 E−/H9280kR−/H9018kR/H9268r/H20849E/H20850−/H9004kR/H9268r/H20849E/H20850. /H208496/H20850 Two different self-energies appear in the Green’s function. First there is the interaction self-energy which can only becalculated approximately for the Hubbard model. We pro-pose a nonequilibrium spectral density approach /H20849NSDA /H20850. The basic idea behind this approach is to choose the self-energy in such a way that the first four spectral moments arereproduced by the theory. Some details of its derivation aregiven in the Appendix. The mean-field /H20849Stoner /H20850solution of the Hubbard model on the other hand satisfies only the firsttwo moments. One finds for the self-energy /H9018 kR/H9268r/H20849E/H20850=URn−/H9268E−T0,R−B−/H9268 E−T0,R−B−/H9268−UR/H208491−n−/H9268/H20850. /H208497/H20850 This expression is coincidentally formally identical to the equilibrium spectral density approach.17The difference is in the spin-dependent band correction B−/H9268which is given byn−/H9268/H208491−n−/H9268/H20850/H20849B−/H9268−T0,R/H20850 =1 2/H9266iN/H20858 kR/H20885 −/H11009/H11009 dE/H20875/H20877/H208752 UR/H9018kR−/H9268r/H20849E/H20850−1/H20876 /H11003/H20851E−T0,R−/H9018kR−/H9268r/H20849E/H20850/H20852+/H208732 UR−1/H20874/H20849/H9280kR−T0,R/H20850 /H11003/H20851E−/H9280kR−/H9018kR−/H9268r/H20849E/H20850/H20852/H20878GkR−/H9268/H11021/H20849E/H20850+2 UR/H9004kR−/H9268/H11021/H20849E/H20850/H20876. /H208498/H20850 It has to be calculated self-consistently since B−/H9268also ap- pears on the right-hand side as part of the lesser Green’sfunction. T 0,Ris the center of gravity of the right ferromag- net. The second self-energy is the transport self-energywhich is due to electrons hopping between the different ma-terials. Its retarded /H20849lesser /H20850component is given by /H9004 kR/H9268r/H20849/H11021/H20850/H20849E/H20850=/H20858 kI/H9280MI2GkI/H9268/H20849L/H20850,r/H20849/H11021/H20850/H20849E/H20850+/H20858 kP/H9280RP2gkP/H9268r/H20849/H11021/H20850/H20849E/H20850, /H208499/H20850 where GkI/H9268/H20849L/H20850,r/H20849E/H20850is the Green’s function of the insulator when it is only coupled to the left ferromagnet, i.e., GkI/H9268/H20849L/H20850,r/H20849E/H20850=1 E−/H9280kI−/H20858kL/H9280MI2gkL/H9268r/H20849E/H20850. /H2084910/H20850 Since we neglected the wave vector dependence of the cou- plings, the transport self-energy is only formally dependent on the wave vector. gkL/H9268r/H20849E/H20850andgkP/H9268r/H20849E/H20850are the equilibrium Green’s functions of the left ferromagnet and the paramag- net, respectively. They can be easily calculated by the equa-tion of motion method. One finds for M=L,P, g kM/H9268r/H20849E/H20850=1 E−/H9280kM−/H9018kM/H9268r/H20849E/H20850. /H2084911/H20850 The paramagnet does not include interactions so that /H9018kP/H9268r /H11013−i0+. Since we are mainly interested in the properties of the right ferromagnet, we assume that the left one is halfmetallic so that its minority states play no role for smallvoltages. This is done by using the mean-field self-energy /H9018 kL/H9268r=ULnL,−/H9268with sufficiently large UL. Thus the retarded Green’s function is known. The lesser Green’s function follows immediately from the Keldysh equation GkR/H9268/H11021/H20849E/H20850=GkR/H9268r/H20849E/H20850/H9004kR/H9268/H11021/H20849E/H20850GkR/H9268a/H20849E/H20850, /H2084912/H20850 where the advanced Green’s function is simply the complex conjugated of the retarded one, GkR/H9268a/H20849E/H20850=/H20851GkR/H9268r/H20849E/H20850/H20852/H11569. Fur- thermore we need the lesser component of the transport self- energy which was already defined in Eq. /H208499/H20850. The lesser part of the insulator Green’s function can again be calculated withthe help of the Keldysh equation G kI/H9268/H20849L/H20850,/H11021/H20849E/H20850=/H20858 kLGkI/H9268/H20849L/H20850,r/H20849E/H20850/H9280MI2gkL/H9268/H11021/H20849E/H20850GkI/H9268/H20849L/H20850,a/H20849E/H20850. /H2084913/H20850 Since the Green’s functions in the left ferromagnet and the paramagnet are equilibrium quantities, their lesser parts readN. SANDSCHNEIDER AND W. NOLTING PHYSICAL REVIEW B 79, 184423 /H208492009 /H20850 184423-2gkL/H20849P/H20850/H9268/H11021/H20849E/H20850=−2 ifL/H20849P/H20850/H20849E/H20850ImgkL/H20849P/H20850/H9268r/H20849E/H20850, /H2084914/H20850 where fL/H20849P/H20850/H20849E/H20850is the Fermi function in lead L/H20849P/H20850with chemi- cal potential /H9262L/H20849P/H20850. They are related by /H9262L−/H9262P=V. Thus we have a closed set of equations for calculating the magnetiza-tion of the ferromagnet. In Fig. 2a typical numerical solution for the voltage- dependent magnetization is shown. We will first discuss theblack curve, which was calculated with a hybridizationstrength of /H9280MI=0.5 eV. For the calculation we started with parallel alignment of the two magnetizations /H20849point Ain the figure /H20850. Then a negative voltage is applied, i.e., the right ferromagnet is shifted to higher energies compared to the leftone. At a critical voltage the parallel alignment becomes un-stable and the magnetization reverses its sign /H20849BtoC/H20850. Thus the magnetizations are now antiparallel. When the voltage isfurther decreased the magnetization stays more or less con-stant until point Dis reached. Then the process is reversed and the voltage is reduced to zero again. The magnetizationfollows the same line as before until the switching point Cis reached. There it does not switch back to parallel alignmentbut rather stays at about the same level. When Eis reached the direction of the voltage is reversed, i.e., the right ferro-magnet will now be shifted to lower energies. For small volt-ages there is only a slight increase until a critical voltage isreached /H20849F/H20850. This voltage has approximately the same value as the first one at point Bbut of course with an opposite sign. There the antiparallel alignment is no longer stable and thesystems returns to its initial parallel state which is not influ-enced by higher voltages /H20849GtoH/H20850. Then the voltage is turned off and the system will be at its starting point Aagain so the hysteresis loop is complete. As another test we start again at point Abut this time we turn on a positive voltage. Then no switching occurs and thesystem will move reversibly to point H. A similar reversible behavior is seen when the alignment is antiparallel /H20849E/H20850and the voltage is decreased. This is shown by the arrows in thefigure. So, one has to conclude that switching of magnetiza- tion from parallel to antiparallel alignment is only possiblefor negative voltages and the reverse process will only ap-pear for positive voltage. The behavior just described is oneof the hallmarks of current-induced switching of magnetiza-tion and thus our proposed model is indeed able to simulatethis effect without leaving the microscopic picture. Now we want to give a short explanation on how exactly our model is able to provide these results. The key to theunderstanding lies in the effect the hybridization parts of theHamiltonian have on the quasiparticle density of states/H20849QDOS /H20850of the switching magnet and in the polarization of the current. A hybridization between two bands generallywill lead to a repulsion between them, i.e., the energetic dis-tance between their respective centers of gravity will in-crease the stronger the effect of the hybridization is. Themagnitude of this shift is mainly influenced by three quanti-ties: the strength of the hybridization itself /H20849 /H9280RPand/H9280MI in our case /H20850, the energetic distance between the two bands /H20849the closer they are to each other, the stronger they will be repelled /H20850, and their spectral weight /H20849higher spectral weight leads to stronger repulsion /H20850. In the upper part of Fig. 3we plotted a typical QDOS for the NSDA without applied volt-age for parallel alignment of the two magnetizations. Thedashed line represents the density of states of the left ferro-magnet. The splitting of the right QDOS into lower andupper Hubbard bands at E/H110150 and E/H11015U Ris clearly visible. Additionally there are contributions of the insulator atE/H11015T 0,I=5 eV which are due to the hybridization. What happens when a voltage is turned on depends on its sign. Theleft ferromagnet FM1 is not influenced by the voltage so itsQDOS will be the same. Let us first discuss the case V/H110220 where both spin bands of the right ferromagnet FM2 areshifted to lower energies. But due to the repulsion betweenthe spin-up bands of FM1 and FM2 the shift of the spin-up-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0. 4 Volta ge (V)-0.8-0.400.40.8Magnetization=0.2 eV =0.5 eVA B C D E FGH εεMI MI FIG. 2. /H20849Color online /H20850Numerical results for the magnetization as a function of applied voltage for two different values of thehybridization strength /H9280MIbetween the metals and the insulator. Arrows indicate in which direction the voltage was changed. Pa-rameters: band occupation n=0.7; band widths: W L=3 eV, WI =1 eV, WR=2 eV, and WP=5 eV; interaction strengths: UR =4 eV, UL=4 eV; band center of the insulator T0,I=5 eV; hybrid- ization strength: /H9280RP=0.05 eV; temperature: T=0 K-3S--10123QDOS -2 0 2 4 6 Ener gy (eV)-3-2-10123QDOSFM1 FM2 FIG. 3. /H20849Color online /H20850Quasiparticle density of states for the parameter set of Fig. 2forV=0 and /H9280MI=0.5 eV. Upper picture for parallel alignment of the magnetizations /H20849point A /H20850; lower picture for antiparallel alignment /H20849point E /H20850. Spin up is shown along the posi- tive; spin down along the negative axis. Black line is the QDOS ofthe right ferromagnet treated within the NSDA and red /H20849gray /H20850bro- ken line shows the QDOS of the left ferromagnet in mean field.MICROSCOPIC MODEL FOR CURRENT-INDUCED … PHYSICAL REVIEW B 79, 184423 /H208492009 /H20850 184423-3band will actually be stronger than for spin-down, where the repulsion is much weaker. So a positive voltage leads to astabilization of the magnetization. This effect is enhanced bythe current. For positive voltages it will flow from left toright. Since the left ferromagnet is fully polarized there areonly spin-up electrons tunneling into the right ferromagnet.These are the reasons for the slight increase in magnetizationin Fig. 2between points A and H and also an explanation why there can be no switching in this case. On the otherhand, if we apply a negative voltage, V/H110210, the right spin bands will be shifted to higher energies. For the same rea-sons as discussed above the shift of the spin-up band will beenhanced by the hydridization. Thus the difference betweenthe centers of gravity of both spin bands will be decreased. Ifthe hybridization strength is sufficiently large this additionalshift together with the self-consistency will be enough topush the spin-up band above the spin-down band and thusthe magnetization changes sign. The self-consistency is im- portant since it will enhance the shift because the occupationin one band depends on the occupation in the other one. Inthis case current flows from the right to the left lead. Sincethere are only spin-up states available for tunneling in theleft ferromagnet, the tunneling current flowing out of theswitching magnet will consist of spin-up electrons only. Thisleads to an additional decrease in the right magnetization.This explains the behavior shown by the parallel alignedcurve in Fig. 2. The antiparallel case can be explained in a very similar way. In the lower part of Fig. 3the corresponding quasipar- ticle density of states is shown, again without applied volt-age. The obvious difference to the case discussed above isthat the center of gravity of the lower spin-down band isbelow the lower spin-up band. This is the reason for thenegative magnetization of course but it is also responsible forthe reversed behavior with respect to the applied voltage. Inthis case a negative voltage cannot push the center of gravityof the spin-up band below the spin-down band. It rather hasthe opposite effect because the repulsion pushes the spin-upband to even higher energies and thus leads to a more stablemagnetization which can also be seen in Fig. 2between points E and D. For a positive voltage the spin-up band ofFM2 moves below the spin-up band of FM1 so that the hy-bridization will shift it to lower energies compared to thespin-down band. Again, if the hybridization strength is largerthan a critical value this additional shift will be enough toreverse the two spin directions such that the magnetizationchanges sign. For the same reasons as discussed for the par-allel case, a positive voltage will increase the magnetizationwhile a negative voltage has the opposite effect. Thereforethe behavior of the magnetization in Fig. 2can be understood in terms of the quasiparticle density of states. In order to prove the explanation based on the hybridiza- tion we plotted a second magnetization curve in Fig. 2with smaller hybridization strength /H9280MI=0.2 eV. Obviously in this case no switching occurs. There is only a slight changein magnetization. Starting from parallel /H20849antiparallel /H20850align- ment the magnetization is reduced for negative /H20849positive /H20850 voltages. This is in agreement with the explanation givenabove. Since the hybridization is weaker the repulsion be-tween the bands is also reduced. It is not strong enough topush the spin-down band above the spin-up band or vice versa. Thus the direction of magnetization is not changed.The current density through the junction is closely linked tothe coupling strength between the materials: 15smaller /H9280MI corresponds to a weaker current. From the results shown inFig.2we can conclude that in order to switch the magneti- zation the current has to exceed a certain value. To summarize, we presented a self-consistent calculation of the voltage-dependent magnetization in a magnetic tunneljunction within a microscopic nonequilibrium framework.The magnetization shows a hysteresis behavior similar tothat seen in experiments. The reason for this effect was ex-plained to be the hybridization between left and right ferro-magnets which could be seen with the help of the quasipar-ticle density of states. It should be noted that the behaviordiscussed above does only appear for very special parametersets /H20849such as low band occupation, small U/H20850when one uses the mean-field approximation for the right ferromagnet. Thisseems reasonable because it is known that mean fieldstrongly overestimates the stability of ferromagnetism. Thusit should be more difficult to switch the direction of magne-tization. We have to conclude that higher correlations seemto be an important factor when describing current-inducedswitching of magnetization within this model. One mightargue that the Kondo peak is missing in the NSDA whichshould have considerable influence on the magnetization.However, we investigated the strong-coupling regime only,where it is known that the Kondo peak does not play a majorrole. On the other hand it would be a very interesting expan-sion of the model to examine its weak-coupling behavior.Another important extension would be the inclusion of spin-orbit coupling which is widely believed to be the micro-scopic origin of phenomenological damping effects 18which play a crucial role in the macroscopic description of switch-ing of magnetization. APPENDIX: NONEQUILIBRIUM SPECTRAL DENSITY APPROACH The basic idea behind the NSDA is to choose the self- energy in such a way that the first four spectral moments MkR/H9268/H20849n/H20850=−1 /H9266/H20885 −/H11009/H11009 dEEnImGkR/H9268r/H20849E/H20850/H20849 A1/H20850 of the spectral density are reproduced exactly. The moments are calculated with the help of the following exact relation: MkR/H9268/H20849n/H20850=1 N/H20858 iRjRe−ikR·/H20849RiR−RjR/H20850 /H11003/H20855/H20853/H20851.../H20851ciR/H9268,H/H20852−, ..., H/H20852−,/H20851H, ..., /H20851H,cjR/H9268+/H20852−.../H20852−/H20854+/H20856, /H20849A2/H20850N. SANDSCHNEIDER AND W. NOLTING PHYSICAL REVIEW B 79, 184423 /H208492009 /H20850 184423-4where the total number of commutators on the right-hand side must be equal to n. Inserting the Hamiltonian /H208491/H20850into this expression yields after some calculation MkR/H9268/H208490/H20850=1 , /H20849A3/H20850 MkR/H9268/H208491/H20850=/H9280kR+URn−/H9268, /H20849A4/H20850 MkR/H9268/H208492/H20850=/H9280kR2+2UR/H9280kRn−/H9268+UR2n−/H9268+/H9280MI2+/H9280RP2, /H20849A5/H20850 MkR/H9268/H208493/H20850=/H9280kR3+2/H9280kR/H20849/H9280MI2+/H9280RP2/H20850+/H9280MI2T0,I+/H9280RP2T0,P +UR/H208533/H9280kR2n−/H9268+2/H20849/H9280MI2+/H9280RP2/H20850n−/H9268/H20854+UR2/H20853/H208492 +n−/H9268/H20850/H9280kRn−/H9268+n−/H9268/H208491−n−/H9268/H20850B−/H9268/H20854+UR3n−/H9268. /H20849A6/H20850 The moments of the transport self-energy /H9004kR/H9268r/H20849E/H20850can be derived in the same way. One gets DkR/H9268/H208490/H20850=0 , /H20849A7/H20850 DkR/H9268/H208491/H20850=/H9280MI2+/H9280RP2, /H20849A8/H20850 DkR/H9268/H208492/H20850=/H9280MI2T0,I+/H9280RP2T0,P. /H20849A9/H20850 The band correction B−/H9268is given by n−/H9268/H208491−n−/H9268/H20850/H20849B−/H9268−T0,R/H20850 =1 N/H20858 iRjR/H20849TiRjR−T0,R/H20850/H20855ciR−/H9268+cjR−/H9268/H208492nˆiR/H9268−1/H20850/H20856 +1 N/H20858 X=I,P/H20858 iRiXTiXiR/H20855diX−/H9268+ciR−/H9268/H208492nˆiR/H9268−1/H20850/H20856,/H20849A10 /H20850 where TiRjRis the hopping integral between lattice sites RiRandRjR. The two higher correlation functions can be reduced to single-particle lesser Green’s functions.17We find /H20855diX−/H9268+ciR−/H9268nˆiR/H9268/H20856=i 2/H9266NU R/H20858 kRkX/H20885 −/H11009/H11009 dEei/H20849kR·RiR−kX·RiX/H20850 /H11003/H20853/H20851−E+/H9280kR+/H9004kR−/H9268r/H20849E/H20850/H20852·GkRkX−/H9268/H11021/H20849E/H20850 +/H9004kR−/H9268/H11021/H20849E/H20850GkRkX−/H9268a/H20849E/H20850/H20854 /H20849 A11 /H20850 and /H20855ciR−/H9268+cjR−/H9268nˆiR/H9268/H20856=−i 2/H9266NU R/H20858 kReikR·/H20849RjR−RiR/H20850 /H11003/H20885 −/H11009/H11009 dE/H9018kR−/H9268r/H20849E/H20850GkR−/H9268/H11021/H20849E/H20850./H20849A12 /H20850 The nondiagonal lesser Green’s function GkRkX/H9268/H11021/H20849E/H20850=i/H20855dkX/H9268+ckR/H9268/H20856is closely related to the right Green’s function and the transport self-energy /H20858 X=I,P/H20858 kXGkRkX/H9268/H11021/H9280XR=GkR/H9268r/H9004kR/H9268/H11021+GkR/H9268/H11021/H9004kR/H9268a./H20849A13 /H20850 Putting all these expressions into the band correction leads to the result in Eq. /H208498/H20850. The Dyson equation of the right ferromagnet reads EGkR/H9268r/H20849E/H20850=1+ /H20851/H9280kR+/H9018kR/H9268r/H20849E/H20850+/H9004kR/H9268r/H20849E/H20850/H20852GkR/H9268r/H20849E/H20850. /H20849A14 /H20850 Inserting the high-energy expansion for both self-energies and the Green’s function GkR/H9268r/H20849E/H20850=/H20858 n=0/H11009MkR/H9268/H20849n/H20850 En+1, /H20849A15 /H20850 /H9004kR/H9268r/H20849E/H20850=/H20858 m=0/H11009DkR/H9268/H20849m/H20850 Em, /H20849A16 /H20850 /H9018kR/H9268r/H20849E/H20850=/H20858 m=0/H11009CkR/H9268/H20849m/H20850 Em, /H20849A17 /H20850 yields a system of equations for the unknown moments CkR/H9268/H20849m/H20850 of the interaction self-energy. It can be solved by sorting according to the order of 1 /Eand the use of the moments given earlier in this appendix. The results are quite simple CkR/H9268/H208490/H20850=URn−/H9268, /H20849A18 /H20850 CkR/H9268/H208491/H20850=UR2n−/H9268/H208491−n−/H9268/H20850, /H20849A19 /H20850 CkR/H9268/H208492/H20850=UR2n−/H9268/H208491−n−/H9268/H20850B−/H9268+UR3n−/H9268/H208491−n−/H9268/H208502./H20849A20 /H20850 These expressions are formally identical to the equilibrium case, therefore the self-energy will also have the same formEq. /H208497/H20850. 17For high energies it is acceptable to neglect higher order terms of the expansion in Eq. /H20849A17 /H20850. Thus /H9018kR/H9268/H20849E/H20850/H11015CkR/H9268/H208490/H20850+CkR/H9268/H208491/H20850 E+CkR/H9268/H208492/H20850 E2/H11015CkR/H9268/H208490/H20850+CkR/H9268/H208491/H20850 E−CkR/H9268/H208492/H20850 CkR/H9268/H208491/H20850 =URn−/H9268E−T0,R−B−/H9268 E−T0,R−B−/H9268−UR/H208491−n−/H9268/H20850. /H20849A21 /H20850MICROSCOPIC MODEL FOR CURRENT-INDUCED … PHYSICAL REVIEW B 79, 184423 /H208492009 /H20850 184423-5*niko.sandschneider@physik.hu-berlin.de 1J. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3J. Grollier, V. Cros, A. Hamzic, J. M. George, H. Jaffres, A. Fert, G. Faini, J. B. Youssef, and H. Legall, Appl. Phys. Lett. 78, 3663 /H208492001 /H20850. 4E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867 /H208491999 /H20850. 5M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 /H208491998 /H20850. 6Y. Huai, F. Albert, P. 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PhysRevB.91.195203.pdf

PHYSICAL REVIEW B 91, 195203 (2015) Manipulating femtosecond spin-orbit torques with laser pulse sequences to control magnetic memory states and ringing P. C. Lingos,1J. Wang,2and I. E. Perakis1,3,* 1Department of Physics, University of Crete, Box 2208, Heraklion, Crete, 71003, Greece 2Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50010, USA 3Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, Heraklion, Crete, 71110, Greece (Received 24 November 2014; revised manuscript received 21 April 2015; published 19 May 2015) Femtosecond (fs) coherent control of collective order parameters is important for nonequilibrium phase dynamics in correlated materials. Here, we propose such control of ferromagnetic order based on usingnonadiabatic optical manipulation of electron-hole ( e-h) photoexcitations to create fs carrier-spin pulses with controllable direction and time profile. These spin pulses are generated due to the time-reversal symmetrybreaking arising from nonperturbative spin-orbit and magnetic exchange couplings of coherent photocarriers. Bytuning the nonthermal populations of exchange-split, spin-orbit-coupled semiconductor band states, we can excitefs spin-orbit torques that control complex magnetization pathways between multiple magnetic memory states. We calculate the laser-induced fs magnetic anisotropy in the time domain by using density matrix equations of motionrather than the quasiequilibrium free energy. By comparing to pump-probe experiments, we identify a “sudden”out-of-plane magnetization canting displaying fs magnetic hysteresis, which agrees with switchings measuredby the static Hall magnetoresistivity. This fs transverse spin-canting switches direction with magnetic state andlaser frequency, which distinguishes it from the longitudinal nonlinear optical and demagnetization effects. Wepropose that sequences of clockwise or counterclockwise fs spin-orbit torques, photoexcited by shaping two-colorlaser-pulse sequences analogous to multidimensional nuclear magnetic resonance (NMR) spectroscopy, can beused to timely suppress or enhance magnetic ringing and switching rotation in magnetic memories. DOI: 10.1103/PhysRevB.91.195203 PACS number(s): 78 .47.J−,75.50.Pp,75.30.Hx,75.78.Jp I. INTRODUCTION Femtosecond (fs) control of switching between condensed matter states [ 1–4] may address challenges posed by multi- functional devices used for information storage and processingon a single chip at up to thousand times faster terahertzspeeds. One of the main obstacles for widespread use of magnetic materials in such applications is the lack of efficient control of magnetization. Fast spin manipulation is one of themain challenges for spin electronics, spin photonics, magneticstorage, and quantum computation [ 5]. To meet this challenge, different magnetic systems must be explored. In diversesystems ranging from ferromagnetic semiconductors [ 6–8] to doped topological insulators [ 9,10], magnetic effects arise from exchange interactions ( ∝S·s) between two distinct subsystems: mobile, spin-orbit-coupled electron spins s, and magnetic local moments S[11]. These interactions couple, for example, magnetic impurity spins with Dirac fermions in topo-logical insulators [ 9] or valence-band holes in (III,Mn)V semi- conductors [ 6]. Such couplings break time-reversal symmetry and result in ferromagnetic states with two distinct but stronglycoupled collective-spin order parameter components [ 6,9]. When brought out of thermodynamic equilibrium, interacting mobile and local collective spins allow more “knobs” for manipulating ultrafast magnetism [ 12] by using fs laser pulses. As is known, in both semiconductors [ 13–17] and met- als [18–20], depending on the time scale, a distinction must be made between e-hquantum excitations, nonthermal eandh populations, and Fermi-Dirac populations [see the schematicin Fig. 1(a)]. Initially, only coherent e-hpairs are photoexcited *Corresponding author: ilias@physics.uoc.gr[left part of Fig. 1(a)], which dephase within a time interval T2.F o r T2shorter than the laser pulse duration, this e-h coherence is only important for determining the photoexcited e andhpopulations. The contribution of such nonthermal (i.e., non-Fermi-Dirac) carrier populations to the spin and chargedynamics must be taken into account when their relaxationtimesT 1are not too short compared to the ∼100 fs time scales of interest [ 18]. Nonthermal population effects are observable in semiconductors [ 13,14] and metals [ 18–20]. Recent pump- probe measurements [ 21] also identified a fs nonthermal hole spin relaxation in (Ga,Mn)As ferromagnetic semiconductors.This temporal regime lasts for 160–200 fs and diminishes withincreasing temperature, together with the ferromagnetic order.It precedes a picosecond (ps) hole energy relaxation, whichoccurs on a time scale of 1–2 ps and is not very sensitive to tem-perature. The above experimental observations [ 21] indicate that the photohole populations redistribute between band stateswith different spin polarizations during T 1∼100 fs prior to relaxation into hot Fermi-Dirac distributions [Fig. 1(a)]. While the quantum kinetics of charge photoexcitations has been studied [ 13,18], fs nonadiabatic magnetic correlation is not well-understood [ 1,3,4,22]. Collective spin dynamics is triggered when coupled magnetic order parameter components are “suddenly” brought out of equilibrium via laser excitation. The relative contributions of spins due to coherent, nonthermal,and hot thermal (Fermi-Dirac) carrier populations, whichinteract with local magnetic moments, [ 1,3,4] depend on laser intensity and frequency, relaxation parameters, materialproperties, and probed time scales. Sequences of fs laser pulsesanalogous to multidimensional NMR spectroscopy [ 15,23,24] offer possibilities for clarifying and controlling such transient magnetic responses. Here, we show that coherent optical 1098-0121/2015/91(19)/195203(20) 195203-1 ©2015 American Physical SocietyP. C. LINGOS, J. W ANG, AND I. E. PERAKIS PHYSICAL REVIEW B 91, 195203 (2015) FIG. 1. (Color online) (a) Schematic of two contributions to the transient magnetic anisotropy: e-hexcitations (nonthermal and coherent carrier contribution, left) and Fermi sea holes (thermalcontribution, right). For /planckover2pi1ω p∼3.1 eV, the holes are excited in high-k, nonparabolic, HH or LH exchange-split valence band states. (b) The thermal hole Fermi sea free energy gives four in-planemagnetic memory states X +,Y+,X−,a n dY−, slightly tilted from the corresponding crystallographic axes. control of nonequilibrium mobile carrier spin induced by laser excitation of a nonthermal population imbalance can be used to suppress or start magnetization ringing andswitching rotation by exerting fs spin-orbit torque sequences in controlled directions. We propose that such a nonadiabaticoptical approach may allow control of magnetic stateswithout relying on magnetic field pulses, circularly-polarizedlight [ 17,25,26], demagnetization [ 8,27–29], quasithermal processes [ 2,30–33], or the precession phase [ 34]. The fs photoexcitation of (Ga,Mn)As has revealed different transient magneto-optical responses, such as ultrafast increase(decrease) of magnetization amplitude under weak (strong)excitation [ 8,28,29,35] and magnetization reorientation due to spin torque [ 17,26] and spin-orbit torque [ 3,36,37]. There is mounting evidence that nonthermal magnetic processesplay an important role in the fs magnetization time evolu-tion [ 3,17,36,37]. (III,Mn)V heterostructures are advantageous for optical control of magnetic order due to their well-characterized optical and electronic properties and their ma-nipulable carrier-induced ferromagnetism. Useful for demon-strating our theoretical predictions is that these systems havefour different in-plane magnetic states ( X +,Y+,X−, andY−), due to biaxial magnetic anisotropy between the [100] and [010]crystallographic axes [see Fig. 1(b) and Appendix A]. While in conventional ferromagnets switching involves spin-flippingbetween two magnetic states (spin-up/spin-down, uniaxialmagnetic anisotropy), the existence of four magnetic statesallows for complex multistate switching pathways and moreelaborate magnetization control schemes. Four-state magneticmemories may be useful for ultrahigh-density magneticrecording applications, as the two equivalent easy axes doublethe recording density by recording two bits of information onthe same spot [ 38]. To take advantage of such multistate mag- netic memories for ultrafast spintronics applications, we mustbe able to selectively access all magnetic states in any desiredsequence. There is no generally accepted scheme on how to dothis. Optical spin manipulation has, however, reached a highlevel of sophistication [ 3,8,17,25,33,34,36,38–42] and control of magnetization on a 100-ps time scale has been demonstratedin various systems, by using magnetic field or laser-generatedmagnetic pulses [ 43–45] or photoinduced effects [ 2,46]. Two outstanding challenges must, nevertheless, be better addressed:(i) how to initiate and stop controlled deterministic switchings during fs time intervals and (ii) how to suppress the magneticringing associated with switchings, which limits the prospectsfor high-speed applications [ 47]. Similar challenges also apply to conventional uniaxial magnetic memories. From amore general perspective, the nonthermal dynamical disen-tanglement, during coherent nonlinear optical excitation, ofdegrees of freedom that are strong-coupled in equilibrium,such as the mobile photocarrier and localized collectivespins here, may lead to a better understanding of correlatedsystems [ 1,4,48,49]. The advantage of using spin-charge quantum kinetics to overcome the limitations of incoherentprocesses for meeting the above challenges is now beginningto be recognized [ 1,3,4,17,41,42,50,51]. This work contributes to the debate of how fs coherent pho- toexcitation could drive and control ultrafast switchings [ 1,12] and magnetic ringing [ 47]. We consider the very early nonthermal and coherent temporal regimes and focus mostlyon magnetization changes that occur during the fs laser pulse and are triggered by the photoexcited carriers. We show that bychoosing appropriate sequences of time-delayed laser pulses,we can control the direction, magnitude, and time-profile of theshort-lived nonthermal photocarrier spin. The latter drives themagnetization away from equilibrium by exerting fs spin-orbittorque on the collective local spin. By coherent manipulationof the e-hphotoexcitations, we photogenerate a controlled population imbalance between spin-orbit-coupled/exchange-split bands. Such photoexcited band carrier population andspin imbalance is not restricted by the chemical potential ortemperature and leads to a controllable “sudden” magnetiza-tion canting in selected directions at desirable times. Based ondirect manipulation of the above nonthermal processes by theoptical field, we propose possible protocols that drive complex360 ◦magnetization pathways, here involving sequential 90◦ deterministic switchings between four different magneticmemory states. Such spin control, as well as suppression ofboth magnetic ringing and switching rotations, are possiblewithout circularly-polarized light due to relativistic spin-orbitcoupling of the photocarriers leading to spin-orbit torque. For linearly-polarized fs optical pulses, we show that the photoexcited carrier spin direction and amplitude is deter-mined by the competition between spin-orbit coupling, withcharacteristic energy /Delta1 so∼340 meV given by the /Gamma1-point energy splitting of the GaAs spin-orbit-split valence band,and the S·smagnetic exchange coupling, with characteristic energy /Delta1 pd=βcS∼100 meV in Ga(Mn)As [ 6], where S andcdenote the Mn spin amplitude and concentration, respectively, and βis the magnetic exchange constant. The time-reversal symmetry breaking can be characterized bythe energy ratio /Delta1 pd//Delta1 so[∼1/3 in (Ga,Mn)As]. It leads to fs photoexcitation of short-lived mobile spin-pulses s, whose direction is controlled by selectively populating thecontinua of exchange-split heavy-hole (HH) or light-hole(LH) spin-orbit-coupled band states with different spin su-perpositions. We model the fs nonlinear photoexcitationprocesses, driven by sequences of time-delayed laser-pulsetrains, with density matrix equations of motion [ 13], which describe photocarrier populations coupled nonperturbativelyto interband coherences and time-dependent local spins. Ourtime-domain calculations describe a nonequilibrium magnetic 195203-2MANIPULATING FEMTOSECOND SPIN-ORBIT TORQUES . . . PHYSICAL REVIEW B 91, 195203 (2015) anisotropy during the laser pulse, which we estimate by treating strong band nonparabolicity and spin-orbit couplingsusing the tight-binding band structure of GaAs with mean-fieldmagnetic exchange interaction [ 6,52]. We relate the calculated photoexcitation of fs spin-orbit torque to existing experimentsand make predictions for new ones to observe switchings byusing pulse-shaping. The paper is organized as follows. In Sec. II, we discuss the symmetry-breaking processes leading to photoexcitationof a 100-fs mobile carrier spin pulse with direction andmagnitude that depend on the ratio /Delta1 pd//Delta1 so. In Sec. III, we compare theory and experiment to demonstrate coherentcontrol of fs spin-orbit torque direction and magnitude bytuning populations of four exchange-split HH and LH valencebands excited by a 100-fs laser pulse. We show that thecanting direction of the excited transverse (out-of-plane)fs magnetization component displays a magnetic hysteresisabsent without pump. In experiment, the above fs spincanting can be distinguished from longitudinal amplitude andnonlinear optical effects by sweeping a perpendicular magneticfield. In Sec. IV, we show that we can initiate controlled switching rotations to any one of the available magnetic statesby shaping a laser-pulse train. In Sec. V, we propose two protocols for controlling four sequential 90 ◦switchings in clockwise or counterclockwise directions. In Sec. VI,w eu s e two time-delayed laser-pulse trains to suppress or enhance thenonlinear switching rotation at any intermediate state and tosuppress magnetic ringing at any time, long or short. Ratherthan relying on the magnetization precession phase, we achievethis coherent control by switching the directions of fs spin-orbittorques. We end with conclusions and a broader outlook. In twoappendices, we present the density matrix equations describingnonlinear coherent excitation of fs spin-orbit torque, distin-guish the nonadiabatic/nonthermal from the adiabatic/thermaltransient magnetic anisotropy, and treat the nonparabolic andanisotropic spin-orbit-coupled band continua. II. FEMTOSECOND SPIN PHOTOEXCITATION In this section, we discuss the processes leading to pho- toexcitation of carrier spin with direction determined by non-perturbative symmetry-breaking interactions. In the systemsof interest, the magnetic effects arise from antiferromagneticinteractions between localized and mobile (delocalized) carrierspins [ 6]. In contrast to magnetic insulators studied before [ 25], the localized electrons do not contribute to the fs magneticanisotropy but mainly determine the magnetization (collective local spin) S=1 cV/summationdisplay i/angbracketleftˆSi/angbracketright, (1) where Vis the volume and Siare the local magnetic moments at positions i, with concentration c. For example, in (III,Mn)V magnetic semiconductors, the local magnetic moments arepureS=5/2 Mn spins with zero angular momentum, L=0, and no spin-orbit interaction. The magnetic anisotropy comesfrom band electrons, which are clearly distinguished fromthe local spins. Unlike for the localized electrons, theseband electrons are subject to spin-orbit interactions andcouple directly to light. The spin-exchange coupling of suchphotoexcited mobile carriers with the local spins induces the magnetization dynamics of interest here. The widely-usedmean-field treatment of the magnetic exchange interaction(Zener model) captures the symmetry breaking of interesthere [ 6]. We thus consider the dynamics of a single-domain macrospin S(t) and neglect spatial fluctuations [ 40,41]. This approximation describes metallic-like (III,Mn)V magneticsemiconductors [ 6]. Our main goal here is to control the nonequilibrium spin of band carriers in order to manipulate the magnetizationmotion during fs time scales. While spin-lattice couplingalso affects the easy axis, lattice heating occurs on longer(picosecond) time scales, following energy transfer from theelectronic system [ 33,37]. Unlike previous demagnetization studies, the optical control scheme proposed here does notrely on population changes due to laser-induced electronicheating [ 8,28,29]. It is based on direct carrier-spin photoexci- tation without circularly-polarized light. The laser excites e-h pairs between different exchange-split valence and conductionbands [Fig. 1(a)]. The magnetic exchange interaction of inter- est mainly involves the photoexcited valence hole collectivespin/Delta1s h(t). Denoting by sh knthe contribution from valence bandnand momentum k, we obtain the total hole spin: sh(t)=1 V/summationdisplay k/summationdisplay nsh kn(t). (2) Below, we demonstrate coherent control of sh kn(t) by exciting a nonthermal imbalance between different band states ( n,k) during the laser pulse. We describe this nonthermal populationimbalance by extending the discrete- kcalculation of fs spin- orbit torque in Ref. [ 3] to include the anisotropic continua of the nonparabolic (Ga,Mn)As bands. This allows us to estimatethe photocarrier density and net spin of different bands asfunction of laser-pulse frequency and intensity for comparisonto experiment. In addition, here we consider sequences of time-delayed laser-pulse trains. The mechanism of Ref. [ 3] is anal- ogous to the current-induced spin-orbit torque [ 53] observed in (Ga,Mn)As [ 54] and other spin-orbit-coupled ferromagnets. Unlike our earlier work [ 17] on fs spin-transfer torque analo- gous to the one induced by spin-polarized currents in spintron-ics applications [ 55,56], which requires circularly-polarized light [ 26], here spin is not conserved due to spin-orbit coupling. As a result, transfer of angular momentum from the photons isnot necessary for carrier spin excitation. Instead, the photoex-cited spin is determined by symmetry-breaking due to the com-petition between spin-orbit and magnetic exchange couplings. To initiate ultrafast spin dynamics, we create a short-lived spin imbalance by optically controlling s h kn(t) from different bands nand Brillouin zone (BZ) directions k. For this, we express the carrier spin in terms of the density matrix /angbracketleftˆh† −knˆh−kn/prime/angbracketrightdefined in terms of an adiabatic basis of band eigenstates created by the operators ˆh† −kn: sh kn=ˆsh knn/angbracketleftˆh† −knˆh−kn/angbracketright+/summationdisplay n/prime/negationslash=nˆsh knn/prime/angbracketleftˆh† −knˆh−kn/prime/angbracketright, (3) where ˆsh kn/primenare the spin matrix elements. The latter describe the direction of the carrier spin for the band states ( n,k). Such spin dependence is determined by spin-mixing due 195203-3P. C. LINGOS, J. W ANG, AND I. E. PERAKIS PHYSICAL REVIEW B 91, 195203 (2015) to the nonperturbative interplay of spin-orbit and magnetic exchange couplings, which is characterized by the energy ratio/Delta1 pd//Delta1 so. The first term on the right-hand side (rhs) of Eq. ( 3) describes the population contribution (coherent, nonthermal,and quasithermal transient populations). The second termdescribes a contribution due to coupling of different bands(intervalence-band coherence). The latter Raman coherencearises when spin is not conserved, ˆs h knn/prime/negationslash=0, and vanishes in equilibrium. We choose as basis ˆh† −knthe eigenstates of the adiabatic Hamiltonian (Appendix A) Hb(S)=H0+Hso+Hpd(S0). (4) H0+Hsodescribes the band structure of the parent material (undoped GaAs here), due to the periodic lattice potential ( H0) and the spin-orbit coupling ( Hso)[52]. The symmetry-breaking is induced by the magnetic exchange interaction Hpd(S0), Eq. ( A1)[6]. Here, S0denotes the slowly-varying contribution to the local macrospin that switches or oscillates during pstime scales (adiabatic contribution). The valence hole and conduction electron basis states, ˆh† −knand ˆe† kmrespectively, were obtained by diagonalizing Hb(S0) using the tight-binding approximation of Ref. [ 52] (Appendix A). In (III,Mn)V semiconductors, a thermal hole Fermi sea bath, characterized by the Fermi-Dirac distribution fnk,i s already present in the ground state [Fig. 1(a)][6]. Similar to ultrafast studies of the electron gas in metals [ 18] and semiconductors [ 57–59], we distinguish this quasiequilibrium contribution to Eq. ( 3) from the non-Fermi-Dirac femtosecond contribution (Appendix A): /angbracketleftˆh† knˆhkn/prime/angbracketright=δnn/primefnk+/Delta1/angbracketleftˆh† knˆhkn/prime/angbracketright. (5) At quasiequilibrium, only the Fermi-Dirac populations con- tribute. These are characterized by a temperature and chemicalpotential and give the adiabatic field due to the thermalizedFermi sea (FS) carriers: [ 6,25,37] γH FS[S]=−∂Eh(S) ∂S, (6) where γis the gyromagnetic ratio and Eh(S)=/summationdisplay knεv nkfnk (7) is the total (free) energy of the relaxed Fermi-Dirac carriers. The latter defines the magnetic memory states of Fig. 1(b) (Appendix A).εv nk(S) are the (valence band) eigenvalues of the adiabatic Hamiltonian Hbfor frozen local spin S.T h e laser-induced heating of the Fermi-Dirac hole distibution ( fnk) is one source of demagnetization [ 28,29], while the subsequent heating of the lattice is also known to thermally alter themagnetic anisotropy fields during ps time scales [ 32,33]. Since the changes of this electronic E hwithSare notoriously small for numerical calculations of the quasiequilibrium magneticanisotropy [ 37,60], while the low-energy states of (III,Mn)V systems are complicated by sample-dependent disorder, im-purity bands, defect states, and strain [ 6,28,39,61], here we approximate E h(S) by using the symmetry-based Eq. ( A9) with parameters extracted from experiment [ 6,39,61]. In this way, we introduce the realistic four-state magnetic memoryof the (III,Mn)V materials. For the low 10–100 μJ/cm 2pumpfluences considered here, we neglect the laser-induced changes in the Fermi-Dirac distribution temperature and chemicalpotential, which add to the predicted effects on the timescale of energy and population relaxation [ 21]. Calculations assuming Fermi-Dirac distributions [ 28,37] gave order-of- magnitude smaller magnetization dynamics than experimentand concluded that the nonequilibrium hole distribution mustbe very broad [ 28]. Here, we study the possible role of short-lived non-Fermi-Dirac populations, which are observedprior to full electronic thermalization [ 21] (we assume T 1∼ 100 fs). We calculate the fs anisotropy due to such nonthermalspin populations in the time domain, by solving the mean- field equations of motion for /Delta1/angbracketleftˆh† knˆhkn/prime/angbracketrightderived with time- dependent Hamiltonian (Appendix A) H(t)=Hb(S0)+/Delta1H exch(t)+HL(t). (8) While the adiabatic Hb(S0) changes during 10’s of ps, the other two contributions to Eq. ( 8) are nonadiabatic and vary during fs time scales. HL(t), Eq. ( A3), describes the dipole coupling of the fs laser Efield [ 13], while /Delta1H exch(t)=1 V/summationdisplay kβkc/Delta1S(t)ˆsh k, (9) where ˆsh kis the hole spin operator and /Delta1S(t)=S(t)−S0, (10) describes the “sudden” changes in magnetization during the fs photoexcitation. We assume exchange constant βk≈βfor the relevant range of k. We describe the non-Fermi-Dirac electronic contribution /Delta1/angbracketleftˆh† knˆhkn/prime/angbracketright,E q .( 5), similar to the well-established semicon- ductor Bloch equation [ 13,62] or local-field [ 16,63] Hartree- Fock treatments of ultrafast nonlinear optical response. Inparticular, we solve coupled equations of motion for the electronic populations and interband coherences /angbracketleftˆh† kmˆhkn/angbracketright, /angbracketleftˆe† kmˆekn/angbracketright, and/angbracketleftˆekmˆh−kn/angbracketright, which are nonperturbatively coupled to the time-dependent local spin S(t). This coupling modifies the electronic dynamics, which, in turn, modifies the motionofS(t) (Appendix A). To obtain meaningful numerical results in the case of switching, the basis defined by the adiabaticH b(S0) is constantly adjusted due to the large changes in S0during the time evolution. Our equations of motion describe, in addition, the nonadiabatic effects of /Delta1S(t) on the time-dependent band states. We consider linearly-polarizedoptical pulses with zero angular momentum. We do not in-clude the carrier-carrier, carrier-phonon, and carrier-impurityinteractions in the Hamiltonian, but treat the photocarrierrelaxation phenomenologically, by introducing e-hdephasing timesT 2and nonthermal population relaxation times T1.O u r calculation thus describes the “initial condition” that bringsthe system out of equilibrium and initiates relaxation [ 28,51]. The latter redistributes the nonthermal carriers among bandstates with different spins and momentum directions k, which leads to spin relaxation. Here, we model this by introducing the relaxation time T 1of the populations /angbracketleftˆh† −knˆh−kn/angbracketrightdetermining the hole spin in Eq. ( 3), which reflects the 100–200-fs hole spin relaxation time measured experimentally in (Ga,Mn)As [ 21]. The latter was calculated in Ref. [ 51] to be several 10’s of fs. On the other hand, momentum scattering and carrier relaxation 195203-4MANIPULATING FEMTOSECOND SPIN-ORBIT TORQUES . . . PHYSICAL REVIEW B 91, 195203 (2015) giveT2’s of few 10’s of fs [ 6,51]. Below we estimate the dependence of the predicted nonthermal effects on T1andT2. The calculations in this paper describe photogeneration of spin that initiates fs dynamics. We describe the averagehole spin /Delta1s h(t) of e-hpairs excited in band continuum states determined by the pump laser frequency ωp.T h e main results were obtained for /planckover2pi1ωp≈3.1e V [ 7,36]. For such pump frequencies, the (Ga,Mn)As disorder-inducedimpurity/defect states [ 28] do not contribute significantly and the photoexcited carriers are initially well separated in energyfrom the Fermi sea holes [see Fig. 1(a)]. We mainly excite HH and LH band states along the eight {111}symmetry lines of the BZ, at high k, where the conduction and valence bands are strongly nonparabolic and almost parallel to eachother [ 7]. As a result, a large number of interband optical transitions are excited simultaneously and a broad continuumof hole band momenta k, inaccessible at quasiequilibrium, is populated during the laser pulse [see Fig. 1(a)]. Such highly anisotropic band continua are accounted for here as describedin Appendix B. Magnetic anisotropy arises since, due to the symmetry-breaking introduced by S(t), the eight photoexcited {111}directions are not equivalent. The calculated hole spin matrix elements ˆs h knn/prime, which determine the photohole spin direction, are fairly constant for each given band over a widerange of high k. Optical transitions at /planckover2pi1ω p≈3.1eV then add constructively to the hole spin from each of the {111} directions and enhance its magnitude, which depends on the total photohole densities1 V/summationtext k/Delta1/angbracketleftˆh† −knˆh−kn/angbracketrightfor each band n assuming a smooth kdependence of the exchange constant βk. By tuning the pump frequency around 3.1 eV , our goal is to control, a short-lived imbalance between the populationsof bands with different spin-admixtures. Our present calcu-lations describe /Delta1s h(t) prior to interband relaxation or large momentum scattering between different kdirections, which occur on a time scale T1of spin relaxation. On the other hand, pump frequencies /planckover2pi1ωp≈1.5e V [ 37] excite smaller kalong {100},{010},{001},{110},{101},{011}, and {111}symmetry directions [ 64], as well as impurity/defect states inside the semiconductor band gap [ 6,28]. Figure 6 shows the quantitative differences between /planckover2pi1ωp≈1.5 eV and ≈3.1e V,which arise from the differences in band structure. In addition to the difference in closely-lying valence bands,disorder-induced states, and density of states at differentenergies, the kdependence of the spin matrix elements ˆs h knn/prime determining the photoexcited spin is stronger for the small wave vectors contributing around /planckover2pi1ωp≈1.5e V . Important for bringing the coupled local and mobile spin subsystems away from equilibrium is their different dynamics.For example, unlike for the band carriers, there is no spin-orbitor optical coupling of the local spins. In equilibrium, the localand mobile collective spins are correlated in the ferromagneticstate, so that S×H FS=0[6]. Within the mean-field approx- imation, S(t) is driven out of this equilibrium configuration by both quasiequilibrium ( HFS) and nonthermal ( /Delta1sh) carrier spins according to a Landau-Lifshitz-Gilbert equation: ∂tS=−γS×HFS[S(t)]−βS×/Delta1sh(t)+α SS×∂tS,(11) where αcharacterizes the slow local spin precession damping [ 33]. The longitudinal magnetization amplitude Δpd__ Δso 1/6 1/3 2/3 5/3 7/33/3[010][010] _ [100]_[100] FIG. 2. (Color online) Maximum of anisotropy spin pulse β/Delta1sh(t), photoexcited by a single 100-fs linearly-polarized laser pulse, as a function of the energy ratio /Delta1pd//Delta1 sothat characterizes the time-reversal-symmetry breaking. The direction of the ground-state magnetization is along the X+easy axis, shown by the black arrow close to [100]. /planckover2pi1ωp=3.14 eV , E0=7×105V/cm,T1=100 fs, andT2=50 fs. changes, due to spin-charge correlations [ 4,28,40,41], are not captured by this mean-field approximation. The dynamics of the mobile carrier spins depends, in addition to magnetic exchange interaction with the local spins,on spin-orbit coupling, direct nonlinear coupling to the opticalfield, and fast carrier relaxation [ 17]: ∂ tsh k=βcS×sh k+i/angbracketleftbig/bracketleftbig Hso,sh k/bracketrightbig/angbracketrightbig +Imhk(t)+∂tsh k/vextendsingle/vextendsingle rel.(12) The above equation is not useful here, as it does not distinguish between different bands in order to treat the spin-orbitcoupling H so. Nevertheless, it demonstrates four processes that determine the nonthermal carrier spin. The first termdescribes spin-torque due to magnetic exchange. The secondterm describes spin-orbit torque , obtained here by calculating the density matrix ( 5). The third term describes the Raman-type coherent nonlinear optical processes that excite the carrierspin [ 17]: h k(t)=2/summationdisplay mn/angbracketleftˆh−knˆekm/angbracketright/summationdisplay m/primed∗ kmm/prime(t)·sh km/primen, (13) where dkmm/prime=μkmm/prime·Eare the Rabi energies of optical transitions between band states ( mk) and ( m/primek) and Eis the laserEfield. The last term describes spin relaxation. The nonperturbative interplay between spin-orbit and magnetic exchange couplings determines the direction andmagnitude of the net spin excited by a fs laser pulse. Figure 2 shows a strong dependence of the maximum and directionof the photoexcited hole-spin-pulse β/Delta1s h(t) on the energy ratio/Delta1pd//Delta1 so. We obtained this result by solving the coupled equations of motion of Appendix A. In the ground state, the magnetization S0points along the X+easy axis (Fig. 2). For /Delta1pd/lessmuch/Delta1so, the net spin /Delta1shis negligible without circularly- polarized light, since all symmetric directions in the BZ areexcited equally. With increasing /Delta1 pd, the magnetic exchange interaction introduces a preferred direction along S(t). This breaks the time-reversal symmetry of GaAs and results ina net /Delta1s h(t) while the laser pulse couples to the magnetic system. With increasing /Delta1pd//Delta1 so,t h i s/Delta1shincreases and its direction changes. For /Delta1pd//Delta1 so∼1/3 [as in (Ga,Mn)As], 195203-5P. C. LINGOS, J. W ANG, AND I. E. PERAKIS PHYSICAL REVIEW B 91, 195203 (2015) Fig. 2shows that the in-plane component of the fs anisotropy fieldβ/Delta1shpoints close to the [ ¯1¯10] diagonal direction for /planckover2pi1ωp=3.14 eV . As discussed below, this result explains the experimental observations. The above /Delta1sh(t) only lasts during the 100-fs laser pulse and drives a “sudden” magnetizationcanting /Delta1S(t) via fs spin-orbit torque. As /Delta1 pdapproaches /Delta1so, /Delta1shis maximized while it changes direction. The photohole spin decreases again for /Delta1pd/greatermuch/Delta1so. III. EXCITING FS SPIN DYNAMICS WITH A SINGLE PULSE: THEORY VERSUS EXPERIMENT Ultrafast magneto-optical experiments in (III,Mn)V semi- conductors have revealed control of magnon oscillationswith frequency /Omega1∼100 ps −1. In these experiments, the magnon excitation is suppressed (enhanced) with a laser pulsedelayed by τsuch that /Omega1τ=π(/Omega1τ=2π)[34]. In this paper, we propose a different optical control scheme, basedon controlling the direction, duration, and magnitude of fsspin-orbit torque sequences photoexcited at any time τ.F i r s t , however, we validate our original prediction [ 3] of fs spin-orbit torque as a source of nonthermal laser-induced spin dynamicsin (III,Mn)V materials. For this, we connect here the numericalresults obtained for anisotropic and nonparabolic band con-tinua with the few existing experiments showing femtosecondnonthermal spin dynamics. In this section, we show that ourcalculations validate the experimental observation in Ref. [ 36] of fs magnetic hysteresis and spin rotations excited by a single100-fs laser pulse in (Ga,Mn)As. We also show that theyare consistent with the observation of “sudden” nonthermal(subpicosecond) magnetization rotation reported in Ref. [ 37]. The fs temporal regime of nonthermal spin dynamics, which isless understood as compared to the extended ps time scales, ismost relevant for the main purposes of this paper, which are to(i) make numerical predictions of all-optical control of spin ro-tation and magnetic ringing, and (ii) propose complex switch-ing protocols similar to multidimensional NMR, but based onfs laser pulse trains with various timing sequences and colors. We start by discussing the experimental technique and (Ga,Mn)As sample used in Ref. [ 36]. We argue that our static and time-resolved experimental curves and their comparisonwith our theory indicate that the measured magneto-opticalresponse for in-plane ground-state magnetization is dominatedby the transverse out-of-plane magnetization component S z and the polar Kerr effect. We performed two-color time- resolved MOKE spectroscopy in order to better discern thegenuine spin dynamics [ 8,65]. Prior to the relaxation time T 1, the high-energy nonthermal carriers excited by the 3.1-eV pump have small effect on the population of the low-energyband states seen by the 1.55-eV probe. By comparing two-color Kerr rotation, ellipticity, and reflectivity pump-probesignals, we distinguish fs magnetization dynamics fromnonlinear optical effects [ 65,66] and identify a fs component displaying magnetic hysteresis induced by a perpendicularmagnetic field. Different magneto-optical effects are observed for different experimental setups. These may be broadly divided based onrotation angles θ K(S) of the linearly-polarized probe electric field that are linear (odd) or quadratic (even) functions ofS. Previous linear magneto-optical spectroscopy experimentsin ferromagnetic (Ga,Mn)As observed a giant magnetic linear dichroism (MLD) signal for probe frequencies between 1.4 and2.4 eV [ 67,68], with quadratic dependence on S. In contrast, the polar Kerr effect signal [ 65] is linear in the perpendicular S z, without contribution from the in-plane spin components. The relative contribution of these two magneto-optical effectsdepends on the direction of light propagation kand linear polarization Ewith respect to the magnetization [ 65,67,68]. Below we discuss the details of our experimental design andmeasured quantities. The main sample studied here was grown by low- temperature molecular beam epitaxy (MBE) and consists of a73-nm Ga 0.925Mn 0.075As layer on a 10-nm GaAs buffer layer and a semi-insulating GaAs [100] substrate. The in-planeground-state magnetization points along the X +easy axis close to the [100] crystallographic axis [Fig. 1(b)]. For probe we used a NIR beam tuned at 1.55 eV , which propagatesalong a direction almost perpendicular to the sample plane(∼0.65 ◦from the normal). The probe linear polarization is along [100], almost parallel to the ground-state magnetization. The pump, on the other hand, was chosen as a UV beam tunedat/planckover2pi1ω p=3.1 eV and was linearly-polarized at an angle ∼12◦ from [100], with ∼10μJ/cm2peak fluence smaller than in previous experiments. Its ∼40-nm penetration depth implies photoexcitation of only the 73-nm-thick magnetic layer. Theduration of the pump and probe pulses was 100 and 130 fs,respectively, while the laser repetition rate was 76 MHz. A de-tailed description of our measurement may be found in section3.1.2 of Ref. [ 65]. We extracted the background-free MOKE rotation angle θ Kby measuring the difference between s- and p-polarized probe light (linear polarization along the [100] and [010] crystallographic axes, i.e., parallel and perpendicular tothe ground-state magnetization). This is achieved by reflectings-polarized light from the sample surface and then passing it through a combination of a half wave plate and Wollastonprism. Further technical details of our setup can be found inRef. [ 8]. The chosen design minimizes the MLD contribution to our measured magneto-optical signals shown in Figs. 3 and4, discussed below. The sweeping of an external magnetic field B almost perpendicular to the sample and easy axes planeproduced the fs magnetic hysteresis shown in Fig. 3(b).T h i s laser-induced hysteresis is consistent with the behavior ofthe static Hall magnetoresistance [inset of Fig. 3(a)], which is known to arise from in-plane magnetization switchingsbetween the four easy axes of Fig. 1(b). However, no magnetic hysteresis is observed in the linear magneto-optical signalwithout pump for the same experimental conditions [compareFigs. 3(a) and 3(b)]. This result implies that the measured signal is dominated by S z(polar Kerr effect) for the linear polarization direction used here. As discussed, e.g., in Refs. [ 65,66], a signature of genuine magnetization dynamics is the complete overlap of thepump-induced transient Kerr rotation /Delta1θ K/θKand ellipticity /Delta1ηK/ηKsignals. Indeed, nonlinear optical effects are expected to contribute differently to /Delta1θKand/Delta1ηK, as determined by the real and imaginary parts of the pump-induced changes inthe Fresnel coefficients [ 65,66]. In our experiment, /Delta1θ K/θK≈ /Delta1ηK/ηKthroughout the fs time-scan range of interest [ 36]. We thus conclude that the measured /Delta1θK/θKprimarily reflects the pump-induced magnetization /Delta1Sz/S. This claim is further 195203-6MANIPULATING FEMTOSECOND SPIN-ORBIT TORQUES . . . PHYSICAL REVIEW B 91, 195203 (2015) FIG. 3. (Color online) By sweeping a perpendicular Bfield, tilted by 5◦from the Zaxis and 33◦from the Xaxis, “up” (red curve) and “down” (blue curve), we measure the Bdependence of the magnetization component perpendicular to the sample plane at 5 K in the polar MOKE geometry (normalized by the ∼4 mrad MOKE angle). (a) Static measurements (no pump). The coinciding “up” and “down” polar Kerr rotation angles θKshow no magnetic hysteresis in the static case. In contrast, the Hall magnetoresistivity (inset)shows 90 ◦in-plane magnetization switchings between the XZand YZplanes, which manifest themselves as a “major” hysteresis loop. This difference implies that the magneto-optical signal in the presentconfiguration is insensitive to the in-plane magnetization components, which switch. (b) Time-dependent measurements (pump on). The pump-induced change /Delta1θ K/θK≈/Delta1Sz/S, measured at probe time delay /Delta1t=600 fs for the same experimental conditions as in (a), shows a magnetic hysteresis similar to the static Hall magnetore- sistivity. In comparison, the ultrafast differential reflectivity /Delta1R/R (inset) is up to thousand times smaller, which points to a magnetic origin of our /Delta1θK/θKsignal. supported by the simultaneous measurent of a differential reflectivity signal /Delta1R/R [inset, Fig. 2(b)] that is up to thousand times smaller than the Kerr rotation and ellipticity signals. Theabove two experimental observations imply that the relativepump-induced change in the Fresnel coefficients, which addsto the magneto-optical response [ 65], is much smaller than /Delta1S z/Sin the studied configuration. As discussed below, the magnetic origin of the measured fs /Delta1θK/θKis further seen when sweeping an external Bfield slightly tilted from the perpendicular direction [Fig. 4(a)], which reveals a magnetic hysteresis absent in the measured linear response. The interpretation of the static θKin the absence of pump [Fig. 3(a)] does not suffer from the complexity of interpreting the fs pump-probe signal. θK(B) switches sign with B field and saturates for |B|>250 mT. It coincides between “up” and “down” sweeps (no magnetic hysteresis). In sharpcontrast, for the same experimental conditions, the static Hallmagnetoresistivity ρ Hallshows in-plane magnetic switchings (planar Hall effect), which manifest themselves as jumps in thefour-state magnetic memory hysteresis (inset of Fig. 3). Since FIG. 4. (Color online) Magneto-optical pump-probe experimen- tal measurements showing development of laser-induced magnetiza- tion canting /Delta1Sz(t) within ∼100 fs. This fs canting displays magnetic hysteresis and switches direction when switching in-plane magneticstate. (a) We sweep a perpendicular Bfield, applied at a small angle ∼5 ◦from the [001] axis. This Bfield tilts the B=0 in-plane easy axes (X± 0andY± 0) out of the plane (Appendix A). (b)–(f): the “sudden” out-of-plane magnetization tilt /Delta1Sz/S, induced by a 100 fs laser pulse with fluence ∼7μJ/cm2, switches direction when sweeping theBfield between B=− 1 and 1 T. The two sweeping directions correspond to increasing (“up”) and decreasing (“down”) Bfield. For each of the measured B=1, 0.2, 0, −0.2, and −1 T, the fs temporal profiles of /Delta1Sz/Sdepend on the equilibrium magnetic state switched byB. the measured static magneto-optical signals show no signature of the above in-plane magnetization switchings between theXZandYZplanes, they are dominated by the polar MOKE Kerr effect that is proportional to S zand thus insensitive to the in-plane magnetization [ 65,67]. In contrast, MLD [ 67] is a second-order effect and includes contributions such asS xSythat are sensitive to the in-plane magnetization switching. Their absence in Fig. 3(a) implies that MLD is not the main origin of our measured magneto-optical signal, which thus isdominated by the polar Kerr effect and S z(B). Furthermore, the probe photon energy (1.55 eV) that we chose gives a MOKEangle of 4 mrad at 5 K. This value is very close to the maximumMOKE angle quoted in the literature and few times largerthan the typical MLD angles observed in (Ga,Mn)As samples.To understand why the polar Kerr effect dominates overMLD in our experimental set-up, we recall that two differentgeometries are used to measure magneto-optical signals:(i) probe linear polarization along [100], almost parallel tothe ground-state magnetization. This is the case here and, asdiscussed, e.g., in Ref. [ 68], only minimal MLD is expected. (ii) The probe linear polarization is close to the [110] directionas in Ref. [ 68]. In this case, one measures a mixed signal with both MLD and polar MOKE contributions [ 67]. While MLD dominates in (Ga,Mn)As when the probe is polarized alongthe [110] or [1 −10] directions [ 67,68], i.e., at ∼45 ◦degrees with respect to the easy axis, our data here was obtained forprobe polarization along [100] or [010]. Unlike previous experiments that measured the dynamics of (III,Mn)V ferromagnets on a ps time scale, Fig. 4shows directly the ∼100 fs temporal profile of the pump-probe 195203-7P. C. LINGOS, J. W ANG, AND I. E. PERAKIS PHYSICAL REVIEW B 91, 195203 (2015) magneto-optical signal as a function of perpendicular Bfield. The pump optical field, with amplitude E0∼2×105V/cm and fluence ∼7μJ/cm2, excites a total photohole density of n∼6×1018cm−3, a small perturbation of the 3 ×1020cm−3 ground-state hole density in our (Ga,Mn)As sample. As seen in Fig. 3, our experimental setup measures the transverse magnetization component /Delta1Sz(t), which is perpendicular to the ground-state magnetization. During fs time scales, Fig. 4 shows a systematic B-field dependence and sign-switching of/Delta1θKthat is absent in θKwithout pump. This behavior correlates with the magnetic switchings observed in the statictransverse Hall magnetoresistivity and demonstrates that thepump-induced out-of-plane magnetization component /Delta1S z(t) switches direction when the in-plane magnetic state switches.Furthermore, the steplike temporal profile of /Delta1θ K/θK≈ /Delta1Sz/Sindicates that such spin reorientation completes during the laser pulse and is therefore driven by e-hphotoexcitation. This fs time dependence is clearly distinguished from subse-quent magnon oscillations during ∼100 ps times [ 21]. We now relate our theory to the observed dependence of /Delta1θ K/θKwith∼100-fs duration on the transverse magnetic fieldBof Fig. 4(a).F o rB=0, the magnetic states X± 0and Y± 0lie inside the plane [Fig. 1(b)]. ForB/negationslash=0, Eq. ( A11)g i v e s an out-of-plane canting of X±andY±easy axes [Fig. 4(a)]. The measured smooth change of static Kerr rotation angle θK as function of Bfield reflects such canting without magnetic hysteresis. As shown by our calculation below, while Szvaries smoothly with increasing or decreasing Bfield, when the magnetization switches between X±andY±the direction of pump-induced fs component /Delta1Szreverses. The above depen- dence of pump-induced magnetization reversal on the easy axiscannot be explained by conventional nonlinear optical effectsor magnetization amplitude longitudinal changes [ 8,27–29]. When the latter dominate, X +(X−)g i v et h e same/Delta1Szas Y+(Y−), as the two in-plane magnetic states are equivalent (symmetric) with respect to the probe propagation directionperpendicular to the X-Yplane. Figure 4(d) (B=0) and Figs. 4(c) and4(e) (B=± 0.2T) clearly show that this is not the case in the experiment. In sharp contrast, for B=± 1T , Figs. 4(b) and4(f)show the same fs changes for both increasing and decreasing B. The fs response is independent of the easy axis for large B, which aligns the magnetization along [001]. Our calculations show that the fs magnetization reorientationdue to fs spin-orbit torque diminishes with increasing perpen-dicular B, consistent with the above behavior. ForB=0, Fig. 4(d) reveals a symmetric and opposite out-of-plane fs canting /Delta1S z(t) between the X0andY0initial states. In this case, the initial magnetization S0lies inside the sample plane [Fig. 4(a)] and thus the observed /Delta1Sz(t) cannot be associated with an amplitude change, as it occurs ina direction [001] perpendicular to S 0.F o rl a r g e B, on the other hand, the magnetization aligns with the Bfield along [001], Sz≈S, and thus /Delta1Sz(t) primarily reflects longitudinal fs changes in magnetization amplitude [ 28,41]. When Sz≈0, as forB=0,/Delta1Sz(t) reflects transverse changes in magnetization direction. We conclude that the observation of opposite signof laser-induced fs /Delta1S z(t) between the X± 0andY± 0states [Fig. 4(d)] can only arise from fs magnetization rotation towards opposite out-of-plane directions. Except for this signdifference, the fs temporal profiles of /Delta1S z/Sin Fig. 4(d) are-0.5 0 0.5 1 t (ps)-0.004-0.00200.0020.004ΔSz/|S| ωp=3.02eV ωp=3.14eV -0.5 0 0.5 1 t (ps)-0.3-0.2-0.100.1βΔs[110] (Tesla) ωp=3.02eV ωp=3.14eV -0.5 0 0.5 1 t (ps)012341018 (carriers/cm3) LH- LH+ HH- HH+ -0.5 0 0.5 1 t (ps)0123451018 (carriers/cm3) LH- LH+ HH- HH+ ωp=3.02eV ωp=3.14eV(a) (b) (c) (d) FIG. 5. (Color online) Frequency dependence of local and mo- bile spin dynamics and photohole populations following excitation by a 100-fs linearly-polarized laser pulse with low pump fluence∼10μJ/cm 2, with initial magnetization along the X+easy axis. (a) Comparison of “sudden” out-of-plane magnetization for /planckover2pi1ωp= 3.14 eV (LH optical transitions) and /planckover2pi1ωp=3.02 eV (HH optical transitions). (b) Comparison of nonadiabatic photoexcited hole spin component along [110] for the two above frequencies. (c) Photoex- cited nonthermal hole total populations of the four exchange-split HHand LH bands for /planckover2pi1ω p=3.02 eV . (d) Same as (c) for /planckover2pi1ωp=3.14 eV . symmetric between X0andY0. This symmetry implies that the out-of-plane /Delta1Szis driven by a laser-induced anisotropy field pulse that points close to the diagonal direction betweenX 0andY0. The steplike temporal profile implies that this field has∼100 fs duration. The above experimental observations are consistent with the direction and duration of the calculated/Delta1s h, shown in Fig. 2for anisotropy parameter /Delta1pd//Delta1 so∼1/3 as in (Ga,Mn)As. Such carrier-spin-pulse, discussed furtherbelow, exerts a fs spin-torque ∝/Delta1s h×S0, whose out-of-plane direction changes sign for S0alongX0orY0, while its magni- tude remains the same. Note here that, although laser-inducedthermal effects due to spin-lattice coupling can also changethe equilibrium easy axis, such changes occur gradually intime, over many picoseconds [ 33,37]. In contrast, here we observe steplike magnetization changes that follow the 100fslaser pulse and are consistent with our predicted fs spin-orbittorque. Note also that the experiment may show, in additionto the predicted magnetic contribution, “coherent antifacts”that appear, e.g., as a small “overshoot” at the very beginingof Fig. 4(d). Such details do not change our conclusion about laser-induced spin canting during the 100-fs pulse. To compare our theory to Fig. 4, we first consider B=0 and show in Fig. 5the calculated spin and charge dynamics for a single linearly-polarized 100-fs pump laser pulse with electricfield amplitude E 0=2×105V/cm similar to the experiment. We compare the spin and charge population dynamics for twodifferent laser frequencies, /planckover2pi1ω p=3.02 and 3.14 eV , tuned to excite different HH and LH band continua. In Fig. 5(a),w e show the development in time of the optically induced out-of-plane local spin component /Delta1S z(t). The calculated steplike fs temporal profile and magnitude for T1=100 fs agrees with Fig. 4. Furthermore, we observe a reversal in the direction of/Delta1Szwhen tuning the photoexcitation frequency. The fs 195203-8MANIPULATING FEMTOSECOND SPIN-ORBIT TORQUES . . . PHYSICAL REVIEW B 91, 195203 (2015) -0,4 -0,2 0 0,2 0,4 Bz (Tesla )-4-2024ΔSz/|S| (10-3) X+ Y+ X- Y- 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 ωp (eV)-1-0.500.51ΔSz/|S| (10-4) 2.7 2.8 2.9 3.0 3.1 3.2 3.3 ωp (eV)-8-6-4-202468ΔSz/|S| (10-3)all HH + HH - LH + LH -(a)(b) (c) ωp=3.14eV t=500fst=1ps t=1ps FIG. 6. (Color online) Calculated fs magnetic hysteresis and frequency dependence of the laser-induced magnetization canting /Delta1Sz/S due to fs spin-orbit torque. (a) The direction of out-of-plane component /Delta1Sz/Satt=500 fs depends on easy axis and magnetic field. This fs magnetic hysteresis diminishes with increasing perpendicular Bfield, which suppresses laser-induced magnetization reorientation, and separates “transverse” from “longitudinal” contributions to spin dynamics. (b) and (c) Frequency dependence of the laser-induced /Delta1Sz/S and its individual contributions from the four exchange-split HH and LH bands, calculated at t=1p sf o r E0=2×105V/cm. We compare between /planckover2pi1ωp∼1.5( b )a n d ∼3 eV (c). Spin-canting at the former frequency is smaller by factor of 10 due to the differences in band structure. The band continua significantly affect the frequency dependence of /Delta1Sz(t) as compared to discrete- kspecial point calculations. spin-orbit torque leading to such /Delta1Sz(t) is exerted by the photohole spin-pulse /Delta1sh(t), whose component along the diagonal [110] direction is shown in Fig. 5(b) for the two above frequencies. The magnitude, direction, and temporalprofile of both local and mobile spin components shownin Figs. 5(a) and 5(b) are consistent with the experimental results of Fig. 4(d). Important for controlling the four-state magnetic memory is that we are able to reverse the directionof the out-of-plane magnetization tilt /Delta1S z,F i g . 5(a), and photoexcited hole spin-pulse, Fig. 5(b), by exciting e-HH (/planckover2pi1ωp=3.02 eV) or e-LH ( /planckover2pi1ωp=3.14 eV) optical transitions. The origin of this spin-reversal can be seen by comparing the total populations1 V/summationtext k/Delta1/angbracketleftˆh† −knˆh−kn/angbracketrightfor the four different exchange-split HH and LH valence bands nin all {111}k directions. These band-resolved total populations are shown inFigs. 5(c) and5(d) as function of time for T 1=100 fs, which is comparable to the measured [ 21] and calculated [ 51] hole spin relaxation time. More than one bands are populated simulta-neously due to the energy dispersion and laser-pulse-width.With frequency tuning, we control a short-lived imbalancebetween these exchange-split bands with different spin-orbitcouplings and spin admixtures. In this way, we coherentlycontrol the superposition of spin- ↑and spin- ↓states prior to spin relaxation, here mostly during the 100 fs pulse. The order of magnitude of the photocarrier densities calculated by including the band continua along all eight{111}kdirections using the GaAs tight-binding parameters of Ref. [ 52] (Appendix B) agrees with the experimentally measured density, n∼6×10 18/cm3, for the same pump fluence. For such photohole populations, we also obtain /Delta1Sz/S with the same order of magnitude and direction as in the exper-iment [compare Figs. 5(a) and4(d)]. The calculated ∼250 mT component of β/Delta1s h(t) along [110], Fig. 5(b), agrees with the 100-fs magnetic anisotropy field extracted from Fig. 4(d) and is larger than typical fields obtained from calculationsthat assume a nonequilibrium Fermi-Dirac distribution [ 37]. This theory-experiment agreement indicates that nonthermalpopulations with lifetimes T 1=100 fs comparable to the hole spin lifetimes in (Ga,Mn)As [ 21,51] can explain the observed impulsive /Delta1Sz(t). Further evidence in support of our proposed fs spin-orbit torque mechanism is obtained from the pump-induced fsmagnetic hysteresis observed in the experiment of Fig. 4.I n Fig. 6(a), we compare the out-of-plane spin canting /Delta1Sz/S calculated at t=500 fs, as function of Bfield pointing along the perpendicular [001] direction for the four B-dependent equilibrium magnetic states X±andY±. Figure 6(a) shows that switching between the XandYinitial magnetic states switches the sign of pump-induced /Delta1Sz(t) (fs magnetic hysteresis). Furthermore, Fig. 6(a) shows that fs magnetization reorien- tation diminishes with increasing B. The above results are consistent with Fig. 4and explain the observed coincidence of /Delta1Szswitchings with static planar Hall effect switchings [ 36], as well as the absence of fs hysteresis at high B.W h i l e nonlinear effects such as dichroic bleaching also contribute tothe fs magneto-optical signal, the observed systematic B-field dependence and magnetic hysteresis in the sign of /Delta1θ K/θK indicate a nonadiabatic physical origin that is consistent with our calculations of fs spin-orbit torque. For high Bfields, the magneto-optical signal comes only from longitudinal changes in the magnetization amplitude [ 29] and from nonlinear optical effects [Figs. 4(b) and 4(f)]. The mean-field density matrix factorization used here does notcapture magnetization amplitude changes, which appear atthe level of electron-magnon spatial correlations [ 40,41]. As discussed in Ref. [ 28], any photoinduced imbalance of spin-↑and spin- ↓states will lead to fs demagnetization and inverse Overhauser effect, which, however, is independent ofeasy axis direction. While such imbalance may arise fromphotoinduced changes in the Fermi-Dirac temperature andchemical potential, a large electronic temperature increaseis required to produce the broad distributions implied bythe magnitude of the experimentally observed effects [ 28]. The broad nonthermal populations photoexcited here createa fs charge imbalance that, for T 1/lessorequalslant100 fs, follows the laser pulse and also contributes to demagnetization. Both “longi-tudinal” (demagnetization) and “transverse” (reorientation) fsspin dynamics arise from the competition of spin-orbit andmagnetic-exchange interactions described here. However, theymanifest themselves differently for different photoexcitationconditions and external magnetic fields. For example, fsdemagnetization (decrease in Mn spin amplitude) throughdynamical polarization of longitudinal hole spins dominatesfor high fluences of 100s of μJ/cm 2[21]. By using pulse 195203-9P. C. LINGOS, J. W ANG, AND I. E. PERAKIS PHYSICAL REVIEW B 91, 195203 (2015) trains, we may achieve spin rotational switching with lower pump intensities, which reduces the fs demagnetization. As already shown in Fig. 5, by coherently controlling the nonthermal population imbalance between the four exchange-split HH and LH bands, we can control the direction ofout-of-plane /Delta1S z/S. This is seen more clearly in Figs. 6(b) and 6(c), which show the frequency-dependence of /Delta1Sz/S and compare its individual contributions obtained by retainingone valence band at a time. The nonequilibrium population ofband states with different spin admixtures leads to differentdirections of laser-induced spin-canting /Delta1S z(t), which allows for magnetization control via pump frequency tuning. Forexample, photoholes excited in the two exchange-split (HHor LH) valence bands may induce opposite out-of-plane tilts.The finite pulse-duration and nonparabolic band dispersion[Appendix Band Fig. 1(a)] lead to different populations of more than one bands and BZ directions at all frequencies. Asalready discussed, such populations and spin-orbit interactionsdiffer between /planckover2pi1ω p∼1.5 and∼3 eV due to the difference in band structure. As seen by comparing Figs. 6(b) and 6(c), the band structure close to the Fermi level, where all {100}, {110}, and {111}symmetry directions are populated, leads to order of magnitude smaller /Delta1Sz/Sas compared to the high- k bands along {111}excited for /planckover2pi1ωp∼3 eV . This theoretical result is consistent with the difference in order of magnitudeof photoexcited spin and populations observed experimentallybetween the two above frequencies [ 36,37]. We conclude that optical control of the photoexcited carrier populationscan be used to switch the directions of photoexcited fsspin-orbit torques and, in this way, control the direction offs magnetization canting at different laser frequencies. The precise magnitude of the proposed effects depends on the relaxation time scales. The nonthermal populations are cre-ated during the 100-fs laser pulse via e-hoptical polarization. Following dephasing after T 2, these photocarriers relax on a time scale T1. The above characteristic relaxation times are expected to be in the 10–200-fs range in (Ga,Mn)As [ 21,51]. For pump fluences of ∼10μJ/cm2, the experiment gives /Delta1Sz/S∼0.5%, reproduced by our theory for T1=100 fs andT2=50 fs. This spin tilt decreases to /Delta1Sz/S∼0.01% as T2decreases to 3 fs with fixed T1=100 fs. For fixed short T2=10 fs, /Delta1Sz/Svaries between 0.05%–0.1% as T1varies between 30 and 100 fs. In all cases, we conclude that the fsspin-orbit torque contribution has the same order of magnitudeas the experimental results unless T 1andT2are few fs or less. From now on we fix T1=100 fs and T2=50 fs. The nonthermal fs spin-orbit torque contribution can be en- hanced by increasing the laser intensity. Figure 7(a) shows that, for easily attainable ∼100μJ/cm2low pump fluences [ 37], the “sudden” magnetization tilt increases to /Delta1Sz/S∼4% (for E0=7×105V/cm). Figure 7(b) then shows that β/Delta1sh(t) along [110] grows into the Tesla range. The precise magnitudeof this fs magnetization canting is sample-dependent anddepends on relaxation. The different intensity dependenceand temporal profiles of the thermal and coherent/nonthermalcarrier-spin components distinguishes these two contributionsto the photoexcited spin. While the quasiequilibrium contribu-tionH FSis limited by the chemical potential, /Delta1shis controlled by the laser frequency. A distinct impulsive component of fastmagnetic anisotropy was observed in the ps magnetization0123 t (ps)-0.04-0.0200.020.04ΔSz/|S|ωp=3.14eV (SGS//X+) ωp=3.02eV (SGS//X+) ωp=3.14eV (SGS//Y+) -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 t (ps)-2-1.5-1-0.500.5βΔ[110] (Tesla )ωp=3.02eV ωp=3.14eV SGS//X+(a) (b) FIG. 7. (Color online) Calculated fs spin dynamics similar to Fig. 5but with order of magnitude higher pump fluence ∼100μJ/cm2. (a) Comparison of out-of-plane magnetization com- ponents for two different initial magnetic states and ωp. (b) Photohole fs anisotropy fields along [110] for the two ωp. trajectory measured in Ref. [ 37] for pump fluences above ∼70μJ/cm2at/planckover2pi1ωp∼1.5 eV. Figure 7(a) also compares the spin canting dynamics for initial magnetization along theX + 0orY+ 0easy axis for B=0. Similar to the experiment of Fig. 4(d), it displays symmetric temporal profiles of /Delta1Sz(t), with opposite signs for the two perpendicular easy axes. In thisway, we can distinguish the two magnetic states within 100 fs.The equal magnitude of /Delta1S zbetween the two perpendicular in-plane easy axes arises from the diagonal direction of /Delta1sh for/Delta1pd//Delta1 so∼1/3 as in (Ga,Mn)As (Fig. 2). The overall agreement between theory and experiment suggest that amagnetic state can be read within 100 fs, by monitoring thedirection of out-of-plane laser-induced magnetization canting. The above theory-experiment comparison of fs magnetism and the connection of Fig. 6(b) to other ps-resolved magneto- optical experiments [ 37] make a case that optical control of a short-lived coherent population imbalance between exchange-split, spin-orbit-coupled anisotropic bands can generate fsspin-orbit torque with controllable direction, temporal profile,and magnitude. The latter initiates “sudden” magnetizationdynamics. This result is not specific to the (Ga.Mn)As four-state magnetic memory but may also apply to other magneticmaterials with strong spin-orbit coupling [ 9,10] and uniaxial magnetic anisotropy. In (III,Mn)V ferromagnets, we are notaware of any experiment so far showing nonthermal 360 ◦ switchings between multiple magnetic states induced by asingle laser pulse. This may be due to the fact that a 100-fslaser pulse not only excites magnon oscillations around theequilibrium easy axis but, even for low ∼10μJ/cm 2fluences, also induces undesired fs electronic heating of spins [ 28,29]. A complete quenching of ferromagnetism in (III,Mn)Vs has beenreported for pump fluences on the mJ/cm 2range [ 8,29]. Our calculations show that, with a single 100fs pulse, similarlyhigh fluences are required to induce a sufficiently strong“initial condition” /Delta1S(t) that achieves switching to a different magnetic state. Below we show that, alternatively, pulseshaping [ 23] can be used to initiate switching in a more controlled way, while keeping the peak laser fluence per pulse as low as possible to reduce fs electronic heating. In this way,we may maximize the “transverse” hole spin excitations whilereducing the “longitudinal” demagnetization by keeping thepump fluence per pulse in the 10–100 μJ/cm 2range. A more general message conveyed by our theory- experiment results is that laser-driven dipolar coupling me-diated by spin-orbit fluctuations in pd-coupled ferromagnetic 195203-10MANIPULATING FEMTOSECOND SPIN-ORBIT TORQUES . . . PHYSICAL REVIEW B 91, 195203 (2015) ground states favors local spin canting during fs optical exci- tation. Interestingly, in strongly correlated electron materialssuch as colossal magnetoresistive manganites, laser-drivendipolar bonding mediated by quantum spin-flip fluctuationswas shown to induce local spin canting in an antiferromagneticground state [ 1,4]. This quantum spin canting was shown to drive a magnetic phase transition during <100 fs laser pulses [ 1]. Such quantum femtosecond magnetism originates from transient modification of the interatomic hopping ofvalence electrons by the laser Efield, which nonadiabatically generates spin-exchange coupling and ferromagnetic correla-tion as photoelectrons hop while simultaneously flipping localspins. Such results point to a more universal behavior: laser-induced dipolar coupling mediated by spin-dependent valencefluctuations favors spin-symmetry-breaking even during thehighly nonequilibrium and nonthermal femtosecond timescales. IV . INITIATING DETERMINISTIC SWITCHINGS WITH A LASER-PULSE TRAIN Results so far imply that a single 100-fs laser pulse with ∼10–100 μJ/cm2fluence excites magnon oscillations around the equilibrium easy axis. Switching of the magnetization to adifferent magnetic state requires photoexcitation of a stronger“initial condition” /Delta1S(t). While switching via thermally assisted processes may be possible by increasing the fluenceto the mJ/cm 2range [ 38], pulse shaping [ 23] can initiate switching in a more controlled way while keeping the laserfluence per pulse in the μJ/cm 2range to reduce fs electronic heating of spins. Here, we coherently control /Delta1sh(t)b y usingMtime-delayed laser pulse-trains, each consisting of N Gaussian pulses with duration τp=100 fs. The optical field is E(t)=M/summationdisplay j=1E0N/summationdisplay i=1exp/bracketleftbig −(t−τj−/Delta1τij)2/τ2 p/bracketrightbig ×exp/bracketleftbig −iω(j) p(t−τj−/Delta1τij)/bracketrightbig . (14) Here, we tune τj, the time delay of the jth laser-pulse-train, and ω(j) p, the pulse-train central frequency, but fix /Delta1τij=500 fs for simplicity. In this section we consider M=1 and control the net duration of the spin-orbit torque with a single trainofNlaser pulses. In Fig. 8, we compare the components of β/Delta1s h(t) and γ/Delta1HFSobtained for N=8 in the coordinate system defined by the [110], [1 −10], and [001] directions. We use the same ∼100μJ/cm2fluence as in Fig. 7.T h e nonthermal contribution β/Delta1sh(t) prevails over the thermal contribution /Delta1HFS(t), which builds-up as /Delta1shdrives /Delta1S(t) and forces the spin of the Fermi sea bath to adjust to the newdirection of S(t)[17]. This /Delta1S(t) builds-up in a step-by-step fashion well before relaxation, driven by a sequence ofsuccessive photoexcited fs spin-orbit torques. /Delta1H FS(t) originates from the spin of the thermal hole Fermi sea and is therefore restricted by the Fermi-Dirac distribution.The latter thermal populations give anisotropy fields of theorder of few 10’s of mT in (Ga,Mn)As [ 6,37], as they are restricted by the equilibrium anisotropy parameters and∼μeV free energy differences with S. On the other hand, the experiments observe anisotropy fields that are at least01234 56 t (ps)-2-1.5-1-0.50TeslaβΔs[1-10] βΔs[110] βΔs[001] ΔHFS[1-10] ΔHFS[110] ΔHFS[001] FIG. 8. (Color online) Comparison of nonthermal and quasither- mal components of laser-induced magnetic anisotropy fields β/Delta1sh(t) and/Delta1HFS(t) during coherent nonlinear photoexcitation with a train ofN=8 100-fs laser pulses separated by 500 fs, with E0= 7×105V/cm and /planckover2pi1ωp=3.14 eV . one order of magnitude larger [ 36,37]. Figure 8compares the thermal anisotropy field /Delta1HFS(t) to the nonthermal photohole contribution β/Delta1sh(t) obtained at /planckover2pi1ωp∼3.1e V for the ∼100μJ/cm2pump fluence used in Ref. [ 37]. This nonthermal photohole spin was calculated in the time-domainby solving density matrix equations of motion after takinginto account the (Ga,Mn)As band structure at 3.1 eV . For/planckover2pi1ω p∼1.5e V,a similar calculation shown in Fig. 6(b) gives smaller photoexcited spin due to the different band structureand populated BZ directions close to the Fermi level. In ourcalculations, the photoexcited populations are not restricted bythe Fermi-Dirac distribution. By tuning the laser frequency,the photocarriers can populate nonparabolic anisotropic partsof the BZ that cannot be accessed close to quasiequilibrium.Our quantum kinetic calculation far from equilibrium givesmore flexibility as compared to assuming quasiequilibriumchanges in the temperature and chemical potential, which areonly established after a short but finite time T 1. As seen in Fig. 8,β/Delta1sh(t) can grow to ∼2 T along [110] for experimen- tally relevant pump fluences and T1∼100 fs. For such fast photocarrier relaxation, /Delta1sh(t) follows the laser-pulse-train temporal profile and the relative phase of consequative pulsesdoes not play a role. However, /Delta1s h(t) is not the same for different pulses, as the nonequilibrium electronic states changenonadiabatically with /Delta1S(t) (Appendix A). We now show that, by increasing N, we can initiate switching rotation to any one of the available magneticstates. Figure 9shows three such magnetization switching trajectories up to long times t=800 ps. These ps trajectories are initiated at t=0b yN=7 [Fig. 9(a)],N=9 [Fig. 9(b)], orN=12 [Fig. 9(c)] pulses with ∼100μJ/cm 2fluence. By increasing N, we can switch from X+to all three other magnetic states Y+,X−, andY−.I nF i g . 9(a),N=7 pulses with /planckover2pi1ωp=3.02 eV (HH photoexcitation) initiate a counterclockwise 90◦switching rotation that stops after reaching the next magnetic state, Y+, within ∼80 ps. The magnetization oscillates around the final state with a significantamplitude that cannot be controlled with a single pulse-train(magnetic ringing) [ 47]. This ringing results from the weak (nanosecond) Gilbert damping of the local-spin precessionobserved in annealed (Ga,Mn)As [ 33,37]. While magnetic ringing can make multiple 90 ◦switchings unstable, below 195203-11P. C. LINGOS, J. W ANG, AND I. E. PERAKIS PHYSICAL REVIEW B 91, 195203 (2015) FIG. 9. (Color online) Magnetization switching trajectories from X+to the other three magnetic states, controlled by tuning the frequency ωpand triggered by a single laser-pulse train with increasing number of pulses NandE0=7×105V/cm. All switchings are followed by pronounced magnetic ringing. (a) Counterclockwise 90◦switching X+→Y+, initiated by HH photoexcitation with N=7 pulses. (b) 180◦ magnetization reversal via clockwise pathway X+→Y−→X−, initiated by LH photoexcitation with N=9 pulses. (c) Photoexcitation as in (a), but with N=12 pulses. By increasing N, the magnetization moves past the Y+andX−intermediate states and accesses the Y−state via the 270◦counterclockwise pathway X+→Y+→X−→Y−. we show that we can suppress it by exerting opposing fs spin-orbit torques. By increasing the number of pulses toN=9, the magnetization continues past Y +to the next available state, X−. Figure 9(b) then shows magnetization reversal via clockwise instead of counterclockwise rotation,since /planckover2pi1ω p=3.14 eV excites e-LH instead of e-HH optical transitions. This X+→Y−→X−pathway completes within ∼150 ps and is again followed by magnetic ringing. By increasing the number of pulses to N=12, the fs spin- orbit torque is sufficient to move the magnetization evenbeyond X −. Figure 9(c) shows 270◦switching to the Y− state within ∼200 ps, following a X+→Y+→X−→Y− pathway initiated by e-HH photoexcitation. V . OPTICAL CONTROL OF SEQUENTIAL 90◦ SWITCHINGS BETWEEN FOUR STATES In this section, we provide an example of how our proposed optical manipulation of fs spin-orbit torques could be used togain full access of a four-state magnetic memory. Figure 10 shows two switching protocols that achieve 360 ◦control of the magnetic states of Fig. 1(b). The upper panel shows the sequences of laser-pulse-trains used to control the foursequential 90 ◦switchings. Two different laser frequencies excite e-HH or e-LH optical transitions, which allow us to stop and restart the magnetization motion at each of the fourmagnetic states as desired. By tuning the laser frequency wechoose the direction of fs spin-orbit torques and multistepswitching process, which takes place via counterclockwise[Fig. 10(a) ] or clockwise [Fig. 10(b) ] magnetization rotations forced to stop at all intermediate states at will. To control thephotoexcited /Delta1s h(t) and fs spin-orbit torques, we turn three experimentally accessible “knobs.” (i) Pulse shaping [23]b y changing N, which controls the net duration and temporal profile of the spin-orbit torques. In this way, we tailor /Delta1S(t) that initiates or modifies the switching rotations with low in-tensity per laser pulse. (ii) Frequency-tuning enables selective photoexcitation of exchange-split LH or HH nonequilibriumpopulations with different superpositions of spin- ↑and spin- ↓ states. In this way, we control the population imbalance thatdecides the directions of /Delta1s h, fs spin-orbit torque, and /Delta1S(t). (iii) By controlling the time delays τj, we exert fs spin-orbit torques at desirable times in order to stop and restart theswitching rotation at all intermediate states and suppress magnetic ringing. This is discussed further in the next section.To understand the role of the twelve laser-pulse trains chosenin Fig. 10, we note the following points: (i) a laser-pulse train initiates switchings or magnon oscillations via fs spin-orbittorque with direction that depends on both laser frequency and magnetic state, (ii) when the magnetization reaches a newmagnetic state, we use a laser-pulse-train to exert opposingfs spin-orbit torques, in a direction that stops the switchingrotation and suppresses the magnetic ringing so that we canaccess the state, and (iii) when we are ready to move on, alaser-pulse train with the appropriate color restarts the 360 ◦ switching process by exerting fs spin-orbit torques in thedesirable direction. Figure 10shows four sequential 90 ◦switchings controlled by/Delta1sh(t). In Fig. 10(a) , a counterclockwise X+→Y− switching is initiated by e-HH photoexcitations with N=12 pulses. After τ=35 ps, the magnetization reaches the vicinitylh hh X-0X+Sx X-0X+Sx Y-0Y+Sy Y-0Y+Sy 0 50 100 150 200 250 300 350 t (ps)Z-0Z+Sz 0 50 100 150 200 250 300 t (ps)Z-0Z+Szhh lh (a) (b) FIG. 10. (Color online) Two protocols for 360◦control of the full four-state magnetic memory via four sequential 90◦switchings that stop and restart at each intermediate magnetic state. (a) Counter- clockwise sequence X+→Y+→X−→Y−→X+, (b) Clockwise sequence X+→Y−→X−→Y+→X+. (Top) Timing sequences and colors of the laser-pulse trains ( N=12) that create the needed fs spin-orbit torque sequences. Blue pulses excite HH optical transitions, magenta pulses excite LH transitions. E0=7×105V/cm (pump fluence of ≈100μJ/cm2). 195203-12MANIPULATING FEMTOSECOND SPIN-ORBIT TORQUES . . . PHYSICAL REVIEW B 91, 195203 (2015) -6-4-20246ΔSz/|S| (10-2)ω1=3.14eV ω1=3.02eV -6-4-20246ΔSz/|S| (10-2)ω1=3.14eV ω1=3.02eV 0 100 200 300 t (ps)-6-4-20246ΔSz/|S| (10-2)ω1=3.14eV ω1=3.02eV 0 100 200 300 t (ps)-6-4-20246ΔSz/|S| (10-2)ω1=3.14eV ω1=3.02eVτ1=74ps τ1=148psτ1=74ps τ1=148ps(a) Sgs//X+(b) Sgs//Y+ FIG. 11. (Color online) Two 100-fs laser pulses, delayed by τ, enhance or suppress magnon oscillations via fs spin-orbit torque. The first pulse, /planckover2pi1ωp=3.14 eV , starts the precession (frequency /Omega1)a tτ=0. The second pulse, /planckover2pi1ωp=3.02 or 3 .14 eV, arrives at τ=74 ps ( /Omega1τ=π) orτ=148 ps ( /Omega1τ=2π). Equilibrium magnetic state: (a) X+and (b) Y+. of the intermediate Y+state. We then stop the switching process by exciting e-HH optical transitions. We restart the motion at τ=85 ps, after waiting for about 50 ps, by using e-LH photoexcitations to switch the magnetization to the X− state. There we again stop the process at τ=160 ps, by exciting e-LH optical transitions. We restart at τ=170 ps with e-HH photoexcitations, which trigger switching to Y−.T h i s switching completes within ∼35 ps, after we stop the motion withe-HH photoexcitations at τ=205 ps. We finish the 360◦ switching loop by using e-LH photoexcitations to restart the counterclockwise motion back to X+,a tτ=250 ps, and later to terminate the process at τ=330 ps. Figure 10(b) shows an opposite clockwise switching sequence X+→Y−→X−→ Y+→X+, obtained by changing the laser-pulse frequencies frome-HH to e-LH excitations and vice-versa. In this case, e-LH optical transitions with N=12 pulses trigger clockwise magnetization rotation, which we suppress at Y−with LH excitations at τ=75 ps. We restart the process with e-HH photoexcitation at τ=85 ps and suppress it again at X−with e-HH optical transitions at τ=120 ps. We restart with e-LH excitation at τ=140 ps and switch to Y+, where we suppress the motion at τ=225 ps with e-LH optical transitions. We complete a closed switching loop to the initial X+state with e-HH photoexcitation at τ=235 ps and suppress the rotation withe-HH optical transitions at τ=275 ps. In the next section, we analyze how tunable fs spin-orbit torque directionoffers more flexibility for controlling switching rotations andmagnetic ringing. VI. CONTROLLING MAGNETIC SWITCHING AND RINGING WITH A LASER-PULSE TRAIN While the optical control scheme via fs spin-orbit torque discussed in the previous section allows for elaborate switchingof a multistate magnetic memory, it may also apply to conven-tional memories exhibiting uniaxial magnetic anisotropy. Itsmain advantage, in addition to initiating selective switchings and flipping the spin between two states, is that it can suppressthe magnetization motion and magnetic ringing at any time,at any intermediate magnetic state. Magnetic ringing arisesfrom the weak damping of the magnetization precessionaround an easy axis following excitation with either opticalor magnetic field pulses and limits the read/write times inmany magnetic materials [ 47]. One known way to reduce it is to take advantage of the phase /Omega1τ of magnetization precession with frequency /Omega1[34,47]. With magnetic field pulses, this can be done by adjusting the duration of a longpulse to the precession period [ 47]. With ultrashort laser pulses, one can suppress (enhance) the precession by exciting when/Omega1τ=π(/Omega1τ=2π)i nt h es a m ew a ya sa t τ=0[34]. Such coherent control of spin precession is possible for harmonicoscillations. Below we show that we can optically controlboth magnon oscillations and nonlinear switching rotations byapplying clockwise or counterclockwise fs spin-orbit torquepulse sequences when needed. We start with the harmonic limit and demonstrate magnon control via fs spin-orbit torque with tunable direction. First,we excite at τ=0 magnon oscillations with frequency /Omega1 (thick solid line in Fig. 11). We thus initiate magnetization precession around the X +[Fig. 11(a) ]o rt h e Y+[Fig. 11(b) ] easy axis with e-LH excitation ( /planckover2pi1ωp=3.14 eV). An impulsive magnetization at τ=0 is observed in the ps trajectory of Fig. 11. The initial phase of these magnon oscillations is opposite between the X+ 0andY+ 0states, due to the opposite directions of the fs spin-orbit torques [Fig. 7(a)]. We then send a control laser pulse at τ=74 ps ( /Omega1τ=π)o ra t τ=148 ps (/Omega1τ=2π), but use either /planckover2pi1ωp=3.14 eV ( e-LH optical transitions) or /planckover2pi1ωp=3.02 eV ( e-HH optical transitions). By controlling the direction of fs spin-orbit torque with suchfrequency tuning, we show that we can both enhance andsuppress the amplitude of the magnetization precession at both /Omega1τ=πand/Omega1τ=2π. While for /Omega1τ=πwe suppress the 195203-13P. C. LINGOS, J. W ANG, AND I. E. PERAKIS PHYSICAL REVIEW B 91, 195203 (2015) X-0X+ X-0X+ X-0X+ X-0X+ Y-0Y+ Y-0Y+ Y-0Y+ Y-0Y+ 0 100 200 300 400 t (ps)Z-0Z+ 0 100 200 300 400 t (ps)Z-0Z+ 0 100 200 300 400 t (ps)Z-0Z+ 0 100 200 300 400 t (ps)Z-0Z+(a) (b) (c) (d) FIG. 12. (Color online) Time-dependence of magnetization components controlled by a time-delayed fs spin-orbit torque pulse train. (a) X+→Y+→X−→Y−switching pathway is initiated at τ=0 with HH photoexcitation (dashed line). After switching completes, the unavoidable magnetic ringing is reduced by a control laser-pulse-train that can exert opposing fs spin-orbit torques at any time (solid line). (b) The X+→Y−switching of (a) is terminated by opposing fs spin-orbit torques after magnetization reversal to X−.( c )T h e X+→Y− switching is terminated by a control laser-pulse-train after 90◦rotation to Y+.( d )T h e X+→Y−switching is stopped immediately after it is initiated, by opposing fs spin-orbit torque at τ=2p s . magnetic ringing when applying the same fs spin-orbit torque as for τ=0(/planckover2pi1ωp=3.14 eV), we can also enhance it by applying an opposite fs spin-orbit torque ( /planckover2pi1ωp=3.02 eV). Similarly, at time /Omega1τ=2π, we enhance the ringing when applying fs spin-orbit torque in the same direction as forτ=0 and suppress it by reversing the direction. We thus gain flexibility in both starting and stopping magnon oscillations. Unlike for harmonic precession, switching also involves nonlinearities and anharmonic effects. In Fig. 12(a) ,aX +→ Y+→X−→Y−switching pathway (dashed line) is initiated atτ=0a si nF i g . 9(c). After about 200 ps, the magnetization switches to Y−, after overcoming the intermediate states Y+ andX−.T h eXcomponent of the magnetization then oscillates with significant amplitude [magnetic ringing, see dashedcurve in Fig. 12(a) ]. Figure 12(a) (solid curve) demonstrates suppression of this ringing by a control laser-pulse-train thatcan arrive at any time after the switching is completed.To accomplish this, we tune the direction, duration, andstrength of the exerted fs spin-orbit torques. Figures 12(b) and 12(c) show that the control pulse-train can also stop the X +→Y+→X−→Y−switching at one of the intermediate magnetic states before reaching Y−. However, we must use different ωpatY+andX−in order to get an opposing fs spin-orbit torque, as the direction of the latter depends onthe magnetic state. In Fig. 12(b) , we stop the switching at theX −magnetic state, after passing through Y+, by exciting with /planckover2pi1ωp=3.14 eV at τ∼100 ps ( e-LH photoexcitation). Figure 12(c) shows that we can stop at Y+after∼35 ps, by exerting a clockwise spin-torque using /planckover2pi1ωp=3.02 eV (HH photoholes). A more dramatic demonstration of theflexibility offered by fs spin-orbit torque is given in Fig. 12(d) . Here, we initiate the X +→Y−switching as above and then stop it immediately, by applying a control laser-pulse trainatτ=2 ps, i.e., long before any oscillations can develop. Instead of relying on the precession phase as in Fig. 11,w e apply a sufficiently strong clockwise fs spin-orbit torque thatopposes the magnetization motion. In this way, we stop themagnetization at its tracks, after a minimal motion withoutoscillations. We conclude that coherent optical control of the mobile spin excited during fs laser pulses allows usto suppress both magnetic ringing and nonlinear switchingrotations, by controlling the direction, duration, and magnitudeof fs spin-orbit torques. VII. CONCLUSIONS AND OUTLOOK In this paper, we used density-matrix equations of motion with band structure to describe photoexcitation and frequency-dependent control of fs spin-orbit torques analogous tothe static current-induced ones in spintronics. In this all-optical way, we initiate, stop, and control multiple magneticswitchings and magnetic ringing. The proposed nonadiabaticmechanism involves optical control of direction, magnitude,and temporal profile of fs spin-orbit torque sequences. Thisis achieved by tuning, via the optical field, a short-livedcarrier population and spin imbalance between exchange-splitbands with different spin-orbit interactions. The photoexcitedspin magnitude and direction depend on symmetry-breakingarising from the nonperturbative competition of spin-orbitand spin-exchange couplings of coherent photoholes. Wevalidated our initial prediction of fs spin-orbit torque [ 3] by comparing our calculations to existing magneto-opticalpump-probe measurements monitoring the very early ∼100 fs temporal regime following excitation with a single linearly-polarized laser pulse. The most clear experimental signatureis the observation of laser-induced fs magnetic hysteresisand switching of the direction of out-of-plane femtosecondmagnetization component with magnetic state. Such magnetichysteresis is absent without pump, while static planar Halleffect measurements observe similar in-plane switchings inthe transverse component of the Hall magnetoresistivity.The observation of switching of laser-induced fs transversemagnetization with magnetic state cannot arise from longitu-dinal nonlinear optical effects and demagnetization/amplitudechanges. The dependence on magnetic state indeed disap-pears with increasing perpendicular magnetic field, which 195203-14MANIPULATING FEMTOSECOND SPIN-ORBIT TORQUES . . . PHYSICAL REVIEW B 91, 195203 (2015) suppresses the magnetization reorientation. In this way, we can separate experimentally longitudinal and transverse fem-tosecond magnetization changes. We discussed two theoreticalresults that may be useful for coherent control of magneticmemory states and magnetic ringing via fs spin-orbit torque:(i) we showed that femtosecond optical excitation can start,stop, and restart switching pathways between the adiabaticfree energy magnetic states in any direction. Based on this, wegave an example of sequences of laser-pulse trains that canprovide controlled access to four different magnetic states viaconsequative 90 ◦switchings, clockwise or counterclockwise. (ii) We demonstrated optical control of magnon oscillationsand switching rotations and suppression of magnetic ringingat any time, long or short. For this we enhance spin-orbit torquevia pulse-shaping and control its direction via laser frequency. The full nonthermal control of a magnetic memory demon- strated here requires the following: (i) The competitionbetween spin-orbit and magnetic exchange couplings breaksthe symmetry while the laser electric field couples to thematerial. As a result, e-hpair excitations are photoexcited with finite spin. There is no need to transfer angular-momentumfrom the photons (no circular polarization) since spin-orbitcoupling does not conserve spin. (ii) The direction, magnitude,and duration of the nonthermal carrier spin-pulse is coherentlycontrolled by the optical field. In particular, the direction ofphotoexcited spin is controlled by the laser frequency, themagnetic state, and the symmetry-breaking. Importantly, itsmagnitude increases with laser intensity and E 2, while its temporal profile follows that of the laser pulse if relaxationis sufficiently fast. Such characteristics of fs spin-orbit torquecan distinguish it from adiabatic free energy effects. (iii) Thephotoexcited spin-pulses exert fs spin-orbit-torques on thecollective local spin and move it “suddenly,” in a control-lable direction that depends on the magnetic state and thelaser frequency. By coherently controlling the nonthermalpopulation imbalance of exchange-split carrier bands withdifferent spin-orbit interactions, we can move the local spinvia nonadiabatic interaction with mobile spins. (iv) Laser-pulseshaping [ 23] and increased pump fluence allow us to access optically the magnetic nonlinearities of the carrier free energy.In this way, we may initiate or modify, during fs time scales,deterministic switchings to any available magnetic state.(v) By using control pulse-trains with appropriate frequencies,we suppress and restart switching rotations at intermediatemagnetic states and suppress magnetic ringing after switchingscomplete. While coherent suppression of magnon oscillationsis possible by taking advantage of the precession phase, herewe mainly rely on controlling the direction of fs spin-orbittorque with respect to the direction of magnetization rotation.In this way, we suppressed and enhanced both switchingrotations and ringing at long and short times. To control the entire four-state memory as in Fig. 10, we had to use time-delayed laser-pulse trains with different frequen-cies at different magnetic states. The first excitation suppressesthe switching rotation/ringing in order to access the state, whilethe second excitation restarts the process and moves the mag-netization to the next magnetic state in the desired direction.While such control of the magnetization trajectory occurs onthe 100-fs time scale of coherent photoexcitation, the initiateddeterministic switchings complete on ∼100-ps time scales, asdetermined by the free energy and micromagnetic parameters. In a massively-parallel memory, we can control ndifferent bits simultaneously on the 100fs time scale without waitingfor each switching to complete. For large n, this would ideally result in memory reading and writing at ∼10 THz speeds. Our proposed fs spin-orbit torque mechanism may be relevant to different unexplored spin-orbit coupled materialswith coexisting mobile and local carriers [ 11], for example, topological insulators doped with magnetic impurities [ 9,10]. Important for practical implementations and experimentalproof of fs spin-orbit torque is to identify materials wherethe quasithermal/adiabatic and nonthermal/nonadiabatic con-tributions to the magnetic anisotropy can be distinguishedexperimentally. It is possible to separate these two based ontheir temporal profiles and their dependence on photoexci-tation intensity, laser frequency, and external magnetic field.In (Ga,Mn)As, Fig. 4shows photogeneration of a “sudden” magnetization reorientation and fs magnetic hysteresis formagnetic field perpendicular to the sample plane. Suchmagnetic field cants the ground-state magnetization out of theplane, from S z=0(B=0) toSz≈±S(large B). When Sz≈ 0 in equilibrium, /Delta1Sz(t) measures transverse magnetization reorientation and magnetic hysteresis correlated with in-planeswitching, while when S z≈Slongitudinal changes dominate /Delta1Sz(t) and there is no hysteresis. In this way, a perpen- dicular magnetic field can be used to elucidate the physicalorigin of the fs magneto-optical pump-probe signal dynamics.Distinct thermal and nonthermal contributions to the psmagnetization trajectory were also observed experimentallyat/planckover2pi1ω p∼1.5e V[ 37]. They were separated based mainly on pump fluence dependence and by controlling the material’smicromagnetic parameters. Qualitative differences in themagnetization trajectory were observed above ∼70μJ/cm 2 pump fluence. Below this, the easy axis rotates smoothly inside the plane, due to laser-induced temperature increaseduring ∼10 ps time scales [ 33,37]. Above ∼70μJ/cm 2,a subpicosecond “sudden” magnetization component is clearlyobserved [ 33,37]. Importantly, while the precession frequency γH FSincreases linearly with equilibrium temperature, it saturates with pump fluence above ∼70μJ/cm2, even though the impulsive out-of-plane magnetization tilt continues toincrease [ 37]. In contrast, the pump-induced reflectivity increases linearly with pump intensity up to much higherfluences ∼150–200 μJ/cm 2[37], which indicates nonthermal photocarriers. Here we suggest that the numerical resultsof Fig. 6(b), which show frequency-dependent fs spin pho- toexcitation for /planckover2pi1ω p∼1.5e V,may explain the “sudden” out-of-plane magnetization canting observed in Ref. [ 37]. This requires ∼100μJ/cm2pump fluences consistent with our theory. Our results describe the initial condition that triggersrelaxation not treated here. In closing, we note that the discussed concepts are of more general applicability to condensed matter systems.The main idea is the possibility to tailor order parameterdynamics via optical coherent control of nonthermal carrierpopulations, as well as via charge fluctuations and interactionsdriven while the optical field couples to the material. Theinitial coherent excitation temporal regime may warrant moreattention in various condensed matter systems [ 1,4]. An analogy can be drawn to the well-known coherent control of 195203-15P. C. LINGOS, J. W ANG, AND I. E. PERAKIS PHYSICAL REVIEW B 91, 195203 (2015) femtosecond chemistry and photosynthetic dynamics, where the photoproducts of chemical and biochemical reactions canbe influenced by creating coherent superpositions of molecularstates [ 69]. Similarly, in condensed matter systems, laser- driven e-hpairs (optical polarization) can tailor nonadiabatic “initial conditions” that drive subsequent phase dynamicsgoverned by the adiabatic free energy. An analogy can alsobe drawn to parameter quenches studied in cold atomicgases. There, quasi-instantaneous quenches drive dynamicsthat, in some cases such as BCS superconductors, can bemapped to classical spin dynamics. Coherent dynamics ofsuperconducting order parameters are now beginning to bealso studied in condensed matter systems [ 70,71], and an analogy to the magnetic order parameter studied here is clear.Other examples include quantum femtosecond magnetismin strongly-correlated manganites [ 1,4], photon-dressed Flo- quet states in topological insulators [ 72], and the existence of nonequilibrium phases in charge-density-wave correlatedsystems [ 48]. Femtosecond nonlinear optical and THz spec- troscopy [ 73] offers the time resolution needed to disentangle different order parameters that are strongly coupled in theground state, based on their different dynamics after “sudden”departure from equilibium [ 48,49]. Multipulse switching protocols based on nonadiabatic quantum excitations cancontrol nonequilibrium phase transitions, by initiating phasedynamics in a controllable way [ 1,4]. Note added to proof. After our paper was submitted, we became aware of a recent preprint on time-resolvedmagneto-optical measurements of the collective magnetiza-tion ultrafast dynamics in (Ga,Mn)As [ 75]. This experiment observed a strong pump-frequency dependence of the mag-netization precession above the semiconductor band gap,which originates from the nonthermal holes photoexcitedin the semiconductor band states similar to our theoreticalpredictions here. The experimental results reveal a systematicbut complex sample-dependent frequency dependence, whichdiffers between annealed and as-grown samples. The observedeffect is consistent with our predictions in Fig. 6(b).F o r example, the quasithermal anisotropy effects predicted here(e.g., /Delta1H FSin Fig. 8) are mainly driven by the fs /Delta1Sz. The latter “sudden” magnetization drives a laser-inducedcontribution to the quasithermal magnetic anisotropy fieldEq. ( A10) determining the precession frequency (especially for in-plane initial magnetization S z≈0, as for small Bfields). While the present theory neglects any laser-induced changes inthe magnetic anisotropy parameters that characterize the freeenergy E h(S), which add to our predicted effects, it suggests that the frequency-dependent initial femtosecond change /Delta1Sz may be important for explaining the frequency dependence of the precession frequency determined by Eq. ( A10). Note that the decay of /Delta1Sphotoinduced during femtosecond time scales due to magnetic exchange interaction with the nonthermalphotohole spin is determined by the sample-dependent Gilbertdamping. The latter differs markedly between annealed andas-grown samples [ 33]. ACKNOWLEDGMENTS This work was supported by the European Union Seventh Framework Programme (FP7-REGPOT-2012-2013-1) underGrant Agreement No. 316165, by the European Union Social Fund and National resources through the THALES programNANOPHOS, by the Greek GSRT project ERC02-EXEL(Contract No. 6260), by the Greek Ministry of EducationARISTEIA-APPOLO, and by the National Science Founda-tion Contract No. DMR-1055352. APPENDIX A: FERMI-DIRAC/ADIABATIC VERSUS NONTHERMAL/NONADIABATIC MAGNETIC ANISOTROPY In this Appendix, we discuss the two contributions to laser- induced anisotropy: nonthermal and quasithermal. The adi-abatic/quasithermal contribution comes from relaxed Fermi-Dirac carriers. The nonadiabatic contribution comes from thecoherent/nonthermal photoexcited carriers, whose populationsincrease with intensity during photoexcitation. In the initialstage, these nonthermal carriers come from the continuum ofe-hexcitations excited by the fs laser pulse, so they follow its temporal profile. At a second stage, they redistribute amongthe different kand band states while also scattering with the Fermi sea carriers. 1. Nonthermal/nonadiabatic magnetic anisotropy We use density matrix equations of motion and band struc- ture to describe the femtosecond photoexcitation of short-livedphotohole spin pulses driven by four competing effects: (i)magnetic exchange interaction between local and mobile spins, (ii) spin-orbit coupling of the mobile carriers, (iii) coherent nonlinear optical processes, and (iv) fast carrier relaxation. Theinterplay of these contributions breaks the symmetry and ex-cites a controllable fs magnetic anisotropy field due to nonther-mal photocarriers. The photoexcited spin, Eq. ( 3), is expressed in terms of the electronic density matrix, which resolves thedifferent band and k-direction contributions. Density matrix equations of motion were derived for the time-dependentHamiltonian H(t), Eq. ( 8), with band structure treated within standard tight-binding and mean-field approximations. ThisHamiltonian has fast and slow contributions. Its adiabaticpartH b(S0), Eq. ( 4), depends on the slowly varying (ps) spinS0. The eigenstates of Hb(S0) describe electronic bands determined by periodic potential, spin-orbit, and adiabaticmagnetic exchange coupling. The latter interaction H pd(S0)=βcS0·ˆsh, (A1) where ˆshis the hole spin operator, leads to exchange-splitting of the HH and LH semiconductor valence bands determinedby the exchange energy /Delta1 pd=βcS. It also modifies the direction of photoexcited spin, by competing with thespin-orbit coupling of the mobile carriers characterizedby the energy splitting /Delta1 soof the spin-orbit-split valence band of the parent material (GaAs) at k=0. By adding to the Hamiltonian carrier-carrier and carrier-phonon interactions,we can also treat relaxation, included here by introducingthe nonthermal population relaxation time T 1and the e-h dephasing time T2. We describe the band eigenstates of the adiabatic electronic Hamiltonian Hb(S0) by using the semiempirical tight-binding model that reliably describes the GaAs band structure [ 52]. Compared to the standard k·peffective mass approximation, 195203-16MANIPULATING FEMTOSECOND SPIN-ORBIT TORQUES . . . PHYSICAL REVIEW B 91, 195203 (2015) this tight-binding approach allows us to also address states with large momenta k. Such anisotropic and nonparabolic band states contribute for laser frequencies away from the bandedge. Following Ref. [ 52], we include the quasiatomic spin- degenerate orbitals 3 s,3p x,3py,3pz, and 4 sof the two atoms per GaAs unit cell and use the tight-binding parameter valuesof the Slater-Koster sp 3s∗model. As in Ref. [ 3], we add to this description of the parent material the mean-field couplingof the Mn spin, Eq. ( A1), which modifies spin-mixing in a nonperturbative way. Similar to Ref. [ 52], we diagonalize the Hamiltonian H b=Hc b+Hv bto obtain the conduction ( Hc b) and valence ( Hv b) bands: Hb(S0)=/summationdisplay knεc knˆe† knˆekn+/summationdisplay knεv −knˆh† −knˆh−kn.(A2) The eigenvalues εc kn(S0) andεv −kn(S0) describe the conduction and valence-band energy dispersions. While S0varies on a ps time scale much slower than the laser-induced electronic fluctuations, the rapidly-varying(fs) part of the Hamiltonian H(t),/Delta1H exch(t)+HL(t), drives “sudden” deviations from adiabaticity. /Delta1H exch(t), Eq. ( 9), describes nonadiabatic interactions of photocarrier spins withthe fs magnetization /Delta1S(t) induced by fs spin-orbit torque. H L(t) describes the optical field dipole coupling within the rotating wave approximation: HL(t)=−/summationdisplay nmkdnmk(t)ˆe† kmˆh† −kn+H.c., (A3) where dnmk(t)=μnmkE(t) is the Rabi energy, E(t) is the pump electric field, and μnmkis the dipole transition matrix element between the valence band nand the conduction band mat momentum k. These dipole matrix elements also depend on S0and are expressed in terms of the tight-binding parameters ofHb(k)a si nR e f .[ 74]: μnmk=i εmk−εnk/angbracketleftnk|∇kHb(k)|mk/angbracketright. (A4) The density matrix /angbracketleftˆρ/angbracketrightobeys the equations of motion i/planckover2pi1∂/angbracketleftˆρ/angbracketright ∂t=/angbracketleft[ˆρ,H (t)]/angbracketright+i/planckover2pi1∂/angbracketleftˆρ/angbracketright ∂t|relax. (A5) The hole populations and coherences between valence bands are given by the equation of motion i/planckover2pi1∂t/angbracketleftˆh† −knˆh−kn/prime/angbracketright−/parenleftbig εv kn/prime−εv kn−i/Gamma1h nn/prime/parenrightbig /angbracketleftˆh† −knˆh−kn/prime/angbracketright =/summationdisplay md∗ mnk(t)/angbracketleftˆh−kn/primeˆekm/angbracketright−/summationdisplay mdmn/primek(t)/angbracketleftˆh−knˆekm/angbracketright∗ +βc/Delta1S/summationdisplay l/bracketleftbig sh kn/primel/angbracketleftˆh† −knˆh−kl/angbracketright−sh∗ knl/angbracketleftˆh† −klˆh−kn/prime/angbracketright/bracketrightbig ,(A6) where n=n/primedescribes the nonthermal populations and n/negationslash= n/primethe coherent superpositions of different valence band states. /Gamma1h nn=/planckover2pi1/T1characterizes the nonthermal population relaxation. /Gamma1h nn/primeare the intervalence-band dephasing rates, which are short and do not play an important role here. Thefirst term on the rhs describes the photoexcitation of holepopulations in band states ( n,k) that depend on S 0. The second term is beyond a simple rate equation approximation anddescribes the nonadiabatic changes in the hole states inducedby their interaction with the rapidly varying (fs) photoinduced magnetization /Delta1S(t), Eq. ( 9). Similarly, i/planckover2pi1∂ t/angbracketleftˆe† knˆekn/prime/angbracketright−/parenleftbig εc kn/prime−εc kn−i/Gamma1e nn/prime/parenrightbig /angbracketleftˆe† knˆekn/prime/angbracketright =/summationdisplay m/primed∗ nm/primek/angbracketleftˆh−km/primeˆekn/prime/angbracketright−/summationdisplay m/primedn/primem/primek/angbracketleftˆh−km/primeˆekn/angbracketright∗,(A7) where the rates /Gamma1e nn/primecharacterize the electron relaxation. In the above equations of motion, the photoexcitation of the carrier populations and coherences is driven by thenonlinear e-hoptical polarization /angbracketleftˆh −knˆekm/angbracketright(off-diagonal density matrix element). This coherent amplitude characterizesthee-hexcitations driven by the optical field, which here only exist during the laser pulse since their lifetime T 2(dephasing time) is short: i/planckover2pi1∂t/angbracketleftˆh−knˆekm/angbracketright−/parenleftbig εc km+εv kn−i/planckover2pi1/T2/parenrightbig /angbracketleftˆh−knˆekm/angbracketright =−dmnk(t)[ 1−/angbracketleftˆh† −knˆh−kn/angbracketright−/angbracketleft ˆe† kmˆekm/angbracketright] +βc/Delta1S(t)·/summationdisplay n/primesh knn/prime/angbracketleftˆh−kn/primeˆekm/angbracketright +/summationdisplay n/prime/negationslash=ndmn/primek(t)/angbracketleftˆh† −kn/primeˆh−kn/angbracketright +/summationdisplay m/prime/negationslash=mdm/primenk(t)/angbracketleftˆe† km/primeˆekm/angbracketright. (A8) The nonlinear contributions to the above equation include phase space filling (first line), transient changes in the nonequi-librium hole states due to the nonadiabatic magnetic exchangeinteraction /Delta1H exch(t) (second line), and coupling to h-h(third line) and e-e(fourth line) Raman coherences. The coupled Eqs. ( A6), (A7), (A8), and ( 11) describe photoexcitation of nonthermal carriers modified by the local spin rotation.They were derived in Refs. [ 3,17] using the Hartree-Fock factorization [ 13,62]. To obtain meaningful numerical results, we re-adjust our basis ˆh −knto reflect the eigenstates of Hb(S0) following large changes in S0during 360◦switching. 2. Adiabatic/Fermi-Dirac anisotropy The equilibrium mobile carriers can be described by Fermi- Dirac populations, fnk, of the eigenstates of the adiabatic Hamiltonian Hb(S0), which determine the quasiequilibrium anisotropy field HFS,E q .( 6)[25,32,37]. We simplify this thermal contribution by neglecting any laser-induced changesin carrier temperature and chemical potential, which add toour predicted effects. A laser-induced thermal field /Delta1H FS(t) develops indirectly from fs spin-orbit torque as the net spin ofthe hole Fermi sea bath adjusts to the new nonequilibriumdirection of S(t)[17]. As already seen from calculations of magnetic anisotropy that assume a Fermi-Dirac distribu-tion [ 6,37], the small ( ∼μeV) free energy differences with Sresult in anisotropy fields of the order of 10’s of mT. The discrepancies between theory and experiment seem to implythat nonequilibrium distributions broad in energy are necessaryto explain the magnitude of the observed effects [ 28]. Our time-domain calculation of laser-induced magnetic anisotropydriven by photoexcited fs population agrees with experimentalmeasurements. However, we must still include the thermalFermi sea anisotropy in order to describe the four-state 195203-17P. C. LINGOS, J. W ANG, AND I. E. PERAKIS PHYSICAL REVIEW B 91, 195203 (2015) magnetic memory. For this we express the free energy in the experimentally observed form dictated by symmetry [ 6,39,61], also obtained by expanding the theoretical expression [ 6]: Eh(S)=Kc/parenleftbigˆS2 xˆS2 y+ˆS2 xˆS2 z+ˆS2 yˆS2 z/parenrightbig +KuzˆS2 z−KuˆSxˆSy, (A9) where ˆS=S/Sis the unit vector that gives the instantaneous magnetization direction. Kcis the cubic anisotropy constant, Kuzis the uniaxial constant, which includes both strain and shape anisotropies , andKudescribes an in-plane anisotropy due to strain. We used measured anisotropy parameter val-ues [ 39]K c=0.0144 meV , Ku=0.00252 meV , and Kuz= 0.072 meV . We thus obtain the thermal anisotropy field γHFS=−2Kc SˆS+1 S/parenleftbig 2KcˆS3 x+KuˆSy, 2KcˆS3 y+KuˆSx,2KcˆS3 z−2KuzˆSz/parenrightbig . (A10) The above expression describes the equilibrium magnetic nonlinearities of the realistic material. By expressing Sin terms of the polar angles φandθ, defined with respect to the crystallographic axes, we obtain the easy axes from thecondition S×H FS=0, by solving the equations 2Kccos3θ−(Kc+Kuz) cosθ+BS 2=0,(A11) sin 2φ=Ku Kcsin2θ, (A12) where we added the external magnetic field Balong the [001] direction. For B=0, the above equation gives θ=π/2, which corresponds to in-plane easy axes as in Fig. 1(b). For small Ku, these magnetic states X+,X−,Y+, andY−are tilted from the [100] and [010] crystallographic directions by few degreesinside the plane [ 33,61]. As can be seen from Eq. ( A11), the Bfield along [001] cants the easy axes out of the plane. In this case,θ/negationslash=π/2 is a smooth function of B, consistent with the behavior of the static polar Kerr rotation angle θ K(B) observed experimentally [see Fig. 3(a)]. Equation ( A12) also shows that the out-of-plane tilt θinduces a magnetization rotation inside the plane. It gives two different values for φ(XandYeasy axes), which can switch due to either B-field sweeping [as seen in the transverse Hall magnetoresistivity, inset of Fig. 3(a)]o r laser-induced fs spin-orbit torque (as predicted here). APPENDIX B: BAND CONTINUUM OF ELECTRONIC STATES The average hole spin sh(t), Eq. ( 3), that triggers the fs magnetization dynamics here has contributions sh kn(t) from an anisotropic continuum of photoexcited nonparabolicband states. At /planckover2pi1ω p∼1.5e V,this continuum also includes disordered-induced states below the band gap of the pure semi-conductor [ 28]. At /planckover2pi1ω p∼3.1e V,photoexcitation of such impurity band/defect states is small, while the almost parallelconduction and valence bands lead to excitation of a wide rangeofkstates. Integration over the BZ momenta, as in Eq. ( 3), presents a well-known challenge for calculating magneticanisotropies and other properties of real materials [ 60]. To sim- plify the problem, one often calculates the quantities of interestat select kpoints and replaces the integral by a weighted sumover these “special points” (special point approximation) [ 60]. In our previous work [ 3], we considered eight special kpoints (/Lambda1point [ 7]) along {111}. While this approximation takes into account the general features of the anisotropic states, it missesimportant details, such as strong band nonparabolicity, densityof states, and photoexcited carrier densities. To comparewith the photocarrier densities in the experiment and toaddress issues such as the frequency dependence of thephotoexcited spins, we must include continua of band statesin our calculation. Here, we integrate over the band momentaalong the eight {111}symmetry lines by using the “special lines approximation” discussed in Ref. [ 64]. At/planckover2pi1ω p≈3.1e V, we approximate the three-dimensional momentum integral bya sum of one-dimensional integrals along the eight kdirections populated by photoexcited carriers. This simple approximationincludes the anisotropic, nonparabolic band continua [ 64]. At/planckover2pi1ω p≈1.5e V,Fig. 6(b) was obtained by calculating the one-dimensional integrals along all symmetry lines {100}, {010},{001},{110},{101},{011}, and {111}as in Ref. [ 64]. Following Ref. [ 64], we first express 1 V/summationdisplay k/Delta1sh k=1 (2π)3/integraldisplay BZ/Delta1sh kdk =/integraldisplayd/Omega1 4π/bracketleftbigg1 (2π)3/integraldisplaykBZ 04πk2dk/Delta1sh k/bracketrightbigg ,(B1) where kBZis the BZ boundary and d/Omega1is the angular integral. To calculate the above angular average, we use the speciallines approximation [ 64] /integraldisplayd/Omega1 4π/Delta1sh k=/summationdisplay αwα/Delta1sh kα, (B2) where αruns over the dominant symmetry directions, kis the wave-vector amplitude, and wαare weight factors. For /planckover2pi1ωp∼ 3.1e V,the dominant contribution comes from the eight {111} symmetry directions, so we approximate 1 V/summationdisplay k/Delta1sh k=1 (2π)3/summationdisplay α={111}wα/integraldisplaykBZ 04πk2/Delta1sh kαdk. (B3) Instead of eight discrete k-point populations as in Ref. [ 3], here we consider continuum distributions along the eight one-dimensional klines. While the estimation of optimum weight factors w αis beyond the scope of this paper [ 60], the order of magnitude of the predicted effects is not sensitive to theirprecise value. We fix w α=wby reproducing the net photohole density nat one experimentally measured intensity: n=1 V/summationdisplay k/summationdisplay n/Delta1/angbracketleftˆh† −knˆh−kn/angbracketright =w (2π)3/summationdisplay n/summationdisplay β={111}/integraldisplaykBZ 04πk2/Delta1/angbracketleftˆh† kβnˆhkβn/angbracketright. (B4) For the results of Fig. 4, the photocarrier density n∼6×1018 cm−3for pump fluence ∼7μJ/cm2givesw∼1/15. The same order of magnitude of nis obtained, however, for all other reasonable values of w[64]. We then used this weight factor for all other laser intensities. 195203-18MANIPULATING FEMTOSECOND SPIN-ORBIT TORQUES . . . PHYSICAL REVIEW B 91, 195203 (2015) [1] T. Li, A. Patz, L. Mouchliadis, J. Yan, T. A. Lograsso, I. E. Perakis, and J. Wang, Nature (London) 496,69(2013 ). [2] J. A. de Jong, I. Razdolski, A. M. Kalashnikova, R. V . Pisarev, A. M. Balbashov, A. Kirilyuk, Th. Rasing, and A. V . Kimel,Phys. Rev. Lett. 108,157601 (2012 ). [3] M. D. Kapetanakis, I. E. 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PhysRevB.69.214409.pdf

Large-angle, gigahertz-rate random telegraph switching induced by spin-momentum transfer M. R. Pufall, W. H. Rippard, Shehzaad Kaka, S. E. Russek, and T. J. Silva Electromagnetics Division 818, National Institute of Standards and Technology, Boulder, Colorado 80305, USA Jordan Katine and Matt Carey Hitachi Global Storage Technologies, San Jose, California 95120, USA (Received 14 January 2004; published 8 June 2004 ) We show that a spin-polarized dc current passing through a small magnetic element induces two-state, random telegraph switching of the magnetization via the spin-momentum transfer effect. The resistances of thestates differ by up to 50% of the change due to complete magnetization reversal. Fluctuations are seen for awide range of currents and magnetic fields, with rates that can exceed 2 GHz, and involve collective motion ofa large volume s10 4nm3dof spins. Switching rate trends with field and current indicate that increasing temperature alone cannot explain the dynamics. The rates approach a stochastic regime wherein dynamics aregoverned by both precessional motion and thermal perturbations. DOI: 10.1103/PhysRevB.69.214409 PACS number (s): 72.25.Pn, 85.75. 2d The recent observations of high-frequency precessional motion and magnetization switching induced by a spin-polarized dc current have further spurred the study of thespin-momentum transfer (SMT )effect as a possible means of efficiently switching the small magnetic elements in mag-netic memory, and as the basis for high-frequency,nanometer-sized microwave oscillators. 1–4In addition to high-frequency precessional motion, the excitation of an un-expected lower-frequency “broadband instability” extendingfrom dc to several gigahertz has also been observed. 1,3Here, we report real-time measurements of this SMT-inducedlower-frequency response of spin-valve structures, both ofpatterned structures, and of lithographic point contacts madeto continuous films. We show that the broadband instabilityis the result of the magnetization in these structures under-going large-angle fluctuations between two distinct states inresponse to the dc current. These fluctuations have the char-acteristics of classic random telegraph switching (RTS), and are observed for a wide range of currents and applied mag-netic fields, and for various device geometries and sizes. Thefluctuations occur on time scales ranging from microseconds to fractions of a nanosecond, and can be observed over thesame range of currents and fields as the coherent high-frequency precessional excitations reported previously. 1,3As the dimensions of magnetoelectronic devices decrease, as forhard-disk drive read heads and spintronic devices, devicevolumes will reach the point where SMT effects such asthose reported here must be addressed. Two-state systems exhibiting RTS—that is, each state having an independent dwell time—have instances across thephysical sciences, from solid-state physics 5–9to biology,10as well as in magnetic systems.11–13Spin-transfer-induced RTS at hertz to kilohertz rates has also been observed previously14 for systems in which two well-defined magnetostatic statesare accessible, i.e., when the applied field H appis less than Hc, the coercive field of a patterned device with uniaxial anisotropy. In this case, the spin-transfer current modifies thethermally activated fluctuation rate between the two states.Here we explore spin-transfer-induced effects for larger ap-plied fields H app.Hc, so that the applied field is of a mag-nitude sufficient to allow only onestable state in the absence of an applied current. It has been previously shown that SMT can induce ran- dom switching15,16of the magnetization between parallel and antiparallel (relative to the applied field )orientations, at rates ,1 MHz. The observations reported here significantly ex- tend these measurements, showing that this spin-transfer-induced random telegraph switching persists in higher fields,and out to very high rates (in excess of 1 GHz ). Conse- quently, these systems approach a dynamical regime inwhich the switching rate approaches the intrinsic dampingrate ofM, and the dynamics of Mhave both precessional and thermal characteristics. Furthermore, we show that thefluctuation amplitude no longer corresponds to 180° reversalof the magnetization: Instead, the device exhibits resistancechanges of up to 50% of complete reversal, indicating thatthe spin torque induces a second (meta )stable configuration of the magnetization, the properties of which are a functionof both applied field and current. These features distinguish spin-transfer-induced telegraph switching relative to other two-state systems: RTS in otherphysical systems frequently results from the fluctuation of asmall defect or region of a larger structure, typically showingcharacteristic rates from subhertz to megahertz before thebreakdown of two-state behavior. In contrast, SMT-inducedtwo-state switching corresponds to the large-amplitude col- lectivemotion of the entire device, a volume of <10 4nm3,a t GHz rates. Though the system is driven by the spin-transfercurrent, we show that nonetheless the general analytical tech-niques developed for the study of two-state switching sys-tems enable us to study in new ways the interaction betweenthe polarized current and the local magnetic moment, and weshow evidence that SMT significantly modifies the energysurface experienced by the free-layer magnetization. Room-temperature, high-bandwidth I-Vresistance mea- surements were made on Cu s100 nm d/IrMn s7n m d/ Cos7.5 nm d/Cus4n m d/Cos3n m d/Cus20 nm d/Aus150 nm d spin-valve structures patterned into 50 3100 nm 2elongated hexagonal pillars. The exchange bias field from the IrMnPHYSICAL REVIEW B 69, 214409 (2004 ) 0163-1829/2004/69 (21)/214409 (5)/$22.50 69214409-1layer s,1m T ddid not have a significant effect on the mag- netics. The thinner Co layer, having a lower total moment, responds more readily to spin torques than the thick Colayer. 4These layers are referred to as the “free” magnetiza- tionMfreeand the “fixed” magnetization Mfixedlayers, re- spectively [see Fig. 1 (a)inset ]. In a spin valve, changes in the relative alignment of the magnetizations of these twolayers result in variations in device resistance through thegiant magnetoresistance (GMR )effect. By dc current biasing the device, these resistance changes are seen as changes inthe voltage across the device. In the following analysis, weassume that all SMT-induced voltage changes in these de-vices are due to GMR. 17,18Measurements were also made on larger s753150 nm2delongated hexagons, circular devices of 50–100 nm radii, and lithographically defined 40 nm point contacts made to unpatterned spin-valve multilayers.3 All devices showed similar RTS, indicating that the bistabil-ity is a generic feature associated with SMT rather than anartifact of a specific device geometry. When a dc current Iis driven through these devices in the presence of an in-plane field H appsufficient to align the mag- netizations of the two layers, we observe a reversible step inthe dc resistance at a critical current I c, as has been describedelsewhere.3,4,19A typical dc resistance trace is shown in Fig. 1(a), withIc<6m A (corresponding to a current density Jc <108A/cm2). This step occurs for only one sign of current, with electrons flowing from the free to fixed layer [see Fig. 1(a)inset ], indicating that the resistance change is due to spin torque rather than current-generated magnetic fields.4,20 The position of the step is a function of applied field, and isalso, as shown in the real-time voltage traces in Fig. 1 (b), correlated as follows with the appearance of two-state tele-graph switching of M free.15BelowIc, the device lies prima- rily in the low resistance state, switching occasionally butrapidly into (and out of )the higher resistance state. As the current approaches I c[see Fig. 1 (b)], this switching occurs with greater frequency. The transient switching time betweenthe states is quite fast, and varies with Ifrom 1 ns to 2 ns for low currents to <0.7 ns for higher I.A sIincreases, the characteristic time spent in each state changes, with the timespent inR low(the dwell time tlow)decreasing, and the time in Rhigh(thigh)increasing. The step in dc resistance occurs where tlow<thigh. Above Ic,Rhighbecomes most likely (a fact also reflected in the dc resistance trace ). The states between which the magnetization switches are functions of both current and field. From triggered real-timevoltage traces taken at a fixed current and field (taken on a different device than for the data shown in Fig. 1 ), we deter- mined the distribution of voltage changes (as it is an ac- coupled measurement ), and from this calculated the resis- tance change DRbetween the two states. As shown in Fig. 2(c),DRvaries with current and applied field, and is not monotonic with Ifor a given H app, having a maximum value aroundIc. On the other hand, the peak value in DRdecreases approximately linearly with Happ, in a manner similar to the dc resistance step [see Fig. 2 (b)]. FIG. 1. (a)dcI-Vcurve for 100 350 nm2spin-valve nanopillar device for m0Happ=0.1 T. SMT-induced step in resistance denoted byIc. Inset graphic shows device structure, field, and current flow directions. Black circles denote currents for real-time traces shownin lower panel. (b)Real-time ac resistance traces taken at specified currents. DRdefined as noted on trace at I=5.2 mA, individual dwell events shown for I=5.7 mA. Device dc response and ac- coupled time-dependent response monitored as functions of currentand applied field, through 40 GHz probe contacts. Real-time tracesacquired with a 1.5 GHz (sampling rate of 8 310 9/s)bandwidth single-shot real-time oscilloscope, used to either capture traces upto 8 ms in length [as in Fig. 1 (b)], or to measure pulse widths and heights of a succession of individual triggered switching events at afixed current and field, as in subsequent figures. Power spectra [see Fig. 3 (g)]acquired with a 50 GHz spectrum analyzer. FIG. 2. (Color online )(a)R-Hcurve of device, showing DRfor 180° rotation of layers. (b)dcI-Vtraces for nanopillar, as a function of field, with current scanned both up and down. Note small hys-teresis between up and down scans, also manifested in dynamic DR traces and Fig. 3 time traces. (c)Measured dynamic DRfor 100 350 nm 2nanopillar, as a function of current and m0Happ. Device structure is the same as for Fig. 1. DRdefined as in Fig. 1 (b).DR determined from 150 triggered switches. Error bars are sx¯from a Gaussian fit to DRdistribution at fixed I,m0Happ. Right axis: esti- mated angular deviations of Mfree, using model described in text.M. R. PUFALL et al. PHYSICAL REVIEW B 69, 214409 (2004 ) 214409-2Field- [see Fig. 2 (a)]and current-induced switching measurements4indicate that these devices are sufficiently small such that the free-layer magnetization vector Mfreebe- haves as a single domain, and that uMfreeuis not reduced by the current.21Assuming this, a cosine dependence of Rsud due to GMR, and taking one of the states as MfreeiMfixed (because DRand angle are not uniquely related ), we esti- mated the dynamic rotation angle of the free layer, shown onthe right axis in Fig. 2 (c). The motions of M freeinduced by the dc current are not simply small perturbations about equi-librium, as is typically observed in thermally driven two-state magnetic systems: 7,11,13For low fields and currents, the magnetization rotates a full 90°, a value that decreases to<40° at higher H appandI. The larger amplitudes previously observed at lower fields and fluctuation rates15are consistent with this trend. The characteristic times thighand tlowat a given Iand Happwere determined from the conditional probability distri- bution functions PsRhighuRhighatt=0dandPsRlowuRlowatt =0d. These probabilities were constructed from multiple trig- gered real-time traces, and show an exponential falloff with time. Fitting the expression pstd=e−t/t(in which tis the elapsed time after a switch, and tisthigh,low )to the measured distributions gives thighandtlow, shown as functions of Ifor several fields in Figs. 3 (a)–3(f). Measurements were made from 4 mA to 12 mA. Dwell times longer than the acquisi-tion wait time (<1 ms before autotriggering )were not re- cordable; consequently very long dwell times (i.e., quasis- table magnetic configurations )are indicated by an absence of dwell times at that current. The tof each state varies by several orders of magnitude over the applied current range,corresponding to states that vary from nominally long-liveds tdwell.8msdto highly unstable stdwell<1n sd. The plots in Fig. 3 represent quantitatively what was described qualita- tively for Fig. 1: Initially tlow@thigh, and asIincreases, the dwell times converge, cross at Ic(at the step in dc resistance15), and diverge, with thighbecoming the longer dwell time. As seen in Figs. 3 (a)–3(f),thighvaries with current in a markedly different fashion from tlow. On a logarithmic scale, tlowis roughly linear with current, whereas thighis a more complicated function of current. Furthermore, the functionalform of thighis a strong function of applied field, whereas tlowremains roughly linear, primarily shifting to higher cur- rents and changing slope slightly. The states, and the poten-tial barrier between them inhibiting fluctuations, are evi-dently distinct functions of current and field. Finally, therange of currents over which fluctuations are observed in-creases with field: The onset current for fluctuations is arelatively weak function of field, while the current at whichfluctuations cease moves rapidly out to higher currents. So,though more field is being applied, the fluctuations are notmore stabilized. For m0Happ.0.19 T, the instability is still observed, but the fluctuation rates exceed the bandwidth of the real-timeoscilloscope. However, with some general assumptions aboutthe system, the dwell times can instead be determined fromthe power spectra. Two state or telegraph noise, in which thedwell times in the two states are independent, produces aLorentzian power spectral density, centered about dc, 9of the form Ssvd=DV2 4pthtl teff21/teff v2+1/teff2, s1d in which DV=1DRis the change in voltage, and 1/ teff =1/thigh+1/tlow. Power spectra for several currents with m0Happ=0.24 T are shown in Fig. 3 (g). Lorentzians fit the data well, as shown for I=12 mA. From the width, a teffof 0.37 ns is determined, corresponding to a single-state switch-ing rate in the range 1.4 GHz–2.7 GHz. 22In other devices, we observed rates in excess of 4 GHz.The lack of deviationsof the spectral shape from a dc-centered Lorentzian indicatesthe validity of the random, two-state assumption. The pre-ponderance of SMT-induced low frequency noise induced inthese devices is due to two-state switching. This signature oftelegraph switching was observed for a wide range of fields:Even fields in excess of 0.5 T were insufficient to suppressthe SMT-induced RTS. It is important to note that these SMT-induced two-state fluctuations are observed for applied fields such that, magne- tostatically , only one orientation of M freeis allowed: In the FIG. 3. (a–f)Measured dwell times thighandtlowvs current, for several applied fields m0Happ. Dwell times determined from mea- sured conditional probabilities of the form Phighslowdstd =PsRhighslowduRhighslowdatt=0d, constructed from 150 triggered events, as described in text. (g)Power spectrum of device at m0Happ=0.24 T for several device currents. Lorentzian fit to I =12 mA shown.LARGE-ANGLE, GIGAHERTZ-RATE RANDOM PHYSICAL REVIEW B 69, 214409 (2004 ) 214409-3absence of current, no second state into which the magneti- zation can switch, and the potential surface is parabolic witha single minimum. The spin-polarized current inducesa sec- ond metastable magnetization configuration at sufficientlyhigh current densities, analogous to an applied field.This is asomewhat surprising conclusion, since typical formulationswrite spin torque as an “antidamping” term in a Landau-Lifshitz-Gilbert dynamical equation, rather than as a field,and damping will ordinarily not affect the energy states of asystem. Indeed, while single-domain numerical models 1,23 based on Slonczewski’s theory of SMT-induced magnetiza-tion dynamics 20have reproduced aspects of the precessional motion observed in Refs. 1 and 3, these models fail to de-scribe this broadband instability. Stochastic switching hasbeen observed in simulations, but either only over small cur-rent ranges near transitions between two different current-dependent precessional states (rather than over a large range ofIas shown here )or between two states with identical GMR values. 24In addition, these simulations describe fluc- tuations between precessional states. However, we have ob-served two-state switching both in the presence of and inde- pendent of the existence of high-frequency precessional signals, up to the bandwidth limit of our system (<12 GHz, for the nanopillar structures, 40 GHz for the lithographicpoint contacts ). The random occurrences of the two-state fluctuations such as those shown in Fig. 1 (b)certainly suggest an Arrhenius- Néel two-state analysis describing a thermally activated pro-cess over a barrier as a way of parametrizing the effects ofthe spin torque and the applied field on the system. Previouswork also used anArrhenius-type analysis to parametrize thelow-frequency RTS, employing an independent magnetictemperature for each state. 15This work used a two- temperature model that does not succeed in describing ourdata over the ranges of currents and fields presented here; 25 here we will use a single-temperature model. A possiblecomplicating factor against using anArrhenius-Néel analysis(a factor that also drives new effects such as the observed high-frequency precession 3)is that torques due to SMT are not conservative, and as a consequence the work done by thetorque is path dependent. 26Therefore, it is not necessarily the case that a “barrier” is well defined for this system. None-theless, previous experimental 14and theoretical26work sup- ports the applicability of a two-state model27if the energy barrier is replaced by an “activation energy” that includes thework done by the spin torque. Subject to these caveats, the expression thigh,low=t0eUsI,Hdhigh,low/kBTs2d describes the characteristic time tfor thermal activation over a barrier, in which UsI,Hdhigh,loware the respective effective barrier heights for the high and low resistance states (shown schematically in Fig. 4 ),1/t0is the “attempt frequency,” kB is Boltzmann’s constant, and Tis the magnetic system tem- perature, which is possibly different from the phonon (sys- tem)temperature. By comparing ln sthigh/tlowd=sUhigh −Ulowd/kBTto ln sthightlow/t02d=sUhigh+Ulowd/kBTwe can estimate a upperlimit on t0of 1 ns,28of the same order as the dwell time determined from the power spectrum in Fig.3(b). The attempt frequency 1/ t0is set by the intrinsic damping rate of Mand is typically several gigahertz for these systems,29consistent with these values. Assuming a value of 0.5 ns for t0, we calculated the ef- fective barrier heights Uhigh/kBTandUlow/kBTas a function ofI, shown for several fields in Fig. 4. These plots show that the current modifies the effective barrier seen by each statefrom roughly 1 k BTto 10kBTover the measurement range. By comparison, the barrier height between the two bistablestates at zero field and current is over hundred times thisvalue. As expected, the barrier heights vary opposite to eachother with current. For low currents, the low resistancesM freeiMfixeddstate is preferred, having a larger barrier im- peding escape, while the high resistance state becomes the more probable state at higher currents with I.Ic. Recall that this high-resistance state does not exist in the absence of thecurrent for these applied fields, as there is no secondary po-tential minimum. In Fig 4, we can see evolution of the state,with the current progressively increasing the barrier heightdefining it. However, though the two barriers have oppositetrends with current, the functional forms are markedly differ-ent:U lowis roughly a linear function of I, whileUhighdis- plays a more complicated dependence. In this sense, theseresults deviate substantially from previous, low-frequencySMT-mediated two-state switching data 14,15in which both states showed a linear dependence on current (though over a smaller field and current range ). In addition, it is also clear that the effective barrier heights do not asymptotically ap-proach zero with increasing current, as would be the case ifan Ohmic or other monotonic temperature increase were thesource of the telegraph noise. Indeed, temperatures far abovethe Curie temperature of the ferromagnet would be requiredto reduce the ratio U/k BTsufficiently to approach the ob- served switching rates. We have shown that a spin-polarized dc current passing through a small magnetic element induces an instability in FIG. 4. (Color online )Plot of calculated effective energy barri- ersUhighandUlow, in units of kBT, as a function of current for several fields. A value of t0=0.5 ns was assumed. Data for up and down scans of current averaged together for clarity. Inset: Diagramshowing definition of effective energy barriers.M. R. PUFALL et al. PHYSICAL REVIEW B 69, 214409 (2004 ) 214409-4the magnetization, resulting in classic two-state, random tele- graph switching for a wide range of currents and for appliedmagnetic fields that magnetostatically allow only one orien-tation ofM.Alarge magnetic volume collectively undergoes large-amplitude fluctuations between two states, at gigahertzrates, resulting in a broadband dc-centered Lorentzian powerspectrum. These fast fluctuation rates suggest that SMT sig-nificantly modifies the potential surface experienced by M.The fluctuation rate trends observed with applied field and current indicate that an increase in temperature — whether ofthe device or magnetic system—does not by itself describethe dynamics. With the rapidly shrinking device dimensionsin magnetoelectronics, device volumes will reach the pointwhere SMT effects are unavoidable. The results presentedhere indicate that attention should be paid to SMT-induceddynamics as a source of broadband, large-amplitude noise. 1S. I. Kiselev et al., Nature (London )425, 380 (2003 ). 2J. Sun, Nature (London )425, 359 (2003 ). 3W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. (to be published ). 4J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000 ). 5K. S. Ralls and R. A. Buhrman, Phys. Rev. Lett. 60, 2434 (1988 ). 6V. D. Ashkenazy, G. Jung, I. B. Khalfin, and B. Ya. Shapiro, Phys. Rev. B 50, 13 679 (1994 ). 7S. Ingvarsson et al., Phys. Rev. Lett. 85, 3289 (2000 ). 8P. D. Dresselhaus, L. Ji, S. Han, J. E. Lukens, and K. K. Likharev, Phys. Rev. Lett. 72, 3226 (1994 ). 9S. Machlup, J. Appl. Phys. 75, 341 (1954 ). 10S. M. Bezrukov and J. J. Kasianowicz, Phys. Rev. Lett. 70, 2352 (1993 ). 11W. Wernsdorfer et al., Phys. Rev. Lett. 78, 1791 (1997 ). 12H. T. Hardner, M. J. Hurben, and N. Tabat, IEEE Trans. Magn. 35, 2592 (1999 ). 13L. S. Kirschenbaum, C. T. Rogers, S. E. Russek, and S. C. Sand- ers, IEEE Trans. Magn. 31, 3943 (1995 ). 14E. B. Myers et al., Phys. Rev. Lett. 89, 196801 (2002 ). 15S. Urazhdin, N. O. Birge, W. P. Pratt, and J. Bass, Phys. Rev. Lett. 91, 146803 (2003 ). 16M. Tsoiet al., Phys. Rev. Lett. 80, 4281 (1998 ). 17M. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988 ). 18While other sources of voltage have been proposed [see L. Berger, Phys. Rev. B 54, 9353 (1996 )], previous work usingGMR to sense time-varying magnetization motion indicates that GMR is the dominant source of ac or dc impedance change inspin-valve devices. For example, see Refs. 1 and 3. The mea-surement geometry makes inductive coupling between the de-vice and waveguide negligible. 19M. R. Pufall, W. H. Rippard, and T. J. Silva, Appl. Phys. Lett. 83, 323(2003 ). 20J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 (1996 ). 21B. Özyilmaz et al., Phys. Rev. Lett. 91, 067203 (2003 ). 22Knowing DR, and also which is the more likely of the two states, one can invert these spectra to determine the respective dwelltimes. However, because DRis a function of both current and field, this is not possible for switching rates faster than 1.5 GHz(our oscilloscope bandwidth ). 23J. Z. Sun, Phys. Rev. B 62, 570 (2000 ). 24Z. Li and S. Zhang, Phys. Rev. B 68, 024404 (2003 ). 25The applied fields and currents used here are too large, so that the “magnetic temperature” will exceed the Curie temperature of themagnet.Also, the energy barriers in this theory are only affectedby the current by the reduction of M sthrough the increase in effective temperature. 26Z. Li and S. Zhang, cond-mat/0302339 (unpublished ). 27This assumes that the damping and the out-of-plane component of Mremain small. Simulations suggest that the second assumption is questionable for SMT-induced precessional modes. 28Equivalently, the shortest dwells we observe are <1 ns. 29N. D. Rizzo, T. J. Silva, and A. B. Kos, Phys. Rev. Lett. 83, 4876 (1999 ).LARGE-ANGLE, GIGAHERTZ-RATE RANDOM PHYSICAL REVIEW B 69, 214409 (2004 ) 214409-5

PhysRevB.89.064412.pdf

PHYSICAL REVIEW B 89, 064412 (2014) Theory of magnon-skyrmion scattering in chiral magnets Junichi Iwasaki,1,*Aron J. Beekman,2and Naoto Nagaosa2,1,† 1Department of Applied Physics, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Received 9 September 2013; published 14 February 2014) We study theoretically the dynamics of magnons in the presence of a single skyrmion in chiral magnets featuring Dzyaloshinskii-Moriya interaction. We show by micromagnetic simulations that the scattering processof magnons by a skyrmion can be clearly defined although both originate in the common spins. We find that (i) themagnons are deflected by a skyrmion, with the angle strongly dependent on the magnon wave number due to theeffective magnetic field of the topological texture, and (ii) the skyrmion motion is driven by magnon scatteringthrough exchange of the momenta between the magnons and a skyrmion: the total momentum is conserved. Thisdemonstrates that the skyrmion has a well-defined, though highly non-Newtonian, momentum. DOI: 10.1103/PhysRevB.89.064412 PACS number(s): 75 .78.−n,75.30.Ds,12.39.Dc,75.78.Cd I. INTRODUCTION The skyrmion is a topological texture of field configu- ration and was first proposed as a model for hadrons innuclear physics [ 1,2] and has been discussed in a variety of condensed-matter systems [ 3–6]. Most recently skyrmions have been found in magnets with Dzyaloshinskii–Moriya(DM) interaction, attracting intensive interest [ 7–10]. Here it is a swirling spin structure characterized by the skyrmionnumber Q, which counts how many times the mapping from the two-dimensional real space to the spin space wraps thesurface of the sphere. The skyrmion has a finite size determinedby the ratio of the ferromagentic exchange interaction J and the DM interaction D, i.e., localized in real space within 3–100 nm, and has very long lifetime because oftopological protection, i.e., any continuous deformation ofthe field configuration cannot change the skyrmion number.Therefore, the skyrmion can be regarded as a particle made outof the spin field. These advantages, i.e., small size and stability,together with ultralow threshold current density for the motion(∼10 6A/m2)[11,12] compared with that for the domain-wall motion ( ∼1010–1012A/m2)[13,14], make the skyrmion an appealing and promising candidate as an information carrierin magnetic devices [ 15–18]. On the other hand, the low-energy excitations in magnets are magnons [ 19]: propagating small disturbances in the under- lying spin texture. In sharp contrast to the skyrmion, a magnonis a propagating wave, and can be created and destroyed easily,i.e., it belongs to the topologically trivial sector. Therefore,an important issue is the interaction between magnons andskyrmions, which offers an ideal laboratory to examine theparticle-field interaction in field theory, and also provides thebasis for the finite temperature behavior of skyrmions. It hasbeen known that the motion of a domain wall in ferromagnetscan be induced by magnons: the domain wall moves againstthe direction of the magnon current [ 20–22]. Recently the skyrmion version of the magnon-induced motion has been *iwasaki@appi.t.u-tokyo.ac.jp †nagaosa@riken.jpstudied [ 23,24], when magnons are produced by a temperature gradient. However, the elementary process involving a singleskyrmion and magnons has not been studied up to now. Theonly work on magnon-skyrmion dynamics we are aware of(Ref. [ 25]) precludes from the outset, in the context of quantum Hall systems, any skew scattering, which does not agree withthe observations in chiral magnets. Another work consideredmagnon scattering off skyrmions in time-reversal invariantsystems [ 26]. The skyrmion is characterized by a spin gauge field aand carries an emergent magnetic flux b=∇× aassociated with the solid angle subtended by the spins. This spin gauge field ais coupled to the conduction electrons, which results in nontrivialeffects such as the spin transfer torque driven skyrmion motionand topological Hall effect. Surprisingly, a tiny current density∼10 6A/m2can drive the motion of the skyrmion crystal via spin transfer torque [ 11,12], which is orders of magnitude smaller than that in domain-wall motion in ferromagnets(10 10–1012A/m2)[13,14]. This has been attributed to the Magnus force acting on the skyrmion and its flexible shapedeformation reducing the threshold current [ 27,28]. An in- teresting recent development is the discovery of skyrmions in an insulating magnet Cu 2OSeO 3[10,29,30], where the electric-field-induced motion is associated with multiferroicbehavior. It is expected that in this insulating system, theonly low-energy relevant excitations are the magnons, and theinteraction between magnon and skyrmion becomes especiallyrelevant. In this paper, we study the scattering process of a magnon by a skyrmion by solving numerically the Landau-Lifshitz-Gilbert (LLG) equation for magnons with the center of wavenumbers kincident on a skyrmion of size ξin Sec. II. The simulations clearly show wave-number-dependent skewscattering of the magnon, and furthermore similar large Hallangle of skyrmion motion due to the back action. This processis well analyzed in terms of momentum conservation, stronglyindicating that the skyrmion is particlelike with a well-definedmomentum as worked out in Sec. III. By mapping the situation to a charged particle scattered by a tube of magnetic fluxwe show in Sec. IVthat the principal contribution to skew scattering is the emergent Lorentz force generated by theskyrmion. 1098-0121/2014/89(6)/064412(7) 064412-1 ©2014 American Physical SocietyJUNICHI IW ASAKI, ARON J. BEEKMAN, AND NAOTO NAGAOSA PHYSICAL REVIEW B 89, 064412 (2014) (a1) t=0 (a2) t=250 (a3) t=1000 (a4) t=2000 (b1) t=0 (b2) t=500 (b3) t=2000 (b4) t=3000 (c1) t=0 (c2) t=2000 (c3) t=4000 (c4) t=8000 kξ=0.52 π kξ=0.83 π kξ=1.87 π Φφ ξ ξ ξλ λ λ FIG. 1. (Color online) Snapshots of scattering processes with three different wave numbers. (a1)–(a4): ¯kξ/similarequal1.87π; (b1)–(b4): ¯kξ/similarequal0.83π; (c1)–(c4): ¯kξ/similarequal0.52π; time steps of the snapshots as indicated. The inset of (a1) shows the color representation of the in-plane spin component in (xy) spin space. In (a1), (b1), and (c1), the wavelengths λ≡2π/¯kof the incident waves are compared with the size of the skyrmion ξ.T h e vertical blue line in (a4), (b4), and (c4) denotes the incoming magnon direction. For the higher wave numbers we can clearly identify the skewscattering of the magnons. In (a4) the white dashed lines indicate the equal phase contour of the scattered magnons, and blue line perpendicular to those defines the scattering skew angle ¯ ϕ. The yellow lines represent the path traversed by the skyrmion, also clearly showing skew scattering over an angle /Phi1. Hence we see that the skyrmion skew angle is nearly half of the magnon skew angle as expected from the conservation of the momentum. II. NUMERICAL RESULTS Our model is the chiral magnet on the 2D square lattice: H=−J/summationdisplay rmr·/parenleftbig mr+aex+mr+aey/parenrightbig −D/summationdisplay r/parenleftbig mr×mr+aex·ex+mr×mr+aey·ey/parenrightbig −B/summationdisplay r(mr)z. (1) Here, mris the unit vector representing the direction of the local magnetic moment and ais the lattice constant. In the following, we measure all physical quantities in units ofJ=/planckover2pi1=a=1, where /planckover2pi1is the reduced Planck constant. For a typical set of parameters J=1 meV and a=0.5 nm, in these units we have the correspondences for time t=1: 6.58× 10 −13s; mass M=1: 2.78×10−28kg; and magnetic field B=1: 17.3 T. We fix DM interaction D=0.18. The ground state for the Hamitonian ( 1) is the helical state for external field B<B c1=0.0075, the ferromagnetic state for B>B c2= 0.0252, and the skyrmion crystal for Bc1<B<B c2[28]. To study the scattering of magnon plane waves off a single skyrmion, we have performed micromagnetic simulationsbased on the Landau-Lifshitz-Gilbert (LLG) equation: dm r dt=−mr×Beff r+αmr×dmr dt, (2)where αis the Gilbert damping coefficient fixed to α=0.04 in the whole paper and Beff r=−∂H ∂mr. We perform the simulation atB=0.0278 ( >B c2), putting a metastable skyrmion at the center of ferromagnetic background [Fig. 1(a1)]. The size of the skyrmion ξin this paper is defined as the distance from the core (mz=− 1) to the perimeter ( mz=0), and ξ=8 for our parameter set. At the lower boundary a forced oscillation offrequency ωwith fixed amplitude A≡/angbracketleftm 2 x+m2 y/angbracketright=0.0669 is imposed on the spins, producing spin waves with wave vectork=(0,k) traveling toward the top. Here, the amplitude of the magnon with wave number kis proportional to 1 ω2−ω2 k+iαω, where ωkis the dispersion of the magnon with energy gap B. We estimated the averaged ¯kfrom the real-space image of the magnon propagation. For ω=0.08,0.04,0.02,0.0125,and 0.01, we find ¯kξ/similarequal1.87π,1.20π,0.83π,0.64π, and 0 .52π, respectively. Note that the latter three frequencies are belowthe magnon gap. Figure 1shows snapshots of the scattering processes with three different wavelengths (see also Supplemental Material,movies 1 and 2 [ 31]). These lead to several remarkable observations. First, one can clearly see that the identity ofthe skyrmion remains intact even though some distortion ofits shape occurs. This originates in the topological protection,and is not a trivial fact since both the skyrmion and magnonsare made out of the same spins. Namely, the skyrmion numberQ= 1 4π/integraltext d2xm·(∂xm×∂ym)i s−1 for the skyrmion while that of magnons is zero, and hence the conservation of 064412-2THEORY OF MAGNON-SKYRMION SCATTERING IN . . . PHYSICAL REVIEW B 89, 064412 (2014) 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 0 0.005 0.01 0.015 0.02Φ (o) v k (π/ξ)skyrmion Hall angle skyrmion velocity (a) (b) (c) FIG. 2. (Color online) The scattering properties obtained by numerical and analytical calculations. (a) The Hall angles /Phi1(red line) and velocities v(blue line) of skyrmion motion are estimated from the numerical results for different wave numbers ¯k. To obtain these values, we traced the center-of-mass coordinate Rof a skyrmion between Y=51 and Y=31. The coordinate Ris defined as R≡/integraltext d2rρtop(r)r//integraltext d2rρtop(r), where ρtop(r)≡m(r)·[∂xm(r)×∂ym(r)]. There is a strong nonmonotonic wave-number-dependent behavior in both quantities. We compare these observations to the idealized cases of magnons scattering off a uniform flux tube by an Aharonov-Bohm-typecalculation: (b) Expectation of the magnon Hall angle ¯ ϕas a function of wave number k. It is strongly peaked around kξ≈1, and vanishes for both low and high wave number, in the latter case as ∼1/k. The relation /Phi1=¯ϕ/2 derived by momentum conservation seems to be well obeyed by comparing images (a) and (b). (c) Magnon scattering amplitude of several wave numbers k. The asymmetry in left or right scattering can be clearly seen and is due to the effective Lorentz force induced by the Berry phase of the skyrmion. For low wave numbers, the scattering amplitude is almost flat, indicating the wave “missing” or “ignoring” the skyrmion; for high wave numbers it is strongly peaked, indicating mostly forward scattering, which is well known in the Aharonov-Bohm effect. the skyrmion number protects the identity of the skyrmion. Second, the incident wave is clearly scattered by the skyrmion,with sizable “skew angle” or “Hall angle.” As the wavenumber ¯kis increased, the diffraction becomes smaller and one can define the trajectory of the scattered magnons clearlyin Figs. 1(a1)–1(a4) for ¯kξ/similarequal1.87π. As shown in the blue lines in Fig. 1(a4), the scattered trajectory has an angle ¯ ϕ compared with the direction of the incident magnons (verticalline). As the wave number ¯kis reduced, the diffraction is enhanced, but the skewness of the scattered waves can still beseen in Figs. 1(b2)–1(b4) for ¯kξ/similarequal0.83πand 1(c2)–1(c4) for ¯kξ/similarequal0.52π. Therefore, the skew angle ¯ ϕstrongly depends on¯kξ. Third, by tracing the center-of-mass position of the skyrmion, it is found that it moves in turn backward andsidewards in the opposite direction as indicated by the yellowlines in Figs. 1(a4), 1(b4), and 1(c4). The skew angle /Phi1of the skyrmion motion is plotted in Fig. 2(a), which shows strong ¯kdependence. Also the speed vof the skyrmion depends on the wave number ¯kfor fixed amplitudes of the magnons, as s h o w ni nF i g . 2(a). This skyrmion motion can be understood by the magnons exerting spin transfer torque on the skyrmion, orequivalently analyzed in the light of momentum conservationas will be discussed below. III. SKYRMION MOMENTUM The dynamic term of a skyrmion particle is Sdyn=/integraltext dtL, where [ 32] L=2πQ(Y∂tX−X∂tY)+M 2[(∂tX)2+(∂tY)2]. (3) Here, X,Y are the skyrmion center-of-mass coordinates, and Mis the mass of the skyrmion. Then the momentum is Px=∂L ∂∂tX=2πQY+M∂tXandPy=∂L ∂∂tY=− 2πQX + M∂tY. Assuming a massless skyrmion ( M=0) and elastic scattering ( p(in) mag=p(out) mag+/Delta1Pskyrm ), we can estimate the skewangle as follows. For the magnon p(in) mag=(0,k) and p(out) mag= (ksin ¯ϕ,kcos ¯ϕ), then /Delta1Pskyrm=(−ksin ¯ϕ,k(1−cos ¯ϕ)). (4) Using Px=2πQY,P y=− 2πQX one finds the skyrmion Hall angle: /Phi1=arctan( /Delta1X//Delta1Y )=¯ϕ/2. (5) The numerics, i.e., /Phi1and ¯ϕin Fig. 1(a4), is consistent with this relation. In the present simulation, the displacement /Delta1R of the skyrmion is about 30, over the time period of 2000forkξ/similarequal1.87π. The velocity vis of the order of 30 /2000∼= 1.5×10 −2.T h em a s s Mis of the order of the number of spins constituting one skyrmion and is of the order of 200 in oursimulation. Therefore, Mv∼3/lessmuch2π/Delta1R ∼200, and hence the assumption of the massless skyrmion above is justified. We can estimate the velocity of the skyrmion purely in terms of momentum transfer of the spin wave to the skyrmion. A plane wave√ Ae−iωt+i¯kyhas momentum p(in)=A¯k. The part of the incident wave that interacts with the skyrmion is of size2ξ, the diameter of the skyrmion. Hence the momentum of the part of the magnon plane wave interacting with the skyrmionisk=2ξA¯k. The magnitude of the transferred skyrmion momentum is |/Delta1P skyrm|=k√2−2 cos ¯ϕ=4ξA¯ksin1 2¯ϕ. Now we are sending in a continuous plane wave instead of a single magnon. The time it takes for the plane wave topass by/through the skyrmion is T k≡2ξ/vkwhere vkis the group velocity of the magnon, given by vk=∂ωk ∂k=2Jk, and ωk=Jk2+Bis the magnon dispersion. Hence in one unit of time, the plane wave interacts with the 1 /Tkpart of the skyrmion. Thus the amount of momentum transferred in oneunit of time is /Delta1˜P≡|/Delta1P skyrm| Tk=4ξA¯ksin1 2¯ϕ 2ξ/2J¯k=4JA¯k2sin1 2¯ϕ. (6) 064412-3JUNICHI IW ASAKI, ARON J. BEEKMAN, AND NAOTO NAGAOSA PHYSICAL REVIEW B 89, 064412 (2014) In our units J=1. The incoming magnons of average wave number ¯kare generated by a forced oscillation with magnitude A≡/angbracketleftm2 x+m2 y/angbracketright=0.0669 per lattice spin. For the case of ¯kξ=1.87π(¯k=1.87π/ξ=1.87π/8≈0.73) we find ¯ϕ/2≈15◦,s os i n ¯ ϕ/2≈0.26 [see Fig. 1(a4)]. In this case we therefore find /Delta1˜P≈0.036 and skyrmion velocity V= /Delta1˜P/2π=0.0058. This is different from the value obtained in the simulations (0 .015) by a factor of ∼=2.5 [Fig. 2(a)], but considering the rough and tentative nature of the estimate, theagreement is rather good. These simple momentum conservation considerations lead us to conclude that the skyrmion is a particle with well-defined momentum, that nevertheless defies the Newtonianintuition. For instance, here an elastic scattering process causesbackwards motion of the skyrmion, which is impossible forNewtonian particles. IV . EFFECTIVE MAGNETIC FIELD To further identify the nature of the magnon skew scattering, we map the situation onto that of a charged particle (themagnon) moving in the background of a static magnetic field(the skyrmion), assuming the disturbances of the magnon onthe emergent fictitious magnetic field are small. The emergentfield corresponds to the skyrmion number, so the sign ofthe scattering direction is fixed, but would be opposite foran antiskyrmion configuration. This situation correspondsprecisely to Aharonov-Bohm (AB) scattering, and usingresults from the extensive literature [ 33–37], we shall derive an exact expression for the scattering amplitude of the magnon. In the continuum limit, the Hamiltonian Eq. ( 1) for the local moments m(x,y) reads H=/integraldisplay d 2x/bracketleftbiggJ 2(∇m)2+Dm·(∇×m)−B·m/bracketrightbigg .(7) We can make a change of variables to a complex 2-vector zρ=(z↑,z↓)( aCP1field) via m=z∗ ρσρσzσ, where σρσare the Pauli matrices and the constraint/summationtext ρ|zρ|2=1m u s tb e imposed [ 38]. The Hamiltonian turns into H=/integraldisplay d2x2J|(∇+ia+iκσ)zρ|2−B·z∗ ρσρσzσ,(8) where κ=D/2Janda=iz∗ ρ∇zρ. The Hamiltonian is invari- ant under gauge transformations zρ→zρeiεanda→a+∇ε, where ε(r) is any smooth scalar field. The gauge field is related to the Berry curvature b=∇× a, and the skyrmion number Q≡1 4π/integraltext d2xbz=1 4π/integraltext d2xm·(∂xm×∂ym) is quantized. We now separate zρ=˘zρ+z0 ρinto magnon and skyrmion contributions, and assume that a static skyrmion a0of size ξ withQ=− 1 has formed while the magnons ˘zρmove in this skyrmion background. A typical skyrmion solution in polarcoordinates is a r=0,a ϕ=r ξ2+r2. For small deviations from this background configuration we need only to consider the exchange term; the DM and Zeeman contributions are constanton this energy scale. Summarizing, we are considering thelow-energy dynamics of H LE=/integraldisplay d2x2J|(∇+ia0)˘zρ|2. (9)This is precisely the Hamiltonian of a charged particle moving in an external magnetic field b0=∇× a0. Notice that the components ˘z↑,˘z↓are now decoupled at this level of the approximation. We are interested in the scattering outcomeof an incoming plane wave, far away from the origin ofthe skyrmion. Then in this ferromagnetic regime, the spinspoint along the out-of-plane zdirection, and we can make the approximation z ↑≈1. In other words, we only consider the field ˘z↓. The problem of a charged particle scattered by a magnetic flux was intensively studied in and after the discovery ofthe Aharonov-Bohm (AB) effect [ 33–37]. There, one is usually interested in the case that the particle does not enterregions of finite magnetic flux, but nevertheless the case of auniform magnetic flux tube of radius ξhas been considered in Refs. [ 34–37]. They also consider an electrostatic shielding potential Vto prevent the particle from entering the region of nonzero flux, but the results are in fact general for any V, and the limit of V→0 may be taken without additional treatment, as we do from now on. To make use of these establishedresults we shall approximate our smooth skyrmion potentiala ϕ=r/(ξ2+r2) with that of a uniform magnetic flux: aϕ=/braceleftbigg1/r, r /greaterorequalslantξ, r/ξ2,r/lessorequalslantξ.(10) One can verify that the total fictitious flux Qis the same for both potentials (in the AB setup, the value of Qcorresponds to the product of the electric charge and magnetic flux). As the magnon will principally scatter due to the fictitious Lorentz force, this approximation will not deviate too much fromthe actual situation, and has the advantage of allowing foran exact solution. The full derivation is quite technical andnot essentially different from the earlier work and is deferredto the Appendix. The wave function can then be expressedin terms of Bessel functions, with coefficients determined bythe properties of the flux tube. The wave function outsideof the flux tube ˘z >can be written as a superposition of an incoming plane wave and a scattered spherical wave, ˘z>= exp(ikx)+F(ϕ)exp(ikr)√rwhere F(ϕ) is called the scattering amplitude. In the Appendix it is derived that the exact solution for the scattering amplitude is F(ϕ)=fAB(ϕ)+e−iπ/4 √ 2πk∞/summationdisplay n=−∞eiπ(n−|n+Q|)(e2i/Delta1n−1)einϕ. (11) Here the AB contribution fAB(ϕ) vanishes for integer skyrmion number Q, and /Delta1nis the phase shift of the nth partial wave. The scattering amplitude is evaluated numerically; the results are shown in Figs. 2(b) and 2(c).W ec l e a r l ys e ea large skew angle at the scattering of the magnon for k≈1/ξ. For both very low and very high wave numbers the skew angletends to zero, and the maximum skew angle is about 60 ◦around ¯kξ≈1.1. V . CONCLUSIONS We have studied the scattering process of magnons and a skyrmion both numerically and analytically. The numerics 064412-4THEORY OF MAGNON-SKYRMION SCATTERING IN . . . PHYSICAL REVIEW B 89, 064412 (2014) show a large skew angle of the magnon scattering, and skyrmion motion as the back action of the scattering. Wehave demonstrated that the principal contribution of the skewscattering is due to the emergent magnetic field generated bythe Berry curvature of the skyrmion. The obtained scatteringamplitude shows that the magnon skew scattering is stronglywave-number dependent, up to 60 ◦around kξ=1.1, which is consistent with the numerical results. This should be comparedwith the case of topological Hall effect of the conductionelectrons coupled to the skyrmions [ 39–42], where the Hall angle is typically of the order of 10 −3because the Fermi wave number kFof the electrons is much larger than ξ−1. For both very low and very high wave numbers the skew angle tends tozero. For large k, the skew angle is reduced and asymptotically behaves as ∝1/k. This indicates that the velocity of the skyrmion induced by the back action should increase linearlyin the large kregion of Fig. 2(a) since the momentum transfer from magnons to skyrmion is ∝k 2×/Phi1∼k2×1/k∼kin that region assuming the elastic scattering. Unfortunately, thislargeklimit was not successfully analyzed in the numerical simulation due to a technical difficulty, which requires furtherstudies. The skyrmion retains its identity during the scattering process as a result of topological protection. Furthermore,the skyrmion can be interpreted as a (semiclassical) particlewith a well-defined momentum which is however highlynon-Newtonian. The observed behavior can then simply beviewed as an elastic scattering process, and the skyrmion isnearly massless in this situation. Due to the topological nature of the interaction, the magnon scattering of a skyrmion is qualitatively different from otherscattering, namely it has a transverse component. Thereforeany experimental signature of transverse motion of magnonswould be evidence of the presence of skyrmions, sincetopologically trivial configurations such as magnetic bubblesor domain walls cannot induce skew scattering. We are ledto think that insulating systems such as Cu 2OSeO 3, in which there are no conduction electrons, would be most suitable forsuch studies. One promising way of inducing spin waves is viathe inverse Faraday effect using laser light [ 43]. ACKNOWLEDGMENTS The authors thank M. Mochizuki for providing us the basic code of the micromagnetic simulation. This work wassupported by Grant-in-Aids for Scientific Research (GrantNo. 24224009) from the Ministry of Education, Culture,Sports, Science and Technology (MEXT) of Japan, StrategicInternational Cooperative Program (Joint Research Type) fromJapan Science and Technology Agency, and by FundingProgram for World-Leading Innovative R&D on Science andTechnology (FIRST Program). A.J.B. was supported by theForeign Postdoctoral Researcher program at RIKEN. APPENDIX: DERIVATION OF THE SCATTERING AMPLITUDE Here we derive Eq. ( 11). For a0=0E q .( 9) describes plane waves of energy E=a2Jk2, where ais the lattice constant andkis the wave number. For nonzero a0the equation ofmotion for a particle of this energy reads in polar coordinates /bracketleftbigg ∂2 r+1 r∂r+1 r2[∂ϕ−ir(−Q)aϕ]2+k2/bracketrightbigg ˘z↓=0.(A1) Here we tentatively allow the skyrmion number Qto deviate from the value −1. The only term dependent on ϕis the one involving ∂ϕ, and we can make a partial wave expansion ˘z↓(r,ϕ)=/summationtext n˘zn=/summationtext nwn(r)einϕ.F o r r/greaterorequalslantξ,t h e wnare eigenfunctions of the equation /bracketleftbigg ∂2 r+1 r∂r+k21 r2(n+Q)2/bracketrightbigg w> n=0. (A2) This is precisely Bessel’s equation, and the general solution is ˘z> n=einϕ[anJ|n+Q|(kr)+bnY|n+Q|(kr)]. (A3) For the region r/lessorequalslantξ, the Schr ¨odinger equation reads /bracketleftbigg ∂2 r+1 r∂r−1 r2/parenleftbigg n+Qr2 ξ2/parenrightbigg2 +k2/bracketrightbigg w< n(r)=0.(A4) We make a change of variables v=Qr2/ξ2andfn(v)= rw< n(r). The above equation is then rewritten as /bracketleftbigg ∂2 v+1/4−n2/4 v2+k2ξ2/4Q−n/2 v−1 4/bracketrightbigg fn(v)=0. (A5) This is known as Whittaker’s equation for the parameters κ=k2ξ2/4Q−n/2 and μ2=n2/4. The solutions, known as Whittaker functions Mκ,μ(v), are not well defined for μ=− 1,−2,..., but for our purposes it suffices to choose μ=|n/2|. These solutions are fn(v)=Mκ,μ(v)=e−z/2zμ+1/2/Phi1/parenleftbig1 2+μ−κ,2μ+1,v/parenrightbig , (A6) where /Phi1is the confluent hypergeometric series, /Phi1(a,c,v )=1+a cv+a(a+1) c(c+1)1 2!v2+··· . (A7) Continuity in the wave function and its first derivative at the matching point r=ξleads to the equalities cnw< n(ξ)=anJ|n+Q|(kξ)+bnY|n+Q|(kξ), (A8) [cn∂rw< n(r)=an∂rJ|n+Q|(kr)+bn∂rY|n+Q|(kr)]r=ξ.(A9) With the notation /Phi1κ,μ(v)=/Phi1(1 2+μ−κ,2μ+1,v) one can derive ∂rw< n/vextendsingle/vextendsingle r=ξ=Mκ,μ(Q) ξ2/parenleftbigg |n|−Q+2Q∂v/Phi1κ,μ(v)/vextendsingle/vextendsingle v=Q /Phi1κ,μ(Q)/parenrightbigg . (A10) Substituting Eq. ( A8)i nE q .( A9) we eventually find bn an=−AnJ|n+Q|−∂¯rJ|n+Q|(¯r)/vextendsingle/vextendsingle ¯r=kξ AnY|n+Q|−∂¯rY|n+Q|(¯r)/vextendsingle/vextendsingle ¯r=kξ, (A11) 064412-5JUNICHI IW ASAKI, ARON J. BEEKMAN, AND NAOTO NAGAOSA PHYSICAL REVIEW B 89, 064412 (2014) where we have defined An=1 kξ/parenleftbigg |n|−Q+2Q∂v/Phi1κ,μ(v)/vextendsingle/vextendsingle v=α /Phi1κ,μ(α)/parenrightbigg . (A12) We expect to retrieve the Aharonov-Bohm result, ˘zAB=∞/summationdisplay n=−∞einϕeiδAB nJ|n+Q|(kr), (A13) where δAB=− |n+Q|π/2, in the limits of vanishing skyrmion size ξ→0 or vanishing flux Q→0. Brown [ 37] has shown that the solution an=cos/Delta1nei/Delta1neiδAB,b n=sin/Delta1nei/Delta1neiδAB,(A14) which defines the partial wave shifts /Delta1nin terms of anand bn, corresponds to an incoming plane wave and an outgoing propagating scattered wave, and this solution does reduce tothe AB results in the mentioned limits, for which all /Delta1 n≡ tan(−bn/an)→0. Brown has also shown that, for any Q, /Delta1n→0a sn→∞ , and in practice the /Delta1nvanish quicklyforn>k ξ . Writing the solution as the superposition of an incoming and a scattered wave, ˘z>=exp(ikx)+F(ϕ)exp(ikr)√r, Brown obtains the scattering amplitude, F(ϕ)=fAB(ϕ) +e−iπ/4 √ 2πk∞/summationdisplay n=−∞eiπ(n−|n+Q|)(e2i/Delta1n−1)einϕ.(A15) HerefABis the Aharonov-Bohm scattering amplitude, fAB(ϕ)eiπ/4 √ 2πksin(π|Q|)eiϕsgn(Q) cos/parenleftbig1 2ϕ/parenrightbig. (A16) The AB scattering amplitude is clearly vanishing for integer Q. We evaluate this exact solution Eq. ( A15) numerically. Here we make use of the fact that the phase shifts /Delta1ntend to zero quickly for n>k ξ , meaning that only the lowest few partial waves contribute to scattering. The scattering amplitude andthe skew angle ¯ ϕ=/integraltext ϕ|F(ϕ)| 2//integraltext |F(ϕ)|2for several values of¯kξare shown in Fig. 2. [ 1 ] T .H .R .S k y r m e , Proc. R. Soc. A 260,127(1961 ). [ 2 ] T .H .R .S k y r m e , Nucl. Phys. 31,556(1962 ). [3] D. C. Wright and N. D. Mermin, Rev. Mod. Phys. 61,385 (1989 ). [4] S. L. Sondhi, A. 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PhysRevB.103.245403.pdf

PHYSICAL REVIEW B 103, 245403 (2021) Interlayer ferromagnetism and high-temperature quantum anomalous Hall effect inp-doped MnBi 2Te4multilayers Yulei Han ,1Shiyang Sun,2Shifei Qi,2,1,*Xiaohong Xu,3,†and Zhenhua Qiao1,‡ 1ICQD, Hefei National Laboratory for Physical Sciences at Microscale, CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 2College of Physics, Hebei Normal University, Shijiazhuang, Hebei 050024, China 3Research Institute of Materials Science, and School of Chemistry and Materials Science, Shanxi Normal University, Linfen, Shanxi 041004, China (Received 1 February 2021; revised 21 May 2021; accepted 21 May 2021; published 2 June 2021) The interlayer antiferromagnetic coupling hinders the observation of quantum anomalous Hall effect in magnetic topological insulator MnBi 2Te4. We demonstrate that interlayer ferromagnetism can be established by utilizing the p-doping method in MnBi 2Te4multilayers. In two septuple layers system, the interlayer ferromagnetic coupling appears by doping nonmagnetic elements (e.g., N, P, As, Na, Mg, K, and Ca), dueto the redistribution of orbital occupations of Mn. We further find that Mg and Ca elements are the mostsuitable candidates because of their low formation energy. Although, the p-doped two septuple layers exhibit topologically trivial band structure, the increase of layer thickness to three (four) septuple layers with Ca (Mg)dopants leads to the formation of the quantum anomalous Hall effect. Our proposed p-doping strategy without introducing additional magnetic disorder not only makes MnBi 2Te4become an ideal platform to realize the high-temperature quantum anomalous Hall effect without external magnetic field, but also can compensate theelectrons from the intrinsic n-type defects in MnBi 2Te4. DOI: 10.1103/PhysRevB.103.245403 I. INTRODUCTION Quantum anomalous Hall effect (QAHE) is a typical topo- logical quantum phenomena with quantized Hall resistanceand vanishing longitudinal resistance in the absence of ex-ternal magnetic field [ 1–3]. It is promising in designing low-power electronic devices due to its dissipationless elec-tronic transport properties. Although it was first theoreticallyproposed by Haldane in 1988 [ 4], the exploration of the QAHE began to attract huge interest ever since the first ex-foliation of monolayer graphene in 2004 [ 5]. After that, there have been various proposed recipes in designing the QAHE[6–17], among which the magnetic topological insulator is the most favorable system by both theoretical and experimentalstudies due to its inherently strong spin-orbit coupling [ 18,19]. To realize the QAHE, the ferromagnetism is prerequisite andcan be engineered by magnetic doping [ 9,20–26]. It was indeed theoretically proposed [ 9] in 2010 and later experimen- tally observed in 2013 in the magnetically doped topologicalinsulator thin films [ 27–31]. However, the major obstacle, hindering the practical applications of QAHE, is the extremelylow QAHE-observation temperature. Therefore, more effortsare being made to increase the QAHE observation temperaturevia various doping schemes in topological insulators [ 24–26]. Alternatively, MnBi 2Te4, composed of septuple-layer (SL) blocks stacking along the [0001] direction via van der Waals *Correspondence author: qisf@hebtu.edu.cn †Corresponding author: xuxh@dns.sxnu.edu.cn ‡Corresponding author: qiao@ustc.edu.cninteraction [see Figs. 1(a) and1(b)], becomes an appealing host material to realize exotic topological phases [ 32–35]. It exhibits intrinsic magnetism, following the A-type anti-ferromagnetic order, where the neighboring ferromagneticMn layers are coupled in an antiparallel manner [ 33,34]. It was reported that the QAHE can be observed at 6.5 Kin a five-SL MnBi 2Te4flake, when an external magnetic field is applied; while the zero-field QAHE can only be observed at 1.4 K with ultrahigh sample quality [ 36,37]. The sensitivity of the QAHE on the sample quality in-dicates that the interlayer antiferromagnetic coupling is acritical obstacle in the QAHE formation, and the interlayerferromagnetism is highly desired. The interlayer magneticcoupling of van der Waals materials is determined by the d- orbital occupation of transition metals [ 38–40]. One approach to manipulate the interlayer ferromagnetism is by stackingdifferent d-orbital occupied van der Waals materials, e.g., MnBi 2Te4/V(Eu)Bi 2Te4[39,40]. As demonstrated in below, another most efficient approach is by directly doping nonmag-netic p-type elements into MnBi 2Te4. In this work, we provide a systematic study on the magnetic and electronic properties of nonmagnetic p-doped MnBi 2Te4multilayers by using first-principles calculation methods. In two-SL MnBi 2Te4, the interlayer ferromagnetic coupling can be realized by doping various nonmagnetic p- type elements (e.g., N, P, As, and Na, Mg, K, Ca) with theCurie temperature up to T C=54 K. The underlying phys- ical origin is the redistribution of d-orbital occupation of Mn element induced hopping channels between t 2gand e g from different SLs. Although it is topologically trivial in the 2469-9950/2021/103(24)/245403(10) 245403-1 ©2021 American Physical SocietyHAN, SUN, QI, XU, AND QIAO PHYSICAL REVIEW B 103, 245403 (2021) FIG. 1. Top view and side views of crystal structures of 2-4 SLs MnBi 2Te4and formation energies of p-type doped systems. (a) Two-SL MnBi 2Te4with one N /P/As substitution at sites Te 1-Te 4, or one Na /Mg/K/Ca substitution at sites Bi 1-Bi 2; (b) The 3-4 SLs MnBi 2Te4with one N/P/As substitution at sites Te 1-Te 6, or one Na /Mg/K/Ca substitution at sites Bi 1-Bi 4. [(c) and (d)] Formation energies of (c) N /P/As or (d) Na /Mg/K/Ca doped two-SL MnBi 2Te4as a function of the host element chemical potentials. p-doped two-SL case, the topological phase transition occurs to harbour the high-temperature QAHE with a Chern numberofC=−1 when the system thickness is increased, i.e. Ca- doped three-SL, and Ca /Mg-doped four-SL MnBi 2Te4, with the interlayer ferromagnetism still being preserved. Our workdemonstrates a p-doping mechanism in producing ferromag- netism in MnBi 2Te4to form the high-temperature QAHE, which is experimentally accessible. II. CALCULATION METHODS Our first-principles calculations are performed by using the projected augmented-wave method [ 41] as implemented in the Vienna ab initio simulation package (V ASP) [ 42,43]. The gen- eralized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof type is utilized to treat the exchange-correlationinteraction [ 44]. In our calculations, the lattice constant of MnBi 2Te4is chosen as the experimental value of a0=4.33 Å [45]. We use zero damping DFT-D3 method [ 46,47] to de- scribe the van der Waals interaction of adjacent SLs ofMnBi 2Te4. All atoms are allowed to move during the struc- tural optimization. The kinetic energy cutoff and energyconvergence threshold are set to be 450 and 10 −6eV , respec- tively. The Hellmann-Feynman force tolerance criterion forconvergence is 0.01 eV /Å. The Gaussian smearing method with a smearing width of 0.01 eV is adopted. A vacuum spaceof 20 Å is considered to avoid interaction between neigh-boring slabs. A /Gamma1-centered 7 ×7×1( 5×5×1)kmesh is adopted for the 2 ×2( 3×3) supercell. The 3 dstates of Mn are treated with GGA +Uapproach [ 48,49], with U =5.34 eV , as in previous studies [ 50,51]. The topological related quantities are calculated by constructing maximally localizedWannier function as implemented in the Wannier90 package[52]. The Curie temperature T Cwas estimated within the mean-field approximation kBTC=2/3Jx[53], where kBis the Boltzmann constant, xis the dopant concentration, and Jis the exchange parameter obtained from the total energy difference between ferromagnetic and antiferromagnetic con-figurations in different heterostructures. The phonon spectrumcalculations are carried out by using the density functionalperturbation theory as implemented in the PHONOPY package [54]. III.p-TYPE DOPING SCHEME IN MnBi 2Te4 It was known that interlayer magnetic coupling in MnBi 2Te4is dominated by p-orbital mediated superexchange interaction, while d-orbital occupation has vital influence on the sign of interlayer magnetic coupling [ 39,40]. Based on the superexchange mechanism, doping p-type nonmagnetic elements can change the d-orbital occupation of Mn in the same SL. With the aid of hopping channel between 3 d-orbital of Mn in undoped SL and virtual 3 d-orbitals of Mn in p-doped SL, the interlayer ferromagnetic coupling becomes possible.In experiments, MnBi 2Te4was found to be electron-doping due to their intrinsic n-type defects [ 33]. Therefore another natural benefit of p-doping is the charge-compensation, which is a prerequisite for realizing the QAHE. We now first study the possibility of pdoping in MnBi 2Te4multilayers. Substituting Te /Bi atoms by 245403-2INTERLAYER FERROMAGNETISM … PHYSICAL REVIEW B 103, 245403 (2021) FIG. 2. Phonon dispersions of monolayer MnBi 2Te4with (a) pristine structure, (b) Ca dopant and (c) Mg dopant. A 2 × 2M n B i 2Te4supercell is used to calculate phonon dispersion of Ca/Mg doped system. nonmagnetic dopants is experimentally feasible, as imple- mented in Bi 2Te3-family topological insulators. The Te and Bi elements in MnBi 2Te4exhibit respectively 2−and 3+ valence states. In order to employ p-doping, the correspond- ing substituted elements can be 3−,1+, and 2+valence states, respectively. Therefore, the typical candidates of p- type nonmagnetic dopants include N /P/As for Te sites, or Na/Mg/K/Ca for Bi sites. As displayed in Fig. 1(a) for at w o - S LM n B i 2Te4, there are four Te substitutional sites (i.e., Te 1,T e 2,T e 3,T e 4), and two Bi substitutional sites (i.e., Bi1,B i 2). The formation energies can be evaluated from the expression [ 55–57]:/Delta1HF=ED tot−Etot−/summationtextniμi, where ED tot, Etotare respectively the total energies of the p-doped and undoped systems, μiis the chemical potential for species i (host atoms or dopants), and niis the corresponding number of atoms added to or removed from the system. Considering the formation energies of N /P/As substitu- tions at Te sites in one SL as displayed in Fig. 1(c), one can find that the Te 4site is preferred. The formation energy of N substitution (2.5–3.0 eV) is larger than that of either P(0.6–1.2 eV) or As (about 0.4–1.0 eV). For Na /Mg/K/Ca substitutions at Bi sites in Fig. 1(d), one can find that the Bi 2 site is preferred, and the formation energies of Bi-site substi- tutions are always lower than those of Te-site substitutions. Inparticular, the formation energies of Na /Mg/Ca-substituted Bi 2site are negative, suggesting that these dopings are ex- perimentally feasible. As far as we know, the C-doped ZnOcan also be experimentally realized, even thought the esti-mated formation energy of C substituted O in ZnO is about5.3 eV [ 58], which is larger than those of aforementioned p-type dopants in MnBi 2Te4. Moreover, phonon dispersions of two most feasible dopants, i.e., Mg and Ca, are calculatedas displayed in Fig. 2, which suggests the stability of p-doped MnBi 2Te4systems. Hereinbelow, we concentrate on the most stable substitional sites (i.e., Te 4and Bi 2) to study the mag- netic and electronic properties of p-doped two-SL MnBi 2Te4. IV . INTERLAYER FERROMAGNETISM FROM p-DOPING Figures 3(a)and3(b) display the energy differences ( /Delta1E= EFM−EAFM) between interlayer ferromagnetic (FM) and antiferromagnetic (AFM) states of the optimal configurationsat different p-doped concentrations in two-SL MnBi 2Te4. FIG. 3. [(a)–(c)] The energy differences between interlayer fer- romagnetic (FM) and interlayer antiferromagnetic (AFM) states ofthe optimal configurations in (a) N /P/As doped two-SL MnBi 2Te4 at 3.13% and 1.39% concentrations, (b) Na /Mg/K/Ca doped two-SL MnBi 2Te4at 6.25% and 2.78% concentrations, (c) Mg /Ca doped 3-4 SLs MnBi 2Te4at 4.17% and 3.13% concentrations. [(d) and (e)] Differential charge density of (d) Ca doped and (e) pristine two-SL MnBi 2Te4. Yellow and green isosurface represent respectively charge accumulation and reduction. (f) Local density of states of Ca doped and pristine two-SL MnBi 2Te4.T e -p,B i -p,a n dM n - dorbitals [t2gand e g] in each SL of MnBi 2Te4are displayed. In the absence of doping, the two-SL MnBi 2Te4indeed exhibits interlayer antiferromagnetism (see Table I). The introduction of p-type dopants leads to /Delta1E<0, strongly indicating that interlayer ferromagnetic state is more stablethan the interlayer antiferromagnetic state. For N /P/As dop- ing at Te 4site [see Fig. 3(a)],/Delta1Echange respectively from −12.4/−11.2/−10.3 meV to −24.5/−19.5/−17.2 meV , along with the increase of doping concentration. ForNa/Mg/K/Ca substitution at Bi 2site [see Fig. 3(b)],/Delta1E are respectively −43.5/−11.8/−35.4/−14.5 meV at 2.78% doping concentration, and −48.6/−24.9/16.7/−19.2 meV at 6.25% doping concentration. Besides the energy difference for optimal doping sites, we also investigate magnetic properties of the remaining dopingsites. Figure 4displays the /Delta1 Eof different configurations in p-doped two-SL MnBi 2Te4. One can find that all p-doped sys- tems prefer interlayer ferromagnetic coupling, and the dopingsites near the van der Waals gap (e.g., Te 4for N/P/As, Bi 2for Na/Mg/K/Ca) exhibit much larger ferromagnetic coupling strength. In addition, the ferromagnetic Curie temperature plays a crucial role in determining the QAHE observation 245403-3HAN, SUN, QI, XU, AND QIAO PHYSICAL REVIEW B 103, 245403 (2021) FIG. 4. The energy differences between interlayer ferromag- netic(FM) and interlayer antiferromagnetic (AFM) states of different configurations in N /P/As doped two-SL MnBi 2Te4at (a) 3.13% and (c) 1.39% doping concentrations. Na /Mg/K/Ca doped two-SL MnBi 2Te4at (b) 6.25% and (d) 2.78% doping concentrations. temperature. The estimated Curie temperature from mean- field theory is listed in Table I, which ranges between 15.7 and 53.7 K depending on the dopants. For example, the Curietemperature of Ca-doped MnBi 2Te4can reach TC=21.2K at 6.25% doping concentration, which can be further raisedwith the increase of doping concentration. Note that the higherdoping concentration may decrease the spin-orbit coupling ofthe whole system. For thicker MnBi 2Te4films (i.e., three-SL and four-SL films), we calculate the /Delta1Eof two most favorable dopants (Mg and Ca). For different substitutional sites, it is foundthat Bi 3(Bi4) site is most stable in three-SL (four-SL) MnBi 2Te4films. And for different magnetic configurations of the most stable doping site, the energy differences showthat the ferromagnetic states are preferred (see Tables IIand III). Figure 3(c) displays the energy difference /Delta1 Eof one Mg or Ca dopant at Bi 3(Bi4) site in 2 ×2 supercells of three-SL (four-SL) MnBi 2Te4. One can see that ferromagnetic coupling strength is dependent on doping concentration thatis determined by the number of layers, i.e., for one dopantthe increase of septuple layers leads to rapidly decrease offerromagnetic coupling strength. Therefore larger ferromag-netic coupling strength in a multilayer system requires higherp-type doping concentration. The formation mechanism of interlayer ferromagnetic cou- pling can be explained from the differential charge densityand local density of states. Let us take the Ca-doped two-SLMnBi 2Te4as an example [see in Fig. 3(d)]. In the pristine case [see Fig. 3(e)], the charge distribution in the top SL is the same as that in the bottom SL. After Ca-doping inbottom SL, the charge of Mn atoms in the same SL is clearlydecreased whereas that in top SL remains nearly unchanged.Such a charge redistribution leads to new hopping channelsbetween Mn atoms in adjacent SLs. In pristine case, the t 2g and e gorbitals are fully occupied, leading to the absence of electron hopping between t 2gand e gorbitals. While as displayed in Fig. 3(f), the decrease of d-orbital occupation in bottom SL generates new hopping channels from t 2g(top SL) to e g(bottom SL) and e g(top SL) to e g(bottom SL),FIG. 5. Band structures and corresponding band gaps of Mg- and Ca-doped 2 SLs MnBi 2Te4with optimal configurations along high-symmetry lines. [(a) and (b)] Doping one Mg or Ca atom in 2×2t w o - S LM n B i 2Te4with the concentration of 6.25%. [(c) and (d)] Doping one Mg or Ca atom in 3 ×3t w o - S LM n B i 2Te4with the concentration of 2.78%. which are allowed for ferromagnetic coupling. In addition, in Fig. 3(f), one can also find that a large spin polarization appears in the Te element after Ca doping, which suggests thatthe interlayer ferromagnetic coupling in Ca-doped two-SLMnBi 2Te4is mediated via the interlayer Te-Te superexchange interaction. V . ELECTRONIC STRUCTURES AND TOPOLOGICAL PROPERTIES Next, we explore the electronic band structures of the Mg and Ca doped multi-SL MnBi 2Te4(see Figs. 10and11 for band structures of other pdopants). Figure 5displays the band structure along high-symmetry lines of the optimalconfigurations of Mg and Ca doped two-SL MnBi 2Te4.A s illustrated in Figs. 5(a) and5(b), a band gap about 53.6 meV (26.9 meV) opens at /Gamma1point with Mg (Ca) dopant for a dop- ing concentration of 6.25%. When the doping concentrationreduces to 2.78%, the band gap decreases to about 35.0 meV(18.9 meV) for Mg (Ca) dopant [see Figs. 5(c) and5(d)]. To verify whether such a gap can host the QAHE or not,one can directly calculate the anomalous Hall conductanceσ xyby integrating Berry curvature of the occupied valence bands [ 59,60]. Unfortunately, we obtained σxy=0e2/hfor all p-doped two-SL MnBi 2Te4, indicating that it is still a topo- logical trivial phase, even though the ferromagnetism is wellestablished. The possible reasons include: (i) the decrease ofspin-orbit coupling originated from the light doping elements,and (ii) the film thickness influence [ 37,51]. To address these concerns, we first choose to dope some heavy metal elements(i.e., Sn, Pb, In, Tl) in two-SL MnBi 2Te4systems. As dis- played in Table I, doping In or Tl results in the interlayer anti-ferromagnetic coupling; whereas although doping Sn orPb gives rise to interlayer ferromagnetic coupling, no bandgap opens at moderate doping concentrations (see Fig. 10). 245403-4INTERLAYER FERROMAGNETISM … PHYSICAL REVIEW B 103, 245403 (2021) FIG. 6. Band structures along high-symmetry lines of MnBi 2Te4 doped with Mg in (a) three-SL and (b) four-SL, doped with Ca in (c) three-SL and (d) four-SL, and (e) doped with two Ca in three-SL.The inset displays the anomalous Hall conductivity as a function of Fermi energy. (f) The dependence of band gap (solid lines) and Chern number (dashed lines) on the doping concentrations of Mg /Ca. The decrease of doping concentration indicates the increase of number of MnBi 2Te4SL. We then consider the influence of film thickness of MnBi 2Te4 in below. Figures 6(a) and6(b) display respectively the band struc- tures of Mg-doped three- and four-SL MnBi 2Te4, where the corresponding band gaps are respectively 24.9 meV (at4.17% doping concentration) and 7.0 meV (at 3.13% dopingconcentration). The Hall conductance σ xyevaluation gives respectively 0 and −1 in the units of e2/hfor three-SL and four-SL Mg-doped systems, strongly signaling a topologicalphase transition from trivial insulator to the QAHE with theincrease of film thickness. For the Ca-doped cases as dis-played in Figs. 6(c) and6(d), one can see that the band gaps are respectively 13.7 meV (three-SL) and 6.8 meV (four-SL).Surprisingly, the Hall conductance in the band gap is σ xy= −e2/hfor both three- and four-SL Ca-doped MnBi 2Te4. Therefore the increase of film thickness can lead to a topo-logical phase transition in p-doped MnBi 2Te4multilayers. which can be attributed to magnetic Weyl semimetal natureof ferromagnetic MnBi 2Te4, as observed in similar systems with thickness dependent Chern number [ 37,39]. For three-SL Ca-doped system, we also investigate the role of doping concentration on the electronic properties byincluding two Ca dopants at different substitutional sites. Wefind that all the calculated configurations display interlayerferromagnetism, and Ca 2Ca4doped configuration is preferred with the Curie temperature of 52 K (see Fig. 12and Table IV). As illustrated in Fig. 6(e), the band gap slightly decreases to 11.2 meV , with the nontrivial topology being preserved,but the Curie temperature is greatly enhanced from 20.3 K(one Ca dopant) to 52.0 K (two Ca dopants). Figure 6(f)FIG. 7. [(a) and (b)] Formation energies of (a) N /P/As or (d) Na /Mg/K/Ca doped two-SL MnBi 2Te4as a function of the host element chemical. (c) The energy differences between interlayer ferromagnetic(FM) and interlayer antiferromagnetic (AFM) states of the most stable configurations in (a) and (b). The red dashedline represents energy difference of pristine two-SL MnBi 2Te4as a reference. summarizes the band gaps and Hall conductance as functions of doping concentration. Compared with Mg dopant, the Cadoped MnBi 2Te4is preferred since that the topological phase appears in system with thinner thickness. Therefore the Mg-and Ca-doped MnBi 2Te4multilayers are beneficial for realiz- ing the high-temperature QAHE. VI. FORMATION ENERGIES AND MAGNETIC PROPERTIES OF ANTISITE SUBSTITUTIONS In above, we have studied formation energies of N /P/As substitutions at Te sites and Na /Mg/K/Ca substitutions at Bi sites. Here we consider two type of representative anti-site substitutions, i.e., N /P/As substitutions at Bi sites and Na/Mg/K/Ca substitutions at Mn sites. Figures 7(a)and7(b) display the corresponding formation energies as a function ofthe host element chemical potentials. We can observe that theBi 1site is preferred for N /P/As substitutions and the positive formation energies is similar to that of N /P/As substitutions at Te sites as displayed in Fig. 1(c).F o rN a /Mg/K/Ca substi- tutions at Mn sites, the formation energies is also negative. It is FIG. 8. Formation energies of Na /Mg/K/Ca doped Bi /Mn in two-SL MnBi 2Te4. The chemical potentials of Na /K/Mg/Ca are evaluated by choosing Na 2Te, K 2Te, MgTe, and CaTe as reference. 245403-5HAN, SUN, QI, XU, AND QIAO PHYSICAL REVIEW B 103, 245403 (2021) FIG. 9. (a) Two-SL MnBi 2Te4with one pair of native antisite defect Mn Biand Bi Mn. The six substitutional sites are labeled as Bi 1-Bi 5 and Mn 6.( b )F o r m a t i o ne n e r g i e so fM g /Ca substitutions at the most stable Bi 5site and at Mn site. The gray dashed line represents formation energy of native antisite defect. (c) The energy differences between interlayer ferromagnetic(FM) and interlayer antiferromagnetic (AFM) states of the six configurations doped with Mg /Ca. The presence of native antisite defect displays weak FM coupling with energy difference of∼− 0.32 meV . worth noting that the chemical potentials from single element are usually larger than that from compound, resulting in aslightly underestimated formation energy. To quantitativelycompare the formation energies of Mg /Ca dopants in Bi /Mn sites, we choose MgTe and CaTe as reference to evaluate thechemical potentials of Mg /Ca. Figure 8displays formation energies of Mg /Ca doped Bi /Mn in two-SL MnBi 2Te4.W e can find that formation energy for dopants at Bi site is lowerthan that at Mn site in almost all range, indicating that Bi siteis preferred for Mg /Ca dopants. Figure 7(c) shows the corresponding energy difference between FM and AFM states. The presence of N /P/As sub- stitutions at Bi sites does not obviously change the AFMcoupling compared with pristine two-SL MnBi 2Te4.F o rN a and K, the substitutions at Mn site displays interlayer FMcoupling due to different valence states between Na /K( 1 +) and Mn (2+) elements induced charge redistribution. For Mg FIG. 10. Band structures and corresponding band gaps of p- doped two-SL (2 ×2) MnBi 2Te4with optimal configurations along high-symmetry lines. (a)–(i) are respectively for N, P, As, Na, Mg, K, Ca, Sn, and Pb doped two-SL MnBi 2Te4. The doping concentrations are respectively 3.13% and 6.25% for substituted Te and Bi sites.and Ca substitutions at Mn site, interlayer AFM coupling is preserved since the same valence states between Mg /Ca and Mn elements. These results indicate that, even if Mg /Ca substitutions at Mn sites, the magnetic coupling strength ofMnBi 2Te4is almost unchanged. Therefore the p-dopant at Bi sites plays a crucial role in determining magnetism ofMnBi 2Te4. Besides above antisite substitutions, we also explore the doping possibility in the presence of native antisite defect, i.e.,Mn Biand Bi Mn. Figure 9(a)displays the six substitutional sites labeled as Bi 1-Bi5and Mn 6. We find that Bi 5site are preferred for Bi substitution. Formation energies of dopants at Bi 5and Mn 6sites are shown in Fig. 9(b), where formation energy of the native antisite defect Mn Biand Bi Mnis depicted in gray for comparison [The MgTe and CaTe are chosen as reference toevaluate the chemical potentials of Mg /Ca]. We can observed that Mg /Ca substitutions at Bi sites have lower formation FIG. 11. Band structures and corresponding band gaps of p- doped two-SL (3 ×3) MnBi 2Te4with optimal configurations along high-symmetry lines. (a)-(i) are respectively for N, P, As, Na, Mg, K, Ca, Sn, and Pb doped two-SL MnBi 2Te4. The doping concentrations are respectively 1.39% and 2.78% for substituted Te and Bi sites. 245403-6INTERLAYER FERROMAGNETISM … PHYSICAL REVIEW B 103, 245403 (2021) TABLE I. The band gap for the ground states, the energy difference /Delta1E=EFM−EAFM, and the estimated Curie temperature T Cofp-doped two-SL MnBi 2Te4with the optimal configurations. For 2 ×2M n B i 2Te4supercell, the doping concentrations are respectively 3.13% and 6.25% for substituted Te and Bi sites. For 3 ×3M n B i 2Te4supercell, the doping concentrations are respectively 1.39% and 2.78% for substituted Te and Bi sites. The red color indicates the antiferromagnetic ground state. 2×2 supercell 3 ×3 supercell Structure Gap (meV) /Delta1E(meV) T C(K) Gap (meV) /Delta1E(meV) T C(K) N4(N1) 40.0 −24.5 27.1 5.4 −2.0 2.3 P4 6.0 −19.5 21.6 0.4 −11.2 12.3 As4 5.3 −17.2 19.0 4.0 −10.3 11.4 Na2 9.3 −48.6 53.7 15.5 −43.5 48.1 Mg2 53.6 −24.9 27.5 35.0 −11.8 13.0 K2 25.9 −16.7 18.4 3.8 −35.4 39.1 Ca2 26.9 −19.2 21.2 18.9 −14.5 16.0 Sn2 Metallic −14.2 15.7 2.8 −14.3 15.8 Pb2 Metallic −17.1 18.9 13.3 −7.5 8.3 In2 197.9 1.7 – 145.5 3.1 – Tl2 163.3 6.1 – 149.5 7.5 – MnBi 2Te4 80.0 1.0 – 2.3 – energy than that at Mn sites, indicating that Bi site is preferred for Mg /Ca dopants in the presence of native antisite defect. Figure 9(c)displays the energy differences between interlayer FM and AFM states of the six configurations doped withMg/Ca. In the presence of native antisite defect, the system demonstrates weak FM coupling ( −0.32 meV), whereas the further inclusion of Mg /Ca dopants at Bi or Mn sites greatly enhances FM coupling strength of MnBi 2Te4. Based on the results displayed in this Section, we can con- clude that (i) Mg /Ca substitutions at Bi sites are preferred than that at Mn sites; (ii) magnetic coupling strength of MnBi 2Te4 is almost unchanged even if Mg /Ca substitutions at Mn sites; (iii) in the presence of native antisite defect, Mg /Ca doped system displays interlayer ferromagnetism regardless of thedoping sites. VII. SUMMARY In conclusion, we propose that a feasible p-type doping strategy in MnBi 2Te4can be used to realize interlayer fer- romagnetism and the high-temperature QAHE. We provideproof-of-principle numerical demonstration that (1) interlayerferromagnetic transition can appear when some nonmagneticp-type elements are doped into MnBi 2Te4; (2) band structures and topological property calculations show that Ca- and Mg-doped MnBi 2Te4multilayer can realize the QAHE with Chern number of C=−1. Experimentally, Mg, Ca and some nonmagnetic elements doped topological insulators have been successfully fabricatedin order to tune carrier type and density [ 61–63]. For example, to compensate the n-type carrier induced by Se vacancies in topological insulator Bi 2Se3, a small concentration of Ca is doped, and insulating behavior is preserved whereas theFermi level is tuned into the band gap [ 61]. For the MnBi 2Te4, from our calculation, the formation energies of Ca substitutionare only −2.5 to −3.0 eV . Hence the p-type Ca dopants in MnBi 2Te4are very feasible in experiment. The merits of p-type doping in MnBi 2Te4is that it can not only result in interlayer ferromagnetic coupling without introducing addi-tional magnetic disorder, but also compensate the intrinsicn-type carrier, which in principle guarantees the insulating state and is beneficial to realize the high-temperature QAHEin MnBi 2Te4. Our work provide a highly desirable scheme to overcome the difficulty of the observing of the QAHE inMnBi 2Te4without applying external magnetic field. ACKNOWLEDGMENTS This work was financially supported by the NNSFC (No. 11974098, No. 11974327 and No. 12004369), TABLE II. The structural and magnetic properties of Ca /Mg doped three-SL MnBi 2Te4with doping concentration of 4.17%. The spin direction of each septuple layer is denoted by the up /down arrow. The ground state of each dopant is denoted by red. The energy differences between the specific structure and ground state are shown. The energy is in unit of meV . Structure /Delta1E(↑↓↑ ) /Delta1E(↑↑↓ ) /Delta1E(↑↓↓ ) /Delta1E(↑↑↑ ) Mg1 91.0 84.1 91.8 84.7 Mg2 46.9 24.5 44.9 24.7 Mg3 25.1 6.3 20.9 0 Ca1 107.3 99.0 108.1 100.3 Ca2 50.6 35.3 48.3 34.2 Ca3 18.3 3.2 15.0 0 MnBi 2Te4 0 1.2 1.1 2.5 245403-7HAN, SUN, QI, XU, AND QIAO PHYSICAL REVIEW B 103, 245403 (2021) TABLE III. The structural and magnetic properties of Ca /Mg doped four-SL MnBi 2Te4with doping concentration of 4.17%. The spin direction of each septuple layer is denoted by the up /down arrow. The ground state of each dopant is denoted by red. The energy differences between the specific structure and ground state are shown. The energy is in unit of meV . Structure /Delta1E(↑↓↑↓ )/Delta1E(↑↑↑↓ )/Delta1E(↑↑↓↑ )/Delta1E(↑↑↓↓ )/Delta1E(↑↓↑↑ )/Delta1E(↑↓↓↑ )/Delta1E(↑↓↓↓ )/Delta1E(↑↑↑↑ ) Mg2 41.3 20.9 23.0 24.2 42.7 43.4 39.2 22.0 Mg3 26.3 1.8 9.3 9.6 26.7 20.5 20.2 3.4 Mg4 20.7 3.5 14.7 11.7 17.9 8.4 6.8 0 Ca2 48.1 33.0 35.2 36.2 48.8 44.9 45.9 34.0 Ca3 44.6 6.7 13.1 13.1 25.6 17.6 18.4 7.4 Ca4 19.6 2.7 11.5 7.7 14.6 7.4 5.2 0 MnBi 2Te4 0 1.9 0.9 1.8 0.9 0.9 1.8 2.8 Natural Science Foundation of Hebei Province (A2019205037), China Postdoctoral Science Foundation(2020M681998) and Science Foundation of Hebei NormalUniversity (2019B08), Fundamental Research Funds for theCentral Universities (WK2030020032 and WK2340000082)Anhui Initiative in Quantum Information Technologies.The Supercomputing services of AM-HPC and USTC aregratefully acknowledged. APPENDIX A: BAND STRUCTURES OF p-DOPED TWO-SL MnBi 2Te4 In Figs. 10and11, we plot the band structures of p-doped two-SL MnBi 2Te4with optimal configurations with different doping concentrations. In Fig. 10, we can observe that finite gaps are opened around /Gamma1point except for Sn- and Pb-doped systems. When reduces doping concentration, as displayed inFig.11, one can find that the gaps for N /P/As/Na/Mg/K/Ca doped systems are decreased whereas small gaps exists forSn/Pb doped systems. In Table I, we summarize the band gap for the ground states, the energy difference /Delta1 E=EFM−EAFM, and the es- timated Curie temperature T Cofp-doped two-SL MnBi 2Te4 with the optimal configurations. Besides the p-doped systems showing interlayer ferromagnetism as discussed above, it isnoteworthy that In /Tl heavy metal-doped MnBi 2Te4exhibits FIG. 12. (a) The substitutional sites for two Ca dopants in three- SL MnBi 2Te4. (b) The comparison of energy differences between FM and AFM states with /without spin-orbit coupling in p-doped two-SL MnBi 2Te4.a enhanced interlayer antiferromagnetic coupling compared with the pristine two-SL MnBi 2Te4. APPENDIX B: MAGNETIC PROPERTIES OF p-DOPED THREE- AND FOUR-SL MnBi 2Te4 Table IIdisplays the structural and magnetic properties of Ca/Mg doped three-SL MnBi 2Te4with doping concentration of 4.17%. The corresponding doping sites are shown in themain text. One can find that Bi 3substitutional site is most stable, and the ferromagnetic ground state is preferred inMg/Ca doped three-SL MnBi 2Te4.F o rC a /Mg doped four- SL MnBi 2Te4, as shown in Table III, we can observe that Bi4substitutional site is most stable, and the ferromagnetic ground state is preserved. APPENDIX C: STRUCTURAL AND MAGNETIC PROPERTIES OF THREE-SL MnBi 2Te4WITH TWO Ca DOPANTS Figure 12(a) displays the possible substitutional sites for two Ca dopants in three-SL MnBi 2Te4, where the sites near van der Waals gap are considered because they are morestable. Due to the inversion symmetry, there are four com-binations of doping sites as shown in Table IV. We can find that the Ca 2Ca4doped configuration is most stable with inter- layer ferromagnetic coupling, and the Curie temperature canapproach 52 K. APPENDIX D: MAGNETIC COUPLING WITH /WITHOUT SPIN-ORBIT COUPLING IN p-DOPED TWO-SL MnBi 2Te4 In above magnetic coupling calculation, the spin-orbit coupling is not included. Fig. 12(b) displays the energy dif- ferences /Delta1Ebetween FM and AFM states with /without TABLE IV . The structural and magnetic properties of two Ca dopants in three-SL MnBi 2Te4. The energy differences between the specific structure and ground state are shown. The ground state is denoted by red. The energy is in unit of meV . Structure /Delta1E(↑↓↑ ) /Delta1E(↑↑↑ )T C(K) Ca2Ca4 47.1 0 52.0 Ca1Ca2 130.0 98.2 35.1 Ca1Ca4 61.5 17.2 48.9 Ca2Ca3 98.0 42.6 61.3 245403-8INTERLAYER FERROMAGNETISM … PHYSICAL REVIEW B 103, 245403 (2021) spin-orbit coupling in p-doped two-SL MnBi 2Te4. 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PhysRevB.98.214428.pdf

PHYSICAL REVIEW B 98, 214428 (2018) Temperature-dependent properties of CoFeB /MgO thin films: Experiments versus simulations H. Sato,1,2,3,4,*P. Chureemart,5,6F. Matsukura,1,2,3,4,7R. W. Chantrell,5H. Ohno,1,2,3,4,7and R. F. L. Evans5,† 1Center for Spintronics Research Network, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan 2Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan 3Center for Spintronics Integrated Systems, 2-1-1 Katahira, Aoba-ku, Tohoku University, Sendai 980-8577, Japan 4Center for Innovative Integrated Electronic Systems, Tohoku University, 468-1 Aramaki Aza Aoba, Aoba-ku, Sendai 980-0845, Japan 5Department of Physics, The University of York, York YO10 5DD, United Kingdom 6Department of Physics, Mahasarakham University, Mahasarakham 44150, Thailand 7WPI-Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan (Received 17 October 2017; revised manuscript received 1 July 2018; published 14 December 2018) CoFeB/MgO heterostructures are a promising candidate for an integral component of spintronic devices due to their high magnetic anisotropy, low Gilbert damping, and excellent magnetoresistive properties. Here, we presentexperimental measurements and atomistic simulations of the temperature and CoFeB thickness dependence ofspontaneous magnetization and magnetic anisotropy in CoFeB/MgO ultrathin films. We find that the thermalfluctuations are different between the bulk and interface magnetizations, and that the interfacial anisotropyoriginates from a two-site anisotropic exchange interaction. These effects lead to a complex temperatureand thickness dependence of the magnetic properties critical to device operation and stability at elevatedtemperatures. DOI: 10.1103/PhysRevB.98.214428 I. INTRODUCTION In recent years, an interfacial anisotropy at ferromagnetic metal (FM)/oxides has been an interesting subject in the fieldof spintronics because of its importance for applications. Forinstance, the interfacial anisotropy reduces the intrinsic crit-ical current for spin-transfer-torque (STT)-induced magneti-zation switching in magnetic tunnel junctions (MTJs) with anin-plane easy axis (i-MTJs) without reduction of the thermalstability factor. In addition, the interfacial anisotropy enablesthe manufacture of MTJs with a perpendicular easy axis(p-MTJs) by reducing the FM layer thickness, which allows ahigher efficiency for the STT switching compared to i-MTJs. A large interfacial perpendicular anisotropy energy den- sityK iof 1.18 mJ/m2at a FM/oxide interface was re- ported in a single-crystal Fe substrate/MgO/Fe/Au struc-ture [ 1]. The presence of the interfacial anisotropy at FM/oxide interfaces was also reported in polycrystalline filmswith Pt /Co(Fe) /MO x(M:A l ,M g ,T a ,o rR u )[ 2,3] and MgO/CoFeB/Pt structures [ 4]. In Ta/CoFeB/MgO structure, a technologically relevant structure owing to its high tun-nel magnetoresistance ratio [ 5,6], the presence of K iwas also found [ 7]. This brought about the demonstration of high-performance p-MTJs with Ta/CoFeB/MgO at a junction *hsato@riec.tohoku.ac.jp †richard.evans@york.ac.uk Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.diameter of 40 nm [ 8], which triggered ongoing intensive studies on p-MTJs with the CoFeB/MgO system at reduceddimensions less than 20 nm [ 9,10]. First-principles calculations indicated that the interfacial perpendicular anisotropy is brought about by the hybridizationof Fe 3 dand O 2 porbitals [ 11]. The calculation is supported by an experimental work on x-ray magnetic circular dichro-ism, which showed that the anisotropy is related to Fe 3 d orbital anisotropy at the CoFeB/MgO interface [ 12]. While the origin of the interfacial anisotropy appears to be wellunderstood, the origin of its temperature dependence, whichis important for further development of p-MTJs, is still to beelucidated. As shown by Callen and Callen, the temperature Tdepen- dence of the anisotropy energy density Kof ferromagnets has a correlation with that of the spontaneous magnetization M S through a power-law scaling relationship [ 13], K(T) K(T∗)=/parenleftbiggMS(T) MS(T∗)/parenrightbiggn , (1) where T* is a normalizing temperature originally taken as 0 K. In this study we choose 10 K as T*, the lowest measurement temperature, at which thermal spin fluctuation is expected tobe small. The exponent nis known to depend on the physical mechanism causing the magnetic anisotropy; nis equal to 3 for materials with a uniaxial single-ion anisotropy [ 13], and is closer to 2 for materials with a dominant two-site anisotropy[14]. However, one may need to care about a mixture of the effects from bulk and surface properties on the anisotropy inthin films and nanoparticles with interfacial anisotropy [ 15]. Hence, the investigation of the correlation between KandM S 2469-9950/2018/98(21)/214428(7) 214428-1 Published by the American Physical SocietyH. SATO et al. PHYSICAL REVIEW B 98, 214428 (2018) -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-2-1012 tCoFeB=1 . 1n mm(Tnm) 0H(T)T=1 0K mS FIG. 1. Magnetic moment mper unit area versus in-plane mag- netic field Hfor 1.1-nm-thick CoFeB film measured at 10 K. From the shaded area, areal magnetic anisotropy energy density KefftCoFeB is evaluated. as a function of Tis expected to provide us an insight of the exchange mechanism relating to the interfacial anisotropy. In this study, we investigate the temperature dependence of MSandKof the thin CoFeB/MgO system. We compare the experimental results with atomistic spin-model simulations[16], and show that their temperature dependence is related to thermal spin fluctuations and the finite thickness of thesystem. II. EXPERIMENT A. Film fabrication and measurement method A stack structure of Ta (5)/Ru (10)/Ta (5)/Co 20Fe60B20 (tCoFeB )/MgO (1.4)/Ta (5) was deposited on a thermally ox- idized Si substrate by rf magnetron sputtering. Numbers inparentheses are nominal thicknesses in nm determined fromthe deposition rate. We prepared five samples with differentCoFeB thicknesses ( t CoFeB=1.1, 1.3, 1.7, 3.0, and 4.0 nm). The CoFeB composition is also nominal, and corresponds tothat of a sputtering target. The boron composition of the film isprobably higher than the nominal one, while the compositionratio of Co to Fe is almost the same [ 17]. The stacks were annealed in vacuum at 300 °C for 1 h under an out-of-planemagnetic field of 0.4 T. We do not expect the formationof a magnetically dead layer in the structures as shownin the previous work [ 8]. We measured the magnetization curve for the stacks along the hard-axis direction at varioustemperatures by a vibrating sample magnetometer. A typicalmagnetization curve is shown in Fig. 1for the stack with t CoFeB=1.1 nm at 10 K. From the curves, we determined the spontaneous magnetic moment per unit area mSand areal effective perpendicular magnetic anisotropy energy densityK efftCoFeB (the area enclosed by the m-H curve and m=mS in Fig. 1)[8]. B. Results Figure 2(a) shows the temperature dependence of MS between 10 and 300 K as a function of tCoFeB , where MSwas determined from MS=mS/tCoFeB .T h eMSexhibits a mono- tonic decreasing tendency in all the CoFeB films with dif-ferent t CoFeB within experimental errors. The thinner CoFeB film (tCoFeB<2 nm) shows a larger variation of MSwith increasing temperature, as noticed from Fig. 2(b), in whichFIG. 2. Temperature Tdependence of (a) spontaneous magne- tization MSand (b) normalized MS(T)/MS(10 K) for CoFeB/MgO stacks as a function of CoFeB thicknesses tCoFeB . the normalized spontaneous magnetization MS(T)/MS(10 K) is presented. Figure 3(a) shows the temperature dependence of the per- pendicular anisotropy energy density K=Keff+MS2/2μ0, where μ0is permeability in free space, as a function of tCoFeB . Because the interfacial anisotropy plays a dominant role forthe perpendicular anisotropy in the CoFeB/MgO system, Kis approximately equal to K i/tCoFeB , where Kiis the interfacial anisotropy energy density. As can be seen, Kdecreases with increasing T, indicating that Kialso decreases with increasing T. The thinner CoFeB film ( tCoFeB<2 nm) shows a larger variation of Kwith change in T, as noticed from Fig. 3(b), in which the normalized anisotropy energy density K(T)/K(10 K) is presented. Figure 4shows the double-logarithm plot of K(T)/K(10 K) versus MS(T)/MS( 1 0K )a saf u n c t i o no f tCoFeB . A linear fit to the data for the samples with tCoFeB<2n mg i v e st h e slopenof 2.2, in agreement with previous experimental mea- surements [ 18–20]. The scaling exponent n∼2 suggests that the anisotropy is not dominated by single-ion anisotropy with 0.00.51.01.5 0 100 200 3000.60.81.01.2(a) 3 41.71.3K(MJ/m3) 1.1tCoFeB (nm)K(T)/K(10 K) T(K)(b) FIG. 3. Temperature Tdependence of (a) perpendicular anisotropy energy density Kand (b) normalized K(T)/K(10 K) for CoFeB/MgO stacks as a function of CoFeB thicknesses tCoFeB . 214428-2TEMPERATURE-DEPENDENT PROPERTIES OF CoFeB /MgO … PHYSICAL REVIEW B 98, 214428 (2018) 0.9 10.70.80.911.1 0.851.1 1.3 1.7 3 4K(T)/K(10 K) MS(T)/MS(10 K)tCoFeB (nm) Linear fit 1.02 FIG. 4. Double-logarithm plot of K(T)/K(10 K) versus MS(T)/MS(10 K) for CoFeB/MgO with different CoFeB thicknesses tCoFeB . Line is a linear fit for the sample with tCoFeB<2 nm. n=3 according to the Callen-Callen theory [ 13]. A similar experimental scaling exponent of n∼2.1 was observed for FePt [ 21], and was explained theoretically by a model based on two-site exchange anisotropy [ 14]. III. SIMULATIONS A. Atomistic spin model with single-ion anisotropy The simulations are based on the atomistic spin model [22], which naturally models the influence of thermal spin fluctuations on the intrinsic magnetic properties such as thespontaneous magnetization and magnetic anisotropy. We use aclassical atomistic spin model utilizing the VA M P I R E software package for the numerical calculations [ 22,23]. The spin Hamiltonian using the Heisenberg form of exchange, H=−/summationdisplay i<jJijSi·Sj−/summationdisplay iku(Si·ei)2, (2) describes the energy of the system, where Jijis an isotropic exchange constant between nearest-neighbor spins as usualin the Heisenberg model, S iandSjare spin unit vectors at local sites iand nearest-neighbor sites j, respectively, kuis the uniaxial anisotropy constant per atom, and eiis a unit vector along the magnetic easy axis. The simulated system is shown schematically in Fig. 5. The system is generated by creating a bulk body-centered- MgO CoFeB 15 nm 15 nmtCoFeB FIG. 5. Schematic of the simulated system incorporating bulk and interfacial CoFeB in contact with the MgO layer. The sys-tem dimensions are 15 nm ×15 nm ×t CoFeB nm, where tCoFeB is the thickness of the CoFeB layer. Boron impurities indicated by dark spheres are randomly distributed through the CoFeB.TABLE I. Adopted model parameters. Bulk Interface ku(J/atom) 0 1 .35×10−22 Jij(J/link) 7 .735×10−211.547×10−20 cubic (bcc) crystal with lattice constant of 0.286 nm. The dimensions of CoFeB layer in the simulation are 15 nm × 15 nm ×tCoFeB nm, with periodic boundary conditions in the in-plane xandydirections. The Fe and Co atoms are treated as an average species with an averaged moment of 1 .6μB(μB is the Bohr magneton). This approximation is not expected to strongly affect the calculated temperature dependence ofM SandKresulting from the thermal spin fluctuations. The boron atoms are included explicitly as nonmagnetic impuri-ties randomly distributed in the CoFeB. Despite the boronbeing nonmagnetic, it strongly affects the magnetic properties,because the presence of the impurities reduces the numberof coordination of the magnetic atoms, and thus reduces theeffective Curie temperature of the whole sample. As withboron, the MgO is nonmagnetic, but has a strong influence onthe magnetic properties of the interfacial Co and Fe atoms.We consider two important effects: one is the presence ofstrong interfacial anisotropy k u[24], and the other is an en- hancement of the exchange interaction Jijat the CoFeB/MgO interface [ 25]. We model the interfacial anisotropy using an effective uniaxial anisotropy for the CoFeB/MgO interfaciallayer guided by previous first-principles calculations showinga localized enhancement of the anisotropy at the CoFeB/MgOinterface [ 11]. The enhancement of the interfacial exchange interaction is treated in the same nearest-neighbor approxi-mation as the bulk CoFeB but with an enhanced exchangeconstant. The adopted values of k uandJijare listed in Table I. Thekuis derived from the experimental results in Fig. 3, and the exchange constant Jijis derived from a mean-field expres- sion of the effective exchange energy [ 22] including a cor- rection for spin-wave excitations [ 26]. The bottom interfacial layer of CoFeB to be in contact with Ta is assumed to have nospecial interfacial qualities other than the usual loss of coordi-nation. We do not consider the bulk anisotropy of the CoFeB,as experimentally it is known to be negligibly small [ 8]. The temperature-dependent properties are calculated us- ing the constrained Monte Carlo method [ 16], in which the direction of the magnetization is fixed, while the net mag-netization can be changed due to thermal fluctuations. Thecomputational approach chooses moves of two spins, whichare rotated in such a way as to conserve the direction of themagnetization. The static properties can be calculated whenthe system achieves thermal equilibrium after many suchmoves. Due to the symmetry of the system (consisting of asingle high-anisotropy interface layer with periodic bound-aries in the plane), the interfacial anisotropy is wholly uniaxialin nature, resulting in the angle-dependent torque followingas i nθform at all temperatures, where θis the angle from the film normal. Applying a quadrature rule, the effectiveanisotropic free energy is calculated directly from the thermo-dynamic average of the total torque on the system [ 16]. Given the sin θform of the torque, we fix θat 45°, at which the 214428-3H. SATO et al. PHYSICAL REVIEW B 98, 214428 (2018) 0.00.51.0 0 500 10000.00.51.01 2 3 4 5MS(T)/MS(0 K)t (nm)(a) Bulk(b)MS(T)/MS(0 K) T(K)Interface t =3nm FIG. 6. Simulated temperature Tdependence of normalized spontaneous magnetization MS(T)/MS(0) for the CoFeB/MgO sys- tem with B composition of 4% (a) as a function of CoFeB thicknessest CoFeB and (b) surface and bulk components for CoFeB with tCoFeB= 3 nm. Lines are fits by Eq. ( 3). torque is largest, to minimize the numerical error. The Monte Carlo (MC) trial moves use the Hinzke-Nowak method using acombination of different trial moves to optimize the relaxationto thermal equilibrium [ 27]. After equilibrating the system by 10 000 MC steps, the thermodynamic averages of the torqueand magnetization are collected averaging over further 10 000MC steps. B. Calculation of the spontaneous magnetization Because of Jij/greatermuchku, the anisotropy has a negligible effect on the calculated MS(T). Figure 6(a) shows the calculated MS(T) normalized by MS(T=0) as a function of tCoFeB of the CoFeB/MgO system with B composition of 4%. The value of4% is extracted from a series of calculations to give the bestagreement with the experimentally measured temperature-dependent magnetization. The reduced boron composition inthe simulation is consistent with an experimental result thatthe boron composition in the CoFeB film is reduced owing toabsorption by Ta adjacent to the CoFeB layer via annealing[28]. Because the CoFeB layers consist of a few monolayers, the magnetization curves in Fig. 6(a) show significant finite-size effects. This is apparent in the reduced criticality of the mag-netization curve compared with a bulk sample, as well as a vis-ible variation among samples with different t CoFeB . It is clear from the simulations that the low-temperature gradient of themagnetization is thickness dependent, as seen also in the ex-perimental data (Fig. 2). Quantitative agreement between the experimental data and simulations is obtained for thin CoFeBafter tuning the boron concentration in the simulations. Both experimental and simulated sets of data show a decrease in the gradient with increasing t CoFeB . In the case of the simulations, this decrease is purely due to finite-sizeeffects. As the film thickness increases, the thermally fluc-tuating surface spins make up a smaller portion of the totalmagnetization, and thus the gradient approaches the classicallimit for the bulk with a slower variation of M S(T). For the experimental result, we see a similar behavior; however, thelarge change of the gradient for the thicker films seen in Fig. 2 may suggest the presence of an additional effect. For bulkCoFe, one would expect the usual Bloch-like behavior, wherethe gradient of the temperature-dependent magnetization issmall due to the quantum nature of the spin-wave spectrum[29]. For the thin-film samples studied in the present work, however, it is clear that the gradient is much closer to that of aclassical system, where the atomic spins are unquantized. Weattribute this fundamental disparity to microstructural disorderand the polycrystalline nature of the films, which disruptthe long-range crystallinity and quantum nature of the spinwaves. One would therefore expect that these effects becomeless important for larger film thicknesses, and so the largerdecrease in the gradient of M S(T) compared to the classical simulations is indicative of a classical-quantum transition.This transition indicates an important finite-size effect, wherethe microstructural disorder can disrupt the quantum natureof the spin-wave spectrum leading to significantly differentM S(T) from the expected bulk behavior. The temperature-dependent magnetization MS(T)f o ra classical system is well described by the expression [ 29] MS(T) MS(0)=/parenleftbigg 1−T TC/parenrightbiggβ , (3) where βis a critical exponent. We fit Eq. ( 3) to simulated MS(T) treating TCandβas free parameters. The obtained TCis∼1100 K, which is not strongly dependent on tCoFeB . However, it is important to note that the finite-size effect isvisible in the larger magnetization fluctuations in the inter-face layers, which has a strong influence on the temperaturedependence of the anisotropy. This is an important genericfeature of ultrathin films, arising from the loss of magnetic coordination at the interface. In the case of CoFeB/MgO, an enhanced exchange interaction at the surface included in thesimulation might be expected to somewhat compensate thiseffect [ 23]. To investigate the difference in the temperature depen- dence between interface and bulk-like magnetization, we havecalculated separate contributions from the MgO-terminatedinterface and bulk atoms to the total magnetization, as shownin Fig. 6(b). The bulk atoms make up the majority of the com- plete system, and so the average magnetization is generallycloser to the net magnetization. The temperature-dependentmagnetizations show a slightly elevated Curie temperature forthe MgO-terminated interface atoms compared with the bulk,owing to the stronger exchange interactions at the interface.The calculated Curie temperature for the interfacial atoms alsoconverges rapidly to an asymptotic value. The temperaturedependence of the interface magnetization is an importantquantity that determines the spin fluctuations of the interfaciallayer, which provides the magnetic anisotropy. C. Calculation of the perpendicular anisotropy b a s e do ns i n g l e - i o na n i s o t r o p y The magnetocrystalline anisotropy in the CoFeB/MgO sys- tem arises almost solely from the MgO-terminated interface.We proceed with calculation of the temperature dependence 214428-4TEMPERATURE-DEPENDENT PROPERTIES OF CoFeB /MgO … PHYSICAL REVIEW B 98, 214428 (2018) 0 500 1000012 1 2 3 4 5K(MJ/m3) T(K)tCoFeB (nm) FIG. 7. Calculated temperature Tdependence of magnetic anisotropy energy densities Kfor the CoFeB/MgO as a function of CoFeB thicknesses tCoFeB based on single-ion anisotropy. of the interface anisotropy using the constrained Monte Carlo method [ 16]. The calculated free anisotropy energy for dif- ferent tCoFeB is shown in Fig. 7(a). All samples have a single MgO-terminated interface with the same anisotropy energy,which leads to a change in the average magnetic anisotropyenergy density, allowing engineering of the magnetic proper-ties through thickness variation [ 8]. Additionally, the different temperature dependence of the surface magnetization leads toa different temperature dependence of the anisotropy for thesamples with different t CoFeB . According to Eq. ( 1), we make double-logarithm plots of K(T)/K(10 K) versus MS(T)/MS(10 K) (Callen-Callen plots) (as shown by dashed line in Fig. 9 shown later) [ 13]. We determine the scaling exponent nto be 2.82–3.26, which is close to 3 expected from single-ionanisotropy but inconsistent with the experimental observationin Fig. 4. This indicates that the magnetic anisotropy in CoFeB/MgO thin films is not single ion in origin. D. Calculation of the perpendicular anisotropy b a s e do nt w o - s i t ea n i s o t r o p y Two-site anisotropy arises from an orientation-dependent exchange interaction. For example, in layered L10alloys such as FePt, the symmetry of the crystal along the caxis causes an asymmetry in the exchange interactions between atoms in thesame plane [ 14,15] leading to a two-site exchange anisotropy. This is also expected to be the case for CoFeB/MgO layers,where the hybridization of the interfacial Fe layer leads toa change in symmetry along the zdirection. The two-site anisotropy can be expressed by an exchange tensor as aperturbation of the usual isotropic exchange, J T ij=⎡ ⎣Jxx 00 0Jyy 0 00 Jzz⎤ ⎦, (4) where subscripts to Jof tensor components denote the compo- nents of the spin direction at iandjsites. For a system with only two-site anisotropy, the spin Hamiltonian is given by Hex=−/summationdisplay i<jSiJT ijSj. (5)In the case of isotropic exchange, all the diagonal exchange components are the same, Jxx=Jyy=Jzz. For anisotropic exchange with asymmetry along the zdirection, Jxx=Jyy/negationslash= Jzz. The value of the effective anisotropy at very low tem- peratures is the same and independent of its physicalorigin, where the origin is only evident from the scaling withrespect to the magnetization. Therefore, in our model wemust translate the value of anisotropy from Sec. II C intoa two-site exchange anisotropy. Considering the exchangeinteractions between two spins S iandSj, we have the total exchange energy Eex=−JxxSixSjx−JyySiySjy−JzzSizSjz, where subscripts for Si(j)denote the x,y, andzcomponents of spin at i(j) site, respectively. In ferromagnets at low temperatures, all spins are well aligned and so it can beassumed that S i≈Sj. For the case with an easy axis along thezdirection and spin rotation in the z-xplane ( Sz=cosθ andSx=sinθ), we obtain Eex=(Jzz−Jxx)sin2θ−Jzz, and thus the anisotropic exchange gives the exchange energy withan identical sin 2θsymmetry to the single-ion anisotropy. The total two-site exchange anisotropy is expressed by thedifference in the diagonal values of the exchange tensorJ zz−Jxx.For spins with several neighbors, the anisotropy energy should be divided amongst each of the interactions togive the same effective anisotropy. In the case of a nearest-neighbor model the coordination number defines the numberof interactions, giving geometric factors of 1 /6f o rs i m p l e cubic, 1 /8 for bcc, and 1 /12 for face-centered-cubic and hexagonal lattices. The CoFeB/MgO interfacial anisotropyis a special case, since the anisotropy arises from the Fe-Ohybridization and the interfacial atoms are only half coordi-nated. In this case, the anisotropic exchange energy should bedivided among the four nearest atoms in the interface, givinga geometric factor of 1 /4. In this case, the exchange values at the interface in terms of the parameters from Table Iare given by J xx=Jij,J yy=Jij,J zz=Jij+ku/4. (6) The bulk parameters are unchanged. Revisiting the simu- lations using the spin Hamiltonian given in Eq. ( 5), we now explicitly exclude any single-ion contribution to the effective 0 500 10000.00.51.0 single ionK(MJ/m3) T(K)tCoFeB =2nm two site FIG. 8. Calculated temperature Tdependence of magnetic anisotropy energy densities Kfor the CoFeB/MgO with 2-nm-thick CoFeB based on two-site anisotropy along with that based on single- ion anisotropy shown in Fig. 7. 214428-5H. SATO et al. PHYSICAL REVIEW B 98, 214428 (2018) 0.9 10.70.80.911.1 1.021.1 1.3 1.7K(T)/K(10 K) MS(T)/MS(10 K)tCoFeB (nm) 0.85 FIG. 9. Double-logarithm plot of K(T)/K(10 K) versus MS(T)/M S(10 K) for CoFeB/MgO. Symbols correspond to the experimental results for the samples with CoFeB thicknesses less than 2 nm. Lines are determined from linear fitting to the simulatedresults (solid line for two-site anisotropy and dashed line for single-ion anisotropy). anisotropy. We note that in both cases the total effective interfacial anisotropy at zero temperature is the same. Theobtained K(T) is shown in Fig. 8. The scaling plots between K(T)/K(10 K) and M S(T)/MS(10 K) give the scaling exponent nof 2.21, which is consis- tent with the experimental observation as shown in Fig. 9. The result indicates that the Callen-Callen plot gives n∼2 for systems with two-site anisotropy [ 14], indicating that the Callen-Callen plot can be used to gain insight into the originof the anisotropy [ 30]. IV . CONCLUSION We have conducted a joint experimental and computa- tional study on the temperature-dependent magnetization andmagnetic anisotropy of CoFeB/MgO ultrathin films. Our ex-perimental measurements have found the strong temperaturedependence of the saturation magnetization in close agree-ment with classical atomistic spin-model simulations. The scaling of the anisotropy with the magnetization providesdirect insight into the physical origin of the anisotropy bycomparison with the Callen-Callen theory. From the Callen-Callen theory, a scaling exponent of n=3 suggests local single-site anisotropy while an exponent of n∼2 suggests two-site exchange anisotropy. Experimental measurementsgive the scaling exponent of n=2.2, indicating the dominant role of two-site exchange anisotropy in this material system.We have also performed atomistic simulations to comparewith experimental observation by considering thermal fluctu-ations due to the finite-size effect and a reduction in atomiccoordination at the interface. The atomistic simulations withsingle-ion anisotropy model result in n=3.03 as expected, which does not agree with the experimental results. The sim-ulations with purely two-site exchange anisotropy reproducethe experimentally observed value ( n=2.2). In this work, we assume identical magnetic atoms in a uniform structureand negligible anisotropy at CoFeB/Ta interface, suggestingthat the temperature-dependent magnetic properties are de-termined mainly by the MgO/CoFeB interface. This studyprovides information of the important factors determiningthe temperature-dependent thermal stability of CoFeB/MgOmagnetic tunnel junctions and to guide the design of structuresfor various applications. ACKNOWLEDGMENTS The work was partly supported by JSPS Core-to-Core Program and RIEC Cooperative Research Projects. The workat Tohoku University was supported by ImPACT program ofCSTI and a Grant-in-Aid for Scientific Research from MEXT(Grant No. 26103002). This work made use of the facilitiesof N8 HPC Centre of Excellence, provided and funded bythe N8 consortium and EPSRC (Grant No. EP/K000225/1),coordinated by the Universities of Leeds and Manchester. P.C.gratefully acknowledges the funding from MahasarakhamUniversity. [1] M. Klaua, D. Ullmann, J. Barthel, W. Wulfhekel, J. Kirschner, R. Urban, T. L. Monchesky, A. Enders, J. F. Cochran, and B.Heinrich, P h y s .R e v .B 64,134411 (2001 ). [2] S. Monso, B. Rodmacq, S. Auffret, G. Casali, F. Fettar, B. Gilles, B. Dieny, and P. 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PhysRevB.95.024409.pdf

PHYSICAL REVIEW B 95, 024409 (2017) Skyrmion oscillations in magnetic nanorods with chiral interactions M. Charilaou*and J. F. L ¨offler Laboratory of Metal Physics and Technology, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland (Received 15 November 2016; published 10 January 2017) We report that in cylindrical nanorods with chiral interactions spin textures corresponding to spatial skyrmion oscillations can be stabilized depending on the initial state, as revealed by micromagnetic calculations. Theskyrmion oscillation, or skyrmion-chain state, occurs when the diameter of the rod is larger than the helical pitchlength of the material, and the number of skyrmions on the chain is proportional to the length of the nanorod. Thetopological charge is localized, breaking translational symmetry, but in the presence of a uniaxial anisotropy, orupon the application of an external field, the localization disappears and a single skyrmion line is formed. Thesefindings provide a deeper understanding of the interplay between geometry and topology, and show how spatialconfinement specifically in curved solids can stabilize skyrmionic spin textures. DOI: 10.1103/PhysRevB.95.024409 I. INTRODUCTION Competition between the symmetric exchange interac- tion and the antisymmetric Dzyaloshinskii-Moriya interaction(DMI) [ 1] can give rise to the formation of complex spin textures in magnetic matter. A fascinating example is theoccurrence of skyrmions [ 2–7] and skyrmion lattices [ 8–11] upon the application of an external field or in the presenceof uniaxial anisotropy. Magnetic skyrmions are topologicalparticlelike spin configurations that are characterized by aninteger topological charge (winding number as defined forspin textures) [ 12] Q=1 4π/integraldisplay m·(∂xm×∂ym)dxdy, (1) which can be either 1 (skyrmion) or −1 (antiskyrmion), where mis the unit vector of the magnetization ( m=M/M S withMSthe saturation magnetization). The intense research on skyrmions is fueled on the one hand by the new fundamental physics related to these complexspin structures, and on the other hand by the potential todevelop new technology for data-storage devices. The latteris motivated by the fact that skyrmions can be moved byrelatively low current densities [ 13–17], promising energy- efficient spintronics, and the compatibility of skyrmion-baseddevices with domain-wall-based technology, which can beachieved by adjusting the geometry of the solid [ 18]. In bulk crystals with free surfaces skyrmions exist in a narrow temperature-field range, close to the Curie temperature[8,19,20], but in thin films the skyrmion phase extends to wider temperature and field ranges [ 7,20], and in nanostructured ma- terials skyrmions are stable even at room temperature [ 21,22]. The stability of skyrmionic states is crucially dependent on thelow dimensionality and symmetry of the solid, as it confinesthe spin structure [ 23–25]. The confinement of skyrmionic spin textures in nanostructures is thus a key element in creating andcontrolling them. As we will discuss in the following, using high-resolution micromagnetic simulations considering B20 FeGe, the geometry of cylindrical nanostructures can give rise *charilaou@mat.ethz.chto nontrivial spin textures, which break translational symmetryin the form of spatially oscillating topological charge. II. THEORETICAL MODEL For the theoretical description of magnetism in FeGe nanorods we consider the following contributions to the totalfree energy density F: (i) ferromagnetic exchange F exc=Aexc(∇m)2, (2) where Aexcis the exchange stiffness; (ii) Dzyaloshinskii- Moriya interaction FDMI=Dm·(∇×m), (3) where Dis the strength of the DMI; (iii) Zeeman energy FZ=−μ0MSHext·m, (4) where Hextis the external field; and (iv) magnetostatic self- energy due to dipolar interactions Fdip=−μ0MS 2m·hdemag, (5) where hdemag is the local demagnetizing field. We start the simulations at a fully polarized configuration (energy maximum), and observe the evolution of the spintexture inside the solid as a function of time by solving theLandau-Lifshitz-Gilbert equation of motion ∂ tm=−μ0γ(m×heff)+α(m×∂tm), (6) where αis the dimensionless damping parameter [ α= G/(γM S)], with Gthe Gilbert damping frequency con- stant and γthe electron gyromagnetic ratio, and heff= −[1/(μ0MS)]∂mFis the effective magnetic field consisting of both internal and external fields. The material parametersfor FeGe taken from literature are: exchange stiffness A exc= 8.78 pJ/m[26], saturation magnetization MS=385 kA /m [27,28], and DMI strength D=1.58 mJ /m2[26]. Since the DMI energy in FeGe is intrinsic and not of interfacial origin,it should not depend on the thickness, hence the DMI strengthwas kept constant throughout this study. For the numericalcalculations the graphics-processing-unit accelerated softwarepackage MuMax3 [29] was used in the high-damping case with α=0.1 (occasional checks with α=0.01 were made to test 2469-9950/2017/95(2)/024409(5) 024409-1 ©2017 American Physical SocietyM. CHARILAOU AND J. F. L ¨OFFLER PHYSICAL REVIEW B 95, 024409 (2017) (a) (b) -10+1mzz FIG. 1. Simulated ultrathin nanodisks with diameter of 120 nm after an external field of 1 T has been applied (a) in-plane and (b) out-of-plane. In (b) a skyrmion core forms. the effects of damping on the simulation findings). Simulations were performed with different cell sizes to test the numericalstability, with the smallest cell tested being 1 nm 3. Even though very small sizes were used, quantum-mechanical effects werenot considered in our simulations. III. RESULTS AND DISCUSSION We begin the discussion by considering ultrathin FeGe nan- odisks. A crucial aspect for the realization of a skyrmion in anultrathin structure is the presence of a perpendicular symmetry-breaking field, either external or internal, e.g., perpendicularmagnetic anisotropy (PMA). Without an external field or PMA,the state with lowest energy corresponds to a helical spintexture [see Fig. 1(a)]. If, however, we prepare the skyrmion state by magnetizing the sample in the out-of-plane directionusing an external field ( μ 0Hz=1 T) and then switching off the field, the resulting spin texture is that of a left-handedskyrmion core [see Fig. 1(b)], in which the zcomponent of the magnetization in the center of the disk is m z=+1 and at the edge of the disk it is mz≈−1. The numerical value of the topological charge for this spin texture is Q≈0.85 (for a perfect skyrmion of Q=±1). The deviation from the integer value is due to the tilting of the spins along the circumferenceof the disk by dipolar interactions [ 23,26], which generally tend to align the moments along the physical edge of the solid[30]. A similar scenario, i.e., where the skyrmion state can be prepared by a sequence of magnetic fields, was experimentallyobserved in artificial skyrmion lattices [ 31]. The formation of the skyrmion core, instead of the helical state, can be explained with topological arguments: in orderto transfer from a collinear perpendicular state (global energymaximum) to a helical state (global energy minimum), the spintexture needs to undergo curling, which begins by a twisting ofthe spins near the edges of the disk. The twisting at the edgeslowers the total energy because it favors both the dipole-dipoleinteraction and DMI energy, and the system is trapped in thisstate (local energy minimum) because there is no continuousway to transfer the spin texture to a helical state due to theconfinement by the solid. In contrast, if the system is fullypolarized in the plane, there is a direct way to transfer the spin texture from the collinear state to the helical state, thusgenerating the texture as seen in Fig. 1(a). For the confinement of the skyrmionic spin texture, the diameter ( d) of the disk needs to be comparable to the skyrmion-core diameter, which depends on the interplaybetween the DMI and magnetostatic interactions [ 23]. For d> 135 nm the spin texture forms concentric rings with alternating m z(not shown), similar to those discussed in Refs. [23,26,32]. While the upper limit is set by the formation of the ring state, defining a lower limit for dis not straightforward, since with decreasing diameter the skyrmion core becomesincreasingly incomplete, and for very small dit corresponds to a conical state. (This again shows that the transformation froma conical spin configuration to a skyrmion is continuous in ananodisk.) We find, however, that a single skyrmion in FeGeultrathin disks is stable for diameters in the range of 70 nm (b) (c) (a) mz +1 -1 FIG. 2. Spin configurations in cylindrical nanowires with a length of 500 nm and a diameter of 120 nm showing (a) the helicoid state and (b) the skyrmion state with oscillating spin texture. The two slices and their respective magnetization profile show the two modes ofoscillation. The dashed line is a fit with the 2 πdomain wall profile (see text). (c) A contour plot of the zcomponent of the magnetization inside the nanowire shown in (b) and the oscillation of the topologicalcharge [see Eq. ( 1)]Qalong the wire. 024409-2SKYRMION OSCILLATIONS IN MAGNETIC NANORODS . . . PHYSICAL REVIEW B 95, 024409 (2017) (Q=0.7) to 135 nm ( Q=0.84), i.e., λ<d< 2λ, where λ=4πA exc/D≈70 nm is the characteristic pitch length for FeGe. Note that without dipolar interactions the upper limit for a single skyrmion core is d< 90 nm. Hence, dipolar interactions are crucial in stabilizing the skyrmion, as they tend to alignthe spins along the physical edge of the disk, thus shrinkingor stretching the skyrmion in order to satisfy this condition.This importance of dipolar interactions was recently discussedwith regards to experiments on Pt/Co/MgO nanostructures [22]. The interplay between dipolar interactions and DMI can be studied by the two characteristic lengths, i.e., λand the exchange length [ 12,33]δ M=2√ Aexc/(μ0M2 S), which for FeGe is 14 nm. When λ> >δ M, the curling period of the spin structure is longer than the exchange length. As, however, theDMI increases and λdecreases, the curling is impeded by the dipolar interactions and the role of magnetostatics becomesmore important. Now we turn to three-dimensional (3D) structures, cylindri- cal nanowires, in order to examine the dimensional evolutionof the skyrmion state in cylindrical geometry. The state wasprepared in the same way as for the nanodisks, i.e., initializingthe system at an energy maximum and observing the resultingstate. Similar to the nanodisks, there are two competing states:the helicoid (distorted helical) [ 34] state and the skyrmion state. The helicoid state, prepared by applying an externalfield perpendicular to the rod axis, is shown in Fig. 2(a) for a nanorod with 120 nm diameter and 500 nm length.If we apply the external field along the rod axis, however,the resulting spin structure is a striking spin texture withbroken translational symmetry, where the core of the rodis magnetized along the zaxis and the outer regions aremagnetized in the opposite directions, with a distinct spatial oscillation [see Fig. 2(b)]. The oscillation of the spin structure is characterized by the oscillation of the topological chargeQ, as shown in Fig. 2(c), which oscillates between Q≈0.9 andQ≈0.2 with a sinusoidal form Q∝sin(πl//Lambda1 ), with a period of /Lambda1=95 nm. The regions with high Qcorrespond to skyrmion formations, whereas the low Qregions resemble a ring formation [ 23,26,32], a mexican-hat-like spin texture. Figure 2(b) shows slices of the spin structure in these two regions, and the corresponding magnetization profile. Thecore profile of both regions can be fitted well by that of a2πdomain, i.e., the profile of a skyrmion, which has the form [ 12]θ m=θs(−r/δ s+R)+θs(−r/δ s−R), with θs(ξ)= 2a r c t a n eξ, where θmis the angle of the magnetic moment at the distance raway from the center (here the distance from the wire center), Ris a variational parameter, and δsis the skyrmion radius. Hence, both regions contain a skyrmion core, but inthe low Qregion the spin texture at the edge has an opposite winding nearly canceling that of the inner skyrmion core, thusreducing Q. Given that the translational symmetry is broken, and despite the fact that the spin texture appears continuous,the state seen in Fig. 2(b) corresponds to a skyrmion chain ,o r a skyrmion stack, since the skyrmions are stacked/displacedvertically from each other. In order to find the rod diameters, for which the skyrmion- chain state is stable, we have simulated conical structures, witha base diameter of 200 nm and a tip diameter of 25 nm, havingan inclination of 2 .5 ◦over a length of 4 μm. Figure 3shows a vector plot of the spin texture and a contour plot of the magne-tization along the zaxis of the cone. On going from a very thin nanowire to a 200 nm thick nanowire we find a number of pos-sible states: for small diameters ( d/lessorequalslant60 nm) we obtain a nearly mz200 nm25 nmskyrmion chain helicoid helicoid conical ring state 60 nm 100 nm 140 nm 160 nm D= 1.0 mJ/m D= 1.4 mJ/m D= 1.8 mJ/m2 2 2(a) (b) (c) FIG. 3. (a) Snapshot of simulation performed on a FeGe cone, showing the possible spin textures at each range of diameters, starting with a conical spin structure for thin rods d/lessorequalslant60 nm, then a helicoid spin structure for 60 <d/lessorequalslant100 nm, entering the skyrmion-chain state for 100 <d/lessorequalslant140 nm, then again a helicoid structure for 140 <d/lessorequalslant160 nm, which then transforms to a ring state for d> 160 nm. (b) Comparison between three systems with different DMI strength and (c) diameter ( dsc) at which the skyrmion-chain state becomes stable as a function of DMI strength; circles are simulation results and solid lines are fits using dsc∝1/D:dscis proportional to the characteristic pitch length [inset to (c)]. 024409-3M. CHARILAOU AND J. F. L ¨OFFLER PHYSICAL REVIEW B 95, 024409 (2017) (b) (a) FIG. 4. Cross section of cylindrical nanowires with diameter 120 nm and length 500 nm with (a) a uniaxial anisotropy ( Ku), and (b) an external field. The spin structure corresponds to that of a single skyrmion line, characterized by the absence of oscillations in Q.T h e dashed line is a fit with the 2 πdomain wall profile. collinear texture with some curling on the edge of the solid, corresponding to a conical state; for 60 /lessorequalslantd/lessorequalslant100 nm we find a helicoid structure, which is the 3D analog of the 2D helicalstructure in Fig. 1(a); then, for 100 /lessorequalslantd/lessorequalslant140 nm we find the skyrmion-chain state, as shown in Fig. 2(b).F o rd> 140 nm a helicoid texture is favored, which transforms to a complex ring-state oscillation that unfolds via the formation of hedgehogs, or Bloch points, similar to those shown by Milde et al. [35]. In order to find the stability range of the skyrmion-chain state, we have simulated nanowires with different DMIstrength and diameter, and we find that the diameters ( d sc) for which the skyrmion-chain state is stable are proportionalto the characteristic length d sc∝λ∝1/D, i.e., when the DMI is increased ( λis decreased) the skyrmion-chain state is stable in thinner nanorods, and vice versa [see Fig. 3(c)]. This shows how these complex spin textures are the result of competitionbetween the different energy contributions (exchange andDMI) and their characteristic lengths. In the skyrmion-chain state the localization of Qcan be suppressed by adding anisotropy in the energy of the system, either with an external field or with uniaxial anisotropy. Let us consider a hypothetical scenario of a system with exactly thesame material parameters ( M S,Aexc, andD) as FeGe, which additionally has a uniaxial magnetocrystalline anisotropy Ku (see Fig. 4). For very small anisotropy ( Ku<103J/m3)t h e skyrmion oscillation remains unchanged, but with increasingK u, the oscillation of Qin the nanowire gradually decreases (we quantify this by measuring /Delta1Q=Qmax−Qmin), and forKu>2×104J/m3the oscillation vanishes. Figure 4(a) shows the spin structure for Ku=105J/m3(we chose this value because it is comparable to the magnetostatic self-energyμ 0M2 S/2), which is a continuous skyrmion line along the rod withQ=0.85.Similarly, if we break the symmetry by an external field along the zdirection opposing the magnetization in the core (without having Ku), the resulting spin configuration is again a single skyrmion line along the rod [see Fig. 4(b)]. For a one-to-one comparison between the effect of internal vsexternal field, the applied field in this example was set equal tothe anisotropy field from the example shown in Fig. 4(a), i.e., (|μ 0Hz|=μ0Han=2Ku/M S=0.52 T). The external field not only generates a single skyrmion line, but also decreasesthe skyrmion radius. In fact, with increasing (opposing) H z, the skyrmion radius decreases monotonically up to the criticalfield, at which Q→0 and m z→− 1. Once mz=−1, if we switch off the external field, the spin configuration will returnto that of a skyrmion chain, but with opposite polarity. Whenthe external field is applied parallel to the polarity in the rod,the skyrmion radius dramatically decreases and vanishes forvery small fields (in this case 100 mT). All the predictions made here may be verified experimen- tally, either by real-space observation, i.e., Lorentz transmis-sion electron microscopy or magnetic force microscopy, or byreciprocal space investigations, such as polarized small-angleneutron scattering. It is expected that the material propertiesmight deviate from the values used in this study, due toinhomogeneities, roughness, and the free surfaces, and thiscould have an effect on the nanowire diameters, for whichthe described magnetic state can be observed, as these areproportional to the characteristic length λ. IV . CONCLUSIONS In summary, we have shown that geometrical confinement in cylindrical nanowires enables the occurrence of nontrivialskyrmion chains with broken translational symmetry, witha distinct oscillation of the topological charge along thewire. The wire thicknesses, for which this state is stable,depend linearly on the characteristic helical pitch-length ofthe material, which in turn depends on the ratio betweenthe strength of the ferromagnetic exchange stiffness and theDzyaloshinskii-Moriya interaction. An external field or uniax-ial anisotropy can turn the structure to that of a single skyrmionline, restoring translational symmetry. These findings providea deeper understanding of the stability of skyrmionic spinconfigurations in nanostructures, where spatial confinementplays a vital role. They may also be of great importance for thefurther development of spintronics towards skyrmion-basedtechnologies, where cylindrical nanostructures can be used toinject/read skyrmions when coupled to other devices. ACKNOWLEDGMENTS We would like to thank H.-B. Braun and C. Moutafis for fruitful discussions. This work was funded by the ETH Zurich. [1] I. E. Dzyaloshinskii, Sov. Phys. JETP 5, 1259 (1957); T. Moriya, Phys. Rev. 120,91(1960 ). [ 2 ] T .H .A .S k y r m e , Proc. R. Soc. London Ser. A 260,127 (1961 ).[3] A. N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP 68, 101 (1989). [4] A. N. Bogdanov and U. K. R ¨ossler, P h y s .R e v .L e t t . 87,037203 (2001 ). 024409-4SKYRMION OSCILLATIONS IN MAGNETIC NANORODS . . . PHYSICAL REVIEW B 95, 024409 (2017) [ 5 ] U .K .R ¨ossler, A. N. Bogdanov, and C. 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PhysRevLett.103.117201.pdf

Ultrafast Path for Optical Magnetization Reversal via a Strongly Nonequilibrium State K. Vahaplar,1,*A. M. Kalashnikova,1,5A. V. Kimel,1D. Hinzke,2U. Nowak,2R. Chantrell,3A. Tsukamoto,4,6A. Itoh,4 A. Kirilyuk,1and Th. Rasing1 1Institute for Molecules and Materials, Radboud University Nijmegen, P .O. Box 9010 6500 GL Nijmegen, The Netherlands 2Fachbereich Physik, Universita ¨t Konstanz, D-78457 Konstanz, Germany 3Department of Physics, University of York, York YO10 5DD, United Kingdom 4College of Science and Technology, Nihon University, 7-24-1 Funabashi, Chiba, Japan 5Ioffe Physical-Technical Institute of the Russian Academy of Sciences, 194021 St. Petersburg, Russia 6PRESTO, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi, Saitama, Japan (Received 10 April 2009; revised manuscript received 28 July 2009; published 8 September 2009) Using time-resolved single-shot pump-probe microscopy we unveil the mechanism and the time scale of all-optical magnetization reversal by a single circularly polarized 100 fs laser pulse. We demonstratethat the reversal has a linear character, i.e., does not involve precession but occurs via a stronglynonequilibrium state. Calculations show that the reversal time which can be achieved via this mechanismis within 10 ps for a 30 nm domain. Using two single subpicosecond laser pulses we demonstrate that for a5/C22mdomain the magnetic information can be recorded and readout within 30 ps, which is the fastest ‘‘write-read’’ event demonstrated for magnetic recording so far. DOI: 10.1103/PhysRevLett.103.117201 PACS numbers: 75.40.Gb, 75.60.Jk, 85.70.Li The fundamental and practical limit of the speed of magnetization reversal is a subject of vital importance for magnetic recording and information processing technolo- gies as well as one of the most intriguing questions of modern magnetism [ 1–8]. The conventional way to re- verse the magnetization Mis to apply a magnetic field Hantiparallel to M. In this collinear M-H geometry the reversal occurs via precession accompanied by damping that channels the associated angular momentum into the lattice. Although this process is perfectly deterministic, it is also unavoidably slow, typically of the order of nano-seconds, due to the required angular momentum transfer [8]. Alternatively, the driving field can be applied orthogonal toM, so that the created torque [ M/C2H] leads to a rapid change of the angular momentum and a possible switching of the magnetization [ 1,3,4,9]. However, such precessional switching requires a magnetic field pulse precisely tuned to half of the precession period. The fastest precessional reversal demonstrated so far using an external magnetic field [ 1,5] or a spin-polarized current [ 6,7,10] is limited to 100 ps. Moreover, it has been shown that for field pulsesshorter than 2.3 ps such a switching becomes nondetermin- istic [ 5,11]. Ultrafast laser-induced heating of a magnetic material is known to stimulate the transfer of angular momentum fromspins to lattice on a femtosecond time scale [ 12,13]. It has been recently demonstrated that a 40 fs, circularly polar- ized, laser pulse is able to reverse magnetization in a col- linear M-H geometry [ 14], as if it acts as an equally short magnetic field pulse H eff/C24½E/C2E/C3/C138(where Eis the electric field of light) pointing along the direction of light [15]. Although this experiment showed the intriguing pos- sibility of triggering magnetization reversal with a subpi-cosecond stimulus, the relevant time scales and mechanism of such an optically induced magnetization reversal arestill unanswered questions, since a precessional switchingwithin 40 fs would require enormous effective magneticfields above 10 2Tand unrealistically strong damping. To address these questions we used femtosecond single-shot time-resolved optical imaging of magnetic structures andmultiscale modeling beyond the macro-spin approxima-tion. The combination of these advanced experimentaland theoretical methods unveiled an ultrafast linear path-way for magnetization reversal that does not involve pre-cession but occurs via a strongly nonequilibrium state. In our experiments the amorphous ferrimagnetic 20 nm Gd xFe100/C0x/C0yCoyfilms with perpendicular anisotropy [ 14] were excited by a single circularly polarized laser pulse (FWHM of about 100 fs, a central wavelength at /C210¼ 800 nm ). A single linearly polarized probe pulse (FWHM ¼100 fs ,/C210¼640 nm ) delayed with respect to the pump was used for ultrafast imaging of the magneticdomain structure by means of the magneto-optical Faradayeffect. Magnetic domains with magnetization parallel(‘‘up’’) or antiparallel (‘‘down’’) to the sample normalare seen as white or black regions, respectively, in an imageon a CCD camera [ 16]. After each ‘‘write-read’’ event, the initial magnetic state was restored by applying a magnetic field pulse. Taking images of the magnetic structure fordifferent delays between the pump and probe pulses wewere able to visualize the ultrafast dynamics of the laser-induced magnetic changes in the material. Figure 1(a) shows images of magnetic domains in a Gd 24Fe66:5Co9:5sample at different delays after excitation by right- ( /C27þ) or left-handed ( /C27/C0) circularly polarized pulses. The images were obtained for both types of do- mains with initial magnetization up and down. In the firstPRL 103, 117201 (2009) Selected for a Viewpoint inPhysics PHYSICAL REVIEW LETTERSweek ending 11 SEPTEMBER 2009 0031-9007 =09=103(11) =117201(4) 117201-1 /C2112009 The American Physical Societyfew hundreds of femtoseconds, pump pulses of both he- licities bring the originally magnetized medium into astrongly nonequilibrium state with no measurable net mag- netization, seen as a gray area in the second column of Fig. 1(a), the size of which is given by the laser beam intensity profile. In the following few tens of picosecondseither the medium relaxes back to the initial state or a small(/C245/C22m) domain with a reversed magnetization is formed. It is thus obvious that (i) the switching proceeds via astrongly nonequilibrium demagnetized state, clearly notfollowing the conventional route of precessional motion, and (ii) the final state is defined by the helicity of the 100 fs pump pulse [last column of Fig. 1(a)]. As one can see from Fig. 1(a), the metastable state corresponding to reversed magnetization is reached within60 ps after /C27 þ(/C27/C0) excitation. This state is, however, slightly different from the final state [the last column inFig.1(a)], as clearly seen from Fig. 1(b). This happens due to the laser-induced heating of the sample followed by slow(/C291n s) heat diffusion [ 17]. To take into account renor- malization of the two metastable states of magnetization at the subnanosecond time scale we introduce two asymptoticlevels [see dashed lines in Fig. 1(b)]. The characteristic time of switching /C28 swcan be identified as the time required to reconstruct 63% ( 1/C0e/C01) the difference between themetastable states [Fig. 1(b)]. For example, in Fig. 1,/C28sw¼ 60 ps . After 1:5/C28swthe difference reaches 80% and, as also can be seen from Fig. 1(a), this time can be reliably assumed as the period required for a write-read event(/C28 w-r¼90 ps for the example in Fig. 1). The switching time is in fact surprising, because in contrast to heat- assisted magnetic recording [ 18], the reversal time is much longer than the effective light-induced magnetic field pulse Heff. The duration of the latter /C1teffis still an open question but can be different from the FWHM of theoptical pulse. However, /C1t effcan be estimated from the spectrum of THz radiation generated by an Fe film when the latter is excited by a subpicosecond visible laser pulse.According to Ref. [ 19], the intensity of the THz emission depends on the polarization of the incoming light and has to be explained in terms of difference-frequency genera- tion. Phenomenologically, this is very similar to the inverse Faraday effect. Based on a half-period oscillation with thelowest frequency in the THz spectrum [ 19], the maximum /C1t effis about 3 ps. The pulse amplitude Heff, for a typical pump fluence of 2:5J=m2and the magneto-optical con- stant of GdFeCo ( /C243/C2105deg=cm), reaches 20 T. To understand this route for magnetization reversal via such a strongly nonequilibrium state we solved the Landau-Lifshitz-Bloch (LLB) equation. This macrospinapproach encapsulates very well the response of a set of coupled atomic spins subjected to rapidly varying tempera- ture changes, including the reduction of the magnitude ofM[20,21]. The temperature dependence of the anisotropy constant K uis introduced in the LLB equation via the temperature dependence of the transverse susceptibility[22]. The temperature-dependent parameters for the LLB equation, i.e., the longitudinal and transverse susceptibili- ties and the temperature variation of the magnetization, are calculated atomistically using Langevin dynamics com- bined with a Landau-Lifshitz-Gilbert equation for eachspin [ 22]. It is well known that, due to the small heat capacity of electrons, optical excitation by a subpicosec- ond laser pulse can cause heating of the electron systemwell above 1000 K, whereafter the electrons equilibrate with the lattice to a much lower temperature on a (sub) picosecond time scale given by the electron-phonon inter-action [ 13]. This laser-induced increase of the kinetic energy (temperature) of the electrons is simulated using a two-temperature model [ 23], the parameters for which were taken to be typical for a metal [ 24] (electron heat capacity C e¼1:8/C2106J=m3Kat room temperature and electron-phonon coupling Gel-ph¼1:7/C21018J=Ks). The simulations show that in the first 100 fs the electron tem- perature Telincreases from 300 K up to T/C3 eland relaxes with a time constant of 0.5 ps down to the vicinity of TC. Simultaneously the spins experience a short pulse of effec- tive magnetic field with amplitude Heff¼20 T and dura- tion/C1teff. The possibility of magnetization reversal under these circumstances has been analyzed numerically for a volume of 30/C230/C230 nm3. The results of the simula- FIG. 1 (color). (a) The magnetization evolution in Gd24Fe66:5Co9:5after the excitation with /C27þand/C27/C0circularly polarized pulses at room temperature. The domain is initiallymagnetized up (white domain) and down (black domain). The last column shows the final state of the domains after a few seconds. The circles show areas actually affected by pumppulses. (b) The averaged magnetization in the switched areas (/C245/C22m) after /C27 þand/C27/C0laser pulses, as extracted from the images in (a) for the initial magnetization up.PRL 103, 117201 (2009) PHYSICAL REVIEW LETTERSweek ending 11 SEPTEMBER 2009 117201-2tions are plotted in Fig. 2(a)as a phase diagram, defining the combinations of T/C3 eland/C1tefffor which switching occurs for the given Heff. The assumed perpendicular anisotropy value was Ku¼6:05/C2105J=m3at 300 K. As can be seen from the diagram, a field pulse as short as /C1teff¼250 fs can reverse the magnetization. For better insight into the reversal process we simulated the latter for/C1t eff¼250 fs andT/C3 el¼1130 K . The result is plotted in Fig.2(b), showing that, already after 250 fs, the effective fields of two different polarities bring the medium into twodifferent states, while the magnetization is nearlyquenched within less than 0.5 ps. This is followed by relaxation either to the initial state or to the state with reversed magnetization, achieved already within 10 ps.The considered pulse duration /C1t effof 250 fs is only 2.5 times larger than the FWHM of the optical pulse inour experiments [ 25] and well within the estimated lifetime of a medium excitation responsible for H eff. Importantly, in simulations /C1teffwas found to be sensitive to the parame- ters of the two-temperature model. In particular, an in-crease of G el-phleads to a reduction of the minimum field pulse duration. This shows that the suggested mechanism may, in principle, explain the experimentally observedlaser-induced magnetization reversal. This magnetizationreversal does not involve precession; instead, it occurs via alinear reversal mechanism, where the magnetization first vanishes and then reappears in a direction of H eff, avoiding any transverse magnetization components, just as seen inFig.1(a). Exactly as in the experiments, the initial 250 fs effective magnetic field pulse drives the reversal process, that takes 1–2 orders of magnitude longer. The state of magnetization after the pulse is critically dependent on the peak temperature T /C3 eland the pulse du- ration. For ultrafast linear reversal by a 250 fs field pulse itis necessary that, within this time, Telreaches the vicinity ofTC. If, however, this temperature is too high and persists above TCfor too long, the reversed magnetization is de- stroyed and the effect of the helicity is lost. This leads to a phase diagram [Fig. 2(a)], showing that the magnetization reversal may occur in a certain range of T/C3 el. Such a theoretically predicted reversal window of electron tem- perature can be easily verified in the experiment when one changes the intensity of the laser pulse. Figure 2(c)shows the switchability, i.e., the difference between the final states of magnetization achieved in the experiment with /C27þ- and /C27/C0-polarized pulses, as a function of T/C3 el, calcu- lated from the laser pulse intensity. It is seen that, indeed, switching occurs within a fairly narrow laser intensityrange [ 26]. For intensities below this window no laser- induced magnetization reversal occurs, while if the inten- sity exceeds a certain level both helicities result in magne- tization reversal, since the laser pulse destroys the magnetic order completely, which is then reconstructed by stray fields [ 27,28]. Such a good agreement between experiment and theory supports the validity of the pro- posed reversal mechanism. Despite this qualitative agreement between simulations and experiments, the experimentally observed reversal time is several times larger than the calculated 10 ps. The latter, however, is calculated for a 30 nm domain, whereas in our experiments the magnetization in a 5/C22mspot is manipulated. This size is defined by (i) the minimum size of the stable domain in the material and (ii) by the area within the laser spot, where the intensity favors the helicity-dependent reversal. Inhomogeneities in the sample and the intensity profile will lead to variations of T /C3 elover the laser spot. If due to these factors every 30 nm element of the 5/C22mspot is reversed with a probability between 50% and 100%, the actual time of magnetization reversal of this large spot will depend on its size and the speed of domain walls. Their mobility increases dramatically in GdFeCo alloys in the vicinity of their angular momenum compensation point ( Tcomp); i.e., the temperature where the angular momenta of the two sublattices cancel each other [29–31]. Therefore, one should expect a dramatic accel- eration of magnetization reversal near Tcomp. Note that this would also perfectly explain the difference between the times required for the formation of the switched domain and the relaxation to the initial state [Fig. 1(b)]. Indeed, in the former case the domain wall motion is additionally accelerated by the demagnetizing field, while in the latter case this field slows the motion down. This hypothesis was verified experimentally by inves- tigating the reversal process as a function of temperature in three alloys Gd22Fe68:2Co9:8,Gd24Fe66:5Co9:5, and Gd26Fe64:7Co9:3that are characterized by different com- pensation temperatures. The observed write-read time /C28w-r is plotted in Fig. 3as a function of the difference between the sample temperature and the compensation point T/C0 Tcomp. The write-read time is the fastest and weakly de-ExperimentTheory Theory FIG. 2 (color). (a) Phase diagram showing the magnetic state of the ð30 nm Þ3volume achieved within 10 ps after the action of the optomagnetic pulse with parameters Heff¼20 T ,/C1teff, and T/C3 el. (b) The averaged zcomponent of the magnetization versus delay time as calculated for 250 fs magnetic field pulses Heff¼ /C620 T andT/C3 el¼1130 K . (c) Switchability versus the pump intensity for Gd22Fe68:2Co9:8at room temperature. We calculated the peak electron temperature T/C3 elusing Ce. Note that in this range of intensities the amplitude of the effective light-induced magnetic field varies within 19.2–20.8 T.PRL 103, 117201 (2009) PHYSICAL REVIEW LETTERSweek ending 11 SEPTEMBER 2009 117201-3pends on temperature below Tcomp. This agrees with the hypothesis that the relaxation time to the metastable state is defined by the domain wall speed averaged over the photo-excited area. If the laser pulse brings the central part of theexcited area from initial temperature T<T comp to the vicinity of TC, somewhere within this area the material is atTcomp, where the domain wall mobility is the largest. Then, it is the mobility at Tcomp which dominates the averaged domain wall speed in the photoexcited area and, thus, determines the write-read time. Above Tcomp, all-optical magnetization reversal can still be realized, but the write-read time increases exponentially with increasingtemperature. For example, while /C28 w-rforGd22Fe68:2Co9:8 (Tcomp¼100 K ) at room temperature is found to be ex- tremely slow (16 ns), a huge decrease of /C28w-rof 2 orders of magnitude is observed as T/C0Tcompdecreases. Finally, at 10 K we succeeded to achieve all-optical magnetization reversal within just 30 ps, which is the fastest write-readevent demonstrated for magnetic recording so far. In conclusion, by time-resolved single-shot microscopy, we found a novel and ultrafast path for magnetizationreversal triggered by a subpicosecond circularly polarizedlaser pulse. The reversal does not involve precession, butinstead has a linear character, proceeding via a strongly nonequilibrium state. This all-optical reversal occurs only in a narrow range of pulse energies. Using two singlesubpicosecond laser pulses we demonstrated the feasibilityof both all-optical recording and reading on an ultrashorttime scale. The magnetic information was recorded by asubpicosecond laser pulse in a 5/C22mdomain and readout by a similarly short pulse within 30 ps, which is the fastestwrite-read event demonstrated for magnetic recording so far. Simulations for 30 nm domains demonstrate the feasi- bility of reversing magnetization within 10 ps. This timecan be even faster for media with a higher magneticanisotropy constant than the one used in our calculations.We thank A. J. Toonen and A. F. van Etteger for techni- cal support and Dr. I. Radu for his help with sample char- acterization and stimulating discussions. This research hasreceived funding from NWO, FOM, NanoNed and EC FP7 [Grants No. NMP3-SL-2008-214469 (UltraMagnetron) and No. 214810 (FANTOMAS)]. *K.Vahaplar@science.ru.nl [1] C. H. Back et al. , Science 285, 864 (1999). [2] B. C. Choi et al. , Phys. Rev. Lett. 86, 728 (2001). [3] Th. Gerrits et al. , Nature (London) 418, 509 (2002). [4] S. Kaka and S. E. Russek, Appl. Phys. Lett. 80, 2958 (2002). [5] I. Tudosa et al. , Nature (London) 428, 831 (2004). [6] T. Devolder et al. , J. Appl. Phys. 98, 053904 (2005). [7] Y. Acremann et al. , Phys. Rev. Lett. 96, 217202 (2006). [8] J. Sto ¨hr and H. C. Siegmann, Magnetism: From Funda- mentals to Nanoscale Dynamics (Springer-Verlag, Berlin, 2006). [9] H. W. Schumacher et al. , IEEE Trans. Magn. 38, 2480 (2002). [10] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [11] A. Kashuba, Phys. Rev. Lett. 96, 047601 (2006). [12] C. Stamm et al. , Nature Mater. 6, 740 (2007). [13] E. Beaurepaire et al. , Phys. Rev. Lett. 76, 4250 (1996). [14] C. D. Stanciu et al. , Phys. Rev. Lett. 99, 047601 (2007). [15] A. V. Kimel et al. , Nature (London) 435, 655 (2005). [16] The angles of incidence for the pump and probe beams were 20/C14and 0/C14, respectively. [17] Note that the heat load brought to the sample by the laser pulse is several orders of magnitude smaller than the oneduring current-induced magnetization reversal used in realdevices. Moreover, the heat diffusion can be accelerated provided a proper nanostructure design. [18] J. Hohlfeld et al. , Phys. Rev. B 65, 012413 (2001). [19] D. J. Hilton et al. , Opt. Lett. 29, 1805 (2004). [20] D. A. Garanin, Phys. Rev. B 55, 3050 (1997). [21] U. Atxitia et al. , Appl. Phys. Lett. 91, 232507 (2007). [22] N. Kazantseva et al. , Phys. Rev. B 77, 184428 (2008). [23] N. Kazantseva et al. , Europhys. Lett. 81, 27004 (2008). [24] G. Zhang, W. Hu ¨bner, E. Beaurepaire, and J.-Y. Bigot, Spin Dynamics in Confined Magnetic Structures I , Topics in Applied Physics Vol. 83 (Springer, New York, 2002). [25] The switching was observed even for laser pulses with a FWHM of /C243p s, which corresponds to a larger /C1t eff. [26] For T<T comp the intensity required for switching was increasing with temperature decrease. For example, forsample Gd 24Fe66:5Co9:5the decrease of Tby 300 K caused an increase of the required intensity by /C248%. [27] T. Ogasawara et al. , Appl. Phys. Lett. 94, 162507 (2009). [28] The slower reversal in heat-assisted recording actually corresponds to Telabove our reversal window, when the whole system is brought above TCand the dynamics is determined by cooling in an external magnetic field [ 18]. [29] T. Miyama et al. , IEEE Trans. Magn. 14, 728 (1978). [30] R. S. Weng and M. H. A. Kryder, IEEE Trans. Magn. 29, 2177 (1993). [31] V. Randoshkin et al. , Phys. Solid State 45, 513 (2003).FIG. 3 (color). The write-read time /C28w-rversus the relative temperature T/C0Tcomp forGd22Fe68:2Co9:8(Tcomp¼100 K ), Gd24Fe66:5Co9:5(Tcomp¼280 K ), and Gd26Fe64:7Co9:3(Tcomp¼ 390 K ). We achieved magnetization reversal within 30 ps for Gd22Fe68:2Co9:8at10 K . The dashed line is guide to the eye.PRL 103, 117201 (2009) PHYSICAL REVIEW LETTERSweek ending 11 SEPTEMBER 2009 117201-4

PhysRevB.83.104422.pdf

PHYSICAL REVIEW B 83, 104422 (2011) Current-induced dynamics of composite free layer with antiferromagnetic interlayer exchange coupling P. Bal ´aˇz1and J. Barna ´s1,2 1Department of Physics, Adam Mickiewicz University, Umultowska 85, PL-61-614 Pozna ´n, Poland 2Institute of Molecular Physics, Polish Academy of Sciences Smoluchowskiego 17, PL-60-179 Pozna ´n, Poland (Received 25 November 2010; published 29 March 2011) Current-induced dynamics in spin valves including a composite free layer with antiferromagnetic interlayer exchange coupling is studied theoretically within the diffusive transport regime. We show that the current-induced dynamics of a synthetic antiferromagnet is significantly different from the dynamics of a syntheticferrimagnet. From macrospin simulations we obtain conditions for switching the composite free layer, as well asfor the appearance of various self-sustained dynamical modes. Numerical simulations are compared with simpleanalytical models of critical current based on a linearized Landau-Lifshitz-Gilbert equation. DOI: 10.1103/PhysRevB.83.104422 PACS number(s): 67 .30.hj, 75.60.Jk, 75.70.Cn I. INTRODUCTION After the effect of spin transfer torque (STT) in thin magnetic films had been predicted1,2and then experimentally proven,3,4it was generally believed that current-controlled spin valve devices would soon replace the memory cells operatedby an external magnetic field. Such a technological progress,if realized, would certainly offer higher data storage densityand faster manipulation with the information stored on ahard drive memory. However, soon it became clear that someimportant issues must be resolved before devices based onspin torque could be used in practice. The most important isthe reduction of current density needed for magnetic excitation(switching) in thin films, as well as enhancement of switchingefficiency and thermal stability. Some progress has been madeby using more complex spin valve structures and/or varioussubtle switching schemes based on optimized current and fieldpulses. 5–8 A significant enhancement of thermal stability can be achieved by replacing a simple free layer (single homogeneouslayer) with a system of two magnetic films separated by a thinnonmagnetic spacer, known as a composite free layer (CFL).The spacer layer is usually thin so there is a strong Ruderman-Kittel-Kasuya-Yoshida (RKKY) exchange coupling betweenmagnetic layers. 9,10In practice, antiferromagnetic configura- tion is preferred as it reduces the overall magnetic moment of the CFL structure and makes the system less vulnerable to external magnetic fields and thermal agitation. When theantiferromagnetically coupled magnetic layers are identical,we call the structure synthetic antiferromagnet (SyAF). If theyare different, then the CFL has an uncompensated magneticmoment, and such a system will be referred to as a syntheticferrimagnet (SyF). Current- and/or field-induced dynamics of CFLs is currently a subject of both experimental and theoreticalinvestigations. 11–14A switching scheme of SyAF by magnetic- field pulses has been proposed in Ref. 11, and then the possi- bility of current-induced switching of SyAF was demonstratedexperimentally. 12In turn, the possibility of critical current reduction has been shown for a CFL with ferromagneticallycoupled magnetic layers. 15However, the reduction of critical current in the case of antiferromagnetically coupled CFLsstill remains an open problem. In a recent numerical studyon switching a SyAF free layer 16it has been shown that the corresponding critical current in most cases is higher than thecurrent required for switching of a simple free layer, and onlyin a narrow range of relevant parameters (exchange coupling,layer thickness, etc.) the critical current is reduced. Hence,proper understanding of current-induced dynamics of CFLsis highly required. We also note that CFL can be used as apolarizer, too. Indeed, it has been shown recently 17that SyAF used as a reference layer (with the magnetic moment fixed to anadjacent antiferromangentic layer due to exchange anisotropy)might be excited due to dynamical coupling 18with a simple sensing layer.19,20 The main objective of this paper is to study current- induced dynamics of a CFL with antiferomagnetic RKKYcoupling in metallic spin valve pillars. We consider a systemAF/F 0/N1/F1/N2/F2, shown in Fig. 1, where AF is an antifer- romagnetic layer (used to bias magnetization of the referencemagnetic layer F 0), F1and F 2are two magnetic layers, while N 1 and N 2are nonmagnetic spacers. The part F 1/N2/F2constitutes the CFL structure with antiferromagnetic interlayer exchangecoupling. We examine current-induced dynamics of both SyAFand SyF free layers. These two structures differ only in thethickness of the F 1layer, while RKKY coupling and other pillar parameters remain the same. We assume that spin-dependent electron transport is diffu- sive in nature, and employ the model described in Refs. 21 and 22. An important advantage of this model is the fact that it enables calculating spin current components and spin accumulation consistently in all magnetic and nonmagnetic layers, as well as current-induced torques exerted on allmagnetic components. The torques acting at the internalinterfaces of CFL introduce additional dynamical couplingbetween the corresponding magnetic layers. Consequently,the magnetic dynamics of CFL has been modeled by two coupled macrospins and described in terms of a Landau- Lifshitz-Gilbert (LLG) equation. In addition, we derive someanalytical expressions for critical currents from the stabilityconditions of a linearized LLG equation in the static points, 23,24 and discuss results in the context of numerical simulations. The paper is organized as follows. In Sec. IIwe describe the assumed models for spin dynamics and STT calculations.In Sec. IIIwe analyze STT acting on CFL and present results 104422-1 1098-0121/2011/83(10)/104422(10) ©2011 American Physical SocietyP. BAL ´AˇZ AND J. BARNA ´S PHYSICAL REVIEW B 83, 104422 (2011) AF FIG. 1. (Color online) Scheme of the spin valve pillar structure with a composite free layer. from numerical simulations on current-induced switching and magnetic dynamics. Some additional information on STTcalculation can be found in the Appendix. Critical currentsare derived and discussed in Sec. IV. Finally, a summary and general conclusions are given in Sec. V. II. MAGNETIZATION DYNAMICS In the macrospin approximation, the magnetization dynam- ics of the CFL is described by two coupled LLG equations, dˆSi dt+αˆSi×dˆSi dt=/Gamma1i, (1) /Gamma1i=− |γg|μ0ˆSi×Heffi+|γg| Msdiτi, fori=1,2, where ˆSistands for a unit vector along the net spin moment of the ith layer, whereas Heffiandτiare the effective field and current-induced torque, respectively, both acting on ˆSi. The damping parameter αand the saturation magnetization Msare assumed to be the same for both magnetic components of the CFL. Furthermore, γgis the gyromagnetic ratio, μ0is the vacuum permeability, and distands for the thickness of the Filayer. The effective magnetic field for the F ilayer is Heffi=−Happˆez−Hani(ˆSi·ˆez)ˆez+Hdemi(ˆSi) +Hinti(ˆS0,ˆSj)+HRKKY iˆSj, (2) where i,j=1,2 and i/negationslash=j. In the latter equation, Happis the external magnetic field applied along the easy axis in thelayers’ plane (and oriented opposite to the axis z),H aniis the uniaxial anisotropy field (the same for both magnetic layers),andH demi=(Ni·ˆSi)Msis the self-demagnetization field of the F ilayer, with Nibeing the corresponding demagnetization tensor. Similarly, Hinti=(N0i·ˆS0)Ms0+(Nji·ˆSj)Ms describes the magnetostatic influence of the layers F 0and Fjon the layer F i, respectively. Here, Ms0is the saturation magnetization of the layer F 0, which might be generally different from Ms. Components of the tensors Ni,N0i, andNijused in our simulations have been determined by the numerical method introduced by Newell et al.25This method was originally developed for magnetic systemswith nonuniform magnetization. To implement it intoa macrospin model we considered discretized magneticlayers with uniform magnetizations, we calculated tensorsin each cell of the layer, and then we averaged themalong the whole layer. Since these tensors are diagonal,the demagnetization and magnetostatic fields can be expressed as H demi=(Hd ixSix,Hd iySiy,Hd izSiz), and Hinti= (H0i xS0x,H0i yS0y,H0i zS0z)+(Hji xSjx,Hji ySjy,Hji zSjz), with Six,Siy, and Sizdenoting the components of the vector ˆSi(i=0,1,2) in the coordinate system shown in Fig. 1. Finally, HRKKY istands for the RKKY exchange field acting onˆSi, which is related to the RKKY coupling constant as HRKKY i=−JRKKY/(μ0Msdi).19 To include thermal effects we add to the effective field (2) a stochastic thermal field Hthi=(Hthix,Hthiy,Hth iz). For both spins its components obey the rules for Gaussianrandom processes /angbracketleftH thiζ(t)/angbracketright=0 and /angbracketleftHthiζ(t)Hthjξ(t/prime)/angbracketright= 2Dδijδζξδ(t−t/prime), where i,j=1,2 and ζ,ξ=x,y,z . Here, Dis the noise amplitude, which is related to the effective temperature, Teff,a s D=αkBTeff γgμ2 0MsVi, (3) where kBis the Boltzmann constant, and Viis the volume of Filayer. In general, the current-induced torques acting on ˆS1and ˆS2can be expressed as a sum of their in-plane and out- of-plane components τ1=τ1/bardbl+τ1⊥andτ2=τ2/bardbl+τ2⊥, respectively. In a CFL structure, the layer F 1is influenced by STT induced by the polarizer F 0, as well as by STT due to the layer F 2. In turn, the layer F 2is influenced by the torques from the layer F 1. Hence we can write τ1/bardbl=IˆS1×/bracketleftbigˆS1×/parenleftbig a(0) 1ˆS0+a(2) 1ˆS2/parenrightbig/bracketrightbig , (4a) τ1⊥=IˆS1×/parenleftbig b(0) 1ˆS0+b(2) 1ˆS2/parenrightbig , (4b) τ2/bardbl=Ia(1) 2ˆS2×(ˆS2׈S1), (4c) τ2⊥=Ib(1) 2ˆS2׈S1, (4d) where Iis the charge current density, which is positive when electrons flow from the layer F 2toward F 0(see Fig. 1), while the parameters a(j) iandb(j) i(i,j=1,2) are independent of current I, but generally depend on magnetic configuration. We write the current density in the spin space as j=j01+ j·σ, where j0is the particle current density ( I=ej0),jis the spin current density (in the units of ¯ h/2),σis the vector of Pauli matrices, and 1is a 2×2 unit matrix. In the frame of the diffusive transport model,21the parameters a(j) iandb(j) iare given by the formulas a(0) 1=−¯h 2j/prime 1y|N1/F1 Isinθ01,b(0) 1=¯h 2j/prime 1x|N1/F1 Isinθ01, a(2) 1=−¯h 2j/prime/prime 2y|F1/N2 Isinθ12,b(2) 1=¯h 2j/prime/prime 2x|F1/N2 Isinθ12, (5) a(1) 2=−¯h 2j/prime/prime/prime 2y|N2/F2 Isinθ12,b(1) 2=¯h 2j/prime/prime/prime 2x|N2/F2 Isinθ12, where the angles θ01andθ12are given by cos θ01=ˆS0·ˆS1 and cos θ12=ˆS1·ˆS2. Here, j/prime 1yandj/prime 1xare transversal (to ˆS1) components of spin current in the layer N 1(taken at the N 1/F1interface) written in the local coordinate system ofˆS1. Thus, j/prime 1yis parallel to the vector ˆS1×(ˆS1׈S0) (lying in the plane defined by ˆS0and ˆS1) and j/prime 1xis aligned with ˆS1׈S0(perpendicular to the plane defined by ˆS0and ˆS1). 104422-2CURRENT-INDUCED DYNAMICS OF COMPOSITE FREE ... PHYSICAL REVIEW B 83, 104422 (2011) Note, that the zcomponent of spin current is aligned along ˆS1and does not contribute to STT.26Similarly, we define the torque amplitudes acting inside the CFL. Here, j/prime/prime 2yandj/prime/prime 2x are the components of spin current in layer N 2(taken at the N2/F1interface), transversal to ˆS1and lying respectively in the plane and perpendicularly to the plane defined by ˆS1and ˆS2. Analogically, j/prime/prime/prime 2yandj/prime/prime/prime 2xare spin current components in N 2 transversal to ˆS2. The relevant spin current transformations are given in the Appendix. Note that the spin current components depend linearly on I, so the parameters a(j) iandb(j) iare independent of the current density I. III. NUMERICAL SIMULATIONS In this section we present results on our numerical simulations of current-induced dynamics for two metal-lic pillar structures including CFL with antiferromagneticinterlayer exchange coupling. As described in the Intro-duction, the considered pillars have the general structureAF/F 0/N1/F1/N2/F2(see Fig. 1). More specifically, we consider spin valves Cu-IrMn(10)/Py(8)/Cu(8)/Co( d1)/Ru(1)/Co( d2)- Cu, where the numbers in parentheses stand for the layer thick-nesses in nanometers. The layer Py(8) is the Permalloy polariz-ing layer with its magnetization fixed due to exchange couplingto IrMn. In turn, Co( d 1)/Ru(1)/Co( d2)i st h eC F L( F 1/N2/F2 structure) with antiferromagnetic RKKY exchange coupling via the thin ruthenium layer. The coupling constant betweenCo layers has been assumed as J RKKY/similarequal−0.6m J/m2, which is close to experimentally observed values.19,20Here, we shall analyze two different geometries of CFL. The first one is aSyAF structure with d 1=d2=2 nm, while the second one is a synthetic ferrimagnet (SyF) with d1=2d2=4n m . Simulations have been based on numerical integration of the two coupled LLG equations ( 1) with simultaneous calculations of STT [see Eq. ( 4)]. We have assumed typical values of the relevant parameters, i.e., the damping parameterhas been set to α=0.01, while the uniaxial anisotropy field H ani=45 kA m−1in both magnetic layers of the CFL. In turn, saturation magnetization of cobalt has been assumed asM s(Co)=1.42×106Am−1, and for Permalloy, Ms(Py)= 6.92×105Am−1. The demagnetization field and magneto- static interaction of magnetic layers have been calculated forlayers of elliptical cross section, with the major and minor axesequal to 130 and 60 nm, respectively. For both structures under consideration we have analyzed the current-induced dynamics as a function of current densityand external magnetic field. The results have been presentedin the form of diagrams displaying time-averaged values ofthe pillar resistance. Numerical integration of Eq. ( 1) has been performed using a corrector–predictor Heun scheme, and theresults have been verified for integration steps in the range from10 −4ns up to 10−6ns. The STT components acting on CFL spins have been calculated at each integration step from thespin currents, which have been numerically calculated fromthe appropriate boundary conditions. 21Similarly, resistance of the studied pillars has been calculated from spin accumula-tion in the frame of the model used also for the STT description(for details see also Ref. 27).-0.20-0.15-0.10-0.050.000.050.10 0 0.5 1 1.5 2Spin-transfer torque N1/F1 interface -0.0010.0000.001 0 0.5 1 1.5 2 θ / πF1/N2 interface -0.002-0.0010.0000.001 0 0.5 1 1.5 2 θ / πN2/F2 interface τx τy τz FIG. 2. (Color online) Angular dependence of the Cartesian components of STT, in units of ¯ hI/|e|, acting at N/F interfaces, when magnetization of the SyAF structure is rotated rigidly with both ˆS1andˆS2remaining in the corresponding layer planes. Here, θ is an angle between ˆS1andˆez.ˆS2is tilted away from the antiparallel configuration with ˆS1by an angle of 1◦. A. Spin transfer torque Let us analyze first the angular dependence of STT components in the structures under consideration. Althoughthe thicknesses of magnetic layers in the studied SyAF andSyF structures are different, the angular dependence of STTcomponents as well as their amplitudes are very similar. Thus,the analysis of STT in SyAF applies also qualitatively to thestudied SyF free layer. First, we analyze STT components in the case when SyAF is rotated as a rigid structure, i.e., the relative configurationofˆS 1and ˆS2is maintained. To have a nonzero torque between F 1and F 2layers, ˆS2has been tilted away from the antiparallel configuration by an angle of 1◦. Figure 2shows all three Cartesian components (see Fig. 1for a definition of the coordinate system) of STT acting at N/F interfaces asa function of the angle θbetween ˆe zand ˆS1. While the y andzcomponents are in the plane of the layers (the spins of CFL are rotated in the layer plane), the component xis normal to the layer plane. However, τxremains negligible at all interfaces of the CFL. The STT acting at N 1/F1reveals a standard (nonwavy28) angular dependence, and vanishes when ˆS1is collinear with ˆS0. Its amplitude is comparable to STT in standard spin valves with a simple free layer. The STT at F 1/N2 and N 2/F2interfaces also depends on the angle θ. However, they are approximately two orders of magnitude smaller, whichis a consequence of a small angle (1 ◦) assumed between ˆS1 andˆS2. As will be shown in the following, CFL is usually not switched as a rigid structure, but generally forms a configura-tion which deviates from the antiparallel one. Figure 3shows how the STT components at the F 1/N2and N 2/F2interfaces vary when ˆS2is rotated from ˆezby an angle θ/prime, while ˆS1 remains fixed and is parallel to ˆS0=ˆez. In such a case, the torque acting at N 1/F1interface remains zero, as ˆS1stays collinear to ˆS0. As before, the out-of-plane components are also negligible in comparison to the in-plane ones. The in-plane 104422-3P. BAL ´AˇZ AND J. BARNA ´S PHYSICAL REVIEW B 83, 104422 (2011) -0.04-0.020.000.020.04 0 0.5 1 1.5 2Spin-transfer torque θ′ / πF1/N2 interface τx τy τz -0.12-0.08-0.040.000.04 0 0.5 1 1.5 2 θ′ / πN2/F2 interface FIG. 3. (Color online) Angular dependence of the Cartesian STT components in the units of ¯ hI/|e|a c t i n go nF 1/N2and N 2/F1interfaces when ˆS0=ˆS1=ˆezandˆS2is rotated from ˆezin the layer’s plane by an angle θ/prime. components of STT reveal a standard angular dependence at both interfaces. The amplitude of STT at the internal interfacesof CFL is comparable to that acting at the N 1/F1interface in the case of noncollinear configuration of ˆS0andˆS1, when ˆS2 is fixed in the direction antiparallel to ˆS0. When ˆS1is noncollinear to ˆS0, the spin accumulation in the N 1layer increases and consequently the amplitude of STT at F 1/N2and N 2/F2decreases. In turn, when ˆS1is antiparallel to ˆS0, the STT inside the CFL structure is reduced by more than a factor of 2. Nevertheless, the STT actingat the internal interfaces of the studied CFL layers mighthave a significant effect on their current-induced dynamicsand switching process, provided the magnetic configurationof CFL might deviate remarkably from its initial antiparallelconfiguration. B. Synthetic antiferromagnet First, we examine the dynamics of the SyAF free layer. From symmetry we have HRKKY 1 =HRKKY 2 ≡HRKKY , and we have set HRKKY=2 kOe, which corresponds to JRKKY∼ −0.6m J/m2. We have performed a number of independent numerical simulations modeling SyAF dynamics induced byconstant current and a constant in-plane external magneticfield. The latter is assumed to be smaller than the critical fieldfor transition to the spin-flop phase of SyAF. Accordingly, eachsimulation started from an initial state close to ˆS 1=− ˆS2= −ˆez. To have a nonzero initial STT for ˆS1, both spins of the SyAF have been tilted by 1◦in the layer plane so that they remained collinear. From the results of numerical simulations we have con- structed a map of time-averaged resistance, shown in Fig. 4(a). The resistance has been averaged in a time interval of 30 nsfollowing an initial 50 ns equilibration time of the dynamics.The diagram shows only that part of the resistance, whichdepends on magnetic configuration, and hence varies withCFL dynamics. 27The constant part of resistance, due to bulk and interfacial resistances of the studied structure, has beencalculated to be as large as R sp=19.74 f/Omega1m2. For the assumed initial configuration, the magnetic dynam- ics has been observed only for a negative current density.When the current is small, no dynamics is observed sincethe spin motion is damped into the closest collinear state(ˆS 1=− ˆS2=− ˆez,m a r k e da s ↓↑) of high resistance. After exceeding a certain threshold value of the current density, 0 1 2 3 4 5 -I / (108 Acm-2)-400-200 0 200 400Happ [Oe] 0.420.440.460.480.500.520.540.56 R [fΩm2](a)➞ ➞ ➞➞➞➞ -1-0.5 0 0.5 1 S1z S2zHapp < H0 (b) 0 0.5 1 1.5 2 m (c) 0.440.480.520.56 234567R [fΩm2] time [ns](d) (e) (i) -0.8-0.400.40.8-1-0.5 00.5 1-1-0.500.51 Sz Sx SySz Happ > H0 (f) (g) 234567 time [ns](h) -0.2 0 0.2-1-0.5 00.5 1-1-0.500.51 Sz Sx SySz FIG. 4. (Color online) (a) Averaged resistance of Cu- IrMn(10)/Py(8)/Cu(8)/Co(2)/Cu(1)/Co(2)-Cu spin valve pillar with aSAF free layer as a function of current density and applied magnetic field. Examples of switching processes at I=−1.0×10 8Ac m−2 andHapp=−400 Oe (b)–(e) and Happ=400 Oe (f)–(i). (b) and (f) show the dynamics of zcomponents of both spin moments, (c) and (g) present the overall magnetization of the free layer, (d) and (h) show the corresponding variation of pillar resistance, and (e) and (i)show the spin trajectories of ˆS 1solid (red) line and ˆS2dashed (black) line in the time interval from 0 to 10 ns, where switching takes place. there is a drop in the averaged resistance, which indicates the current-induced dynamics of the SyAF free layer. Figures 4(b) and4(f)show that this drop is associated with switching of the whole SyAF structure into an opposite state ( ˆS1=− ˆS2=ˆez, marked as ↑↓). From Fig. 4(a) it follows that the threshold current for the onset of the dynamics markedly depends on the appliedfield and reaches a maximum at a certain value of H app, Happ=H0. Furthermore, it appears that the mechanisms of the switching process for Happ<H 0andHapp>H 0are qualitatively different. To distinguish these two mechanisms,we present in Figs. 4(b)–4(i) the basic characteristics of switching, calculated for I=−1.0×10 8Ac m−2and for fieldsHapp=−400 Oe, which is below H0[Figs. 4(b)–4(e)], andHapp=400 Oe, which lies above H0[Figs. 4(f)–4(i)]. 104422-4CURRENT-INDUCED DYNAMICS OF COMPOSITE FREE ... PHYSICAL REVIEW B 83, 104422 (2011) Figures 4(b) and 4(f) present the time evolution of the z components of both spins. To better understand the SyAFdynamics, in Figs. 4(c) and4(g) we plotted the amplitude of overall SyAF magnetization, defined as m=|ˆS 1+ˆS2|. This parameter vanishes for antiparallel alignment of bothspins of CFL, but becomes nonzero when the configurationdeviates from the antiparallel one. Magnetization of SyAF isalso a measure of the CFL coupling to external magnetic field.Furthermore, Figs. 4(d) and4(h) show the corresponding time variation of the resistance, R, which might be directly extracted from experimental measurements as well. In addition, inFigs. 4(e) and 4(i) we show the trajectories of ˆS 1and ˆS2 in real space taken from the time interval from t=0t o 10 ns. In addition, from Fig. 4(a) it has been found that the point where the threshold current reaches its maximumis located at H 0/similarequalH02 z, which indicates its relation to the magnetostatic interaction of F 2and fixed polarizer. This also has been confirmed by analogical simulations disregardingthe magnetostatic coupling between magnetic layers, whichresulted in a similar diagram, but with H 0=0 (not shown). This fact significantly facilitates understanding the mechanismof SyAF switching. The initial configuration assumed above was −ˆS 1=ˆS2/similarequal ˆS0with ˆS0=ˆez(↓↑). When the magnitude of the current density is large enough and I<0, the orientation of ˆS1 becomes unstable and ˆS1starts to precess with a small angle around −ˆez. Initial precession of ˆS1induces precession of ˆS2—mainly via the RKKY coupling. Generally, the response to the exchange field is slower than current-induced dynamics.Therefore, a difference in the precession phase of ˆS 2and ˆS1appears, and the configuration of SyAF deviates from the initial antiparallel one. This, in turn, enhances the STT actingon F 2, which tends to switch ˆS2. Its amplitude, however, is small in comparison to the strong RKKY coupling. Afurther scenario of the dynamics depends then on the externalmagnetic field. When H app<H 0[Figs. 4(b)–4(e)] the Zeeman energy of ˆS2has a maximum in the initial state and the external magnetic field tends to switch ˆS2to the opposite orientation. Competition between the torques acting on SyAFresults in out-of-plane precessions of both spins. After severalprecessions ˆS 1reaches the opposite static state, which is stable due to STT. In turn, ˆS2is only slightly affected by STT, and its dynamics is damped in the external magneticand RKKY exchange fields. In contrast, when H app>H 0 [Figs. 4(f)–4(i)], the Zeeman energy of F 2has a local minimum in the initial state, which stabilizes ˆS2. Therefore, in a certain range of current density, SyAF does not switch but remains inself-sustained coherent in-plane precessions [boundary areabetween ↓↑and↑↓(red) in the upper part of Fig. 4(a)]. For a sufficient current density, the SyAF structure becomesdestabilized and the precessional angle increases until the spinspass the ( x,y) plane. Consequently, the precessional angle decreases and spins of the SyAF are stabilized in the oppositestate ( ↑↓). Moreover, as shown in Fig. 4(c), the switching process for H app<H 0is connected with a high distortion of the SyAF configuration, where min a certain point reaches its maximum value (corresponding to a parallel orientationof both spins). Contrary, the mremains small for H app>H 0 [Fig. 4(g)], and the effective magnetic moment of the free layer stays smaller than magnetic moment of a single layer. Thismight play an important role in applications of spin-torque devices based on CFLs. The two switching mechanisms described above dominate the current-induced dynamics when the current density is closeto the dynamics threshold. For higher current densities, thenonlinearities in SyAF dynamics become more pronounced,which results in a bistable behavior of the dynamics, especiallyforH app<H 0andI/greaterorsimilar108Ac m−2. In that region, the number of out-of-plane precessions before SyAF switching increaseswith the current density. However, their precessional angleincreases in time and consequently ˆS 1might reach an out-of- plane static point slightly tilted away from the ˆexdirection while ˆS2=ˆezremains in the layers’ plane. The out-of-plane static states (marked as ←↑) have a small resistance and appear as dark (red) spots in the diagram shown in Fig. 4(a). In addition, from the analysis of the dispersion of pillar re- sistance (not shown) one finds that, except for a narrow regionclose to the dynamics threshold with persistent in-plane pre-cessions, no significant steady-state dynamics of the SyAF el-ement appears. As will be shown below, such a behavior mightbe observed when CFL becomes asymmetric (SyF free layer). C. Synthetic ferrimagnet Let us study now the spin valve with SyF as a free layer, assuming d1=4 nm and d2=2 nm. Accordingly, HRKKY 2 remains at 2 kOe while HRKKY 1 is reduced to 1 kOe. As in the case of SyAF, from the averaged time-dependent part of thepillar resistance, we have constructed a diagram presentingcurrent-induced dynamics [see Fig. 5(a)]. The static part of resistance is now R sp=19.80 f/Omega1m2. The diagram has some features similar to those studied in the previous section. However, the maximum criticalcurrent is shifted toward negative values of H app,e v e ni f the magnetostatic interaction between magnetic layers isneglected. This asymmetry is caused by the difference inexchange and demagnetization fields acting on layers F 1and F2. Moreover, this difference leads to more complex dynamics of the CFL’s spins than that in the case of SyAF. Generally, there are several dynamic regimes to be dis- tinguished. The first one is the region of switching fromthe↓↑configuration to the opposite one, ↑↓, which is located at the largest values of H appin the diagram. The mechanism of the switching is similar to that of SyAF shownin Figs. 4(f)–4(i), where CFL changes its configuration just via in-plane precessional states with a small value of m (weak distortion of the antiparallel alignment of ˆS 1and ˆS2). Furthermore, the darker area above H0indicates one of the possible self-sustained dynamic regimes of SyF, i.e., the in-plane precessions (IPP); see Figs. 5(b)–5(e). This precessional regime starts directly after the SyF switching, and ˆS 1and ˆS2precess around ˆezand−ˆez, respectively. Due to different effective fields in F 1and F 2, and energy gains due to STT, the spins precess with different precessional angles [Fig. 5(e)] and consequently different frequencies. Because of the stronginterlayer coupling and spin transfer between the layers, theamplitudes of their precessions are periodically modulated intime. This modulation appears also in the time dependenceof the pillar resistance. Conversely, below H 0the dynamics is dominated by large-angle out-of-plane precessions (OPPs) 104422-5P. BAL ´AˇZ AND J. BARNA ´S PHYSICAL REVIEW B 83, 104422 (2011) 0 1 2 3 4 5 -I / (108 Acm-2)-400-200 0 200 400Happ [Oe] 0.460.480.500.520.540.560.580.60 R [fΩm2](a)➞ ➞ ➞➞ IPP OPP -1-0.5 0 0.5 1 S1z S2zIPP (b) 0 0.5 1 1.5 2 m (c) 0.480.520.560.60 02468 1 0R [fΩm2] time [ns](d) (e) (i) -0.4-0.200.20.4-0.8-0.400.40.8-1-0.500.51 Sz SxSySz-1-0.500.51 OPP (f) 00.511.52 (g) 0.440.480.520.560.60 4 8 12 16 20 time [ns](h) SxSy-1-0.500.51 Sz -1-0.500.51-1-0.500.51Sz FIG. 5. (Color online) (a) Averaged resistance of Cu- IrMn(10)/Py(8)/Cu(8)/Co(4)/Cu(1)/Co(2)-Cu spin valve pillar with a SyF free layer, presented as a function of current density and appliedmagnetic field. Examples of current-induced dynamics for I= −3×10 8Am−1andHapp=200 Oe (b)–(e) and Happ=−400 Oe (f)–(i). (b) and (f) show the dynamics of zcomponents of ˆS1andˆS2, (c) and (g) present the overall magnetization of the free layer, (d) and (h) show the corresponding variation of pillar resistance, and (e) and (i) show spin trajectories of ˆS1solid (red) line and ˆS2dashed (black) line taken from a time interval as large as 30 ns after 100 ns of initial equilibration. of both spins, as shown in Figs. 5(f)–5(i). This dynamic state is connected with a strong distortion of the antiparallel CFLconfiguration, i.e., large value of m, and a large variation of the resistance. From Fig. 5(i)one can see that trajectories of ˆS 1andˆS2are rather complicated, including both IPP and OPP regimes with dominant OPP. D. Power spectral density From the analysis of current-induced dynamics we found that self-sustained dynamics in structures with a SyF free layeris much richer than that in systems with a SyAF free layer[see Figs. 5(b)–5(i)]. Therefore, in this section we restrict ourselves to dynamic regimes of the SyF free layer only. Morespecifically, we shall examine the power spectral density (PSD)as a function of current density and external magnetic field. 2 3 4 5 -I / (108 Acm-2) 10 20 30 40 50frequency [GHz](a) 1 2 3 4 5 -I / (108 Acm-2) 30 40 50 60 10-610-510-410-310-210-1100 PSD [pW/MHz](b) -1-0.500.51 S1z (c) -1-0.500.51 2468 1 0S2z time [ns](d) (g)(h)0.440.480.520.56 R [fΩm2] (e) 0.460.470.480.49 4 8 12 16 20R [fΩm2] time [ns](f) -0.8-0.400.40.8-1-0.500.51 -1-0.500.51 Sz SxSySz -1-0.6-0.20.2 -0.8-0.400.40.8-1-0.500.51 Sz SxSySz FIG. 6. (Color online) PSD calculated for the spin valve with a SyF free layer at Teff=5Ka n d Happ=−400 Oe (a) and 200 Oe (b). (c) and (d) show the steady time evolution of the spins’ zcomponents in a time window of 10 ns after equilibration at Happ=−400 Oe and I=−2.8×108Ac m−2. (e) and (f) show the steady time evolution of the time-dependent part of the spin valve resistance in a timewindow of 20 ns after equilibration at H app=−400 Oe and I= −3.6×108Ac m−2(both ˆS1and ˆS2precess out of the layer plane) andI=−3.8×108Ac m−2(ˆS2performs out-of-plane precessions while ˆS1precesses in the layer’s plane), respectively. (g) and (h) depict trajectories of ˆS1solid (red) line and ˆS2dashed (black) line corresponding to resistance oscillations (e) and (f), respectively. In the simulation we started from I=0 and changed the current density in steps /Delta1I=106Ac m−2at a fixed applied field. As before, to protect the SyF dynamics fromcollapsing into a collinear static state, we assumed smallthermal fluctuations corresponding to T eff=5 K. At each step we simulated the dynamics of the coupled CFL’s spins andcalculated PSD. As in Ref. 29, we assumed that the input current is split between a load with resistance R Land a pillar with resistance Rsp+R(t). Hence, the voltage on the pillar has been calculated as U(t)=IR(t)/[1+Rsp/(RLS)], where we assumed RL=50/Omega1, andSis the cross section of the pillar (ellipsoid with the major and minor axes equal to 130 and60 nm, respectively). Then, at a given I, we calculated the voltage in the frequency domain, U(f), using fast Fourier transformation over the period t FFT=50 ns following an equilibration time of teq=30 ns. The PSD has been defined as PSD( f)=2U2(f)/(RL/Delta1f), where /Delta1f=1/tFFT. Figures 6(a) and6(b) show the PSD calculated at Happ= −400 and 200 Oe, respectively. The former case corresponds to the part of the diagram in Fig. 5(a), which includes OPP modes, while in the latter case we observed IPP only. Let us 104422-6CURRENT-INDUCED DYNAMICS OF COMPOSITE FREE ... PHYSICAL REVIEW B 83, 104422 (2011) analyze first the situation in Fig. 6(a). When the current passes through the corresponding threshold value, both spins startprecessing in the layers’ plane, similarly as shown in Fig. 5(b). Apart from the main peak in the PSD at f/similarequal40 GHz, two additional minor peaks close to f/similarequal20 GHz are visible. We attribute them to the oscillations of precessional amplitudesof both spins. With an increasing amplitude of the currentdensity, the precessional angles of both spins increase andtheir precessional frequencies slightly decrease. Moreover,with increasing current the frequencies of the minor peaksbecome increasingly closer, until they finally coincide. At thispoint the PSD becomes widely distributed along the wholerange of observed frequencies, which is evidence of noisyvariation of the resistance. An example of spin dynamics in thisregion is shown in Figs. 6(c)and6(d), which have been taken in a time window as large as 10 ns after the equilibration periodforH app=−400 Oe and I=−2.8×108Ac m−2[within the broad feature of PSD in Fig. 6(a)]. First, the figures show that ˆS2starts to perform out-of-plane precessions as a result of the competition between STT and RKKY coupling. Second, onecan note thermally activated random transitions of ˆS 1between the OPP and IPP modes. These random transitions modify OPPprecessions of ˆS 2as well. The simultaneous dynamics of both spins causes a chaotic variation of spin valve resistance andbroadens the PSD. The quasichaotic feature of the spin dynam-ics in this range of current densities can be seen also on the spintrajectories, which cover almost the whole sphere (not shown). A further increase in current density leads to stabilization of the OPP mode of ˆS 1. Hence the spin valve resistance becomes more periodic [see Fig. 6(e)] and PSD again reveals a narrow peak. Since both spins perform rather complicated dynamicswhen including IPP but dominated by the OPP regime [seeFig. 6(g)], we observe a blueshift in the PSD with current, which is connected with a decrease in the precessional angles.However, at a certain value of Iwe notice an abrupt drop in the peak’s frequency. At this current density the STT acting on theleft-hand interface of layer F 1starts to dominate the dynamics ofˆS1and enables only small-angle IPPs along the ˆS0direction, which modifies the trajectory of ˆS2.ˆS2still remains in the OPP regime [see Fig. 6(h)] and hence the blueshift with current ap- pears. The fact that the IPP of ˆS1still influences the dynamics of the whole SyF is also shown in Fig. 6(f), which presents the dynamic part of the spin valve resistance at I=−3.8× 108Ac m−2andHapp=−400 Oe. As a result of the IPPs of ˆS1, the amplitude of the resistance varies periodically. In addi- tion, a comparison of Figs. 6(e)and6(f)shows that the simul- taneous OPPs of both spins lead to a stronger variation of theresistance than in the case when the layers are in the IPP state. Contrary to this, at H app=200 Oe one observes only IPP modes of both spins similar to those shown in Fig. 5(e).T h e in-plane precessional angle increases with current density andhence the peak frequency in PSD decreases and becomesbroader. In real systems, however, one might expect narrowerpeaks than those obtained in the macrospin simulations, asobserved in standard spin valves with a simple free layer. 30,31 IV . CRITICAL CURRENTS First, we derive some approximate expressions for the critical current density needed to induce the dynamics ofCFL, derived from a linearized LLG equation. In metallic structures, the out-of-plane torque components are generallymuch smaller than the in-plane ones, and therefore willbe omitted in the analytical considerations of this section (b (0) 1,b(2) 1,b(1) 2→0). The coupled LLG equations in spherical coordinates can be then written as d˜S dt=1 1+α2M·˜v, (6) where ˜S=(θ1,φ1,θ2,φ2)Tis a four-dimensional column vector which describes spin orientation in both layers constitutingthe CFL, and ˜v=(v 1θ,v1φ,v2θ,v2φ)Tstands for the torque components, viθ=/Gamma1i·ˆeiθandviφ=/Gamma1i·ˆeiφ, with ˆeiφ= (ˆez׈Si)/sinθiand ˆeiθ=(ˆSi׈eiφ)/sinθidenoting unit vectors in local spherical coordinates associated with ˆSi.I n turn, the 4 ×4m a t r i x Mtakes the form M=⎛ ⎜⎜⎜⎝1 α 00 −α/sinθ11/sinθ1 00 001 α 00 −α/sinθ21/sinθ2⎞ ⎟⎟⎟⎠.(7) The static points of the CFL dynamics have to satisfy viθ=0 andviφ=0 for both i=1 and i=2. These conditions are obeyed in all collinear configurations, i.e., θi=0,π. Additionally, four trivial static points can be found in theout-of-plane configurations with θ i=π/2 andφi=0,π. Following Ref. 23, we linearize Eq. ( 6) by expanding ˜vinto a series around the static points, which leads to d˜S dt=1 1+α2M·J·˜δv, (8) where Jis a Jacobian matrix of ∂˜vi/∂˜Sjcomponents. The matrix product M·Jdefines here the dynamic matrix D= M·J. This matrix allows one to study the stability of the CFL’s spins in their static points. If Tr {D}is negative, the static point is stable, otherwise it is unstable. Hence, the condition for thecritical current is 32Tr{D}=0. To obtain threshold current for the dynamics onset of individual spins in the CFL, we assume first that one of thespins is fixed in its initial position, and investigate the stabilityof the second spin. The dynamic matrix Dthen reduces to a2×2 matrix. Considering the initial position of SyAF with ˆS1=− ˆS2=− ˆez(i.e.,θ1=πandθ2=0), marked as ↓↑, and polarizer ˆS0=ˆez, the stability condition leads to the following critical currents I↓↑ c1andI↓↑ c2forˆS1andˆS2, respectively: I↓↑ c1=−αμ0Msd1 a(0) 1+a(2) 1/bracketleftbig −H1↓↑ ext+Hani−HRKKY 1 +Hd 1/bracketrightbig ,(9) with H1↓↑ ext=Happ−H01 z−H21 z , and I↓↑ c2=−αμ0Msd2 a(1) 2/bracketleftbig H2↓↑ ext+Hani−HRKKY 2 +Hd 2/bracketrightbig ,(10) 104422-7P. BAL ´AˇZ AND J. BARNA ´S PHYSICAL REVIEW B 83, 104422 (2011) withH2↓↑ ext=Happ−H02 z−H12 z. In both of the above expres- sionsa(0) 1,a(2) 1, anda(1) 2are taken in the considered static point, while the demagnetization field for the ith layer is given by Hd i=Hd ix+Hd iy 2−Hd iz. (11) Analogically, one can derive similar formulas for critical currents in the opposite ( ↑↓) magnetic configuration of the CFL. Now we relax the assumption that one of the spins is fixed, and consider both spins of the CFL as free. Then, wecalculate the trace of the whole 4 ×4 matrix, which leads to the following expression for critical current destabilizing thewhole CFL: I ↓↑ c CFL=−αμ0Msd1d2 d2/parenleftbig a(0) 1+a(2) 1/parenrightbig +d1a(1) 2/bracketleftbig H↓↑ ext+2Hani −HRKKY 1 −HRKKY 2 +Hd 1+Hd 2/bracketrightbig , (12) where H↓↑ ext=H01 z−H02 z+H21 z+H12 z. Since the spins of CFL are antiparallel in the considered static point, I↓↑ c CFL is independent of the external magnetic field. The above equationdescribes the critical current at which the CFL is destabilizedas a rigid structure (unaffected by external magnetic field alongthezaxis). Numerical calculations presented below show that the critical current is usually smaller than that given by Eq. ( 12). Apparently, as shown by numerical simulations, there is aphase shift in the initial precessions of ˆS 1andˆS2. Such a phase shift slightly perturbs the initial antiparallel configuration andmight reduce the critical current for the onset of the dynamics. A similar formula also holds for the opposite configuration (↑↓), where the critical current is given by I↑↓ c CFL=αμ0Msd1d2 d2/parenleftbig a(0) 1−a(2) 1/parenrightbig −d1a(1) 2/bracketleftbig −H↑↓ ext+2Hani −HRKKY 1 −HRKKY 2 +Hd 1+Hd 2/bracketrightbig , (13) withH↑↓ ext=H01 z−H02 z−H21 z−H12 z. Now we discuss the theoretical results on critical currents in the context of those following from numerical simulations. Letus consider first the critical currents for individual spins of theSyAF free layer, assuming that the second spin remains stablein its initial position, Eqs. ( 9) and ( 10). The corresponding results obtained from the formula derived above are presentedin Table I, where we have omitted a weak dependence on H app. For the studied structure with a SyAF free layer, I↓↑ c1is negative while I↓↑ c2is positive. From our analysis, it follows that the current density at which the dynamics appears in the simulations [Fig. 4(a)] is higher than that given by I↓↑ c1. However, we checked numerically that the critical value I↓↑ c1 agrees with the critical current obtained from simulations when assuming ˆS1as free and fixing ˆS2along ˆez. Following the above discussion of the CFL dynamics, one can understand the shift of the threshold current as follows.Initially, when the current density exceeds I↓↑ c1,ˆS1becomes destabilized. Then, ˆS2responses to the initial dynamics of ˆS1with a similar coherent precession. However, ˆS2should still be stable in its initial position at this current density andTABLE I. Critical current densities in units of 108Ac m−2, calculated according to Eqs. ( 9), (10), and ( 12) for both SyAF and SyF free layers. SyAF SyF ↓↑ ↑↓ ↓↑ ↑↓ Ic1 −0.31 0.43 −0.54 0.63 Ic2 0.98 −0.46 2.33 −0.78 IcC F L −0.86 0.87 −2.37 −9.18 common precessions of the two coupled spin moments damp the initial dynamics. Accordingly, SyAF ends up in the closeststatic state ( ↓↑). However, as the current density increases, the initial precessions of ˆS 1become more pronounced, which in turn means that the initial antiparallel configuration becomesdistorted and ˆS 2becomes destabilized. This results in a coupled dynamics of both spins and finally leads to switchingof the SyAF structure. On the other hand, we have also calculated the critical current for the whole SyAF structure according to Eq. ( 12), and for the given structure we got I↓↑ c CFL, shown in Fig. 4(a)by the dashed vertical line (see also Table I). Equation ( 12) describes the stability of the whole CFL, and since the interlayer coupling is strong, I↓↑ c CFL corresponds to the current density at which both spins become destabilized, simultaneously preserving theirantiparallel orientation. As can be seen in Fig. 4(a),t h i si s the maximum threshold current density for current-induceddynamics. Because the rigid structure consisting of twoantiparallel spins is not influenced by an external homogeneous magnetic field, there is no dependence of I ↓↑ c CFL onHapp. Nevertheless, from our numerical simulation, it follows thatthe threshold current for the SyAF dynamics, I thr, obeys the condition |I↓↑ c1|<|Ithr|<|I↓↑ c CFL|, provided that |I↓↑ c1|<|I↓↑ c2| or (as in our case) I↓↑ c2has a different sign. When the SyAF is in the ↑↓configuration, the spin accumulation and spin current are different from those in the↓↑configuration (at the same voltage). This in turn leads to different spin torques, which is the reason why the criticalcurrents destabilizing the ↑↓state are different from those for↓↑, as shown in Table I. From the critical currents one can expect a relatively symmetric hysteresis with an applied current in the structures with SyAF. In contrast, I ↑↓ c CFL for the SyF is negative, similarly as I↓↑ c CFL, but it is significantly larger, which indicates a lack of hysteresis. To compare theswitching of the SyAF and SyF free layers from the ↓↑to ↑↓configurations with the opposite ones ( ↑↓to↓↑), we have simulated the dynamics of the corresponding CFLs assumingH app=0 and varying current density. The simulations have been performed in the quasistatic regime, i.e., for each valueof current density the spin dynamics was first equilibrated for50 ns, and then averaged values of spin components and pillarresistance were calculated from the data taken for the next30 ns of the dynamics. In order to prevent the system fromcollapsing into a static state with zero torque, we have includeda thermal stochastic field corresponding to T eff=5 K [see Eq. ( 3)]. Starting from I=0 and going first toward negative currents, we have constructed the current dependence of the 104422-8CURRENT-INDUCED DYNAMICS OF COMPOSITE FREE ... PHYSICAL REVIEW B 83, 104422 (2011) 0.48 0.5 0.52 0.54 R [fΩm2] (a)SyAF SyF ↑↓↓↑ -1 0 1 S1z (b) -1 0 1 -1 -0.5 0 0.5 1S2z I / (108 Acm-2)(c) 0.5 0.52 0.54 0.56 0.58 (d)↑↓↓↑ -1 0 1 (e) -1 0 1 -2 -1 0 1 2 I / (108 Acm-2)(f) 0.48 0.5 0.52 0.54 -1 -0.5 0 0.5 1R [fΩm2] I / (108 Acm-2)(g) 0.5 0.52 0.54 0.56 0.58 -2 -1 0 1 2 3 I / (108 Acm-2)(h) FIG. 7. (Color online) Hysteresis loops of the resistance for the studied pillars with (a) SyAF and (d) SyF free layers. (b) and (c) depict the spin dynamics of ˆS1andˆS2in SyAF, respectively, corresponding to resistance loop (a). (e) and (f) show the dynamics of ˆS1and ˆS2 in SyF, respectively, corresponding to resistance loop (d). The initial point of each hysteresis loop is marked with a dot. The arrows indicatethe direction of the current change. (g) and (h) correspond to the upper parts of (a) and (d), in which, however, the effects due to the magnetostatic field of the reference layer to the CFL spins have beenomitted. averaged resistance and related zcomponents of both spins, as shown in Fig. 7. For both SyAF [Figs. 7(a)–7(c)] and SyF [Figs. 7(d)–7(f)] free layers, one can see a relatively symmetric hysteresis withthe current density. In both cases direct switching from the ↓↑ to↑↓state occurs at a current density comparable to I↓↑ c CFL.I n contrast, in the case of the SyF free layer, the second transition(↑↓to↓↑) appears at a current density which is very different from that predicted by the linearized LLG model. Moreover,in both cases, switching from the ↑↓to↓↑state does not appear directly, but through some precessional states. Moreprecisely, as the positive current density increases, both spinsstart precessing in the layers’ plane prior to switching. Thein-plane precessions are connected with a significant drop inthe resistance and with a reduction of the s zcomponents. The range of the IPP regime is particularly large in the case of SyF.From the analysis of the spins’ trajectories one may concludethat the angle of the IPPs increases with increasing currentdensity, and after exceeding a certain threshold angle CFLswitches to the ↓↑configuration. The other factor giving rise to the the difference in switching from↑↓to↓↑and from ↓↑to↑↓follows from the fact that the magnetostatic interaction of the CFL’s layers with the polarizeris different in the ↓↑and↑↓states. To prove this we have constructed analogical hysteresis loops for SyAF and SyF freelayers disregarding magnetostatic interaction with the F 0layer;see Figs. 7(g) and7(h). For both SyAF and SyF free layers we observe now a large decrease in Rfor both switchings. This implies that both switchings are realized via in-planeprecessions, in contrast to the case when F 0influences the CFL dynamics via the corresponding magnetostatic field. While thehysteresis loop for SyAF remains symmetric, the one for SyFbecomes highly asymmetric. The asymmetry of the SyF loop isdue to a significant asymmetry of STT in the ↑↓and↓↑states, which was previously shaded by the magnetostatic couplingwith the layer F 0. V . DISCUSSION AND CONCLUSIONS We have studied current-induced dynamics of SyAF and SyF composite free layers. By means of numerical simulationswe identified a variety of dynamical regimes. The mostsignificant difference between the dynamics of SyAF andSyF free layers concerns the evidence of the self-sustaineddynamics of both CFL spins. While in the case of SyAFonly coupled in-plane precessions in a narrow window ofexternal parameters ( H appandI) are observed, the SyF free layer reveals a more complex and richer dynamics, with thepossibility of coupled out-of-plane precessions which mightbe interesting from an application point of view. Furthermore,as shown by numerical simulations, both SyAF and SyF areswitchable back and forth without the need of an externalmagnetic field. For the SyAF element two possible ways ofswitching have been identified. Since they lead to differentswitching times, their identification might be crucial for theoptimization of switching in real devices with SyAF freelayers. However, one has to note that the diagrams shownin Figs. 4and5may be changed when magnetization in CFL becomes nonhomogeneous. A disadvantage of the studied structures is their relatively low efficiency of switching, i.e., high amplitude of criticalcurrent and long switching time. In order to show moresophisticated ways of tuning the CFL devices, we haveanalyzed critical currents derived from the linearized LLGequation. Formula ( 12) has been identified as the maximum value of critical current at which the dynamics of the CFLstructure should be observed. This formula reveals some basicdependence of critical current on spin valve parameters, andtherefore might be useful as an initial tool for its tuning.However, in some cases nonlinear effects in the CFL dynamicsmight completely change the process of CFL switching, asshown by the presented numerical simulations. But the effectsof nonlinear dynamics go beyond the simple approach of alinearized LLG equation, and their study requires more so-phisticated nontrivial methods and/or numerical simulations. ACKNOWLEDGMENTS This work was supported by the Polish Ministry of Science and Higher Education as a research project in the years2010–2011, and partly by the EU through the Marie CurieTraining Network SPINSWITCH (MRTN-CT-2006-035327).The authors thank M. Gmitra for helpful discussions. One ofus (P.B.) also thanks L. L ´opez D ´ıaz, E. Jaromirska, U. Ebels, and D. Gusakova for valuable suggestions. 104422-9P. BAL ´AˇZ AND J. BARNA ´S PHYSICAL REVIEW B 83, 104422 (2011) APPENDIX: TRANSFORMATIONS OF SPIN CURRENT The torque acting on the left-hand interface of F 1is calcu- lated from the xandycomponents of j/prime 1=T(θ1,φ1)·j1, where j1is the spin current vector in the N 1layer (written in a global frame; shown in Fig. 1), and T(θ1,φ1)=Rx(−θ1)Rz(φ1− π/2), where Rq(α) is the matrix of rotation by angle αalong the axisqin the counterclockwise direction when looking toward the origin of the coordinate system. Hence the j/prime 1components can be written as j/prime 1x=j1xsinφ1−j1ycosφ1, (A1a) j/prime 1y=(j1xcosφ1+j1ysinφ1) cosθ1−j1zsinθ1,(A1b) j/prime 1z=(j1xcosφ1+j1ysinφ1)s i nθ1+j1zcosθ1,(A1c) where ( θ1,φ1) are the spherical coordinates of ˆS1in the global frame. Similarly, we define the torques’ amplitudes on the left-hand interface of F 2from the components of the transformed spin current vector j/prime/prime/prime 2=T(θ2,φ2).j2. In this case, however, j2 is not written in the global frame, but in the local coordination system coordinate system connected with ˆS1. To rotate the local coordinate system of ˆS1to the local coordinate system of ˆS2, we need to know the spherical angles θ2andφ2of vector ˆS2in the local coordinate system of ˆS1. This might be done by transforming first the ˆS2vector to the local coordinate systemofˆS1asˆS/prime 2=T(θ1,φ1)·ˆS2and calculate its angles θ2andφ2. Then we can calculate the components of j/prime/prime/prime 2similarly as for the left-hand interface, j/prime/prime/prime 2x=j2xsinφ2−j2ycosφ2, (A2a) j/prime/prime/prime 2y=(j2xcosφ2+j2ysinφ2) cosθ2−j2zsinθ2,(A2b) j/prime/prime/prime 2z=(j2xcosφ2+j2ysinφ2)s i nθ2+j2zcosθ2,(A2c) The equations in N 2, which is an adjacent nonmagnetic interface from the right-hand side of F 1, are written in the local coordinate system of ˆS1. To apply the definition of a(1) 2andb(1) 2 we need to rotate the local coordinate system so that its yaxis will lie in the layer given by vectors ˆS1andˆS2. This might be done by a single rotation of the local coordinate system arounditszaxis by an angle φ 2−π/2,j/prime/prime 2=Rz(φ2−π/2)·j2, where j/prime/prime 2x=j2xsinφ2−j2ycosφ2, (A3a) j/prime/prime 2y=j2xcosφ2+j2ysinφ2, (A3b) j/prime/prime 2z=j2z. (A3c) Note that angle φ2is calculated for vector ˆS2transformed into a coordinate system of ˆS1as in previous case. 1J. C. Slonczewski, J. Magn. Magn. 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PhysRevB.83.174424.pdf

PHYSICAL REVIEW B 83, 174424 (2011) Fractional locking of spin-torque oscillator by injected ac current Dong Li,1Yan Zhou,2Changsong Zhou,1,*and Bambi Hu3 1Department of Physics, Centre for Nonlinear Studies and The Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex Systems (Hong Kong), Hong Kong Baptist University, Kowloon Tong, Hong Kong 2Department of Information Technology, Hong Kong Institute of Technology, Mid-levels west, Hong Kong 3Department of Physics, University of Houston, Houston, Texas 77204-5005, USA (Received 4 January 2011; revised manuscript received 21 March 2011; published 11 May 2011) Fractional synchronization is one of the most interesting collective behaviors in coupled or driving-response oscillators system, very important for both a deep understanding of a particular oscillator and for its applications.We numerically investigate the fractional synchronization of a spin-torque oscillator by injected ac current.Multiple p:qlocking regions are found, which display some sophisticated overlaps. The system can be analyzed as a perturbed heteroclinic cycle rather than a phase oscillator. Both the modulations on the output frequencyand power are mainly due to the modulation by the external signals on the distance between the dynamical orbitand the saddle point in phase space. By using this dynamical picture, we can well understand all the numericalresults, including the variation of the locking region with the amplitude |J a|or frequency fof the injected signal, the influence by noise, and the difference among the output powers of coexisting locking attractors.These understandings are significant for both potential applications in electronic communications and a deepinvestigation into this novel device. DOI: 10.1103/PhysRevB.83.174424 PACS number(s): 75 .40.Gb, 85 .75.Bb, 05 .45.Xt I. INTRODUCTION The spin-torque oscillator (STO)1currently is receiving a rapidly growing interest thanks to its significant advantages,such as its extremely small footprint (without the need ofa large inductor), ultrawide-band-frequency operation, andeasy on-chip integration, which make it very promisingfor broadband high-quality microwave generator. 2It is of significance to study the interaction between a STO and anexternal stimuli both for a deep understanding of the dynamicalcharacteristics of a STO device and for potential applications.For example, for the purpose of utilizing that STO in mostof the existing communication technology, it is imperativethat it be readily modulated by an external stimuli since apure microwave resonance of the STO does not carry anyinformation. 3When the external frequency fis close to the intrinsic (free-evolving) STO frequency F, it is possible to get a 1 : 1 phase locking, where the locking ratios p:q= f:˜F=1 : 1, and ˜Fis the oscillation frequency of the STO under external forces. This effect has been studied both exper-imentally and theoretically for the purpose of understandingand realization of mutual synchronization of two or moreSTOs. 4 Understanding the response of a STO to a wider range of injected frequencies can be obtained by investigating thefractional synchronization, which means p:qis a rational number. Urazhdin et al. 5experimentally demonstrated that the STO can be fractionally locked to a microwave magneticfield and pointed out that the nontrivial observation was dueto the complex nonlinear characteristics of STO. Microwavemagnetic field and microwave current are the two externalstimuli that are usually used to interact with STOs. However,it will be much more convenient to modulate the STO with anexternal microwave current instead of a microwave magneticfield since the incorporation of a large magnetic field sourcefor the generation of a microwave field will outweigh anyadvantages of STO-based nanosized devices. 6In this paper,we will study the fractional synchronization of the STO with a wide range of injected ac currents.5 Previously, such a driven STO has usually been analyzed by phase oscillator model, where the synchronization is usually associated with a 1 : 1 locking when the driving frequency isclose to the free-evolving frequency. 4,7But the phase oscillator model cannot be used to explain some important nonlinearcharacteristics. 8Therefore, it is not expected to be suitable to analyze most of the phenomena relating to p:qfractional locking, since the appearance of p:qlocking usually is as- sociated with some highly nonlinear characteristics. The STO is a nonlinear oscillatory system with some saddle-connectionstructures, and its dynamics is more suitably analyzed as aperturbed heteroclinic cycle. 8In this type of dynamical system, there is a good degree of flexibility of locking in a wide range of both subharmonics and superharmonics, which comes from the sensitivity of its dynamical state near the heteroclinic orbit, especially near the saddle points.9This is the theoretical foundation basis on which we analyze the p:qfractional locking of STO. Three problems are of special importance in studying locking behavior. First, how do these p:qlocking regions change with the parameters of the driving signals? They can highly reflect the nonlinear characteristics of a particularsystem. 5,10Second, how do these regions change if noise effect is taken into consideration? It is important becausenoise plays a unique role in such a dynamical system witha saddle-connection structures 9,11and noise is inevitable in the experiments.12Third, what is the output power of these locking attractors? The output power is of importance for itsapplications. 2 Our studies in this paper will be focused on the afore- mentioned three problems. Multiple p:qlocking regions are found, and their rich behaviors are shown. The STO systemhas a great degree of flexibility of locking to a wide rangeof injected frequencies than that of a typical driven phase 174424-1 1098-0121/2011/83(17)/174424(8) ©2011 American Physical SocietyDONG LI, Y AN ZHOU, CHANGSONG ZHOU, AND BAMBI HU PHYSICAL REVIEW B 83, 174424 (2011) oscillator. Different p:qfractional locking regions can have complicated overlaps, and multiple p:qlocking attractors can coexist at the same system parameters. Noise can destroy theattracting basins of the locking attractor having the smallestdistance to its bifurcation point in parameter space or the onehaving the smallest attracting basin. Noise can also makethe STO lock to a slower injected frequency. The outputpower is different among the coexisting locking attractors.These nontrivial features must be taken into considerationwhen developing STOs into applications. Importantly, all theobservations can be well understood by the nonlinearity ofthe perturbed heteroclinic cycle structure, i.e., the results ofthe modulation on the frequency or on the output powerindeed comes from the modulation on the distance between thedynamical orbit and the saddle point in the phase space, andthe larger the distance, the slower the frequency. Through thesedetailed studies of fractional locking, the complex nonlinearcharacteristics of this nontrivial STO system can therefore bewell understood. II. MODEL A macrospin approximation treats the magnetization of a sample as a single macroscopic spin and has been extensivelyused to capture some features of magnetic materials qualita-tively as well as for studying the fundamental aspects of thespin-torque-induced magnetization precession and switching,albeit with some limitations, such as are typically used forstudying nanopillar with lateral sizes below 30 nm where thedominant spin wave is a zero-order coherent spin wave mode;and it becomes invalid for the cases where the large-amplitude,higher-order spin wave modes excitations become importantsuch as magnetic nanocontacts, vortex oscillators, and mostof spin-torque-driven domain-wall motions. In this paper weutilize a macrospin approximation for studying the qualitativefeatures of microwave-current-driven fractional locking of themostly commonly investigated in-plane spin-torque nanopillardevice injected with ac current [Fig. 1(a)]. The unit vector of the free-layer magnetization mis described by the Landau– Lifshitz–Gilbert–Slonczewski (LLGS) equation, 13 dm dt=− |γ|m×Heff+αm×dm dt+|γ|βJm×(m×M), (1) where γis the gyromagnetic ratio, αis the Gilbert damping parameter, and βcontains material parameters and funda- mental constants.14The electrical current Jis defined as positive when electrons flow from the fixed layer to the freelayer. The effective field H effcarries the contribution of an external applied magnetic field Ha, an anisotropy (easy axis) fieldHkalong the xaxis, and a demagnetization (easy plane anisotropy) field μ0Hdz=4πMs, where Msis the saturation magnetization of the free-layer material. Thus we get Heff= Haˆex+(Hkmxˆex−Hdzmzˆez)/|m|. In most experiments, due to the utilization of a synthetic antiferromagnetic stucture suchas a pinned layer, the stray field acting on the free layer canbe significantly compensated. In the framework of macrospinsimulation employed in the current study, the stray field actingon the free (sensing) layer is assumed to be zero.FIG. 1. (Color online) (a) Schematic of the in-plane STO investi- gated in this paper. The free-layer magnetization mis separated from the fixed layer Mby a nonmagnetic layer. In the spherical coordinate system (b), the three possible oscillatory states in a free-evolving STO are shown respectively in (c) a attracting heteroclinic cycle, (d) global oscillation, and (e) local oscillation. The bold lines are theattractors. Their trajectories in the three-dimensional configuration space are shown respectively in (f) a homoclinic orbit, (g) out-of-plane oscillation, and (h) in-plane oscillation. Lin (d) indicates the distance between the orbit and saddle. ˆLin (g) demonstrate this distance in the configuration space. In spherical coordinate system, we can express Eq. ( 1)a s dθ dt=γ 1+α2{Ucosθcosφ+α(Hdz+Hkcos2φ) ×sinθcosθ−Vsinφ−Hksinφcosφsinθ}, (2) dφ dt=γ 1+α2/braceleftbigg1 sinθ[−Usinφ−(Hdz+Hkcos2φ)s i nθ ×cosθ−Vcosφcosθ−αHksinφcosφsinθ]/bracerightbigg ,(3) where U=αHa−β(Jd+Ja) and V=Ha+αβ(Jd+Ja). Jdis the dc current and Ja=|Ja|cos 2πf t represents the injected ac current with the amplitude |Ja|and the injected frequency f. The values of some parameters used in the calculation are as follows: |γ|=1.885×1011Hz/T,α= 10−2,Ha=0.2T ,β=10/3,Hdz=1.6T ,Hk=0.05 T, andJd=10 mA, so that the free-evolving frequency F approximates to 21 .7 GHz. We call it a free-evolving state when |Ja|=0. In the pa- rameter region where the STO can itself sustain an oscillation,the equilibrium state /Pi1 (1)thatmis antiparallel to Mmust be a saddle point in the dynamical phase space, whereas theother equilibrium state /Pi1 (2)thatmis parallel to Mmust be an unstable focus. There may be a homoclinic orbit (a heterocliniccycle in spherical coordinate system) connecting the saddle,but it can just happen at a particular subset with zero measurein the parameter space. The heteroclinic cycle in spherical 174424-2FRACTIONAL LOCKING OF SPIN-TORQUE OSCILLATOR ... PHYSICAL REVIEW B 83, 174424 (2011) coordinate [Fig. 1(b)] is shown in Fig. 1(c), and its trajectory in the three-dimensional configuration space is shown inFig. 1(f). The dynamics of the system with parameters deviating from that for the heteroclinic cycle can be dealt withas aperturbed heteroclinic cycle . The asymptotic stable states can be classified into two cases: (i) global oscillation [Figs. 1(d) and 1(g)] which means the free layer mrotates around the current J, and (ii) local oscillation (LO) [Figs. 1(e) and1(h)] which means mjust vibrates near a particular direction. 8Such three types of oscillation have also been numerically and/oranalytically given in others’ work. 15In this paper, we will focus on the physically more relevant global oscillatory state exceptfor special notes. Such a state has a free-evolving frequencyFdepending on the distance Lbetween the orbit and the saddle point, 8as shown in Fig. 1(d). While the free-evolving state studied in this paper is always a global oscillatorystate, the orbit could be driven across the saddle point andbecomes a local oscillatory one when the injected current islarge. III. FRACTIONAL SYNCHRONIZATION In the presence of an injected current, the system is driven to a new orbit with a different distance ˜Lfrom the saddle, and the frequency is modulated to ˜Fmainly due to ˜L. When the injected ac current Jais small, it influences the system as another perturbation. The case of small Jahas been studied in Ref. 8, showing that the 1 : 1 locking region is proportional to the injected |Ja|. Here, we consider a wider range of driving frequencies and stronger currents; the synchronizationregions are shown in Fig. 2. In the case of a wider range of driving frequencies, several fractional locking regions could beobserved even when |J a|is small. For example, in Fig. 2(below the dashed line), the regions of locking ratios p:q=1:2 , 1 : 1, 2 : 1, and 3 : 1 are obvious and well separated. In thefollowing, we investigate these fractional locking regions andfocus on the three aforementioned problems. A.p:qlocking regions Before investigating how the p:qlocking regions change with system parameters, we show that it is relatively easierto fractionally lock this system and get larger locking regionsthan typical driven phase oscillators. The reason is that thedynamical orbit in this type of system is very sensitive toexternal perturbation near the saddle points and easier tobe driven to the target orbit of the synchronized frequency. 9 Typically, an oscillator which could be fractionally locked to a driving signal, when simplified to the form of phase oscillator,would take the following form: dφ/dt =2πF+μ/summationdisplay p,qgp,qsin (2πqft−pφ), (4) where Fandfare the free-evolving and driving frequencies, respectively, μis the driving strength, and gp,qis the weight of the p:qdriving component. Usually, gp,qis bigger with smaller pandq, so that the smaller pandqlocking attractor is more stable and its locking region is usually larger. When the other p:qcomponents could be ignored com- pared with 1 : 1, this phase oscillator could be simplified as dφ/dt =2πF+μsin (2πf t−φ), (5) whose locking region is 2π|F−f|/lessorequalslantμ. (6) If we regard the driven STO as a phase oscillator, the 1 : 1 driving strength has the same order asβγ 1+α2|Ja|.8,16The 1 : 1 locking region given by Eq. ( 6) (dashed line in Fig. 3,l o w e r panel) is much smaller than the 1 : 1 locking region in Fig. 2 at the same injected amplitude |Ja|. Here, our aim is to show a simple comparison, so we preserve three terms of p:q=1 : 1, 2 : 1, and 3 : 1 and assume that g1,1=g2,1=g3,1=1 (in physical case, usually g1,1>g 2,1>g 3,1).μis still assigned the value asβγ 1+α2|Ja|. In Fig. 3, we show the locking region of this simplified driven phase oscillator: dφ/dt =2πF+μ[sin (2 πf t−φ) +sin (2πf t−2φ)+sin (2πf t−3φ)]. (7) Two significant differences are observed between the real driven STO system (Fig. 2) and the driven phase oscillator system (Fig. 3). First, this driven STO system shows a much larger locking region of each fractional locking attractor, whencomparing Fig. 2with the lower panel of Fig. 3, both having t h es a m es c a l eo f |J a|. To get a similar size of locking region in the driven phase oscillators, the driving current |Ja|needs to be enlarged by 2–3 orders of magnitude, as seen in upper panel ofFig. 3. Second, there could be many p:qlocking regions in the real STO system whereas the number of fractionallocking regions in the driven phase oscillator depends on FIG. 2. (Color online) Probabilities of synchronization from different initial conditions in 1 : 3, 1 : 2, 1 : 1, 2 : 1, and 3 : 1 regions. Black indicates the 100% synchronization regions. We use different color scales to help distinguish different locking regions. In each region, strips ofthe lighter color indicate smaller synchronization probabilities. Some other locking regions, such as 3 : 2, 4 : 3, etc., are also observed, but not shown in the figure for the sake of clarity). |J a|=2 mA is on the dashed line. The probability in this figure and also other places in this paper represents the ratio of the phase volume of an attracting basin to the whole phase space. 174424-3DONG LI, Y AN ZHOU, CHANGSONG ZHOU, AND BAMBI HU PHYSICAL REVIEW B 83, 174424 (2011) FIG. 3. (Color online) Fractional locking regions of the phase oscillator in Eq. ( 7). The lower panel has the same scale of |Ja| as that of Fig. 2, but the locking regions are much smaller. To get a similar size of locking region to Fig. 2, the injected current |Ja| needs to be enlarged greatly, as shown in the upper panel. The dashedlines delimit the 1 : 1 locking region of the driven phase oscillator, dφ/dt =2πF+β|J a|sin (2πf t−φ). the form of the coupling term [in the case of Eq. ( 5), only 1 : 1 is found; in the case of Eq. ( 7), 2 : 1 and 3 : 1 locking emerge]. This comparison shows that the degree of flexibilityof fractional synchronization of the driven STO system is quitehigh, consistent with the expectation based on the perturbedheteroclinic cycle structure. Several typical representations ofthe locking dynamics of the driven STO system are shown inFig. 4. In a typical oscillator which can be well described by a phase oscillator, increasing J aof the driving signal will generally enlarge the locking region and make an lockingattractor more stable, which can be beneficial for many FIG. 4. (Color online) Time evolution of cos φ(solid lines) and the injected signal cos (2 πf t) (dashed lines) of several typical locking examples: (a) 1 : 3 ( f=7.2 GHz), (b) 1 : 2 ( f=10.8 GHz), (c) 1 : 1 (f=21.6 GHz), (d) 2 : 1 ( f=45.0 GHz), (e) 3 : 1 ( f=66.0 GHz), (f) 4 : 3 ( f=30.0 GHz), and (g) 2 : 1 local oscillation ( f= 30.0 GHz). |Ja|=10 mA, and other parameters are the same as in Fig. 2. LO stands for local oscillation.applications. However, this is not always the case in the driven STO system. Comparing Fig. 2and Fig. 3, we can see that when different p:qlocking regions meet, they may have some complicated overlaps. It is totally different from the p:q locking in the driven phase oscillator, where the fractionallocking regions are separated by obvious boundaries. Thisdifference has significant implications in applications, becausethe overlapping of locking regions means that the attractormay not become more stable but rather could shift to anotherunder some particular initial conditions when |J a|is increased. Here, multiple p:qlocking attractors can coexist for the same set of system parameters, for example the coexistenceof 4 : 3 locking attractor and 2 : 1 locking attractor with[2 : 1(LO)], shown in Figs. 4(f) and 4(g). The problem happens when an expected 100% synchronization region is pierced bythe locking regions of other attractors. For example, in theenvelope of the 100% synchronization region of 1 : 1 lockingin Fig. 2, synchronization may be expected to be achieved from any initial conditions, and then this synchronization regioncould be robustly employed in applications. However, thisregion is in fact pierced by other regions, e.g., 2 : 1, 3 : 1locking, so the 1 : 1 synchronization is not always achievable.An example is shown in Fig. 5. When f=24.2 GHz and |J a|=3 mA, 1 : 1 locking will be always achieved from any initial condition because it is the only attractor. But whenincreasing to |J a|=5 mA, the synchronization is invaded by 2 : 1 locking. From some particular initial conditions, 2 : 1locking is achieved. When |J a|becomes larger, the overlaps of the coexisting multiple locking attractors and even some newlyemergent ones will make the asymptotical dynamics dependstrongly on initial conditions. The sophisticated overlapping of different locking regions is a result of the saddle-node bifurcation of synchronizationin the perturbed heteroclinic cycle system. 16W h e nan e w locking attractor emerges, an unstable orbit emerges at thesame parameter point, becoming the boundary of the attractingbasin. So there is no need for the other attractors to lose FIG. 5. (Color online) Time series of cos φof different locking attractors with the identical injected frequency f=24.2 GHz. (a) 1 : 1 locking when |Ja|=3 mA (unfilled squares), (b) 1 : 1 locking when |Ja|=5 mA (filled squares), and (c) 2 : 1 locking when |Ja|=5 mA (diamonds) compared with the driving signal cos 2πf t (solid line). (d) The dynamical orbits of these three attractors in phase space {θ,φ}. The filled circles and stars are respectively the saddle point /Pi1(1)and unstable focus /Pi1(2). The slower locking attractor (diamonds) shows an obviously shorter distance from the saddle point. 174424-4FRACTIONAL LOCKING OF SPIN-TORQUE OSCILLATOR ... PHYSICAL REVIEW B 83, 174424 (2011) stability at the same time and multiple attractors can coexist as the overlap of the locking regions. On the other hand, adriven phase oscillator model in Eq. ( 4) is not relevant to explain this phenomenon, because the synchronization anddesynchronization in phase oscillators are usually associatedwith supercritical Hopf bifurcation. When the synchronizationattractor becomes stable, the other attractors must disappear,so that there are clear boundaries between locking regions. According to our calculations within the framework of the perturbed heteroclinic cycle structure, the STO can demon-strate fractional locking to an ac current, which is similarto its fractional locking to an ac magnetic field, observed inexperiments. 5However, compared with the fractional locking by an ac field,5our studies show that the STO will exhibit different locking characteristics by an ac current, e.g., theoverlapping of the locking regions, which has not beenreported in the experiments by the ac magnetic field. Anotherdifference refers to the size of the synchronization regions.As shown in Fig. 2,1 : 1 ,2 : 1 ,a n d3 : 1h a v es i m i l a r synchronization regimes, different from the case of the ac fieldp: 1 phase locking, where the sizes of the synchronization regions are significantly different for even and odd p. It should also be noticed that we simulate the probability in Fig.2and elsewhere by using uniform distribution of the initial conditions, so that the exact meaning of probability in fact reflects the ratio of the phase volume of an attracting basin tothe whole phase space. The probabilities of these attractors inexperiments is equivalent to the probabilities in this work with a weighting factor, which originates from the initial conditions’distribution affected by a variety of stochastic factors in reality,including, but not limited to, the temperature. B. Noise effect What the effect of noise is on these p:qlocking phe- nomena of the STO is another important problem. The greatdegree of flexibility of STO locking to a wide range is dueto its sensitivity to the perturbation near saddle points. 9Noise thus usually plays an important role in such a system.11And noise is inevitable in experiments. In the driven STO system,the influence of noise is more complex and interesting. It caninfluence both the locking frequency and the attracting basinsof the coexisting locking attractors. Let us first study the impact of noise on the locking frequency. Usually noise tends to drive the orbit far awayfrom the saddle points in a pure heteroclinic cycle system, soas to speed up the oscillations. 11Therefore it is often expected that the system tends to lock with a higher driving frequencywhen noise strength is increased. But in this system, thereis already a perturbation induced by the system parameters,which makes the system have a higher opportunity of beingdriven close to the saddle in the presence of noise. As a result,the STO can be locked to a smaller driving frequency withsome noise. But this phenomenon of noise-induced slowingdown is not easily observed in a 1 : 1 locking region, becausethis locking region is usually too wide where the variationinduced by noise is easily ignored. Thanks to the fractionalsynchronization states, we can demonstrate this nontrivialphenomenon obviously in a thinner locking region. As shownin the 1 : 2 region in Fig. 6(a), increasing Gaussian white noise can make the STO obviously lock to a smaller drivingFIG. 6. (Color online) Probabilities of synchronization of (a) 1 : 2, (b) 1 : 1, and (c) 2 : 1 regions while noise added as /angbracketleftHa(t)/angbracketright= 0.2T,/angbracketleftHa(t)Ha(s)/angbracketright=2Dδ(t−s).|Ja|=2mA and the other param- eters are the same as in Fig. 2. frequency. The variation of locking frequency induced by noise is a typical phenomenon of synchronization in such adynamical system with a saddle-connected structure, but theslowing-down phenomenon induced by noise is nontrivial. Inthese systems, the modulation by noise on the distance betweenthe dynamical orbit and the saddle point results in a modulationon the frequency, one of the mechanisms underlying anotherwell-known phenomenon named stochastic resonance. 17In this way, noise can sometimes contribute to the locking of aSTO to a different frequency. This effect is especially obviouswhen the locking region is small [comparing Fig. 6(a) with Figs. 6(b) and6(c)]. It is therefore very important to consider the effect of noise in potential applications of STOs, especiallyin the case where the locked or modulated frequency is usedto encode information. The effect of noise on attracting basins is easier to compre- hend. In the region of coexistence of multiple attractors, noisetends to destroy the basin of the locking attractor that has thesmallest distance to its bifurcation point in parameter space,or the one with the smallest attracting basin. We demonstratethe meanings of the smallest distance to its bifurcation point andsmallest attracting basin schematically in Fig. 7, where initial conditions refer to all the state variables, and ωcould be any system parameter, e.g., the injected frequency fin our STO system. Along with the increasing ω, one can see the emergence of D and A and the disappearance of B, A, and Cin order. At ω 1andω2, A and B have the smallest distance to their bifurcation points S1andS2, respectively. D has the smallest attracting basin at most region of the parameter ω. In the driven STO system, by increasing the driving frequency f, different p:qlocking attractors can emerge and disappear in a way similar to that of Fig. 7. We show one of such examples in Fig. 8. Figures 8(a) and8(b) respectively show the probabilities of getting each locking attractor withthe uniform distribution of initial conditions in the absenceand in the presence of noise, when the amplitude of the drivingsignal is fixed as |J a|=10 mA; Figs. 8(c),8(e),8(g), and 8(h) show the distributions of those attracting basins at f=29.1 GHz, f=30.0 GHz, f=30.3 GHz, and f=15.9 GHz; and Figs. 8(d) and 8(f) show two realizations of simulation in the presence of noise at f=29.1 GHz and f=30.0 GHz to demonstrate the change of the distributions of attractingbasins and the smearing of the basins’ boundaries. Noise is absent at first. When ffalls within the region between about 29 .1 GHz and 30 .0 GHz (1 .38–1.43 times ofF), the coexistent locking attractors are 1 : 1, 4 : 3, 3 : 2, and 2 : 1(LO) [Fig. 8(a)]. Similar to other fractional locking problems, the locking attractors with small pandqusually 174424-5DONG LI, Y AN ZHOU, CHANGSONG ZHOU, AND BAMBI HU PHYSICAL REVIEW B 83, 174424 (2011) FIG. 7. (Color online) Schematic drawing of the bifurcation process versus ωin a dynamical system with the coexistence of multiple attractors, where ωcould be any system parameter, e.g. the injected frequency fin our STO system. A, B, C, and D indicate the attracting basins of four p:qlocking attractors. S1indicates the bifurcation point of the emergence of A; S2indicates the bifurcation point of the disappearance of B. At ω1andω2,Aa n dBh a v et h e smallest distance to their bifurcation point, respectively. D has the smallest attracting basin in most regions of the parameter ω. exist in a wider region of system parameters, and they are usually more stable (having large basins),10e.g., C in Fig. 7and the 1 : 1 locking in Fig. 8. When fincreases to 30.3 GHz, the desynchronization attractor appears [Fig. 8(g)]. The output frequency of the 1 : 1 locking has the largestdifference from the free-evolving frequency F. However, the 1 : 1 locking still has a longer distance to its bifurcation pointof disappearance than 4 : 3 locking. The attracting basin of thedesynchronization attractor does not invade the basin of the1 : 1, but the basin of 4 : 3. With further increasing of f,4:3 locking will disappear completely at about 30 .6 GHz. The similar invasion happens to 3 : 2 locking when fdecreases below 28 .8 GHz. These observations show clearly that 1 : 1 locking appears to be more stable, whereas 4 : 3 is least stableatf=30.0 GHz and 3 : 2 is least stable at f=29.1 GHz. The next question is, when noise is taken into consideration, which basin will be invaded first? The answer is that noise tendsto invade the basin of 4 : 3 at f=30.0 GHz or the basin of 3 : 2 atf=29.1 GHz. It always invades the one having the smallest distance to its bifurcation point, because the existence of anattractor is sensitive to external perturbation when the systemparameters are so close to the bifurcation point. Thereforethe probabilities of achieving 4 : 3 locking at f=30.0 GHz or 3 : 2 at f=29.1 GHz significantly decrease, as shown in Fig.8(b). We show a realization of simulation at f=29.1 GHz with noise in Fig. 8(d), where the basin of 3 : 2 locking disap- pears completely due to noise. A similar phenomenon happensto the 4 : 3 locking at f=30.0 GHz, shown in Fig. 8(f). In other ranges of system parameters, the coexistent attractors are different; for example, we show in Fig. 8(h) that, at f=15.9 GHz, the coexistent attractors turn out to be 1 : 1, 1 : 1(LO) and desynchronization state (DS), but thechange of probabilities under noise effect is always similar. Noise also tends to destroy the smallest attracting basin. In Fig. 7, D has the smallest basin and so does the 2 : 1 locking attractor of local oscillation in Fig. 8(a). Comparing Fig. 8(f)with Fig. 8(e), one can easily see that 2 : 1(LO) is noFIG. 8. (Color online) Attracting basins for the coexisting multi- ple attractors when |Ja|=10 mA. (a) In the absence of noise, when fis between about 29 .1 GHz and 30 .0 GHz (the vertical dashed lines), locking attractors of 1 : 1, 4 : 3, 3 : 2 obviously coexist. [2 : 1 (LO) also exists, but its probability is too small.] (b) In the presence of noise of /angbracketleftHa(t)/angbracketright=0.2T ,/angbracketleftHa(t)Ha(s)/angbracketright=0.01δ(t−s)T2,t h e probabilities of these attractors significantly change: the 3 : 2’sturn out to be negligible at about f=29.1 GHz, the 4 : 3’s turn out to be negligible at about f=30.0 GHz, and the 2 : 1(LO)s completely disappear for the whole region. (c) The distribution ofattracting basins at f=29.1 GHz, and (d) one of its realizations of simulation in the presence of noise demonstrates the change of the distribution and the smearing of the basins’ boundaries. (e) Theattracting basins at f=30.0 GHz and (f) one of their realizations in the presence of noise. (g) The attracting basins at f=30.3 GHz. (h) An example of the distribution of attracting basins far from theaforementioned parameter region, while f=15.9 GHz. DS stands for the desynchronization state. longer present under the noise effect. This change can also be observed by comparing Fig. 8(b) with Fig. 8(a).I nF i g . 8(b), the probability of 2 : 1(LO) is zero, whereas in Fig. 8(a),i t is not zero. But this comparison is not clear enough since theprobability of 2 : 1(LO) is quite tiny in Fig. 8(a). 174424-6FRACTIONAL LOCKING OF SPIN-TORQUE OSCILLATOR ... PHYSICAL REVIEW B 83, 174424 (2011) Now let us get back to discussing Figs. 6(b) and 6(c). Note that 100% synchronization state is most significante inapplications because it is independent of initial conditions.An interesting question is whether noise can contributeto increasing the 100% synchronization state if the basinsof other attractors are very small. The answer is yes. InFig. 6(b) and Fig. 6(c), it is seen that, when the probability of synchronization is quite close to 100%, increasing noisecan enhance it to 100% and make the 100% region widerand wider until noise strength is too large and it destroyssynchronization again. On the other hand, when probabilitiesof synchronization are low, noise will always destroy thesynchronization state. These effects of noise on the attracting basins resulted from the particular bifurcation process as demonstrated inFig. 7. The saddle-node bifurcation of synchronization in this perturbed heteroclinic cycle system is therefore crucial. All thenontrivial effects of noise originate from the role of noise tomodulate the dynamical orbits near the saddle points in phasespace in such a system with a perturbed heteroclinic cyclestructure. Thus, a driven phase oscillator model is not relevantto explain these effects. C. Output power Besides all aforementioned dynamical behaviors, the output power is another important issue, since it is tied toapplications of the system. The emitted microwave powerspectra of the STO depends on a wide range of materialparameters. Here we study how the output power is influencedby external driving signals. We have performed the Fourier FIG. 9. (Color online) (a) Fourier transformation amplitudes of the cosine function of the relative angle between mandMof different attractors and (b) the positions of their orbits in phase space {θ,φ}. Parameters are the same as in Fig. 8(e), except for that in the free-evolving state |Ja|=0 and in the desynchronization statef=30.3 GHz. The dashed lines in (a) demonstrate the com- parison among the output powers of the coexisting synchronization state.transformations of sin θcosφ, the cosine function of the relative angle between mandM, which is proportional with the STO output signal, to reflect the microwave power. Theperturbed heteroclinic cycle structure can help us easily knowthe difference among the output powers of the coexistinglocking attractors. The faster attractor oscillates farther awayfrom the saddle point, usually with a smaller amplitude ofoscillation, 8leading to a smaller output power. Figure 9(a) shows the simulation results. The dashed line demonstratesthe comparison we analyzed (which can be simply markedas 3 : 2 >4:3>1 : 1). The positions of these attractors in phase space {θ,φ}are shown in Fig. 9(b). When q/negationslash=1, the distances between an orbit and the saddle point /Pi1 (1)have a little difference each time it gets close to the saddle point/Pi1 (1)since one p:qattractor gets back to its original position after it passes over the saddle point for qtimes (rotates q cycles in configuration space). However, one can still easilynotice the significant difference among the distances betweeneach orbit and the saddle points. The significant differenceamong the distances induces different locking frequencies,and oscillatory amplitudes, leading to different output power. IV . DISCUSSION To gain a deeper understanding of how the STO device responses to a wide range of injected frequencies, we studythe fractional synchronization of STO by an injected ac current.Multiple p:qlocking regions are observed. Our studies focus on three important problems: how do the locking regionschange with driving parameters, how does noise affect thep:qlocking phenomenon, and what is the output power of thesep:qlocking attractors? First, we found that the system has a great degree of flexibility of locking to a wide rangeof driving frequencies. The locking regions can have somesophisticated overlaps, where multiple p:qattractors can coexist at the same system parameters. Even some 100%synchronization regions can be merged by other lockingregions. Second, noise plays a nontrivial role. It can makethe STO lock to relatively slower frequencies, and it can alsodestroy the attracting basin of the locking attractor having thesmallest distance to its bifurcation point, or the one having thesmallest attracting basin. Finally, we showed that the outputpower of the coexistent locking attractors depends on theoscillating frequencies. All these novel dynamical behaviors were well explained by the perturbed heteroclinic cycle structures. Our studies aresignificant both for understanding the nonlinear characteristicsof the STO system and for potential applications. Our work can also help understand better about the dynamical behavior of the LLGS equation and thus shedlight on a broad range of magnetism problems which can bedescribed by this equation. 18 ACKNOWLEDGMENTS This work is support by Hong Kong Baptist University and conducted using the resources of the High Performance ClusterComputing Centre, Hong Kong Baptist University, which re-ceives funding from Research Grant Council, University GrantCommittee of the HKSAR, and Hong Kong Baptist University. 174424-7DONG LI, Y AN ZHOU, CHANGSONG ZHOU, AND BAMBI HU PHYSICAL REVIEW B 83, 174424 (2011) *cszhou@hkbu.edu.hk 1J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, P h y s .R e v .L e t t . 84, 3149 (2000); D. Houssameddine, U. Ebels, B. Delaet, B. Rodmacq, I. Firastrau, F. Ponthenier, M.Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda, M.-C. Cyrille,O. Redon, and B. Dieny, Nat. Mater. 6, 447 (2007); J. Katine and E. Fullerton, J. Magn. Magn. Mater. 320, 1217 (2008); D. Ralph and M. Stiles, ibid.321, 2508 (2009); Y . Zhou and J. Akerman, Appl. Phys. Lett. 94, 112503 (2009). 2M. Pufall, W. Rippard, S. Kaka, T. Silva, and S. Russek, Appl. Phys. Lett. 86, 082506 (2005); T. J. Silva and W. H. Rippard, J. Magn. Magn. Mater. 320, 1260 (2008). 3M. R. Pufall, W. H. Rippard, S. Kaka, T. J. Silva, and S. E. Russek, Appl. Phys. Lett. 86, 082506 (2005). 4W. H. Rippard, M. R. Pufall, S. Kaka, T. J. Silva, S. E. Russek, and J. A. Katine, P h y s .R e v .L e t t . 95, 067203 (2005); B. Georges, J. Grollier, M. Darques, V . Cros, C. Deranlot, B. Marcilhac, G.Faini, and A. Fert, ibid.101, 017201 (2008); Y . Zhou, J. Persson, S. Bonetti, and J. Akerman, Appl. Phys. Lett. 92, 092505 (2008). 5S. Urazhdin, P. Tabor, V . Tiberkevich, and A. Slavin, Phys. Rev. Lett.105, 104101 (2010). 6T. J. Silva, Nat. Phys. 3, 447 (2007). 7R. Adler, Proc. IEEE 34, 351 (1946). 8D. Li, Y . Zhou, C. Zhou, and B. Hu, Phys. Rev. B 82, 140407(R) (2010). 9M. I. Rabinovich, R. Huerta, and P. Varona, Phys. Rev. Lett. 96, 014101 (2006). 10S. E. Brown, G. Mozurkewich, and G. Gr ¨uner, Phys. Rev. Lett. 52, 2277 (1984); J. Teki ´c, D. He, and B. Hu, Phys. Rev. E 79, 036604 (2009).11E. Stone and P. Holmes, SIAM J. Appl. Math. 50, 726 (1990). 12M. W. Keller, M. R. Pufall, W. H. Rippard, and T. J. Silva, Phys. Rev. B 82, 054416 (2010). 13J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); L. Berger, Phys. Rev. B 54, 9353 (1996). 14J. Grollier, V . Cros, and A. Fert, P h y s .R e v .B 73, 060409(R) (2006); J. Persson, Y . Zhou, and J. Akerman, J. Appl. Phys. 101, 09A503 (2007). 15S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J.Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature (London) 425, 380 (2003); Y . Zhou, J. Persson, S. Bonetti, and J. Akerman, Appl. Phys. Lett. 92, 092505 (2008); J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 72, 014446 (2005); Z .L i ,Y .C .L i ,a n dS . Zhang, ibid.74, 054417 (2006); J. Z. Sun, ibid.62, 570 (2000); G. Bertotti, C. Serpico, I. D. Mayergoyz, A. Magni, M. d’Aquino, andR. Bonin, Phys. Rev. Lett. 94, 127206 (2005); O. Boulle, V . Cros, J. Grollier, L. G. Pereira, C. Deranlot, F. Petroff, G. Faini, J. Barnas,and A. Fert, Nat. Phys. 3, 492 (2007); M. Gmitra, D. Horvath, M. Wawrzyniak, and J. Barnas, Phys. Status Solidi 243, 219 (2006). 16C. Sperpico, R. Bonin, G. Bertotti, M. d’Aquino, and I. Mayergoyz, IEEE Trans. Magn. 45, 3441 (2009). 17G. Hu, T. Ditzinger, C. Z. Ning, and H. Haken, P h y s .R e v .L e t t . 71, 807 (1993); L. Gammaitoni, P. H ¨anggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998); B. Hu and C. Zhou, Phys. Rev. E 63, 026201 (2001). 18T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004); X. R. Wang and Z. Z. Sun, Phys. Rev. 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PhysRevB.91.064421.pdf

PHYSICAL REVIEW B 91, 064421 (2015) Nuclear magnetic resonance study of thin CihFeAIo.s Sio.5 Heusler films with varying thickness A. Alfonsov,1 ’* B. Peters,2 F. Y. Yang,2 B. Buchner,1 ,3 and S. Wurmehl1 ,3 ,3 1 Leibniz Institute for Solid State and Materials Research IFW Dresden, D-01171 Dresden, Germany 2 Department o f Physics, The Ohio State University, Columbus, Ohio 43210, USA 3 Institute for Solid State Physics, Technische Universitdt Dresden, D-01062 Dresden, Germany (Received 13 August 2013; revised manuscript received 5 February 2015; published 20 February 2015) Type, degree, and evolution of structural order are important aspects for understanding and controlling the properties of highly spin-polarized Heusler compounds, in particular, with respect to the optimal film growth procedure. In this work, we compare the structural order and the local magnetic properties revealed by nuclear magnetic resonance (NMR) spectroscopy with the macroscopic properties of thin Co2FeAlo.5Sio .5 Heusler films with varying thickness. A detailed analysis of the measured NMR spectra presented in this paper enables us to find a very high degree of L2r type ordering up to 81% concomitantly with excess Fe of 8%-13% at the expense of A 1 and Si. We show that the formation of certain types of order depends not only on the thermodynamic phase diagrams as in bulk samples, but also that the kinetic control may contribute to the phase formation in thin films. It is an exciting finding that Co2FeAl05Si 0.5 can form an almost ideal L2, structure in films, though with a considerable amount of Fe-Al/Si off stoichiometry. Moreover, the very good quality of the films as demonstrated by our NMR study suggests that the technique of off-axis sputtering used to grow the films sets the stage for the optimized performance of Co2FeAl0. 5S i0.5 in spintronic devices. DOI: 10.1103/PhysRevB.91.064421 PACS number(s): 75.30.-m , 71.20.Be, 61.05.Qr, 76.60.Jx I. INTRODUCTION Spintronics is considered a potential follow-up technology to purely charge-based electronics. In spintronic devices, both charge and spin of electrons are used as information carriers, leading to faster switching at lower energy consumption com pared to charge-based electronics. Half-metallic ferromagnets (HMFs) are the optimal materials to be implemented in spintronic devices [1-5], as their conduction electrons are expected to be 100% spin polarized. Heusler compounds with L2]-type structure represent an especially favorable family of predicted HMF compounds and seem to offer all the necessary ingredients for their implementation in spintronic devices, qualities such as high spin polarization [ 1,3,5,6], high Curie temperatures [7,8], and a low Gilbert damping constant [9], However, the observation of the required key spintronic properties in Heusler compounds crucially depends on the type and degree of structural ordering [4,5,10]. NMR spectroscopy allows one to probe the local envi ronments of 59Co nuclei in Co-based Heusler bulk and film samples, and thus enables characterization of local order and quantification of different structural contributions concomi tantly with an off-stoichiometric composition [10-18]. Such a local probe of structure and composition is very useful since compounds comprising elements from the same periodic row (e.g., Co and Fe) have very similar scattering factors for x rays, and thus x-ray diffraction (XRD) only may not be sufficient to resolve the structural ordering unambiguously, particularly if both disorder and deviations from the 2:1:1 stoichiometry are present [19]. In addition to information on the chemical, crystallographic environments, the NMR technique is useful to determine the magnetic state of a ferro- or ferrimagnetic material. The *a.alfonsov@ifw-dresden.de 3 s. wurmehl@ifw-dresden.derestoring field (//re st) is an effective magnetic field originating from a resistance to magnetic oscillations and therefore is proportional to the square root of the optimal power (i.e., the power producing the maximum spin-echo intensity) of the applied rf pulses during an NMR experiment. //rcst derived in NMR experiments provides a measure of magnetic stiffness or magnetic anisotropy on a local scale, compared with the macroscopic domain wall stiffness contributing to the coercive fields from superconducting quantum interference device (SQUID) magnetometry [11,13]. The advantage of NMR is that we can measure at a given frequency and can thus relate the magnetic stiffness to a specific local magnetic environment (e.g., phase or structure). A particular interesting Heusler compound to be mentioned in the context of HMF is C02FeAlo.5Sio.5- Band-structure calculations predict a high stability of the minority band gap in this compound [3,20], a prediction which is experimentally supported [ 2 1]. Co 2FeAlo. 5Sio.s has been implemented in thin films and magnetic tunnel junctions [22-30], Recently, we have epitaxially grown Q^FeAl o.sSi 0.5 films on lattice- matched MgAl 20 4 (0 0 1) substrates by an off-axis sputtering technique, yielding films with an exceptionally high quality [31]. In this work we characterize the local crystallographic and magnetic structure of these films using NMR. We were able to relate the macroscopic physical properties of these Co2FeAl o.sSi 0.5 films to the local ordering. II. EXPERIMENTAL DETAILS Epitaxial Q^FeAl o.sSi 0.5 films were grown on MgAl 20 4 (001) substrates by off-axis sputtering in a UHV system with a base pressure as low as 7 x 10“ 1 1 Torr using ul- trapure Ar (99.999 9%) as sputtering gas. Optimal-quality Co2FeAlo. 5Sio .5 epitaxial films were obtained at an Ar pressure of 4.5 mTorr, a substrate temperature of 600 °C, and dc sputtering at a constant current of 12 mA, which results in a deposition rate of 5.6 A/min. The Co 2FeAl o.sSi 0.5 1098-0121/2015/91 (6)/064421 (7) 064421-1 ©2015 American Physical Society ALFONSOV, PETERS, YANG, BUCHNER, AND WURMEHL 20 (deg) 65.0 65.5 66.0 66.5 67.0 FIG. I. (a) High-resolution 6/26 XRD scan of a 45-nm Co2 FeAl o ,5Si o.5 (CFAS) film grown on MgAL04 (001) substrates, (b) XRD rocking curve of the (004) peaks of the Co2FeAl 0 5Si 0.5 film gives a FWHM of 0.004 3°. epitaxial films were characterized by a Bruker D8 Discover high-resolution triple-axis x-ray diffractometer. Details about growth and characterization are found elsewhere [31]. The NMR experiments were performed at 5 K in an automated, coherent, phase-sensitive, and frequency-tuned spin-echo spectrometer (NMR Service, Erfurt, Germany). We used a manganin coil wrapped around the sample to apply and pick up the rf pulses. This coil is implemented in an LC circuit with three capacitors. The NMR spectra were recorded at 5 K in the frequency (v) range from 104 to 254 MHz in steps of 0.5 MHz in zero magnetic field. All NMR spectra shown here were corrected for the enhancement factor as well as for the v2 dependence, resulting in relative spin-echo intensities which are proportional to the number of nuclei with a given NMR resonance frequency [11,13], III. RESULTS AND DISCUSSION Figure 1 shows the 6/26 scan of a 45-nm-thick Co2FeAl 0.5Si 0.5 epitaxial film on MgAl20 4 (001). The clear Laue oscillations near the Co2FeAl o.sSi 0 .5 (004) peak demonstrate the high crystalline uniformity as well as smooth surface and sharp interface with the substrate. Figure 1(b) presents a rocking curve of the (400) peak with a FWHM of 0.004 3°, which is at the instrumental resolution limit of our high-resolution XRD system, revealing exceptional crystalline quality. In order to further characterize the structural quality of our films, we measured the 59Co NMR spectra for different thin-film samples with varying thickness (20, 45, 84, 120, and 200 nm). Figure 2 exemplarily shows the normalized 59Co NMR spectra of films with thickness t = 20, 84, and 200 nm in comparison with that of a Co2FeAl 0.5Si 0.5 bulk sample (data taken from Ref. [32]). All spectra share the main line around 163 MHz with one shoulder on the low-frequency side and two pronounced satellites on the high-frequency side with spacing of about ~33 MHz between them.PHYSICAL REVIEW B 91, 064421 (2015) l.O ----.---- -------- 1 -------------. ------------ -- -r-A feT T I 100 150 200 250 frequency (MHz) FIG. 2. (Color online) Normalized 5 9 Co NMR spectra of Co2FeAl o.jSi 0.5 thin films with thicknesses of t = 20, 84, and 200 nm in comparison with the NMR data of a Co2FeAl0 .5Sio .5 bulk sample [32], Note that missing data points in the middle of the spectra are due to the increased spectrometer noise for this frequency. The observation of low- and high-frequency satellite lines with a spacing of about 33 MHz suggests a contribution from 62-type ordering of the films, in line with the interpretation of the NMR data for Co2FeAlo.5Sio.5 and Co2FeAl bulk samples [15,32], Partial 62-type ordering of the films is consistent with the ternary thermodynamic Co-Fe-Al phase diagram at 650°C (Ref. [33]) and with experimental results for the Co2FeAlo.5Sio.5 compound from Umetsu et al. [34]. Neglecting other contributions to the high-frequency satellite, its higher intensity in the films may be understood in terms of a higher degree of 62-type contributions. This interpretation, however, is in strong contrast to the significantly smaller NMR echo intensity at ~ 130 MHz in films compared to the bulk sample. In fact, the NMR spectra of the film samples do not exhibit a clear satellite but rather a shoulder, on the low- frequency side, which complicates the qualitative comparison of the spectra. Taking into account both observations, the larger high-frequency satellites and the poorly resolved low- frequency satellite in the films compared to the bulk sample, it seems natural to interpret this observation as a deviation from the 2:1:0.5:0.5 stoichiometry, and more specifically, to assume that the films are more Fe-rich and Al/Si-poor than the expected 2:1:0.5:0.5 stoichiometry (compare Refs. [10], [16], and [32]). The formation of Fe-rich environments may also be responsible for the slightly higher than expected magnetic saturation moment as, according to the Slater- Pauling rule, the expected value for magnetic moment in the case of Co2FeAl0.5Sio,5 compound is 5.5 /aB/f.u., whereas the measured one for the 45-nm film is about 5.6 /zB/f.u. (see Ref. [31]). Thereby, already a qualitative analysis of the thin-film NMR spectra suggests a contribution from both L2{ and 62 types of order, as well as a presence of an Fe to Al/Si off stoichiometry. Kozakai et al. [33] report that 62 is the thermodynamic stable phase at 600 °C in Co2FeAl, whereas Umetsu et al. report that the L2r type phase is thermodynamically stable below 1125 K and Co2FeAl05Sio.5 undergoes a transition to 62-type order 064421-2 NUCLEAR MAGNETIC RESONANCE STUDY OF THIN Co ... PHYSICAL REVIEW B 91, 064421 (2015) at 1125 K [34], Please note that even for the bulk sample annealed below the ordering temperature no full L2\ order is realized [32,34], In the present case of a thin film, both 62 and the higher-ordered L 2r type phases are found, suggesting additional influence of, e.g., the substrate, strain, and/or kinetic contributions upon cooling. In order to perform a detailed quantitative analysis of all the contributions to the NMR spectra, we fitted the NMR spectra of all samples using a sum of Gaussian lines. The corresponding parameters of these lines, such as resonance frequency, linewidth, and intensity, were constrained according to a model similar to the one described in detail in Ref. [32]. For L2i -type order only one NMR line is expected, while 52-type order yields several NMR lines [13,15]. Hence, in the presence of both L2\-and 62-type order and off stoichiometry, the relative area of the NMR spectra can be represented as a sum of several lines originating in different structural and compositional contributions. The spacing between adjacent resonance lines, A 62, may be assumed to be a constant while their relative contribution to the NMR spectrum is given by the amount of random mixing of Fe and Al/Si on one crystallographic site (52-type structure), as well as by the Fe-to-Al/Si ratio. The off stoichiometry between Fe and Al/Si contributes to NMR lines on the high-frequency side only due to the extra Fe at the Al/Si sites in the first Co shell. From the relative areas of these lines, the amount of off stoichiometry and L2i/52-type order in the films can be quantified. Due to the random mixing of A 1 and Si on one (L2i plus off-stoichiometry) or two (52 plus off-stoichiometry) crystallographic sites, each NMR line further broadens or splits into a set of sublines with equal spacing A ^ between them. This splitting originates in the small difference in the hyperfine field seen by Co nuclei depending on which atom, either A1 or Si, is located in the first coordination shell [20,32]. Compounds with Si have one extra valence electron with respect to the compounds with Al. This extra valence electron increases the magnetic moment of the compound, which in turn changes the transferred contributions to the hyperfine field and concomitantly the resonance frequency [20,21,34], Hence, each specific configuration with particular Al and Si neighbors in the first shell of Co arising from the random distribution expected for a quaternary compound will have a different resonance frequency. For details see Supplemental Material Ref. [35] and Refs. [21] and [32]. The relative contributions of the Gaussian lines in the fit can be compared to the probabilities calculated from a random atom model [32], which is mathematically expressed in the form of a binomial distribution function: P(n,m,l,k,x,u,y,CB 2,C L 2 l) n\(N — n — m)\m\ (1 _ x )N - {m+k)x m+ky k(\ - y ) (Ar- n)- * + Cl2,--------- —--------- (1 — u)L~lulyk( 1 — y)(L~l)~kS „ 4' 1 \{L - 1 - k)\k \ with S nAif n ^ 4 if n < 4.The first term in Eq. (1) represents the 5 2 contributions with a random distribution of Fe and Al/Si, where Cb2 is the degree of 52-type order. This random distribution involves both the 4 a and 4b Wyckoff positions of the respective L2| lattice, which correspond to the 1 b position in the 52 notation. In Eq. (1), x represents the Fe to (Al+Si) stoichiometry, enabling us to calculate the probability of finding Fe atoms on the Z (Al and Si) sites, and, hence, to quantify the Fe- Al,Si off stoichiometry; y denotes the Al to Si stoichiometry (AlvSi 1— y). For stoichiometric Co 2FeAlo.sSio .5 films with complete 62-type order, x — 0.5 while x > 0.5 indicates off stoichiometry with Fe excess. Here, N = 8 is the number of possible sites for atoms in the first Co shell, while n, m, and k are the corresponding numbers of Fe, Si, and Al atoms, respectively, in the first Co shell (note n + m + k = N). The second term in Eq. (1) represents the contribution from L 2 r type order, u is the amount of Fe to (Al + Si) off stoichiometry (u = 0 for stoichiometric composition), L = 4 is the number of possible sites for Fe atoms on the Al/Si sites in the first Co shell, and / and k are the numbers of Fe and Si/Al atoms in the first shell, respectively. Since both x and u represent the off stoichiometry, there is a relation between these two parameters x = 0.5 u + 0.5. The coefficients Cb2 and Ci2, represent the relative contribution from 59Co nuclei with a 6 2 and L2\ first shell environment, respectively, and Cb2 + Ci 2, = 1. There are two ways to realize the presence of both L2\ and 6 2 in a given sample: Case (i) deals with large 62-type domains within a L2i matrix, where the number of Co nuclei located at the interfaces between both phases is negligible compared to the number of Co nuclei within a certain phase region in line with the recent report on Co 2MnSi films by Miyajima et al. [36]. In that case, the coefficients obtained from our binomial model immediately give the ratio between L2i and 52 phases. In the second case (ii) both L2i and 6 2 phases are so finely dispersed that the number of Co nuclei at the interfaces is not negligible anymore. In that case, the Co nuclei at the interface experience surroundings similar to that of B2, and therefore the overall degree of order is even underestimated by our model. Moreover, in this case the distribution is no longer described by Eq. (1) (see Supplemental Material [35] for details). Since our binomial model [Eq. (1)] well fits the measured NMR data (see below), scenario (i) seems to be valid in the present case, which is also in line with the recent report on Co 2MnSi films by Miyajima et al. [36]. Figure 3 exemplarily shows the fitting result of the NMR spectrum for the 84-nm Co 2FeAlo. 5Sio.5 films where the respective fitting parameters are NMR resonance frequencies, the spacing between lines A 62, y, u, Cb2, AAl/Si, and the linewidths of individual Gaussian lines. The residual fit mismatch for all spectra does not exceed 15%, which is quite good for such a rather simple model. The fit yields the average spacing between the main line and the high-frequency satellites of 33 MHz, which is slightly larger than in the corresponding bulk samples (31 MHz) [32], The spacing AAl/Si between lines due to the mixing of Al and Si is found to be about 7 MHz, which is very similar to the bulk sample. In addition, the fit yields the Al-to-Si ratio of 0.5(±0.01):0.5(±0.01), as expected from the nominal composition. In general, Al and Si may not be homogeneously distributed in the Co 2FeAl l-^Si* 064421-3 ALFONSOV, PETERS, YANG, BUCHNER, AND WURMEHL PHYSICAL REVIEW B 91, 064421 (2015) 120 140 160 180 200 220 240 frequency (MHz) FIG. 3. (Color online) Normalized 59Co NMR spectrum of an 84- nm-thick Co2FeAI o.sSi 0 .5 film, shown together with a fitted curve (solid line). Analysis of the data gives a degree of L2, order of 81 %. series. We have seen such an inhomogeneous distribution by NMR in the Co2Mn i_^Fe vSi series, where Fe in the the Fe-rich samples is not entirely randomly distributed. Obviously, such a preferential order will not follow the random atom model as described in Eq. (1) (also see Ref. [37]). Such an inhomogeneity is likely not present in the Co2FeAl 0 5Si 0 .5 films for two reasons: (i) for the corresponding bulk system Al and Si are fairly homogeneously distributed [32], and (ii) we found no deviations from our random atom model hinting on such an inhomogeneous distribution of Al and Si, with the exception that the relatively large mismatch of the fit and the measured data at frequencies near ~140 and ~215 MHz may be related to the additional contributions from Co2FeSi and CoAl or fee Co impurities, as suggested in Ref. [32] for the bulk sample. Our results confirm a quite high degree of order for all film thicknesses (Fig. 4). The highest degree of L2\ order is found to be as high as 81% for the 84-nm film. In order to further 0 ) TJ < UQ . ■ S ' 70 film thickness (nm) FIG. 4. (Color online) L2r type order contribution (red open circles) and NMR linewidth (black squares) as a function of film thickness.NX cr (1) Dt164.0 163.5 163.0 - 162.5 162.0 50 100 150 film thickness (nm)200<D5o CL Q .O FIG. 5. (Color online) Thickness dependence of NMR resonance frequency (blue squares, left side) and square root of the optimal power (red circles, right side) for the main line (~ 163 MHz) of the 59Co NMR spectrum. validate our analysis, we compare the trend in NMR linewidth for all films as a function of thickness. Figure 4 shows that the linewidth decreases with increasing film thickness, indicating an improvement of ordering in thicker films and/or release of strain. (Please note that the linewidth axis is inverted to allow for a more direct comparison between evolution of linewidth and degree of order.) Generally, the dependence of the NMR linewidth reflects the evolution of L 2i ordering, as expected. Interestingly, the 84-nm film sticks out, demonstrating the smallest linewidth along with the highest degree of order of about 81%. We will come back to the peculiarities of the 81 -nm-thick film at a later point. In order to further shed light on the relation between structural quality and film thickness, we analyzed the thickness dependence of the square root of the optimal power (Fig. 5, red circles, right side), since the measurement of optimal power (i.e., the power producing the maximum spin-echo intensity) of the applied rf pulses allows us to indirectly investigate the magnetic stiffness or magnetic anisotropy on a local scale via monitoring the restoring field [11,13]; specifically, the square root of the optimal power is proportional to the restoring field. The analysis of the local restoring field is interesting for the investigation of the quality of thin films with respect to their thickness as explained in the following: Typically, there is a critical thickness for films with an epitaxial relation between film and substrate. While films below a certain thickness show uniform full strain—either tensile or compressive, depending on the ratio between the lattice constants—films above the critical thickness release strain. The critical thickness may depend on several parameters, such as the ratio of lattice constants between the film and substrate and the elastic properties of the film material [38]. The release of strain in films with their thickness exceeding the critical value may lead to dislocations, defects, and disorder accompanied by a change in magnetic anisotropy [38]; we may monitorthis effect by measuring the restoring field by NMR. In the present case of C^FeAlo.sSio.s films, we find that the restoring field of the 200-nm film reaches the value of the bulk sample consistent with a lull release of strain and negligible magnetic anisotropy consistent with a cubic system. Interestingly, there is a clear 064421-4 NUCLEAR MAGNETIC RESONANCE STUDY OF THIN Co ... transition for the optimal power at thicknesses between 84 and 120 nm (open circles in Fig. 5). This transition for the optimal power at thicknesses between 84 and 120 nm may be related to the fact that the C^FeAlo.sSio.s films are mostly strained at thicknesses below 100 nm and start to relax above 100 nm, as observed by XRD [31]. In the following, let us combine the information on thickness dependence of the linewidth, amount of L2\ order, and local magnetic anisotropy to understand the evolution of film quality in films with different thickness. The C^FeAlo.sSio.s films under 100 nm thick are fully strained with a tetragonal distortion and remain, hence, structurally uniform, while above 100 nm the films start to relax, which leads to a lower quality of thicker films. Since for Co 2FeAlo. 5Sio .5 the critical thickness is about 100 nm, the best structural quality with the highest L2i ordering is observed in the largest available thickness below 100 nm, i.e., an 84-nm Q^FeAlo.sSio.s film. In general, the resonance frequencies of the films are higher than in the bulk sample and closer to the average value (165 MHz) between highly ordered Co 2FeSi (139 MHz) and 62-ordered Co 2FeAl (190 MHz). Three factors may contribute to the evolution of resonance frequencies: (i) strain, and hence the frequency may scale with the film thickness, (ii) stoichiometry, and (iii) degree of order, (ii) and (iii) are both based on the fact that the stoichiometry, particularly the Fe stoichiometry, and order are both closely linked to the magnetic moment of local neighbors and, hence, in turn to the local hyperfine field and frequency via the transferred fields. Since our analysis of the local magnetic anisotropy revealed a relation between strain and thickness in our Co 2FeAl 0.5 Si 0.5 films, we study the impact of strain on the NMR resonance frequencies using their evolution as a function of film thickness (Fig. 5). The resonance frequencies are more or less constant for films thinner than 84 nm and significantly increase with increasing film thickness. Interestingly, a matching inverse trend between the local restoring field and film thickness, or, in other words, a similar transition between resonance frequencies and optimal rf power at thicknesses between 84 film thickness (nm) FIG. 6. Fe/(A1+Si) off stoichiometry as a function of film thickness.PHYSICAL REVIEW B 91, 064421 (2015) and 120 nm is observed, confirming that the release of strain contributes to the evolution of resonance frequencies. Besides 62-type ordering, we also observe Fe excess at the expense of A 1 and Si. We were able to quantify this off stoichiometry by fitting the data with Eq. (1); as a result, we obtain about 8%-13% excess Fe in the films, indicating that the film composition differs from that of the target. This may be understood as follows: In the off-axis sputtering geometry, the substrate is positioned at an angle of =55° with respect to the normal direction of the sputter target. This arrangement is crucial in minimizing the energetic bombardment damage of the sputtered atoms on the film. Due to this angled deposition, sometimes there is a difference in the stoichiometry of the arriving atoms at the substrate as com pared to the target composition. Film compositions different from the corresponding target when using on-axis sputtering were already reported previously, as, e.g., stoichiometric Co2MnSi films are obtained from stoichiometry adjusted targets [39,40]. For further analysis of the contribution of Fe stoichiometry on the resonance frequencies, we plotted the relation between resonance frequency (MHz) resonance frequency (MHz) FIG. 7. Fe/(A1+Si) off stoichiometry (a), L2\ ordering (b), and their ratio (c) of the Co2FeAl o.sSi 0.5 films as a function of resonance frequency of the main NMR line. 064421-5 ALFONSOV, PETERS, YANG, BUCHNER, AND WURMEHL the Fe to Al/Si off stoichiometry and the corresponding resonance frequency of the main NMR line [Fig. 6(a)], The NMR resonance frequency monotonously increases with a decreasing amount of off stoichiometry. A lower level of off stoichiometry implies a lower macroscopic magnetic moment of the sample, which in turn, due to a negative hyperfine constant of Co [21,41], yields a higher resonance frequency consistent with our observations [Fig. 6(a)], Interestingly, the Fe stoichiometry scales also with the film thickness (see Fig. 7), with the thinnest film being the exception from the trend. This observation may also be related to the thick films being more similar to the bulk samples. The binomial analysis shows much larger contributions from L2i order in the present films (70%-81 %) than in the bulk sample (59%, see Ref. [32]). Hence, the shift of the resonance frequencies towards the mean frequency value between highly ordered Co 2FeSi and B2-ordered CoiFeAl may relate, at least partially, to the higher order in the films compared to the bulk. This working hypothesis is confirmed by the dependence of the L2| order as a function of resonance frequency [see Fig. 6(b)], where a higher degree of L2\ order yields a higher frequency, with the 84-nm film being the only exception. Please note that a similar shift of the resonance line of Co2FeAl 0.5Si 0.5 films in response to the degree of order has been shown by Inomata et al. but is not commented [10]. Since our films always have a higher degree of L2i-type order than the bulk, we may conclude that the transformation to the higher-ordered L2\ -type phase depends rather on kinetic than thermodynamic control. This competition between thermodynamic and kinetic control has a strong dependence on the film thickness, since the ordering depends not only on the thermal history of a given sample and hence on the substrate temperature (which is constant and is 600 °C in all films), the cooling rate, but also on the defect concentration and the length of the meanPHYSICAL REVIEW B 91, 064421 (2015) free path, viz. the diffusion length of atoms during the ordering process. Additionally, if we plot the ratio between L2\ order and Fe to Al/Si off stoichiometry [see Fig. 6(c)] as the function of resonance frequency, we clearly see a monotonous shift of the NMR frequency towards the mean value between Co2FeSi and Co2FeAl, which further proves the scenario of the interplay between strain, stoichiometry, and structural order. IV. SUMMARY We presented a detailed NMR analysis of the structural and local magnetic properties of Co2FeAl0. 5Si0.5 films with varying thickness. Our findings are classified into two categories, (i) We confirm an off-axis sputtering growth technique to yield Heusler films of high quality, which may open the way to enhance the performance of Heusler compounds in spintronic devices in general, if this technique is established. We prove the quality of the films by detailed NMR analysis of the film properties, (ii) We use the NMR technique to disentangle different contributions to the film quality, namely, film thickness, its impact on strain and local anisotropy, stoichiometry, and degree of order. ACKNOWLEDGMENTS This work is supported by a Materials World Network grant from the National Science Foundation (DMR-1107637) and from Deutsche Forschungsgemeinschaft DFG (WU595/5-1). Partial support is provided by the NanoSystems Laboratory at the Ohio State University. S.W. gratefully acknowledges funding by DFG in the Emmy Noether Program (Project No. WU595/3-1). We thank P. Woodward and W. Loser for discussion and C.G.F. 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Jungblut, G. Giintherodt, and B. Hillebrands, Phys. Rev. B 54, 4075 (1996). [39] Y . Sakuraba, T. Miyakoshi, M. Oogane, Y . Ando, A. Sakuma, T. Miyazaki, and H. Kubota, Appl. Phys. Lett. 89, 052508 (2006). [40] M. Oogane, Y. Sakuraba, J. Nakata, H. Kubota, Y. Ando, A. Sakuma, and T. Miyazaki, J. Phys. D: Appl. Phys. 39, 834 (2006). [41] H. Akai, M. Akai, S. Bliigel, B. Drittler, H. Ebert, K. Terakura, R. Zeller, and P. H. Dederichs, Prog. Theor. Phys. Suppl. 101. 11 (1990). 064421-7 Copyright of Physical Review B: Condensed Matter & Materials Physics is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.

PhysRevB.81.125444.pdf

Response of nanoparticle structure to different types of surface environments: Wide-angle x-ray scattering and molecular dynamics simulations Hengzhong Zhang,1,*Bin Chen,1Yang Ren,2Glenn A. Waychunas,3and Jillian F. Banfield1,3 1Department of Earth and Planetary Science, University of California, Berkeley, California 94720, USA 2Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA 3Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA /H20849Received 25 August 2009; revised manuscript received 22 February 2010; published 31 March 2010 /H20850 The structure of nanoparticles is nonstationary and changes in response to the surface environment where the nanoparticles are situated. Nanoparticle-environment interaction determines the nature of the structure change,an important consideration for evaluating subsequent environmental impact. In this work, we used ZnS nano-particles to interact with surface environments that contain different inorganic salts, water, and organic mol-ecules. From analysis of the pair-distribution function /H20849PDF /H20850derived from wide-angle x-ray scattering experi- ments, we found that a stronger surface interaction results in a thicker crystalline core and a thinner distortedshell, corresponding to PDF curves having larger peaks and more peaks at longer radial distances. Plane-waveelectronic calculations were used to quantify the interaction strength. An analogous atomic view of thenanoparticle-environmental interactions and structures was provided by molecular dynamics simulations. Theextent of response of the nanoparticle structure to various surface environments is used as a measure of theinteraction strength between them. DOI: 10.1103/PhysRevB.81.125444 PACS number /H20849s/H20850: 61.46.Df, 61.05.cf, 02.70.Ns I. INTRODUCTION When coupled with surface ligands, nanoparticles can be used to target cancer-specific receptors and other maliciouscells. 1–3Functionalized nanoparticles can be used as sensors for detection of deoxyribonucleic acid targets and explosivematerials. 4–6Realization of these nanotechnologies relies on detailed understanding of the nanoparticle structure as wellas the interactions between nanoparticles and the surround-ing molecules/ions. These interactions include several typesof electrostatic interactions /H20849ionic and dipole interactions /H20850, covalent bonding, hydrogen bonding, and van der Waalsforces. Though atomic force microscopy can measure theforces between nanoparticles and solid surfaces/substrates, 7,8 it is difficult to characterize the details of the interactions between nanoparticles and surface molecules and ions. Inparticular, the manner in which the nanoparticle structureresponds to the surface environments is yet to be explored. Inthis work, we used atomic pair-distribution function /H20849PDF /H20850 analysis to study the structural changes in ZnS nanoparticlesdue to interactions with different surface species. Molecularsimulations were carried out to validate the experimental re-sults and to provide an atomic view of the interaction pro-cesses. The approach developed in this work may be used toanalyze interaction strengths and effects in nanoparticles thatare difficult to measure using other methods. Both the newapproach and the acquired knowledge will be advantageousin the development of highly specific nanomaterials for ap-plications in nanomaterial-environment interactions, such asmedical/cosmetic products and environmental remediationmechanisms that utilize nanoparticles. 9,10 II. EXPERIMENTAL A. Sample preparation ZnS /H20849sphalerite /H20850nanoparticles with an average diameter of/H110113 nm were synthesized in anhydrous methanol. A sus-pension of 0.09 M ZnS nanoparticles in methanol was pre- pared by reacting anhydrous zinc chloride /H20849ZnCl 2/H20850and so- dium sulfide /H20849Na2S/H20850in anhydrous methanol, followed by purification and redispersion in methanol.11For interactions with ZnS nanoparticles, ionic salts /H20849sodium chloride NaCl, calcium chloride CaCl 2, and sodium sulfate Na 2SO4/H20850and molecules having different structures /H20849water H 2O, methanol CH 3OH, thiophenol C 6H6S, and chlorobenzene C 6H5Cl/H20850 were used as the surface species. Specific concentrations of different surface species were added to the as-synthesized nano-ZnS suspensions, produc-ing methanol suspensions of 0.07 M ZnS+13.9 MH 2O, 0.06 M ZnS+3.3 M C 6H5Cl, 0.06 M ZnS+3.3 M C6H6S, 0.09 M ZnS+0.03 M NaCl, 0.09 M ZnS +0.015 M CaCl 2, and 0.09 M ZnS+0.015 M Na 2SO4. The new suspensions equilibrated for /H1101124 h before per- forming wide-angle x-ray scattering /H20849WAXS /H20850measurements. B. X-ray diffraction X-ray diffraction /H20849XRD /H20850was used to identify the phase of the synthesized ZnS and to estimate the crystallite size. XRDspecimens were prepared by dispersing a thin layer of thenano-ZnS sample in methanol on to a low-scattering back-ground silicon plate which were then allowed to dry natu-rally. The plate was loaded immediately into the sampleholder of an x-ray diffractometer /H20849PANalytical X’Pert PRO /H20850 operated at 40 kV and 40 mA with a Co K /H9251radiation x-ray source /H20849wavelength 1.7903 Å /H20850. The XRD pattern was col- lected at room temperature in the 2 /H9258range of 20° –80° with a scanning rate of 1 ° /min. C. Wide angle x-ray scattering WAXS measurements were performed at room tempera- ture at the high-energy beamline station 11-ID-C, AdvancedPHYSICAL REVIEW B 81, 125444 /H208492010 /H20850 1098-0121/2010/81 /H2084912/H20850/125444 /H208496/H20850 ©2010 The American Physical Society 125444-1Photon Source, Argonne National Laboratory /H20849USA /H20850, with an x-ray wavelength of 0.10770 Å and a sample-to-camera dis-tance of /H11011272 mm. The exact distance was calibrated using a CeO 2standard. Small volumes of the ZnS suspensions /H20849containing different concentrations of surface species /H20850were encapsulated in 1.5 mm diameter quartz capillary tubes.These tubes were then put into a capillary tube sample holderfor the WAXS experiments. WAXS patterns were acquired atroom temperature, with a q/H20849scattering vector /H20850range of 0.3–30 Å −1, and a step size of 0.01 Å−1. The WAXS signal was captured by an image plate detector. The WAXS patternsof several identical capillary tubes filled with methanol andvarious surface species /H20849without ZnS nanoparticles /H20850were ac- quired for background subtraction from the sample patterns. III. COMPUTATIONAL A. Electronic structure calculations To gain insight into the nature of the interactions /H20849such as binding energy, bond length, and electron sharing /H20850,w ed i d first-principle calculations of the interactions between onemolecular unit of surface species and a bulk ZnS /H20849100 /H20850sur- face. The electronic-structure calculations were carried outusing the CPMD package,12employing a plane-wave basis /H20849energy cutoff=80 AU /H20850 with Troullier-Martins pseudopotentials13and a local-density approximation ex- change correlation using the high accuracy Padéapproximation. 14A periodic slab of ZnS /H20849/H1101111/H1100311 /H1100311 Å3, 32 ZnS molecular units /H20850with a CPMD geometri- cally optimized /H20849100 /H20850surface was used as the basis for inter- action with one molecular unit of NaCl, Na 2SO4,H 2O, C6H6S, C 6H5Cl, and CH 3OH, respectively. Initial setup for a surface species interacting with the ZnS /H20849100 /H20850surface for the CPMD electronic calculation was obtained from a classical molecular dynamics /H20849MD /H20850simulation performed using the Forcite module of MATERIALS STUDIO 4.0 .15In a CPMD geo- metric optimization, atoms of the interacting surface specieswere allowed to move in any direction. The coordinates ofatoms of the surface species interacting with the ZnS surfacewere optimized and the energy of the system was minimizedusing standard criteria /H20851convergence of both orbital /H20849energy and gradient /H20850and geometry to within certain preset variations /H20852. 12 B. Molecular dynamics simulations In order to correlate the ZnS nanoparticle structure and the strength of the interaction with surface species, we per-formed MD simulations of three representative systems,nano-ZnS in vacuum /H20849to approximate methanol, see Sec. IV E below /H20850, nano-ZnS interacting with water, and nano-ZnS interacting with Na +/Cl−. MD simulations o fa3n mZ n S /H20849sphalerite /H20850particle in vacuum and with adsorption of differ- ent numbers of water molecules have been done previously.16 MD simulation of adsorption of Na+/Cl−ions ont oa3n m ZnS nanoparticle was performed using the code DL_POLY /H20849Ref. 17/H20850in this work. A shell model with Buckingham-type interatomic potential functions was used to describe the ZnSstructure. 18Using the shell model, a Zn or S atom is treatedas a core and a massless /H20849or, very light /H20850shell that are con- nected by a spring, accounting for ionic polarity inducedunder a local electric field. The Zn and S atoms have theelectrical charges of +2 and −2, respectively. The short-rangevan der Walls interaction between two atoms iandjtakes a Buckingham form, u ij/H20849short range /H20850=Aijexp/H20873−Rij /H9267ij/H20874−Cij Rij6, /H208491/H20850 where uijis the interaction potential, Rijthe distance between atoms iand j, and Aij,/H9267ij, and Cijare three model param- eters. An angle-bending form of three-body interactions isconsidered for nearest S-Zn-S atoms, u ijk=1 2kijk/H20849/H9258−/H9258ijk/H208502, /H208492/H20850 where uijkis the interaction potential, kijka model parameter, /H9258the angle formed by atoms i/H20849S/H20850,j/H20849Zn, center /H20850andk/H20849S/H20850, and/H9258ijkthe equilibrium value of the angle /H20849109.4° /H20850. The interatomic interactions in NaCl were described by the Born-Huggins-Mayer potential functions, which has thesame form as Eq. /H208491/H20850but with an additional term of −D ij//H20849Rij/H208508/H20849Dijis a model parameter /H20850.19For detailed values of the model parameters for ZnS and NaCl, the readers arereferred to Refs. 18and19. Suitable potential functions for the interaction between ZnS and NaCl are not available fromthe literature. Thus, we developed interatomic potential func-tions for the NaCl-ZnS system as described below /H20849Sec. IV D /H20850. In the MD simulation, 24 Na +/Cl−ions /H20849to simulate the low concentration of NaCl due to the low solubility inmethanol in the experiment /H20850were placed over the surface /H20849with a initial distance of /H110113 Å between them /H20850o fa3n m ZnS nanoparticle constructed from the bulk structure ofsphalerite. The MD simulation was carried out at 300 K for aMD time of 20 ps with a step of 10 −5ps first and then for another 60 ps with a step of 10−4ps. Potential-energy evo- lution showed that after /H1101145 ps, the system has converged to a steady state. IV. RESULTS AND DISCUSSION A. Sample characterization Inspection of the XRD pattern /H20849Fig. 1/H20850shows that the as-synthesized ZnS is nanosphalerite with very broad diffrac-tion peaks. The XRD line profile was analyzed using a nu-merical method 20for separation of the overlapping sphalerite /H20849111 /H20850,/H20849200 /H20850,/H20849220 /H20850, and /H20849311 /H20850peaks which show significant broadening at the nanoscale size. The Scherrer size21of the nano-ZnS was estimated to be /H110111.5 nm in diameter using the full width at the half maximum of the /H20849111 /H20850peak /H20849with a Scherrer constant of 0.90 /H20850. This size corresponds to the crys- talline core of the ZnS nanoparticles while the physical di-ameter determined by transmission electron microscopy andUV-vis spectroscopy was /H110113 nm. 11These values show that the as-synthesized ZnS nanoparticles are highly distortedcore-shell structures with a weakly diffracting surface layer/H110110.75 nm in thickness.ZHANG et al. PHYSICAL REVIEW B 81, 125444 /H208492010 /H20850 125444-2B. PDF analysis PDF analysis has been used to study structures of nano- materials, including metals /H20849e.g., nano-Au /H20850, semiconductors /H20849e.g., nano-CdSe /H20850, metal sulfides /H20849e.g., nano-MoS 2/H20850, and metal oxides /H20849e.g., nano-TiO 2/H20850, as reviewed in Refs. 22–24. However, systematic study of environmental response ofnanoparticle structure using PDF analysis was not reportedpreviously. In this work, the PDFs of ZnS nanoparticles upon inter- action with various surface species were derived from theWAXS data. First, structure factors of the nanoparticles werederived from the WAXS patterns after data reduction. 25 Then, the PDFs, or G/H20849r/H20850functions, were obtained through Fourier transform of the structure factors S/H20849q/H20850,25 G/H20849r/H20850=2 /H9266/H20885 0/H11009 q/H20851S/H20849q/H20850−1/H20852sin/H20849qr/H20850dq. /H208493/H20850 PDF patterns reflect the atomic correlations in a material. The theoretical PDF of a bulk crystalline material consists ofa series of peaks extending to arbitrarily large radial distance.Experimental PDF peaks are damped with increasing radialdistance because of limited experimental resolution /H20849for beamline 11-ID-C at APS, this is at a radial distance of/H110117n m /H20850or because the particles themselves have dimen- sions smaller than the resolution limit. For our ZnS nanopar-ticles interacting with various surface ligands, we observedenhanced damping at radial distances smaller than the nano-particle diameter /H20849Fig.2/H20850, which indicates that the outer parts of the nanoparticles have less-crystalline character than thecores. This must be due to structural distortion of the tetra-hedral coordination of Zn or S atoms near the surface. 11,26As there is a correspondence between different ligand type andthe degree of PDF damping, we theorize that the structuraldistortion /H20849Fig.2/H20850is specific to the surface ligand and hence the nanoparticle environment. We now quantify how theligand-surface interaction induces this structural disorder.The PDF curves in Fig. 2can be classified into three groups. In group I, there are PDF patterns characterized by afew small peaks at r/H11021/H110118 Å. The relative magnitude of the second broad peak /H20849r/H110113.74–3.77 Å /H20850is lower than or com- parable to that of the first peak /H20849r/H110112.33 Å, the average Zn-S bond length /H20850. This is the case for nano-ZnS in methanol and methanol+chlorobenzene. In group II, the PDF peaks arelarger and more numerous than in group one /H20849rextends to /H1101114 Å /H20850. The relative magnitude of the second broad peak /H20849r/H110113.76–3.80 Å /H20850is higher than that of the first peak /H20849r/H110112.33 Å /H20850. In this category are patterns from nano-ZnS in methanol+water and methanol+thiophenol. In group III, thePDF patterns contain many medium to large PDF peaks atr/H11021/H1101110 Å /H20849beyond which the minor peaks are noises from data reduction /H20850. This group includes nano-ZnS in methanol +dilute inorganic salts /H20849NaCl, CaCl 2, and Na 2SO4/H20850. The rela- tive magnitude of the second broad peak /H20849r /H110113.72–3.76 Å /H20850varied from almost comparable to higher than that of the first peak /H20849r/H110112.33 Å /H20850. Compared to group I, the peaks in /H110115.5–7.5 Å are more significant in this group even though the salt concentrations are much lower thanthose of methanol and chlorobenzene in group I. The unequal structural responses /H20849as shown by the PDF curves in Fig. 2/H20850of the ZnS nanoparticles to different surface species are expected to be determined by the interactionstrength between the nanoparticles and the surface species.Results from the plane-wave electronic structure calculations/H20849below /H20850are used to determine the binding strength of the surface species and to correlate these with the different in-duced PDF characters. C. Nature of surface binding by plane-wave electronic calculations Table Isummarizes results from the electronic-structure calculations. Results show that surface species interact withZnS /H20849100 /H20850surfaces mainly via interactions between their high-electronegativity atoms /H20849O, Cl, and S /H20850and the Zn atoms111 200 0100200300400500600700 30 40 50 60 70 2θθθθ(degree)Intensity ( CPS) fitted220 311 FIG. 1. XRD pattern /H20849dots /H20850of nano-ZnS synthesized in metha- nol. Overlapping sphalerite /H20849111 /H20850,/H20849200 /H20850,/H20849220 /H20850, and /H20849311 /H20850peaks were separated /H20849lines /H20850using a numerical decomposition method /H20849Ref. 20/H20850for estimation of the crystalline core size.2 4 6 8 10 12 14 r(Å)G D: + 3. 3 M C 6H6S C: + 13. 9 M H 2OG: + 0. 03 M NaCl F: + 0. 015 M CaCl 2 E: + 0. 015 M Na 2SO4 B: + 3. 3 M C 6H5Cl A: in CH 3OHIIIIII FIG. 2. Atomic pair-distribution function /H20849G/H20850of/H110113 nm ZnS nanoparticles /H208490.06–0.09 M ZnS /H20850suspended in methanol /H20849A/H20850and methanol plus various surface species /H20849B–G /H20850. Group I represents surface species with weak interactions with nanoparticles, group IIwith enhanced interactions, and group III with strong interactions.RESPONSE OF NANOPARTICLE STRUCTURE TO … PHYSICAL REVIEW B 81, 125444 /H208492010 /H20850 125444-3on the ZnS surface /H20849Fig. 3/H20850. The equilibrium distance be- tween the interacting atoms is one indication of the strengthof the interaction between the surface species and the ZnSsurface. A shorter distance usually corresponds to a strongerinteraction as the latter usually corresponds to a higher de-gree of electron sharing and hence a shortening of the inter-nucleus distance. The binding energy represents the energyreleased when a surface species is adsorbed on the ZnS sur-face. In general, the higher the binding energy, the greaterthe structural interaction between the surface species and theZnS surface. Table Ishows that ionic salts, water and thiophenol, and methanol have binding energies of /H11011125–135 kJ /mol, 67–90 kJ/mol, and 37 kJ/mol, respectively. These ranges cor-respond to the different sets of PDF profiles we measured/H20849Fig.2/H20850. For the ionic salts we expect strong binding and thus most surface Zn species is bonded with strengths similar tothe underlying Zn-S bonding. A crystal-chemical analysis,e.g., using the Pauling bond valence principle, would holdthat these Zn ions are largely valence satisfied. In contrast,for methanol we have weak surface binding, leading to un-satisfied valence contributions on the uppermost Zn ions,which creates structure distortion due to changes in the Zn-Sbonding network. For chlorobenzene, though the binding en- ergy is close to those of water and thiophenol, the equilib-rium distance between the interacting atoms/H20849Cl-Zn:2.869 Å /H20850is longer than those of the latter two /H20849O-Zn:2.140 Å; S-Zn:2.446 Å /H20850. This would decrease the electron sharing between the ZnS surface and chlorobenzeneand thus lead to an effect similar to that of methanol on thestructure of ZnS. Finally, water and thiophenol represent anintermediate case, with intermediate strength binding, andintermediate size structural distortion. The effects of thesestrained/distorted surface layers propagate into the nanopar-ticle interior, causing remarkably distinct PDF characteris-tics. Molecular dynamics simulations of representative sys-tems provide more details of these structural changes/H20849below /H20850. D. Development of ZnS-NaCl interatomic potential functions For MD simulation of the interaction between ZnS-NaCl, we developed a set of potential functions by fitting to theinteraction energies calculated using the electronic code CPMD /H20849see Sec. III A and Fig. 4/H20850. We found that the Morse potential function /H20851Eq. /H208494/H20850/H20852describes well the pairwise atomic interactions in the ZnS-NaCl system, E/H20849r/H20850=a⌊/H208531 − exp /H20851−b/H20849r−c/H20850/H20852/H208542−1 ⌋, /H208494/H20850 where Eis the interaction energy, ris the interatomic dis- tance, and a,b, and care adjustable model parameters. Pa- rameter crepresents the equilibrium distance between the atoms in a diatomic cluster. The calculations used the interaction geometry shown in Fig.4/H20849a/H20850. For the NaCl-ZnS interaction, there are four atomic pairs /H20849Na-S, Na-Zn, S-Cl, and Zn-Cl /H20850and hence there are 4/H110033=12 Morse potential parameters. Simultaneous deter- mination of the 12 parameters could not be achieved from asingle fitting of the Morse potential functions to the calcu-lated interaction energies due to the many number of un-known parameters. Thus, as a first step, the three Morse po-tential parameters /H20849a,b, and c/H20850were derived for an isolated diatomic cluster /H20849Na-S, Na-Zn, S-Cl, or Zn-Cl /H20850by fitting Eq.TABLE I. First-principle calculated binding energies of surface species interacting with ZnS /H20849100 /H20850surface. Surface speciesMajor interacting atomic pairaInteratomic distance /H20849Å/H20850Binding energy /H20849kJ/mol species /H20850 Na2SO4 O-Zn 2.083 134.7 NaCl Cl-Zn 2.332 124.7C 6H6S S-Zn 2.446 89.8 H2O O-Zn 2.140 66.7 C6H5Cl Cl-Zn 2.869 78.5 CH3OH O-Zn 2.963 36.8 aThe atomic pair with the shortest distance between the atoms of the surface species and the ZnS /H20849100 /H20850surface. S ZnClNaO H SNaO SHCCl CH HC ONaCl Na2SO4 H2O C6H6SC6H5Cl CH3OHSS SSSZnZn Zn ZnZn2.332ÅZnS(100)2.083Å2.140Å 2.446Å 2.869Å2.963ÅZnS(100) FIG. 3. /H20849Color online /H20850Optimized configurations of surface spe- cies interacting with a ZnS /H20849100 /H20850surface. Dotted networks are the charge-density isosurfaces of the valence electrons at density valueswhen the isosurfaces of the major interacting atoms /H20849shown in ball- and-stick model /H20850begin to overlap. z ∆z=0 ZnS(100)Cl Na S Zn2.73Å1.89Å2.56Å -2024681012 - 1 0123 4 ∆∆∆∆z(Å)Interaction energy (eV)CPMD fitted (b) (a) FIG. 4. /H20849Color online /H20850Interaction between a ZnS /H20849100 /H20850surface and a NaCl cluster. /H20849a/H20850Structure setup for the calculation of the interaction energy. /H20849b/H20850The interaction energy as a function of the ZnS /H20849100 /H20850-NaCl separation. Circles are from CPMD electronic cal- culations and diamonds from fitting using Morse potential functions/H20851refer Eq. /H208494/H20850and Table II/H20850.ZHANG et al. PHYSICAL REVIEW B 81, 125444 /H208492010 /H20850 125444-4/H208494/H20850to the interaction energy of the diatomic cluster /H20849calcu- lated using CPMD separately /H20850as a function of the atomic separation. Here, the interaction energy is the energy changeby bringing two ions /H20849e.g., Na +and Cl−/H20850from infinitely far away to a designated distance. The obtained parameters arelisted in Table II. We then used the following method to get the Morse po- tential parameters for the ZnS-NaCl system. As a first ap-proximation, it was assumed that the three parameters /H20849a,b, andc/H20850of the ZnS-NaCl system are proportional to the cor- responding ones, respectively, of the diatomic clusters. Theproportional coefficients then were optimized such that thedifferences between the interaction energies calculated fromthe Morse potential functions and the energies calculated us-ing CPMD became minimal for the ZnS /H20849100 /H20850-NaCl cluster system /H20851Fig.4/H20849b/H20850/H20852. Here, the interaction energy is the energy change by bringing a NaCl cluster to the ZnS /H20849100 /H20850surface from infinitely far away to a designated distance. The derivedparameters are listed in Table II. One notes that the equilib- rium distances care similar in both the diatomic cluster and the ZnS-NaCl system. The derived parameters /H20849Table II/H20850 were used for the MD simulation of the interaction betweenNaCl an da3n mZ n S particle. E. Insight from molecular dynamics simulations Because the binding energy of methanol on a ZnS surface is low /H20849Table I/H20850, the interaction between methanol and ZnS nanoparticles is weak. The weak interaction can be approxi-mated by the MD simulation of a 3 nm ZnS particle invacuum. 16The binding energy of water on a ZnS surface islarger than that of methanol /H20849Table I/H20850. Hence, the interaction between water and ZnS nanoparticles is stronger and can bestudied with MD simulations using sufficient water mol-ecules /H20849e.g., 362 H 2O/H20850to saturate sorption sites o na3n m ZnS particle.16The MD simulation of the stronger interaction o fa3n mZ n S particle with 24 Na+/Cl−ions was similarly performed in this work /H20849see Sec. III B /H20850. Figure 5shows /H20849a/H20850snapshots of the MD structures of a 3 nm ZnS particle in vacuum, /H20849b/H20850after adsorption of 362 H 2O molecules, and /H20849c/H20850after adsorption of 24 Na+/Cl−ions. Fig- ure6shows the comparisons between the PDF curves calcu- lated from the MD structures /H20849Fig. 5/H20850with those from the WAXS determinations /H20849Fig.2/H20850. Results show that the calcu- lated PDF are in good agreement with the experimental PDF.This indicates that the MD simulations can generate atomicstructures that are consistent with the WAXS experiments. Figure 5/H20849a/H20850shows that the 3 nm ZnS nanoparticle in vacuum has a highly distorted shell and a small crystallinecore /H20849/H110111.6 nm in diameter /H20850. The core size is close to that /H20849/H110111.5 nm /H20850estimated from XRD determination for the as- synthesized ZnS nanoparticles in methanol. In contrast, the 3nm ZnS nanoparticle after adsorption of water molecules/H20851Fig.5/H20849b/H20850/H20852is more crystalline due to the strong binding of water molecules /H20849energy change due to adsorption of water isTABLE II. Morse potential parameters. Atomic pairFor diatomic cluster For ZnS /H20849100 /H20850-NaCl cluster a /H20849eV/H20850b /H20849Å−1/H20850c /H20849Å/H20850a /H20849eV/H20850b /H20849Å−1/H20850c /H20849Å/H20850 Na-S 2.0444 1.2145 2.3899 0.4197 1.7098 2.4820 Na-Zn 0.2443 1.2617 2.9917 0.0502 1.7764 3.1071S-Cl 4.7858 1.4652 2.0511 0.9826 2.0627 2.1301Zn-Cl 3.8863 1.2503 2.2823 0.7979 1.7602 2.3703 (a)( c) (b) FIG. 5. /H20849Color /H20850Snapshots of the equilibrated structures of a 3 nm ZnS /H20849sphalerite /H20850particle in molecular dynamics simulations. /H20849a/H20850 MD in vacuum. /H20849b/H20850MD of nano-ZnS with adsorption of 362 H 2O molecules. /H20849c/H20850MD of nano-ZnS with adsorption of 24 Na+and 24 Cl−ions. Zn: gray; S: dark yellow; O: red; H: light gray; Na: blue; and Cl: green.2 4 6 8 10 12 14 r(Å)G B: + 13. 9 M H 2OC: + 0. 03 M NaCl A: in CH 3OH FIG. 6. Comparisons between experimental /H20849thin lines /H20850and cal- culated PDF /H20849thick lines /H20850of/H110113 nm ZnS nanoparticles suspended in methanol /H20849A/H20850, methanol plus water /H20849B/H20850, and methanol plus NaCl /H20849C/H20850. The experimental data are from WAXS determinations and the calculated data are from molecular dynamics simulations.RESPONSE OF NANOPARTICLE STRUCTURE TO … PHYSICAL REVIEW B 81, 125444 /H208492010 /H20850 125444-5182 kJ/mol H 2O/H20850and the full coverage of the surface by water molecules. The average Zn-O bond length is 1.988 Å.The 3 nm ZnS nanoparticle after adsorption of 24 Na +/Cl− ions /H20851Fig.5/H20849c/H20850/H20852is also more crystalline than that in vacuum due to the even stronger binding of the ions on the ZnSnanoparticle /H20849energy change due to adsorption of NaCl is 346 kJ/mol NaCl /H20850despite the low number of bound ions. The average Zn-Cl bond length is 2.353 Å and the average Na-Sbond length is 2.600 Å. The MD simulation results show that stronger surface binding and more surface coverage by surface species canlargely compensate for the disruption of the periodic struc-ture of the ZnS nanoparticles at the surfaces. This results in amore crystalline nanostructure and hence more well definedand larger R peaks in the PDF. V. CONCLUSIONS In this study, we demonstrate that the distortion and core- shell structures of nanoparticles in various chemical environ-ments are determined largely by surface interactions, andthat the structural responses to different surroundings can be analyzed by PDFs obtained using high-energy WAXS meth-ods. More and larger PDF peaks at longer radial distancesare indicative of stronger surface interactions, as confirmedby first-principle calculations and molecular dynamics simu-lations. PDF analysis is a sensitive probe of these structuralchanges, and hence is a general method for the identificationand characterization of nanoparticle-surface environment in-teractions. ACKNOWLEDGMENTS We thank B. Gilbert for helpful discussions. Use of the Advanced Photon Source is supported by the U.S. Depart-ment of Energy, Office of Science, under Contract No. DE-AC02-06CH11357. Computations were carried out in theGeochemistry Computer Cluster, Lawrence Berkeley Na-tional Laboratory. Financial support was provided by theU.S. Department of Energy /H20849Grant No. DE-FG03- 01ER15218 /H20850and the National Science Foundation /H20849Grant No. EAR-0123967 /H20850. *Corresponding author; heng@eps.berkeley.edu 1G. J. Kim and S. Nie, Mater. Today 8/H208498, Suppl. 1 /H20850,2 8 /H208492005 /H20850. 2T. M. Fahmy, P. M. Fong, A. Goyal, and W. M. Saltzman, Mater. Today 8/H208498, Suppl. 1 /H20850,1 8 /H208492005 /H20850. 3M. C. Woodle and P. Y. Lu, Mater. Today 8/H208498, Suppl. 1 /H20850,3 4 /H208492005 /H20850. 4N. C. Tansil and Z. Gao, Nanotoday 1/H208491/H20850,2 8 /H208492006 /H20850. 5F. Patolsky and C. M. Lieber, Mater. Today 8/H208494/H20850,2 0 /H208492005 /H20850. 6L. Senesac and T. G. Thundat, Mater. Today 11/H208493/H20850,2 8 /H208492008 /H20850. 7K. L. Chen and S. E. Mylon, Langmuir 23, 5920 /H208492007 /H20850. 8Q. K. Ong and I. Sokolov, J. Colloid Interface Sci. 310, 385 /H208492007 /H20850. 9A. D. Maynard, Nanotoday 1/H208492/H20850,2 2 /H208492006 /H20850. 10P. G. Tratnyek and R. L. Johnson, Nanotoday 1/H208492/H20850,4 4 /H208492006 /H20850. 11H. Zhang, B. Gilbert, F. Huang, and J. F. Banfield, Nature /H20849Lon- don /H20850424, 1025 /H208492003 /H20850. 12The CPMD Consortium, CPMD /H20849v3.11 /H20850An ab initio Electronic Structure and Molecular Dynamics Program /H20849Copyright IBM Corp, 1990–2006; Copyright MPI fuer FestkoerperforschungStuttgart, 1997–2001 /H20850. 13N. Troullier and J. L. Martins, Phys. Rev. B 43, 8861 /H208491991 /H20850.14S. Goedecker, M. Teter, and J. Hutter, Phys. Rev. B 54, 1703 /H208491996 /H20850. 15MATERIALS STUDIO , version 4, Accelrys Inc., San Diego, 2006. 16H. Zhang, J. R. Rustad, and J. F. Banfield, J. Phys. Chem. A 111, 5008 /H208492007 /H20850. 17W. Smith and T. R. Forster, The DL_POLY v2.13 User Manual, Daresbury Laboratory, Daresbury, Warrington, 2001. 18K. Wright and A. Jackson, J. Mater. Chem. 5, 2037 /H208491995 /H20850. 19R. Bahadur, L. M. Russell, S. Alavi, S. T. Martin, and P. R. Buseck, J. Chem. Phys. 124, 154713 /H208492006 /H20850. 20H. Zhang, B. Chen, B. Gilbert, and J. F. Banfield, J. Mater. Chem. 16, 249 /H208492006 /H20850. 21R. Jenkins and R. L. Snyder, Introduction to X-Ray Powder Dif- fractometry /H20849Wiley, New York, 1996 /H20850. 22S. J. L. Billinge and I. Levin, Science 316, 561 /H208492007 /H20850. 23S. J. L. Billinge, J. Solid State Chem. 181, 1695 /H208492008 /H20850. 24T. Proffen, Rev. Mineral. 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PhysRevB.93.054411.pdf

PHYSICAL REVIEW B 93, 054411 (2016) Two-body problem of core-region coupled magnetic vortex stacks Max H ¨anze,1,*Christian F. Adolff,1Sven Velten,1Markus Weigand,2and Guido Meier3,4 1Institut f ¨ur Angewandte Physik und Zentrum f ¨ur Mikrostrukturforschung, Universit ¨at Hamburg, 20355 Hamburg, Germany 2Max-Planck-Institute for Intelligent Systems, Stuttgart, Germany 3The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany 4Max-Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany (Received 4 August 2015; revised manuscript received 26 January 2016; published 9 February 2016) The dynamics of all four combinations of possible polarity and circularity states in a stack of two vortices is investigated by time-resolved scanning transmission x-ray microscopy. The vortex stacks are excited byunidirectional magnetic fields leading to a collective oscillation. Four different modes are observed that dependon the relative polarizations and circularities of the stacks. They are excited to a driven oscillation. We observe arepulsive and attractive interaction of the vortex cores depending on their relative polarizations. The nonlinearityof this core interaction results in different trajectories that describe a two-body problem. DOI: 10.1103/PhysRevB.93.054411 I. INTRODUCTION Magnetic structures found in thin ferromagnetic layers [1,2], such as magnetic bubbles [ 3], domain walls [ 4], skyrmions [ 5], and vortices [ 6], have been studied intensively over the past few decades. Their characteristic magnetizationstructures result from the minimization of different energycontributions. For instance, the formation of magnetic bubblesoriginates from a uniaxial anisotropy determined by the ferro-magnetic layer. Magnetic skyrmions are commonly stabilizedin the presence of external magnetic fields and an asymmetryin the boundary layers that leads to the Dzyaloshinskii-Moriyainteraction [ 5]. Magnetic vortices emerge when the geometry of the ferromagnet is confined to dimensions on the micrometerscale [ 7], e.g., in micron-sized disks. Here, stray fields at the edges of the structures are minimized. The magnetic vortexconstitutes a magnetization circulating in the plane aroundthe center position of the disk where it points out of plane.The sense of circulation ccan be either clockwise or counter- clockwise ( c=±1), whereas the out-of-plane component of the center region points either up or down (polarization p= ±1). Due to the four possible states, vortices are promising candidates for applications in potential storage devices [ 8,9]. In order to realize a storage device with a high storage densityone needs to incorporate many vortices in a finite volume.Since neighboring vortices couple due to stray fields emergingat the surfaces of the ferromagnetic elements [ 10], the motions of closely packed vortices are strongly influenced by theirsurrounding ferromagnetic structures [ 11,12]. The interaction between laterally arranged elements has been studied forpairs [ 13,14], chains [ 15], and two-dimensional arrangements [16–18] of vortices. In laterally coupled arrangements it has been shown recently that memorylike writing processes arepossible based on the excitation of the gyrotropic mode [ 19], where bits are stored as polarization patterns. The gyrotropicmode corresponds to a gyration of the vortex core aroundthe center of the disk and can be compared to the oscillationof a harmonic oscillator [ 20]. Even for closely packed two- dimensional arrays of vortices, the storage density is expected *max.haenze@physnet.uni-hamburg.deto be below that of conventional storage devices [ 21]. We introduce an additional dimension to the collective gyrationsof vortices known from spin-torque oscillators [ 22]. Stacking the vortices allows for a strongly increased packing densityand has thus stimulated recent studies [ 23–30]. While for two-dimensional arrangements the minimization of the strayfields at the side surfaces creates the vortices and mediatestheir interaction, we observe a second coupling mechanism forthree-dimensional stacks that has been investigated theoreti-cally [ 26,31,32]. Due to core coupling the collective motions in an elementary stack of two vortices become a two-bodyproblem. Here, we study the vortex core motions emerging in a stack of magnetic vortices depicted in Fig. 1using time-resolved scanning transmission x-ray microscopy. In the first step weexcite the vortices by a short magnetic field pulse leading toa collective motion of the core regions. Both vortex cores areimaged along their damped oscillation. We observe a strongdependence of the resonance frequency and the trajectoriesof the gyration on the relative circulations and polaritieswithin the stack. In the second step we excite the differentnondegenerate states close to their resonance frequency.Stationary trajectories are observed that are comparable tomotions of the gravitational two-body problem. This effectoriginates in the proximity of the vortex core regions withinthe stack. Calculations elucidate their functional dependence. II. SAMPLE PREPARATION AND METHODS Figure 1(a) depicts schematics of the measurement setup and x-ray measurements of the investigated stack of vortices.Magnetic contrast is provided via the x-ray magnetic circulardichroism (XMCD) at the Ni L 3-absorption edge (852.7 eV). The spatial resolution with the zone plate used in the presentexperiment is 25 nm. The maximum temporal resolutionis 40 ps. Stacks of polycrystalline permalloy (Ni 80Fe20) disks separated by an interlayer of silicon are prepared withelectron-beam lithography, in situ thermal evaporation of permalloy/silicon/permalloy layers, and liftoff processing ona 100-nm-thick silicon nitride membrane. The disks have adiameter of 1 μm and a thickness of 40 nm. The silicon spacer has a thickness of 20 nm. A coplanar waveguide is 2469-9950/2016/93(5)/054411(6) 054411-1 ©2016 American Physical SocietyH¨ANZE, ADOLFF, VELTEN, WEIGAND, AND MEIER PHYSICAL REVIEW B 93, 054411 (2016) 100 nm 200 nm(a) (b) j H detectorx-ray photons c1c2 = 1 c1c2 = -1(c)1μm FIG. 1. (a) Schematics of the measurement setup along with scanning electron micrographs of the investigated ferromagnetic microstructures. The subset depicts micrographs of the vortex stacks with in-plane magnetic contrast using scanning transmission x-raymicroscopy. The stray fields emerging (b) at the side surfaces and (c) the center regions of the ferromagnets are obtained from micromagnetic simulations. deposited on top of the stacks via thermal evaporation of 250 nm of copper and a protection layer of 5 nm of gold.A sinusoidal current is driven through the signal line of thecoplanar waveguide, leading to an alternating magnetic fieldon the order of several tenths of a millitesla acting in the planeof the ferromagnetic elements. The provided in-plane magneticcontrast of the microscope yields the relative configuration ofthe circulations. Here, the sample is tilted by 60 ◦relative to the incident x-ray beam. As shown in the inset we observe bothpossible combinations ( c 1c2=1 and c1c2=−1), indicating that the silicon spacer decreases interlayer exchange couplingbetween the vortices. For interlayer exchange-coupled vorticesonly one state would occur [ 33]. Stray fields at the side surfaces of the disks emerge when the vortices are deflected from their equilibrium position indicated in Fig. 1(b).I n addition, the vortex core regions exhibit a coupling due tothe out-of-plane component of the magnetization [Fig. 1(c)]. Both interaction effects are crucial to understand the collectivebehavior in a stack of vortices. Here, only the stray fieldsfor equal circulations ( c 1c2=1) of the vortices are depicted. They have been obtained from micromagnetic simulations ofthe investigated structures as described in the last part of thiswork. III. EXPERIMENTS In principle two coupled oscillators have two eigenmodes that describe all possible motions of the system. For stacksof magnetic vortices both modes have been observed inspin-torque oscillators [ 22]. In the experiments the dynamic behavior of a stack of vortices is investigated using a shortmagnetic field pulse (1 ns, 3 mT) pointing in the ydirection. The field pulse allows for the excitation of only one of thetwo modes [ 17]. The different circularities and polarities in the stacks then yield different frequencies and motions of the one excited mode. Due to symmetry considerations there are four possible nondegenerate states. All four combinations ofthe relative circulations and polarizations are imaged usingout-of-plane magnetic contrast. Here, the sample is tilted by90 ◦relative to the incident x-ray beam. The trajectories of the core regions are shown in Fig. 2(a)and can be found as a movie in the Supplemental Material [ 34]. While for static magnetic fields the deflection depends only on the circularity, the initialdeflection due to a nanosecond magnetic field pulse dependson the handedness ( cp=±1) of the isolated magnetic vortex [35]. Subsequent to the initial deflection the vortex performs a damped gyration around the center of the disk, where thesense of gyration is determined by the polarity pof the core. Isolated vortices with a positive polarity ( p=1) gyrate counterclockwise, while vortices with a negative core polaritygyrate ( p=−1) clockwise. The oscillation frequency of the isolated vortex is 240 MHz for the investigated structures. Inthe stack the external field pulse individually deflects bothvortices depending on their handedness. Thus, two identical vortex states ( p 1p2=1,c1c2=1) are deflected in identical directions and gyrate on equal trajectories after the fieldpulse. Due to the additive contrast of the two structures, thedifferent vortex cores cannot be distinguished in this case.Changing the circularity of one of the vortices results inopposite initial deflections yielding different trajectories ofthe vortex cores. The two vortex cores have the same senseof gyration. They gyrate around a common barycenter untilthey reach their equilibrium position and merge into a singleblack dot due to their direct superposition. This motion hasalso been described theoretically [ 23,31]. For both circularity combinations the frequency of gyration is approximately410 MHz. It is comparable to the frequency of an isolated disk with the combined thickness of the stacked vortices. The frequencies have been obtained by the sum of the Fouriertransforms of the two vortex core motions in the stack. Theyare depicted in Fig. 2(b). In the next step, the relative polarities of the vortex stacks are changed from equal ( p 1p2=1) to opposite ( p1p2=−1) polarizations using self-organized state formation [ 19,36,37]. Thereby, the remaining nondegenerate states of opposite polarization are accessible. Two isolatedvortices with opposite polarizations have a different sense ofgyration that is also observed within the stack. The collectivemotions have a lower frequency of about 175 MHz compared 054411-2TWO-BODY PROBLEM OF CORE-REGION COUPLED . . . PHYSICAL REVIEW B 93, 054411 (2016)p1p2 = 1 p1p2 = -1 c1c2 = - 1 c1c2 = 1(a) (b) 100 nm 0101 500 300 100 frequency (MHz)xyFFT mag. (arb. u.)12p p = 1 p1p2 = -1 c1c2 = 1 c1c2 = -1 12p p = 1 p1p2 = -1 FIG. 2. (a) Collective trajectories of the vortex cores within the stack, indicated by red and blue lines. The out-of-plane component of each magnetic vortex is either black ( p=−1) or white ( p=1). It is obtained from the out-of-plane measurements. The vortices areexcited by a short magnetic field pulse pointing in the ydirection. The motions depend on the indicated combination of the relative circulations and polarizations. (b) Sum of the Fourier transform ofthe two vortex core motions subsequent to a short magnetic field pulse. The relative circulations are indicated by dashed ( c 1c2=1) and solid ( c1c2=−1) lines. The relative polarizations are depicted in black ( p1p2=1) and gray ( p1p2=−1). to the resonance frequency of the isolated disks (240 MHz). When the vortex cores approach each other, we observe aslight evasion that is attributed to the repulsion of the coreregions. The two different relative polarities ( p 1p2=±1) have a strong influence on the resonances of the oscillations.The relative circulations yield slight variations. For the caseof opposite polarities and equal circularities ( p 1p2=−1, c1c2=1) a splitting of the resonances can be observed. It could be attributed to a change in the oscillation regime fromlarge to small vortex core trajectories. This mode splitting canbe observed for a critical core distance of about 50 nm in the movie (at 52 ns) in the Supplemental Material [ 34]. Since the distance between the cores decreases over time, the influenceof the core interaction increases. Further measurements of thesteady-state motions elucidate this dependence. The vortex stacks are excited near resonance by a sinusoidal magnetic field. Two different frequencies of 175 and 410 MHzfor the two relative polarizations ( p 1p2=±1) are used. Figure 3(a) illustrates the motions of the vortex cores for all(b)(a) 100 nmp1p2 = 1 p1p2 = -1 c1c2 = - 1 c1c2 = 1p1p2 = -1 reduced amplitudestrong amplitude FIG. 3. (a) Stationary trajectories of all nondegenerate-state combinations in a stack of vortices, indicated by red and blue lines.The vortices are excited by a sinusoidal magnetic field pointing in theydirection. Stacks with p 1p2=1 are excited with a frequency of 410 MHz and a field amplitude of 0.4 mT; a frequency of 175 MHzis used for p 1p2=−1 with an amplitude of 0.5 mT for c1c2=1a n d 0.8 mT for c1c2=−1. (b) When the amplitude of the excitation is reduced by about 40% the type of the trajectory changes for one ofthe two relative circulations. four state combinations. The vortices oscillate on constant trajectories that resemble possible motions of the gravitationaltwo-body problem. V ortices with the same polarities gyrateon circular trajectories around a common barycenter. Forequal circularities ( c 1c2=1) the vortices gyrate on the same trajectory, whereas a phase shift of 180◦emerges for the case of opposite circularities ( c1c2=−1). In both cases the interaction is mediated by coupled in-plane dipoles rotating in the samedirection. For equal circularities the attractive force of the twocores is zero since they gyrate on the same lateral positions.For opposite circularities the interaction of the cores can alsobe neglected. This is due to the large interdistance ( ∼100 nm) of the cores. V ortices with opposite polarities gyrate in an opposite sense. Here, the relative phase and thereby the crossing point of thecores depend on the relative circulation. One of the cores isconstrained to a lower radius. This asymmetry could be pro-voked by the Oersted field that slightly varies in its amplitudefor the two disks. Then, the intrinsic repulsion of the coresstrengthens this effect. However, as the oscillation radii areconstant for the case of identical polarizations the asymmetry 054411-3H¨ANZE, ADOLFF, VELTEN, WEIGAND, AND MEIER PHYSICAL REVIEW B 93, 054411 (2016) of the Oersted field is rather small. The amplitude of the excitation has been adjusted to maintain large-trajectory radii.By reducing the amplitude of the excitation the type of gyrationchanges for vortices with different polarities [Fig. 3(b)]. We observe a gyration of the vortices around individualbarycenters for the case of equal circularities ( c 1c2=1). Note that the barycenter is defined as the center of the core trajectoryduring one oscillation period of an individual disk. The vorticesare repelled by the core interaction. This behavior cannot beobserved for the case of opposite circularities ( c 1c2=−1). Slight oscillations of the barycenters could not be observeddue to the stroboscopic measurement method that integratesover millions of oscillation periods. IV . THEORETICAL MODEL AND DISCUSSION The strong frequency splitting of the two polarity states can be understood within the model presented in Ref. [ 23]. Within this model the splitting between the two polaritystates sums up to 310 MHz for the investigated structures.This value is larger than the experimental splitting reportedin Fig. 2, which can be explained by the overestimation of the rigid vortex approach [ 38]. Still, the observed repulsion of the vortex cores has to be taken into account [ 26,31,32]. Therefore, we performed calculations where the interactionof the vortex cores is modeled by an additional potential.We calculate the total energies of two deflected magneticvortices using the Thiele approach [ 39], which considers the magnetic vortex as a rigid particle. Figure 4(a)depicts the stray fields of two magnetic dipoles, the vortex cores, in a stack ofmagnetic vortices. The energy of these dipoles is modeledby the assumption of two interacting point dipoles. It can beexpressed as a function of their lateral deflection in oppositedirections /Delta1r. The amplitude of the dipole moment is derived from the actual size of the magnetic vortex core. Here, weassume that all magnetic moments in a cylindrical shape witha radius r cand the thickness tof the disks point into the same out-of-plane direction. The energy is given by Ecore=−μ0πM2 sp1p2r4 ct2 4[/Delta1r2+(t+/Delta1z)2]3 2/parenleftbigg3(t+/Delta1z)2 /Delta1r2+(t+/Delta1z)2−1/parenrightbigg , (1) where ( t+/Delta1z) is the distance between the vertical centers of the two disks, Msis the saturation magnetization of permalloy, and /Delta1zis the thickness of the silicon spacer. The core interaction is repulsive for opposite ( p1p2=−1) and attractive for equal polarities ( p1p2=1). Reasonable values of rc=12 nm [ 6] andMs=800 kA /m are assumed. The energy contribution of the confinement of the disks ismodeled by a harmonic potential [ 20,38] with a curvature κ=1.72×10 −3kg/s2and corresponds to a frequency of ω0/(2π)=240 MHz. For vortices the interaction of the stray fields at the side surfaces of the disks is also describedby a harmonic potential when equally deflected in oppositedirections. The coupling coefficient η=1.45×10 −3kg/s2is obtained in analogy to Ref. [ 10] from numerical integration of the magnetostatic energy between the side surfaces. The strayfield energy depends on the relative circulations of the disksas depicted in Fig. 4(b). The sum of all energy contributions∆r∆z(a) (b) (c) (d) 60 40 20 0 40 30 20 10 ∆z (nm)∆req (nm)Esum (10-18 J)8 6 4 2 0 -2 120 80 40 0 ∆r (nm) c1c2=1 c1c2=-1 FIG. 4. Stray fields emerging (a) in the core regions and (b) at the side surfaces of vortices that are equally deflected in oppositedirections. The stray fields have been calculated using dipoles indicated by black and white arrows. (c) Energy contributions as a function of the lateral distance of the vortex cores /Delta1r within the Thiele approach. All four nondegenerate combinations of the circulations (dashed: c 1c2=1, solid: c1c2=−1) and the polarizations (black: p1p2=1, gray: p1p2=−1) are shown. (d) Resulting equilibrium position for opposite polarities as a function of the thickness /Delta1zof the silicon spacer. Micromagnetic simulations depicted by dots and crosses are in agreement with the calculations. Esumreads Esum=Ecore(p1p2,/Delta1r)+(κ−c1c2η)/Delta1r2 4. (2) Figure 4(c) depicts the total energy Esumfor all four possible nondegenerate state combinations. For large deflections /Delta1r the energy contribution of the stray fields at the side surfacesdominates. For the case of equal circularities and oppositepolarities ( c 1c2=1,p1p2=−1) the minimum energy, i.e., the equilibrium deflection /Delta1req, is obtained for /Delta1r=30 nm. All other state combinations reach their equilibrium positionat/Delta1r=0. This behavior explains the collective oscillations observed in Fig. 3(b). Depending on the circularity, the two vortices oscillate around different or equal barycenters. Forthe case of equal circularities the distance between the twobarycenters is approximately /Delta1r≈50 nm, whereas for the case of opposite circularities the two vortices gyrate aroundthe same barycenter. These values are in good agreement withthe analytical model. Slight variations can be explained by themeasurement method that yields only the superposed contrastof both vortex cores. The energy between the vortex corescan also be calculated using the model of magnetic surfacecharges that emerge at the top and bottom surfaces of thedisks. The proceeding is described in Ref. [ 26] and yields similar results. To gain insight into the strength of the observeddipolar coupling we performed further calculations to predictthe limits of strong and weak core interactions. Figure 4(d) 054411-4TWO-BODY PROBLEM OF CORE-REGION COUPLED . . . PHYSICAL REVIEW B 93, 054411 (2016) illustrates the dependence of the equilibrium deflection on the thickness of the silicon spacer. The analytical calculations areconfirmed by micromagnetic simulations [ 40]. The dimensions of the simulated stacks are identical to the experiments. Theequilibrium deflection is obtained by a relaxation of the systeminto its energetic minimum. Therefore, we use a cell size of4×4×4 nm, a saturation magnetization of M s=800 kA /m, an exchange stiffness constant of A ex=1.3×10−11J/m, and a Gilbert damping constant of α=0.01. In agreement with Ref. [ 32] the case of equal circularities provokes a larger static displacement of the cores. Depending on the geometry of thedisks, the displacement is expected to emerge for the case ofdifferent circularities as well. V . CONCLUSION We conclude that the collective magnetic excitation in a stack of vortices is dominated by the relative polaritiesof the vortex cores. A strongly increased splitting of theresonance frequencies compared to laterally coupled structuresis observed. The proximity of the disks results in a couplingof the vortex cores. This coupling yields a displacementof the equilibrium positions for vortex stacks with equal circulations and opposite polarities. Its nonlinear influenceleads to different types of steady-state motions observed byscanning transmission x-ray microscopy that yield a two-bodyproblem. The access to the third dimension in stacked vorticesovercomes the limitations concerning the storage density inpotential memory devices. ACKNOWLEDGMENTS We thank U. Merkt for fruitful discussions and M. V olkmannfor superb technical assistance. We acknowledge the support ofthe Max Planck Institute for Intelligent Systems (formerly MPIfor Metals Research), Department Sch ¨utz, and the MAXY- MUS team, particularly, M. Bechtel and E. Goering. We thankthe Helmholtz-Zentrum Berlin f ¨ur Materialien und Energie for the allocation of synchrotron radiation beam time. Financial support of the Deutsche Forschungsgemeinschaft via the Sonderforschungsbereich 668 and the Graduiertenkolleg 1286is gratefully acknowledged. This work has been supportedby the excellence cluster “The Hamburg Centre for UltrafastImaging (CUI): Structure, Dynamics and Control of Matter atthe Atomic Scale” of the Deutsche Forschungsgemeinschaft. [1] V . V . Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D 43,264001 (2010 ). [2] A. Hoffmann and S. D. Bader, Phys. Rev. Appl. 4,047001 (2015 ). [3] A. H. Eschenfelder, Magnetic Bubble Technology , Springer Series in Solid-State Sciences V ol. 14 (Springer, Berlin, 1981). [4] S. S. P. Parkin, M. Hayashi, and L. 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PhysRevB.100.224411.pdf

PHYSICAL REVIEW B 100, 224411 (2019) Theory for shift current of bosons: Photogalvanic spin current in ferrimagnetic and antiferromagnetic insulators Hiroaki Ishizuka1and Masahiro Sato2 1Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan 2Department of Physics, Ibaraki University, Mito, Ibaraki 310-8512, Japan (Received 5 July 2019; published 12 December 2019) We theoretically study the optical generation of dc spin current (i.e., a spin-current solar cell) in ordered antiferromagnetic and ferrimagnetic insulators, motivated by a recent study on the laser-driven spinon spincurrent in noncentrosymmetric quantum spin chains [H. Ishizuka and M. Sato, P h y s .R e v .L e t t . 122,197702 (2019 )]. Using a nonlinear response theory for magnons, we analyze the dc spin current generated by a linearly polarized electromagnetic wave (typically, terahertz or gigahertz waves). Considering noncentrosymmetrictwo-sublattice magnets as an example, we find a finite dc spin-current conductivity at T=0, where no thermally excited magnons exist; this is in contrast to the case of the spinon spin current, in which the optical transition ofthe Fermi degenerate spinons plays an essential role. We find that the dc spin-current conductivity is insensitiveto the Gilbert damping, i.e., it may be viewed as a shift current carried by bosonic particles (magnons). Ourestimate shows that an electric-field intensity of E∼10 4–106V/cm is sufficient for an observable spin current. Our theory indicates that the linearly polarized electromagnetic wave generally produces a dc spin current innoncentrosymmetric magnetic insulators. DOI: 10.1103/PhysRevB.100.224411 I. INTRODUCTION Materials subject to an intense incident light show rich be- haviors which are studied in the context of nonlinear responseand nonequilibrium phenomena. An example of such is elec- tric shift current in noncentrosymmetric semiconductors and ferroelectrics [ 1–7], where a nontrivial shift of electron po- sition during its optical transition produces a macroscopicelectric current. Recent studies revealed that the shift cur-rent exhibits strikingly different behaviors from the ordinaryphotocurrent; the shift current shows unique light-positiondependence when it is excited locally [ 8–10], and propagates faster than the Fermi velocity of electrons [ 10–12]. On the other hand, in correlated materials, lower-energy excitationsoften emerge due to the interaction effect; a typical example ismagnetic excitations in Mott insulators. The optical transitionof these emergent particles may produce nontrivial phenom-ena, especially, transport phenomena, related to the nonlinearresponse of the emergent excitations. Several recent studies in optospintronics and magneto- optics [ 13–15] imply that the intensity and coherence of cur- rently available electromagnetic waves are sufficient for thecontrol of magnetic excitations or magnetism. Typical resultsare the following: magnetization switching by a circularlypolarized laser in ferrimagnets [ 16–19], laser-driven demag- netization [ 20–22], the spin pumping by gigahertz (GHz) or terahertz (THz) waves [ 23,24], focused-laser-driven magnon propagation [ 25,26], intense THz-laser-driven magnetic res- onance [ 27,28], spin control by THz-laser-driven electron transitions [ 29], dichroisms driven by THz vortex beams [ 30], angular-momentum transfer between photons and magnons incavities [ 31–35], an ultrafast detection of spin Seebeck effect [36], a phonon-mediated spin dynamics with THz laser [ 37],etc. Moreover, recent theoretical works have proposed several ways of optical control of magnetism: GHz /THz-wave-driven inverse Faraday effect [ 38,39], Floquet engineering of mag- netic states such as chirality ordered states [ 40,41] and a spin-liquid state [ 42], generation of magnetic defects with laser-driven heat [ 43,44], applications of topological light waves to magnetism [ 44–47], control of exchange couplings in Mott insulators with high- [ 48] and low-frequency [ 49] waves, optical control of spin chirality in multiferroic mate-rials [ 50], rectification of dc spin currents in magnetic insu- lators with electromagnetic waves [ 51–53]. These studies are partly supported by recent developments in THz laser science[54,55] which realized high-intensity light beams with the photon energy comparable to those of magnetic excitations.Despite these developments, the optical control of the currentcarried by magnetic excitations is limited to some theoreticalproposals. Among the proposals, a recent theory proposes a mecha- nism for producing a dc spin current in quantum spin chainswithout the angular-momentum transfer [ 52]; it is distinct from the known mechanisms in which the angular momentumof photons is transferred to the magnet [ 23,24,51,53,56]. The mechanism in Ref. [ 52] is analogous to that of the shift- current photogalvanic effect [ 2]. The close relation between two phenomena are clear from the Jordan-Wigner fermionrepresentation of spin chain; the ground state of the spin chainis a band insulator of Jordan-Wigner fermions, and the pho-togalvanic response is related to the optical transition of thefermions by the linearly polarized GHz /THz light. However, the relation of this mechanism to the fermion excitations castsdoubt on the generality because the low-energy excitationsof the ordered magnets are usually magnons, i.e., bosonicexcitations. 2469-9950/2019/100(22)/224411(12) 224411-1 ©2019 American Physical SocietyHIROAKI ISHIZUKA AND MASAHIRO SATO PHYSICAL REVIEW B 100, 224411 (2019) In this work, we theoretically show that a dc spin current similar to that of the spin chain [ 52] also appears in ordered antiferromagnetic (AFM) and ferrimagnetic (FRM) insulatorsby applying a linearly polarized electromagnetic wave. Thesymmetry argument in Sec. IIIshows that the creation of dc spin current with linearly polarized waves is possibleonly if both site- and bond-center inversion symmetries arebroken. AFM and FRM insulators violate the bond-centerinversion symmetry and thereby they naturally satisfy halfof the required symmetry condition. The staggered momentis an advantage of considering AFM /FRM insulators for generating a dc spin current. As an example, we considertwo-sublattice models with Néel-type ground state. Bosonicparticles describe the low-energy excitations of these models,i.e., magnons; the ground state is the zero-magnon state. Thisground state is very different from that of noncentrosymmetricS= 1 2spin chains [ 52] which are described by a Fermi degen- erated state of spinons. Despite the difference, our calculationusing a nonlinear response theory finds a finite photogalvanicspin current similar to that of the spinons. We discuss that it isrelated to the zero-point fluctuation of the quantum magnets.Our theory also indicates that the magnon spin current is shift-current like, i.e., it is insensitive to the magnon lifetime as inthe spinon case [ 3,57,58]. This mechanism allows generation of spin current using a linearly polarized electromagneticwave and ordinary AFM or FRM insulators. The remaining part of the paper is organized as follows. In Sec. II, we introduce the nonlinear response theory for two-species magnons, which we will use in the followingsections. The main results of this paper are in Secs. IIIandIV. Section IIIfocuses on the photoinduced spin current in AFM and FRM insulators with a strong one dimensionality, whilewe study the three-dimensional (3D) magnets in Sec. IV. Effective experimental setups and signatures for investigatingthe proposed mechanism are discussed in Sec. V. Section VI is devoted to the summary and discussions. II. NONLINEAR RESPONSE THEORY We calculate the nonlinear response coefficients for the photoinduced spin current by extending the linear responsetheory to the quadratic order in the perturbation. A similarmethod for fermions is used to calculate the photogalvaniccurrent in semiconductors [ 57,58] and the spin current of spinons [ 52]. The derivation of the formula is summarized in Appendix A. We here summarize the outline of the derivation. We also discuss the physical implications. We consider a two-sublattice AFM /FRM insulator with two species of magnons. The effective Hamiltonian for themagnons is H=/summationdisplay kεα(k)α† kαk+εβ(k)βkβ† k, (1) where αk(α† k) andβk(β† k) are the boson annihilators (creators) for the magnons with the momentum k=(kx,ky,kz) and εa(k)(a=α,β) is the energy of the magnons in the a= α,β branch with momentum k. We here consider a generalperturbation (spin-electromagnetic-wave coupling) H/prime=−/summationdisplay μ,k/integraldisplaydω 2π/Gamma1μ ωeiωtψ† k/parenleftBigg/parenleftbig Bμ k/parenrightbig αα/parenleftbig Bμ k/parenrightbig αβ/parenleftbig Bμ k/parenrightbig βα/parenleftbig Bμ k/parenrightbig ββ/parenrightBigg ψk +H.c., (2) and spin-current operator J=/summationdisplay kψ† k/parenleftbigg (Jk)αα (Jk)αβ (Jk)βα (Jk)ββ/parenrightbigg ψk. (3) Here, ωis the frequency of ac light, /Gamma1μ ωis the spin-light coupling constant for the μdirection, and ψk=(αk,β† −k)T. The nonlinear conductivity is defined by /angbracketleftJ/angbracketright(/Omega1)=/summationdisplay μ,ν/integraldisplay dωσμν(/Omega1;ω,/Omega1−ω)/Gamma1μ ω/Gamma1ν /Omega1−ω, (4) where /angbracketleftJ/angbracketright(/Omega1)≡/integraltext dt/angbracketleftJ/angbracketright(t)e−i/Omega1tis the Fourier transform of the expectation value of the spin current /angbracketleftJ/angbracketright(t). For the two-sublattice model, the formula for nonlinear spin currentconductivity reads as σ μν(/Omega1;ω,/Omega1−ω) =1 2π/summationdisplay k,ai=α,βsgn(a3)/bracketleftbig ˜ρk,a1sgn(a2)−sgn(a1)˜ρk,a2/bracketrightbig/parenleftbig Bμ k/parenrightbig a1a2 ω−˜εa2(k)+˜εa1(k)−i/(2τk) ×/bracketleftbigg/parenleftbig Bν k/parenrightbig a2a3(Jk)a3a1 /Omega1+˜εa1(k)−˜εa3(k)−i/(2τk) −(Jk)a2a3/parenleftbig Bν k/parenrightbig a3a1 /Omega1+˜εa3(k)−˜εa2(k)−i/(2τk)/bracketrightbigg , (5) where ˜εa(k)=sgn(a)εa(k), (6) sgn(a)=/braceleftbigg 1( a=α), −1( a=β),(7) ˜ρk,a=/braceleftbigg/angbracketleftα† kαk/angbracketright0 (a=α), /angbracketleftβ−kβ† −k/angbracketright0(a=β).(8) The relaxation time of magnons τkwas introduced in Eq. ( 5), and/angbracketleft.../angbracketright0is the expectation value of ...in the equilibrium state of the Hamiltonian in Eq. ( 1). The conductivity for dc spin current corresponds to the /Omega1=0 case, σμν(0;ω,−ω). In the rest of this work, we focus on the case Bμ k=Bν k= Bkbecause we are interested in the response to a linearly polarized light. Hence, we abbreviate the subscripts in thenonlinear conductivity σ μν(0;ω,−ω)=σ(0;ω,−ω). We note that the conductivity in Eq. ( 5) remains nonzero at T=0. The substitutions of ˜ ρk,α=0 and ˜ ρk,β=1i nE q .( 5) 224411-2THEORY FOR SHIFT CURRENT OF BOSONS: … PHYSICAL REVIEW B 100, 224411 (2019) reduce the formula to σ(0;ω,−ω)=/summationdisplay k−1 π/bracketleftbigg(1+i2τkω)|(Bk)βα|2[(Ak)αα+(Ak)ββ] (ω−i/2τk)2−[εα(k)+εβ(k)]2/bracketrightbigg +1 2π(Bk)βα(Ak)αβ[(Bk)ββ+(Bk)αα] [ω−i/2τk−εα(k)−εβ(k)][εα(k)+εβ(k)+i/2τk] +1 2π(Bk)αβ(Ak)βα[(Bk)ββ+(Bk)αα] [ω−i/2τk+εα(k)+εβ(k)][εα(k)+εβ(k)−i/2τk]. (9) Here, Ais an observable; it is the spin-current operator in the rest of this paper. Because of ˜ ρk,β=1, the terms involving the off-diagonal component of Bkremain at T=0. In other words, the two-magnon creation /annihilation process plays a crucial role as shown in Fig. 1(d). We focus on the T=0 case in the rest of this paper as this process is dominant in thelow-temperature limit. From a different viewpoint, Eq. ( 9) implies the zero-point fluctuation plays a key role in the photogalvanic response ofmagnons. In our formalism, the zero-point fluctuation is mani-fested in the Bogoliubov transformation of Holstein-Primakovbosons. This transformation creates β kβ† kandα† kβ† −kterms which contribute to the photogalvanic response in the groundstate. ˜ ρ k,β=1 is another consequence of the Bogoliubov transformation. The importance of the zero-point fluctuationresembles the spinon spin current [ 52], in which the Fermi degeneracy of spinons represents the quantum fluctuation of (a) (b)ω ω 001234 0E π −π −π π(c) (d) kkαk, βkαk+ β−k αkαk+ β−k βk ω FIG. 1. Schematic pictures of the noncentrosymmetric magnets. A quasi-one-dimensional magnet consisting of weakly coupled spin chains (a) and a three-dimensional magnet with two-sublattice order (b). Each sublattice (blue and orange) has a different environment,e.g., different gfactors, uniaxial anisotropy, etc., and with the bond dimerization (shown by the thick bond). The two-sublattice order and bond dimerization, respectively, break the inversion symmetry on thebond center and sites. Magnetic excitation and nonlinear spin-current conductivity of the spin chain. The magnon band dispersions of the model in Eq. ( 11a)f o r( c ) h +=0a n d( d ) h+=1 100. Parameters h± are defined in Eq. ( 17). When h+=0, two magnon dispersions are degenerate. The GHz /THz light produces two magnons, one on each branch as schematically shown in (d).spins. A crucial difference in the current case is the absence of Fermi degeneracy. However, in the case of the AFMs /FRMs, the condensate of Holstein-Primakov bosons plays a simi-lar role to the Fermi degeneracy. The pair-creation processrepresented by α † kβ† kgenerates photogalvanic response of the magnons which is manifested in the denominator ofEq. ( 9); the sum of eigenenergies, ε α(k)+εβ(k), represents creation /annihilation of a magnon pair. These features imply that the zero-point fluctuation is necessary for the shift-currentresponse at T=0. The first term in Eq. ( 9) vanishes when the ground state has a certain symmetry. For example, collinear magnetic orderswith the moments parallel to S zaxis are often symmetric with respect to G=TMs x, which is the product of time-reversal operation ( T) and the mirror operation for the spin degrees of freedom about the xaxis ( Ms x). In this case, the real part of σ(0;ω,−ω) reads as Re[σ(0;ω,−ω)] =−1 π/summationdisplay kRe/braceleftbigg(Bk)βα(Ak)αβ[(Bk)ββ+(Bk)αα] ω2−[εα(k)+εβ(k)+i/2τk]2/bracerightbigg .(10) The conductivities for the models considered in the following sections are calculated using this formula. III. SPATIALLY ANISOTROPIC MAGNET In this section, we apply the above formula to a spin chain with AFM or FRM order, which corresponds to aquasi-one-dimensional (quasi-1D) magnetic compound with anegligible interchain interaction. The spins are coupled to theelectromagnetic wave through the Zeeman coupling. To makethe problem theoretically well defined, we consider a modelwhich conserves the spin angular momentum S z; the model has an easy axis and the applied ac magnetic field is parallelto the ordered moments. The conservation of S zallows us to unambiguously define the spin-current operator from thecontinuity equation. This setup is in contrast to those of usualmagnetic resonances and spin pumping [ 23,24], in which the ac field is perpendicular to the magnetic moment. We usethe standard spin-wave approximation to describe magneticexcitations (magnons). A. Model We consider an ordered noncentrosymmetric spin chain with a two-sublattice unit cell [Fig. 1(a)], whose Hamiltonian is given by Htot=H0+H(ω) Z, (11a) H0≡/summationdisplay ry,rzH1D(ry,rz), (11b) 224411-3HIROAKI ISHIZUKA AND MASAHIRO SATO PHYSICAL REVIEW B 100, 224411 (2019) H1D(ry,rz)≡/summationdisplay rxJ(1+δ)SA(r)·SB(r) +J(1−δ)SA(r+ˆx)·SB(r) −(D+Ds)/bracketleftbig Sz A(r)/bracketrightbig2−(D−Ds)/bracketleftbig Sz B(r)/bracketrightbig2 −B/bracketleftbig gASz A(r)+gBSz B(r)/bracketrightbig , (11c) H(ω) Z=− (Bωeiωt+H.c.)/summationdisplay rgASz A(r)+gBSz B(r), (11d) where H1Dis the spin-chain Hamiltonian with the staggered nearest-neighbor exchange interaction (i.e., dimerization)along the xdirection, H 0is the bundle of all the chains, and H(ω) Zis the Zeeman coupling between the spins and the exter- nal electromagnetic wave. Here, Sa(r)≡(Sx a(r),Sy a(r),Sz a(r)) (a=A,B)i st h es p i n - Saoperator on the asublattice of the unit cell at position r=(rx,ry,rz). Symbols ˆ x,ˆy, and ˆ z stand for the unit vectors along the x,y, and zdirections, respectively. The parameters in the Hamiltonian H1Dare as follows: J>0 is the antiferromagnetic exchange interaction along the spin-chain ( x) direction, δis the dimerization, D>0 (Ds) is the uniform easy-axis (staggered) anisotropy, gA(gB) is the gfactor for the spins on A(B) sublattice, and his the external static magnetic field along the Szaxis. In the spin-light coupling H(ω) Z,|Bω|and arg( Bω) are, respectively, the magnitude and the phase of the ac magnetic field of thelinearly polarized electromagnetic wave. We assume |D s|< D, and|δ|<1. When SA/negationslash=SB, the ground state of the model in Eq. ( 11c)i s a FRM-ordered state with magnetization |SA−SB|per a unit cell [ 59,60]. The ground state is Néel ordered when SA=SB. The classical ground state of H0has a collinear order with spins pointing along the Szaxis because of the easy-axis anisotropy D[Fig. 1(a)]. The anisotropy also produces the spin gap in the excitation spectrum [Figs. 2(a) and1(b)]. We discuss the effect of the gap and its relation to the frequencydependence of the nonlinear spin conductivity in the nextsection. Here, we define the spin current for S z. Since the model H0 conserves the zcomponent of total spin angular momentum, the spin current for Szcan be defined from the continuity equa- tions∂tSz A=Jz x(rx−1,B;rx,A)−Jz z(rx,A;rx,B) and∂tSz B= Jz x(rx,A;rx,B)−Jz z(rx,B;rx+1,A), in which Jα β(r,a;r/prime,b) is the local spin- Sαcurrent operator between two neighboring sites ( r,a) and ( r/prime,b) and it flows along the βdirection. The above continuity equation is obtained from Heisenbergequation of motion for local spins. With these procedures, wefind the uniform current operator for H 1Dreads as Jz x=J 2N/summationdisplay r(1+δ)/braceleftbig Sx B(r)Sy A(r)−Sy B(r)Sx A(r)/bracerightbig +(1−δ)/braceleftbig Sx A(r+ˆx)Sy B(r)−Sy A(r+ˆx)Sx B(r)/bracerightbig ,(12) where Nis the total number of unit cells. B. Linear spin-wave approximation We here study the shift current of magnons using linear spin-wave approximation. Hereafter, we assume that in the(a) σ(0;ω,−ω ) (a.u.) ω/J(b)-0.15-0.10-0.050.000.05 10-410-310-210-1100101 0 1 2 3 4σ(ω) α=5x10-3 α=1x10-2 α=1x10-10.01 0.10 1 0.0010.0100.1001 J┴=0 J┴=1/4 J┴=1δω|σ(0;ω,−ω )| (a.u.) FIG. 2. Frequency dependence of the nonlinear spin-current con- ductivity σ(0;ω,−ω). (a) Analytic result for the small Gilbert damping limit α→0 and (b) numerical results for a finite α.T h e inset in (a) is the δω≡ω−ωc1for different J⊥. The calculations are done using a chain with N=2048–32 768 unit cells. All results are for J=1,δ=1/4,SA=SB=1,gA=1,gB=1/2,h−=0, and B+=1/100 unless noted explicitly. ground state of H1D, the spins on the Asublattice point up while those on Bsublattice are down [see Fig. 1(a)]. The in- teraction between the low-energy excitations of H0(magnons) are negligible when the temperature is sufficiently lower thanthe magnetic transition temperature. Therefore, we neglectthe interaction between the magnons, i.e., we study the spincurrent within the linear spin-wave approximation. Using the Holstein-Primakov bosons, the spin operators are given by S z A=SA−ˆnA(r), (13a) S+ A(r)=/radicalbig 2SA/parenleftbigg 1−ˆnA(r) 2SA/parenrightbigg1 2 a(r), (13b) S− A=/radicalbig 2SAa†(r)/parenleftbigg 1−ˆnA(r) 2SA/parenrightbigg1 2 (13c) for the Asublattice, and Sz B=ˆnB(r)−SB, (14a) S+ B(r)=/radicalbig 2SBb†(r)/parenleftbigg 1−ˆnB(r) 2SB/parenrightbigg1 2 , (14b) S− B=/radicalbig 2SB/parenleftbigg 1−ˆnB(r) 2SB/parenrightbigg1 2 b(r) (14c) 224411-4THEORY FOR SHIFT CURRENT OF BOSONS: … PHYSICAL REVIEW B 100, 224411 (2019) for the Bsublattice. Up to the linear order in SAandSB,H0 reads as H0∼/summationdisplay k/parenleftbiggak b† −k/parenrightbigg†/parenleftBigg h0 k+hz khx k−ihy k hx k+ihy kh0 k−hz k/parenrightBigg/parenleftbiggak b† −k/parenrightbigg +const, (15) where the wave number along the chain ( x) direction is simply represented by k,ak≡(1/√ N)/summationtext ra(r)eik·r,bk≡ (1/√ N)/summationtext rb(r)eik·(r+ˆx/2)are the Fourier transformation of Holstein-Primakov bosons. The matrix elements of themagnon Hamiltonian ( 15) are calculated as h 0 k=B++J(SA+SB), (16a) hx k=2J√ SASBcos(k/2), (16b) hy k=− 2Jδ√ SASBsin(k/2), (16c) hz k=B−−J(SA−SB), (16d) where B+=D(SA+SB−1)+Ds(SA−SB)+B 2(gA−gB),(17a) B−=D(SA−SB)+Ds(SA+SB−1)+B 2(gA+gB).(17b) We note that, in general, the magnon Hamiltonian for two- sublattice ordered system has a 4 ×4 matrix form, but that of the present system can be reduced to a 2 ×2 form as shown in Eq. ( 15). The quadratic Hamiltonian ( 15) is diagonalized by the Bogoliubov transformation: ak=cosh/Theta1kαk+sinh/Theta1kβ† −k, (18) b† −k=sinh/Theta1kei/Phi1kαk+cosh/Theta1kei/Phi1kβ† −k, (19) where αk(α† k) andβk(β† k) are bosonic annihilation (creation) operators. By choosing ei/Phi1k=hx k+ihy k/radicalBig/parenleftbig hx k/parenrightbig2+/parenleftbig hy k/parenrightbig2(20a) and cosh(2 /Theta1k)=h0 k/radicalBig/parenleftbig h0 k/parenrightbig2−/parenleftbig hx k/parenrightbig2−/parenleftbig hy k/parenrightbig2, (20b) sinh(2 /Theta1k)=−/radicalBig/parenleftbig hx k/parenrightbig2+/parenleftbig hy k/parenrightbig2 /radicalBig/parenleftbig h0 k/parenrightbig2−/parenleftbig hx k/parenrightbig2−/parenleftbig hy k/parenrightbig2, (20c)the Hamiltonian becomes H0=/summationdisplay kεα(k)α† kαk+εβ(k)β† −kβ−k, (21) where εα(k)=hz k+/radicalBig/parenleftbig h0 k/parenrightbig2−/parenleftbig hx k/parenrightbig2−/parenleftbig hy k/parenrightbig2, (22a) εβ(k)=−hz k+/radicalBig/parenleftbig h0 k/parenrightbig2−/parenleftbig hx k/parenrightbig2−/parenleftbig hy k/parenrightbig2. (22b) Here, we ignored the constant term in H0. We note that the dispersions εα,β(k) and the phases ( /Theta1k,/Phi1 k) are all indepen- dent of kyandkzbecause we now consider the 1D model H0. Using the same transformation, we find H(ω) Z=B/summationdisplay k(gAcosh2/Theta1k−gBsinh2/Theta1k)α† kαk +(gAsinh2/Theta1k−gBcosh2/Theta1k)β−kβ† −k +gA−gB 2sinh(2 /Theta1k)(α† kβ† −k+β−kαk) +const (23) and Jz x=J√ SASB/summationdisplay ksinh(2 /Theta1k)/parenleftbigg sink 2cos/Phi1k+δcosk 2sin/Phi1k/parenrightbigg ×(α† kαk+β−kβ† −k) +/bracketleftbigg/braceleftbigg cosh(2 /Theta1k)/parenleftbigg cos/Phi1ksink 2−δsin/Phi1kcosk 2/parenrightbigg +i/parenleftbigg sin/Phi1ksink 2+δcos/Phi1kcosk 2/parenrightbigg/bracerightbigg α† kβ† −k+H.c./bracketrightbigg . (24) C. Spin-current conductivity Combining the magnon representation of ( αk,βk) with the formula ( 10), we compute the nonlinear dc spin-current con- ductivity for the model Htotunder the application of GHz wave or THz laser. We first study the nonlinear conductivity in theclean limit with infinite relaxation time τ k→∞ . The analytic solution for the conductivity Re[ σ(0;ω,−ω)] obtained from Eq. ( 10) reads as Re[σ(0;ω,−ω)]=(gA−gB)2δ[B++J(SA+SB)](ω2−4[B++J(SA+SB)]2−2J2SASB(1+δ2)) 8π(1−δ2)ω2/radicalBig 4J4S2 AS2 B(1−δ2)2−{(ω/4)2+2J2SASB(1+δ2)−]B++J(SASB)]2}2, (25) whenω∈[ωc1,ωc2] and zero otherwise. Here, ωc1≡εα(0)+εβ(0) =2/radicalbig [B++J(SA+SB)]2−4J2SASB (26)corresponds to the energy for the band bottom of the pair excitation and ωc2≡εα(π)+εβ(π) =2/radicalbig [B++J(SA+SB)]2−4δ2J2SASB (27) 224411-5HIROAKI ISHIZUKA AND MASAHIRO SATO PHYSICAL REVIEW B 100, 224411 (2019) is that for the top of the pair excitation [see Figs. 1(c) and (d)]. The frequency dependence of the conductivity is shownin Fig. 2(a). One finds from Eq. ( 25) that the spin-current generation disappears in the case g A=gB. However, this comes from our simple setup. If we consider a larger-sublattice magnetor noncollinear-ordered one, a finite spin current is expectedeven in g A=gB. In addition, in our previous study [ 52], we show that other types of spin-light couplings such as inverseDyzaloshinskii-Moriya and magnetostriction couplings pro-duce a spin current in a 1D quantum spin chain. Equation ( 25) is an odd function of δ. This reflects the fact that the inversion-symmetry breaking is necessary forthe spin current. H 0has two inversion centers when δ=0, Ds=0,gA=gB, and SA=SB: one at the center of the bond and the other on the site. The inversion center on the siteis broken by the dimerization δ. To see the dependence of σ(0;ω,−ω) on the model parameters, we explicitly write the nonlinear conductivity as a function of the parameters, i.e.,σ(0;ω,−ω)=σ(ω;δ,g A−gB,Ds,m), where m=/angbracketleftSz r∈A/angbracketright− /angbracketleftSz r∈B/angbracketrightis the order parameter of the AFM or FRM insula- tors. A symmetry argument on the transport coefficient findsσ(ω;δ,g A−gB,Ds,m)=−σ(ω;−δ,gA−gB,Ds,m)f o rt h e site-center inversion operation. This result is identical to thespinon case in Ref. [ 52]. On the other hand, the magnetic order changes the param- eter dependence of σ(0;ω,−ω), which is related to the bond- center inversion operation. The inversion operation about thecenter of the bonds is broken by the Néel ordering D s/negationslash= 0o r gA/negationslash=gB. Therefore, the symmetry operation indicates σ(ω;δ,gA−gB,Ds,m)=−σ(ω;δ,−gA+gB,−Ds,−m). In addition, the translation operation about half a unit cellswitches Aand Bsublattices and m→− m;σ(ω;δ,g A− gB,Ds,m)=−σ(ω;δ,gA−gB,Ds,m). Hence, the conduc- tivity in the ordered phase is an even function of gA−gBand Ds. This is a different behavior from the spinon case, in which the conductivity is an odd function of the staggered magneticfield (corresponds to g A−gBin our case). The conductivity diverges when ωapproaches ωc1.T h e asymptotic form reads as Re[σ(0;ω,−ω)]≈−(gA−gB)2J2δSASB[B++J(SA+SB)] 8πJ{[B++J(SA+SB)]2−4J2SASB}5 4 ×1/radicalbig (1−δ2)SASBδω, (28) where δω≡ω−ωc1. A similar feature is also found in the spinon case, in which the divergence is related to the sin-gularity of the density of states [ 52]. On the other hand, the asymptotic form around ω=ω c2reads as Re[σ(0;ω,−ω)]≈(gA−gB)2J2δSASB[B++J(SA+SB)] 8πJ{[B++J(SA+SB)]2−4J2δ2SASB}5 4 ×1/radicalbig (1−δ2)SASB|δω|. (29) The sign of the conductivity is the opposite of that in the lower-frequency regime. This is in contrast to the spinon case[52], in which the sign of the nonlinear conductivity remains the same for all frequencies ω∈[ω c1,ωc2].D. Relaxation-time dependence We next study the damping (relaxation-time) dependence of the spin current. Different mechanisms of the photogal-vanic effect are classified by their relaxation-time dependence[2,3,57,58]: it is called shift current when σ(0;ω,−ω)∝τ 0 while is injection current when σ(0;ω,−ω)∝τ. In bosonic systems, a slight difference appears in the momentum de-pendence of the single-particle relaxation time [ 61]; it is inversely proportional to the momentum for the Goldstonemodes. Therefore, we assume the momentum dependence ofdamping term as τ k=1/(α0εβ(k))so that the momentum dependence is consistent with the field-theoretic requirement(α 0is the damping factor). Physically, the assumed form of τk corresponds to the phenomenological Gilbert damping. We substitute τk=1/(α0εβ(k))in Eq. ( 10) in order to estimate the relaxation-time dependence of the spin-current conductivity. Figure 2(b) shows the α 0dependence ofσ(0;ω,−ω). Our numerical result shows σ(0;ω,−ω)i s insensitive to the damping. A slight difference, however,appears in the high-frequency region, where the smearing dueto the damping is more distinct than that in the low-frequency region. This behavior is related to the momentum dependence ofτ k, which is inversely proportional to the energy of the magnon. The insensitivity shows the spin current is a shift-current type photoinduced current [ 2]; this is a similar feature to the spinon case [ 52]. IV . THREE-DIMENSIONAL MAGNETIC INSULATORS In this section, we consider a three-dimensional (3D) magnet which consists of coupled spin chains H1Dwith a non-negligible interchain interaction [see Fig. 1(b)]. We par- ticularly focus on the limit in which ωis close to the band gap of two-magnon excitations. The procedure of the calculationis the same as the 1D case in the previous section. The staticpart of the Hamiltonian reads as H (3D) 0≡/summationdisplay rJ(1+δ)SA(r)·SB(r) +J(1−δ)SA(r+ˆx)·SB(r) −(D+Ds)/bracketleftbig Sz A(r)/bracketrightbig2−(D−Ds)/bracketleftbig Sz B(r)/bracketrightbig2 −J⊥[SA(r)·SA(r+ˆy)+SA(r)·SA(r+ˆz) +SB(r)·SB(r+ˆy)+SB(r)·SB(r+ˆz)] −B/bracketleftbig gASz A(r)+gBSz B(r)/bracketrightbig . (30) The spin chains are parallel to the xdirection, while the yandz directions are perpendicular to the chains. The ferromagneticcoupling J ⊥>0 denotes the strength of the interchain ex- change interaction. We study this model within the linear spin-wave approximation using Holstein-Primakov transformationin Sec. III B . Focusing on the lower edge of the magnon dispersion, we first expand the matrix elements h a kof the magnon Hamiltonian [see Eq. ( 16)] up to second order in k: h0 k/similarequalB++J(SA+SB)+J⊥(SA+SB) 2/parenleftbig k2 y+k2 z/parenrightbig ,(31a) hx k/similarequalJ√ SASB/parenleftbigg 2−1 4k2 x/parenrightbigg , (31b) 224411-6THEORY FOR SHIFT CURRENT OF BOSONS: … PHYSICAL REVIEW B 100, 224411 (2019) FIG. 3. Schematic figure of the magnon dispersion for the 3D model in Eq. ( 30). We set ky=0. The blue and orange planes are the dispersions of two magnon branches and the green transparentplane is that of two-magnon excitation. The plot is for J=1,J ⊥=1, δ=1/4,SA=1,SB=1,h+=1/100, and h−=1/10. hy k/similarequal−J√ SASBδkx, (31c) hz k/similarequalB−+J(SA−SB)+J⊥(SA−SB) 2/parenleftbig k2 y+k2 z/parenrightbig .(31d) We note that the magnon dispersions depend on both in- trachain and interchain wave numbers differently from the1D case. The dispersion around the /Gamma1point k=0is shown in Fig. 3. Using the momentum gradient of the low-energy Hamiltonian with h0,x,y,z k, we can define the spin-current oper- ator; this approximation is essentially equivalent to expandingthe lattice spin-current operator in Eq. ( 24) up to the linear order in k: J z z=J√ SASB/summationdisplay ksinh(2 /Theta1k)/parenleftbiggkx 2cos/Phi1k+δsin/Phi1k/parenrightbigg ×(α† kαk+β−kβ† −k) +/braceleftbigg cosh(2 /Theta1k)/parenleftbigg cos/Phi1kkx 2−δsin/Phi1k/parenrightbigg +i/parenleftbiggkx 2sin/Phi1k+δcos/Phi1k/parenrightbigg/bracerightbigg α† kβ† −k+H.c. (32) These equations correspond to the k·pexpansion of the lattice model. Therefore, it should be a good approximationfor the lattice model when ωis close to the gap for two- magnon excitations. The spin-current conductivity is calculated using the for- mula of Eq. ( 10). A calculation similar to the 1D model considered in Sec. IIIgives Re[σ(0;ω,−ω)] =−J 2δSASB(gA−gB)2 (4π)22J⊥ω2(SA+SB)/parenleftbig 8kx−k3 x/parenrightbig kx=KX, (33)where KX=/bracketleftbigg 8(1−δ2) −4/radicalBigg [B++J(SA+SB)]2−(ω/2)2 J2SASB+δ2(δ2−2)/bracketrightbigg1 2 . (34) When ωis close to the lower edge, i.e., ω∼ωc1≡2/radicalbig [B++J(SA+SB)]2−4J2SASB, (35) KXbecomes KX≈/radicalBigg 2/radicalbig [B++J(SA+SB)]2−4J2SASBδω (1−δ2)J2SASB, (36) where δω=ω−ωc1. Therefore, the asymptotic form of Re[σABB(0;ω,−ω)] is Re[σ(0;ω,−ω)]≈−(gA−gB)2δ√SASB 16π2√ 1−δ2(SA+SB) ×J√ δω J⊥{[B++J(SA+SB)]2−4J2SASB}3 4. (37) Unlike the 1D case, in which the conductivity diverges at the band edge ωc1, the 3D result in Eq. ( 37) decreases proportionally to√ δωwhen approaching ωc1. The result is plotted in the inset of Fig. 2(a) with the results for the 1D limit. This difference is a consequence of the difference inthe density of states: it diverges in the 1D model while it isproportional to√ δωin the present 3D case. The approximation we used in this section is accurate when ωis close to the magnon gap at the /Gamma1point in the Brillouin zone. In our model, the band bottom for the two-magnonexcitations are at the /Gamma1point, and the bandwidth of two- magnon excitation along the xandydirections is in the order ofJ ⊥and that for zdirection is in the order of J. Therefore, our approximation is accurate when δω/lessmuchJ,J⊥. This condition is manifested in J⊥in the denominator of Eq. ( 37), which implies the divergence of Re[ σ(0;ω,−ω)] at J⊥→0. When J⊥is very small, we expect Re[ σ(0;ω,−ω)] to behave like that of the 1D case. On the other hand, Re[ σ(0;ω,−ω)] looks like Eq. ( 37) when J⊥is sufficiently large, e.g., when J⊥∼J. Therefore, the 1D result and the result in this section correspond to the two limits of the 3D magnet. V . EXPERIMENTAL OBSERV ATION In this section, we discuss experimental methods for de- tecting signatures of a directional spin current in our mecha-nism. A. Setup We here discuss experimental setups for the observation of the spin current generated by linearly polarized light. Themechanism studied here produces a directional flow of thespin current, which is a distinct feature from the spin pumping 224411-7HIROAKI ISHIZUKA AND MASAHIRO SATO PHYSICAL REVIEW B 100, 224411 (2019) FIG. 4. Schematic figure of the experimental setups for measur- ing photoinduced spin current: all-optical setup [(a) and (b)] and two-terminal setup (c). (a) The all-optical setup irradiates the isolated magnet using GHz /THz light. The optically induced spin current accumulates the angular momentum at the end of the magnet which is depicted by the clouds; it produces the asymmetric distribution of the angular momentum in the magnet. (b) A similar observationby attaching a thin layer of a soft ferromagnet at the two ends. The photogalvanic spin current is injected to or absorbed from the soft ferromagnets. (c) The two-terminal setup observes the directionalflow of spin current using the inverse spin Hall effect. The optically induced spin current flows along a certain direction of the system. Therefore, inverse spin Hall voltage of the two leads has the samesign. These setups are different from that of spin pumping of (d), in which a transverse ac field is applied to the magnet and the spin current is diffusively expanded. [23,24]. Therefore, the observation of the directional flow should provide an evidence for our mechanism. In addition,our theory in the previous sections corresponds to the case inwhich the ac magnetic field is parallel to the magnetization.Therefore, the angle dependence of the photocurrent providesinformation on the origin of the spin current. We discusstwo different mechanisms: first one is an all-optical setupusing Kerr rotation or Faraday effect, and the second is atwo-terminal setup using inverse spin Hall effect. Observation of the spatial distribution of angular momen- tum in the open-circuit setup provides a direct evidence forthe optically generated spin current [see Fig. 4(a)]. In an isolated magnet, the spin current produced by a GHz /THz light flows along a direction defined by the magnetic orderand the crystal symmetry. Therefore, if the system becomesclose enough to a laser-driven nonequilibrium steady state, theangular momentum accumulates at the two ends in an open-circuit setup in Fig. 4(a); positive angular momentum on one end and negative on the other end. The angular-momentumdistribution is antisymmetric along the direction of the spincurrent. This distribution is strikingly different from the spin-pumping case in which the distribution is symmetric and itsdifference from the equilibrium state is larger at the focal areaof the laser than at the ends.An all-optical setup using Kerr rotation or Faraday effect would be a useful setup for the observation of such a spatialdistribution. Measurement of magnetic moments and its spa-tial distribution using the optical probe is a commonly usedtechnique for observing the spin current. For instance, thismethod is used to observe the spin Hall effect [ 62]. Similarly, observing the magnetization of soft magnet layers attachedto the two ends is another possible setup for the experiment[Fig. 4(b)]. The observation of spin current in a two-terminal setup in Fig. 4(c) also enables us to see the directional flow of spin current and to distinguish it from the spin-pumpingeffect. This setup consists of a noncentrosymmetric magneticinsulator which is sandwiched between two metallic leads;the two leads detect spin current via inverse spin Hall effect[63–65]. In the photogalvanic mechanism, the spin current in the two leads flows toward the same direction. Therefore, theinverse spin Hall voltage of the two leads has the same sign.In contrast, in the spin pumping, the spin current diffusivelyflows outward from the magnet; the inverse spin Hall voltageis positive on one side and negative on the other. Therefore,the relative sign of the inverse spin Hall voltage of the twoleads can make a distinction between the spin pump and ourmechanism. Finally, we shortly comment on heating effect of applied electromagnetic waves. When we try to detect the photo-galvanic spin current with the above setups, spin pumpingmight also occur due to the heating effect of the appliedlaser. For such a case, extracting the asymmetric part of theangular-momentum distribution or inverse spin Hall voltageis important to detect an evidence for our mechanism. B. Required intensity of ac field We next estimate the required ac electromagnetic field for generating an observable spin current. We here assumea spin current of J s=10−16J/cm2is observable. This es- timate is based on a Boltzmann theory calculation for spinSeebeck effect in a ferromagnet [ 52,74]. The details of the estimate are briefly explained in Appendix B.W eu s et h e following parameters as a typical value for 1D insulating mag-nets: J=100k BJ,δ=0.1,SA=SB=1,gA−gB=0.1μB J/T,B+=10kBJ, and a the light with a frequency which is ¯hδω=6π¯h×1011Hz above the band gap. Here, ¯ his the Planck constant. With these parameters, the conductivityfor the 1D AFM /FRM chain is Re[ σ(0;ω,−ω)]∼10 −14 J/(cm2T2). Therefore, the required magnitude of oscillating magnetic field to produce a spin current of Js=10−16J/cm2 isB∼/radicalBig Js |Re[σ(0;ω,−ω)]|∼0.1 T. This corresponds to the elec- tric field E=cB∼104–105V/cm under the assumption ofc=108m/s which is a typical value of speed of light in insulators. Similar estimate for the 3D magnetwith J=100k BJ,J⊥=10kBJ,δ=0.1,SA=SB=1,gA− gB=0.1μBJ/T,h+=10kBJ, andω=2π×1012Hz gives Re[σ(0;ω,−ω)]∼10−11J/(cm2T2) and E=cB∼105–106 V/cm. The difference in the magnitude for 1D and 3D cases is ascribed to the difference of the density of states; the 1Dsystem has a larger density of states due to the divergenceat the band edge. Our estimate predicts that the photogal- 224411-8THEORY FOR SHIFT CURRENT OF BOSONS: … PHYSICAL REVIEW B 100, 224411 (2019) vanic spin current is experimentally observable by using a moderate-intensity GHz /THz light. C. Candidate material We believe the photogalvanic spin current should be seen generically in noncentrosymmetric magnets. In a recent work[52], the authors find three kinds of spin-light couplings induce the spin current in a spin chain, and this work presentsphotogalvanic spin current in ordered magnets. These resultsimply the generation of photogalvanic spin current is a uni-versal phenomenon in noncentrosymmetric magnetic insu-lators. Various kinds of such noncentrosymmetric materialshave been synthesized or discovered [ 66]: a magnetoelectric material Cr 2O3[67], ferrimagnetic diamond chains [ 68–70], multiferroic materials [ 71,72], and a polar ladder magnet BaFe 2Se3[73]. As we showed in the previous sections, a large density of states for the magnon excitations is advantageous for alarge dc spin current. Therefore, quasi-1D noncentrosymmet-ric magnets such as the ferrimagnetic diamond chain andBaFe 2Se3would be promising candidates for studying the spin current. VI. SUMMARY AND DISCUSSION To summarize, we studied the spin-current generation through the shift-current mechanism inferrimagnetic /antiferromagnetic insulators. Our theory uses a nonlinear response theory, which is a generalizationof the linear response theory. Based on this method,we find that the linearly polarized light produces themagnon current in noncentrosymmetric magnets withantiferromagnetic /ferrimagnetic order. The photogalvanic spin current appears even at the zero temperature where nomagnon excitation exists; the current is related to excitingtwo magnons from the ground state, not to the opticaltransition of existing (thermally excited) magnons. Therelaxation-time dependence of the spin current indicates thatour photogalvanic effect is a “shift current,” i.e., the nonlinearconductivity is insensitive to the damping. Our theory clearlyshows that the shift-current mechanism, which is well knownin electron (fermion) systems, is also relevant to systemswith bosonic excitations, whose ground state is a vacuum ofbosons (zero-boson state). Our result implies the zero-point quantum fluctuation is a key for the shift-current type photocurrent. In the spinon spincurrent [ 52], the optical transition of a fermionic excitation plays a crucial role for the photocurrent. In contrast to thesecases, the ground state of the ordered magnets is the zero-magnon state. Therefore, there is no optical transition of theexisting magnons. Despite the crucial difference, we find afinite photogalvanic spin current at the zero temperature. Themagnon photocurrent we found is ascribed to the opticaltransition of the “condensed” Holstein-Primakov bosons. Inthe antiferromagnets /ferrimagnets, the ground state is a con- densate of Holstein-Primakov bosons, which is technicallyrepresented by the Bogoliubov transformation. The opticaltransition of the condensed Holstein-Primakov bosons allowsgeneration of the shift-current type photocurrent even at the Photon− Unit cellABAB−ℏ +ℏ FIG. 5. Schematic picture of the two-magnon excitation process. The laser creates a pair of magnons with up and down spins, respectively, β−kandαk. Blue and yellow waves, respectively, denote typical density profiles of photoexcited magnons β−kandαk.T h e center of mass of the magnons deviates from the center of the unit cell owing to the noncentrosymmetry of the system. zero temperature. On the other hand, we find that the non- linear conductivity is zero at T=0 for the ferromagnetic version of the model considered here. From this viewpoint,the two-magnon creation is similar to the particle-hole paircreation in semiconductors; the optical transition of fermionsfrom the valence band to the conduction band is equiva-lent to the pair creation (Fig. 5). As the condensation of the Holstein-Primakov bosons is a manifestation of zero-point fluctuation, the zero-point fluctuation is the essence forthe shift-current type photogalvanic effects in the magneticinsulators. Experimental setups for experimental observation of this phenomenon include the two-terminal inverse-spin Hallmeasurements and magneto-optical Kerr effect. Our estimatesuggests that an GHz /THz light of E∼10 4–106V/cm is sufficient for experimental observation; the response is rela-tively larger by tuning the frequency to the two-magnon ex-citations with larger density of states. The estimation impliesthis phenomenon is observable within the currently availableexperimental techniques. ACKNOWLEDGMENTS We thank R. Matsunaga and Y . Takahashi for fruitful dis- cussions. We also thank W. Murata for providing Fig. 4.H . I . was supported by JSPS KAKENHI Grants No. JP18H04222,No. JP19K14649, and No. JP18H03676, and CREST JSTGrant No. JPMJCR16F1. M.S. was supported by JSPS KAK-ENHI (Grant No. JP17K05513), and Grant-in-Aid for Sci-entific Research on Innovative Area “Nano Spin ConversionScience” (Grant No. 17H05174) and “Physical Properties ofQuantum Liquid Crystals” (Grant No. 19H05825). 224411-9HIROAKI ISHIZUKA AND MASAHIRO SATO PHYSICAL REVIEW B 100, 224411 (2019) APPENDIX A: DERIV ATION OF KRAUT–VON BALTZ FORMULA FOR BOSONS Here, we shortly explain the derivation of the nonlinear conductivity in two-band boson systems. We used the formulain Eq. ( A10) for the analytic calculations and Eq. ( A5)f o r numerical results with a finite Gilbert damping. We calculate the nonlinear response coefficients using a formalism similar to the linear response theory. We as-sume a system with a time-dependent perturbation H /prime= −/summationtext μˆBμFμ(t), where ˆBμis an operator and Fμ(t)i sa time-dependent field; the Hamiltonian reads as H=H0+H/prime. The expectation value of an observable ˆAreads as /angbracketleftˆA/angbracketright(t)= Tr[ ˆρ(t)ˆA]/Z,where ρ(t) is the density matrix at time tand Z≡Trρ(t). By expanding ρ(t) up to the second order in Fμ(t), the Fourier transform of /angbracketleftA/angbracketright(t),/angbracketleftA/angbracketright(/Omega1), reads as /angbracketleftA/angbracketright(/Omega1)=/summationdisplay μ,ν/integraldisplay dωσμν(/Omega1;ω,/Omega1−ω)Fμ(ω)Fν(/Omega1−ω), (A1) with the nonlinear conductivity σμν(/Omega1;ω,/Omega1−ω) =1 2π/summationdisplay n,m,l(ρn−ρm)(Bμ)nm ω−Em+En−i/(2τmn) ×/bracketleftbigg(Bν)mlAln /Omega1+En−El−i/(2τmn) −Aml(Bν)ln /Omega1+El−Em−i/(2τmn)/bracketrightbigg . (A2)Here, Enis the eigenenergy of the many-body eigenstate n,τmn is the relaxation time, and Onm(O=A,Bμ,Bν)i st h em a t r i x element of ˆOin the eigenstate basis of H0. We here consider a periodic free-boson system in which all matrices A,Bμ, and Bνhave the following form: ˆO=/summationdisplay k(α† kβ−k)Ok/parenleftbiggαk β† −k/parenrightbigg , (A3) =/summationdisplay k(α† kβ−k)/parenleftbigg (Ok)αα (Ok)αβ (Ok)βα (Ok)ββ/parenrightbigg/parenleftbiggαk β† −k/parenrightbigg ,(A4) where αk(α† k) andβk(β† k) are the annihilation (creation) op- erators of the boson eigenstates with momentum k, and Ok= Ak,Bμ k,Bν k. The theory for spin-wave excitations of many antiferromagnetic models with a Néel-type order reduces tothe above form by using Holstein-Primakov and Bogoliubovtransformations. For the two-band system, we can express Eq. ( A2)u s - ing single-particle eigenstates. We note that A,B μ, and Bν for the two-band system above do not conserve the par- ticle number. However, all operators are quadratic in theannihilation /creation operators and consist of only four terms: α † kαk,β−kβ† −k,β−kαk, andα† kβ† −k. Therefore, only few terms out of the possible Wick decomposition remain nonzero,similar to that of the systems with conserved particle number.Using these features, we find σ(/Omega1;ω,/Omega1−ω)=1 2π/summationdisplay k,ai=α,βsgn(a3)/bracketleftbig ˜ρk,a1sgn(a2)−sgn(a1)˜ρk,a2/bracketrightbig/parenleftbig Bμ k/parenrightbig a1a2 ω−˜εa2(k)+˜εa1(k)−i/(2τk) ×/bracketleftBigg /parenleftbig Bν k/parenrightbig a2a3(Ak)a3a1 /Omega1+˜εa1(k)−˜εa3(k)−i/(2τk)−(Ak)a2a3/parenleftbig Bν k/parenrightbig a3a1 /Omega1+˜εa3(k)−˜εa2(k)−i/(2τk)/bracketrightBigg . (A5) Here, sgn(a)=/braceleftbigg 1( a=α), −1( a=β),, (A6) ˜εa(k)=sgn(a)εa(k), (A7) ˜ρk,a=/braceleftbigg/angbracketleftα† kαk/angbracketright0 (a=α), /angbracketleftβ−kβ† −k/angbracketright0(a=β),(A8) and we assumed the relaxation time only depends on k. It is worth noting that the conductivity remains finite at T=0 despite there are no excitations. Technically, this is a consequence of ˜ ρk,β, which is 1 at T=0. Physically, this is because the pair creation /annihilation processes contribute to the spin current even at T=0. We here focus on the T=0 limit. In this limit, ˜ ρk,α=0 and ˜ρk,β=1. Using these results, we obtain σ(0;ω,−ω)=−1 π/summationdisplay k,ai=α,β/bracketleftbigg(1+i2τω)|Bβα|2(Aαα+Aββ) (ω−i/2τk)2−[εα(k)+εβ(k)]2/bracketrightbigg +1 2π/summationdisplay k,ai=α,β/braceleftbigg(Bk)βα(Ak)αβ[(Bk)ββ+(Bk)αα] [ω−i/2τk−εα(k)−εβ(k)][εα(k)+εβ(k)+i/2τk] +(Bk)αβ(Ak)βα[(Bk)ββ+(Bk)αα] [ω−i/2τk+εα(k)+εβ(k)][εα(k)+εβ(k)−i/2τk]/bracerightbigg . (A9) 224411-10THEORY FOR SHIFT CURRENT OF BOSONS: … PHYSICAL REVIEW B 100, 224411 (2019) As we discussed in the main text, certain symmetries restrict the first term to be zero; this is the case for the models we consider in the main text. Assuming the first term vanishes, we find Re[σ(0;ω,−ω)]=−1 πRe/braceleftbigg(Bk)βα(Ak)αβ[(Bk)ββ+(Bk)αα] ω2−[εα(k)+εβ(k)+i/2τ]2/bracerightbigg . (A10) We used this formula for the calculation of nonlinear conductivity in the main text. APPENDIX B: BOLTZMANN THEORY FOR SPIN SEEBECK EFFECT The magnitude of spin current Js=10−16J/cm2is the esti- mate for the spinon spin current produced by the spin Seebeckeffect in a recent experiment [ 74]. We here summarize the method and result discussed in the Supplemental Material ofa recent work [ 52]. The spin current is estimated from the Seebeck effect of magnons whose dispersion is given by ε(k)=JSk 2+2DS+h (B1) near the /Gamma1point of k=0. This magnon dispersion corre- sponds to that of a ferromagnetic Heisenberg model with ex-change interaction J, uniaxial anisotropy D, and the magnetic field hparallel to the anisotropy. The current is calculated using the semiclassical Boltzmann theory, in which the currentreads as J s(r)=¯h/integraldisplaydk (2π)3vzfk(r). (B2) Here, fk(r) is the density of magnons with momentum kat position randvz≡∂kzε(k) is the group velocity of magnons. fk(r) is calculated from the Boltzmann equation with temper- ature gradient vk·∇rfk(r)=−fk(r)−f(0) k(r) τk, (B3)where f(0) k(r) is the density at the equilibrium. Here, the relaxation-time approximation is used to simplify the calcu-lation of collision integral on the right-hand side. The spincurrent induced by the spin Seebeck effect is estimated bysubstituting the solution of f k(r)i nE q .( B3) into the current formula in Eq. ( B2) In the Boltzmann theory, the spin current by the spin Seebeck effect reads as Js(r)∼3(6π2)2 3J2 HS2 2αkBaT(r)/Delta1T T(r)F/parenleftbiggJHSa2/Lambda12 2kBT(r),2DS+h 2kBT(r)/parenrightbigg ,(B4) where /Lambda1=(6π2)1/3/ais the cutoff for magnon dispersion and F(a,b)=/integraldisplay1 0x4csch2(ax2+b)dx. 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PhysRevB.76.224430.pdf

Microwave photovoltage and photoresistance effects in ferromagnetic microstrips N. Mecking,1,2,*Y . S. Gui,1and C.-M. Hu† 1Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2 2Institut für angewandte Physik und Zentrum für Mikrostrukturforschung, Universität Hamburg, Jungiusstraße 11, 20355 Hamburg, Germany /H20849Received 23 August 2007; revised manuscript received 4 November 2007; published 27 December 2007 /H20850 We investigate the dc electric response induced by ferromagnetic resonance in ferromagnetic Permalloy /H20849Ni80Fe20/H20850microstrips. The resulting magnetization precession alters the angle of the magnetization with respect to both dc and rf current. Consequently the time averaged anisotropic magnetoresistance /H20849AMR /H20850 changes /H20849photoresistance /H20850. At the same time the time-dependent AMR oscillation rectifies a part of the rf current and induces a dc voltage /H20849photovoltage /H20850. A phenomenological approach to magnetoresistance is used to describe the distinct characteristics of the photoresistance and photovoltage with a consistent formalism, whichis found in excellent agreement with experiments performed on in-plane magnetized ferromagnetic microstrips.Application of the microwave photovoltage effect for rf magnetic field sensing is discussed. DOI: 10.1103/PhysRevB.76.224430 PACS number /H20849s/H20850: 76.50. /H11001g, 75.30.Gw, 07.57.Kp I. INTRODUCTION The fact that macroscopic mutual actions exist between electricity and magnetism has been known for centuries asdescribed in many textbooks of electromagnetism. 1Now, this subject is transforming onto the microscopic level, as re-vealed in various spin-charge coupling effects studied in thenew discipline of spintronics. Among them, striking phenom-ena are the dc charge transport effects induced by spin pre-cession in ferromagnetic metals, which feature both aca-demic interest and technical significance. 2,3Experiments have been performed independently by a number of groupson devices with different configurations. 4–16Most works were motivated by the study of spin torque,17,18which de- scribes the impact of a spin-polarized charge current on themagnetic moment. In this context, Tulapurkar et al. made the first spin-torque diode, 4and Sankey et al. detected the spin- torque-driven ferromagnetic resonance /H20849FMR /H20850electrically.5 Both measured the vertical transport across nanostructured magnetic multilayers. Along a parallel path, a number ofworks 19–21have been devoted to study the effect of spin pumping. One of the interesting predictions is that injectionof a spin current from a moving magnetization into a normalmetal induces a dc voltage across the interface. To detectsuch a dc effect induced by spin pumping, 20experiments have been performed by measuring lateral transport in hybriddevices under rf excitation. 6–8 From a quite different perspective, Gui et al. set out to explore the general impacts of the high frequency responseon the dc transport in ferromagnetic metals, 9based on the consideration that similar links in semiconductors have beenextensively applied for electrical detection of both spin andcharge excitations. 22Guiet al. detected, subsequently, pho- toresistance induced by bolometric effect,9as well as photocurrent,10photovoltage,11and photoresistance12caused by the spin-rectification effect. A spin dynamo10was thereby realized for generating dc current via the spin precession, andthe device was applied for a comprehensive electrical studyof the characteristics of quantized spin excitations in micro-structured ferromagnets. 11The spin-rectification effect wasindependently investigated by both Costache et al.13and Yamaguchi et al.14and seems to be also responsible for the dc effects detected earlier by Oh et al.15A method for dis- tinguishing the photoresistance induced by either spin pre-cession or bolometric effect was recently established, 12 which is based on the nice work performed by Goennenweinet al. , 16who determined the response time of the bolometric effect in ferromagnetic metals. While most of these studies, understandably, tend to em- phasize the different nature of dc effects investigated in dif-ferent devices, it is perhaps more intriguing to ask the ques-tions of whether the seemingly diverse but obviously relatedphenomena could be described by a unified phenomenologi-cal formalism and whether they might arise from a similarmicroscopic origin. From a historical perspective, these twoquestions reflect exactly the spirit of two classic papers 23,24 published by Juretscheke and Silsbee et al. , respectively, which have been often ignored but have shed light on the dceffects of spin dynamics in ferromagnets. In the approachdeveloped by Juretscheke, photovoltage induced by FMR inferromagnetic films was described based on a phenomeno-logical depiction of magnetoresistive effects. 23While in the microscopic model developed by Silsbee et al. based on the combination of Bloch and diffusion equations, a coherentpicture was established for the spin transport across the in-terface between ferromagnets and normal conductors underrf excitation. 24 The goal of this paper is to provide a consistent view for describing photocurrent, photovoltage, and photoresistanceof ferromagnets based on a phenomenological approach tomagnetoresistance. We compare the theoretical results withexperiments performed on ferromagnetic microstrips in de-tail. The paper is organized in the following way: In Sec. II,a theoretical description of the photocurrent, photovoltage,and photoresistance in thin ferromagnetic films under FMRexcitation is presented. Sections II A–II D establish the for-malism for the microwave photovoltage /H20849PV /H20850and photore- sistance /H20849PR /H20850based on the phenomenological approach to magnetoresistance. These arise from the nonlinear couplingof microwave spin excitations /H20849resulting in magnetization M precession /H20850with charge currents by means of the anisotropicPHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 1098-0121/2007/76 /H2084922/H20850/224430 /H2084914/H20850 ©2007 The American Physical Society 224430-1magnetoresistance /H20849AMR /H20850. Section II E compares our model with the phenomenological approach developed by Ju-retscheke. Section II F provides a discussion concerning themicrowave photovoltage and photoresistance based on othermagnetoresistance effects /H20851like anomalous Hall effect /H20849AHE /H20850, giant magnetoresistance /H20849GMR /H20850, and tunneling mag- netoresistance /H20849TMR /H20850/H20852. Experimental results on microwave photovoltage and photoresistance measured in ferromagnetic microstrips arepresented in Secs. III and IV , respectively. We focus in par-ticular on their characteristic different line shapes, which canbe well explained by our model. In Sec. V conclusions andan outlook are given. II. MICROWA VE PHOTOVOLTAGE AND PHOTORESISTANCE BASED ON PHENOMENOLOGICAL AMR A. AMR coupling of spin and charge The AMR coupling of spin and charge in ferromagnetic films results in microwave photovoltage and photoresistance.The photovoltage can be understood regarding Ohms law/H20851current I/H20849t/H20850and voltage U/H20849t/H20850/H20852 U/H20849t/H20850=R/H20849t/H20850·I/H20849t/H20850. /H208491/H20850 We consider a time-dependent resistance R/H20849t/H20850=R 0 +R1cos /H20849/H9275t−/H9274/H20850which oscillates at the microwave frequency /H9275=2/H9266fdue to the AMR oscillation arising from magnetiza- tion precession. /H9274is the oscillations phase shift with respect to the phase of the rf current I/H20849t/H20850. For the sake of generality /H9274will be kept as a parameter in this work and will be dis- cussed in detail in Sec. III C. I/H20849t/H20850takes the form I/H20849t/H20850 =I1cos /H20849/H9275t/H20850and is induced by the microwaves. It follows that U/H20849t/H20850consists of time-dependent terms with the frequency /H9275, 2/H9275and a constant term /H20849time independent /H20850which corre- sponds to the time average voltage and is equal to the pho-tovoltage: U MW=/H20855R1I1cos /H20849/H9275t−/H9274/H20850cos /H20849/H9275t/H20850/H20856=/H20849R1I1cos/H9274/H20850/2 /H20849/H20855 /H20856denotes time-averaging /H20850. A demonstrative picture of the microwave photovoltage mechanism can be seen in Fig. 1. The second effect we investigate which is also based on AMR spin-charge coupling is the microwave photoresistance/H9004R MW. This has been reported recently13with the equilib-rium magnetization M0of a ferromagnetic stripe aligned to a dc current I0. Microwave induced precession then misaligns the dynamic magnetization Mwith respect to I0and thus makes the AMR drop measurably. In this work, we presentresults which also show that if M 0lies perpendicular to I0 the opposite effect takes place: Microwave induced preces- sion causes Mto leave its perpendicular position which in- creases the AMR /H20849see Fig. 2/H20850. After this qualitative introduction we want to go ahead with a quantitative description of the AMR induced micro-wave photovoltage and photoresistance. Therefore, we definean orthogonal coordinate system /H20849x,y,z/H20850/H20849see Fig. 3/H20850. The y axis lies normal to the film plane and the zaxis is aligned with the magnetic field Hand hence with the magnetization Mwhich is always aligned with Hin our measurements because of the sample being always magnetized to satura-tion. Geometrically our samples are thin films patterned to stripe shape, so that d/lessmuchw/lessmuchl, where d,w, and lare the thickness, width, and length of the sample. We apply Hal- ways in the ferromagnetic film plane. For calculations basedon the stripes geometry the coordinates x /H11032andz/H11032are defined. These lie in the film plane. x/H11032is perpendicular and z/H11032parallel to the stripe. The following coordinate transformationapplies: /H20849x,y,z/H20850=/H20851x /H11032cos /H20849/H92510/H20850−z/H11032sin /H20849/H92510/H20850,y,z/H11032cos /H20849/H92510/H20850 +x/H11032sin /H20849/H92510/H20850/H20852where /H92510is the angle between Hand the stripe. FIG. 1. /H20849Color online /H20850Mechanism of the AMR-induced micro- wave photovoltage: Mprecesses /H20849period P/H20850in phase with the rf current I./H20849a/H20850Mlying almost perpendicular to Iresults in low AMR. /H20849b/H20850Mlying almost parallel to Iresults in high AMR. The time average voltage Ubecomes nonzero. FIG. 2. /H20849Color online /H20850Mechanism of the AMR-induced photo- resistance. /H20849a/H20850Without microwaves /H20849MW /H20850Mlies perpendicular to the dc current Iand the AMR is minimal /H20849b/H20850With microwaves M precesses and is not perpendicular to Ianymore. Consequently the AMR increases /H20849higher voltage drop U/H20850. FIG. 3. /H20849Color online /H20850/H20849x,y,z/H20850and /H20849x/H11032,y,z/H11032/H20850coordinate systems in front of a layout of our Permalloy film stripe /H20849200 /H110032400/H9262m2/H20850with two contacts and six side junctions.MECKING, GUI, AND HU PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-2For the microwave photovoltage and photoresistance the longitudinal resistance R/H20849t/H20850=R0+RAcos2/H9258/H20849t/H20850of the film stripe matters. It consists of the minimal longitudinal resis- tance R0and the additional resistance RAcos2/H9258/H20849t/H20850from AMR. /H9258/H20849t/H20850is the angle between the z/H11032-axis /H20849parallel to the stripe /H20850and M.Mmoves on a sphere with the radius M0, which is the saturation magnetization of our sample. /H9258/H20849t/H20850can be decomposed into the angle /H9251/H20849t/H20850in the ferromagnetic film plane and the out-of-plane angle /H9252/H20849t/H20850/H20849see Fig. 4/H20850. Conse- quently, cos/H9258/H20849t/H20850= cos/H9251/H20849t/H20850cos/H9252/H20849t/H20850. /H208492/H20850 Precession of the magnetization then yields oscillation of /H9251/H20849t/H20850,/H9252/H20849t/H20850, and/H9258/H20849t/H20850. In our geometry the equilibrium magne- tization M0encloses the in-plane angle /H92510with the stripe. Hence in time average /H20855/H9252/H20849t/H20850/H20856=0 and /H20855/H9251/H20849t/H20850/H20856=/H92510. In general the magnetization precession is elliptical. Its principle axis lie along the xandyaxis and correspond to the amplitudes /H92511and/H92521of the in- and out-of-plane angles /H92511tand/H92521tof the rf magnetization: /H9251/H20849t/H20850=/H92510+/H92511t/H20849t/H20850=/H92510+/H92511cos /H20849/H9275t−/H9274/H20850and /H9252/H20849t/H20850=/H92521t/H20849t/H20850=−/H92521sin /H20849/H9275t−/H9274/H20850/H20851see Fig. 4/H20852. Using Eq. /H208492/H20850we approximate cos2/H9258/H20849t/H20850to second order in /H92511tand/H92521t: cos2/H9258/H20849t/H20850/H11015 /H20841cos2/H9258/H20841/H92511t=/H92521t=0+/H92511t/H20879dcos2/H9258 d/H92511t/H20879 /H92511t=/H92521t=0+0 +/H92511t2 2/H20879d2cos2/H9258 d/H92511t2/H20879 /H92511t=/H92521t=0+/H92521t2 2/H20879d2cos2/H9258 d/H92521t2/H20879 /H92511t=/H92521t=0. /H208493/H20850 The first order in /H92521tvanishes because it is proportional to /H20849sin/H9252/H20850/H20841/H92521=0=0. It follows that cos2/H9258/H20849t/H20850/H11015cos2/H92510−/H92511sin 2/H92510cos /H20849/H9275t−/H9274/H20850 −/H925112cos 2/H92510cos2/H20849/H9275t−/H9274/H20850 −/H925212cos2/H92510sin2/H20849/H9275t−/H9274/H20850. /H208494/H20850 This equation is now used to calculate the longitudinal stripe voltage. To consider the general case an externallyapplied dc current I 0and a microwave induced rf current I1 are included in I/H20849t/H20850=I0+I1cos /H20849/H9275t/H20850. It follows from Eq. /H208491/H20850 thatU/H20849t/H20850=/H20851R0+RAcos2/H9258/H20849t/H20850/H20852/H20851I0+I1cos /H20849/H9275t/H20850/H20852. /H208495/H20850 Consequently U/H20849t/H20850can be written as U/H20849t/H20850=U0 +U1cos /H20849/H9275t−/H92741/H20850+U2cos /H208492/H9275t−/H92742/H20850+U3cos /H208493/H9275t−/H92743/H20850. For the photovoltage and photoresistance only the constant term U0, which is equivalent to the time average voltage /H20855U/H20849t/H20850/H20856, matters. Combining Eqs. /H208494/H20850and /H208495/H20850,w efi n d U0=I0/H20849R0+RAcos2/H92510/H20850−I1RA/H92511sin 2/H92510cos /H20849/H9274/H20850/2 −I0/H20849/H925112cos 2/H92510+/H925212cos2/H92510/H20850RA/2. /H208496/H20850 Note that /H20855sin2/H20849/H9275t−/H9274/H20850/H20856=/H20855cos2/H20849/H9275t−/H9274/H20850/H20856=1/2 and /H20855cos/H9275tcos /H20849/H9275t−/H9274/H20850/H20856=cos /H20849/H9274/H20850/2. The first term in Eq. /H208496/H20850is independent of the rf quantities I1,/H92511and/H92521and represents the static voltage drop of I0. The second term is the micro- wave photovoltage UMW. It shows no impact from the dc current I0. The third term represents the microwave photore- sistance /H9004RMW. It is proportional to I0and depends on the microwave quantities /H92511and/H92521. It can be seen now that the rf resistance amplitude R1used in the beginning of this para- graph corresponds to R1=RA/H92511sin 2/H92510. To analyze the magnetization’s angle oscillation ampli- tudes /H92511and/H92521it is necessary to express them by means of the corresponding rf magnetization Re /H20849me−i/H9275t/H20850.mis the complex rf magnetization amplitude. Its phase is defined with respect to I1, so that Re /H20849mxe−i/H9275t/H20850is in phase with I1cos/H9275tat the FMR. Because M=M0+m,m=/H20849mx,my,0/H20850 can /H20849in first order approximation /H20850only lie perpendicular to M0because MandM0have the same length /H20849M0/H20850. Hence /H20841mx/H20841/M0=sin/H92511/H11015/H92511 and /H20841my/H20841/M0=sin/H92521/H11015/H92521 for /H92511,/H92521/lessmuch90°. The microwave photovoltage and photoresistance appear whenever magnetization precession is excited. This means ifthe microwaves are in resonance with the FMR, with stand-ing exchange spin waves perpendicular to the film 10,11,25or with magnetostatic modes.11In this article we will analyze the FMR induced microwave photoresistance and photovolt-age. B. Magnetization dynamics To understand the impact of the applied rf magnetic field Re/H20849he−i/H9275t/H20850on the microwave photovoltage and photoresis- tance the effective susceptibilities /H9273xx,/H9273xy, and /H9273yy, which link me−i/H9275tinside the sample with the complex external rf magnetic field he−i/H9275t=/H20849hx,hy,hz/H20850e−i/H9275toutside the sample, have to be calculated. Here /H9274is encoded in the complex phase of m. The susceptibility inside the sample /H20849magnetic field hine−i/H9275t=/H20849hxin,hyin,hzin/H20850e−i/H9275t/H20850is determined by the Polder tensor26/H9273ˆ/H20849received from solving the Landau-Liftshitz- Gilbert equation28/H20850: m=/H9273ˆhin=/H20898/H9273Li/H9273T0 −i/H9273T/H9273L0 00 0 /H20899hin, /H208497/H20850 with FIG. 4. /H20849Color online /H20850Sketch of the magnetization precession. The magnetic field Hencloses the angle /H92510with the current I. The magnetization oscillation toward Ihas the amplitude /H92511and that perpendicular to I:/H92521.MICROWA VE PHOTOVOLTAGE AND PHOTORESISTANCE … PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-3/H9273L=/H9275M/H9275r /H9275r2−/H92752,/H9273T=/H9275/H9275 M /H9275r2−/H92752, where /H9275M=/H9253M0with the gyromagnetic ratio /H9253/H11015/H92620/H11003e/m =2/H9266/H92620/H1100328 GHz /T/H20849electron charge eand mass me/H20850and /H9275r=/H9253Hwithout damping. Approximation of our sample as a two-dimensional film results in the boundary conditions thath xand byare continuous at the film surface meaning hx =hxinandby=/H92620hy=/H92620/H20851/H208491+/H9273L/H20850hyin−i/H9273Thxin/H20852. Hence m=/H20898/H9273xx i/H9273xy0 −i/H9273xy/H9273yy0 00 0 /H20899h, /H208498/H20850 with /H9273xx=/H9275r/H9275M+/H9275M2 /H9275r/H20849/H9275r+/H9275M/H20850−/H92752, /H9273xy=/H9275/H9275 M /H9275r/H20849/H9275r+/H9275M/H20850−/H92752, /H9273yy=/H9275r/H9275M /H9275r/H20849/H9275r+/H9275M/H20850−/H92752. /H9273xxis identical to the susceptibility describing the propaga- tion of microwaves in an unlimited ferromagnetic medium inV oigt geometry 29/H20849propagation perpendicular to M0/H20850./H9273xx, /H9273xy, and /H9273yyhave the same denominator, which becomes resonant /H20849maximal /H20850when /H9275=/H20881/H9275r2+/H9275r/H9275M. This is in accor- dance with the FMR frequency of the Kittel formula for in-plane magnetized infinite ferromagnetic films. 30 This relatively simple behavior is due to the assumption thathinis constant within the film stripe. This assumption is only valid if the skin depth1/H9254of the microwaves in the sample is much larger than the sample thickness. During ourmeasurements we fix the microwave frequency fand sweep the magnetic field H. Consequently we find the FMR mag- netic field H 0with /H92752=/H92532/H20849H02+H0M0/H20850/H20849 9/H20850 and H0=/H20881M02/4+/H92752//H92532−M0/2. /H2084910/H20850 Now we introduce Gilbert damping27/H9251Gby setting /H9275r ª/H92750−i/H9251G/H9275with now /H92750=/H9253Hinstead of /H9275r=/H9253H. We sepa- rate the real and imaginary part of /H9273xx,/H9273xy, and/H9273yy: /H9273xx=/H20849/H9275r/H9275M+/H9275M2/H20850F, /H9273xy=/H9275/H9275 MF, /H9273yy=/H9275r/H9275MF, /H2084911/H20850 withF=/H92750/H20849/H92750+/H9275M/H20850−/H9251G2/H92752−/H92752+i/H9251G/H9275/H208492/H92750+/H9275M/H20850 /H20851/H92750/H20849/H92750+/H9275M/H20850−/H9251G2/H92752−/H92752/H208522+/H9251G2/H92752/H208492/H92750+/H9275M/H208502 /H11015/H20849H+H0+M0/H20850/H20849H−H0/H20850+i/H208492H+M0/H20850/H9251G/H9275//H9253 /H20849H+H0+M0/H208502/H20849H−H0/H208502+/H208492H+M0/H208502/H9251G2/H92752//H92532. The approximation was done by neglecting the /H9251G2/H92752cor- rection to the resonance frequency /H92752=/H92750/H20849/H92750+/H9275M/H20850−/H9251G2/H92752 /H11015/H92750/H20849/H92750+/H9275M/H20850which is possible if /H9251G/lessmuch1. Hence /H9273xx,xy,yy/H11015Axx,xy,yy/H9004H/H20849H−H0/H20850+i/H9004H2 /H20849H−H0/H208502+/H9004H2, /H2084912/H20850 with/H9004H=/H20851/H208492H+M0/H20850//H20849H+H0+M0/H20850/H20852/H9251G/H9275//H9253. This can be ap- proximated as /H9004H/H11015/H9251G/H9275//H9253if/H20841H−H0/H20841/lessmuchH0.Axx,Axy, and Ayydetermine the scalar amplitude of /H9273xx,/H9273xy, and/H9273yy. To analyze the FMR line shape in the following, we will call the Lorentz line shape which is proportional to/H9004H//H20851/H20849H−H 0/H208502−/H9004H2/H20852symmetric Lorentz line shape and the line shape proportional to /H20849H−H0/H20850//H20851/H20849H−H0/H208502−/H9004H2/H20852anti- symmetric Lorentz line shape. A linear combination of both will be called asymmetric Lorentz line shape. /H20841H−H0/H20841/lessmuchH0 allows us to approximate Axx/H11015/H9253/H20849H0M0+M02/H20850 /H9251G/H9275/H208492H0+M0/H20850, Axy/H11015M0 /H9251G/H208492H0+M0/H20850, Ayy/H11015/H9253H0M0 /H9251G/H9275/H208492H0+M0/H20850. /H2084913/H20850 These are scalars which are independent of the dc mag- netic field Hand hence characteristic for the sample at fixed frequency. Indeed the assumption of Gilbert damping is notessential for the derivation of Eq. /H2084913/H20850. In the event of a different kind of damping, /H9004Hcan also be directly input into Eq. /H2084913/H20850replacing /H9251G/H9275. However, because of the common- ness of Gilbert damping, its usage here can provide a betterfeeling for the usual frequency dependence of A xx,xy,yy. Going ahead, Eq. /H208498/H20850becomes m/H11015/H9004H/H20849H−H0/H20850+i/H9004H2 /H20849H−H0/H208502+/H9004H2/H20898Axx iAxy0 −iAxyAyy0 00 0 /H20899h. /H2084914/H20850 TheH-field dependencies has Lorentz line shape with an- tisymmetric /H20849dispersive /H20850real and symmetric /H20849absorptive /H20850 imaginary part, the amplitudes Axx,±iAxy, and Ayy, respec- tively, and the width /H9004H. Note that AxxAyy/H11015Axy2for /H20841H−H0/H20841/lessmuchH0. Consequently, the susceptibility amplitude tensor can be simplified to /H20898Axx iAxy0 −iAxyAyy0 00 0 /H20899h/H11015/H20898/H20881Axx −i/H20881Ayy 0/H20899/H20900/H20898/H20881Axx i/H20881Ayy 0/H20899h/H20901 and Eq. /H2084914/H20850becomesMECKING, GUI, AND HU PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-4m=/H9253M0 /H9251G/H9275/H208492H0+M0/H20850/H9004H/H20849H−H0/H20850+i/H9004H2 /H20849H−H0/H208502+/H9004H2/H20898/H208811+M0/H0 −i 0/H20899 /H11003/H20900/H20898/H208811+M0/H0 i 0/H20899h/H20901. /H2084915/H20850 It is visible that the ellipticity of mis independent of the exciting magnetic field h. Only the amplitude and phase of mare defined by h. The reason is the weak Gilbert damping /H9251Gfor which much energy needs to be stored in the magne- tization precession to have a compensating dissipation.Hence little energy input and impact from happears. From Eq. /H2084915/H20850follows that m xandmyhave cardinally the ratio mx/my=i/H208811+M0/H0. /H2084916/H20850 Therefore, myvanishes for /H9275→0 and mx=imyfor/H9275→/H11009. This means that the precession of Mis elliptical and becom- ing more circular for high frequencies and more linear /H20849along thexaxis /H20850for low frequencies. This description applies for the case of an in-plane magnetized ferromagnetic film. How-ever, in the case that the sample has circular symmetry withrespect to the magnetization direction /H20849e.g., in a perpendicu- lar magnetized disk or infinite film 10,11/H20850:/H92511=/H92521. This is the same as in the case that /H9275→/H11009. Only in these cases the mag- netization precession can be described in terms of one pre-cession cone angle. 13Otherwise, distinct attention has to be paid to /H92511and/H92521/H20849see III B /H20850. Additionally, it can be seen in Eq. /H2084915/H20850thatmy/mxis also the ratio of the coupling strength ofmtohyandhx, respectively. C. Microwave photoresistance The microwave photoresistance /H9004RMWcan be deduced from Eq. /H208496/H20850. First the microwave photovoltage is excluded by setting the rf current I1=0. Then we only regard the mi- crowave power dependent terms which depend on /H92511and/H92521: /H9004RMW=/H20849/H20841U0/H20841I1=0−/H20841U0/H20841I1=0,/H92511=0,/H92521=0/H20850/I0 =RA/H20849−/H925112cos 2/H92510−/H925212cos2/H92510/H20850/2. /H2084917/H20850 If the magnetization lies parallel or antiparallel to the dc current vector I0along the stripe /H20849/H92510=0° or /H92510=180° /H20850the AMR is maximal. In this case magnetization oscillation /H20849/H92511 and/H92521/H20850reduces /H20849−cos 2 /H92510=−1 /H20850the AMR by /H9004RMW=− /H20849/H925112 +/H925212/H20850RA/2/H20849negative photoresistance /H20850. In contrast, if the mag- netization lies perpendicular to I0/H20849/H92510=90°, see Fig. 2/H20850the resistance is minimal. In this case magnetization oscillationcorresponding to /H92511will increase /H20849−cos 2 /H92510=+1 /H20850the AMR /H20849positive photoresistance /H20850by/H9004RMW=+/H925112RA/2/H20851oscillations corresponding to /H92521leave /H9258/H20849t/H20850constant in this case and do not change the AMR /H20852. The next step is to calculate /H92511and/H92521. The dc magnetic field dependence of /H92511=/H20841mx/H20841/M0=/H20841/H9273xxhx+i/H9273xyhy/H20841/M0and /H92521=/H20841my/H20841/M0=/H20841−i/H9273xyhx+/H9273yyhy/H20841/M0is proportional to that of /H20841/H9273xx/H20841,/H20841/H9273xy/H20841, and /H20841/H9273yy/H20841given in Eq. /H2084912/H20850/H20849imaginary symmetricand real antisymmetric Lorentz line shape /H20850. Squaring this results in symmetric Lorentz line shape: /H925112/H11008/H925212/H11008/H20879/H9004H/H20849H−H0/H20850+i/H9004H2 /H20849H−H0/H208502+/H9004H2/H208792 =/H9004H2 /H20849H−H0/H208502+/H9004H2. Hence /H925112=/H20841Axxhx+iAxyhy/H208412 M02/H9004H2 /H20849H−H0/H208502+/H9004H2, /H925212=/H20841Ayyhy−iAxyhx/H208412 M02/H9004H2 /H20849H−H0/H208502+/H9004H2. /H2084918/H20850 Using Eqs. /H2084915/H20850and /H2084918/H20850, Eq. /H2084917/H20850transforms to /H9004RMW=RA /H20849/H9251G/H9275//H9253/H208502/H208492H0+M0/H208502/H20851−/H20849H0+M0/H20850cos 2/H92510 −H0cos2/H92510/H20852/H9004H2 /H20849H−H0/H208502+/H9004H2 /H11003/H20841hx/H20881H0+M0+ihy/H20881H0/H208412. /H2084919/H20850 The strength of the microwave photoresistance is propor- tional to 1 //H9251G2. Weak damping /H20849small /H9251G/H20850is therefore critical for a signal strength sufficient for detection. The magneticfield dependence shows symmetric Lorentz line shape. The dependence of /H9004R MWon/H92510in Eq. /H2084919/H20850reveals a sign change and hence vanishing of the photoresistance at cos2/H92510=1 2/H208731−H0 3H0+2M0/H20874. /H2084920/H20850 This means that the angle at which the photoresistance van- ishes shifts from /H92510= ±45° and /H92510= ±135° /H20849for/H9275→0/H20850to /H92510= ±54.7° and /H92510= ±125.3° respectively /H20849for/H9275→/H11009/H20850when increasing /H9275. The reason for this frequency dependence is the frequency dependence of the ellipcity of mdescribed at the end of Sec. II B. D. Microwave photovoltage The most obvious difference in appearance between the microwave photoresistance discussed in Sec. II C and themicrowave photovoltage discussed in this paragraph is thatthe photoresistance is proportional to the square of the rf magnetization /H20851see Eq. /H2084917/H20850, /H925112/H11015/H20841mx/H208412/M02and /H925212 /H11015/H20841my/H208412/M02/H20852while the photovoltage UMWis proportional to the product of the rf magnetization and the rf current. Con-sequently, the photovoltage has a very different line shape:While the rf magnetization depends with Lorentz line shapeonH/H20851see Eq. /H2084912/H20850/H20852,I 1is independent of H. The line shape is hence determined by the phase difference /H9274between the rf magnetization component Re /H20849mxe−i/H9275t/H20850and the rf current I1cos/H9275t. This effect does not play a role in the case of photoresistance because there only one phase matters namelythat of the rf magnetization. In contrast in photovoltage mea-surements a linear combination of symmetric and antisym-metric Lorentz line shapes is found. This will be discussed indetail in the following.MICROWA VE PHOTOVOLTAGE AND PHOTORESISTANCE … PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-5To isolate the microwave photovoltage in Eq. /H208496/H20850the dc current I0is set to 0: UMW=/H20841U0/H20841I0=0=−I1/H92511RAsin 2/H92510cos/H9274 2. /H2084921/H20850 From Eq. /H208498/H20850we follow with /H92511cos/H9274=R e /H20849mx/H20850=R e /H20849/H9273xxhx+i/H9273xyhy/H20850. /H2084922/H20850 We split hx=hxr+ihxiandhy=hyr+ihyiinto real /H20849hxr,hyr/H20850and imaginary /H20849hxi,hyi/H20850part. This enables us to isolate the real part in Eq. /H2084921/H20850using Eq. /H2084914/H20850: UMW=I1RAsin 2/H92510 2M0/H20877/H20849Axyhyr+Axxhxi/H20850/H9004H2 /H20849H−H0/H208502+/H9004H2 +/H20849Axyhyi−Axxhxr/H20850/H9004H/H20849H−H0/H20850 /H20849H−H0/H208502+/H9004H2/H20878. /H2084923/H20850 Conclusively in contrast to the microwave photoresistance /H20851/H9004RMW/H110081//H9251G2, see Eq. /H2084919/H20850/H20852the photovoltage is only pro- portional to 1 //H9251G/H11008Axx,xy,yy. Thus good damping is less im- portant for its detection.31 To understand the measurement results it will be neces- sary to transform the coordinate system of Eq. /H2084923/H20850to /H20849x/H11032,y,z/H11032/H20850. In this coordinate system the rf magnetic field his constant during rotation as described in Eq. /H2084933/H20850. To better understand the photovoltage line shape we have a closer look on /H9274: When sweeping Hthe rf magnetization phase is shifted by /H9274mwith respect to the resonance case /H20849H=H0/H20850. The rf current has a constant phase /H9274Iwhich is defined with respect to the magnetization’s phase at reso- nance. The impact of the dc magnetic field Hon the rf cur- rent /H20849I1,/H9274I/H20850via the FMR is believed to be negligible: cos/H9274= cos /H20849/H9274m−/H9274I/H20850= cos/H9274mcos/H9274I+ sin/H9274msin/H9274I. /H2084924/H20850 /H9274is determined by the /H20849complex /H20850phase of /H9273xx,/H9273xy, and/H9273yy with respect to the resonance case /H20851Re/H20849/H9273xy,yy/H20850=0 at H=H0/H20852 during magnetic field sweep /H20851asymmetric Lorentz line shape; see Eq. /H2084912/H20850/H20852: tan/H9274m=Im/H20873/H9004H/H20849H−H0/H20850+i/H9004H2 /H20849H−H0/H208502+/H9004H2/i/H20874 Re/H20873/H9004H/H20849H−H0/H20850+i/H9004H2 /H20849H−H0/H208502+/H9004H2/i/H20874=H0−H /H9004H. /H2084925/H20850 It should be noted that according to the Landau-Liftshitz equation28happlies a torque on the magnetization and hence excites mttransversal. That is why at resonance mxshows a phase shift of 90° with respect to hx. Consequently in Eq. /H2084925/H20850division by iis necessary /H20849/H9273xxand/H9273xybecome imagi- nary at resonance /H20850. Equation /H2084925/H20850means that in case that the applied micro- wave frequency is higher than the FMR frequency /H20849H0 /H11022H/H20850/H9274m/H110220/H20849note that mt=me−i/H9275t/H20850,mtis delayed with re- spect to the resonant case. The other way around /H20849H0/H11021H/H20850 the FMR frequency is higher than that of the applied micro- wave field and mtis running ahead compared to the reso- nance case. Using Eq. /H2084925/H20850we findcos/H9274m=/H9004H /H20881/H20849H−H0/H208502+/H9004H2. /H2084926/H20850 Inserting Eqs. /H2084918/H20850and /H2084924/H20850–/H2084926/H20850into Eq. /H2084921/H20850gives UMW=−RAI1sin 2/H92510 2/H20841Axxhx+iAxyhy/H20841 M0/H20873/H9004H2cos/H9274I /H20849H−H0/H208502+/H9004H2 −/H20849H−H0/H20850/H9004Hsin/H9274I /H20849H−H0/H208502+/H9004H2/H20874. /H2084927/H20850 The dependence on Htakes the form of a linear combi- nation of symmetric and antisymmetric Lorentz line shapewith the ratio 1:tan /H9274I. The symmetric line shape contribu- tion /H20849/H11008/H9004H/H20850arises from the rf current contribution that is in phase with the rf magnetization at FMR and the antisymmet- ric from that out-of-phase. This gives a nice impression ofthe phase /H9274Iof the rf current determining the line shape of the FMR. E. Vectorial description of the photovoltage To complete the discussion of the microwave photovolt- age we want to return to the approach used by Juretschke23 to demonstrate that it is consistent with the descriptionabove. In Sec. II A we started with Ohm’s law /H20851scalar equa- tion /H208491/H20850/H20852. There we integrate an angle- and time-dependent resistance. Here we want to start with the vectorial notationof Ohm’s law used in Juretschke’s publication /H20851Eq. /H208491/H20850/H20849Ref. 23/H20850/H20852. This integrates AMR and anomalous Hall effect AHE. /H9267 is the resistivity of the sample and /H9004/H9267that additionally aris- ing from AMR. RHis the anomalous Hall effect constant: E=/H9267J+/H20849/H9004/H9267M2/H20850/H20849J·M/H20850M−RHJ/H11003M. /H2084928/H20850 We split M=M0+mtand the current density J=J0+jtinto their dc /H20849M0and J0/H20850and rf contributions /H20851mt=Re /H20849me−i/H9275t/H20850 andjt=jcos/H9275t/H20852. Constance of /H20841M/H20841allows mt=/H20849mxt,myt,0/H20850in first order approximation only to lie perpendicular to M0 =/H208490,0, M0/H20850. To select the photovoltage we set J0=0 and ap- proximate equation /H2084928/H20850to second order in jtand mt. The terms of zeroth order in both jtandmtrepresent the sample resistance without microwave exposure and are not discussedhere. The terms of first order in either j tormt/H20849but not both /H20850 have zero time average and do not contribute to the micro-wave induced dc electric field E MW. Only the terms that are simultaneously of first order in jtandmtcontribute to EMW /H20851compare Eq. /H208494/H20850from Juretschke23/H20852: EMW=/H9004/H9267 M02/H20855/H20849jtmt/H20850M0+/H20849jtM0/H20850mt/H20856−RH/H20855jt/H11003mt/H20856./H2084929/H20850 The/H9004/H9267dependent term represents the photovoltage con- tribution arising from AMR and the RHdependent term that arising from AHE. Note that a second order of mtappears when applying a dc current J0/HS110050. It represents the photore- sistance discussed in Sec. II C. However, it we will not bediscussed here. In the following we will calculate the photovoltage in our Permalloy film stripe considering its geometry which fixes the current direction. j t=jz/H11032tz/H11032along the stripe /H20849z/H11032is the unitMECKING, GUI, AND HU PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-6vector along the Permalloy stripe /H20850. The small dimensions perpendicular to the stripe /H20849/lessmuchL/H20850will prevent the formation of a perpendicular rf current. A similar approximation of a metal grating forming a linear polarizer has been consideredpreviously. 9The photovoltage UMWis also measured along the stripe /H20849length vector L=z/H11032/H110032.4 mm /H20850. When fluctuations ofEMWalong the stripe are neglected considering the large microwave wavelength, /H9261/H1101520 mm/greatermuch2.4 mm= L,w efi n d UMWby multiplying EMWwith L: UMW=/H20885 0L EMWdz/H11032/H11015EMW·L =/H9004/H9267L M02/H20855jz/H11032t/H20849z/H11032mt/H20850/H20849M0z/H11032/H20850+jz/H11032t/H20849z/H11032M0/H20850/H20849mtz/H11032/H20850/H20856−0 =/H9004/H9267L M0/H20855jz/H11032tmxt/H20856sin /H208492/H92510/H20850. /H2084930/H20850 This is equivalent to Eq. /H2084921/H20850which can be verified by replacing /H9004/H9267jz/H11032tL=RAI1cos /H20849/H9275t/H20850and mxt=/H92511M0cos /H20849/H9275t−/H9274/H20850. Time averaging results in the additional factor cos /H20849/H9274/H20850/2. As discussed in Sec. II F the contribution belonging to the anomalous Hall effect has no impact in this geometry be-cause it can only generate a photovoltage perpendicular tothe rf current, i.e., perpendicular to the stripe. Comparing our results to those of Juretschke and Egan, 23,31we note that an equation similar to Eq. /H2084930/H20850has been derived in the formula for ey0in Eq. /H2084931/H20850in Juretsch- ke’s publication.23There the photovoltage is measured par- allel to the rf current as done in our stripe. However, it has tobe noted that the coordinate system is defined differently.The major difference compared to our system is that we usea stripe shaped film to lithographically define the direction ofthe rf current I 1, while the direction of his left arbitrary. In contrast to that, Juretschke and Egan23,31define the direction of the rf magnetic field and rf current by means of theirmicrowave setup. In Eq. /H2084931/H20850/H20849e y0/H20850from Juretschke23this results in the additional factor cos /H9258/H20849which is equivalent to cos/H92510in our work /H20850compared to Eq. /H2084930/H20850. This arises from the definition of hfixed parallel to the rf current /H20851compare Eq. /H2084933/H20850/H20852. F. Other magnetoresistive effects that couple spin and charge current In this section we present other magnetoresistive effects which can generate photovoltage and photoresistance like theAMR. This selection gives a broader view on the range ofeffects for which the photovoltage and photoresistance canbe discussed in terms of the analysis presented in this work.In principle every magnetoresistive effect can modulate thesample resistance and thus rectify some of the rf current tophotovoltage. One magnetoresistive effect is the anomalous Hall effect AHE in ferromagnetic metals that was /H20849together with the AMR /H20850the basis for the discussion of Juretschke. 23There a current with perpendicular magnetization generates a voltageperpendicular to both. Under microwave exposure this alter-nates with the microwave frequency but in an asymmetricway due to the modulated AHE arising from magnetization precession. The asymmetric voltage has a dc contribution/H20849photovoltage /H20850, 31which can be measured using a two- dimensional ferromagnetic film with the magnetization nei-ther parallel nor perpendicular to it. The photovoltage in-duced by AHE appears in the film plane perpendicular to therf current and is small 25for Permalloy /H20849Ni80Fe20/H20850. Also a photoresistive effect which alters the AHE can be expected if the magnetization lies out-of-plane. Other examples for magnetoresistive effects are GMR and TMR structures which exhibit a photovoltage mechanismsimilar to that in AMR films. The difference is that there thedirection of the ferromagnetic layer magnetization with re-spect to the current does not matter. Effectively instead thedirection of the magnetization Mof one ferromagnetic layer with respect to that of another layer is decisive /H20849see Fig. 5/H20850. Exciting the FMR in one layer yields again oscillation of thesample resistance R/H20849t/H20850and thus gives the corresponding rf voltage U/H20849t/H20850a nonzero time average /H20849photovoltage /H20850. 4,32This is usually stronger than that from AMR films due to the generally higher relative strength of GMR and TMR com-pared to AMR. It should be noted that in current studies of the microwave photovoltages effect in multilayer structures, the focus is oninterfacial spin transfer effects. 4–8,19–21,32It remains an in- triguing question whether interfacial spin transfer effects andthe effect revealed in our approach based on phenomenologi-cal magnetoresistance might be unified by a consistent mi-croscopic model, as Silsbee et al. have demonstrated for de- scribing both bulk and interfacial spin transport under rfexcitation. 24 Multilayer structures also provide a nice example that photovoltage generation can also be reversed when the oscil-lating magnetoresistance, transforms a dc current into an rfvoltage, 33instead of transforming an rf current into a dc voltage /H20849photovoltage /H20850. This gives a new kind of microwave source and seems—although weaker—also possible in AMRand AHE samples. It can be reasoned that like microwave photovoltage the microwave photoresistance can also be based on GMR orTMR instead of AMR: When aligning the two magnetiza- FIG. 5. /H20849Color online /H20850Microwave photovoltage in a GMR/TMR heterostructure /H20851ferromagnetic /H20849M/H20850/nonferromagnetic/ferromag- netic /H20849Mf/H20850/H20852: The dynamic magnetization Mprecesses /H20849period P/H20850in phase with the current I./H20849a/H20850Mlies almost perpendicular to Mf: High GMR/TMR. /H20849b/H20850Mlies almost parallel to Mf: Low GMR /TMR ⇒nonzero time average of the voltage U.MICROWA VE PHOTOVOLTAGE AND PHOTORESISTANCE … PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-7tions of both ferromagnetic layers in a GMR or TMR struc- ture microwave induced precession of one magnetization isexpected to increase the GMR/TMR because of the arisingmisalignment with the other magnetization. With the magne-tizations initially antiparallel the opposite effect, a micro-wave induced resistance decrease, is expected. Further workdemonstrating these effects would be interesting. III. PHOTOVOLTAGE MEASUREMENTS A. Measurement setup The sample we use to investigate the microwave photo- voltage consists of a thin /H20849d=49 nm /H20850Permalloy /H20849Ni 80% ,Fe 20% /H20850film stripe /H20849200/H9262m wide and 2400 /H9262m long /H20850with 300 /H11003300/H9262m2bond pads at both ends /H20849see Fig. 3/H20850. These are connected via gold bonding wires and coaxial cables to a lock-in amplifier. For auxiliary measurements/H20849e.g., Hall effect /H20850six additional junctions are attached along the stripe /H20849see Fig. 3/H20850. The resistance of the film stripe is R 0+RA=85.0 /H9024for parallel and R0=83.6 /H9024for perpendicular magnetization. Hence the conductance is /H9268=1//H9267=2.9/H11003106/H9024−1m−1and the relative AMR is /H9004/H9267//H9267=1.7%. The absolute AMR is RA =1.4/H9024. This is in good agreement with previous publications.9–11 The film is deposited on a 0.5 mm thick GaAs single crys- tal substrate, and patterned using photolithography and liftoff techniques. The substrate is mounted on a 1 mm polyeth-ylene print circuit board which is glued to a brass plate hold-ing it in between the poles of an electromagnet. This pro-vides the dc magnetic field B= /H92620H /H20849maximal /H110151T /H20850. The sample is fixed 1 mm behind the end of a WR62 /H2084915.8 /H110037.9 mm /H20850hollow brass waveguide which is mounted nor- mal to the Permalloy film plane. The stripe is fixed along the narrow waveguide dimension. In the Kuband /H2084912.4–18 GHz /H20850, that we use in our measurements, the WR62 waveguide only transmits the TE 01mode.1The stripe was fixed with respect to the waveguide but was left rotatablewith respect to H. This allows the stripe to be parallel or perpendicular to H, but keeps the magnetic field always in the film plane. A high precision angle readout was installedto indicate /H92510./H20849See Fig. 6/H20850. The waveguide is connected to an HP83624B microwave generator by a coaxial cable supplying frequencies of up to20 GHz and a power of 200 mW. The power is howeverlater significantly reduced by losses occurring within the co-axial cable, during the transfer to the hollow waveguide andby reflections at the end of the waveguide. Microwave pho-tovoltage measurements are performed sweeping the mag-netic field while fixing the microwave frequency. The sampleis kept at room temperature. To avoid external disturbances the photovoltage was de- tected using a lock-in technique: A low frequency /H2084927.8 Hz /H20850 square wave signal is modulated on the microwave CW out- put. The lock-in amplifier, connected to the Permalloy stripe,is triggered to the modulation frequency to measure the re-sulting square wave photovoltage across the sample. Insteadof the photovoltage also the photocurrent can be measured. 10Its strength I0can be found when setting U0=0 in Eq. /H208496/H20850 /H20849instead of I0=0 /H20850. B. Ferromagnetic resonance The measured photovoltage almost vanishes during most of the magnetic field sweep but shows one pronounced reso-nance of several /H9262V . The strength and line shape of this resonance are strongly depending on /H92510and will be dis- cussed in Sec. III C. A line shape dependence of the photo-voltage on the microwave frequency is also found. The pho-tovoltage with respect to the strength of the externalmagnetic field Hand the microwave frequency f= /H9275/2/H9266can be seen in a gray scale plot in Fig. 7, in which the resonance can be identified with the FMR by the corresponding fits/H20849dashed line /H20850because the Kittel equation /H208499/H20850/H20849Ref. 30/H20850for ferromagnetic planes /H20849our Permalloy film /H20850applies. The mag- netic parameters found are /H92620M0/H110151.02 T and /H9253/H110152/H9266/H92620 /H1100328.8 GHz /T. They are in good agreement with previous publications.9,10 The exact position of the FMR is obscured by its strongly varying line shape. We overcome this problem by the pro- FIG. 6. /H20849Color online /H20850Sketch of the measurement geometry. A 1 mm thick polyethylene plate is glued on a brass holder. On top ofthe polyethylene a GaAs substrate is glued. On the substrate thePermalloy /H20849Py/H20850stripe is defined. This is electrically wired to a volt- age amplifier for photovoltage measurements. For photoresistancemeasurements an additional current source is connected parallel tothe voltage amplifier, which is not shown explicitly here. FIG. 7. /H20849Color online /H20850Gray scale plot of the measured fre- quency and magnetic field dependence of the microwave photovolt-age at /H92510=47°. The dashed line shows the calculated FMR fre- quency /H20851see Eq. /H208499/H20850/H20852. The photovoltage intensity is strongly frequency dependent because of the frequency dependent wave-guide transmission.MECKING, GUI, AND HU PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-8ductive line shape analysis in Sec. III C. It is found that H0is slightly dependent on /H92510. This can be attributed to a small demagnetization field perpendicular to the stripes but withinthe film plane arising from the finite stripe dimensions in thisdirection. So, when M 0lies perpendicular to the stripe, H0 slightly increases compared to the value fulfilling the Kittel equation for a plane /H20851see Eq. /H208499/H20850/H20852. In the parallel and perpen- dicular case we use the approximation of our film stripe as anellipsoid, where we can use the corresponding Kittelequation 30/H20849demagnetization factors Nx,Ny, and Nzwith re- spect to the dc magnetic field /H20850: /H9275=/H9253/H20881/H20851H0+/H20849Nx−Nz/H20850M0/H20852/H20851H0+/H20849Ny−Nz/H20850M0/H20852. /H2084931/H20850 The difference of the resonance field between the case that M0lies in the film plane parallel to the stripe and per- pendicular is 1.6 mT /H208490.7% /H20850atf=15 GHz. From this we can calculate the small demagnetization factor Nx/H11032=0.085% per- pendicular to the Permalloy stripe within the film plane using Eq. /H2084931/H20850. From the sum rule34follows: Ny=1− Nx/H11032−Nz/H11032=1 −0.085%−0=99.915%. Nz/H11032/H20849parallel to the stripe /H20850can be assumed to be negligibly small. This matches roughly with the dimension of the height to width ratio /H2084949 nm:200 /H9262m/H20850 of the sample. For the stripe presented in Sec. IV similar but stronger demagnetization effects are found. Now we will have a closer look on the magnetic proper- ties of the investigated film. Again at f=15 GHz we find using Eq. /H2084910/H20850:H0=0.219 T. Using asymmetric Lorentz line shape fitting as described in Sec. III C we get /H9251G=0.0072. Consequently, Axx=231.1, Axy=97.1, and Ayy=40.8 accord- ing to Eq. /H2084913/H20850. Because of /H9251G=0.0072 the magnetization precession does impressive n/H1101522 turns before being damped to 1 /eof its initial amplitude /H20849n=1/2/H9266/H9251G/H20850. Therefore the ellipcity of m is almost independent of h/H20849see Sec. II B /H20850. It can be calcu- lated from Eq. /H2084916/H20850thatmx/my=2.38 iat/H9275/2/H9266=15 GHz. To check the validity of our approximation /H20849d/lessmuch/H9254, see Sec. II B /H20850we will now regard the skin depth /H9254atf =15 GHz in our sample /H20849d=49 nm /H20850. For/H9262=/H92620/H20849away from the FMR /H20850we find /H9254=/H208812//H9275/H9262/H9267=2.4/H9262m. Hence /H9254/greatermuchd. This is in accordance with our approximation that his almost constant within the Permalloy film /H20849see II B /H20850. However, in the vicinity of the FMR: /H20841/H9262/H20841/greatermuch/H92620and for the same fre- quency and conditions as above: /H9262L=/H208491+/H9273L/H20850/H92620=133 i/H92620at the FMR. Thus we approximate /H9254FMR=/H208812//H9275/H20841/H9262L/H20841/H9267 =210 nm. Hence /H9254FMRis still significantly larger than dand our approximation is still valid. Finally we can summerize that for samples with weak damping /H20849/H9251G/lessmuch/H9275//H9275M/H20850like ours the approximation H/H11015H0 gives results with impressive precision /H20849see Fig. 8/H20850because its discrepancies are limited to the unimportant magneticfield ranges with /H20841 /H9273xx/H20841,/H20841/H9273xy/H20841, and /H20841/H9273yy/H20841/lessmuch1, which are far away from the FMR. C. Asymmetric Lorentz line shape Although in Sec. III B the frequency dependence of the FMR field is verified with the gray scale plot in Fig. 7,i ti s still desirable to receive a more accurate picture of the cor-responding line shape which is found to be strongly angulardependent /H20849see Fig. 8/H20850.I nE q . /H2084927/H20850it is shown that the mag- netic field dependence of U MWexhibits asymmetric Lorentz line shape around H=H0. Hence UMWtakes the form UMW=UMWSYM+UMWANT =U0SYM /H9004H2 /H20849H−H0/H208502+/H9004H2+U0ANT/H9004H/H20849H−H0/H20850 /H20849H−H0/H208502+/H9004H2. /H2084932/H20850 This is used to fit the magnetic field dependence of the photovoltage in Fig. 8. For clearness the symmetric /H20849absorp- tive /H20850and antisymmetric /H20849dispersive /H20850contributions are shown separately in Fig. 9. A small constant background is found and added to the antisymmetric contribution. The back-ground could possibly arise from other weak nonresonantphotovoltage mechanisms. The fits agree in an unambiguous manner with the mea- sured results. Hence they can be used to determine the Gil-bert damping parameter with high accuracy: /H9251G/H11015/H9253/H9004H//H9275 /H11015/H208490.72% ±0.015% /H20850. However, if the magnetization lies par- allel or perpendicular to the stripe the photovoltage vanishes /H20851see Eq. /H2084921/H20850/H20852. Hence we can only verify /H9251Gwhen the mag- netization is neither close to being parallel nor perpendicularto our stripe. The corresponding /H9251G=1//H9275/H9270in the Nickel sample of Egan and Juretschke,31can be estimated using the ferromag- netic relaxation time /H9270from their Table II. It lies in between /H9251G=0.12 and 0.18, so being more than 16 times higher than the value in our sample. This makes the line shape approxi-mation of Sec. II D invalid for their case. Consequently, a FIG. 8. /H20849Color online /H20850Fitting /H20849black line /H20850of the microwave photovoltage signal /H20849dots /H20850for different angles /H92510atf=15 GHz. The black horizontal bars indicate zero signal.MICROWA VE PHOTOVOLTAGE AND PHOTORESISTANCE … PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-9much more elaborated line shape analysis23appears neces- sary. In Fig. 8the photovoltage along the stripe is presented at four different angles /H92510. The signal to noise ratio is about 1000 because of the carefully designed measurement system,where the noise is suppressed to less than 5 nV. Because ofthis good sensitivity we can verify the matching of ourtheory from Sec. II with the measurement results in greatdetail. /H20849See Fig. 10./H20850 In the following we want to investigate the angular depen- dence in detail. Therefore, we transform the coordinate sys-tem of Eq. /H2084923/H20850according to the transformation presented inSec. II A. Doing so we can separate the contributions from h x/H11032,hy, and hz/H11032: UMW=RAI1sin /H208492/H92510/H20850 2M0/H20877/H20851Axyhyr+Axx/H20849hx/H11032icos/H92510 −hz/H11032isin/H92510/H20850/H20852/H9004H2 /H20849H−H0/H208502+/H9004H2+/H20851Axyhyi +Axx/H20849hz/H11032rsin/H92510−hx/H11032rcos/H92510/H20850/H20852/H9004H/H20849H−H0/H20850 /H20849H−H0/H208502+/H9004H2/H20878. /H2084933/H20850 hx/H11032,hy, and hz/H11032are fixed with respect to the hollow brass waveguide and its microwave configuration and do not change when /H92510is varied. We find that the angular dependence of the line shape in Eq. /H2084933/H20850exhibits two aspects: An overall factor sin /H208492/H92510/H20850and individual factors /H20849sin/H92510, cos/H92510, and 1 /H20850for the terms be- longing to the different spatial components of h. The overall factor sin /H208492/H92510/H20850arises from the AMR photovoltage mecha- nism and results in vanishing of the photovoltage signal at /H92510=0°, 90°, 180°, and 270°. This means if M0lies either parallel, antiparallel, or perpendicular to the stripe axis. Thisis illustrated in Fig. 11and is clearly observed in our mea- surements /H20849see Fig. 10/H20850. We take this as a strong support for the photovoltage being really AMR based. Another support comes from the similarity with the planar Hall effect. 35The planar Hall effect generates a voltage UPHE perpendicular to the current in ferromagnetic samples /H20849width W/H20850when the magnetization M0lies in the current-voltage plane. It arises as well from AMR and vanishes when M0lies either parallel or perpendicular to the current axis. The similarity arises because of the AMR only generating a transversal resistance when the current is not lying alongthe principle axis of its resistance matrix /H20849parallel or perpen- FIG. 9. /H20849Color online /H20850Symmetric and antisymmetric contribu- tions to the asymmetric Lorentz line shape fit from Fig. 8/H20849black /H20850.A small constant background is found and added to the antisymmetriccontribution. FIG. 10. /H20849Color online /H20850Bars show the angular dependence of the amplitude of the symmetric /H20849U0SYM, thin bars /H20850and antisymmet- ric /H20849U0ANT, thick bars /H20850contribution to the microwave photovoltage atf=15.0 GHz. Note that both the symmetric and antisymmetric contribution vanish for /H92510=0°, 90°, 180°, and 270°. The lines rep- resent the corresponding fits by means of Eq. /H2084934/H20850. The inlet shows the geometry of the investigated Permalloy stripe and the coordi-nate systems from Fig. 3/H20849note: z /H20648H/H20850. (a) (b) FIG. 11. /H20849Color online /H20850When the magnetic field Hlies parallel or perpendicular to the stripe, the time average voltage vanishes. /H20849a/H20850 Hlies perpendicular to I: Precession of the magnetization Mleaves /H20849after half a period P/2/H20850the angle /H9258between the axis of MandI unchanged. Hence the AMR /H20849and so voltage U/H20850is also unchanged. The photovoltage vanishes. /H20849b/H20850His parallel to I:/H9258and the AMR stay constant during the precession of Mand the time average of I is zero. This means that only when His neither parallel nor perpen- dicular to the stripe a photovoltage is generated.MECKING, GUI, AND HU PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-10dicular to the magnetization /H20850. This is the same geometrical restriction as shown above for the microwave photovoltage/H20851see Eq. /H2084921/H20850and Fig. 11/H20852. We want to emphasize the importance that in any of these microwave photovoltage experiments, due to the unusuallystrong angle dependence, it is important to pay attention tothe exact angle adjustment of the sample with respect to thedc magnetic field Hwhen measuring under high symmetry conditions /H20849Hparallel or perpendicular to the stripe /H20850to avoid involuntary signal changes due to small misalignments. Asfound in 90° out-of-plane configuration 10already a misalign- ment as small as a tenth of a degree can yield a tremendousphotovoltage change in the vicinity of the FMR. Finally we want to come back to the individual angular dependencies of the photovoltage contributions arising fromthe different external magnetic field components. In additionto the sin /H208492 /H92510/H20850proportional dependence of UMWonmx, also the strength with which mxis excited by hdepends on /H92510. This is displayed in Fig. 12and reflected by the three terms in Eq. /H2084933/H20850depending on hx/H11032,hy, and hz/H11032with cos /H92510,1 ,a n d sin/H92510factors, respectively. Hence the symmetric U0SYMand antisymmetric U0ANTLorentz line shape contribution to UMW are fitted in Fig. 10with U0SYM=/H20851Uz/H11032Ssin /H20849/H92510/H20850+Ux/H11032Scos /H20849/H92510/H20850+UyS/H20852sin /H208492/H92510/H20850, U0ANT=/H20851Uz/H11032Asin /H20849/H92510/H20850+Ux/H11032Acos /H20849/H92510/H20850+UyA/H20852sin /H208492/H92510/H20850./H2084934/H20850 From Uz/H11032S,Ux/H11032S, and UyAthe dynamic magnetic field com- ponents hz/H11032i,hx/H11032i,hyiwhich are 90° out-of-phase with respect to the rf current I1can be determined using Eq. /H2084933/H20850and from Uz/H11032A,Ux/H11032A, and UySwe find hz/H11032r,hx/H11032r, and hyrwhich are in phase with I1. In principle I1can be separately deduced using the bolo- metric effect12as discussed in Sec. IV A. However, for the sample used here our usage of multiple stipes does not allowus to address the bolometric heating to one single stripe.Consequently the strength of I 1is unknown so that we can not determine h, but only hI1.Besides, considering the special dynamic magnetic field configuration in our rectangular hollow waveguide no rfmagnetic field component h z/H11032is expected to be generated along the waveguides narrow dimension /H20849z/H11032axis /H20850by the TE01mode1/H20849which is the microwave configuration of our waveguide /H20850. It follows that the sin /H20849/H92510/H20850terms in Eq. /H2084934/H20850 vanishes. This results in the additional symmetry UMW /H20849/H92510/H20850 =−UMW /H20849−/H92510/H20850, which is clearly observed in our measure- ments /H20849see Fig. 10/H20850. This symmetry was broken when we used a round waveguide. The vanishing of hz/H11032in our waveguide will allow us to plot the direction of htwo-dimensional /H20849instead of three- dimensional /H20850in Fig. 13. A small deviation from the symme- tryUMW /H20849/H92510/H20850=−UMW /H20849−/H92510/H20850is however found and arises from a small hz/H11032component /H20849see Table I/H20850which is not displayed in Fig. 13. It might arise from the fact that the rf microwave magnetic field hat the waveguide end already deviates from the TE 01mode. TABLE I. Determination of the rf magnetic field hat the 200/H9262m wide stripe at 1 mm distance from the waveguide end by means of Eq. /H2084933/H20850.Ux/H11032,y,z/H11032S,Ux/H11032,y,z/H11032A: Measured amplitudes of the contributions to the symmetric and antisymmetric Lorentz line shape of UMW /H20851see Eq. /H2084934/H20850/H20852with the angular dependence belong- ing to x/H11032,y, and z/H11032, respectively /H20849taken from the fitting in Fig. 10/H20850. Axx,xy: Corresponding amplitudes of /H9273xx,xy.hx/H11032,y,z/H11032r,hx/H11032,y,z/H11032i:r fm a g - netic field strength calculated from Ux/H11032,y,z/H11032S,Ux/H11032,y,z/H11032A/H20849in-phase and 90° out-of-phase contribution with respect to the current /H20850. Ux/H11032,y,z/H11032SUx/H11032,y,z/H11032AAxx Axy I1hx/H11032,y,z/H11032rI1hx/H11032,y,z/H11032i /H20849/H9262V/H20850/H20849 mA/H9262T//H92620/H20850 x/H11032 +2.60 +2.55 231.1 −15.7 +16.4 y +0.95 +0.30 97.1 +14.0 +4.4 z/H11032 +0.12 0.00 231.1 0.0 −0.7 FIG. 12. /H20849Color online /H20850Angular dependent coupling of the mag- netization Mto the dynamic magnetic field h=/H20849hx/H11032,hy,hz/H11032/H20850. Only the components of hperpendicular to M0can excite precession of Mand therefore generate a dynamic m.hyis always exciting m. The excitation strength of hx/H11032andhz/H11032is angular dependent /H20851com- pare Eq. /H2084933/H20850/H20852. Here the two symmetry cases are shown: M /H20849a/H20850 perpendicular /H20849only hz/H11032andhycan excite M/H20850and /H20849b/H20850parallel /H20849only hx/H11032andhycan excite M/H20850to the stripe. FIG. 13. /H20849Color online /H20850Direction and ellipticity of the rf mag- netic field hdisplayed by showing the path I1·hpasses during one cycle. This is shown at the location of the three stripes /H20849these lie normal to the picture on top of the gray GaAs substrate; the200 /H9262m wide stripe to the right /H20850for two sample positions. I1·hwas determined by means of Eq. /H2084933/H20850. The upper right path corresponds to the I1·hfrom Table I. The hatched edges indicate metal surfaces reflecting microwaves. Within the waveguide the rf magnetic field h corresponding to the TE 10mode is displayed in the background.MICROWA VE PHOTOVOLTAGE AND PHOTORESISTANCE … PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-11D. Determination of the rf magnetic field direction Using the different angular dependencies of the three symmetric and three antisymmetric terms in Eq. /H2084933/H20850hI1can be determined. We make the assumption that the stripe itselfdoes not influence the rf magnetic field configuration, what isat least the case when further reducing its dimensions. Thusthe film stripe becomes a kind of detector for the rf magneticfield h. To test this an array of 36 additional 50 /H9262m wide and 20/H9262m distant Permalloy stripes of the same height and length as the 200 /H9262m wide stripe described above /H20849see Sec. III A /H20850was patterned beside this one. The 50 /H9262m wide stripes were connected with each other at alternating ends to form along meandering stripe. 9Four stripes were elongated on both ends to 300 /H11003300/H9262m2Permalloy contact pads. For the outer two stripes and the single 200 /H9262m stripe hI1is calculated from the measured photovoltage using Eq. /H2084923/H20850. Table I shows the measured voltage and the corresponding hI1for the 200 /H9262m stripe at 1 mm distance from the waveguide. hI1 for all three stripes is displayed in Fig. 13, while positioning the sample at two distances /H208491 and 3.5 mm, respectively /H20850 from the waveguide end. For comparison the rf magneticfield hconfiguration of the TE 01mode is displayed in the background. From other measurements we can estimate thatI 1lies somewhere in the 1 mA range. It is worth noting that possible inhomogeneities of the rf magnetic field hwithin the Permalloy stripes will be aver- aged because UMWis linear in h. Determining the sign of the rf magnetic field components from the photovoltage contri-butions signs exhibits a certain complexity because a lot ofattention has to be paid to the chosen time evolution /H20849e i/H9275tor e−i/H9275t/H20850and coordinate system /H20849right hand or left hand /H20850. How- ever, the sign only reflects the phase difference with respectto the rf current. The rf current is admittedly not identical fordifferent stripe positions. Consequently the comparison ofthe rf magnetization phase at different stripe locations is ob-scured. It is a specially interesting point concerning microwave photovoltage that the phase of the individual components ofthe rf magnetic field with respect to the rf current, and there-fore also with respect to each other can be determined. Thephase information is encoded in the line shape, which is aparticular feature of the microwave photovoltage describedin this work. At this point only determining hI 1is possible because I1is unknown. However, in Sec. IV A, an approach to determineI 1using the bolometric effect is presented. Using this ap- proach the bolometric photoresistance is the perfect supple-ment for the photovoltage. It delivers unknown I 1with al- most no additional setup. IV . PHOTORESISTANCE MEASUREMENTS The principle difficulties when detecting the AMR in- duced photoresistance are to increase the microwave powerfor a sufficient signal strength and to reduce the photovoltagesignal, which is in general much stronger and superimposeswith the photoresistance. We overcome the microwavepower problem by using high initial microwave power/H20849316 mW /H20850and a coplanar waveguide /H20849CPW /H20850, 10which emits the microwaves as close as possible to the Permalloy film stripe /H208490.137/H1100320/H110032450/H9262m3/H20850with which we detect the photoresistance. Its resistance is found to be R=880 /H9024and the AMR RA=15/H9024. Its magnetic properties /H20849/H9253,M0/H20850are al- most identical to that of the sample investigated in Sec. III.We use again lock-in technique like in III A with now anadditional dc current from a battery to measure resistanceinstead of voltage. The strong microwave power results instrong rf currents within the sample which give a speciallystrong photovoltage signal /H20851see Eq. /H2084927/H20850/H20852. To achieve a suf- ficiently strong photoresistance signal the dc current I 0and rf current I1have to be increased to the maximal value that does not harm the sample /H20849a few mA, hence I0/H11015I1/H20850. Ignoring the trigonometric factors sin 2 /H92510, cos 2 /H92510, and cos/H9274as well as the photoresistance term depending on /H92521 /H20849that is always smaller than /H92511/H20850the photovoltage signal /H20851UMW=/H92511sin /H208492/H92510/H20850cos/H9274RAI1/2, Eq. /H2084921/H20850/H20852and the photore- sistance signal /H20851/H9004RMWI0/H11015−/H925112cos /H208492/H92510/H20850RAI0/2, Eq. /H2084917/H20850/H20852be- come almost identical. But the major difference is that the photoresistance is multiplied by /H925112and the photovoltage only by /H92511.A s/H92511is particularly small /H20849/H110211°/H20850in our experi- ments, this means that /H9004RMWI0is much smaller than UMW. However, suppressing UMWis possible because it vanishes for/H92510=0°, 90°, 180°, 270° /H20851see Eq. /H2084921/H20850/H20852. A very precise tuning of /H92510with an accuracy below 0.1° is necessary to suppress UMWbelow /H9004RMWI0. Fortunately in contrast to /H20841UMW /H20841,/H20841/H9004RMW /H20841is maximal for /H92510=0°, 90°, 180°, 270°. In the following, we will first discuss the bolometric photore-sistance arising from microwave heating of the sample andafterwards the AMR induced photoresistance that is dis-cussed above. A. Bolometric (nonresonant) The AMR-induced /H9004RMWis not the only photoresistive effect present in our Permalloy film stripe. Also nonresonantheating by the microwave rf current I 1results in a /H20849bolomet- ric/H20850photoresistance. The major difference compared to the AMR-based photoresistance is that the bolometric photore-sistance is almost independent of the applied dc magneticfield Hand that its reaction time to microwave exposure is much longer /H20849in the order of ms /H20850than that of the AMR-based photoresistance /H208491/ /H9251G/H9275, in the order of ns /H20850.12The nonreso- nant bolometric photoresistance is found with a typicalstrength of /H20849/H9004R/R/H20850/P=0.2 ppm /mW /H20849see Fig. 14/H20850. The bolometric heating power P bolarises from resistive dissipation of the rf current I1in the sample /H20849Pbol=/H20855RI2/H20856 =RI02+RI12/2/H20850. This can hence be used to determine I1, which is otherwise an unknown in Eq. /H2084927/H20850.I1can be determined for example by finding the corresponding dc current I0with the same bolometric resistance change. However, especiallyin the sample we use the thermal conductivity of the GaAscrystal on which our Permalloy stripes were deposited is sohigh /H2084955 W /mK /H20850that the different stripes are strongly ther- mally coupled. Thus we cannot address the bolometric signal of one stripe solely to the rf current of the same stripe. Thiseffect was verified comparing the resistance changes fromone stripe while applying a dc current through an otherstripe. Hence determination of /H20841I 1/H20841by means of Eq. /H2084927/H20850isMECKING, GUI, AND HU PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-12only possible when using a substrate material with low heat conductance /H20849e.g., glass /H20850or by not depositing more than one stripe. B. AMR based (resonant) In contrast to the nonresonant bolometric photoresistance in Sec. IV A, the typically 50 times weaker resonant AMR-based photoresistance is very hard to detect. After visualizingit by using the CPW and turning the sample into a highsymmetry position /H20849parallel or perpendicular to H/H20850it is still necessary to regard the difference of the photoresistancemeasured with the same current strength but with reversedcurrent sign instead of measuring with only one current di-rection. This eliminates the remaining still significant photo-voltage signal, which depends on the absolute currentstrength possibly due to bolometric AMR change. Measurement results are presented in Fig. 14for f =3.8 GHz. There it can be seen that /H20849as deduced in Sec. II C /H20850, if the stripe lies parallel to the magnetization, the AMR is maximal and the resistance decreases when the FMR isexcited /H20849negative photoresistance /H20850. In contrast in the perpen- dicular case the AMR is minimal and we measure a resis-tance increase /H20849positive photoresistance /H20850. This behavior is schematically explained in Fig. 15. The curves in Fig. 14 show the photoresistance at the FMR with symmetric Lor-entz line shape as predicted in Sec. II C. Using Eq. /H208499/H20850we calculate /H92620H0=16.6 mT. However, a deviation of H0is found in both, parallel /H20849/H92620H0=11.1 mT /H20850 and perpendicular /H20849/H92620H0=25.3 mT /H20850, configuration. This is due to demagnetization which gives rise to an FMR shift with respect to the result from the infinite film approximation/H20851compare Eq. /H2084931/H20850/H20852.N x=0.7% can be assumed because of this shift.Using Eq. /H2084916/H20850we find that for our conditions mx/my =7.9 i. Consequently, we can neglect the contribution from /H92521=/H20841my/H20841/M0in Eq. /H2084917/H20850and find /H20841mx/H20841=13 mT using /H9004RMW=/H20849/H9004R/R/H20850R=1.23 m /H9024 /H20849from Fig. 14/H20850and thus /H92511 =/H208812/H9004RMW/RA=0.73° and /H92521=/H92511//H20841mx/my/H20841=0.09°. The smallness of /H92521is the reason for the resonant photoresistance strength being almost identical for M /H20648IandM/H11036I/H20849although the sign is reversed /H20850. We must expect /H20841mx/H20841,/H92511, and/H92521to be even a little bit larger due to our lock-in measurement tech-nique only detecting the sinusoidal contribution to the squarewave signal from the microwaves. The photoresistive decrease is in accordance with that found by Costache et al. 13There the magnetization is aligned with the current /H20849/H92510=0 /H20850. Thus applying an rf magnetic field decreases the AMR from RAtoRAcos2/H9258c. This is used to determine the precession cone angle /H9258cby assuming /H9258c=/H92511 =/H92521. The height to width ratio of the strip is 35 nm to 300 nm. Because of the magnetization lying along the stripe,13the magnetization precession strongly deviates from being circu-lar. Using the corresponding parameters /H92620M0=1.06 T, /H9253 =2/H9266/H92620/H1100328 GHz /T, and /H9275/2/H9266=10.5 GHz /H20850, we find from Eq. /H2084916/H20850that the ratio of the amplitudes is mx/my=3.15 i. This indicates strongly elliptical precession and suggests thatdistinguishing /H92511and/H92521would provide a refined description compared to that using the cone angle /H9258c, as discussed in Sec. II C. V . CONCLUSIONS We have presented a comprehensive study of dc electric effects induced by ferromagnetic resonance in Py micro-strips. A theoretical model based on a phenomenological ap-proach to magnetoresistance is developed and compared with FIG. 14. /H20849Color online /H20850Photoresistance /H9004RMWmeasurement /H20849stripe resistance R/H20850. The curves show the difference between the signals /H9004Uwith I0= +5 mA and I0=−5 mA at P=316 mW: /H9004R =/H20851/H9004U/H20849I0=+5 m A /H20850−/H9004U/H20849I0=−5 mA /H20850/H20852/10 mA. The subtraction suppresses the photovoltage dependence on absolute /H20841I0/H20841/H20849for ex- ample from bolometric AMR change /H20850. For both curves the dc mag- netic field H/H20849and so M/H20850was applied within the film plane, but for /H20849a/H20850parallel to the stripe /H20849and hence to the dc current I0/H20850and for /H20849b/H20850 perpendicular. A nonresonant background of about 70 ppm frombolometric photoresistance is found. It is decreases by about1.2 ppm when the sample is turned from parallel to perpendicularconfiguration. This is caused by the 1.7% AMR which changes R and the bolometric signal proportionally. The FMR signal has al-most Lorentz line shape and its position is significantly changingwhen the sample is turned from parallel to perpendicular position/H20849see Sec. IV B /H20850. FIG. 15. /H20849Color online /H20850Demonstration of the angular depen- dence of the microwave photovoltage: Without microwaves /H20851/H20849a/H20850, /H20849c/H20850/H20852the AMR is /H20849a/H20850minimal in perpendicular configuration of M andIand /H20849c/H20850maximal in parallel configuration. When the micro- waves are switched on the resistance /H20849b/H20850increases in parallel con- figuration and /H20849d/H20850decreases in perpendicular configuration.MICROWA VE PHOTOVOLTAGE AND PHOTORESISTANCE … PHYSICAL REVIEW B 76, 224430 /H208492007 /H20850 224430-13experiments. These provide a consistent description of both photovoltage and photoresistance effects. We demonstrate that the microwave photoresistance is proportional to the square of magnetization precession am-plitude. In the special case of circular magnetization preces-sion, the photoresistance measures its cone angle. In the gen-eral case of arbitrary sample geometry and ellipticalprecession, we refine the cone angle concept by defining twodifferent angles, which provide a precise description of themicrowave photoresistance /H20849and photovoltage /H20850induced by elliptical magnetization precession. We show that the micro-wave photoresistance can be either positive or negative, de- pending on the direction of the dc magnetic field. In contrast to the microwave photoresistance, we find that the microwave photovoltage is proportional to the product ofthe in-plane magnetization precession component with the rfcurrent. Consequently, it is sensitive to the magnetic fielddependent phase difference between the rf current and the rfmagnetization. This results in a characteristic asymmetricphotovoltage line shape, which crosses zero when the rf cur-rent and the in-plane component of the rf magnetization areexactly 90° out of phase. Therefore, the microwave photo-voltage provides a powerful insight into the phase of magne- tization precession, which is usually difficult to obtain. We demonstrate that the asymmetric photovoltage line shape is strongly dependent on the dc magnetic field direc-tion, which can be explained by the directional dependenceof the magnetization precession excitation. By using themodel developed in this work, and by combining such asensitive geometrical dependence of the microwave photo-voltage with the bolometric photoresistance which indepen-dently measures the rf current, we are now in a position todetect and determine the external rf magnetic field vector,which is of long standing interest with significant potentialapplications. ACKNOWLEDGMENTS We thank G. Roy, X. Zhou, and G. Mollard for technical assistance and D. Heitmann, U. Merkt, and the DFG for theloan of equipment. N.M. is supported by the DAAD. 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Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 /H208492006 /H20850. 8M. V . Costache, M. Sladkov, S. M. Watts, C. H. van der Wal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603 /H208492006 /H20850; J. Grollier, M. V . Costache, C. H. van der Wal, and B. J. van Wees, J. Appl.Phys. 100, 024316 /H208492006 /H20850. 9Y . S. Gui, S. Holland, N. Mecking, and C.-M. Hu, Phys. Rev. Lett. 95, 056807 /H208492005 /H20850. 10Y . S. Gui, N. Mecking, X. Zhou, Gwyn Williams, and C.-M. Hu, Phys. Rev. Lett. 98, 107602 /H208492007 /H20850. 11Y . S. Gui, N. Mecking, and C.-M. Hu, Phys. Rev. Lett. 98, 217603 /H208492007 /H20850. 12Y . S. Gui, N. Mecking, A. Wirthmann, L. H. Bai, and C.-M. Hu, Appl. Phys. Lett. 91, 082503 /H208492007 /H20850. 13M. V . Costache, S. M. Watts, M. Sladkov, C. H. van der Wal, and B. J. van Wees, Appl. Phys. Lett. 89, 232115 /H208492006 /H20850. 14A. Yamaguchi, H. Miyajima, T. Ono, Y . Suzuki, S. Yuasa, A. Tulapurkar, and Y . Nakatani, Appl. Phys. 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PhysRevB.80.134401.pdf

Influence of three-dimensional dynamics on the training effect in ferromagnet-antiferromagnet bilayers Paolo Biagioni, *Antonio Montano,†and Marco Finazzi LNESS, Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy /H20849Received 10 April 2009; revised manuscript received 19 July 2009; published 1 October 2009 /H20850 Training effect in exchange-bias systems consists of a variation in coercivity and symmetry between the first reversal after field cooling and the following loops. It has been shown, in the frame of a two-dimensionalcoherent-rotation approach, that training might be explained in terms of an initial noncollinear arrangement ofthe antiferromagnetic spins after field cooling, which relaxes to a collinear arrangement during the first reversal/H20851A. Hoffmann, Phys. Rev. Lett. 93, 097203 /H208492004 /H20850/H20852. In this paper, we extend the model to three dimensions, by numerically solving the Landau-Lifshitz-Gilbert equation describing the precession motion of magneticmoments. We are thus able to discuss the validity of Hoffmann’s model within a three-dimensional approach,with parameter values similar to those in the original publication, and to enlighten the role of out-of-planeanisotropies and Gilbert damping in determining the occurrence of training. Moreover, when realistic valuesare considered for the magnetocrystalline anisotropy of the system, we find that no training is reproducedwithin our extended model, suggesting that symmetry-driven irreversibilities might not be as relevant aspreviously believed for training effect. DOI: 10.1103/PhysRevB.80.134401 PACS number /H20849s/H20850: 75.40.Mg, 75.60.Ej, 75.60.Jk, 75.70.Cn I. INTRODUCTION Magnetization reversal in bilayer systems constituted by a ferromagnet /H20849FM /H20850and an antiferromagnet /H20849AFM /H20850is often characterized by a shift and by enhanced coercivity in thehysteresis loop. This effect, known as exchange bias /H20849EB/H20850, finds fundamental applications in the field of magnetic-datastorage and has originated a large debate and a flourishing ofpublications in the attempt to give it a firm description. 1 While it is now widely recognized, both experimentally and theoretically, that EB mechanisms must be described at amicroscopic level by taking the detailed spin structure at theinterface and inside the AFM into account, 2–9some peculiar features of magnetization reversal have been proposed to de-pend mainly on the average exchange and anisotropy ener-gies, which are well described even within the frame of mac-roscopic coherent-rotation models. An important example is given by the training effect /H20849TE/H20850, resulting in a different coercivity and a change in symmetrybetween the first reversal after field cooling and the follow-ing loops. 10–19It has been suggested that two mechanisms can contribute to this phenomenon: on the one hand, in somepolycrystalline samples, TE seems to be connected with thedomain microstructure in the EB system and with thermallyactivated depinning of AFM spins, as corroborated by ex-perimental and numerical results. 15,19On the other hand, ex- periments reveal that in some systems TE can be indepen-dent on the crystalline quality of the film. 13Hoffmann has shown20that this observation might be related to the aniso- tropy symmetry properties of the magnetic films and inter-preted in terms of a difference in the arrangement of themacroscopic magnetic moments in the AFM between the ini-tial condition right after field cooling /H20849noncollinear arrange- ment /H20850and all the following configurations /H20849collinear arrange- ment /H20850: while the first loop begins with the system in its minimum-energy configuration, which could be reached bysurmounting energy barriers during field cooling, all the fol-lowing loops lead to a metastable configuration, which modi- fies the symmetry and coercivity of reversal in the FM layer.This mechanism is fully determined by exchange and aniso-tropy energies in the system, within a macroscopic descrip-tion of magnetic moments, and has been reproduced by mini- mizing the total energy of the system as a function of theapplied field. However, two approximations represent a pos-sible limit to the application of such a model: 20/H20849i/H20850the system is treated by assuming an infinite in-plane anisotropy, fullyconfining the moments in the plane of the film and /H20849ii/H20850the values chosen for the in-plane magnetocrystalline anistropyof the AFM, when compared with the exchange energies, areroughly two to three orders of magnitude larger than those inrealistic EB systems. One of the most intriguing aspects of hysteresis-loop simulation in EB systems are the very different results thatare sometimes obtained when minimization algorithms arecompared with calculations where the full Landau-Lifshitz-Gilbert /H20849LLG /H20850equation is solved to describe precession of the magnetic moments. A remarkable example is given in apaper by Schulthess and Butler, 21who showed how Koon’s model for FM-AFM interfaces22is not a good description for EB when moment precession, rather than energy minimiza-tion, is taken into account. There is a fundamental reason forthis: EB reversal dynamics, as also evident in Hoffmann’smodel for TE, develops in an energy landscape which showsmany local energy minima. In this situation, the transientdynamics of magnetic moments, i.e., their path toward equi-librium, can largely influence the final local energy minimumwhere the system falls. Different paths lead to a differentability of overcoming energy barriers and therefore to differ-ent final steady states. 23The simulation of a magnetic mo- ment preceding around an effective field therefore allows thesystem to reach new final states, which could not be reachedby means of simple in-plane rotation. In the frame of this discussion, an important issue is to extend Hoffmann’s model for symmetry-driven TE, wherePHYSICAL REVIEW B 80, 134401 /H208492009 /H20850 1098-0121/2009/80 /H2084913/H20850/134401 /H208497/H20850 ©2009 The American Physical Society 134401-1the interplay between local and global minima plays a key role, to a three-dimensional /H208493D/H20850description for the evolu- tion of the FM-AFM system by means of LLG equations. Arecent paper by Saha and Victora 24already applies LLG dy- namics to a polycrystalline FM-AFM bilayer composed ofnoninteracting, randomly oriented grains. Their paper high-lights the role of micromagnetic domain evolution on EB andTE. However, the presence of many grains while making thesystem more realistic, partially conceals the role of aniso-tropy in the TE. Indeed, for some parameter values, they findtraining even in the case of uniaxial magnetocrystalline an-isotropy in the AFM, at variance with Hoffmann’s model,probably due to the many degrees of freedom made availableby the randomly oriented grains. In this work we simulate the behavior of an FM-AFM bilayer by solving the LLG equation. In the first part of thepaper, we show that the LLG equation can indeed reproduceTE within a three-dimensional extension of Hoffmann’smacroscopic model, when an initial in-plane noncollinear ar-rangement of the AFM moments is considered and as long asparameter values similar to those in the original manuscriptare chosen. In doing this, we also enlighten some differenceswhich emerge in the magnetic-moment configurations. Inparticular, the presence of a finite out-of-plane anisotropyopens a new channel for AFM spin relaxation by out-of-plane reorientation, which turns out to be strictly connectedwith the occurrence of training. To further enlighten the keyrole of the precession motion, we also show how changes inthe Gilbert damping constant can as well rule the occurrenceof training, by determining different paths towardequilibrium. In the second part of the paper, we choose the system parameters, particularly the magnetocrystalline anisotropy inthe AFM, in order to better adhere to the properties of real-istic EB bilayers. In doing so we find that, although noncol-linear initial conditions can still be obtained, they now pos-sess a large out-of-plane component. When hysteresis loopsare then simulated by solving the LLG equations, no trainingis observed anymore, a hint that symmetry-driven effectsmight be responsible for TE only in the limit of very largemagnetocrystalline anisotropy. II. MODEL The system under study is an FM/AFM bilayer, modeled following Ref. 20in the frame of a coherent-rotation ap- proach as an ensemble of three magnetic moments MF, MAF1, and MAF2, the first one describing the FM layer and the other two for the two sublattices representing the AFMlayer /H20849see Fig. 1/H20850. The total energy of the system can be written as the sum of Zeeman, anisotropy, and exchange/H20849AFM exchange and interface exchange /H20850contributions E tot=EZeeman +Eanisotropy +Eexchange . /H208491/H20850 The temporal evolution of each magnetic moment Miis de- scribed by the LLG equation25–27dMi dt=−/H9253Mi/H11003Hi+/H9251 /H20841Mi/H20841Mi/H11003dMi dt, /H208492/H20850 where /H9253is the gyromagnetic ratio of the electron spin, /H9251is the Gilbert damping constant, and Hiis the effective field acting on the ith magnetic moment, defined as Hi=−/H11509Etot /H11509Mi. /H208493/H20850 A normalized LLG equation can then be written by substi- tuting mi=Mi//H20841Mi/H20841and/H9270=/H9253t. Hence the system dynamics is fully determined once the damping constant /H9251and the total energy Etotare provided. The latter can be written by consid- ering the following expressions: EZeeman =−/H20858 iH0·Mi; /H208494a/H20850 Eanisotropy =/H20858 i/H20841Mi/H20841/H20875−1 2k1,i/H20849mi,x4+mi,y4+mi,z4/H20850 +k2,imi,y2+k3,imi,z2/H20876; /H208494b/H20850 Eexchange =−/H20858 i/HS11005jJi,jMi·Mj, /H208494c/H20850 where H0is the external applied field, k1,i/H110220 and k2,i/H110210 are anisotropy constants describing cubic and uniaxial magneto-crystalline anisotropy, respectively, k 3,i/H110220 describes in- plane anisotropy due to both shape anisotropy /H20849for the FM /H20850 and interface anisotropy associated to the removal of inver-sion symmetry in a layered structure /H20849for both FM and AFM moments /H20850, and finally J i,jis the exchange coupling constant between the ith and the jth magnetic moment. The exchange coupling energy contains the AFM exchange coupling/H20849J AF1,AF2 /H110210/H20850and the interface exchange coupling of the FM layer with the first /H20849JF,AF1/H110220/H20850and the second /H20849JF,AF2/H110220/H20850 AFM sublattice. As the effect of a finite temperature is notincluded in the model, results must be interpreted as a zero-temperature limit. In order to implement a numerical solution for the LLG equation, a suitable constraint must be imposed to numeri- z xy/CID2F /CID1F mAF1 mAF2mF FIG. 1. /H20849Color online /H20850Sketch of the simulated FM-AFM system with the polar coordinate system used throughout the paper.BIAGIONI, MONTANO, AND FINAZZI PHYSICAL REVIEW B 80, 134401 /H208492009 /H20850 134401-2cally ensure conservation of the magnitude of magnetic mo- ments during their evolution. A natural choice is to rewritethe normalized LLG equation in polar coordinates, whichautomatically guarantees /H20841m/H20841=1. The vectorial LLG equation /H20849three equations, three unknowns for each moment /H20850is thus replaced by the following system /H20849two equations, two un- knowns for each moment, see Fig. 1/H20850: d /H9277 d/H9270+/H9251d/H9272 d/H9270sin/H9277=−hxsin/H9272+hycos/H9272; /H208495a/H20850 −/H9251d/H9277 d/H9270+d/H9272 d/H9270sin/H9277=hzsin/H9277−/H20849hxcos/H9272+hysin/H9272/H20850cos/H9277. /H208495b/H20850 This finally yields a system of six nonlinear, strongly inter- twined ordinary differential equations, which is solved bymeans of a multistep adaptive algorithm based on numericaldifferentiation formulas of order 5. 28 In order to provide the numerical code with suitable initial conditions, mimicking the state of the system after field cool-ing, we find the absolute minimum-energy configuration, fora given set of parameters, by means of a global search heu-ristic method, namely, a genetic algorithm, because of theoccurrence of many local minima. 29,30After each iteration, a fast deterministic algorithm is used to refine the search be-fore fitness evaluation. III. SIMULATIONS FOR LARGE MAGNETOCRYSTALLINE ANISOTROPY In this section we simulate the FM-AFM system under study with parameter values in the range of those used byHoffmann. 20As a suitable initial condition for each loop simulation, the minimum-energy configuration of the mag-netic moments for a given set of parameters must be calcu-lated, in order to reproduce the state of the system after fieldcooling. Such an initial condition has already been derivedby Hoffmann in the two-dimensional limit of very largeAFM cubic magnetocrystalline anisotropy /H20849AFM moments always aligned along an easy axis /H20850and no FM magnetocrys- talline anisotropy /H20849FM moment always aligned with the ap- plied field /H20850. 20His results show the occurrence of three differ- ent regimes as a function of magnitude and direction of theapplied field, namely, parallel, antiparallel, and noncollinear/H20849perpendicular /H20850in-plane arrangements of the two AFM mo- ments M AF1and MAF2. It is also shown that if the cubic anisotropy term is replaced by a uniaxial term in the AFMthen the noncollinear phase disappears. It seems to be im-plicit in the paper that whenever the system is found in thenoncollinear phase after field cooling then its evolution ischaracterized by training in the FM hysteresis loop. We first test our genetic algorithm within a two- dimensional energy description /H20849i.e., fixing /H9277=90° in our model /H20850in order to reproduce the phase-diagram analytically calculated by Hoffmann but with finite anisotropy values. Weuse J F,AF1 =JF,AF2 =−0.4 JAF1,AF2 ,k1,F=−0.1 JAF1,AF2 MF, k1,AF1 =k1,AF2 =−0.4 JAF1,AF2 MF, no uniaxial anisotropy /H20849k2,i =0/H20850, and in-plane anisotropy only for the FM layer /H20849k3,F=−JAF1,AF2 MF,k3,AF1 =k3,AF2 =0/H20850. We indeed reproduce the trend already obtained by Hoffmann, just with slightlyshifted boundaries between different phases /H20849see Fig. 2/H20850.W e also find, in agreement with Ref. 20that the presence of uniaxial anisotropy prevents the stabilization of a noncol-linear AFM configuration /H20849not shown /H20850. We then include the /H9277degree of freedom in our descrip- tion and maintain the same parameters as above. In doingthis, we find again that three phases are present /H20849see Fig. 3/H20850, however the antiparallel phase now corresponds to a con-figuration where the two AFM moments have their main pro-jections along the out-of-plane anisotropy axis with just asmall canting /H20849/H1102130° with respect to the polar axis /H20850. This out-of-plane configuration can be attributed to the inherentin-plane frustration determined by the competition between0 5 10 15 20 25 30 35 40 450.51.01.52.02.53.03.54.0 |(+)/ | JH JF,AF 0 AF1,AF2 Field azimuth (degrees ) FIG. 2. Results from two-dimensional energy minimization with the genetic algorithm /H20849gray-scale boxes /H20850and comparison with Hoff- mann’s analytical model /H20849solid line /H20850. The arrows represent the ar- rangement of the two AFM sublattices. Each box is the result of onesimulation with parameter values corresponding to the center of thebox. White, gray, and black boxes correspond to in-plane noncol-linear, parallel, and antiparallel arrangements, respectively. 0 5 10 15 20 25 30 35 40 450.51.01.52.02.53.03.54.0 |(+)/ | JH JF,AF 0 AF1,AF2 Field azimuth (degrees ) FIG. 3. Results from three-dimensional energy minimization with the genetic algorithm. The arrows represent the arrangement ofthe two AFM sublattices. Each box is the result of one simulationwith parameter values corresponding to the center of the box.White, gray, and black boxes correspond to in-plane noncollinear,in-plane parallel, and out-of-plane antiparallel arrangements, re-spectively. In the out-of-plane antiparallel arrangement, a smallcanting is present, as described in the text.INFLUENCE OF THREE-DIMENSIONAL DYNAMICS ON … PHYSICAL REVIEW B 80, 134401 /H208492009 /H20850 134401-3AFM exchange coupling and interface exchange coupling, which is relaxed in the out-of-plane arrangement. Also thetwo in-plane parallel and antiparallel phases show a small canting with respect to the anisotropy axes. Within our extension of Hoffmann’s model to three di- mensions, we simulate training by first applying the minimi-zation genetic algorithm to find the system energy minimum,in order to describe the configuration of magnetic momentsafter field cooling. We then cover the whole hysteresis looptwice /H20849from negative fields to positive fields and back /H20850in order to evaluate the occurrence of training. For each fieldvalue, the LLG equations are solved numerically by takingthe configuration obtained at the end of the previous step asinitial condition and finally obtaining the new steady-statearrangement. The typical integration time for each step, cho-sen in order to fully reach a steady state, is /H9270/H1122910 000, while we use /H9251=0.1 as a damping constant. This damping value will be modified later on in order to discuss its influence onthe simulation results. Representative hysteresis loops are shown in Fig. 4, cal- culated starting from an initial condition of noncollinear ar-rangement for the two AFM sublattices /H20849white area of the phase diagram in Fig. 3/H20850. We indeed find that, for parameter values similar to those presented by Hoffmann in his ex-amples, TE is well reproduced /H20849see panel m F/H20648in Fig. 4/H20850. All the situations where we have occurrence of training do notqualitatively differ from this one. Loops are simulated withthe same set of parameters used for Fig. 3. The field is ap- plied in the plane of the sample with a /H9272=20° tilt with re- spect to the cubic anisotropy axis. In the figure we show thethree components for each of the three magnetic momentsinvolved in the simulations, namely, the two components inthe plane of the sample /H20849parallel and perpendicular to the applied field, respectively /H20850and the one perpendicular to the sample surface. By looking at the out-of-plane component ofthe two AFM moments /H20849see panels m AF1zandmAF2zin Fig. 4/H20850, it is clearly seen that during the first half loop they lay inthe plane of the sample, while their main projection is along the surface normal during the whole following evolution.This relaxation from an in-plane to an out-of-plane arrange-ment takes place during the first FM reversal and can beattributed to the already mentioned in-plane frustration deter-mined by the interplay between AFM and interface ex-change. Such an out-of-plane relaxation is a peculiar feature emerging from our model and it appears to be strictly con-nected with the occurrence of training. In order to prove this,we show in Fig. 5simulation results obtained with the same parameters and same initial conditions as in Fig. 4but with an in-plane anisotropy term added to the two AFM sublat-tices /H20849k 3,AF1 =k3,AF2 =−JAF1,AF2 MF/H20850, preferentially confining them in the sample plane. It is clearly seen that now theevolution of the AFM moments is fully confined in the planeof the sample /H20849see panels m AF1zandmAF2zin Fig. 5/H20850and that this is accompanied by no training /H20849see panel mF/H20648in Fig. 5/H20850. Such a behavior highlights that not only the symmetry of thein-plane anisotropy but also its out-of-plane component 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.30.40.5 Dampin g/c97KJ1,AF/|AF FM|Training No training FIG. 6. Graph showing the occurrence of training for a FM- AFM bilayer as a function of the damping coefficient /H9251and for three different values of the AFM cubic anisotropy constantsk 1,AF1 =k1,AF2. All other simulation parameters are the same as those used for the simulation shown in Fig. 4.-1.0-0.50.00.51.0mF/CID2/CID2 mFmFzmF/CID3 mAF1 mAF2-1.0-0.50.00.51.0 -1.0-0.50.00.51.0 H0 AF1,AF2/J-0.3 -0.15 0 0.15 0.3 H0/JAF1,AF2-0.3 -0.15 0 0.15 0.3 H0/JAF1,AF2-0.3 -0.15 0 0.15 0.3mAF1 /CID2/CID2 mAF2 /CID2/CID2mAF2 /CID3mAF1 /CID3 mAF2 zmAF1 z1st loop 2nd loop FIG. 4. /H20849Color online /H20850Results from LLG simulations of TE. For each magnetic moment the three components are shown, namely,the in-plane component parallel /H20849 /H20648/H20850to the applied field, the in-plane component perpendicular /H20849/H11036/H20850to the applied field, and the out-of- plane /H20849z/H20850component. The hysteresis loop is covered twice.-1.0-0.50.00.51.0mF/CID2/CID2 mFmFzmF/CID3 mAF1 mAF2-1.0-0.50.00.51.0 -1.0-0.50.00.51.0 H0 AF1 ,AF2 /J-0.3 -0.15 0 0.15 0.3 H0/JAF1 ,AF2-0.3 -0.15 0 0.15 0.3 H0/JAF1 ,AF2-0.3 -0.15 0 0.15 0.3mAF1 /CID2/CID2 mAF2 /CID2/CID2 mAF2 /CID3mAF1 /CID3 mAF2 zmAF1 z1st loop 2nd loop FIG. 5. /H20849Color online /H20850Results from LLG simulations with in- plane anisotropy for the AFM layers. All other parameters are thesame as in Fig. 4. For each magnetic moment the three components are shown, namely, the in-plane component parallel /H20849 /H20648/H20850to the ap- plied field, the in-plane component perpendicular /H20849/H11036/H20850to the ap- plied field, and the out-of-plane /H20849z/H20850component. The hysteresis loop is covered twice.BIAGIONI, MONTANO, AND FINAZZI PHYSICAL REVIEW B 80, 134401 /H208492009 /H20850 134401-4might play a significant role in determining the occurrence of training. It should then be stressed that, within our approach based on LLG equations, an in-plane noncollinear arrangement ofthe two AFM magnetic moments after field cooling is a nec-essary but not sufficient condition for the occurrence of train-ing. We indeed find that for several combinations of param-eter values, for which we can find a cooling field leading toan initial AFM noncollinear arrangement, nontrained loopsare nevertheless obtained. This can be attributed to the pecu-liar spiral-like path of the transient moment dynamics, lead-ing to a different ability of overcoming energy barriers com-pared with simulations based on energy minimization. Thisfinding is in full analogy with the analysis by Schulthess andButler 21about Koon’s model for FM-AFM interfaces,22 where the introduction of LLG equations extended the origi-nal results showing new possible regimes. As already pointed out, in such complex systems, such as FM-AFM interfaces, where the interplay between local andglobal minima plays an important role in the system dynam-ics, different transient spatial paths can lead to very different final steady states. This is also true when the evolution ischanged by modification in the damping constant /H9251. A larger damping value shrinks the spiral-like evolution of the mag-netic moments and therefore makes again different final-energy minima available. This is clearly shown in Fig. 6, where we analyze the occurrence of training as a function ofthe damping constant /H9251for three different values of the AFM cubic anisotropy constants k1,AF1 =k1,AF2. As a lower value for/H9251determines a longer characteristic evolution time for the system, we increase the value of /H9270accordingly, in order to ensure that a steady-state configuration is always reached.All other simulation parameters are the same as the onesused for the simulation in Fig. 4. The relevance of this find- ing is evident when considering that many common factorscan influence the damping constant, for example, the size ofa magnetic device, 31impurities,32–34or its operating temperature.35,36It should also be pointed out that, in FM/ AFM LLG simulations, care is often taken in order to ensurethat the results are independent of the value of the damping FIG. 7. /H20849Color online /H20850Results from three-dimensional energy minimization with the genetic algorithm, for a system with realistic magnetocrystalline anisotropy /H20849see text /H20850:/H20849a/H20850azimuthal angle /H20841/H92721−/H92722/H20841between the two in-plane projections of the AFM moments and /H20849b/H20850 average polar tilt /H20841/H92771−/H92772/H20841/2 of the two AFM moments. The arrows nearby the color bar are a sketch of the AFM moment geometry.INFLUENCE OF THREE-DIMENSIONAL DYNAMICS ON … PHYSICAL REVIEW B 80, 134401 /H208492009 /H20850 134401-5parameter.21,24While this might be the case for a single FM structure, our findings show that in the dynamics of a FM/AFM bilayer the damping constant might play a relevant rolein determining the local minimum reached during the rever-sal dynamics. IV . SIMULATIONS FOR SMALL MAGNETOCRYSTALLINE ANISOTROPY As briefly discussed in the introduction, the parameter values used in Hoffmann’s model, where all exchange andanisotropy energies are of the same order of magnitude,might be a poor description for many experimentally rel-evant systems showing EB and TE. If we restrict ourselvesto the case of CoO/Co bilayers, as in Ref. 20, the magnetocrystalline anisotropy constant takes a value ofabout 2 /H1100310 5erg /cm3,37corresponding to roughly 2/H1100310−6eV /atom once the lattice parameter of CoO is taken into account. On the other side, typical valuesfor the exchange integrals are 2 /H1100310 −4eV /atom and 2/H1100310−3eV /atom for the nearest-neighbor 90° exchange and the second-neighbor 180° exchange, respectively.38 Therefore, in a realistic model the exchange energy shouldbe two to three orders of magnitude larger than the magne-tocrystalline anisotropy. As for the AFM coupling at the in-terface between the FM layer and the two AFM sublattices, ithas been evaluated, assuming Heisenberg exchange acrossthe interface, to be on the order of 1 meV /nm 2.39In order to be used in our model, where all the spins of each sublatticeare represented by a single magnetic moment, such a valueshould be scaled down by the number of atomic layers con-stituting the film, which might be of some tens to some hun-dreds. Therefore, the interface exchange energy is also ex-pected to be two to three orders of magnitude lower than theAFM exchange coupling. According to the discussion above, we run new simula- tions for the initial conditions after field cooling, by means ofthe genetic algorithm. All parameter values are the same asbefore, except for the AFM magnetocrystalline anisotropyand the interface exchange coupling, which are set tok 1,AF1 =k1,AF2 =−0.01 JAF1,AF2 MF and JF,AF1 =JF,AF2 =−0.01 JAF1,AF2 , respectively. The results are shown in Fig. 7. Due to the low magneto- crystalline anisotropy, the phase diagram now shows a num-ber of configurations where the AFM moments are notaligned close to any of the anisotropy axes. Therefore, tobetter convey the complex 3D arrangement, we plot both theangle /H20841 /H92721−/H92722/H20841between the in-plane components of the two AFM moments /H20851panel /H20849a/H20850/H20852and the angle /H20841/H92771−/H92772/H20841/2/H20851panel /H20849b/H20850/H20852, which for AFM moments laying on opposite sides withrespect to the equatorial plane /H20849which is the case with our set of parameters /H20850provides the average polar tilt of the AFM moments with respect to such a plane. A close inspection of the results from energy minimiza- tion reveals that many noncollinear situations are again ob-tained but mostly with moment orientation not aligned withany of the anisotropy axes. We have extensively analyzed thehysteresis loops simulated with LLG equations starting fromsuch initial conditions and found that no sign of TE is everobtained. This result is a hint that, for realistic systems wherethe magnetocrystalline anisotropy is much lower than theAFM exchange, symmetry-driven contributions to TE mightbe less relevant than previously believed. V . CONCLUSIONS In conclusion, we have extended Hoffmann’s model for symmetry-driven TE in FM-AFM bilayers to three dimen-sions, by numerically solving the LLG precession equationfor the magnetic moments. For the same parameter values asthose used by Hoffmann, we verify that even within ourextended three-dimensional model the occurrence of trainingis strictly connected with the configuration of AFM magneticmoments after field cooling. Some peculiar new features ofthe training dynamics anyway emerge in our analysis. Firstof all, the transition during the first FM reversal is accompa-nied by an out-of-plane relaxation of the two AFM moments,driven by the inherent in-plane frustration between interfaceand AFM exchange. This enlightens that the out-of-planeanisotropy can play a key role in the occurrence of training.Moreover, an initial noncollinear AFM arrangement is a nec-essary but not sufficient condition for training, whose dy-namics strongly depend also on other system parameters. Inparticular, when realistic values are chosen for the exchangeand anisotropy energies, TE is not reproduced anymorewithin our model, suggesting that symmetry-driven irrevers-ibilities might not be as relevant as previously believed forTE in realistic systems where the magnetocrystalline aniso-tropy is much lower than the AFM exchange. All such considerations confirm that the behavior of FM- AFM interfaces can be very complex even within a coherent-rotation approach based only on three magnetic momentsand that therefore not only the symmetry-driven initial stateafter field cooling plays a role for TE but also the dynamicsof the magnetic moments as governed by anistropy, interfacecoupling, and damping. ACKNOWLEDGMENTS We warmly acknowledge L. Duò for discussions and for his continuous support. *paolo.biagioni@polimi.it †Present address: Edison s.p.a., Foro Buonaparte 31, 20121 Milano, Italy. 1For review articles, see J. Nogués and I. K. Schuller, J. Magn.Magn. Mater. 192, 203 /H208491999 /H20850; A. E. Berkowitz and K. Takano, ibid. 200, 552 /H208491999 /H20850; M. Kiwi, ibid. 234, 584 /H208492001 /H20850. 2A. P. Malozemoff, Phys. Rev. B 35, 3679 /H208491987 /H20850;37, 7673 /H208491988 /H20850; J. Appl. Phys 63, 3874 /H208491988 /H20850.BIAGIONI, MONTANO, AND FINAZZI PHYSICAL REVIEW B 80, 134401 /H208492009 /H20850 134401-63U. Nowak, K. D. Usadel, J. Keller, P. Miltényi, B. Beschoten, and G. Güntherodt, Phys. Rev. B 66, 014430 /H208492002 /H20850. 4J. Keller, P. Miltényi, B. Beschoten, G. Güntherodt, U. Nowak, and K. D. Usadel, Phys. Rev. B 66, 014431 /H208492002 /H20850. 5U. Nowak, A. Misra, and K. D. Usadel, J. Magn. Magn. Mater. 240, 243 /H208492002 /H20850. 6F. Nolting, A. 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PhysRevLett.104.146802.pdf

Inverse Spin-Galvanic Effect in the Interface between a Topological Insulator and a Ferromagnet Ion Garate1,2and M. Franz1 1Department of Physics and Astronomy, The University of British Columbia, Vancouver, BC V6T 1Z1, Canada 2Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada (Received 2 November 2009; published 9 April 2010) When a ferromagnet is deposited on the surface of a topological insulator the topologically protected surface state develops a gap and becomes a two-dimensional quantum Hall liquid. We demonstrate that the Hall current in such a liquid, induced by an external electric field, can have a dramatic effect on the magnetization dynamics of the ferromagnet by changing the effective anisotropy field. This change isdissipationless and may be substantial even in weakly spin-orbit coupled ferromagnets. We study thepossibility of dissipationless current-induced magnetization reversal in monolayer-thin, insulating ferro-magnets with a soft perpendicular anisotropy and discuss possible applications of this effect. DOI: 10.1103/PhysRevLett.104.146802 PACS numbers: 73.43. /C0f, 75.30.Gw, 75.70. /C0i, 72.15.Gd Introduction.— Understanding the electric-field control of magnetization and harnessing its technological potential are among the most important objectives of spintronics. Current-induced spin torques can reverse the magnetiza-tion of conducting ferromagnets (FM) and move magneticdomain walls [ 1]. However, the Joule heating generated by transport currents remains a handicap from a practicalviewpoint. An electric field can also reorient the magneti-zation of insulating compounds with broken inversionsymmetry via the magnetoelectric coupling [ 2]. While they overcome the issue with Joule heating, these multi- ferroic materials are fewer and more difficult to engineerthan common metallic ferromagnets. Recently, a novelmagnetoelectric effect has been discovered [ 3] in topologi- cal insulators that are coated with ferromagnetic films.Topological insulators (TIs) are bulk insulators with ananomalous band structure that supports topologically ro-bust gapless states at the surfaces [ 4]. These materials are predicted to display a variety of unconventional spintronics effects [ 5]. One unique feature is the universal quantized topological magnetoelectric effect [ 3], described by M top¼/C0C1e2 2/C25E: (1) HereMtopis the induced magnetization, C1is a half- integer topological invariant that depends solely on the sign of the time-reversal-symmetry-breaking perturbation,Eis the applied electric field, and C 1e2=2/C25/C17/C27His the Hall conductance ( @/C171throughout). Unfortunately, the prospects for manipulating the magnetization of real ferro-magnets via Eq. ( 1) are limited because below the thresh- old Hall current density ( j H<1A=m, see Ref. [ 6]) the topological magnetic field Btop¼/C220Mtop&10/C06Tis very small compared to typical coercive fields ( *0:01 T) in a ferromagnet. In this Letter we unveil a qualitatively new contribution to the topological magnetoelectric effect, which stems from the current-induced spin polarization of the TI sur- face states. Unlike Eq. ( 1), this effect depends on materialparameters and is not related to Ampere’s law; instead it is the topological counterpart of the inverse spin-galvanic effect (ISGE) [ 7], which has only recently been exploited in conducting ferromagnets with spin-orbit interaction [ 8]. The topological variant of ISGE is unique in that it isquantized, topologically protected, and occurs in insulatingferromagnets without spin-orbit interaction. In ultrathin(thickness &1n m ) ferromagnetic insulators deposited on a surface of TI (Fig. 1) the topological ISGE produce torques that may be comparable to the coercive field, thus opening an unprecedented avenue for current-induced control of magnetization without Joule heating. Functional integral formalism.— We begin by reviewing the equation of motion for the magnetization M/C17M^/C10of a classical ferromagnet (in units of 1=volume). At low en- ergies the magnitude Mis approximately constant and the only dynamical variable is the direction ^/C10¼ð/C10 x;/C10y;/C10zÞ. FIG. 1. Corbino-disk-shaped TI coated with an ultrathin ferro- magnet. (a) Top view: In the absence of electric fields, the magnetization of the ferromagnet points outside the page (dottedcircles). When a voltage difference is applied between the inner and outer circles, a dissipationless Hall current flows at the interface between the two materials (solid arrows). This currentmagnetizes the surface states of the TI (inverse spin-galvanic effect) along the radial direction (dashed arrows), thus exerting a spin torque on the magnetization of the ferromagnet. (b) Crosssectional view: The shaded region is the TI, whereas the un-shaded region is the ferromagnetic film. H FMis the anisotropy field in electric equilibrium. HCS;1is a topological magnetic field proportional to (and parallel to) the applied electric field.PRL 104, 146802 (2010) PHYSICAL REVIEW LETTERSweek ending 9 APRIL 2010 0031-9007 =10=104(14) =146802(4) 146802-1 /C2112010 The American Physical SocietyThe time dependence of ^/C10may be determined using the functional integral approach [ 9], which is built on the partition function Z¼Z0Z D^/C10ðx;tÞe/C0SFM½^/C10/C138: (2) Z0is the partition function corresponding to the equilib- rium magnetic configuration ^/C10¼^/C10eq.SFM¼SB/C0Eis the action for small (quadratic) spin fluctuations, where SB¼MRdxdt^/C10/C1ð^/C10eq/C2_^/C10Þis the Berry phase and E½^/C10/C138¼Rdxdt^/C10/C31/C01^/C10is the micromagnetic energy func- tional. /C31is the spin-spin response function. The semiclas- sical equation of motion can be derived from /C14SFM=/C14^/C10¼0, _^/C10¼^/C10eq/C2/C18 /C01 M/C14E /C14^/C10/C19 : (3) A gradient expansion [ 10]o f /C31yields the venerable Landau-Lifshitz-Gilbert-Slonczewski equation for magne-tization dynamics in the presence of damping and transportcurrents, _^/C10¼^/C10 eq/C2H/C0/C11^/C10eq/C2_^/C10/C0vs/C1r^/C10 /C0/C12^/C10eq/C2vs/C1r^/C10þ/C1/C1/C1 : (4) His an effective magnetic field (in energy units) that includes the anisotropy field, the exchange field, as wellas external magnetic fields. Hdetermines the easy axis along which the magnetization of a single-domain ferro-magnet points in equilibrium. v sis the adiabatic spin trans- fer velocity and is proportional to the transport current. /C11 and/C12characterize dissipative processes in which energy is transferred from magnetic to nonmagnetic (e.g., lattice) degrees of freedom. Topological effective magnetic field.— We now address the magnetization dynamics of an insulating ferromagnetsitting on top of a TI. The low-energy effectiveHamiltonian for the surface states of the TI is [ 3,4] H¼v F/C28/C1ð/C25/C2^zÞ/C0/C1/C28/C1^/C10; (5) where vFis the Fermi velocity, /C28i(i2fx; y; z g) are Pauli matrices denoting real spin of the surface states, /C25¼ /C0ir/C0eA,Ais the electromagnetic vector potential, ^z is the unit vector normal to the interface between the TI andthe ferromagnet, and /C1is the exchange coupling between the surface states and the local moments of the ferromagnet(/C1>0for ferromagnetic coupling). We consider a ferro- magnet with perpendicular anisotropy ( /C10 eq¼^z) so that in equilibrium a gap opens in the energy spectrum of the surface states. The partition function for this composite system is Z¼Z0Z D^/C10ðx;tÞe/C0SFM½^/C10/C138Z D2/C9ðx;tÞe/C0STI½/C22/C9;/C9;^/C10/C138;(6) where SFMis the ferromagnetic action discussed above andSTI¼Z d2xdt/C22/C9½@0/C0/C22/C0H/C138/C9 (7) is the action for the surface states. /C9is a fermionic spinor, @0¼@t/C0eA0,/C22is the chemical potential (located in the gap), and A0is the electrostatic potential. After rotating the spins by an angle /C25=2around ^z, Eq. ( 7) may be rewritten as STI¼Rd2xdt/C22c½@0/C0/C22/C0~H/C138cwith ~H¼vF/C28xð/C25x/C0eaxÞþvF/C28yð/C25y/C0eayÞ/C0/C10z/C1/C28z;(8) wherecis the rotated fermion field. In this new basis, a/C17 /C1=ðevFÞð^/C10/C2^zÞappears as an additional contribution to the effective vector potential. /C10z/C1acts as a mass term. These massive Dirac fermions may be integrated out in the standard manner [ 11], whereby Z¼RD^/C10ðx;tÞe/C0Seff½^/C10/C138. To second order in ^/C10the effective action is Seff’SFMþ SCSþSEB, where SCS¼e2 2/C25C1Z d2xdt/C15/C22/C23/C21A/C22@/C23A/C21; (9) ~A¼ðA0;Axþax;AyþayÞis the effective vector poten- tial and /C22¼t; x; y . The Chern-Simons (CS) action ( 9) arises in ( 2þ1)-dimensional systems with broken time reversal symmetry and nontrivial topology. The topology of the band structure is encoded in the Thouless, Kohmoto, Nightingale, den Nijs (TKNN) [ 12] invariant C1. For fer- mions described by a single Dirac Hamiltonian ( 8), we have [ 13] C1¼/C01 2sgnð/C10z/C1Þ: (10) SEBis quadratic in spatial and temporal derivatives of A/C22and encodes the ordinary dielectric or diamagnetic response of the gapped surface state. Herein we focus on SCS, which is first order in the derivatives of A/C22(and ^/C10), and thus outweighs SEBin the description of ^/C10ðx;tÞat long length and time scales. It also produces the effective mag- netic field that underlies the inverse spin-galvanic effectwhich is central to this study. The semiclassical magnetization dynamics follows from /C14S eff=/C14^/C10¼0, _^/C10¼^/C10eq/C2ðHFMþHCSÞþ/C1/C1/C1 ; (11) where HFM¼/C0/C14SFM=ðM/C14^/C10Þis the effective magnetic field that collects the anisotropy or exchange fields of theisolated ferromagnet and H CS¼/C01 M2D/C14SCS /C14^/C10¼/C0/C27H M2D/C1 evF/C20 Eþ/C1 evFð^z/C2_^/C10Þ/C21 (12) is an additional (topological) contribution to the magnetic field that results from the exchange coupling between theferromagnet and the TI. M 2Dis the areal magnetization at the interface (in units of 1=area). HCSdepends on materialPRL 104, 146802 (2010) PHYSICAL REVIEW LETTERSweek ending 9 APRIL 2010 146802-2parameters ( vF,/C1,M2D) and is proportional to the Hall conductivity /C27H¼C1e2=2/C25. Because the exchange cou- pling between the surface states and the localized moments of the ferromagnet is local in space, the influence of HCS weakens as the thickness of the ferromagnetic film increases. HCS;1/C17/C0/C1=ðevFM2DÞ/C27HEcan be interpreted as an electric-field induced change of magnetic anisotropy. The underlying cause of this effect is that the electric field spin polarizes the surface states along a direction ( E=E) which is misaligned with the equilibrium easy axis ( ^z). We illus- trate this point by computing the magnetization induced bya static and uniform electric field: /C14 EMi 2D/C17/C31ij M;EEj; (13) where /C31ij M;E¼lim !!0e i!1 AX kX n;n0/C28i n;n0vj n0;nfk;n/C0fk;n0 Ek;n0/C0Ek;nþ!(14) is the linear magnetoelectric response function [Fig. 2(a)]. n; n0are the band indices of the surface states, Ek;nare the band energies, fk;nare the Fermi distributions, /C28i n;n0¼ hn;kj/C28ijn0;ki, and Ais the area of the interface. From Eq. ( 5), the velocity operator is related to the spin operator viav¼@H=@k¼/C0vF/C28/C2^z, which allows us to use the TKNN formula for conductivity [ 12] and write /C31ij M;E¼/C0/C27H evF/C14ij; (15) where /C14ijis the Kronecker delta and we have used the fact that the longitudinal conductivity is zero. Hence, Hi CS;1¼ ð/C1=M 2DÞ/C14EMi 2D. This result is reminiscent of the current- induced effective field in single-domain metallic ferromag- nets that belong to the gyrotropic crystal class [ 14]. Some significant differences between Ref. [ 14] and the present work are that HCS;1(i) does not depend on the strength of spin-orbit interactions in the ferromagnet or at the interface [Eq. ( 5) involves vFrather than a ‘‘spin-orbit velocity’’],(ii) reverses sign when /C10z!/C0 /C10zand vanishes when /C10z¼0, and (iii) exerts a dissipationless torque provided that the ferromagnet is insulating. HCS;2/C17/C0 ð /C27H=M 2DÞð/C1=ev FÞ2^z/C2@t^/C10is associated with the change in the spin response function under a magnetic field [Fig. 2(b)]: /C31ij M;BðqÞ¼/C1 AX kX n;n0/C28i n;n0/C28j n0;nfk;n/C0fkþq;n0 Ekþq;n0/C0Ek;nþ!;(16) where q¼ð!;qÞis the energy-momentum of the magnon and/C28i n;n0¼hn;kj/C28ijn0;kþqi.A tq¼0we get /C31ij M;B¼ ð/C1=ev FÞ2ð/C0i!Þ/C27H/C15ij, where /C15xy¼/C0/C15yx¼1and/C15xx¼ /C15yy¼0. Thus, Hi CS;2¼ð/C1=M2DÞ/C31ij M;B/C10jsimply increases (if/C1>0) or decreases (if /C1<0) the Berry phase of the isolated ferromagnet [ 15], thereby renormalizing the pa- rameters entering Eq. ( 4). When the magnetization of the ferromagnet is uniform, Eq. (12) captures the entire current-induced spin torque for weak electric fields. In the presence of inhomogeneousmagnetic textures, one must add the ordinary spin transfer torque. The microscopic theory for v s/C1r^/C10amounts to evaluating the change of the xyspin-spin response function [10] under an electric field [Fig. 2(c)]. Starting from /C31xy M;BðqÞ, perturbing the matrix elements of the spin opera- tors to first order in E[16] and expanding the resulting expression to first order in q, we find (numerically) that vs/C1q//C10zðExqx/C0EyqyÞ. Furthermore, for realistic pa- rameters the torque exerted by HCSis found to dominate overvs/C1qby an ample margin even when jqj/C24nm/C01 (note that HCSdoes not vary as ^/C10is slightly tilted away from ^z). Current-induced magnetization switching.— As ex- plained above, HCS;1modifies the anisotropy field of the ferromagnet in the presence of a Hall current jH¼ /C27H^z/C2E: Han¼K M2D/C10z^zþ/C1 evFM2D^z/C2jH; (17) where Kis the anisotropy energy per unit area for the magnetic ultrathin film in electric equilibrium. When E¼ 0the magnetization of the ferromagnet points along ^z. After turning on the electric field, the magnetization beginsto precess around H anand (assisted by the damping) equilibrates along the modified easy axis. For instance, in a Corbino disk geometry depicted in Fig. 1the electric field produces a crown-shaped magnetization. Provided thatquantum coherence is preserved, this configuration hosts[17] a circulating spin-current proportional to Mð/C30Þ/C2 Mð/C30þ/C14/C30Þ//C10 zðjH/C2^zÞþOðE2Þ, which is radially po- larized and persistent (dissipationless). /C30is the azimuthal angle around the disk. IfjH*evFK=/C1,^/C10reaches the interface ( /C10z¼0)i n the course of the precession. At that moment, according to Eq. ( 10),C1¼0and hence @t^/C10¼0; yet this is an un- stable fixed point and an infinitesimal in-plane magnetic FIG. 2. Feynman diagrams for (a) the electric-field-induced magnetization (inverse spin-galvanic effect), (b) the xycompo- nent of the spin-spin response function, (c) the xycomponent of the spin-spin response function in the presence of an electric current (it yields the adiabatic spin transfer torque vs/C1r^/C10). The solid straight lines are propagators for massive Dirac quasipar- ticles (quasiholes). The solid wavy lines are magnons that coupleto the spin operator, and the dashed straight lines are photons that couple to the velocity operator.PRL 104, 146802 (2010) PHYSICAL REVIEW LETTERSweek ending 9 APRIL 2010 146802-3field suffices to kick the magnetization towards /C10z<0. Once this occurs the electric field may be turned off and the magnetization will equilibrate towards /C0^z. Thus a 180/C14 magnetization switching may be completed by combining a dissipationless Hall charge current with a very small magnetic field. Nevertheless, achieving jH*evFK=/C1in real materials presents challenges. First, jH(E) cannot be larger than /C241A=m(0:5m V =nm) because otherwise the dissipationless quantum Hall effect will break down [ 6]. Second, we require relatively small coercive fields: Hcoer¼K=M 2D&0:02 T. While such a soft perpendicu- lar anisotropy is inadequate for the magnetic recordingindustry, it may find applications in magnetic random access memories and magnetic field sensors [ 18]. Third, the thickness of the ferromagnet needs to be comparable tothe penetration depth of the Dirac fermions into the ferro- magnetic insulator ( &1n m ). While ultrathin films are commonplace in metallic ferromagnets [ 19], insulating ferromagnets such as EuO or EuS present additional ex- perimental difficulties (but see Ref. [ 20] for recent progress). Alternatively, one could electrically manipulate the spin textures caused by magnetic impurities placed on the surface of the TI [ 21,22]. Using /C1¼JM 2D,jH¼ 1A=m,vF¼5/C2105m=s, and Hcoer¼0:01 T, we esti- mate J*50 meV nm2as the condition for magnetization switching. Hence J=a2*0:2e V , where a’0:5n m is a typical lattice constant for the topological insulator. J=a2’ 0:2e V is an a priori reasonable value [ 21] for the exchange integral between the localized moments of the ferromag-netic insulator and the surface states of the TI. For stronger perpendicular anisotropies (say, H coer*0:05 T) the ex- change integral would need to be of the order of a few eV, and at such strong coupling the surface states of the TIwould be altered in a way not captured by Eq. ( 5). From the precession frequency ! prec’/C22BHan=@’1 GHz we infer switching times of the order of a nanosecond. There has been some recent work along the lines of the above discussion, albeit in topologically trivial materials[23]. There are two salient differences between Ref. [ 23] and the present work. (i) The microscopic origin of the change in magnetic anisotropy: in our case it is the current-induced spin-polarization of massive Dirac fermions (the topological inverse spin-galvanic effect), whereas Ref. [ 23] concentrates on the electrical manipulation of the atomicpositions and distortions of the charge distribution. (ii) Symmetry of the anisotropy mechanism: in our case it is odd under time reversal (because j His odd), whereas in Ref. [ 23] it is even under time reversal (because Eand charge density are even). Conclusions.— When a ferromagnetic film with perpen- dicular anisotropy is placed on top of a topological insu-lator, a quantum Hall current induces a spin torque which substantially modifies the magnetic easy axis. The origin of this new torque can be traced to a topological variant of the inverse spin-galvanic effect. In Corbino disk geometries this effect may be exploited to generate crown-shapedmagnetic textures and to switch the magnetization of a ferromagnet by 180 /C14without Joule heating. We thank I. Affleck, J. Folk, A. H. MacDonald, and G. Sawatzky for helpful comments and questions. This re- search has been supported by NSERC and CIfAR. I. 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PhysRevB.80.144427.pdf

Gilbert damping and current-induced torques on a domain wall: A simple theory based on itinerant 3 delectrons only L. Berger Physics Department, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA /H20849Received 23 April 2009; revised manuscript received 11 September 2009; published 30 October 2009 /H20850 Many electronic theories of Gilbert damping in ferromagnetic metals are based on the s-dexchange model, where localized 3 dmagnetic spins are exchanged-coupled to itinerant 4 selectrons, which provide the needed spin relaxation. Recently, Tserkovnyak et al. have obtained Gilbert damping from itinerant 3 delectrons alone, which have their own spin relaxation. We show that simple semiclassical equations of motion for precessingitinerant 3 dspins predict exactly the same formula /H9251=1 //H20849/H9275d/H9270srd/H20850for the Gilbert damping constant as the full Green’s function quantum treatment by Tserkovnyak et al. Here,/H9275dis the precession frequency of 3 dspins in thed-dmutual exchange field, and /H9270srdthe 3 dspin-relaxation time. A correct form for the spin-relaxation torque is crucial for success: The spins relax toward an instantaneous direction which is that of the vector sum ofexternal field and d-dexchange field. Remarkably, d-dexchange torques disappear completely from the equations of motion for the total 3 dmagnetization, and exchange plays only an indirect role through the spin relaxation. This purely 3 dmodel is simpler than the traditional s-dmodel. We also present a theory of current-induced torques on a domain wall, based on the 3 dmodel. We find equivalents to the so-called adiabatic and nonadiabatic torques. They are given by formulas similar to those holding for the s-dexchange model. DOI: 10.1103/PhysRevB.80.144427 PACS number /H20849s/H20850: 75.60.Ch, 85.75. /H11002d, 75.75. /H11001a, 75.47. /H11002m I. INTRODUCTION Damping of the motion of magnetic spins in ferromagnets is of the kind described by Gilbert, where the damping rate isproportional to the spin precession frequency. Many elec-tronic theories for metallic ferromagnets are based on the s-d exchange model, 1where localized 3 dmagnetic spins Sare coupled to itinerant 4 selectron spins sby an interaction E =−2JsdS·s, where Jsd/H112290.1–0.2 eV. Because of the momentum gap existing2between spin-up and spin-down Fermi surfaces, no damping is obtained at T =0 unless spin relaxation,3connected with a combination of spin-orbit interaction and electron scattering, is introducedfor the 4 selectrons. It is represented by a spin-relaxation time /H9270srs/H1122910−12–10−13s. One exception is the theory of Mills et al. ,4who showed that spin orbit can be replaced here bys-dexchange itself. Using s-dexchange and coupled semiclassical equations of motion for Sands, Turov5derived the value of the ferro- magnetic resonance linewidth. It is directly related to thedimensionless Gilbert damping parameter /H9251. In the limit /H9275s/H9270srs/H112711, this reduces to /H9251=s /H20849S+s/H20850/H20849/H9275s/H9270srs/H20850. /H208491/H20850 Here, Sandsare the magnitudes of Sands, with units of atom−1. The quantity /H9275s=2JsdS//H6036would represent the spre- cession frequency in the s-dexchange field set up by S,i f that field had a constant direction. Later, Heinrich et al.6treated this problem with a Green’s function formalism. Remarkably, this quantum treatmentyields exactly the same expression for /H9251/H20851Eq. /H208491/H20850/H20852as the simple equations of Turov5for the classical precession of S ands.Recently, Tserkovnyak et al.7obtained Gilbert damping from itinerant 3 delectrons alone, assumed to have their own spin-relaxation time /H9270srd.This purely 3 dmodel leads to /H9251=1 /H9275d/H9270srd, /H208492/H20850 where /H9275dwould be the 3 d-spin precession frequency in the Stoner exchange field generated by all the other 3 ditinerant spins, if that field had a fixed direction. The purpose of the present paper is to show that a simple classical equation of motion for a precessing 3 dspin predicts exactly the same formula for /H9251/H20851Eq. /H208492/H20850/H20852as the full quantum treatment by Tserkovnyak et al.7which uses Keldysh Green’s functions combined with the Boltzmann equation.The present approach also provides a clear physical picture/H20851Fig. 1/H20849b/H20850/H20852of processes involved in Gilbert damping. Actually, the fact that the present model uses only one kind of electron is more important than the exact dors nature of such electrons. II.S-DEXCHANGE MODEL The equations of motion for the localized 3 dmagnetic spinSand the itinerant 4 sconduction-electron spin s/H20851Fig. 1/H20849a/H20850/H20852are5 /H6036ds dt=−g/H92620/H9262Bs/H11003/H20849H+Hsd/H20850−/H6036/H20849s−s0/H20850 /H9270srs /H6036dS dt=−g/H92620/H9262BS/H11003/H20849H+Hds/H20850. /H208493/H20850 Here, His the external static field, Hsd=−2JsdS/g/H92620/H9262B the exchange field exerted by Sons,/H92620the vacuum perme-PHYSICAL REVIEW B 80, 144427 /H208492009 /H20850 1098-0121/2009/80 /H2084914/H20850/144427 /H208495/H20850 ©2009 The American Physical Society 144427-1ability in the SI system of units /H20849Systeme International /H20850, and Hds=−2Jsds/g/H92620/H9262Bthe field exerted by sonS. Also, /H9262Bis the Bohr magneton. The g-factor gis assumed for simplicity to have the same value for Sand for s. In Eqs. /H208493/H20850,s0is the instantaneous direction toward which sis relaxing. As discussed a long time ago by Hasegawa,8 this direction should be antiparallel to the total field H +Hsdacting on s/H20851Fig. 1/H20849a/H20850/H20852 s0=−sH+Hsd /H20841H+Hsd/H20841. /H208494/H20850 This choice of s0represents the instantaneous direction where the total Zeeman energy of swould be minimum. It is a logical choice since, during spin relaxation, the Zeemanenergy is lost to the lattice through emission of phonons. Choices which differ from Eq. /H208494/H20850would lead 5,8to shifts in the Sprecession frequency, away from the usual value; such shifts are not observed in actual resonance experiments.Note also that Walker 9has derived Eq. /H208494/H20850on the basis of Fermi-liquid theory. We introduce coordinates x,y, and z, with zantiparallel to H/H20851Fig.1/H20849a/H20850/H20852, and look for solutions of Eqs. /H208493/H20850and /H208494/H20850of the form s+/H20849t/H20850=s+/H208490/H20850e−/H20849/H9003+i/H9275/H20850t,S+/H20849t/H20850=S+/H208490/H20850e−/H20849/H9003+i/H9275/H20850t, where s+=sx +isy. We assume H/H11270Hsd,Hdsand /H20841s+/H20841/H11270s,/H20841S+/H20841/H11270S. We intro- duce the quantity /H9275s=2JsdS//H6036. It would represent the spre- cession frequency around Hsdif the latter had a fixed direc- tion. We obtain in the limit /H9275s/H9270srs/H112711 /H9275/H11229g/H92620/H9262BH /H6036;/H9003/H11229s/H9275 /H20849s+S/H20850/H9275s/H9270srs. /H208495/H20850 Then, the Gilbert damping parameter, defined as /H9251=/H9003//H9275, is given by Eq. /H208491/H20850in agreement with Refs. 5and6. Inter-estingly, starting with a Bloch-type spin-relaxation term in the equations of motion /H20851Eqs. /H208493/H20850/H20852, we arrived nevertheless /H20851Eq. /H208495/H20850/H20852to a Gilbert form for the damping rate /H9003, i.e., with /H9003/H11008/H9275. The Hterm in Eq. /H208494/H20850is responsible for this. III. ITINERANT D-DMODEL In this model, we consider only itinerant 3 delectrons, in Bloch waves with various wave vectors and spin states, la-beled with the index n=1,2,3,.... Paired spin-up and spin- down electrons of same wave vector give zero total spin, andcan be ignored. Only the remaining unpaired spin-up statesmatter. Since they all have different wave vectors, they canhave nonorthogonal spin parts while still being orthogonaland obeying the exclusion principle. This makes possible aclassical picture of individual precessing 3 dspins, pointing in different directions, with increased exchange energy. As mentioned before, the fact that only one kind of elec- tron appears in the model is more important than the exact d orsnature of such electrons. Actually, the two kinds of states are significantly mixed through s-dhybridization. This ques- tion will be discussed in more detail in Sec. VI. As in the last section, we write a classical equation of motion for the spin s n/H20851Fig.1/H20849b/H20850/H20852of an individual 3 delectron /H6036dsn dt=−g/H92620/H9262Bsn/H11003/H20849H+Hdd/H20850−/H6036sn−s0 /H9270srd. /H208496/H20850 Here, Hddis the d-d/H20849Stoner /H20850exchange field /H20851Fig. 1/H20849b/H20850/H20852 acting on sn, generated by all other itinerant 3 delectrons, and /H9270srd/H1122910−13–10−14s the 3 dspin-relaxation time. Also, Sis the total spin of 3 delectrons in the system, with S=/H9018nsn. The total exchange energy is −2 Jdd/H9018n/H11022m/H9018msn·sm. Then, Hdd is given by Hdd=−JddS/g/H92620/H9262B. For simplicity, we assume thed-dexchange integral Jddto have the same value between all pairs of 3 dstates. Band-structure calculations are consistent10with Jdd/H112290.5 eV. Similarly to the last section, and for the same reasons, sn relaxes /H20851see Fig. 1/H20849b/H20850/H20852toward the direction s0=−sH+Hdd /H20841H+Hdd/H20841. /H208497/H20850 The remarks about 1 //H9270srsmade in that section also apply to 1 //H9270srd. The mechanism of spin relaxation of Ref. 3works for 3 delectrons, since these are now assumed itinerant. We sum Eq. /H208496/H20850over n, to obtain an equation of motion for the total 3 dspinS /H6036dS dt=−g/H92620/H9262BS/H11003H−/H6036 /H9270srd/H20849S−S0/H20850, /H208498/H20850 where S0=−S/H20849H+Hdd/H20850//H20849/H20841H+Hdd/H20841/H20850. We see that exchange torques have disappeared from Eq. /H208498/H20850. The reason is that these are internal to the 3 d-electron system, not external as in the case of the s-dexchange model of last section. Exchange appears only indirectly, through S0in the spin-relaxation term. We define the quantity /H9275d=g/H92620/H9262BHdd//H6036. It would rep- resent the snprecession frequency around Hddif the latter had a fixed direction. Similarly, we define /H9275=g/H92620/H9262BH//H6036andso sdHdsH H Hddos snz z Hs a) b)S SS0 FIG. 1. /H20849a/H208504sconduction-electron spin sand 3 dmagnetic- electron spin Sprecessing around the magnetic field H. The s-d exchange field Hsdis antiparallel to Sand acts on s; and vice versa forHds. The vector s0is antiparallel to the total field H+Hsdacting ons, and is the direction toward which sis relaxing. /H20849b/H208503dindi- vidual spin snand total 3 dspinS=/H9018nsnprecessing around the mag- netic field H. The d-dmutual exchange field Hddis antiparallel to S and acts on sn. The vector s0is antiparallel to H+Hddand is the direction toward which snis relaxing.L. BERGER PHYSICAL REVIEW B 80, 144427 /H208492009 /H20850 144427-2S+=Sx+iSy, with zantiparallel to H/H20851Fig.1/H20849b/H20850/H20852. After assum- ingH/H11270Hdd,/H20841s+/H20841/H11270s,/H20841S+/H20841/H11270S, Eq. /H208498/H20850gives dS+ dt=−i/H9275S+−/H9275 /H9275d/H9270srdS+ dSz dt/H112290. To first order in the precession amplitude /H20841S+/H20841, the modu- lus of Sis constant. Again, we look for a solution of the form S+/H20849t/H20850=S+/H208490/H20850e−/H20849/H9003+i/H9275/H20850t, and find immediately /H9003=/H9275 /H9275d/H9270srd. /H208499/H20850 Then,/H9251=/H9003//H9275is given by Eq. /H208492/H20850in agreement with Ref. 7. Again, and for the same reasons, /H9003is of the Gilbert form. Even when taking into account s-dhybridization, we have /H9275s/H11021/H9275dbut/H9270srs/H11022/H9270srd. Thus, the dimensionless parameters /H9275srs/H9270srsin Eq. /H208491/H20850and/H9275srd/H9270srdin Eq. /H208492/H20850may have comparable values /H1122910–100. IV. CURRENT-INDUCED TORQUES ON A DOMAIN WALL, IN THE 3 dMODEL We consider a tail-to-tail wall in a nanowire /H20851Fig. 2/H20849a/H20850/H20852. The spatial coordinate Xruns along the length of the nano- wire. The total 3 dspinSat location Xmakes an angle /H9258/H20849X,t/H20850 with the − Xaxis. As an approximation,11we assume that the vector Sin the wall is everywhere contained in a plane P parallel to the Xdirection, which makes an angle /H9274with thesubstrate plane /H20851Fig.2/H20849b/H20850/H20852. In a static wall at zero current, we have/H9274=0. The sign convention for /H9274is such that it increases when Sturns toward the − xdirection. We introduce local spin coordinates x,y, and zwith zparallel to Sandxnormal toXand to plane P/H20851Fig. 2/H20849a/H20850/H20852. When /H9274differs from zero, the canted magnetization creates11in the wall a demagnetizing field HD. If the nano- wire thickness is much less than the width, this field is nor-mal to the substrate plane. The component of H Dalong the normal to plane PisHDx=−HDcos/H9274=−Msin/H9258sin/H9274cos/H9274. The torque exerted by HDxon the total 3 dspinSis in plane P and is /H9270y=/H20849/H92620Ms2/2/H20850sin/H208492/H9274/H20850sin/H9258. /H2084910/H20850 The usual energy eigenstates of an itinerant electron are plane waves where the spin direction is the same at all loca-tions. However, more general “spiral states” have beenintroduced 12to represent itinerant electrons in domain walls. As long as the wall width is much larger than an electronwavelength, the spatial variation in the direction of Sis slow and there is no difference with the usual theory of domainwalls based on localized electrons. The structure of a simpletransverse wall is given 13by/H9258=f/H20849X−vwt/H20850//H9004/H20850where vwand /H9004are the wall speed in the laboratory frame and the wall width, and f /H20849u/H20850is a certain function. In earlier sections, there was no electric current. We intro- duce now the current density j↑carried by spin-up 3 delec- trons, as seen from the laboratory frame. The existence ofsuch a 3 dcurrent will be discussed further in Sec. VI. The effect on Sof torque /H9270yis evaluated in the simplest manner14in a moving frame where the electron gas is at rest and, therefore, the spin current vanishes and causes no addi-tional torque. The torque itself is the same in all frames. Inthe case of spin-up electrons, the speed of that moving frame is ve↑=−j↑/ne↑e, where ne↑is the spin-up electron density. In that frame, the spin-up parts of /H9270yandSare related by /H9270y↑=/H6036Sz↑/H11509/H9258//H11509t=−/H6036Sz↑/H20849f/H11032//H9004/H20850/H20849vw−ve↑/H20850, /H2084911/H20850 where f/H11032/H20849u/H20850=df /du, and where vw−ve↑is the apparent speed of the wall as seen from the moving frame. It is also possible to derive Eq. /H2084911/H20850in the laboratory frame. In that frame, the apparent wall speed is vw, not vw −ve↑. Also, the current density j↑present in that frame gener- ates a 3 dspin current js↑, leading to an extra term − divjs↑in Eq. /H2084911/H20850. These two changes cancel each other, so that we obtain the same Eq. /H2084911/H20850as before. By working in the moving frame, we have shown that the case with current can be reduced to the case without current,by a simple change in frame. Also, we have avoided theintroduction of the spin current. We also write a expression similar to Eq. /H2084911/H20850for the contribution /H9270y↓of spin-down electrons. Because of the exclu- sion principle and of orthogonality, the spins S↑andS↓of spin-up and spin-down electrons stay closely antiparallel. By equating /H9270y↑+/H9270y↓to/H9270yof Eq. /H2084910/H20850, and using the fact13that f/H11032=sin/H9258for a uniaxial anisotropy, we obtain finally /H208491/2/H20850sin/H208492/H9274/H20850=− /H20851vw−/H20849P/Pn/H20850ve/H20852//H9275D/H9004z y ddnadXSS S S H HM zx Msubstrate planeS S HH ddX X Da) b)0plane P ψθ FIG. 2. /H20849a/H20850Simple tail-to-tail domain wall in a nanowire. The X axis runs along the length of the nanowire. The total 3 dspinS makes an angle /H9258/H20849X,t/H20850with the − Xaxis. The plane of the picture is plane Pwhich contains all the spins Sand makes an angle /H9274with the plane of the substrate. Local spin coordinates x,y, and zhave thezaxis parallel to S, and xnormal to plane Pand to the Xaxis. /H20849b/H20850View of the same domain wall, with the plane of the picture normal to the Xaxis. Plane P, which contains the spins S,i sa ta n angle/H9274to the plane of the substrate. The vector S0is antiparallel to the total field Hdd+HDand is the direction toward which Sis relaxing.GILBERT DAMPING AND CURRENT-INDUCED TORQUES … PHYSICAL REVIEW B 80, 144427 /H208492009 /H20850 144427-3P=j↑−j↓ j↑+j↓;Pn=ne↑−ne↓ ne↑+ne↓ ve=−j/nee;/H9275D=g/H92620/H9262BMs//H6036. /H2084912/H20850 Here,/H9262Bis the Bohr magneton, and all carriers are as- sumed electronlike. And veis the average electron drift speed. Also, ne=ne↑+ne↓and j=j↑+j↓. Note that /H9258has dropped out of the expression for /H9274, thus justifying our as- sumption of a constant /H9274. The demagnetizing-field torque /H9270yof Eq. /H2084910/H20850and, de- pending on the frame, the − divjsterm are the only external torques along y acting on the 3 dspins Sof the wall. The −divjsterm plays the same role in our 3 dmodel as the so- called adiabatic torque in the s-dexchange model.15,17In the latter theory, that torque had the nature of an s-dexchange torque. By Eq. /H2084912/H20850, the maximum stable value of /H9274is/H9266/4, and the corresponding critical value of the current density is15 j/H9274=/H11006/H92620Ms2e/H9004 P/H6036. /H2084913/H20850 Field HDalso has a component in the plane P, which has the same effect on Sas an additional anisotropy with easy axis along X. This tends to reduce the wall width below the value/H9004holding at /H9274=0. This effect varies like /H92742at small /H9274, and we will ignore it. V. NONADIABATIC TORQUE As before /H20851Eq. /H208497/H20850/H20852, each 3 dspinsnrelaxes toward the instantaneous direction of the total field acting on it. Here,this field is H dd+HD/H20851Fig. 2/H20849b/H20850/H20852. After summing over nand assuming /H20841/H9274/H20841/H112701 rad and HD/H11270Hdd,Sis found to relax to- ward S0=−Sz/H20849Hdd+HD/H20850/Hdd. The spin-relaxation torque act- ing on Sis /H9270x=/H6036/H20849S0/H20850x /H9270srd=−/H6036SHDx Hdd/H9270srd=/H6036S/H9275D/H9274sin/H9258 /H9275d/H9270srd. /H2084914/H20850 This spin-relaxation torque plays the same role in the present 3 dtheory as the so-called nonadiabatic torque in theories16,17based on the s-dexchange model. Contributions to/H9270xfrom interatomic-exchange and anisotropy torques can- cel each other as long as the wall has the structure discussedin the last section. We substitute /H9274from Eq. /H2084912/H20850into Eq. /H2084914/H20850. Also, torque /H9270xis equivalent to the torque /H92620MsHnadXsin/H9258of a fictitious field Hnadalong the easy axis X. From all this, we obtain finally HnadX=−/H6036ne/H20849Pnvw−Pve/H20850 2/H92620Ms/H9004/H20849/H9270srd/H9275d/H20850, /H2084915/H20850 where /H9258has dropped out. The term in vwrepresents Gilbert damping. The positive sign of its coefficient Pn/H20851Eq. /H2084912/H20850/H20852is required by the second law. In real magnetic materials, it is important to take into account domain-wall pinning, caused by lattice defects. It ischaracterized 13by the coercivity Hc. The wall will movewhenever HnadX=/H11006Hc. Combining this condition with Eq. /H2084915/H20850, we obtain vw=P Pn/H20849ve/H11007vec/H20850;vec=2/H92620Ms/H9004/H20849/H9270srd/H9275d/H20850 Pn/H6036neHc. /H2084916/H20850 Because of the existence of the coercivity, a minimum electron drift speed vecis needed before wall motion can start /H20851Eq. /H2084916/H20850and Fig. 3/H20852. For 3 delectrons, P/Pnis on the order of unity. Then, Eq. /H2084916/H20850shows that vwis on the order of the electron drift speed ve, whenever /H20841ve/H20841exceeds the critical value vec/H20851Fig. 3/H20852. VI. APPLICABILITY OF 3 dMODEL The equilibrium physical and magnetic properties of Ni, Co, and Fe depend primarily10on the 3 dband. By them- selves, 3 delectrons are already itinerant, with a bandwidth18 of several electron volts. As shown by Hodges et al.19for Ni, the addition of the 4 sband causes only minor changes in the structure and bandwidth of that 3 dband. Despite significant hybridization of 3 dand 4 sstates, 3 delectrons retain distinct physical properties, such as high density of states and lowvelocity. These electrons are the basis of the present d-d model. This model applies best to the problem of Gilbert damp- ing /H20849Sec. III/H20850in transition-metal materials. The best example is that of Permalloy thin films, studied experimentally 20by Ingvarsson. For Ni and Co, it has to be complemented by theKambersky Fermi surface breathing mechanism 21of damp- ing, which depends in opposite fashion on electron relaxationtime. On the other hand, band-structure calculations for ferro- magnetic Ni 19,22all show that the spin-up Fermi level is lo- cated above the top of the 3 dband, in a region with the low density of states and high electron velocity characteristic of4selectrons. The spin-up Fermi surface of Ni even has 23 necks similar to those of Cu. This is confirmed by ordinary Hall effect data24for Ni, Ni-Fe, Ni-Fe-Cu, and Ni-Co, which show that a small number /H112290.3 el. /at. of carriers carry most of the current. Also by deviations from Matthiessen rule,25(P/P)v v vw e ec vecn FIG. 3. Normalized wall speed vwversus average electron drift speed ve, according to Eqs. /H2084916/H20850. Here, PandPnare the current- polarization and electron-density polarization factors. These are de-fined in Eqs. /H2084912/H20850.L. BERGER PHYSICAL REVIEW B 80, 144427 /H208492009 /H20850 144427-4which indicate a large ratio 3–20 of spin-up to spin-down conductivities. Again, despite s-dhybridization, it is these distinct properties which justify giving the name 4 sto these spin-up electrons at the Fermi level. They are responsible formost of the electrical conductivity. It appears, therefore, that the s-dexchange model /H20849Sec. II/H20850 would be more reasonable 15,17for the problem of current- induced torques on domain walls, in many materials. Oneexception is iron-rich Fe-Mn, Fe-Cr, Fe-V , and Fe-Ti, wheredeviations from Matthiessen rule 25show conduction by spin- down 3 dcarriers to be dominant. Hall effect data for Fe-Cr /H20849Ref. 26/H20850show these carriers to be holelike. There, our purely 3 dmodel may apply even for current-induced torques /H20849Sec. IV/H20850. VII. CONCLUSIONS AND FINAL REMARKS The model based on 3 ditinerant electrons only, used by Tserkovnyak et al.7for their original derivation of Eq. /H208492/H20850is conceptually simpler than the s-dexchange model, which uses two different kinds of electrons. Also, it is less plaguedby uncertainties arising from s-dhybridization. Our present treatment of Gilbert damping in this model achieves maximum mathematical simplicity, as well as maxi-mum physical clarity and insight /H20851Fig. 1/H20849b/H20850/H20852, through the useof a semiclassical equation /H20851our Eq. /H208496/H20850/H20852for the precession of a3dspins n. This method was pioneered by Turov5in con- nection with the s-dexchange model, but has almost been forgotten since. Further simplification happens because we do not try, like Tserkovnyak et al. , to rederive known results about spin re- laxation /H20849see Refs. 3and8/H20850. Instead, we just focus on the Gilbert damping part of the problem. The most important and least trivial ingredient for our calculation is the choice8of the direction s0toward which the spins relax /H20851Eq. /H208494/H20850and /H208497/H20850/H20852, also made by Turov for the s-d exchange model. In the case of current-induced torques on a domain wall, the formulas obtained for the angle /H9274/H20851Eq. /H2084912/H20850/H20852and for the fictitious field Hnad/H20851Eq. /H2084915/H20850/H20852are the same as they would be in a similar theory14,15,17based on s-dexchange, even though exchange plays a much less explicit role in the equations. Ofcourse, the values of parameters such as P,P nandnemay be somewhat different. Our results are consistent with those ofTserkovnyak et al. ; 7for example, the dimensionless coeffi- cient/H9252, used by these authors to describe the intensity of the nonadiabatic torque, can be shown in the 3 dmodel to be equal to the Gilbert constant /H9251, itself given by our Eq. /H208492/H20850. On the other hand, /H9252/H11022/H9251holds in the s-dexchange model. 1S. V . V onsovskii, Zh. Eksp. Teor. Fiz. 16, 981 /H208491946 /H20850. 2E. A. Turov, Izv. Akad. Nauk SSSR, Ser. 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Dreesen, and E. M. Pugh, ibid. 133, A407 /H208491964 /H20850. 25A. Fert and I. A. Campbell, J. Phys. F: Met. Phys. 6, 849 /H208491976 /H20850. 26G. C. Carter and E. M. Pugh, Phys. Rev. 152, 498 /H208491966 /H20850.GILBERT DAMPING AND CURRENT-INDUCED TORQUES … PHYSICAL REVIEW B 80, 144427 /H208492009 /H20850 144427-5

PhysRevB.75.174430.pdf

Reliability of Sharrocks equation for exchange spring bilayers D. Suess, *S. Eder, J. Lee, R. Dittrich, and J. Fidler Institute of Solid State Physics, Vienna University of Technology, Vienna, A-1040 Austria J. W. Harrell MINT Center and Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama 35487–0209, USA T. Schrefl and G. Hrkac Department of Engineering Materials, The University of Sheffield, Sheffield, S-10 2TN United Kingdom M. Schabes, N. Supper, and A. Berger San Jose Research Center, Hitachi Global Storage Technologies, San Jose, California 95135, USA /H20849Received 19 January 2007; published 22 May 2007 /H20850 A Monte Carlo approach and a modified nudged elastic band method are used to study the dynamic coercivity of interacting particle arrays in particular perpendicular recording media and exchange spring bi-layers. Monte Carlo simulations are performed to study the effect of the interactions on the dynamic coercivityof interacting particle arrays. It is shown that the interactions in magnetic recording media only slightlyinfluence the dynamic coercivity. The reliability of energy barrier measurements based upon Sharrock’s equa-tion for frequency-dependent coercivity data is investigated using a modified nudged elastic band method. It isshown that the extrapolated energy barrier at zero field may deviate from the correct one by up to 18% if theconventional exponent n=1.5 is assumed. Our micromagnetic simulations furthermore indicate that the accu- racy of the extrapolated energy barrier can be improved by about a factor of 3 upon measuring the dynamiccoercivity at an angle of 45° and using the exponent nas an additional fit parameter. DOI: 10.1103/PhysRevB.75.174430 PACS number /H20849s/H20850: 74.25.Ha I. INTRODUCTION With increasing areal density in magnetic recording, new concepts have to be introduced in order to obtain a highthermal stability and a good writeability and, at the sametime, a good signal-to-noise ratio. For example, a break-through technology in longitudinal recording was introducedwith the concept of antiferromagnetic coupled /H20849AFC /H20850 media. 1,2Recently, perpendicular recording was introduced into products, which allows a further increase in areal den-sity. With increasing areal density the grains in the recordingmedia have to be decreased in diameter. However, in themost simple picture where the thermal stability depends onthe volume of one grain, a minimum volume is required toobtain the required thermal stability. In future perpendicularrecording media it will be a trade-off between areal densityand thermal stability. Therefore it is important to be able tomeasure the thermal stability of advanced magnetic record-ing media with high accuracy. The lifetime of stored infor-mation /H20849thermal stability /H20850in granular recording media is ob- viously connected to the stability of the magnetization statesin each grain, which can be estimated by the Arrhenius-Néelformula /H9270=/H92700e/H9004E/kBT. /H208491/H20850 Here, /H9004Eis the energy barrier which separates the two mag- netic lowest-energy states in a recording media grain. /H92700is the inverse of the attempt frequency. A commonly usedmethod to determine this energy barrier as well as the short-time coercive field /H20849switching field /H20850H 0of longitudinal re- cording media was proposed by Sharrock.3More recently,the validity of Sharrock’s equation for more complex mag- netic recording materials, such as AFC media was investi-gated by Margulies et al. 4 In this paper the validity of Sharrock’s equation for the case of perpendicular recording media and, more specifically,for the case of exchange spring media is investigated. Ex-change spring media consist of strongly-exchange-coupledhard and soft layers. Exchange spring magnets were initiallyintroduced by Coehoorn et al. 5and Kneller and Harwig6for permanent magnet applications. The optimal tuning of thefraction of the soft magnetic phase and the hard magneticphase allowed the design of materials with a high remanenceand at the same time a high coercive field. 7Experiments on exchange spring films, in particular on a bilayer structureconsisting of a soft magnetic NiFe layer, coupled to a CoSmlayer, were done by Mibu et al. 8and Fullerton et al.9for the scope of a high-energy product for hard magnetic materials. Recently, the compositions of hard and soft magnetic lay- ers were introduced theoretically10,11to reduce the write field requirements in magnetic recording. Experimental work on exchange spring media was done by Wang et al.12and Sup- peret al.13The influence of the interface coupling on the coercive field and the compression of the domain wall at thehard-soft interfaces can be found in Refs. 14–17. In a multilayer structure with continuously increasing an- isotropy from layer to layer it was shown theoretically thatthe coercive field can be decreased to an arbitrarily smallvalue while keeping the energy barrier /H20849thermal stability /H20850 constant. 18 Studies of exchange spring structures show that extremely hard magnetic films can be written with a limited head fieldif they are coupled to softer magnetic layers. InterestinglyPHYSICAL REVIEW B 75, 174430 /H208492007 /H20850 1098-0121/2007/75 /H2084917/H20850/174430 /H2084911/H20850 ©2007 The American Physical Society 174430-1the scope of exchange spring media /H20849ESM /H20850in magnetic re- cording is opposite to the scope of exchange spring magnetsfor permanent magnets. In magnetic recording ESM shoulddrastically decrease the coercive field, while it should bemaintained high in permanent magnet applications. The paper is structured as following. In Sec. II the basic concept of measuring the thermal stability using Sharrock’sequation is given. In Sec. III micromagnetic models are dis-cussed that allow one to simulate magnetic structures at finitetemperature. In particular the introduction of a Monte Carlomethod is given that allows one to simulate the hysteresisloop of recording media at finite temperature. Furthermore,Sec. III deals with the nudged elastic band method, whichallows for the calculation of energy barriers of magneticstructures. In Sec. IV validation of Sharrock’s equation is investi- gated for perpendicular recording media and exchange springmedia. First, the effect of the interactions field is investigatedusing the Monte Carlo method introduced in the previoussection. Finally, energy barrier calculations on a single grainof various exchange spring media are performed. II. BACKGROUND OF SHARROCK’s EQUATION Let us start with a quick review of Sharrock’s equation. In the following it is assumed that each grain of the recordingsystem can be described by a two-level system. One levelcorresponds to the state with magnetization up; the otherlevel corresponds to the state with magnetization down. Oneis interested in the average lifetime of the state with magne-tization pointing up. The occupation probabilities of the twoenergy levels P 1andP2satisfy the normalization condition P1+P2=1 and the master equationdP1 dt=−w12P1+w21P1, where w12is the transition rate from the up state to the down state and w21the transition rate from the down to the up state. w12is the inverse of the average lifetime of the up state, w12=1 /H9270. The magnetization as a function of time, which depends on the occupation probability P1and P2, can be written as M/H20849t/H20850=Ms/H20849P1−P2/H20850=Ms/H208492P1−1 /H20850. /H208492/H20850 For sufficiently large downward fields the up state has a much larger energy than the down state. In this limit w21is much smaller than w12and can be set to zero. Under this assumption, it follows that P1=e−w12t. For macroscopic particle assemblies, such as recording media, in an accurate approach the energy barrier has to bereplaced by a distribution of energy barriers, which results inthe fact that the decay of the magnetization no longer followsan exponential decay. Instead, one finds that for a distribu-tion of energy barriers the magnetization decreases accordingto a logarithmic law as a function of time. 19 In the following the simple case of only one energy bar- rier height is investigated. The average life time /H9270can be extracted by substituting the equation for P1into the equa- tion for M/H20849t/H20850and calculating the time t0, when M/H20849t0/H20850=0. It follows that t0=/H9270ln/H208492/H20850. Therefore, applying a field and mea- suring the time t0until the magnetization becomes zero al-lows one to determine /H9270, which depends on the system, par- ticularly on the energy barrier separating the state up fromthe state down. From the measurement of the time /H9270the energy barrier can be extracted using Eq. /H208491/H20850. However, the energy barrier of recording media at zero field cannot beextracted from /H9270, because the average lifetime /H9270for media is usually several years and cannot be accessed experimentally.A way to enhance the decay of the magnetization is to applyan external field that opposes the magnetization. From thedecay of the magnetization at finite fields one tries to esti-mate the thermal stability at zero field. For Stoner-Wohlfarthparticles the energy barrier at finite opposing field is con-nected to the energy barrier at zero field by the relation /H9004E=/H9004E 0/H208731−H H0/H20849/H9258/H20850/H20874n , /H208493/H20850 where H0/H20849/H9258/H20850is the Stoner-Wohlfarth switching field, when the external field is applied at an angle /H9258with respect to the anisotropy axis. Upon applying the external field exactly par-allel to the easy axis, the exponent nis found to be 2. How- ever, the exponent will deviate significantly if /H9258/H110220. Victora expressed the energy barrier as a series expansion as /H9004E =C1/H208491−H/H0/H208503/2+O/H208495/2 /H20850.20Harrell investigated in detail the exponent nas a function of the external field Hand the angle between the external field and the easy axis of singledomain particles. 21He found that for an angle /H9258=15.9° the exponent nis very close to 1.5 for all external field values. For/H9258=1° the exponent depends on the external field and decreases from 1.85 to 1.62 as the external field increasesfrom zero to the coercive field. For /H9258=45° the value of the exponent nis between 1.4 and 1.5. For the analysis of ex- periments, different values of the exponent nare used, such asn=2 /H20849Ref. 22/H20850andn=3/2 /H20849Ref. 23/H20850. Substituting the Arrhenius-Néel formula into Eq. /H208493/H20850leads to Sharrock’s equation Hc,dyn=H0/H208771−/H20875kBT /H9004E0ln/H20873t0 ln/H208492/H20850/H92700/H20874/H208761/n/H20878. /H208494/H20850 Therefore applying different external fields Hc,dynand mea- suring for every field the time t0until the magnetization be- comes zero allows one to determine /H9004E0andH0. This pro- cedure is usually called measuring the time dependence ofthe coercivity. III. MICROMAGNETIC THEORY A. Energy barriers In magnetic storage applications thermal switching events determine the long-term stability of the stored information.The main difficulty in the computation of transition pro-cesses is caused by the disparity of the time scales. If thethermal energy k BTis comparable to the energy barrier /H9004E separating two local energy minima, direct simulations of theescape over the energy barrier using Langevin equation arepossible. 24,25However, this is usually not the case in magnet recording applications where kBT/H11270/H9004E. Due to the granular- ity in magnetic recording simulations, it is a good approxi-mation that switching occurs grain by grain. Therefore, theSUESS et al. PHYSICAL REVIEW B 75, 174430 /H208492007 /H20850 174430-2thermal stability can be estimated if the energy barrier of each grain is known. Henkelman and Jónsson proposed the nudged elastic band method to calculate minimum-energy paths.26A path of the nudged elastic band method is represented by a sequence ofimages. One image represents one magnetization state of themagnetic system. An initial path is assumed which connectsthe initial magnetization state M /H20849i/H20850with the final magnetiza- tion state M/H20849f/H20850. In the work of Henkelman and Jónsson chemical pro- cesses are simulated. Therefore the coordinates of thenudged elastic band method denote the position of particles.In contrast to space coordinates, the magnetization in micro-magnetics has to fulfill the constraint that the magnitude re-main constant with time. Therefore it is not possible to di-rectly use the formulation of the nudged elastic band methodas proposed by Henkelman and Jónsson. Dittrich et al. suc- cessfully applied the nudged elastic band to micromagneticsusing polar coordinates in order to fulfill the constraint of aconstant magnetization. 27However, convergence problems can occur because of the problem of a good definition of thedifference vector between two magnetization states in polarcoordinates. The difference vector is required in the nudgedelastic band method in order to relax the initial path towardsthe minimum-energy path. In order to avoid this problem the magnetization of the nudged elastic band method is represented by Cartesian co-ordinates in the following. A modified relaxation procedurein the nudged elastic band method is proposed. Every imageconsists of Mdiscretization points /H20849e.g., node points of the finite-element mesh or cells of a finite-difference scheme /H20850. On each discretization point the magnetic polarization is described by a three-dimensional vector. The magnetizationof the image iand the discretization point kis given by J i,k=/H20849Jx,Jy,Jz/H20850. /H208495/H20850 The optimal path can be found by solving the following par- tial differential equation for the magnetization Ji,kon every node point on each image: /H11509Ji,k /H11509t=−/H20841/H9253/H20841 JsJi,k/H11003/H20851Ji,k/H11003Di,k/H20849J/H20850/H20852. /H208496/H20850 The three-dimensional vector Di,kcan be regarded as an ef- fective field. The right-hand side of Eq. /H208496/H20850has the same form as the damping term of the Landau-Lifshitz-Gilbertequation. As a consequence Eq. /H208496/H20850conserves the magnitude of the magnetization in time. The vector D iis composed of three-dimensional vectors Di,kon every discretization point of each image i, Di=/H20849Di,1,Di,2, ..., Di,M/H20850. /H208497/H20850 This vector, which governs the relaxation of the images to- wards the minimum-energy path, is calculated using Eq. /H208498/H20850, Di=/H20853Heff,i/H20849Ji/H20850−/H20849Heff,i/H20849Ji/H20850·ti/H20850ti/H20854+Fi. /H208498/H20850 The effective field is the negative functional derivative of the total Gibbs’ energy density of the image i,Heff,i=−/H9254/H9255Gibb /H9254J=2A JS/H9004Ji+2Ku JS2/H20849Ji·u/H20850u+HS+Hext. /H208499/H20850 The first term denotes the exchange energy contribution with Aas the exchange constant. The second term is the aniso- tropy term with Kuas the magnetocrystalline anisotropy con- stant and uthe unitary direction vector of the easy magneti- zation axis. HSandHextare the stray field and the external field, respectively. Care has to be taken when calculating the local tangent ti at an image i. The single use of either a forward-difference approximation, backward-difference approximation, or acentral-difference approximation develops kinks in thepath. 26The kinks prevent the string from converging to the minimum-energy path. The optimal choice of the appropriatedifference approximation depends on the energy differencebetween successive images. In a first approach, forward dif-ferences climbing up a hill, backward differences goingdown a hill, and central differences at energy minima andmaxima are used. The tangent t ican be calculated using ti=Ji+1−Ji /H20648Ji+1−Ji/H20648ifE/H20849Ji−1/H20850/H11021E/H20849Ji/H20850/H11021E/H20849Ji+1/H20850, /H2084910/H20850 ti=Ji−Ji−1 /H20648Ji−Ji−1/H20648ifE/H20849Ji−1/H20850/H11022E/H20849Ji/H20850/H11022E/H20849Ji+1/H20850, /H2084911/H20850 ti=Ji+1−Ji−1 /H20648Ji+1−Ji−1/H20648ifE/H20849Ji−1/H20850/H11021E/H20849Ji/H20850/H11022E/H20849Ji+1/H20850 or if E/H20849Ji−1/H20850/H11022E/H20849Ji/H20850/H11021E/H20849Ji+1/H20850. /H2084912/H20850 This prevents the formation of kinks. A detailed analysis of this topic and the motivation for this choice of the tangentcan be found in the work of Henkelman and Jónsson. 26The norm which is used in all expressions is the L2norm. The last term of Eq. /H208498/H20850denotes the spring force. It pre- vents the images from moving towards the end points andlocal minima of the path, giving a low resolution near saddlepoints and a high resolution near energy minima. This prob-lem is known as “sliding-down” and can be solved by intro-ducing spring forces between the images which make themstay equally spaced in the L 2norm: Fi=k /H92620/H20849/H20648Ji+1−Ji/H20648−/H20648Ji−Ji−1/H20648/H20850/H9270i /H20648/H9270i/H20648. /H2084913/H20850 The direction of the spring force is given by the difference of the magnetization state of two images, /H9270k+=Jk+1−Jk, /H2084914/H20850 /H9270k−=Jk−Jk−1, /H2084915/H20850 /H9270k=/H9270k+ifE/H20849Jk−1/H20850/H11021E/H20849Jk/H20850/H11021E/H20849Jk+1/H20850, /H2084916/H20850RELIABILITY OF SHARROCKS EQUATION FOR … PHYSICAL REVIEW B 75, 174430 /H208492007 /H20850 174430-3/H9270k=/H9270k−ifE/H20849Jk−1/H20850/H11022E/H20849Jk/H20850/H11022E/H20849Jk+1/H20850. /H2084917/H20850 One problem is the choice of the strength of the spring con- stant k. The optimal value for kdepends on the number of images used, on the number of finite elements, and on thesize of the model. It is difficult to give a general rule for thevalue of the spring constant. It should be strong enough toprevent images from falling down into the energy minima,but not too strong as to dominate by orders of magnitude inEq. /H208498/H20850. Fortunately, the absolute value of kis usually not very critical and can be varied over several orders of magni-tude without losing speed within the time integrationscheme. B. Monte Carlo simulations The nudged elastic band method is a powerful tool to estimate the thermal stability for systems with small numbersof energy minima and saddle points. However, for calcula-tion of the magnetization decay of a granular recording me-dia the sole application of the nudged elastic band methoddoes not provide the magnetization as a function of time orfield. For these systems a better approach is the use of MonteCarlo methods. Bortz et al. investigated activated reversal processes of Ising spin systems with Monte Carlo methods. 28 Charap et al. used Monte Carlo methods in order to estimate the areal density limit of longitudinal recording.29The time increment in the Monte Carlo method was adjusted accord-ing to the average time between successful reversals. There-fore the method could describe magnetization reversal pro-cesses of any time span of interest. However, the methodused by Charap et al. is not suitable to calculate hysteresis loops with different field sweep rates. Chantrell et al. used a Monte Carlo method to model the low-field susceptibility ofa cobalt granular system. 30Standard Monte Carlo steps are performed in order to achieve a correct thermodynamic de-scription of the magnetization states close to an energy mini-mum. In order to model thermal activations over larger en-ergy barriers the Arrhenius-Néel model is applied. For eachgrain of the recording media the probability of switchingwithin the measuring time t m/H20849time step of the Monte Carlo method /H20850is given by Pr=1− e−tm//H9270, /H2084918/H20850 where /H9270is the relaxation time given by Eq. /H208491/H20850. The Monte Carlo simulations performed in this paper are based on thework by Chantrell et al. A granular microstructure was con- structed using Voronoi tessellations. The equilibrium magne-tization state is described with one magnetic polarizationvector. For every magnetization state the finite-elementmethod is used to calculate the effective field on every grainof the media. The effective field contains the demagnetizingfield of the neighboring grains, the exchange field, and theexternal field. For the Monte Carlo method one grain iof the media is chosen at random. The switching probability withinthe time step t mwas calculated according to Eq. /H2084918/H20850.O n average all grains are chosen one time within the time tm.I n the following simulations the time step tmwas chosen suffi- ciently small that the results do not depend on tm. The energybarrier in Eq. /H2084918/H20850depends on the effective field acting on grain i. In order to calculated the energy barrier two different approached are used. In the first approach we followed thework of Chantrell et al. 30The energy barrier is calculated using the Pfeiffer approximation.31In the second approach the energy barriers for the system were precomputed usingthe nudged elastic band method. In order to calculate theenergy barriers for an arbitrary grain iof the media the fol- lowing procedure was applied. A finite-element model wasconstructed to model a standard grain with a basal plane of1n m/H110031 nm and a thickness that equals the film thickness. The obtained energy barrier was multiplied by the area of thebasal plane of the grain i. In order to save computational time in a preprocessing step a table was constructed thatcontains the energy barrier for discrete values of the effectivefield and the angle /H9258between the external field and the easy axis. For every field value and angle /H9258the energy barrier was calculated using the nudged elastic band method as describedin the previous section. Figure 1compares the precomputed energy barriers using the nudged elastic band method withthe Pfeiffer approximation for a grain with a rectangularbasal plane with an edge length of 1 nm. The film thicknessis 20 nm, the anisotropy constant K 1=3/H11003105J/m3, and the exchange constant A=10−11J/m. The magnetic polarization Js=0.5 T. The demagnetizing field of the grain which leads to a shape anisotropy was not taken into account. For theprecomputed barriers the external field was discretized be-tween zero and the switching field using 20 mesh points. Theangle /H9258was discretized between 0 and 90° using 14 discreti- zation points. Figure 1shows that the Pfeiffer approximation is well suited to estimate the energy barriers even for a grainwith a thickness of 20 nm. For the Monte Carlo simulation a second-order interpola- tion scheme was used to evaluate the energy barrier for anyarbitrary point E/H20849H, /H9258/H20850. This method allows for the calcula- tion of the thermal stability of recording structures where the thermally activated reversal mechanism occurs via a forma-tion of a nucleation. This is particularly important for ex-change spring media. IV . MICROMAGNETIC RESULTS A. Monte Carlo simulations of single-phase media Sharrock’s equation /H20851Eq. /H208494/H20850/H20852was derived under the as- sumption that no interaction fields act on the media. How-ever, if the time-dependent coercivity is measured for agranular recording media, this assumption may not be justi-fied. The internal field that acts on one grain changes duringthe measurement. At the beginning of the measurement allgrains point up. The full demagnetizing field adds to theexternal field. At M z=0 the demagnetizing field is zero /H20849at least within the mean-field approximation /H20850, leading to zero demagnetizing field. However, the field Hc,dynin Eq. /H208494/H20850is assumed to be constant. If the external field is applied at afinite angle with respect to the film normal, apart from themagnitude of the internal field, also the angle of the internalfield changes during the measurement. A similar problemoccurs if the intrinsic hysteresis loop of a tilted recordingmedium is measured. The internal field angle /H20849sum of theSUESS et al. PHYSICAL REVIEW B 75, 174430 /H208492007 /H20850 174430-4external field, the exchange field, and the demagnetizing field /H20850changes along the hysteresis loop even if the angle of the applied field is kept constant. This problem was dis-cussed by Richter who suggested an iterative procedure tocompensate for the error. 32Recently, the iterative procedure was used to measure intrinsic hysteresis loops of perpendicu-lar recording media at different angles between the easy axisand the film normal. 33 In order to investigate the influence of the interaction field on the dynamic coercivity, Monte Carlo simulations as de-scribed in Sec. III are performed. A recording media with20/H1100320 grains is simulated. The grain diameter is 6.5 nm and the film thickness is 20 nm. The magnetic polarization is0.5 T and the exchange constant is A=10 −11J/m. The aniso- tropy constant is K1=3/H11003105J/m3. No distribution of the easy axis is assumed in order to clearly separate the effect ofthe interaction field on the dynamic coercivity. The externalfield is applied at an angle of 15.9° with respect to the easyaxis. In a first set of simulations the exchange field and thestray field were not taken into account. The dashed lines inFig.2show the remanent hysteresis loops for different wait- ing times tat a temperature of T=300 K. The remanent hys- teresis loops are obtained by first saturating the sample. Anexternal field His applied for a time t. After the time tthe field is removed and the remanence is measured. This is donefor different fields Hin order to obtain the remanent hyster- esis loop. The different dashed curves in Fig. 2denote simu- lations for different waiting times t. The numerically obtained values of the dynamic coerciv- ity are plotted as a function of ln /H20849 /H9260t/H20850in Fig. 3.tis the waiting time and /H9260=1/ /H20851/H92700ln/H208492/H20850/H20852, where /H92700=10−9s. The curves in Fig. 3are fitted using Eq. /H208493/H20850in order to determine the energy barrier /H9004E0andH0. Equivalently the energy bar- rier/H9004E0andH0can be obtained by fitting H(ln/H20849/H9260,t/H20850)data to Sharrock’s equation. Equation /H208493/H20850is the inverse function of Sharrock’s equation. For the simulations neglecting the demagnetizing field and the exchange field the dynamic coercivities /H20849circles inFig.3/H20850agree very well with the values obtained from Shar- rock’s equation /H20849solid line in Fig. 3/H20850./H9004E0andH0in Shar- rock’s equation were calculated using the micromagnetic in-put parameters. The differences between the analyticallyobtained dynamic coercive fields /H20849from Sharrock’s equation /H20850 and the numerical values were smaller than 10 −3T /H20849/H110210.4% /H20850for all simulations. The numerical obtained curves of Fig. 3were fitted with Sharrock’s equation in order to obtain the energy barrier /H9004E andH0. The exponent nin Eq. /H2084919/H20850was assumed to be n =1.55 which follows from the Pfeiffer approximation. Asexpected for zero interactions, the fitted values of /H9004EandH 0 agree very well with the calculated ones. The fitted values are/H9004E0,fitt=48.32 kBT300and/H92620H0,fitt=0.918 T. The Stoner- FIG. 1. /H20849Color online /H20850Energy barrier as a function of the external field and the angle /H9258between the easy axis and the external field. The grain has a rectangular basal plane with edge length of 1 nm. The film thickness is 20 nm. Left image: the energy barrier was calculatedusing the nudged elastic band method. The barrier is calculated for 20 different values of the external field and 14 values of the angle /H9258. Right image: the Pfeiffer approximation is used to estimate the energy barrier. FIG. 2. Remanent hysteresis loops obtained by Monte Carlo simulations. The temperature T=300 K. No intergranular exchange field is assumed in the calculations. The waiting time is t=1 s, 10−1s, 10−2s,...,10−9s for the curves a,b,c,..., h, respectively. The angle between the easy axis and the external field is 15.9°.Dashed lines a-h: the stray field of neighboring grains is not taken into account. Solid lines A-H: same as a-hbut the demagnetizing field is taken into account.RELIABILITY OF SHARROCKS EQUATION FOR … PHYSICAL REVIEW B 75, 174430 /H208492007 /H20850 174430-5Wohlfarth theory gives for the micromagnetic input param- eters/H9004E0,SW=48.7 kBT300and/H92620H0,SW=0.914 T. In order to investigate the influence of the stray field in- teractions, the simulations were repeated taking the demag-netizing field into account. The self-demagnetizing field ofone grain of the recording media that leads to a shape aniso-tropy was not taken into account in order to be able to com-pare the results directly with the simulations where the de-magnetizing field was neglected. The demagnetizing fieldleads to a small reduction of the dynamic coercive field asshown in Fig. 2/H20849solid lines /H20850and Fig. 3/H20849dashed line /H20850. This is in contrast to a simple mean-field theory where at coercivityno mean field acts on the grains. Again Sharrock’s equationwas used to fit the numerical obtained values of the dynamiccoercivity leading to /H9004E 0,fitt=46.47 kBT300and/H92620H0,fitt =0.88 T. Finally, simulations were performed taking into account the demagnetizing field and an exchange field between thegrains with a mean exchange field of 0.16 T. Interestingly,the dynamic coercive field increases as the exchange inter-actions are introduced leading to /H9004E 0,fitt=48.8 kBT300and /H92620H0,fitt=0.9 T as shown in Fig. 3/H20849dotted line /H20850. In all previous simulations the energy barriers in the Monte Carlo simulations where calculated using the Pfeifferapproximation. Simulations with precomputed energy barri-ers using the nudged elastic band method are shown by thedotted dashed line in Fig. 3. The simulations show that for single-phase media and a film thickness of 20 nm the resultsonly slightly deviate from the simulations using the Pfeifferapproximation /H20849dotted line /H20850. B. Energy barriers of bilayers with a perfectly soft layer In the last section it was shown that an extrapolation using Sharrock’s equation leads to values of H0and/H9004E0,fittthat are not significantly influenced by the interaction fields. This canbe understood by the following argument. The state of the film where the values of H0and/H9004E0,fittare measured /H20849fitted /H20850 is the demagnetized state. Therefore, 50% of the grains arepointing up and the other 50% are pointing down, leading tozero mean field in first order. The measured /H9004E 0,fittalso has a physical meaning for magnetic recording. It approximatesthe energy barrier of a grain at the transition. At the transitionit is justified to assume that no demagnetizing field and noexchange field /H20849this is only true in the limit for weak ex- change /H20850act on the grain. However, usually the most unstable grains in magnetic recording are the grains close to the center of a bit. Here, alarge demagnetizing field acts on the grains. In order to es-timate the thermal stability of a grain at the center of a bit,care has to be taken because the extrapolated value of /H9004E 0,fitt does not take demagnetizing fields into account. The influ- ence of neighboring grains /H20849demagnetizing field and ex- change field /H20850on the energy barrier in a saturated film is investigated in Ref. 17. It is shown that the influence of the demagnetizing field and the exchange field can be treatedwith a mean-field approach. /H9004E 0,fittonly corresponds to the energy barrier of a grain in the demagnetized film if theexponent nof the energy barrier as a function of the external field is known in detail. In order to calculate the exponent n for exchange spring media the energy barrier is calculatednumerically using the nudged elastic band method. The ex-change constant and the magnetic polarization are the sameas in the last section. In contrast to the last section only onegrain of the exchange spring media is modeled. This effec-tive mean field can be added to the external field. The graindiameter of Fig. 4shows the energy barrier as a function of the external field for exchange spring media with differentsoft layer thicknesses. If not stated otherwise, in all the simu-lations the following parameters are used. The magnetic po-larization in the hard layer and the soft layer is J s=0.5 T. The exchange constant A=1/H1100310−11J/m. The anisotropy in the hard layer is K1=1/H11003106J/m3. In Fig. 4the external field is applied at an angle /H9258=0.5° with respect to the easy axis. In the limit of an infinitely thicksoft and an infinitely thick hard magnetic layer an analyticexpression for the energy barrier as a function of an appliedfield /H20849 /H9258=0 /H20850was derived by Loxley and Stamps.34The pre- dictions of the analytical formula are compared with values obtained from the nudged elastic band method for a soft-layer thickness of 36 nm and an angle /H9258of 0.5°. As shown in Fig.4the agreement is excellent, especially for values of the external field larger than about 0.5 times the dynamic coer-cive field. For large field values the external field stronglypushes the domain wall against the hard-soft interface, lead-ing to a small width of the domain wall at the saddle point,which is the state along the minimum-energy path with thelargest energy. In terms of the domain wall width, the exter-nal field can be thought of an effective anisotropy in theorder of J sH. For smaller field values, the width of the do- main wall at the saddle point is larger than the thickness ofthe soft magnetic layer, leading to deviations from the ana-lytical formula due to the finite soft-layer thickness. Since Eq. /H208493/H20850was derived for single-domain particles, it is not obvious at all if it can be used for exchange spring mediawhere highly nonuniform states are formed during reversal. FIG. 3. /H20849Color online /H20850Compilation of the dynamic coercivity obtained from the waiting time experiments of Fig. 2. The tempera- ture is T=300 K. Instead of the waiting time tthe logarithm ln /H20849/H9260t/H20850 is used as the yaxis. The constant /H9260=1/ /H20851/H92700ln/H208492/H20850/H20852. The effect of interaction fields /H20849stray field and exchange field /H20850on the dynamic coercivity is investigated.SUESS et al. PHYSICAL REVIEW B 75, 174430 /H208492007 /H20850 174430-6In order to check whether Eq. /H208493/H20850is valid or not, the expo- nent nis calculated as a function of the external field for various exchange spring media. Similar to experimentswhere the external field is applied perpendicular to the filmplane, the exponent nis calculated for an angle /H9258of 0.5°. The exponent nis locally fitted in a field range of about 0.1 T /H20849five data points /H20850. For the fit the numerically calculated values for /H9004EandH0/H20849/H9258/H20850are used.Figure 5shows that nstrongly depends on the value of the external field. It is interesting to note that even the single-phase media without a soft layer show a nonconstant expo-nent n. This is different to the Stoner-Wohlfarth theory where the exponent ndoes not exceed a value of 2 /H20849Ref. 21/H20850. The reason is that in the investigation the hard-layer thickness is18 nm. Slightly inhomogeneous states are formed that leadto different results from the Stoner-Wohlfarth theory. Micro-magnetic simulations of a sample with a thickness of 10 nmlead to very similar results as reported by Harrell. 21 The numerical results for a soft layer thickness ts =36 nm very well agree with the analytical results. The ana- lytical formula34shows that in the limit of zero external field ngoes towards infinity. Even for larger values of Hthe ex- ponent nis significantly larger than 1.5. This indicates that an experimental fit with n=1.5 leads to significantly wrong results and that the energy barrier as a function of the exter- TABLE I. Compilation of the error of the extrapolated energy barrier /H9004E=Efitted−Erealand the extrapo- lated H0using Sharrock’s equation for different soft-layer thicknesses ts. The thickness of the hard layer is 18 nm. /H9258=0.5°. nglobal is determined by fitting the exponent nwith Sharrock’s equation in the whole field range /H208490/H11021H/H11021H0/H20850. In all other columns Sharrock’s equation is fitted in the range 5 kBT300/H11021/H9004E/H20849H/H20850 /H1102120kBT300. The columns “fit n” determine the error of H0and/H9004Eifnis used as a free fit parameter. The columns n=1.5 denote the error if nis set constant to 1.5 which is done in most experimental measurements. In the columns n=nglobal,H0and/H9004Eare determined from the fits using nglobal of the second columns. In the last row the standard deviation is calculated of the six lines above. ts/H20849nm /H20850 nglobalError /H92620H0/H20849T/H20850 Error /H9004E/H20849kBT300/H20850 Fitnn =1.5 n=nglobal Fitnn =1.5 n=nglobal 0 1.90 −0.01 − 0.01 0.21 −15.97 − 15.97 −7.19 3 1.87 0.02 − 0.01 0.08 −9.39 − 11.72 −4.80 5 1.51 −0.04 0.03 0.01 −9.74 2.68 −0.61 7 1.47 −0.02 − 0.03 −0.05 10.93 6.91 3.98 9 1.62 0.05 − 0.05 −0.03 15.58 − 5.26 0.28 11 1.78 0.07 − 0.06 0.02 3.61 − 14.61 −2.99 Standard deviation 0.04 0.03 0.09 12.72 9.47 3.96 FIG. 4. Energy barrier of exchange spring media for different soft-layer thicknesses as a function of the external field strength.The grain diameter is 6 nm; the hard-layer thickness is 18 nm. Theanisotropy constant in the hard layer is K 1,hard=1/H11003106J/m3. The numbers in the plot /H208490–36 /H20850denote the soft-layer thickness in nm. The solid lines are fitted to numerically calculated energy barriersusing E 0,H0, and nas fit parameters. The angle between the easy axis and the external field is /H9258=0.5°. The dotted line shows the results of the analytical formula that is valid for /H9258=0° and infinite thick layer thicknesses. Exchanging the xaxis and the yaxis in the above plot gives a curve which is usually called “time dependenceof the remanent coercivity.” FIG. 5. Field dependence of the fitting parameter n./H9258=0.5°. The hard-layer thickness th=18 nm. The numbers next to the curves denote the soft-layer thickness.RELIABILITY OF SHARROCKS EQUATION FOR … PHYSICAL REVIEW B 75, 174430 /H208492007 /H20850 174430-7nal field can hardly be described by Eq. /H208493/H20850using a constant value of n. In the following, the error is estimated that occurs by the determination of the energy barrier /H9004E0and H0by fitting H(ln/H20849/H9260,t/H20850)data to Sharrock’s equation. Equivalently, /H9004E0 andH0can be determined by fitting /H9004E/H20849H/H20850with Eq. /H208493/H20850. The range used of the fit is 5 kBT300/H11021/H9004E/H20849H/H20850/H1102120kBT300. The per- formance of the fit using Sharrock’s equation is measured by comparing the extrapolated energy barrier at zero field aswell as the extrapolated H 0with the numerically calculated energy barrier and switching field. In Table Ithe perfor- mances of different fits using Sharrock’s equation are com-piled. The actual energy barrier at zero field is 85 k BT300. The external field is applied at an angle of 0.5° off the film nor-mal. For a constant value of n=1.5 the extrapolated energy barriers show significant errors. For zero soft-layer thicknessthe energy barrier is underestimated by 18%. For a soft-layerthickness of 7 nm the energy barrier is overestimated by 8%.For the case of a large soft-layer thickness of 36 nm thelargest error of an underestimation of about 40% occurs. Us- ing the exponent nas an additional fit parameter the extrapo- lated energy barrier is even more inexact. Calculating thestandard deviation of the error of the energy barrier for allinvestigated soft-layer thicknesses leads to /H9268=12.72 kBT300 and/H9268=9.47 kBT300for using nas an additional fit parameter and a constant nof 1.5, respectively. In contrast to the energy barrier the extrapolation to determine H0is very good. The standard deviation of H0is just 0.03 T. To summarize, E0cannot be extrapolated accurately for perpendicular recording media and exchange spring mediafrom the dynamic coercivity /H20849pulse duration is assumed to change from 10 −7s to about 1 s /H20850if the field angle is close to the easy axis and a constant exponent nis assumed. This is an important fact because fitting the /H9004E/H20849H/H20850loops of Fig. 4 with Eq. /H208493/H20850in the whole range from 0 /H11021/H9004E/H20849H/H20850/H1102185kBT300 and using E0,H0, and nas free fit parameters leads to fits that do not seem too bad /H20849solid lines in Fig. 4/H20850. Only for small energy barriers /H9004E/H20849H/H20850/H110155kBT300can clear misfits be ob- served. In the following a method is presented that allows one to increase the accuracy of the measurement of the energy bar-rier of perpendicular recording media and exchange springmedia. The origin of the wrong extrapolation of Sharrock’sequation for exchange spring media can be found in the fielddependence of the exponent n. The idea is to measure the energy barrier as a function of the applied field in such a way that/H9004E/H20849H/H20850can be well described by Eq. /H208493/H20850. This can be realized as will be shown later by applying the field at a large angle of 45° with respect to the film normal. In Fig. 6the simulated energy barriers as a function of the external field are fitted with Eq. /H208493/H20850. The field angle is 45° with respect to the easy axis. For soft-layer thicknesses of0–9 nm the fits are very good for the whole field range. The /H92732values of the fits are 0.3, 0.02, 0.38, and 2 for soft-layer thickness of 3, 5, 7, and 9 nm, respectively. For very largesoft-layer thicknesses the field dependence of the energy bar-rier can hardly be described with a simple power law. Asshown in Fig. 6fort s=36 nm the fit is very bad, leading to a /H92732value of 170. In order to investigate the quality of the fit in more detail the exponent nis plotted as a function of the field strength H. FIG. 6. Same as Fig. 4except that the external field is applied at an angle of /H9258=45°. FIG. 7. Field dependence of the fitting parameter n./H9258=45°. The hard-layer thickness th=18 nm. The numbers next to the curves denote the soft-layer thickness. FIG. 8. Same as Fig. 3. However, th=10 nm.SUESS et al. PHYSICAL REVIEW B 75, 174430 /H208492007 /H20850 174430-8Figure 7shows that for this field angle the exponent nonly weakly depends on the strength of the external field for soft-layer thicknesses relevant for practical media in the rangefrom 3 nm to 11 nm. Due to the insensitivity of nonH, the energy barrier as a function of the external field can be ex-cellently fitted with Eq. /H208493/H20850. This insensitivity of the exponent nonHfor /H9258=45° also remains if other parameters of the exchange spring mediasuch as the hard-layer thickness are changed as shown in Fig.8. The thickness of the hard layer is 10 nm. The simulation for zero soft-layer thickness shows an exponent nthat is in very good agreement with the Stoner-Wohlfarth theory. Ingeneral for different soft-layer thicknesses the exponent nis not 1.5 but varies from about 1.4 to about 2 depending on theactual sample. Since nmainly depends on the design of the particular exchange spring media, one can expect that ncan be determined by using nas a free fit parameter in Sharrock’s equation of using nas a fit parameter. If the accuracy of the vibrating sample magnetometer /H20849VSM /H20850measurement is not sufficient to use nas a free fit parameter, it might be possible to determine a global nby fitting results obtained using a contact tester, which can span many decades of time. ForVSM measurements, modelers would need to suggest to ex-perimentalists an appropriate nto use. The improved accu- racy using nas a free fit parameter in the simulation could be confirmed as summarized in Table II. Measuring the rema- nent coercivity in a field range of 5 k BT300/H11021/H9004E/H20849H/H20850 /H1102120kBT300at an angle 45° and using nas a free fit parameterdrastically increases the quality of the extrapolated energy barrier. The standard deviation of the error decreases to about /H9268=2.8 kBT300. Table IIIcompiles the standard deviations of the error of the energy barrier and H0for the different measurements and different hard-layer thicknesses. For both th=18 nm and th =10 nm, the standard deviation of the error of /H9004E0is about 3 times smaller than for the measurement with /H9258=0.5° and a constant value of n=1.5. Using nas a free fit parameter increases the quality of the fit only if nweakly depends onH, which is the case if the field is applied at an angle /H9258=45°. C. Exchange spring media with finite K1in the soft layer In the previous sections the energy barrier was investi- gated for exchange spring media where the soft magneticlayer was perfectly soft. However, the assumption of a finitevalue of the anisotropy in the soft layer is more realistic. Theshape anisotropy alone contributes considerably to the aniso-tropy of a granular grain with a large aspect ratio. Interest-ingly, a finite anisotropy in the soft layer is not only morerealistic but also beneficial for magnetic recording because itfurther decreases the coercive field. 35In Fig. 9,/H9004E/H20849H/H20850is investigated for bilayers with a finite value of the anisotropy in the soft layer /H20849K1,soft=2/H11003105J/m3/H20850. The anisotropy in the hard layer is K1,hard=1/H11003106J/m3. The hard layer thickness is 18 nm. The numbers in Fig. 9denote the soft-layer thick-TABLE II. Same as Table Iexcept that /H9258=45°. ts/H20849nm /H20850 nglobalError /H92620H0/H20849T/H20850 Error /H9004E/H20849kBT300/H20850 Fitnn =1.5 n=nglobal Fitnn =1.5 n=nglobal 0 1.39 − 0.02 0.08 0.01 − 6.17 8.17 −0.62 3 1.47 − 0.01 0.00 0.00 − 0.31 1.76 0.15 5 1.54 0.00 −0.01 0.00 − 0.48 −2.33 0.06 7 1.68 0.01 −0.02 0.02 − 3.07 −7.77 −1.66 9 1.85 0.02 −0.02 0.05 − 6.55 −12.14 −2.87 11 2.04 0.03 −0.02 0.07 − 9.93 −15.54 −5.45 Standard deviation 0.02 0.04 0.03 3.80 8.89 2.15 TABLE III. Compilation of the standard deviations of the error in H0and/H9004Efor different exchange spring media. The standard deviation is calculated from data of six different soft-layer thicknesses as shownin Tables IandII.t his the hard-layer thickness of the bilayer. th/H20849nm /H20850 /H9258Error /H92620H0/H20849T/H20850 Error /H9004E/H20849kBT300/H20850 Fitnn =1.5 n=nglobal Fitnn =1.5 n=nglobal 10 0.5 0.06 0.11 0.04 9.98 9.24 2.68 10 5.0 0.04 0.03 0.03 6.09 6.30 1.74 10 45.0 0.02 0.02 0.04 3.86 6.78 2.39 18 0.5 0.04 0.03 0.09 12.72 9.47 3.96 18 5.0 0.04 0.06 0.07 8.94 6.55 2.28 18 45.0 0.02 0.04 0.03 3.80 8.89 2.15RELIABILITY OF SHARROCKS EQUATION FOR … PHYSICAL REVIEW B 75, 174430 /H208492007 /H20850 174430-9nesses. It is interesting to note that for a soft-layer thickness of 11 nm, H0for/H9258=0.5° is similar to H0for/H9258=45°. This is in contrast to the Stoner-Wohlfarth theory, which predicts aminimum of H 0at/H9258=45°. This effect is also in contrast to the predictions of a pure pinning behavior, where the coer-cive field follows H 0/H110081/cos /H20849/H9258/H20850according to Kondorsky.36 The observed angular dependence of H0is summarized in the inset of Fig. 10. The angular dependence can be under- stood if one keeps in mind that the reversal process in ex-change spring media occurs in two steps. In a first step anucleation is formed into the soft layer. This nucleationprocess shows an angular dependence similar to the pre-diction of the Stoner-Wohlfarth theory, H N/H11008/H20851sin2/3/H20849/H9258/H20850 +cos2/3/H20849/H9258/H20850/H20852−3/2. In a second step the domain wall that was nucleated propagates towards the soft-hard interface. The angular de-pendence of the /H20849pinning /H20850field to push the domain wall in the hard layer follows HP/H110081/cos /H20849/H9258/H20850. The switching field H0 is determined by H0=max /H20849HN,HP/H20850. Since the nucleation field and the pinning field show a different angular depen- dence, it may depend on the angle /H9258if the switching field is determined by HNor by HP. In the investigated sample the angular dependence of H0/H20849/H9258/H20850follows a Stoner-Wohlfarth- like behavior for small angles /H9258. For larger angles the nucle- ation field becomes smaller than the pinning field. Hence forlarge angles the switching field H 0/H20849/H9258/H20850is determined by HP, leading to pinninglike behavior H0/H110081/cos /H20849/H9258/H20850. In Fig. 11the exponent nis calculated by fitting /H9004E/H20849H/H20850data to Eq. /H208493/H20850. Similar to the results for a perfectly soft layer, the exponent nstrongly depends on the applied field strength for /H9258=0.5°. Even values of nlarger than 2 are observed. Similar to the results of the last section, the exponent nbecomes less de- pendent on Hif the external field is applied at an angle /H9258 =45° /H20849see Fig. 12/H20850. Values close to n=3/2 are observed. To find a physical argument that explains why for a variety ofsamples the exponent nbecomes almost constant if the angle is applied at 45° will be a task of future research. FIG. 9. Energy barrier of exchange spring media for different soft-layer thicknesses and different angles /H9258between the external field and the easy axis. The anisotropy constant in the hard layerand in the soft layer is K 1,hard=1/H11003106J/m3and K1,soft=2 /H11003105J/m3, respectively. The numbers in the figure denote the soft- layer thicknesses. FIG. 10. Hysteresis loops of a bilayer with an 18-nm-thick hard layer and an 11-nm-thick soft layer /H20849K1,hard=106J/m3,K1,soft=2 /H11003105J/m3/H20850. The angle /H9258between the external field and the easy axis is varied. The inset shows the angular dependence of H0as a function /H9258. FIG. 11. Same as Fig. 5/H20849/H9258=0.5°, th=18 nm /H20850. However, the anisotropy in the soft layer is K1=2/H11003105J/m3. FIG. 12. Same as Fig. 11except that /H9258=45°.SUESS et al. PHYSICAL REVIEW B 75, 174430 /H208492007 /H20850 174430-10V . CONCLUSION AND OUTLOOK Monte Carlo simulations of dynamic coercivity simula- tions have shown that the interaction fields such as the ex-change field and the strayfield do not significantly change thedynamic coercivity. Although Sharrock’s equation was de-rived without taking interaction fields into account, it is wellsuited to describe interacting grains of magnetic structures. The analysis of the paper shows that the accuracy of mea- surements of energy barriers of exchange spring media canbe improved by changing the experimental conditions. Wepropose that the external field is applied at 45° with respectto the film normal. For measurements under 45° the expo- nent nis almost constant which is in contrast to measurement parallel to the film normal. This makes it possible to use nas an additional fit parameter along with the fitted energy bar-rier at zero field /H9004E 0and the fitted H0. ACKNOWLEDGMENTS The financial support of the Austrian Science Fund P19350 is acknowledged. I would like to thank P. Visscherfor helpful discussions. *Electronic address: dieter.suess@tuwien.ac.at 1E. E. Fullerton, D. T. Margulies, M. E. Schabes, M. Carey, B. Gurney, A. Moser, M. Best, G. Zeltzer, K. Rubin, H. Rosen, andM. Doerner, Appl. Phys. Lett. 77, 3806 /H208492000 /H20850. 2E. N. Abarra, A. Inomata, H. Sato, I. Okamoto, and Y. Mizoshita, Appl. Phys. Lett. 77, 2581 /H208492000 /H20850. 3M. P. Sharrock, J. Appl. Phys. 76, 6413 /H208491994 /H20850. 4D. T. Margulies, A. Berger, A. Moser, M. Schabes, and E. Ful- lerton, Appl. Phys. Lett. 82, 3701 /H208492003 /H20850. 5R. Coehoorn, D. B. de Mooij, and C. de Waard, J. Magn. Magn. Mater. 80, 101 /H208491989 /H20850. 6E. F. Kneller and R. Harwig, IEEE Trans. Magn. 27, 3588 /H208491991 /H20850. 7T. 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PhysRevLett.97.117601.pdf

Origin of Increase of Damping in Transition Metals with Rare-Earth-Metal Impurities A. Rebei *and J. Hohlfeld Seagate Research Center, Pittsburgh, Pennsylvania 15222, USA (Received 18 May 2006; published 11 September 2006) The damping due to rare-earth-metal impurities in transition metals is discussed in the low concen- tration limit. It is shown that all established damping mechanisms based on spin-orbit and/or spin-spininteractions cannot explain experimental observations even qualitatively. We introduce a different relaxation channel due to the coupling of the orbital moments of the rare-earth-metal impurities and the conduction pelectrons that leads to good agreement with experiment. Using an itinerant picture for the host ions, i.e., write their magnetization in terms of the electronic degrees of freedom, is key to the success of our model. DOI: 10.1103/PhysRevLett.97.117601 PACS numbers: 76.30.Kg, 72.25.Rb, 76.60.Es Magnetization dynamics has become one of the most important issues of modern magnetism. This developmentis driven by the technological demand to tailor magneticresponses on ever smaller length and shorter time scales.The importance of this issue manifests itself in a com-pletely new area of research, spintronics, and a huge lit- erature that cannot be cited here. Selected highlights include precessional switching by tailored field pulses[1,2], spin-torque [ 3,4], and laser-induced magnetization dynamics [ 5,6]. In general, magnetization dynamics is described via the Landau-Lifshitz-Gilbert equation (LLG) [ 7] including ad- ditional terms to incorporate spin-torque effects [ 8]o r those due to pulsed optical excitations [ 9]. All these de- scriptions account for energy dissipation via a phenome- nological damping parameter /.0011which governs the time needed for a nonequilibrium magnetic state to return toequilibrium. Recently it has even been suggested that /.0011 determines the magnetic response to ultrafast thermal ag-itations [ 10]. Technological applications call for the ability to tailor /.0011 [11]. The most systematic experimental investigation on this topic was published by Bailey et al. [12] who studied the effect of rare-earth-metal doping on the damping inpermalloy. Most rare-earth-metal ions induced a large in-crease of/.0011, but neither Eu nor Gd altered the damping of permalloy (cf. Fig. 2). Since Gd 3/.0135andEu2/.0135have no orbital momentum, this points immediately to the impor-tance of the angular momentum in the damping process.Bailey et al. determined damping by reproducing their data via the LLG equation using /.0011as a fit parameter. This widely used procedure points to a fundamental problemof this phenomenological approach. Though the LLGequation describes data well, a more microscopic approachis needed to understand the origin of damping. It was Elliott [ 13] who first studied damping in semi- conductors due to spin-orbit coupling. Later Kambersky[14] argued that the Elliot-Yafet mechanism should be also operable in magnetic conductors. Korenman and Prange [15] developed a more microscopic treatment and foundthat spin-orbit coupling should be important at low tem- perature in transition metals. Recent measurements ofdamping in magnetic multilayers at room temperature[16] suggest that the s-dinteraction might also be at the origin of damping [ 17,18]. However, all of the present models fail to reproduce the data of Ref. [ 12]. In this Letter, we explain the increase of damping in rare-earth-metal-doped transition metals via a novel orbit-orbit coupling between the conduction electrons and theimpurities. The well-known s-finteraction [ 19] gives rise to a/.0133g J/.02551/.01342dependence of the damping that is in contra- diction to experimental observations [ 12]. In contrast, the orbit-orbit coupling considered here reproduces the mea-sured/.0133g J/.02552/.01344dependence of the damping. Both depen- dencies on the Lande gfactorgJfollow directly from the fact that the rare-earth-metal ions are in their ground state.Hence, their angular momentum L f, spin Sf, and total angular momentum Jfare related by the Wigner Eckard theorem: Lf/.0136/.01332/.0255gJ/.0134JfandSf/.0136/.0133gJ/.02551/.0134Jf. Deriving the magnetic moments of the transition-metal ions from theelectronic degrees of freedom is essential to capture thecorrect behavior of damping as a function of J f. For the uniform mode, the damping due to orbit-orbit coupling is of Gilbert form in the low frequency limit. Taking the wave functions of the d-,f-, and conduction electrons orthogonal, the Hamiltonian for the rare-earth-metal-doped transition metal in an external field His H/.0136He/.0135Hf/.0135Hd: (1) This approximation should be valid for the heavy rare- earth metals but probably fails for elements like cerium where valence fluctuations are important. The conductionelectron Hamiltonian H eis the usual one, He/.0136P k;/.0027/.0015k;/.0027ak;/.0027yak;/.0027, whereay k;/.0027andak;/.0027are the creation and annihilation operators of a conduction electron with momentum kand spin/.0027./.0015k;/.0027is the energy of the con- duction electrons including a Zeeman term. Hfis the Kondo Hamiltonian [ 20] of the localized rare- earth-metal momentPRL 97,117601 (2006)PHYSICAL REVIEW LETTERSweek ending 15 SEPTEMBER 2006 0031-9007=06=97(11)=117601(4) 117601-1 ©2006 The American Physical Society Hf/.0136/.0255Se/.0001Sf/.0135/.0021Le/.0001Lf/.0255/.0022f/.0001H: (2) Se=fandLe=fare the spin and angular momentum of conduction and felectrons, respectively. Le=fare taken with respect to the position of the impurity. The spin-spin term is the well-known s-fcoupling used by de Gennes to reproduce the Curie temperatures in rare-earth metals with/.0255being of the order 0.1 eV [ 19]. The last term is again a Zeeman term. The middle term is the essential orbit-orbitinteraction needed in our discussion. To get a nonzeroorbit-orbit term due to a single impurity at the center, it is essential to include higher terms of the partial wave expansion for the wave functions of the conduction elec-trons: k/.0133r/.0134/.01364/.0025/.0129/.0129/.0129 VpP1 l/.01360Pm/.0136l m/.0136/.0255lilf/.0133r/.0134jl/.0133kr/.0134Ylm/.0133/.0018k;/.0030k/.0134/.0002 Y/.0003 lm/.0133/.0018;/.0030/.0134. The first nontrivial contribution for l/.01361is [20] HLL/.0136i/.01332/.0255gJ/.0134X k;k0/.0021/.0133k;k0/.0134^k/.0002^k0/.0001Jfay kak0;(3) where the orbit-orbit coupling /.0021will be assumed to be a function of the relative angles of the kvectors and is almost everywhere zero except for kclose to the Fermi levelkF. The magnitude of /.0021is not known but is expected to be of the same order as the spin-spin coupling constant /.0255 [21,22]. The crystalline electric field effect in transition metals is less than 0.1 meV which is small and hence thespin-orbit term S e/.0001Lfis neglected. At room temperature all the rare-earth-metal ions studied in Ref. [ 12] are in their ground state making the term Sf/.0001Lfineffective as damp- ing mechanism. This follows immediately from the Wigner-Eckart theorem. The Hamiltonian for the host transition-metal ions is based on the Anderson Hamiltonian with explicit spinrotational invariance in the absence of a Zeeman term[15,23,24]. It is Hd/.0136/.0015ddy/.0027d/.0027/.0135X kVkd/.0133ay k;/.0027d/.0027/.0135dy/.0027ak;/.0027/.0134/.0135U 8/.00262 /.0255U 2Sd/.0001Sd/.0255/.0022d/.0001H; (4) where Sdis the spin operator of the local delectrons while their orbital angular momentum is assumed quenched. /.0026is the charge density operator of the delectrons. In transition- metal ions such as Ni, Vkd/.00251:0–10:0e V is comparable to the Coulomb potential U. The hybridization term between the conduction and delectrons is essential to establish a spin-independent orbit-orbit coupling between the dand thefions. The degree of localization of the magnetic moments increases with decreasing Vkd[25] and controls the extent to which rare-earth-metal impurities enhancedamping. The orbit-orbit coupling [cf. Eq. ( 3)] gives no contribu- tion forGd 3/.0135/.01334f7/.0134as observed in the experiment [ 12]. As for the element Eu, it is believed from measurements of theparamagnetic susceptibilities that the ionic state isEu 2/.0135/.01334f7/.0134and notEu3/.0135/.01334f6/.0134[19,26]. If this is the casethen clearly this is a state with Lf/.01360and it is the same as that ofGd3/.0135. Yb is also present in a double-ionized state [27] and therefore doping with Yb2/.0135/.01334f14/.0134should not increase damping. This result remains to be confirmed byexperiment. For Eu there is an additional reason why its angular momentum is quenched. The first excited state of this latter element lies only about 400 K above the groundstate [ 27] and this can lift the degeneracy of the ground state. The average orbital angular momentum will there-fore be zero even though L 2remains a good quantum number [ 28]. Hence our Hamiltonian from the outset re- produces the experimental results for Eu and Gd andpredicts that doping with Yb should not change the damp- ing. We next address the remaining rare-earth elements. First, we outline the steps to derive the damping due to the orbit-orbit coupling term. We are only interested in thedamping of the dmoments of the transition metal; there- fore, it is advantageous to adopt a functional integralapproach. Since our system is near equilibrium and farfrom the Curie point, we use the spin wave approximationand expand the spin operators of the fmoments in terms of Boson operators f /.0006, wheref/.0006/.0136Sy f/.0006iSx f. We keep only the first nontrivial terms. The integration of the conduction electrons is carried out exactly. Afterward we integrate theimpurity variables, fandf y, also exactly but keep only quartic terms in dandd/.0135. The remaining effective action has now only the fields danddyand from their equations of motion the spin propagator hm/.0255/.0133/.0028/.0134m/.0135/.0133/.00280/.0134iof thed moments,m/.0006/.0136Sx d/.0006iSy d, can be determined. We use a Stratonovich-Hubbard transformation to write this effec- tive Lagrangian in terms of m/.0006. Then a stationary phase approximation of the functional generator allows us to determine the desired propagator and hence the damping.We finally compare the functional form of this result to thatof LLG and discuss why the electronic (itinerant) picture ofthe host transition-metal ions is essential. The fundamental quantity in our calculation is the gen- erating functional Z/.0137/.0017/.0003;/.0017/.0138/.0136Tre/.0255R/.0012 0d/.0028fH/.0255/.0017/.0003/.0133/.0028/.0134m/.0255/.0133/.0028/.0134/.0255/.0017/.0133/.0028/.0134m/.0135/.0133/.0028/.0134g; (5) where/.0017and/.0017/.0003are external sources and /.0012is inverse temperature. The propagator, i.e., the connected two-- point Green’s function, of the volume mode of thetransition-metal ions is found by functional differentia-tions with respect to the external sources /.0017 /.0003and/.0017, hm/.0135/.0133/.0028/.0134m/.0255/.0133/.00280/.0134ic/.0136/.00142lnZ/.0137/.0017/.0003;/.0017/.0138=/.0014/.0017/.0133/.0028/.0134/.0014/.0017/.0003/.0133/.00280/.0134. It is cal- culated within a double random phase approximation(RPA2) method. The true single particle propagator of thedbands is first found within a RPA in the presence of an effective field due to the conduction electrons and the impurities. In turn, the effect of the fimpurities on the conduction electrons is calculated within RPA. The result-ing effective Lagrangian is now written in terms of monly L/.0136/.02551 2mijKijklmkl/.0255Tr ln/.0137G/.02551 d/.0135Km/.0138; (6) whereG/.02551 d/.0133/.00271;/.00272/.0134/.0136@/.0028/.0255/.0022/.0015d/.0135V2Gc/.0135TrkfGfGcBGcAgPRL 97,117601 (2006)PHYSICAL REVIEW LETTERSweek ending 15 SEPTEMBER 2006 117601-2is the propagator of the delectrons in the presence of the conduction electrons and the rare-earth-metal impurity ( /.0027i/.01361,2 for spin-up and spin-down, respectively). The quadratic term in mrepresents effective anisotropy and spin-charge interactions and is given by K/.00271/.00272/.00273/.00274/.0136/.0255U 4/.0133/.0014/.00271/.00272/.0014/.00273/.00274/.02552/.00141/.00271/.00142/.00272/.00141/.00273/.00142/.00274/.0134/.025522V4Gf/.0133GcBGcAGc/.0134/.00271/.00272Gf/.0133GcBGcAGc/.0134/.00273/.00274/.0014/.00271/.00274/.0014/.00272/.00273 /.0255V4GcAGcGfGcBGc: (7) Integrations over momentum and spin are implied in all these expressions. The different terms that appear in Kare as follows:Gcis the Green’s function of the conduction electrons in the mean field approximation G/.02551c/.0133k;/.00271;k0;/.00272;/.0028/.0134/.0136/.0133@/.0028/.0135/.0022"k/.00271/.0255/.0022F/.0134/.0014kk0/.0014/.00271/.00272/.0135i/.0021/.0133k;k0/.0134/.01332/.0255gJ/.0134hJz fi/.0133k0xky/.0255k0ykx/.0134/.0014/.00271/.00272; (8) which is off diagonal in momentum due to the orbit-orbit coupling. /.0022"k;/.0027now includes Zeeman terms due to the external field and the zcomponent of the field due to impurity. The propagator Gfis that of the fions in the presence of both the conduction electrons and the transition-metal ions, G/.02551 f/.0133/.0028/.0134/.0136@/.0028/.0135/.0022fH/.0135Trk;/.0027fGcAGcBg. TheAandBmatrices are solely due to the presence of the impurity and represent the indirect coupling between the transition-metal ions and the f ions A/.0133k0;/.00271;k;/.00272/.0134/.0136B/.0133k;/.00271;k0;/.00272/.0134/.0003/.0136/.02550/.0027/.0135/.00271/.00272/.0255i/.00210/.0001/.0135 k0k; (9) where we have set /.02550/.0136/.0255/.0129/.0129/.0129/.0129/.0129/.0129 2Jfp 4/.0133gJ/.02551/.0134,/.00210/.0136/.0021/.0129/.0129/.0129/.0129/.0129/.0129 2Jfp 2/.01332/.0255gJ/.0134, and/.0001/.0006 kk0/.0136/.0133^k0/.0002^k/.0134/.0006. In the trace log term of the effective Lagrangian, the first nontrivial contribution is of order V4and is given by Fig. 1. The diagram with a single insertion of an f propagator does not contribute due to the antisymmetry of the orbit-orbit coupling in the momentum space. Varying theeffective action with respect to m ijgives four equations which can be averaged and differentiated with respect to the external sources to get the mpropagators. We are only interested in C/.01331221/.0134/.0136hm12m21iwhich is given by fG/.02551 d11/.0135K11ijhmijigC/.01331221/.0134/.0135K11ijC/.0133ij21/.0134hm12i/.0136/.0255 hm22i/.0255K21ijC/.0133ij21/.0134hm22i/.0255K21ijhmijiC/.01331221/.0134: (10) In the absence of impurities, these equations are to lowest order the time-dependent generalization of the Hartree-Fock equations derived by Anderson [ 23]. Using the RPA2 method, we solve for C/.01331221/.0134 C1221/.0133!l/.0134/.0136X nm11/.0133!n/.0134m22/.0133!n/.0135!l/.0134/.0030/.0020 1/.0135X n;mK2112/.0133!m/.0134m11/.0133!n/.0135!m/.0134m22/.0133!n/.0135!m/.0135!l/.0134/.0021 ; (11) where!l/.0136/.01332l/.01351/.0134/.0025=/.0012 for integerl. If we ignore the impurity interaction and replace the average values of the mijby the Anderson solution, we recover the RPA result for the propagator of the magnetization. To include theimpurities, we evaluate the dpropagators, m ij, within RPA. In the low frequency limit, !/.0028/.0001/.0028!c, we find that the (retarded) propagator CRof the theory is proportional to /.0133!/.0255!0/.0135i/.0011!/.0134/.02551. Here,/.0001/.02551is the lifetime of the virtual dstates [ 23],!cdenotes the frequency of the conduction electrons, and !0is the ferromagnetic resonance frequency of the transition metal. This low frequency limit for thedamping is similar to that of the LLG result [ 15]. The damping/.0011in the spin-conserving channel is proportional toJ f/.0133Jf/.01351/.0134/.0137/.0133gJ/.02552/.0134jVj/.01384and is given by /.0011/.0136cj/.0021Vj4Jf/.0133Jf/.01351/.0134/.01332/.0255gJ/.01344 /.0002/.0020U/.0001E 25/.00253/.0133E/.0255/.0001E/.01342/.0133E/.0135/.0001E/.01342/.0133nmkF/.01342 18!4c/.0135Q/.0133!f/.0134/.0021 : (12) Herenis the density of conduction electrons, cis the concentration of the fimpurities, and E/.0006/.0001Eis the energy of the up-down dstates. These latter energies can be determined self-consistently as in the Anderson solution[23] and hence their form is not expected to dependstrongly on the atomic number of the rare-earth-metal impurity at low concentrations. The explicit form of the functionQis not needed here but it represents contribu- tions beyond the ‘‘mean’’ field approximation of the f impurities and is given by Fig. 1. In Fig. 2, we show that the leading coefficient of the damping due to non-spin-flipscattering (solid curve) is in very good agreement with the experimental results of Bailey et al. [12]. Finally we point out the reasons behind insisting on using the itinerant electrons explicitly instead of the sim- plers-dexchange interaction which accounts well for damping in permalloy [ 16]. Using a localized-type Hamiltonian for the dmoments d d d dVV VVff cc FIG. 1. The first diagram that is contributing to the damping of thedelectrons due to the fimpurities through the conduction electrons.PRL 97,117601 (2006)PHYSICAL REVIEW LETTERSweek ending 15 SEPTEMBER 2006 117601-3 Hd/.0136/.0255JSe/.0001Sd/.0255/.0022d/.0001Sd (13) instead of Eq. ( 4), leads to a damping which differs sig- nificantly from experiment (dashed curve in Fig. 2). This localized moment Hamiltonian, however, appears to de-scribe well damping in insulators such as heavy rare-earth-metal-doped garnets [ 29]. In garnets, the hybridization coupling is smaller than in metals. Hence our result alsoexplains why the damping in rare-earth-metal-doped gar-nets is not as strong as in the rare-earth-metal-doped tran-sition metals. The experimental measurements (triangles) clearly show that at room temperature non-spin-flip scat- tering is more important than spin-flip scattering whichonly becomes important close to the critical temperature.Again, the data are well reproduced by the orbit-orbitcoupling and the relatively large increase in damping isdue to the large virtual mixing parameter V kd. In contrast, thes-fcoupling (squares in Fig. 2) is in conflict with experiment. In summary, we have shown that the damping in rare- earth-metal-doped transition metals is mainly due to anorbit-orbit coupling between the conduction electrons andthe impurity ions. For near equilibrium conditions and inthe low frequency regime this leads to damping for theuniform mode that is of Gilbert form. The orbit-orbitmechanism introduced here is much stronger than thespin-orbit based Elliott-Yafet-Kambersky mechanism since the charge-spin coupling at the host ion is of the order of 1–10 eV compared to 0.01 eV for spin-orbitcoupling. The predicted increase of damping is propor-tional toV 4which in transition-metal ions is of the same order asUthe Coulomb potential. A localized model for thedmoments based on the s-dexchange is unable to account for the increase in damping in these doped systemsas a function of the orbital moment of the rare-earth-metal impurities. An additional test of this damping theory would be to measure the effect of a single rare-earth element onthe damping in various transition metals. Such experimentswill provide further insight into the dependence of damp-ing onVand will improve our understanding of the itin- erant versus localized pictures of magnetism. We acknowledge fruitful discussions with P. Asselin, W. Bailey, O. Heinonen, P. Jones, O. Myarosov, and Y . Tserkovnyak. *Electronic address: arebei@mailaps.org [1] Th. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. Ba ¨r, and Th. Rasing, Nature (London) 418, 509 (2002). [2] H. W. Schumacher, C. Chappert, R. C. Sousa, P. P. Freitas, and J. Miltat, Phys. Rev. Lett. 90, 017204 (2003). [3] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature(London) 425, 380 (2003). [4] S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, Nature (London) 437, 389 (2005). [5] E. Beaurepaire, J. C. Merle, A. Daunois, and J. Y . Bigot, Phys. Rev. Lett. 76, 4250 (1996). [6] J. Hohlfeld, E. Matthias, R. Knorren, and K. H. Bennemann, Phys. Rev. Lett. 78, 4861 (1997). [7] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935); T. L. Gilbert, Phys. Rev. 100, 1243 (1955). [8] Z. Li and S. Zhang, Phys. Rev. B 69, 134416 (2004). [9] M. V omir, L. H. F. Andrade, L. Guidoni, E. Beaurepaire, and J. Y. Bigot, Phys. Rev. Lett. 94, 237601 (2005). [10] B. Koopmans, J. J. M. Ruigrok, F. D. Longa, and W. J. M. de Jonge, Phys. Rev. Lett. 95, 267207 (2005). [11] E. M. Ryan et al. , B22.00003, March APS Meeting, Baltimore, Maryland, 2006. [12] S. G. Reidy, L. Cheng, and W. E. Bailey, Appl. Phys. Lett. 82, 1254 (2003). [13] R. J. Elliott, Phys. Rev. 96, 266 (1954). [14] V . Kambersky, Can. J. Phys. 48, 2906 (1970). [15] R. E. Prange and V . Korenman, Phys. Rev. B 19, 4691 (1979); 19, 4698 (1979). [16] S. Ingvarsson et al. , Phys. Rev. B 66, 214416 (2002). [17] B. Heinrich, D. Fraitova, and V . Kambersky, Phys. Status Solidi 23, 501 (1967). [18] A. Rebei and M. Simionato, Phys. Rev. B 71, 174415 (2005). [19] P.-G. de Gennes, Compt. Rend. 247, 1836 (1958); J. Phys. Radium 23, 510 (1962); P.-G. de Gennes, C. Kittel, and A. M. Portis, Phys. Rev. 116, 323 (1959). [20] J. Kondo, Prog. Theor. Phys. 27, 772 (1962). [21] J. H. Van Vleck and R. Orbach, Phys. Rev. Lett. 11,6 5 (1963); R. C. LeCraw, W. G. Nilsen, and J. P. Remeika, and J. H. Van Vleck, Phys. Rev. Lett. 11, 490 (1963). [22] P. Levy, Phys. Rev. Lett. 20, 1366 (1968). [23] P. W. Anderson, Phys. Rev. 124, 41 (1961). [24] C. A. Macedo, M. D. Coutinho-Filho, and M. A. de Moura, Phys. Rev. B 25, 5965 (1982). [25] J. R. Schrieffer, J. Appl. Phys. 38, 1143 (1967). [26] B. T. Thole et al. , Phys. Rev. B 32, 5107 (1985). [27] R. J. Elliott, Magnetic Properties of Rare Earth Metals (Plenum, New York, 1972). [28] C. Kittel, Introduction to Solid State Physics (Wiley, New York, 2005). [29] P. E. Seiden, Phys. Rev. 133, A728 (1964).Z-57Normalized Damping Prefactor 1 2 3 4 5 6 7 8 910 11 12 13 1400.20.40.60.81 Eu Gd Ho Yb Ce FIG. 2. Comparison of the normalized leading factor in the damping as a function of the rare-earth-metal impurity in Eq. ( 12) (solid line) and Eq. ( 13) (dashed line) to the data of Ref. [ 12]. The squares represent damping due to s-fcoupling only, Eq. ( 2), without the orbit-orbit coupling.PRL 97,117601 (2006)PHYSICAL REVIEW LETTERSweek ending 15 SEPTEMBER 2006 117601-4

PhysRevB.85.184409.pdf

PHYSICAL REVIEW B 85, 184409 (2012) Harmonic transition-state theory of thermal spin transitions Pavel F. Bessarab,1,2Valery M. Uzdin,2,3and Hannes J ´onsson1 1Science Institute and Faculty of Science, VR-III, University of Iceland, 107 Reykjav ´ık, Iceland 2Department of Physics, St. Petersburg State University, St. Petersburg, 198504, Russia 3St. Petersburg State University of Information Technologies, Mechanics and Optics, St. Petersburg, 197101, Russia (Received 14 November 2011; revised manuscript received 17 February 2012; published 9 May 2012) A rate theory for thermally activated transitions in spin systems is presented. It is based on a transition-state approximation derived from Landau-Lifshitz equations of motion and quadratic expansion of the energy surfaceat minima and first order saddle points. While the flux out of the initial state vanishes at first order saddle points,the integrated flux over the hyperplanar transition state is nonzero and gives a rate estimate in good agreementwith direct dynamical simulations of test systems over a range in damping constant. The preexponential factorobtained for transitions in model systems representing nanoclusters with 3 to 139 transition metal adatoms is onthe order of 10 11to 1013s−1, similar to that of atomic rearrangements. DOI: 10.1103/PhysRevB.85.184409 PACS number(s): 05 .20.Dd, 75 .10.−b I. INTRODUCTION Metastable magnetic states have been studied experimen- tally in small systems of various kinds, in particular, molec-ular magnets 1and supported2–4and free-standing5transition metal clusters. The stability of such states with respect tothermal fluctuations is an important issue in many contexts,for example, when assessing the limit to which magneticrecording devices can be miniaturized. Although the systemsmentioned above are quite different, they are all characterizedby two or more magnetic states which correspond to differentorientations and/or different values of the magnetic moments. A preparation of a system in a particular magnetic state can be destroyed by thermally activated transitions to otherstates. For long-lived magnetic states, the separation intime scale between fast precession of magnetic momentsand slow transitions between states make direct dynamicalsimulation of spin dynamics 6impractical. This, however, opens the possibility for the use of a statistical approachfor estimating spin transition rates as well as determiningthe transition mechanism. Statistical approaches have beenpresented previously for a single macrospin, 7,8but we are not aware of a previous presentation of a statistical ratetheory for systems with multiple spins. Even in small clusters,transitions can involve nucleation and propagation of a domainwall rather than coherent rotation of magnetic moments. 2,3A macrospin approximation would in such cases give the wrongactivation barrier height and a poor estimate of the rate. Inthis paper, a method for finding the mechanism and rate ofthermal spin transitions is developed by adapting transitionstate theory (TST) 9to multiple spin degrees of freedom. It gives an Arrhenius law for the transition rate, which can beevaluated using only the input that would be needed for a directsimulation of the spin dynamics—a simulation that would,however, be impossibly long in the cases of interest. II. THEORY TST9has been used extensively for estimating the rate of thermally activated atomic rearrangements such as chemicalreactions and diffusion. 10The separation of time scale mentioned above makes it possible to estimate the rate fromthe probability of finding the system in the most restrictive andleast likely region separating the initial state from possible final states—the transition state. Given a transition statedividing surface, f(x)=0, where xrepresents all dynamical variables in the system, the reaction rate constant, k TST, can be estimated for spin systems in a way that is analogous toatomic systems 11as kTST=1 C/integraldisplay Re−E(x)/kBTδ[f(x)]v⊥(x) (1) ×H[v⊥(x)]J(x)/productdisplay idxi, where Rdenotes the region associated with the initial state up to and including the dividing surface, J(x) is a Jacobian determinant, Eis the energy of the system, v⊥(x)=∇f(x)·˙x is a projection of the velocity onto the normal of the dividingsurface, and Cis a normalization constant given by C=/integraldisplay Re−E(x)/kBTJ(x)/productdisplay idxi. A central approximation is that a trajectory only crosses the dividing surface once,12and this is taken into account by inserting a Heaviside step function Hinto Eq. (1).F o rs p i n systems, the relevant variables are taken to be spherical anglesθ iandφidefining the direction of the ith magnetic moment. The set of variables for a system of spins is denoted as x≡ {θ,φ}≡{θ1,θ2,..., θ N,φ1,φ2,..., φ N}. The magnitude of the magnetic moments Mii sa s s u m e dt ob eaf u n c t i o no ft h ea n - gles,Mi(θ,φ), i.e., an adiabatic approximation is invoked. The Jacobian determinant is then J(θ,φ)≡/producttext iM2 i(θ,φ)s i nθi. The normal projection of the velocity, v⊥(θ,φ), needs to be estimated at each point on the dividing surface. The equationof motion is taken to be the Landau-Lifshitz equation (seeRef. 13) dM i dt=γMi×∂E ∂Mi, (2) where γis a gyromagnetic ratio. In the adiabatic limit, this equation can be split into two equations: ˙φi=γ Misinθi∂E ∂θiand ˙θi=−γ Misinθi∂E ∂φi.(3) 184409-1 1098-0121/2012/85(18)/184409(4) ©2012 American Physical SocietyBESSARAB, UZDIN, AND J ´ONSSON PHYSICAL REVIEW B 85, 184409 (2012) FIG. 1. Comparison of the rate of transitions in a spin trimer obtained directly from dynamics given by the Landau-Lifschitz-Gilbert equation of motion as a function of the damping constant αatT=23 K (solid line) and a harmonic TST estimate (dotted line). Inset: the energy surface near a first order saddle point, representation of a hyperplanar transition state dividing surface (thick line) and the spin velocity (arrows). The length of the velocity vector is proportional to the magnitude of the energy gradient but the velocity and gradientvectors are perpendicular. The TST expression for the rate constant can be simplified by introducing quadratic approximations to the energy surfacearound the critical points to give a harmonic TST (HTST)approximation. The transition state dividing surface is thenchosen to be a hyperplane going through a first order saddlepoint on the energy ridge separating the initial state fromproduct states (see Fig. 1). The hyperplane normal is chosen to point in the direction of the unstable mode, the eigenvector ofthe Hessian matrix along which the saddle point is a maximum.If second order saddle points on the ridge are high enoughabove first order saddle points, then each first order saddlepoint corresponds to a specific transition mechanism and acertain product state. For each possible final state, one or moreminimum energy paths (MEP) can be found. Following anMEP means advancing each degree of freedom of the system insuch a way that the energy is minimal with respect to all degreesof freedom perpendicular to the path. The nudged elastic band(NEB) method 14can be used to find MEPs between a given pair of initial and final states. A maximum along an MEPcorresponds to a first order saddle point on the energy surfaceand the highest one gives an estimate of the activation energy. Unlike atomic systems, the velocity in spin systems is zero at a saddle point because the gradient is zero. In the vicinityof the saddle point, the energy surface, E(θ,φ), can in general be approximated as a parabolic function and the magnitudeof the energy gradient and, thereby, the velocity increasesas one moves away from the saddle point. Moreover, sincethe energy gradient lies within the dividing surface at pointson the dividing surface, the velocity is perpendicular to thedividing surface. The expansion of the energy at the minimum ( β=m) and at the saddle point ( β=s)i s E β(q)=Eβ(0)+1 2D/summationdisplay j=1/epsilon1β,jq2 β,j, (4)where Dis twice the number of spins. The expansion is in terms of normal mode coordinates, displacements alongeigenvectors of the Hessian matrix. The Landau-Lifshitzequations become linear with this quadratic approximation tothe energy surface. At the saddle point, one of the eigenvectors,the one corresponding to the unstable mode, is orthogonal tothe dividing surface. Labeling this mode as q s,1, the velocity v⊥(θ,φ)=˙qs,1can according to Eqs. (2)–(4)be written as a linear combination of normal mode coordinates, v⊥=D/summationdisplay i=2aiqs,i. (5) The direction of each eigenvector at the saddle point is chosen so that ai>0 leads to a positive contribution to v⊥, i.e., pointing away from the initial state. With these quadratic approximations to the energy surface, the HTST expression for the rate constant becomes kHTST=/integraltext e−/summationtextD j=2/epsilon1s,jq2 s,j/2kBT/summationtextD i=2aiqidq2...dq D/integraltext e−/summationtextD j=1/epsilon1m,jq2 m,j/2kBTdq1...dq D ×Js Jme−(Es−Em)/kBT, (6) where Jβ≡J(θβ,φβ). The denominator is simply a product of Gaussian integrals. The numerator is more complicatedbecause the integrals involved are carried out over the regionwhere v ⊥/greaterorequalslant0, i.e., over the half-plane a2q2+a3q3+···+ aDqD/greaterorequalslant0. After some algebra (which will be published elsewhere) one obtains kHTST=1 2πJs Jm/radicaltp/radicalvertex/radicalvertex/radicalbtD/summationdisplay j=2a2 j /epsilon1s,j/producttextD i=1√/epsilon1m,i/producttextD i=2√/epsilon1s,ie−(Es−Em)/kBT,(7) which agrees with an Arrhenius expression with an activation energy Ea=Es−Emand a temperature independent preex- ponential, ν. The theory presented here is classical and makes use of harmonic approximations to the energy surface. An extensionto full transition-state theory involving statistical samplingwithin the dividing surface as well as the inclusion of quantumtunneling by use of Feynman path integrals, analogous to whathas been formulated for particle systems (see, for example,Refs. 15and16), is an ongoing project. III. APPLICATIONS Below, this rate theory is applied to transitions in three different systems. First, three spins are considered and HTSTresults compared with direct simulations of the dynamics.Then, the method is applied to a cluster of 139 Fe adatomson a W(110) surface. These first two systems are describedby a Heisenberg-type Hamiltonian. The third example is athree atom Fe cluster on a substrate described by a Alexander-Anderson Hamiltonian for itinerant electrons. 184409-2HARMONIC TRANSITION-STATE THEORY OF THERMAL ... PHYSICAL REVIEW B 85, 184409 (2012) The Heisenberg-type Hamiltonian can be written in a general form: E=−/summationdisplay nKn/summationdisplay i(Mi·en)2−1 2J/summationdisplay /angbracketlefti,j/angbracketrightMi·Mj −D/summationdisplay i/negationslash=j3(rij·Mi)(rij·Mj)−r2 ij(Mi·Mj) r5 ij.(8) The magnitude of the magnetic moments Miis independent of angles. The first term represents anisotropy, Jdenotes the exchange coupling, Dis dipolar coupling constant, and rij is the vector between sites iandj. Exchange interaction is only between nearest neighbors (indicated by the angularbrackets). The HTST rate constant estimate, Eq. (7), was tested by comparing it with the rate of transitions observed in a directsimulation of the dynamics of a multidimensional systeminvolving three spins which are coupled through the exchangeinteraction. Parameters of the Hamiltonian [Eq. (8)]w e r e chosen to include easy-axis K ⊥and easy-plane K/bardblanisotropies which could result from the interaction with a substrate. Asa result, minima and saddle points are formed on the energysurface. Parameter values and temperature were chosen so as tomake the transitions frequent enough to obtain good statisticsin dynamics simulations spanning a long time interval butinfrequent enough for the system to be able to thermalize atthe bath temperature in between transitions. The parameterswereM=2.7μ B,K⊥M2=5m e V , K/bardblM2=−10 meV, and JM2=15 meV. Two equivalent minima exist on the energy surface: at θi=π/2 and φi={π/2,3π/2}. There are two equivalent saddle points between the minima: at θi=π/2 andφi={0,π}. The activation energy was found to be Es− Em=15 meV. The dynamics of the spins were calculated numerically from the Landau-Lifshitz-Gilbert equations wheredissipation and random force terms are included to couplethe spins to a thermal heat bath. 17,18Long simulations were performed, spanning 10−5s and 109steps at a temperature of 23 K. This gave satisfactory statistics in the counting oftransitions. The dynamical simulations were carried out forvarious values of the damping constant, α, as shown in Fig. 1. The agreement with the HTST estimate is good, within a factorof 2, for a wide range in the damping constant. While theparameters and temperature have been chosen here to make itpossible to obtain the transition rate from direct simulation ofthe spin dynamics, the usefulness of the rate theory presentedhere becomes clear when the temperature is lowered, theactivation energy barrier increased, and/or the number of spinsincreased. Then, the direct calculation become difficult, oreven impossible, while the evaluation of the rate expressionremains relatively straightforward. To demonstrate the methodology presented here on a more challenging system, we applied it to a larger, rectangular islandof 139 Fe atoms (see Fig. 2). The parameters in Eq. (8) were chosen to mimic roughly Fe on W(110) substrate: 2,19 dipole-dipole interactions were included as well as anisotropy in the [1 ¯10] direction (which is perpendicular to the long axis of the cluster) resulting from the interaction with the sub-strate. The parameter values were KM 2=0.55 meV, JM2= 12.8m e V ,2andD/J=10−3.19Two degenerate states with-3.50-3.48-3.46-3.44 0 10 20 30 40 50 FIG. 2. Minimum energy path (solid line) for a magnetic transi- tion in a rectangular shaped 139 Fe atom island on W(110) surface.A relaxation starting from a straight line interpolation (dotted line) representing a uniform rotation of the spins between spin up and spin down states revealed an intermediate metastable state, as shown bythe insets. The discretization points used in the NEB calculation are shown with filled circles but the minimum for the metastable state and a saddle point obtained by subsequent optimization are marked withX.Insets: noncollinear spin configurations at various points along the path. spins parallel to the anisotropy axis represent the most stable states. A NEB calculation starting from a uniform rotationrevealed a more complicated transition mechanism involvinga metastable intermediate state, as shown in Fig. 2.T h e metastable state can be seen as the emergence of a new domain, π π θ2ππθ3 π π θ2ππθ3 FIG. 3. Minimum energy path for a transition between a parallel (P) and antiparallel (AP) state of a Fe trimer on a metal substrate described with the Alexander-Anderson model. Spin configurations corresponding to locations marked with X on the path are shown witharrows denoting magnitude and direction of the magnetic moments. The direction of spin 1 is taken to be fixed but the relative angles, θ 2 andθ3, are variable. The energy is given in units of the d-level width, /Gamma1, due to s-dhybridization. Inset: energy surface showing minima corresponding to P and AP states, and the calculated minimum energy path for the transition. Saddle point (SP) is also shown. 184409-3BESSARAB, UZDIN, AND J ´ONSSON PHYSICAL REVIEW B 85, 184409 (2012) but the cluster is too small for it to form fully. The activation energy for transitions out of the metastable state is 14 meVand the calculated preexponential, according to Eq. (7),i s ν=7.4×10 12s−1. The lifetime of the intermediate state can be estimated as τ(T)=1/kHTST. Although the Landau-Lifshitz equation was first formulated in order to describe the precessional motion of classicalmagnetization, it has proven to be an equation of motionalso for quantum systems. 13,20This expands the range of applicability of the rate theory presented here. We demonstratethis on a triatomic Fe island described within the noncollinearAlexander-Anderson model (see Ref. 21) which captures the main features of magnetic ordering in 3 dtransition metal clusters on a metal surface. In particular, several differentmagnetic states close in energy have been found for supportedtrimers of Fe, Cr, and Mn. 21The question is how large an energy barrier separates these states and how long their lifetimeis at a given temperature. While a triatomic island is too smallto support long-lived metastable states, we use this as anillustration of the methodology because the energy surfacecan be visualized easily. The implementation of the model within a mean field approximation and the parameter values used here to representFe trimer are given in Ref. 21. The interaction of d-electrons with the itinerant s-andp-bands is included, but not spin-orbit interaction so spin space and the real space are not connected.There is no energy barrier to uniform rotation of magneticmoments and only relative orientation of spins is relevant. The quantization axis for the system is chosen to be alongthe magnetic moment of one of the trimer atoms, atom 1. Theconfiguration is then specified by only four degrees of freedom.There are two energy minima and both of them correspondto in-plane orientation of the spins. The global minimumrepresents a state with parallel (P) magnetic moments, buta metastable state with antiparallel (AP) moments also exists.Figure 3shows a contour graph of the energy as a function of two angles θ 2andθ3, while for all atoms we set φi=0. The minimum energy path between AP and P states turns outto lie in-plane and is also shown in Fig. 3. The activation energy for leaving the metastable state is E a=ESP−EAP= 0.007/Gamma1, where /Gamma1is the width of the d-level due to s-d hybridization.21If/Gamma1=1 eV, the calculated preexponential isν=2.4×1011s−1. A study of the effect of island size and shape on such metastable states and the rate of transitionsbetween the states is ongoing and the results will be presentedat a later time. ACKNOWLEDGMENTS This work was supported by The Icelandic Research Fund, Nordic Energy Research, and The University of IcelandScholarship Fund. We thank Professor Pieter Visscher for sev-eral constructive comments on the manuscript and ProfessorBj¨orgvin Hj ¨orvarsson for stimulating discussions. 1R. Sessoli et al. ,Nature (London) 365, 141 (1993). 2S. Krause, G. Herzog, T. Stapelfeldt, L. Berbil-Bautista, M. Bode, E. Y . Vedmedenko, and R. Wiesendanger, Phys. Rev. Lett. 103, 127202 (2009). 3S. Rohart, P. Campiglio, V . Repain, Y . Nahas, C. Chacon, Y . Girard,J. Lagoute, A. Thiaville, and S. Rousset, Phys. Rev. Lett. 104, 137202 (2010). 4N. N. Negulyaev, V . S. Stepanyuk, W. Hergert, and J. Kirschner,P h y s .R e v .L e t t . 106, 037202 (2011). 5X. Xu, S. Yin, R. Moro, A. Liang, J. Bowlan, and W. A. deHeer, P h y s .R e v .L e t t . 107, 057203 (2011). 6J. Fidler and T. Schrefl, J. Phys. D: Appl. Phys. 33, 135R (2000). 7D. M. Apalkov and P. B. Visscher, P h y s .R e v .B 72, 180405(R) (2005). 8Y . P. Kalmykov, W. T. Coffey, and S. V . Titov, Phys. Rev. B 77, 104418 (2008). 9E. Wigner, Trans. Faraday Soc. 34, 29 (1938). 10D. G. Truhlar, B. C. Garrett, and S. J. Klippenstein, J. Phys. Chem. 100, 12771 (1996); H. J´onsson, Proc. Natl. Acad. Sci. USA 108, 944 (2011).11P. Pechukas, in Dynamics of Molecular Collisions B , edited by W. H. Miller (Plenum, New York, 1976). 12J. C. Keck, Adv. Chem. Phys. 13, 85 (1967). 13V . P. Antropov, M. I. Katsnelson, B. N. Harmon, M. van Schilfgaarde, and D. Kusnezov, Phys. Rev. B 54, 1019 (1996). 14G. Henkelman, B. Uberuaga, and H. J ´onsson, J. Chem. Phys. 113, 9901 (2000); 113, 9978 (2000). 15G. K. Schenter, G. Mills, and H. J ´onsson, J. Chem. Phys. 101, 8964 (1994). 16G. Mills, G. K. Schenter, D. Makarov, and H. J ´onsson, Chem. Phys. Lett.278, 91 (1997). 17T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 18See Eqs. (2)–(5)in C. Schieback et al. ,E u r .P h y s .J .B 59, 429 (2007). 19E. Y . Vedmedenko, A. Kubetzka, K. vonBergmann, O. Pietzsch,M. Bode, J. Kirschner, H. P. Oepen, and R. Wiesendanger, Phys. Rev. Lett. 92, 077207 (2004). 20V . Korenman and R. E. Prange, J. Appl. Phys. 50, 1779 (1979). 21S. Uzdin, V . Uzdin, and C. Demangeat, Europhys. Lett. 47, 556 (1999); Comput. Mater. Sci. 17, 441 (2000); Surf. Sci. 482, 965 (2001). 184409-4

PhysRevB.92.054434.pdf

PHYSICAL REVIEW B 92, 054434 (2015) Influence of uniaxial anisotropy on domain wall motion driven by spin torque P. Chureemart,1,*R. F. L. Evans,2I. D’Amico,2and R. W. Chantrell2 1Computational and Experimental Magnetism Group, Department of Physics, Mahasarakham University, Mahasarakham 44150, Thailand 2Department of Physics, University of York, York YO10 5DD, United Kingdom (Received 19 April 2015; revised manuscript received 15 July 2015; published 25 August 2015) Magnetization dynamics of a bilayer structure in the presence of a spin-transfer torque is studied using an atomistic model coupled with a model of spin accumulation. The spin-transfer torque is decomposed into twocomponents: adiabatic and nonadiabatic torques, expressed in terms of the spin accumulation, which is introducedinto the atomistic model as an additional field. The evolution of the magnetization and the spin accumulationare calculated self-consistently. We introduce a spin-polarized current into a material containing a domain wallwhose width is varied by changing the anisotropy constant. It is found that the adiabatic spin torque tends todevelop in the direction of the magnetization whereas the nonadiabatic spin torque arising from the mistrackingof conduction electrons and local magnetization results in out-of-plane magnetization components. However, theadiabatic spin torque significantly dominates the dynamics of the magnetization. The total spin-transfer torqueacting on the magnetization increases with the anisotropy constant due to the increasing magnetization gradient. DOI: 10.1103/PhysRevB.92.054434 PACS number(s): 75 .78.Fg,75.60.Ch,75.70.Kw,87.15.hj I. INTRODUCTION The ability to manipulate the magnetization in a domain wall (DW) using a spin-polarized current has significant poten-tial for novel spintronic devices and has attracted considerableattention from both experimental and theoretical researcherssince its first introduction by Berger [ 1] and Slonczewski [ 2]. The spin-transfer torque resulting from the exchange in-teraction between the conduction electrons and the localmagnetization is an important phenomenon with potentialapplications as spin-torque oscillators for telecommunicationsapplications, in DW-based magnetic devices such as racetrackmemory [ 3,4], and in the switching of magnetic random access memory (MRAM) elements [ 5,6]. It provides an exciting technological advance, coupling fast speed, nonvolatility, andlow power requirements [ 7–9]. The physics of the spin-torque phenomenon can be described in terms of a spin accumulation,which interacts with the local magnetic moments via aquantum mechanical exchange interaction. The mechanism of spin-transfer torque in slowly varying magnetization, i.e., a domain wall, can be theoretically described by consideringthe spin current carried by the conduction electrons intothe magnetic domain wall. The spin-transfer torque arisingfrom the s-dexchange interaction acts on the spin current to adiabatically align it in the direction of the local magnetization.Simultaneously, a reaction torque proportional to the spincurrent density is created on the local magnetization withinthe DW. For sufficiently high spin current density, thespin injection causes the magnetization reorientation, resultingin the DW motion in the direction of conduction electron flow. The spin-transfer torque can be decomposed into adiabatic and nonadiabatic contributions. The former, known as theSlonczewski torque, accounts for the conduction electrons’following the direction of local magnetization whereas thelatter occurs with a rapid change in the magnetization, whichhas been explained by spin mistracking, momentum transfer, orspin-flip scattering [ 10]. In general, the nonadiabatic torque is *phanwadeec@gmail.comassumed to be much weaker than the adiabatic torque [ 11,12]. The magnitudes of the adiabatic and nonadiabatic torquesstrongly depend on the DW width, which is determinedby the anisotropy constant and exchange interaction of thematerial. The DW width becomes a significant factor for thespin-torque efficiency due to the fact that for a material withhigh anisotropy the resulting strong magnetization gradientwithin the DW subsequently gives rise to a high DW velocity.Therefore, the materials with a large anisotropy, which inturn allow a low current density to initiate DW motion,are promising candidates for the application in spintronicdevices. Spin torques are generally introduced into micromagnetic models via adiabatic and nonadiabatic terms proportional,respectively, to coefficients μ xandβx[11,13–15]. The magnitudes of both coefficients are generally taken as (un-known) constants (i.e., spatially independent) in the usualformalism and their magnitude is still a matter of discussion.Interestingly, a recent study by Claudio-Gonzalez et al. [11] has demonstrated that the magnitude of these coefficientsstrongly depends on the spatial variation of the magnetizationgradient giving rise to the nonuniform behavior throughoutthe layer. This results in the divergence of the coefficients atpositions with small gradient of magnetization. To avoid theconsequent nonphysical behavior of the empirical constantsμ xandβxClaudio-Gonzalez et al. [11] evaluated an effective nonadiabatic coefficient βdiffproviding the description of nonadiabatic torque by averaging |∂M/∂x |2with a weight function. In our recent work [ 16], the spin-torque coefficients are calculated directly from the spin accumulation. The resultsshow that the dynamic micromagnetic approach based onadiabatic and nonadiabatic terms with constant coefficientsis valid only for systems with slow spatial variations of themagnetization. In the present work, we calculate the spatial spin-transfer torque within the DW directly from the spin accumulationbased on our recent work in Ref. [ 16] instead of calculating from the conventional model [ 11,13–15] through the spin- torque coefficients, which are unknown. Spin accumulationnaturally includes the effect of both adiabatic and nonadiabatic 1098-0121/2015/92(5)/054434(9) 054434-1 ©2015 American Physical SocietyCHUREEMART, EV ANS, D’AMICO, AND CHANTRELL PHYSICAL REVIEW B 92, 054434 (2015) torques. This provides the new route of spin-torque calcula- tion to get insight into the physical description behind themechanism of current-induced domain wall motion and avoidsvarious limitations of the conventional model in estimating theempirical constants, i.e., μ xandβx. Using this approach we have investigated the DW motion driven by the spin torqueusing the self-consistent solution of the spin accumulation andmagnetization coupled with atomistic model. We find that theDW displacement and initial DW velocity strongly depend onthe strength of the magnetocrystalline anisotropy and currentdensity, and that the adiabaticity of the spin torque is dependenton the domain wall width. II. METHODOLOGY To investigate the domain wall motion driven by injecting spin-polarized current, the generalized spin accumulationmodel coupled with atomistic model is employed. Semian-alytical solution of spin accumulation is applied to a series oflayers representing the spatial variation of the magnetization ina domain wall to calculate the spin torque at any position of thesystem. Meanwhile, the atomistic model is used to investigatethe dynamics of magnetization caused by spin torque. Thesetwo models are detailed in the following. A. Spin accumulation model The spin-transfer torque originating from the spin accu- mulation can be described via the s-dexchange interaction model [ 13,16,17]. The s-dmodel has been used to present a qualitative description of current-induced torque acting onspin moment. The exchange energy due to the interaction of thespin accumulation and the local spin moment is conventionallydescribed by the Hamiltonian given by H sd=−Jm·S, (1) where mdenotes the spin accumulation. Sis the unit vector of the local spin moment and Jis the exchange coupling strength between the spin accumulation and the local spin moment. Toinvestigate the spin accumulation in the ferromagnet for anyarbitrary direction of the spin moment, the general solution ofthe spin accumulation based on a transfer matrix approach asdetailed in Ref. [ 16] will be employed. In the rotated basis system ˆb 1,ˆb2, and ˆb3, the component of spin accumulation is parallel and perpendicular to the localspin moment. The longitudinal component will be parallel to ˆb 1and the two components of the transverse spin accumulation along the directions ˆb2and ˆb3. The general solution of spin accumulation can then be expressed in the following form m/bardbl(x)=m/bardbl(0)e−x/λ sdlˆb1 m⊥,2(x)=2e−k1x[ucos(k2x)−vsin(k2x)]ˆb2 m⊥,3(x)=2e−k1x[usin(k2x)+vcos(k2x)]ˆb3, (2) with ( k1±ik2)=/radicalBig λ−2 trans±iλ−2 J, where λJ=√2/planckover2pi1D0/J. Hereλsdlis the spin diffusion length, λtransis the transverse damping and D0the diffusion constant. m/bardbl(0),u, andvare constants, which can be obtained by imposing continuity ofthe spin current at the interface. The spin accumulation inthe rotated basis system can be expressed in the Cartesian coordinate system by using a transformation matrix [ 16]. The effect of the spin torque can be considered as an addi- tional effective field arising from the s-dexchange interaction between the local spin moment and the spin accumulation,given by H ST=−∂Hsd ∂S=Jm. (3) B. Atomistic spin model We model the magnetization dynamics induced by the spin- transfer torque using an atomistic spin model coupled with thespin accumulation. The energetics of the system are describedusing a classical spin Hamiltonian with the Heisenberg formof the exchange interaction [ 18] written as H=−/summationdisplay i/negationslash=jJijSi·Sj−ku/summationdisplay i(Si·e)2−|μs|/summationdisplay iSi·Happ, (4) where Jijis the nearest-neighbor exchange integral between spin sites iandj,Siis the local normalized spin moment, Sj is the normalized spin moment of the neighboring atom at site j,kuis the uniaxial anisotropy constant, eis the unit vector of the easy axis, and |μs|is the magnitude of the spin moment. The parameters of the model are representative of Co witha simplified simple cubic discretization, with an interatomicexchange energy J ij=11.2×10−21J/link and μs=1.44μB at 0 K. The demagnetizing field is calculated at the micromagnetic level using the macrocell approach [ 18,19]. Each macrocell contains a predefined number of atomic unit cells and the netmagnetization within the cell is determined by the average ofthe atomic spins in the cell. Macrocell moments ( k,l) then interact using the dipole-dipole interaction including the self-demagnetizing term [ 18] given by H dip,k=μ0 4π/summationdisplay l/negationslash=k/bracketleftbigg3(μl·ˆrkl)ˆrkl−μl |rkl|3/bracketrightbigg −μ0 3μkˆμk V,(5) where μl=μs/summationtextnatom i=1Siis the vector of the magnetic moment in the macrocell site l, which is found from the summation of spin moments in the macrocell l,μ0is the permeability of free space, Vis the volume of the macrocell, rklis the distance and ˆrklthe corresponding unit vector between macrocell sites k andl, andnatomis the number of atoms in each macrocell. The self-interaction term in Eq. ( 5) neglects the configurational anisotropy of the (cubic) macrocells. We approximate thedipole field as constant over the cell κcontaining spin i.T h e effective local field acting on spin iis therefore given by H eff,i=−1 |μs|∂H ∂Si+Hdip,κ. (6) The dynamics of the spin system under the action of the spin-transfer torque can be modeled using the standardLandau-Lifshitz (LL) equation of motion with the inclusion ofan additional spin-torque field ( J sdm)[13,20,21] as follows: ∂S ∂t=−γS×(Heff+Jsdm)+α μsS×∂S ∂t. (7) 054434-2INFLUENCE OF UNIAXIAL ANISOTROPY ON DOMAIN . . . PHYSICAL REVIEW B 92, 054434 (2015) For convenient numerical integration, we cast Eq. ( 7) into the Landau-Lifshitz-Gilbert (LLG) form, giving the finalequation of motion ∂S ∂t=−γ (1+λ2)S×(Heff+Jsdm) −γλ (1+λ2){S×[S×(Heff+Jsdm)]}, (8) where γis the absolute gyromagnetic ratio, λ=0.1i st h e intrinsic Gilbert damping constant applied at the atomic level,Sis the normalized spin moment, and H effis the effective field given by Eq. ( 6). The local effective field Heffleads to damped precessional motion into the direction of the local effectivefield. Interestingly, the additional field due to the presence ofthe injected spin current Jmgives rise to the contribution of adiabatic and nonadiabatic torques. This term describes thespin-torque effect on the spin motion and indicates that theadditional field due to the spin-transfer torque can be anothersource of precession and damping [ 22,23]. We note that all simulations are done without thermal fluctuations, that is, atzero K using the VA M P I R E software package [ 18]. C. Spin-torque calculation To calculate the adiabatic (AST) and nonadiabatic spin torques (NAST), let us consider the rotated basis system wherethe local spin moment in the current layer ( S)i sa l o n gt h e ˆb 1direction whereas that in the previous layer is oriented in the plane ˆb1ˆb2. The spin moment in the previous layer can be rotated into the basis coordinate system as illustratedin Fig. 1,S p=Sp/bardblˆb1+Sp⊥ˆb2, by using the transformation matrix given by [Sbasis]=[T]−1[Sglobal], (9) and the transformation matrix is as follows [T]=⎡ ⎢⎢⎢⎣SxX/prime/prime+SyY/prime/prime D2D3X/prime/prime D2 2D3−SxSyY/prime/prime D1D2D3SzY/prime/prime D1D3 −SxY/prime/prime+SyX/prime/prime D2D3Y/prime/prime D2 2D3−SxSyX/prime/prime D1D2D3SzX/prime/prime D1D3 Sz D1−SxSz D1D2−Sy D1⎤ ⎥⎥⎥⎦(10) FIG. 1. (Color online) Schematic representation of the spin- transfer torque consisting of the adiabatic (AST) and nonadiabatic(NAST) torques in the rotated basis system.with ⎡ ⎢⎣X/prime/prime Y/prime/prime Z/prime/prime⎤ ⎥⎦=⎡ ⎢⎢⎢⎣Sp,xD2 1−Sx(SySp,y+SzSp,z) D1D2 SzSp,y−SySp,z D1 SxSp,x+SySp,y+SzSp,z D2⎤ ⎥⎥⎥⎦, (11) where [ S basis] and [ Sglobal] are the spin moments in the basis coordinate system and in the global coordinate system respec-tively. S x,Sy, andSzare the x,y, andzcomponents of spin moment in the current layer, respectively. D1=/radicalBig S2y+S2z, D2=/radicalBig S2x+S2y+S2z, andD3=√ X/prime/prime2+Y/prime/prime2. In the basis system as shown in Fig. 1, the adiabatic and nonadiabatic torques can be determined from the total spintorque τ STvia the s-dexchange interaction as follows τST=S×Jsdm =ˆb1×Jsd(m/bardblˆb1+m⊥,2ˆb2+m⊥,3ˆb3) =−Jsdm⊥,3ˆb2+Jsdm⊥,2ˆb3. (12) In general, the AST is the in-plane torque whereas the NAST is introduced as the fieldlike torque or the out-of-planetorque. Therefore, the spin moments in the rotated basis systemas illustrated in Fig. 1results in the AST and NAST along the directions of ˆb 2and ˆb3, respectively. As a consequence, the AST and NAST in the rotated basis system are given by τAST=−Jsdm⊥,3ˆb2 τNAST=Jsdm⊥,2ˆb3. (13) The above equation shows that the AST and NAST can be accessed directly via the spin accumulation. Subsequently,the dynamics of spin motion including the effect of the spin-transfer torque can be investigated by employing Eq. ( 8). III. CURRENT-INDUCED DOMAIN WALL MOTION In this work we investigate the dynamics of the magnetiza- tion in a bilayer system consisting of two ferromagnets (FMs).The current-induced domain wall motion is studied by inject-ing a spin current perpendicular to the plane of the bilayer.In this computational study, the investigation is presented intwo sections. First, the effect of the spin-transfer torque onthe DW dynamics is studied, followed by an investigationof the time evolution of DW displacement and DW velocity.Furthermore, the effect of the current density ( j e)i sa l s o studied by injecting currents with different magnitudes. Thisallows the investigation of the critical current density which isthe minimum spin current required to move the domain wall.Second, the effect of the DW width on the time evolution ofthe DW displacement and DW velocity is considered. A. Time evolution of magnetization and spin torque The system consists of a bilayer structure with a pinned layer (PL) providing a spin-polarized current (which is notmodeled explicitly) and a free layer (FL) with dimensionsof 60 nm ×30 nm ×1.5 nm. In order to calculate the spin accumulation and spin torque the system is discretized intomacrocells 1 .5n m×1.5n m×1.5 nm in size. 054434-3CHUREEMART, EV ANS, D’AMICO, AND CHANTRELL PHYSICAL REVIEW B 92, 054434 (2015) FIG. 2. (Color online) The the tail-to-tail domain wall contained in the second ferromagnet of the bilayer system with the uniaxialanisotropy constant of k u=2.52×106J/m3: The arrows indicate the direction of magnetization. The magnetization along the ydirection is represented by the blue coloring. In contrast, the red color showsthe orientation of magnetization in the −ydirection. The magnetic moment in each macrocell is then calculated by averaging over the spins within the cell. The pinned layeris not considered explicitly; its role is simply to provide aspin-polarized current through the layer under investigation.A domain wall is forced into the free layer by fixing anantiparallel magnetization configuration at the boundaries ofthe system as illustrated in Fig. 2. The DW profile is transverse in thexyplane. The studied system is based on a material with a uniaxial anisotropy constant of k u=2.52×106J/m3≡ 2.7×10−23J/atom with the ydirection as the easy axis and a lattice constant of a=3.49˚A. The transport parameters of Co used in spin accumulation calculation are takenfrom Ref. [ 21] as the following values, β=0.5,β /prime=0.9, D0=0.001 m2/s,λsdl=60 nm, and λJ=3n m . We first investigate the effect of the spin-transfer torque on the domain wall motion by introducing a current densityof 50 MA/cm 2into the bilayer system. The current-induced domain wall motion can be observed through the componentsof magnetization. Figure 3shows the time evolution of the magnetization after the application of the current induced bythe spin-torque. In the absence of the spin-transfer torque att=0 ns, the DW is situated centrally and the position of the DW center is defined by the maximum magnetization of thexcomponent and zero of the ycomponent. In interpreting the numerical results it is necessary to stress that the DWcan initially move freely but, due to the finite system size,after some time interacts with the strong pinning sites atthe boundaries, which are used to inject the DW into thesystem. The DW initially moves when the spin current isinjected above the critical value. The DW has a translationalmotion to the right, which is the direction of the injectedcurrent and it tends to stop moving at the equilibration timet=0.6 ns with a finite DW displacement, as expected given its interaction with the boundary pinning sites. Specifically,a small out-of-plane or zcomponent develops during the propagation time. Its appearance comes from the fact thatthe domain wall interacts with the strong pinning site. Thisis evidence of DW deformation due to interaction with thepinning site. The DW deformation and the occurrence of z component exhibited in this study are in good agreement withthe recent experimental and theoretical studies [ 24–26]. We next consider the time variation of the spin-transfer torque via self-consistent solution of the magnetization andspin accumulation, naturally including the adiabatic andnonadiabatic torques, to understand its dependence on the 0 0.01 0.02 0.03 0 5 10 15 20 25 30 35 40Mz Layer number0.0 ns 0.2 ns 0.4 ns 0.6 ns-1-0.5 0 0.5 1My 0 0.2 0.4 0.6 0.8 1Mx FIG. 3. (Color online) Schematic representation of the magneti- zation component with time evolution from 0 ns to the equilibration time of 0 .6 ns: The current density injected into the bilayer system containing the DW is 50 MA/cm2. magnetic structure and its time evolution. The xandy components give the adiabatic torque, which tends to developtowards the direction of magnetization. On the other hand thezcomponent arises from the contribution of the nonadiabatic torque, which acts out of the plane. The spin torque actingon the local magnetization due to the spin-polarized currentresults in the translation of the DW. As a consequence, thespatial spin torque at different times as illustrated in Fig. 4 reflects the spatial magnetization configuration of Fig. 3, which is translated due to the spin torque. It is found thatthe motion ceases after 0.6 ns as the DW contacts with theboundary pinning sites. In addition, in this case the magnitudeof the adiabatic and nonadiabatic torques remain constant withtime and the domain wall width is not significantly affectedas the wall contacts the boundary pinning sites, suggesting thatthe spin current density of 50 MA/cm 2is not high enough to distort the pinned DW. B. DW displacement and velocity We next consider the effect of the current density on the domain wall motion. This leads to the investigation of thecritical spin current density ( j e), required to initiate domain wall motion and also spin-torque driven oscillations of themagnetization of the DW fixed at the strong boundary pinningsites. It is first noted that the calculation in this sectionobserved the domain wall motion in the bilayer system with theanisotropy constant of k u=2.52×106J/m3giving rise to the 054434-4INFLUENCE OF UNIAXIAL ANISOTROPY ON DOMAIN . . . PHYSICAL REVIEW B 92, 054434 (2015) -0.01 0 0.01 0 5 10 15 20 25 30 35 40STz [T] Layer number0.0 ns 0.2 ns 0.4 ns 0.6 ns-0.03-0.02-0.01 0STy [T] -0.02-0.01 0 0.01 0.02STx [T] FIG. 4. (Color online) The time evolution of the spatial spin- transfer torque with je=50 MA/cm.2 domain wall width of approximately 6.86 nm. The application of the spin-polarized current induces a displacement of theDW position with time, as shown in Fig. 5(top panel). The DW displacement is monitored by observing the shiftof the DW center from the initial position at each time step.It can be seen that the DW displacement is time dependentand increases linearly in the first time period before reaching asteady state with finite displacement due to the interaction withthe boundary pinning sites [ 24,27,28]. The equilibration time of DW displacement tends to decrease with increasing spincurrent density, consistent with the increased DW velocity. To describe the behavior of the DW displacement with different regimes of the current density, it is important toconsider the critical current density, which can be evaluatedthrough the initial DW velocity. The initial velocity iscalculated by determining the rate of change of the DWdisplacement in the first 0.1 ns as the DW shows uniformtranslational motion during that period. The relation betweenthe initial DW velocity as a function of the current density isplotted on a semilogarithmic scale in Fig. 5(bottom panel). It is found that the critical current density is 0.5 MA/cm 2.T h i s behavior is also found in the previous studies [ 26,29–33]. On increasing the current density above the critical value, the domain wall moves uniformly without any precessionalong the direction of the injected spin current. This motioninduced by the spin current is due to the conservation ofthe angular momentum. At high spin current density, thedomain wall motion is accompanied by oscillatory behavior,which tends to be observed with a high current density over100 MA/cm 2. Interestingly, at extremely high values of current 0 4 8 12 16 20 0 0.5 1 1.5 2DW displacement [nm] Time [ns]0.5 MA/cm2 10 MA/cm2 30 MA/cm2 50 MA/cm2 100 MA/cm2 500 MA/cm2 1000 MA/cm2 0 40 80 120 160 200 0.0001 0.001 0.01 0.1 1 10 100 1000DW velocity [m/s] je [MA/cm2]je, critical FIG. 5. (Color online) (Top) The time variation of domain wall displacement with different current densities. (Bottom) The initial DW velocity as a function of current density: The critical current den-sity, minimum current density required to move DW is 0 .5 MA/cm 2. density je=1000 MA/cm2, the DW reaches the boundary inning sites and the translational motion stops. At this point thedynamic behavior of the DW is becomes oscillatory, exhibitinga stable precessional state around a finite wall displacement[27]. At equilibrium, the DW displacement oscillates at a high frequency of 300 GHz since the pinned DW essentially actsas a spin-torque oscillator. This also implies the appearanceof an out-of-plane component of magnetization resultingfrom the nonadiabatic torque, consistent with the previousstudies [ 34,35]. Our result with the current density of j e= 1000 MA/cm2yielding the initial velocity at approximately 200 m/s is in good agreement with the analytical results ofthe one-dimensional (1D) Walker ansatz model in Ref. [ 36] where the domain displacement at equilibrium is about 18 nm.However, the oscillatory behavior cannot be observed in 1Dmodel because the effect of nonadiabatic spin torque is nottaken into account. In order to understand the origin of the oscillatory behavior, the magnetization component at the initial DW center isinvestigated in its time evolution after the introduction ofthe spin-transfer torque. Figure 6clearly shows that the spin-transfer torque causes the deformation of the DW leadingto precessional motion of the xandzcomponents. This is the precession of the equilibrium magnetization about the effectivefield determined by the interaction with the pinning site. Thenonadiabatic torque driving the DW in the stable precessionalstate is strong enough to deform the N ´eel wall so as to have a significant out-of-plane component, which results in theoscillatory behavior. The domain wall motion accompaniedby the precessions due to the nonadiabatic torque has beenconfirmed by recent studies [ 10,25,27,35,37]. Interestingly, 054434-5CHUREEMART, EV ANS, D’AMICO, AND CHANTRELL PHYSICAL REVIEW B 92, 054434 (2015) 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2Mx, y, z Time [ns]MxMyMz FIG. 6. (Color online) The magnetization component of the ini- tial DW center with time evolution after injecting the spin current with the density of 1000 MA/cm2. the observed oscillatory motion of the domain wall for large current is similar to the behavior of domain walls at currentsabove the Walker threshold. IV . CURRENT-INDUCED DW MOTION: EFFECT OF THE DOMAIN WALL WIDTH We now turn to the effect of the domain wall width on the magnetization dynamics. This is investigated by introducinga spin-polarized current into a bilayer system containing adomain wall whose width is varied by changing the anisotropyconstant. The domain wall profile with different anisotropyconstants can be seen in Fig. 7. The magnetization is allowed to vary continuously throughout, constrained by pinning sitesat the boundaries. The width of the domain wall is variedby increasing the uniaxial anisotropy constant to investigatethe influence of the magnetic anisotropy on the spin-transfer -1-0.5 0 0.5 1 0 5 10 15 20 25 30 35 40My Layer NumberKu2Ku4Ku6Ku10Ku 0 0.2 0.4 0.6 0.8 1Mx FIG. 7. (Color online) The domain wall profile transverse in the xyplane with various anisotropy constants: The uniaxial anisotropy constant of cobalt is Ku=4.2×105J/m3. The distance between layer is given in units of supercells, corresponding to five atomic spacings.torque on the domain wall. The anisotropy constant is varied from the typical anisotropy value of cobalt ku=4.2×105J/m3 up to 10 times that value. The xandycomponents of magnetization can be used to characterize the center of DWand the DW width. The zcomponent of the magnetization is zero according to the usual properties of the N ´eel wall for the thin sample. A detailed qualitative investigation ofthe current-induced DW motion with the effect of anisotropyconstant will be discussed in the following. A. DW displacement and velocity First, a spin current with the density of 50 MA/cm2is injected into the bilayer system along the xdirection in order to observe the manipulation of the magnetization within theDW with different anisotropy constants. The magnetizationconfiguration after the introduction of the spin current for 1 nsis illustrated in Fig. 8. It shows that the DW motion is initiated after injecting the spin current into the system. The centerof the domain wall moves from the initial position along thedirection of the spin current. The system with high anisotropyis easily displaced due to a larger gradient of magnetizationwithin the DW giving rise to a high magnitude of spin torqueacting on it. Interestingly, the DW center position of the systemwith the anisotropy constant of k uis unchanged. This implies that the density of spin current injected to the system is below 0 0.01 0.02 0.03 0 5 10 15 20 25 30 35 40Mz Layer numberKu6Ku8Ku10Ku-1-0.5 0 0.5 1My 0 0.2 0.4 0.6 0.8 1Mx FIG. 8. (Color online) The component of magnetization in the second FM with various anisotropy constants after the introductionof the spin current for 1 ns: The center of the DWs are displaced in the direction of injected spin current. The system with high anisotropy constant leading to a large gradient of magnetization within domainwall results in a large displacement of the DW. 054434-6INFLUENCE OF UNIAXIAL ANISOTROPY ON DOMAIN . . . PHYSICAL REVIEW B 92, 054434 (2015) 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3DW displacement [nm] Time [ns]Ku2Ku6Ku8Ku10Ku40Ku80Ku100Ku 0 10 20 30 40 4 6 8 10 12 14 16DW velocity [m/s] δ [nm] FIG. 9. (Color online) (top) The time-dependent variation of the domain wall displacement and (bottom) the initial domain wall velocity of different uniaxial anisotropy systems with the spin currentdensity of 50 MA/cm 2. the critical value, which depends on the DW width [ 38–40]. In addition, the out-of-plane component is likely to be large forhigh anisotropy. Furthermore, it is also worthwhile to observe the dynamic behavior of the DW motion via the DW displacement andthe initial DW velocity. As illustrated in Fig. 9(top panel), the DW displacement is not noticeable for a very wide wall,specifically for uniaxial anisotropy constants of k uand 2ku. The DW exhibits transient oscillatory behavior back to itinitial position. Hence, higher spin current density is neededin order to initiate the translation of DW for these cases. Onthe other hand, displacement of the narrow DW tends to bemore easily initiated than the wide DW. This is because ofthe strong interaction between the spin current and the localmagnetization gradient within the DW giving rise to a largespin-transfer torque. For a low anisotropy, the linear responseof the DW displacement occurs in the first 0 .1 ns and then reaches the equilibrium state after reaching the boundarypinning sites. For a high anisotropy, the DW displacementdeviates from linear behavior and the precessional motion isenhanced for several cycles in the first ns before reaching theequilibrium state. The deviation from the linear behavior inthe first period becomes stronger for higher anisotropy. In thecase of this spin current density, the stable precessional stateis not established as the current density is not high enough topush the DW against the boundary pinning sites. Finally, we consider the initial DW velocity as a function of the DW width. Clearly the initial DW velocity dependssensitively on the DW width as can be seen in Fig. 9(bottom panel). The initial DW velocity decreases with increasing DWwidth as a result of the decreasing magnetization gradient. Thesimilar result has been shown in Ref. [ 41]. This relation can be used to evaluate the critical DW width for each spin currentdensity. The current density of 50 MA/cm 2is able to move a DW along the direction of the injected spin current in case ofthe DW width less than 11 .2n m . B. Spin-transfer torque We now consider the spin-transfer torque consisting of adiabatic (AST) and nonadiabatic (NAST) components. Asmentioned before, the total spin-transfer torque is mainlycontributed by the AST resulting from the spin accumulationcomponent following the direction of the local magnetizationwhereas the out-of-plane torque comes from the NAST arisingfrom the electron mistracking. The strength of the spin-transfertorque on the DW can be represented by considering themaximum value occurring at any position over the DWregion given that its contribution is nonuniform throughoutthe DW. In addition, the degree of nonadiabatic torque orthe so-called nonadiabaticity ( D NAST), which characterizes the relative influence of the NAST on the DW compared with theAST, is also evaluated by the following equation, D NAST=|NAST max| |AST max|. (14) |NAST max|and|AST max|are the maximum value of adiabatic and nonadiabatic torques within DW. Clearly, as shown in Fig. 10, both adiabatic and nonadia- batic torques tend to be more effective in narrow domain wallsdue to the large gradient of the magnetization. It can also beseen that the nonadiabaticity factor becomes more significantfor a small DW width. This is schematically shown in Fig. 10. In contrast, the pure adiabatic torque is likely to dominate thetotal torque, with negligible nonadiabatic torque, for a largeDW width. This is consistent with previous studies [ 14,41,42]. The results also indicate that the nonadiabatic torque, whichis represented by the value of βused in the micromagnetic approach is directly dependent on the DW width. In the case 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 0.05 0.1 0.15 0.2 0.25 0.3 0.2 0.25 0.3 0.35 0.4 0.45Magnitude of ST(T) DNAST δ/λsdl|ASTmax| |NASTmax| DNAS T FIG. 10. (Color online) The thickness dependence of a maximum of adiabatic spin torque (AST), nonadiabatic spin torque (NAST) and the degree of nonadiabaticity ( DNAST): The spin diffusion length ( λsdl) is 60 nm. 054434-7CHUREEMART, EV ANS, D’AMICO, AND CHANTRELL PHYSICAL REVIEW B 92, 054434 (2015) ofδ/lessmuchλsdl, the nonadiabatic torque becomes significant to the system. V . CONCLUSION In this work we have applied the modified formalism of spin accumulation with an atomistic spin model to studythe dynamics of a DW in the presence of a spin-transfertorque. The model uses a transfer matrix approach to determinedirectly the equilibrium spin accumulation avoiding the needfor computationally expensive time stepping of the equationof motion. Domain wall dynamics under the influence of aspin-polarized current are studied by self-consistent calcula-tions of the spin accumulation and magnetization. The spinaccumulation is calculated in a rotated basis, which givesaccess to both the adiabatic and nonadiabatic contributions tothe spin torque, which arise naturally in the model. The total spin torque contributed by adiabatic and nonadiabatic torques at any position within the DW is considered. The resultsindicate that both torques are inversely proportional to domainwall width. Furthermore, it is found that the adiabatic torquedominates the total spin torque; meanwhile the nonadiabatictorque controls the out-of-plane component of spin torque. Thedependence of spin torque on the DW width is consistent withthe proportionality of the spin torque to the gradient of the magnetization. However, it is important to note that the self-consistent solution of spin accumulation and magnetizationleads to a further contribution to the effect of the DW width.Specifically, we show that the adiabatic and nonadiabaticcomponents of the spin torque reduce with increasing DWwidth relative to the spin diffusion length, which becomes animportant characteristic length in the calculation of the spintorque. Both components decrease at different rates, with theresult that the nonadiabaticity factor, indicative of the relativestrength of the nonadiabatic torque, tends to decay to zero asthe DW width increases. 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PhysRevB.97.134421.pdf

PHYSICAL REVIEW B 97, 134421 (2018) Thickness-dependent enhancement of damping in Co 2FeAl/β-Ta thin films Serkan Akansel,1Ankit Kumar,1,*Nilamani Behera,1Sajid Husain,2Rimantas Brucas,1 Sujeet Chaudhary,2and Peter Svedlindh1 1Department of Engineering Sciences, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden 2Thin Film Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016, India (Received 5 March 2018; revised manuscript received 6 April 2018; published 23 April 2018) In the present work Co 2FeAl (CFA) thin films were deposited by ion beam sputtering on Si (100) substrates at the optimized deposition temperature of 300 °C. A series of CFA films with different thicknesses ( tCFA), 8, 10, 12, 14, 16, 18, and 20 nm, were prepared and all samples were capped with a 5-nm-thick β-Ta layer. The thickness- dependent static and dynamic properties of the films were studied by SQUID magnetometry, in-plane as wellas out-of-plane broadband vector network analyzer–ferromagnetic resonance (FMR) measurements, and angle-dependent cavity FMR measurements. The saturation magnetization and the coercive field were found to be weaklythickness dependent and lie in the range 900–950 kA /m and 0.53–0.87 kA /m, respectively. The effective damping parameter ( α eff) extracted from in-plane and out-of-plane FMR results reveals a1 tCFAdependence, the values for the in-plane αeffbeing larger due to two-magnon scattering (TMS). The origin of the αeffthickness dependence is spin pumping into the nonmagnetic β-Ta layer and in the case of the in-plane αeff, also a thickness-dependent TMS contribution. From the out-of-plane FMR results, it was possible to disentangle the different contributionstoα effand to the extract values for the intrinsic Gilbert damping ( αG) and the effective spin-mixing conductance (g↑↓ eff)o ft h eC F A / β-Ta interface, yielding αG=(1.1±0.2)×10−3andg↑↓ eff=(2.90±0.10)×1019m−2. DOI: 10.1103/PhysRevB.97.134421 I. INTRODUCTION Half-metallic ferromagnetic materials having a small Gilbert damping parameter ( α), which describes the relaxation of the magnetization, are of immense interest for spin transfertorque devices since a low αvalue results in a low value for the critical current density required to switch the magneti-zation [ 1,2]. Co-based Heusler alloys, e.g., Co 2FeAl (CFA), have unique properties such as half-metallicity [ 3–5], large magnetization, and high Curie temperature ( Tc=1000 K) [ 5]. Their use as electrode material in magnetic tunnel junctions,due to giant tunneling magnetoresistance (360%) at roomtemperature, has been reported [ 6,7]. Full Heusler alloys, X 2YZ, can exhibit three different crystallographic phases—the fully ordered L21phase, the partially ordered B2 phase, and the fully disordered A2 phase. In the L21phase, the different types of atoms occupy their assigned sites, while in the partially ordered phase the YandZ atoms randomly share sites. For the A2 phase, all available sites are occupied at random [ 8]. Unfortunately, crystallographic disorder reduces half-metallicity and increases the value of theGilbert damping parameter. There exist a number of studieswhere postannealing has been utilized in order to obtain lowerdamping parameters. Damping parameters in the range from0.001 to 0.004 have been obtained by postannealing of CFAfilms deposited on MgO substrates [ 8–14]. In the case of CFA films deposited on Si substrates [ 8,15,16] comparably few studies have been reported and the B2 phase is rarely achieved [ 15]. However, using ion beam sputtering and *ankit.kumar@angstrom.uu.seoptimizing the growth temperature, it has recently been shown that the B2 phase can be obtained without any postannealing process [ 17]. Besides obtaining the B2 phase, a record low value for the damping parameter for CFA films deposited onSi was reported. The objective of this work is to investigate the thickness dependence of the magnetic relaxation for B2 phase Si/CFA/Ta thin films. Spin transfer torque devices typically require ul-trathin magnetic electrode layers. One problem for devicesis that decreasing the thickness of the magnetic layer oftenresults in a concomitant increase of the magnetic dampingparameter due to effects such as surface anisotropy [ 18]. The explanation for the increase of damping can be linked to theexcitation of both uniform and nonuniform precession modesof the magnetization and, if a nonmagnetic layer with largespin-orbit coupling is used together with the magnetic layer,spin pumping into the nonmagnetic layer also contributes[19,20]. Uniform modes give rise to intrinsic or Gilbert type of relaxation, while nonuniform modes are known as extrinsicmodes which can be caused by magnetic inhomogeneity orsurface anisotropy fields. The nonuniform precession of themagnetic moments results in two-magnon scattering (TMS),where magnons are created. The TMS increases the linewidthof the ferromagnetic resonance (FMR) absorption as wellas the effective damping parameter of the material [ 21,22]. The influence of surface/interface anisotropy is expected toincrease by decreasing film thickness [ 23]. However, in out- of-plane FMR measurements, TMS is avoided and hencethe effective damping parameter has a contribution from theintrinsic relaxation and can in addition exhibit a thickness-dependent spin-pumping contribution. The latter enhancementof the damping parameter has from theory been shown to 2469-9950/2018/97(13)/134421(8) 134421-1 ©2018 American Physical SocietySERKAN AKANSEL et al. PHYSICAL REVIEW B 97, 134421 (2018) be large for a ferromagnetic layer in direct contact with a nonmagnetic metal layer with large spin-orbit coupling (largespin-flip probability) [ 19,20]. Enhancement is thus expected for heavier elements with panddelectrons in the conduction band, while the enhancement will be absent for lighter elementsas well as for heavier elements with only selectrons in the conduction band. In this study CFA thin films capped with a high spin-orbit coupling 4 dβ-Ta layer have been investigated in terms of dynamic magnetization properties using both in-plane andout-of-plane FMR techniques to be able to distinguish betweenintrinsic and extrinsic contributions to the magnetic relaxation.Besides broadband FMR studies, angle-resolved cavity FMRmeasurements have also been utilized. The intrinsic dampingparameter and the enhancement of damping due to TMSand spin pumping can be disentangled by angle-resolved andbroadband FMR measurements, which is quite enlightening interms of understanding the fundamental dynamical propertiesof these promising materials for future spintronic applications. II. SAMPLES AND METHODS CFA thin films were deposited on Si(100)/SiO 2sub- strates by utilizing an ion beam sputtering deposition sys-tem (Nordiko-3450). Prior to deposition substrates were heattreated in situ at 620 °C for 2 h to remove surface contamina- tions. A base pressure of about 7 ×10 −7Torr was achieved by using cryo and turbo pumps. The Ar gas pressure wasmaintained at 2 .4×10 −4Torr and the rf ion source was operated at 75 W during deposition. The film depositiontechnique is explained in detail in our previous work [ 17]. Films were grown at 300 °C. A series of films was preparedwith the stacking Si/CFA( t CFA)/Ta(5 nm). Nominal tCFAvalues were 8, 10, 12, 14, 16, 18, and 20 nm. All films were coveredwith a 5-nm-thick capping layer of β-Ta. The β-Ta layers were grown at room temperature and their quality was ascertainedby x-ray diffraction and resistivity measurements [ 23]. Film thickness and surface/interface roughnes were ob- tained by x-ray reflectivity (XRR) measurements. The scanscovered the 2 θrange 0°–4°, and the XRR results were ana- lyzed using the PANALYTICAL X ’PERT REFLECTIVITY software package (ver. 1.2 with segmented fit). Layer thicknesses,densities, and surface/interface roughness were obtained fromthis analysis. Magnetic hysteresis loop measurements were performed using a Magnetic Property Measurement System (MPMS,Quantum Design). Dynamic magnetic properties were investigated both by fixed frequency cavity and broadband ferromagnetic resonancemeasurements. In the X-band cavity FMR measurements the frequency was kept constant at 9.8 GHz and an in-planemagnetic field was scanned during the measurement. The setupwas equipped with a goniometer making it possible to performangular-dependent in-plane as well as out-of-plane FMR mea-surements, providing information about in-plane anisotropyfields, and two-magnon scattering (TMS) contribution to therelaxation of the magnetization. Besides cavity FMR measurements, the samples were inves- tigated by broadband FMR measurements. In-plane broadbandFMR measurements were performed using a transmissiongeometry coplanar waveguide (CPW) where a lock-in am- plifier detection technique was used. A pair of homemadeHelmholtz coils generating a low-frequency, low-amplitudemagnetic field (211.5 Hz and 0.25 mT magnetic field ampli-tude) was used to modulate the rf signal which was detectedby the lock-in amplifier. As in cavity FMR, each measurementwas performed varying the dc magnetic field while keeping themicrowave frequency constant. FMR spectra were recorded inthe frequency range from 5 to 20 GHz in steps of 1 GHz. A setup enabling out-of-plane FMR measurements was also utilized. Recording the FMR signal by applying the field outof plane with respect to the sample surface provides a TMS-free FMR signal. For out-of-plane measurements a broadbandvector network analyzer (VNA) was utilized. Two ports of theVNA were connected to a coplanar waveguide mounted in theair gap of an electromagnet. III. RESULTS AND DISCUSSION XRR measurements were performed to accurately de- termine thickness and roughness of the different layers inthe Si/CFA( t CFA)/Ta samples. Figure 1shows XRR spectra (symbols) together with simulated spectra (solid lines) forsamples with different nominal CFA thickness. A three-layermodel CFA/Ta/Ta 2O5was used in the simulations, since previous studies using x-ray photoelectron spectroscopy [ 17] and transmission electron microscopy [ 24] have shown that the top part of the Ta layer becomes oxidized, yielding Taand Ta 2O5layers with thicknesses of about 2.5 and 2.2 nm, respectively. The results of the simulations are summarizedin Table I. The results of the simulations show that the CFA thickness matches quite well with the nominal thickness andthat differences in interface roughness between samples aresmall. Figure 2shows in-plane magnetization versus field curves for samples with different CFA thickness; for sake of clarityresults are only shown for three samples. All samples exhibitrectangular hysteresis curves with small coercivity values; thecoercivity was found to be weakly CFA thickness dependent FIG. 1. X-ray reflectivity spectra recorded for Si/CFA( tCFA)/ Ta/Ta 2O5thin films. Symbols correspond to experimental spectra and solid lines represent simulated curves. 134421-2THICKNESS-DEPENDENT ENHANCEMENT OF DAMPING IN … PHYSICAL REVIEW B 97, 134421 (2018) TABLE I. Thickness and roughness values ( σ) of the different layers in CFA/Ta/Ta 2O5films extracted from the simulation of the experimental XRR data. tCFA(nm) σ(nm)±0.06 tCFA(nm)±0.03 σ(nm)±0.06 tTa(nm)±0.03 σ(nm)±0.03 tTa2O5(nm)±0.06 σ(nm)±0.03 8 0.21 8.54 0.34 2.37 0.20 2.26 0.28 10 0.35 10.88 0.60 2.62 0.19 2.42 0.34 12 0.38 12.20 0.55 2.59 0.37 2.16 0.50 14 0.28 14.32 0.46 2.42 0.47 2.15 0.4116 0.20 16.03 0.55 2.50 0.45 2.19 0.31 20 0.22 20.22 0.43 2.45 0.31 2.25 0.13 and varied in the range 0.53–0.87 kA /m (0.65–1.10 mT). The inset shows the magnetization curve for one sample ( tCFA= 20 nm) applying the magnetic field out of plane with respectto the film surface; all samples exhibit similar out-of-planemagnetization curves. The saturation magnetization is best de-termined from the saturation field; the saturation magnetizationdetermined in this way indicates a weakly CFA thickness-dependent value for the saturation magnetization ( μ 0Ms) with values of about 1.10 T. These values are in good agreementwith previously determined values for CFA films deposited byion beam sputtering on Si substrates. The thickness-dependentsaturation magnetization clearly demonstrates the absence ofinterfacial dead layers in these samples. The in-plane angle-dependent cavity FMR data were ana- lyzed using the following equation [ 25]: f=γμ 0 2π([Hrcos(φH−φM)+Hccos 4(φM−φC) +Hucos 2(φM−φu)]{Hrcos(φH−φM)+Meff +Hc 4[3+cos 4(φM−φC)]+Hucos2(φM−φu)})1/2, (1) FIG. 2. In-plane magnetization normalized with the saturation magnetization versus field for CFA films with different thickness. The inset shows the normalized out-of-plane magnetization versusfield for the CFA film with 20 nm thickness.where Hris the resonance field, fis the cavity microwave frequency, and γ=gμB ¯his the gyromagnetic ratio. Here, gis the Landé spectroscopic splitting factor, μBthe Bohr magneton, and ¯ his the reduced Planck´s constant. With respect to the [100] direction of the Si substrate, in-plane directionsof the magnetic field, magnetization, uniaxial, and cubicanisotropies are given by φ H,φM,φuandφC, respectively. Hu=2Ku μ0MsandHc=2Kc μ0Mscorrespond to the uniaxial and cubic anisotropy fields, respectively, with KuandKcbeing the uniaxial and cubic magnetic anisotropy constants, respectively.M eff=Ms−H⊥ kis the effective magnetization, where H⊥ kis the perpendicular anisotropy field of the film. Here Meff,Hc, andHuare used as fitting parameters. Figure 3shows Hrversus φHextracted from the angular-dependent FMR measurements together with fits according to Eq. ( 1), clearly revealing a dom- inant twofold uniaxial in-plane magnetic anisotropy. Usingg=2.10, a value which is in accord with values extracted from broadband FMR measurements, μ 0Meffshows small variation between samples taking values in the range 1.00–1.02 T. Theresults for the effective magnetization are close to the valuesextracted for the saturation magnetization (cf. inset in Fig. 2), showing that the perpendicular anisotropy field is negligiblysmall for the samples studied here. The uniaxial anisotropyfieldμ 0Huexhibits a decreasing trend with increasing CFA thickness, with values in the range 2.20–3.90 mT, while thecubic anisotropy field values are less than one-tenth of theuniaxial anisotropy field values. FIG. 3. Resonance field μ0Hrversus magnetic field rotation angle φHobtained from cavity FMR measurements. Symbols are experimental data points and lines are fits to Eq. ( 1). 134421-3SERKAN AKANSEL et al. PHYSICAL REVIEW B 97, 134421 (2018) The recorded FMR spectra linewidth have the following different contributions: μ0/Delta1H=μ0/Delta1Hinh+μ0/Delta1HG+sp+μ0/Delta1Hmosaic +μ0/Delta1HTMS. (2) In the following we will discuss all four contributions in the linewidth. μ0/Delta1Hinhis the frequency-independent sample in- homogeneity contribution, while μ0/Delta1HG+sp=4πα efff/γ/Phi1 is the Gilbert and spin-pumping damping contribution. Here,α effand/Phi1are the effecive damping constant and a correc- tion factor due to the field dragging effect. For the in-planeconfiguration /Phi1=cos(φ M−φH) and for the out-of-plane, /Phi1=cos(θM−θH), where φHis the magnetic field azimuthal angle with respect to the in-plane crystallographic [100]direction, and θ His the polar angle of the magnetic field. φM(θM) is the azimuthal (polar) angle of sample magnetization. This field dragging term enhances damping, but vanishesalong the easy and hard axes (its presence in our studiedsamples is minute and will not be discussed further). The thirdtermμ 0/Delta1Hmosaic=∂Hr ∂φH/Delta1φH+∂Hr ∂θH/Delta1θHis due to sample mosaicity [ 14,26]. This contribution to the linewidth originates from variation of crystallite orientations in the films, and fromthickness variations. These microscopic variations result inspatial variations of the anisotropy fields and consequentlyslight variations in the resonance field for different regions. This contribution is present in our studied samples. The lasttermμ 0/Delta1HTMSis the two-magnon scattering (TMS) contri- bution. The TMS is a process where the q=0 magnon scatters into a degenerate magnon with wave vector /vectorq/negationslash=0. Arias and Mills have formulated a theoretical model where latticegeometrical defects induce magnetic inhomogeneity and yieldtwo-magnon scattering [ 27]. Later Woltersdorf and Heinrich formulated a model including both isotropic and anisotropicangle-dependent TMS contributions to the linewidth [ 28]. For the in-plane configuration, which is discussed here, the TMSdepends on the in-plane direction of the applied magnetic fieldrelative to the principal in-plane crystallographic direction ofthe film. Angle-dependent TMS contributions appear when thescattering centers are anisotropic, e.g. self-assembled networksof misfit dislocations result in a fourfold angular dependencedue to effective channeling of scattered spin waves. Moreover,rectangular surface defects cause a (cos 2 φ H)2angular depen- dence. A slightly different symmetry of surface defects resultsin a/vectorqwave-vector-dependent (cos 2 ϕ) 4angular dependence, where ϕ=φM+ψ;ψis the angle between the magnetization vector and /vectorq. Therefore, combining the Arias and Mills, and Woltersdorf and Heinrich models of TMS, the angular dependence of TMS can be expressed as μ0/Delta1HTMS∝/Gamma1 /Phi1sin−1/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftbigg/radicalBigg ω2+/parenleftbiggω0 2/parenrightbigg2 −ω0 2/parenrightbigg/slashbigg/parenleftbigg/radicalBigg ω2+/parenleftbiggω0 2/parenrightbigg2 +ω0 2/parenrightbigg/integraldisplay w(ψ)(cos 2 φH)2(cos 2ϕ)4dψ, (3) where /Gamma1is the intensity of the TMS, w(ψ) is a weighting parameter along the path of the TMS scattering lobes /vectorq(ψ), ω0=γμ 0Meff, andωis angular frequency. The angle-dependent linewidth obtained from cavity FMR measurements was fitted using Eqs. ( 2) and ( 3) to extract the thickness-dependent TMS linewidth contribution, shown inFig. 4. Since broadband in-plane FMR results do not indicate any field dragging effect, implying φ M=φH, both /Delta1Hinh and/Delta1HG+spcorrespond to isotropic contributions to the linewidth and the exact value of both cannot be extracted fromthis analysis. However, the weighting-factor-dependent TMSintensity /Gamma1can be extracted from the fitting. /Gamma1increases from 2.6 to 4.5 mT, decreasing the CFA thickness from 20 to 8 nm,clearly indicating the presence of a thickness-dependent TMSlinewidth contribution in our studied samples. In-plane broadband FMR measurements were performed with the magnetic field applied along the easy axis of the films.Recorded FMR spectra were fitted to the expression [ 29] dA dH∝2(H−Hr)/Delta1H 2/bracketleftBig/parenleftbig/Delta1H 2/parenrightbig2+(H−Hr)2/bracketrightBig2−/bracketleftbig/parenleftbig/Delta1H 2/parenrightbig2−(H−Hr)2/bracketrightbig /bracketleftbig/parenleftbig/Delta1H 2/parenrightbig2+(H−Hr)2/bracketrightbig2, (4) wheredA dHis the magnetic field derivative of the microwave absorption signal. The full width at half maximum linewidth/Delta1H and resonance field H rwere used as fitting parameters. Figure 5shows FMR spectra at different frequencies forthetCFA=16 nm sample and Fig. 6(a) shows /Delta1H versus frequency for CFA samples with different thickness (leaving out results for two samples for the sake of clarity). The insetin Fig. 5shows H rversus frequency for the tCFA=16 nm sample together with a fit of the experimental data to Eq. ( 1); the results for other samples are very similar and plotting more FIG. 4. Linewidth μ0/Delta1H versus magnetic field rotation angle φH obtained from cavity FMR measurements. Symbols are experimental data points and lines are fits to Eqs. ( 2)a n d( 3). The extracted TMS contributions to the linewidth are 2.6, 3.8, and 4.5 mT for the 20-,14-, and 8-nm-thick CFA samples, respectively. 134421-4THICKNESS-DEPENDENT ENHANCEMENT OF DAMPING IN … PHYSICAL REVIEW B 97, 134421 (2018) FIG. 5. In-plane FMR spectra for the 16-nm-thick CFA film. Symbols are experimental data and solid lines are fits to Eq. ( 4). The inset shows frequency versus resonance field for the same sample. Symbols are experimental data and the solid line is a fit to Eq. ( 1). than one curve in the graph it is very difficult to distinguish one curve from the other by eye. Using gas a free parameter, one obtains g=2.10, while μ0Mefftakes values in the range 1.00–1.05 T. The /Delta1H versus frequency data were fitted to the expression [ 29] μ0/Delta1H=4πα eff γf+μ0/Delta1H 0, (5) where αeffis the effective damping parameter, which for the in-plane configuration, in addition to the intrinsic Gilbertdamping, contains both a TMS contribution and a con-tribution due to spin pumping into the Ta layer, and/Delta1H 0(=/Delta1Hinh+/Delta1Hmosaic) is a sum of the frequency- independent inhomogeneity and mosaicity contributions tothe linewidth. Extracted α effvalues versus CFA thickness are shown in Fig. 6(b). The extracted values of αeffincrease with decreasing CFA layer thickness, indicating a1 tCFAdependence. The extracted μ0/Delta1H0values vary in the range 1.2–2.5 mT, being smaller for tCFA<12 nm (1.2–1.6 mT).Broadband out-of-plane FMR measurements were per- formed in the frequency range from 5 to 17 GHz. During thesemeasurements the VNA was utilized to record the frequencyand magnetic field dependence of the complex transmissionparameter S 21of the microwave signal. Typical results for the real and imaginary parts of S21for the tCFA=20 nm sample are given in Fig. 7. Recorded S21spectra were fitted to the following set of equations [ 30]: S21(H,t)=S0 21+Dt+χ(H) ˜χ0, χ(H)=Meff(H−Meff) (H−Meff)2−H2 eff−i/Delta1H (H−Meff).(6) In these equations S0 21is the nonmagnetic contribution to S21,χ(H) is the complex susceptibility of the magnetic film, and ˜χ0is an imaginary function of the frequency and film thickness. The term Dtaccounts for a linear drift of the recorded S21signal and Heff=2πf/γ μ 0. Meffand the Landé gfactor can be extracted by fitting the Hrversus frequency results to the expression μ0Hr=2πf γ+μ0Meff. (7) Typical results are shown in Fig. 8(a) for the tCFA=8- and 20-nm samples. Following the method outlined in Ref. [ 30], μ0Meffandgincrease slightly and take values in the range 1.15–1.20 T and 2.07–2.13, respectively. Figure 8(b) shows /Delta1H versus fextracted from out-of-plane FMR results, again indicating an increase of αeffwith decreasing thickness of the CFA layer. The damping parameter extracted in thisway includes the intrinsic Gilbert contribution ( α G) and the contribution due to spin pumping ( αsp);αeff=αG+αsp.H e r e we have ignored the radiative and eddy current contributions tothe damping, which are expected to give a contribution /lessorsimilar3× 10 −4. The theoretical framework describing the relaxation of injected spins in the nonmagnetic layers, including thebackflow of spin angular momentum from the nonmagneticlayers into the magnetic layer, was presented in Refs. [ 19,20]. The theory as derived is restricted to metals with a ratioof the spin-conserved to spin-flip scattering times (the spin-flip probability) /epsilon1=τ el/τSF=(λel/λSD)2/3/greaterorsimilar10−3, where λelandλSDare the mean free path and spin-diffusion length, FIG. 6. (a) μ0/Delta1Hversus f(a) from in-plane FMR measurements for samples with different CFA thickness. Symbols correspond to experimental data and lines are fits to Eq. ( 5). (b)αeffversus tCFA. 134421-5SERKAN AKANSEL et al. PHYSICAL REVIEW B 97, 134421 (2018) FIG. 7. Out-of-plane FMR spectra for 8- and 20-nm-thick CFA films showing (a) real and (b) imaginary parts of S21. Symbols are experimental data and lines are fits to Eq. ( 6). respectively. For a nonmagnetic metal to be an efficient spin sink, the requirement is /epsilon1/greaterorsimilar10−2[20]. Using values for λelandλSDderived for ferromagnetic/ β-Ta bilayers ( λel= 0.5 nm and λSD=2.5n m ) [ 31], the value for the spin-flip probability becomes /epsilon1=1.3×10−2indicating that the model is applicable to ferromagnetic/ β-Ta bilayers and that β-Ta acts as an efficient spin sink. In the simplest case, with onlyone interface, the extra contribution to the damping can beexpressed as α sp=gμBg↑↓ eff 4πMs1 tCFA, (8) where g↑↓ effis the real part of interfacial mixing conductance g↑↓in series with the nomal-metal resistance. For the samples discussed here there are two interfaces, one between the CFAand Ta layers and one between the Ta and Ta 2O5layers. This implies that g↑↓ effwill be a function of the conduc- tance at both interfaces, since spin relaxation is expectedboth in the Ta and Ta 2O5layers. Figure 9(a) shows αeff extracted from out-of-plane FMR measurements versus1 tCFAtogether with a fit of the experimental data to Eq. ( 8). UsingMs=915 kA /m(μ0Ms=1.15 T) and g=2.10, one obtains g↑↓ eff=(2.90±0.1)×1019m−2, which is comparable to the value obtained for a Pd/CoFe/Pd multilayer structure [ 32]. Using this value for the effective mixing conductance, it isnow possible to disentangle the two contributions to α eff; Fig. 9(b) shows αefftogether with αspandαGversus CFA layer thickness. The extracted value for the intrinsic Gilbertisα G=1.1±0.2×10−3, which is in good agreement with previously determined values [ 17]. Moreover, assuming that the spin current is reflected at the β-Ta/Ta2O5interface and using the relation between the intrinsic and effective spin- mixing conductance g↑↓ eff=g↑↓(1−e−2tTa/λSD), where tTais the thickness of the β-Ta layer and the exponential term within the brackets accounts for the backflow of spin angular momentum,one obtains g ↑↓≈3.35×1019m−2for the intrinsic spin- mixing conductance. The low Gilbert damping ( /lessorequalslant1×10−3) and high spin- mixing conductance ( /greaterorequalslant1×1019m−2) observed for the CFA/β-Ta bilayer system are key requirements for spin transfer torque magnetization switching and spin logic devices. How-ever, efficient switching of magnetic memory and spin logicdevices also requires a large interface transparency ( T). The FIG. 8. fversus Hr(a) and μ0/Delta1Hversus f(b) from out-of-plane FMR measurements for samples with different CFA thickness. Symbols correspond to experimental data and lines are fits to Eqs. ( 7)a n d( 5). Since error in Hris negligible, no error bars are shown in (a). 134421-6THICKNESS-DEPENDENT ENHANCEMENT OF DAMPING IN … PHYSICAL REVIEW B 97, 134421 (2018) FIG. 9. (a) αeffversus1 tCFAusing data extracted from out-of-plane FMR measurements. Squares correspond to the experimental data and solid line fit to Eq. ( 8). (b)αeff, spin-pumping contribution αspto damping and intrinsic Gilbert damping αGversus tCFA. interface transparency in the CFA/ β-Ta bilayer system controls the flow of spin angular momentum across the interface anddepends on the microscopic intrinsic and extrinsic interfacialfactors, such as band structure mismatch, Fermi velocity, andinterface imperfections, and can be expressed as [ 33] T=g ↑↓ efftanh/parenleftbigtCFA 2λSD/parenrightbig g↑↓ effcoth/parenleftbigtTa λSD/parenrightbig +σTah λSD2e2, (9) where σTa(=5×105/Omega1−1m−1) is the conductivity of the β-Ta layer. The estimated value of the transparency for the CFA/ β- Ta interface is ∼68%. This Tvalue is even higher than for FM/Pt interfaces [ 33], clearly indicating the significance of using the CFA/ β-Ta structure in innovative spin transfer torque devices. IV . CONCLUSIONS The effects of Co 2FeAl thickness covering the range 8– 20 nm on the static and dynamic properties of Si/ Co 2FeAl/β- Ta multilayers have been investigated. It was found that staticproperties like the saturation magnetization and coercivitywere weakly thickness dependent, with values covering therange 900–950 kA /m and 0.53–0.87 kA /m, respectively. The in-plane uniaxial anisotropy field was determined from angle-dependent cavity FMR measurements, indicating a decreasingtrend with increasing CFA thickness, with values covering therange 2.20–3.90 mT. Both in-plane and out-of-plane broadbandFMR measurements show that the effective damping parameter increases with decreasing thickness, indicating an enhance-ment of damping due to spin pumping into the nonmagneticcap layer. The in-plane damping parameter is also affected byspin relaxation due to two-magnon scattering, resulting in alarger effective damping parameter as compared to the out-of-plane damping parameter. The out-of-plane effective dampingparameter, being free from spin relaxation due to two-magnonscattering, was further analyzed to extract information aboutthe effective spin-mixing conductance of the multilayer as wellas to disentangle the contributions to the effective damping parameter, yielding g ↑↓ eff=(2.90±0.10)×1019andαG= (1.1±0.2)×10−3. The high value of g↑↓ efffor the CFA/ β-Ta structure, at par with that of FM/Pt bilayers, in conjunctionwith∼68% interface transparency and low Gilbert damping (/lessorequalslant1.1×10 −3) of CFA clearly makes the CFA/ β-Ta structure a promising building block for spin transfer torque devices. ACKNOWLEDGMENTS This work is supported by the Knut and Alice Wallen- berg (KAW) Foundation, Grant No. KAW 2012.0031. S.H.acknowledges the Department of Science and TechnologyIndia for providing the the INSPIRE Fellowship (Grant No.IF140093) grant. S.A. and A.K. contributed equally to this work. [1] J. C. 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Hono, Coherenttunneling and giant tunneling magnetoresistance in Co 2FeAl/ MgO/CoFe magnetic tunneling junctions, P h y s .R e v .B 81, 140402(R) (2010 ). [7] W. Wang, H. Sukegawa, and K. Inomata, Temperature depen- dence of tunneling magnetoresistance in epitaxial magnetictunnel junctions using a Co 2FeAl Heusler alloy electrode, Phys. Rev. B 82,092402 (2010 ). [8] M. Belmeguenai, H. Tuzcuoglu, M. S. Gabor, T. Petrisor, C. Tiusan, F. Zighem, S. M. Chérif, and P. Moch, Co 2FeAl Heusler 134421-7SERKAN AKANSEL et al. PHYSICAL REVIEW B 97, 134421 (2018) thin films grown on Si and MgO substrates: Annealing temper- ature effect, J. Appl. Phys. 115,043918 (2014 ). [9] B. Sun, K. Kim, N. Leibing, and S. Serrano-Guisan, Structural and magnetic properties of epitaxial Co 2FeAl films grown on MgO substrates for different growth temperatures, Acta Mater . 60,6714 (2012 ). [10] S. Mizukami, D. Watanabe, M. Oogane, Y . Ando, Y . Miura, M. Shirai, and T. Miyazaki, Low damping constant for Co 2FeAl Heusler alloy films and its correlation with density of states,J. Appl. Phys. 105,07D306 (2009 ). [11] G. Ortiz, M. S. Gabor, T. Petrisor, Jr., F. Boust, F. Issac, C. Tiusan, M. Hehn, J. F. Bobo, and T. Petrisor, Static and dynamicmagnetic properties of epitaxial Co 2FeAl Heusler alloy thin films, J. Appl. Phys. 109,07D324 (2011 ). [12] H. Sukegawa, Z. Wen, K. Kondou, S. Kasai, S. Mitani, and K. Inomata, Spin-transfer switching in full-Heusler Co 2FeAl-based magnetic tunnel junctions, Appl. Phys. Lett. 100,182403 (2012 ). [13] W. Wang, E. Liu, Y . Du, J. Chen, G. Wu, H. Sukegawa, S. Mitain, and K. Inomata, Thickness-dependent structural, magnetic andtransport properties of epitaxial Co 2FeAl Heusler alloy thin films, arXiv:1210.5807 . [14] M. Belmeguenai, H. Tuzcuoglu, M. S. Gabor, T. Petrisor, C. Tiusan, D. Berling, F. Zighem, T. Chauveau, S. M. Chérif, and P. Moch, Co 2FeAl thin films grown on MgO substrates: Corre- lation between static, dynamic, and structural properties, Phys. Rev. B 87,184431 (2013 ). [15] X. G. Xu, D. L. Zhang, X. Q. Li, J. Bao, Y . Jiang, and M. B. A. Jalil, Synthetic antiferromagnet with Heusler alloyCo 2FeAl ferromagnetic layers, J. Appl. Phys. 106,123902 (2009 ). [16] M. Belmeguenai, H. Tuzcuoglu, M. Gabor, T. Petrisor, C. Tiusan, D. Berling, F. Zighem, and S. Mourad Chérif, Magneticand structural properties of Co 2FeAl thin films grown on Si substrate, J. Magn. Magn. Mater. 373,140(2015 ). [17] S. Husain, S. Akansel, A. Kumar, P. Svedlindh, and S. Chaud- hary, Growth of Co 2FeAl Heusler alloy thin films on Si(100) having very small Gilbert damping by ion beam sputtering,Sci. Rep. 6,28692 (2016 ). [18] X. Liu, W. Zhang, M. J. Carter, and G. Xiao, Ferromagnetic resonance and damping properties of CoFeB thin films as freelayers in MgO-based magnetic tunnel junctions, J. Appl. Phys. 110,033910 (2011 ). [19] Y . 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Physics.14.92.pdf

VIEWPOINT ANewDriftinSpin-Based Electronics Asymmetry-breakingmechanismallowsresearcherstoproduceand observeadirectedcurrentofmagnonsinamagneticinsulator,opening newpossibilitiesinmagnon-basedelectronics. ByArabindaHaldar andAnjanBarman Demandforminiaturized,energy-efficient,andultrafast information-processingdevicescontinuestorise,but manufacturersareapproachingthefundamentallimits oftheprevailingtechnology,whichisbasedoncomplementary metal-oxidesemiconductors(CMOS).Oneofthemainbarriers toprogressisJouleheating: asCMOScomponentsbecome fasterandsmaller,theysuffermorefromtheheatthatbuildsup becauseoftheflowofelectroniccharge. Researchershave proposedcircumventingthisproblembydoingawaywith movingchargesaltogether. Instead,informationprocessing couldbeaccomplishedbymanipulating “magnons”—quasiparticlesofelectron-spinexcitations—ina magneticmedium. Now,RichardSchlitzattheSwissFederal InstituteofTechnology(ETH),Zurich,andcolleagueshave takenanimportantsteptowardsuchatechnologybyshowing thatamagnondriftcurrentcanbeinducedtoflowina magneticheterostructure[ 1]. Justasphononsrepresentthecoherentpropagationoflattice vibrations,magnonsrepresentthecollectiveprecessionof electrons’magneticmoments. Inbothcases,these quasiparticlesmovethroughamaterialeventhoughthe excitationsthatsustainthemremainfixedwithinthelattice. Thepossibilityofon-chipdataprocessingbasedonmagnonsis stimulatinganewfrontierinphysicscalled“magnonics,”where amagnoncurrentreplacesthespinorchargecurrentusedin spintronicorelectronicdevices,respectively[ 2,3]. Thisisnota straightforwardreplacement,however. Whereasanelectron driftcurrentisthephysicalmotionofcharge,amagnoncurrent representsonlythepropagationofthephaseofcollectivespin precession. Thisdifferencemeansthattheinteractionofmagnonswithamagneticfieldisnotanalogoustothe interactionofelectronswithanelectricfield. Ratherthan drivingmagnonsthroughacircuit,amagneticfieldonlycauses achangeintheirfrequency. Instead,magnondriftcurrents mustbemanipulatedbymechanismsthatresultfrombreaking theinversionsymmetryofthemagneticmedium. OnesuchmechanismistheDzyaloshinskii-Moriyainteraction (DMI),whichcanoccuratthecontactbetweenaferromagnetic layerandanonmagneticlayerwithlargespin-orbitcoupling. ThisinterfacialDMI(iDMI)isanantisymmetric,three-site exchangeinteraction,wherethespinsoftwoferromagnetic atomsinteractviaanonmagneticatombelongingtothe nonmagneticlayerattheothersideoftheinterface. Asaresult, theDMIvectorliesintheplaneoftheinterface,producingan asymmetricmagnondispersionthatcanbedirectlyprobed usinganopticalmagnon-spectroscopytechnique[ 4]. However, thisprobingtechniquecannotdifferentiate“pure”magnondrift currentsinducedbytheiDMIfromdiffusivemagnoncurrents drivenbymagnonchemicalpotential. Asaresult,clean observationsofmagnondriftcurrentshavenotbeenreported. Inanewstudy,Schlitzandhiscolleaguesproposeatheoryof magnontransportinwhichthedriftcurrentcontributionis addedtothediffusivemagnoncurrentbyincludingan additionalasymmetrictermintheequationthatdescribesthe system. Theeffectofthisadditionaltermistocreatean anisotropyinthemagnonvelocity,whichtheteamusedasa wayofdisentanglingthetwocontributionsexperimentally. Theysputter-depositedaY 3Fe5O12(YIG)thinfilmona (111)-orientedGd 3Ga5O12(GGG)substrate. YIGisapopular physics.aps.org | ©2021AmericanPhysicalSociety | June23,2021 | Physics14,92 | DOI:10.1103/Physics.14.92 Page1VIEWPOINT Figure1: Schematicshowingtheexperimentalsetupusedto measurethemagnondriftcurrent. Acurrent(I)flowinginthe centralplatinum(Pt)wiregeneratesaspincurrentintheYIGlayer viathespinHalleffect. Dzyaloshinskii-Moriyainteractionatthe YIG-GGGinterfaceproducesanasymmetryinthepropagation velocityofdriftmagnoncurrent(V DMI),whichcanbecontrolledby aligningthemagnetizationvector( M).Theasymmetryisrevealed asanunequalresistancemeasuredbythetwodetectorwireson eithersideofthecurrent-carryingwire. Credit: A.Barman/S.N.BoseNationalCentreforBasicSciences; A.Haldar/IndianInstituteofTechnologyHyderabad;adaptedby APS/AlanStonebraker ferrimagneticoxideforsuchstudies,asitallowslong-range spin-wavepropagation,whiletheYIG-GGGinterfacehasbeen showntogenerateiDMI[ 5]. Wheretheteam’ssetupdeviated frompriorexperimentsisintheirinnovativenonlocal measurementtechnique. Usually,suchnonlocalelectrical measurementsofamaterialaremadeusingtwophysically separatedcontactpads. Currentissentintothematerial throughoneofthecontactpads,andthematerial’sresistanceis calculatedbymeasuringthevoltageattheothercontactpad. Asthevoltageismeasuredawayfromthecurrent-carrying contactpad,thecalculatedresistancecarriesinformationon thetransportpropertiesofthematerialuponwhichthecontact padsarefabricated. Theproblemisthatthenonlocal resistance,asdescribedbyasimplifiedmagnontransport model,arisesbecauseofthecombinationofbothdiffusionand driftmagnoncurrentinthematerial. Assuch,thereisnowayto teasethetwoeffectsapartinexperiments.Toavoidthelimitationsofthisconventionaltwo-contact approach,Schlitzandhiscolleaguesfabricatedthreeequally spaced,parallelplatinumwiresontopoftheYIGfilm(Fig. 1). By drivinganoscillatingcurrentinthecentralwire,theyinduceda magnoncurrentinthelowerYIGlayerthroughthespinHall effect(SHE).TheSHEgeneratesapurespincurrentasaresultof theflowofchargecurrentinmaterialswithlargespin-orbit coupling,suchasplatinumandotherheavymetals. Itsinverse effect—knownasISHE—isthegenerationofavoltageduetothe conversionofspincurrenttochargecurrent. Intheabsenceofa magnondriftcurrent,thediffusivemagnoncurrentwould generateanequalvoltageateachwire,withthesizeofthe voltagedependentonthemagnetic-fieldstrengthand orientation. Butadriftcurrentwouldproduceavoltage asymmetry. Indeed,theteamfoundthattheISHE-induced voltageateachwirewasdifferentandthatthesevoltagesvaried asymmetricallyasthemagnetic-fieldorientationwaschanged. Fromthisasymmetry,theresearcherswereabletocalculatethe driftcurrentcontributioninisolation. Schlitzandcolleagues’cleardemonstrationofamagnondrift currentisaproof-of-principleofanimportantphenomenonin magnonics. Assuch,theresultopensupnewpossibilities,such asimprovedmagnon-basedlogicandcommunicationdevices. Agoalforthefuturewillbetofindmaterialcombinationsthat generateastrongerDMIeffectthantheYIG-GGGinterface, whichwouldenhancethemagnondriftcurrent. Suchmaterials mustalsoexhibitlowlevelsofGilbertdamping—a phenomenonthatcausesspinexcitationstodissipate—toallow forsignificantmagnonpropagation. ButDMImightnotbethe onlygameintown: Othermechanismsofinversionsymmetry breaking—suchasanasymmetricornonuniformfield[ 6]and theRashbaeffect[ 7]—canalsobeutilizedtorealizelarge magnondriftcurrents,andwelookforwardtothe demonstrationofsuchideas. ArabindaHaldar: DepartmentofPhysics,IndianInstituteof TechnologyHyderabad,Telangana,India AnjanBarman: DepartmentofCondensedMatterPhysicsand MaterialSciences,S.N.BoseNationalCentreforBasicSciences, Kolkata,India REFERENCES 1.R.Schlitz etal.,“Controlofnonlocalmagnonspintransportvia physics.aps.org | ©2021AmericanPhysicalSociety | June23,2021 | Physics14,92 | DOI:10.1103/Physics.14.92 Page2VIEWPOINT magnondriftcurrents,” Phys.Rev.Lett. 126,257201(2021). 2.A.V.Chumak etal.,“Magnonspintronics,” Nat.Phys. 11,453 (2015). 3.ABarman etal.,“The2021magnonicsroadmap,” J.Phys. Condens.Matter (2021). 4.H.T.Nembach etal.,“LinearrelationbetweenHeisenberg exchangeandinterfacialDzyaloshinskii–Moriyainteractionin metalfilms,” Nat.Phys. 11,825(2015).5.H.Wangetal.,“Chiralspin-wavevelocitiesinducedbyall-garnet interfacialDzyaloshinskii-Moriyainteractioninultrathinyttrium irongarnetfilms,” Phys.Rev.Lett. 124,027203(2020). 6.J.H.Kwon etal.,“Giantnonreciprocalemissionofspinwavesin Ta/Pybilayers,” Sci.Adv2,e1501892 (2016). 7.K.-W.Kim etal.,“Predictionofgiantspinmotiveforcedueto Rashbaspin-orbitcoupling,” Phys.Rev.Lett. 108,217202 (2012). physics.aps.org | ©2021AmericanPhysicalSociety | June23,2021 | Physics14,92 | DOI:10.1103/Physics.14.92 Page3

PhysRevB.101.205407.pdf

PHYSICAL REVIEW B 101, 205407 (2020) Editors’ Suggestion Spin caloritronics in a CrBr 3-based magnetic van der Waals heterostructure Tian Liu ,*,†Julian Peiro ,†Dennis K. de Wal , Johannes C. Leutenantsmeyer, Marcos H. D. Guimarães , and Bart J. van Wees Zernike Institute for Advanced Materials, Nijenborgh 4, 9747 AG Groningen, The Netherlands (Received 3 March 2020; revised manuscript received 11 April 2020; accepted 14 April 2020; published 6 May 2020) The recently reported magnetic ordering in insulating two-dimensional (2D) materials, such as chromium tri- iodide (CrI 3) and chromium tribromide (CrBr 3), opens new possibilities for the fabrication of magnetoelectronic devices based on 2D systems. Inevitably, the magnetization and spin dynamics in 2D magnets are strongly linkedto Joule heating. Therefore, understanding the coupling between spin, charge, and heat, i.e., spin caloritroniceffects, is crucial. However, spin caloritronics in 2D ferromagnets remains mostly unexplored, due to theirinstability in air. Here we develop a fabrication method that integrates spin-active contacts with 2D magnetsthrough hBN encapsulation, allowing us to explore the spin caloritronic effects in these materials. The angulardependence of the thermal spin signal of the CrBr 3/Pt system is studied, for different conditions of magnetic field and heating current. We highlight the presence of a significant magnetic proximity effect from CrBr 3on Pt revealed by an anomalous Nernst effect in Pt, and suggest the contribution of the spin Seebeck effect fromCrBr 3. These results pave the way for future magnonic devices using air-sensitive 2D magnetic insulators. DOI: 10.1103/PhysRevB.101.205407 I. INTRODUCTION The search for magnetism in 2D systems has been a nontrivial topic for decades. Recently, 2D magnetism wasdemonstrated in an insulating material CrI 3[1], which shows antiferromagnetic exchange between the layers, resulting inzero (nonzero) net magnetization for even (odd) number oflayers. It has been shown that CrBr 3exhibits ferromagnetism when exfoliated down to a few layers [ 2] and monolayers [ 3] while preserving its magnetic order. This discovery offers us a platform to explore magnonics in 2D systems. Magnonics refers to spintronics based onmagnons, which are quantized spin waves, i.e., collectiveexcitations of ordered electron spins in magnetic materials[4–6]. Magnonic spin transport has been extensively studied in various ways in 3D magnetic insulators, e.g., spin pumping[7], Spin Seebeck effect (SSE) [ 8], and electrical injection and detection of magnons [ 9]. The outstanding magnon trans- port properties of the ferrimagnetic insulator yttrium irongarnet (YIG) and the robustness and fast dynamic of AFMmaterials like iron oxide [ 10] and nickel oxide [ 11]t r i g - gered the development of the first magnon transport deviceprototypes for application using these materials [ 9,12,13]. The predicted novel physical phenomena [ 14–18] hosted by low-dimensional magnon systems represent a strong potentialfor 2D magnonics. Thermally excited magnon transport wasreported recently in an AFM vdW 2D material MnPS 3[19]. However, magnonics in 2D van der Waals magnetic systemsstill remains mostly unexplored, especially for 2D ferromag-netic (FM) systems. *tian.liu@rug.nl †These authors contributed equally to this work.One of the difficulties to study such phenomena is the easy degradation in air of the magnetic 2D materials, bringing extratechnical challenges for integrating magnonic circuits withthese materials. Here, we introduce a technique of bottommetallic contacts on an air-sensitive material CrBr 3, aiming at preliminary study of magnonics in 2D ferromagnetic materi-als. We select CrBr 3as a medium for 2D magnonics study [ 20] as its FM order is independent on the number of layers andthus it simplifies the device fabrication. The Curie temperatureof CrBr 3is about Tc=37 K [ 20] in bulk, reducing to 27 K for monolayers [ 3]. CrBr 3presents perpendicular magnetic anisotropy (PMA) [ 2] with an out-of-plane coercive field of 4 mT and an in-plane saturation field of 400 mT for a fewlayers [ 3]. The saturation magnetization of about 271 kA /mi s reported nearly equal for in-plane and out-of-plane orientationin bulk and differs by less than 20% for three-layer CrBr 3 [2,21]. II. DEVICE GEOMETRY AND MEASUREMENTS In this work we employ nonlocal angular-dependent magnetoresistance (nlADMR) measurements on a hBN-encapsulated CrBr 3flake contacted by Pt strips. ADMR mea- surements have been widely used to characterize the spin Hallmagnetoresistance (SMR) in local geometries [ 22]o rt h es p i n Seebeck effect (SSE) in nonlocal geometry [ 9] and identify them out of other caloritronics effects [ 23–26]. We fabricated a device where Pt strips (5.5 nm thick) are deposited into apreetched 16.6-nm-thick hBN flake on top of a silicon oxidesubstrate. A 6.5-nm-thick top hBN flake is used to pick up andfully cover a 7-nm-thick CrBr 3flake (about 10 layers) [ 27]. A schematic of the device and nonlocal measurement geometryis shown in Fig. 1(a). 2469-9950/2020/101(20)/205407(8) 205407-1 ©2020 American Physical SocietyTIAN LIU et al. PHYSICAL REVIEW B 101, 205407 (2020) FIG. 1. (a) Schematic of the device and the circuit for the nonlocal measurements. A 7-nm-thick CrBr 3flake placed on top of 5.5-nm-high Pt strips is fully encapsulated by two layers hBN. The x,ydirections are defined to be in-plane (Pt strips parallel to the yaxis), where the magnetic field is rotated over the azimuthal angle ϕ(IP) and polar angle theta (OOP). (b), (d) Principle of generation and detection of respectively electrically and thermally generated magnons. (c), (e) Measured corresponding first (c) and second (e) order harmonic NL resistances with 20 μA are fitted with the cos2(ϕ)a n dc o s ( ϕ) function, respectively. The small red arrows in (b) and (d) indicate spin polarization direction. For (e) the sign of the fitted cosine for the ISHE from the SSE agrees with this spin polarization and therefore with the standard definition of the spin Hall angle [ 28]. For the measurement in (e), the offset R2ω 0=16.3±0.8V/A2. The error bars represent the standard deviation from the fits. In this system, a gradient of temperature is created by the Joule heating from a remote Pt heater which generatesa magnon-mediated spin flow due to the magnon densitydependence on the temperature [ 29], i.e., the SSE. At the interface between a magnet and a nonmagnetic material, atransfer of magnon spin ( +¯h) from the CrBr 3to the Pt is possible by spin flip of a −¯h/2s p i nt oa +¯h/2 spin in the Pt. The spin current generated this way in the Pt contactconverts into a charge current by inverse spin Hall effect(ISHE) and can be measured as a voltage difference. In thegeometry defined in Fig. 1, the ADMR is then sensitive to the xcomponent of the magnetization of CrBr 3,Mx.I n the in-plane ADMR configuration [Figs. 1(a) and1(d)], the orientation of the magnetization with regard to the detectioncontact drives the angular dependence; therefore, a cos( ϕ) dependence is expected. All data shown in the main text was measured on a pair formed of a 310-nm-wide injector and a 520-nm-wide detec-tor, spaced by 500 nm edge to edge, and at a base temperatureof 5 K under a reference magnetic field of 4 T, unless specif-ically mentioned. We separate different harmonics by usingstandard low frequency (6 Hz to 13 Hz) lock-in techniques.The voltage response is composed of different orders that areexpanded as V(t)=R 1I(t)+R2I(t)2+··· [9], where Riis theith-order response [ 30] to the applied ac current I(t). As the electrical magnon injection scales linearly with current, itsresponse is expected in the first harmonic signal. The thermalinjection depends quadratically on the applied current and theresponse appears in the second harmonic signal. First and second harmonic responses of the nonlocal signal have been measured simultaneously all along this study. Thefirst order angular dependence is expected to obey the relationR 1ω=V/I=R1ω 0+R1ω nlcos2(ϕ)[9], where R1ω 0is an offset resistance and R1ω nlis the magnitude of the first harmonicsignal. However, we do not observe the expected cos2(ϕ) modulation in the first harmonic signal, as the fitted firstorder resistance R 1ω nlis only detected in the order of 0.01 m /Omega1 which is comparable to the standard deviation. An example ofmeasured first harmonic signal can be found in Fig. 1(c).Y e t , this value is at least three orders smaller than the R 1ω nlreported for the Pt /YIG system [ 9]). The measurements are carried out over a wide range of applied currents and magnetic fields,and with the maximum lock-in detection sensitivity. A typicalmeasurement of first harmonic nonlocal signal is shown inFig. 1(c), for a current of 20 μA at 5 K. In contrast, the nonlocal second harmonic signals exhibit a clear sinusoidalbehavior [Fig. 1(e)] under an in-plane rotating magnetic field. The magnitudes of nonlocal signals were fitted with R 2ω=V I2=R2ω 0+R2ω nlcos(ϕ), (1) where R2ω 0is the offset resistance for the second harmonic signal. A nonzero offset R2ω 0is always present, possibly from unintended Seebeck contribution in the detector [ 31].R2ω nlis the magnitude of the second harmonic signal. For the corre-sponding second harmonic measured in Fig. 1(e), we extract an amplitude R 2ω nl=−36±1V / A2, which is comparable to the magnitude of room-temperature nonlocal SSE measuredon bulk Pt /YIG samples [ 9] with equal angular dependence. If we compare to the typical top contact geometry used todetect SSE from YIG [ 9], the same SSE detected here in bottom contact geometry should produce a spin current inthe opposite direction. Therefore, the ISHE induced in Pt isreversed compared to the top Pt on YIG; hence we expect anopposite sign of the signal. The negative sign observed herewould correspond to the positive sign measured in [ 9] and, if attributed to SSE, reveals a transfer of magnon spin fromCrBr 3to the Pt top surface. However, at this point, we cannot 205407-2SPIN CALORITRONICS IN A CrBr 3-BASED … PHYSICAL REVIEW B 101, 205407 (2020) FIG. 2. Dependence of second harmonic signals on applied cur- rent through the injector. (a) Top panel: low bias signals with cos( ϕ) fitting measured at 20 μA, with a fitted amplitude ( −29±1V/A2); bottom panel: high bias signals with cos( ϕ) fitting measured at 140μA, with a fitted amplitude (0 .64±0.03 V/A2). (b) Bias dependence of R2ω nl. Bias dependence shown in these graphs were measured at 5 K under a magnetic field of 4 T. The inner figure presents the zoom-in data of R2ω nl, for the applied current from 100 μA to 300 μA. rule out other effects like proximity induced anomalous Nernst effect (pANE) in Pt [ 32]. We discuss relevant effects later [see Fig. 4(c), rotation of out-of-plane magnetic field]. The current dependence of R2ω nlis plotted in Fig. 2,f o ra contact pair with distance of 950 nm center to center (edgeto edge distance of 500 nm). R 2ω nldepends on the applied current nonlinearly, and a sign reversal of R2ω nloccurs between 40 and 100 μA. For data measured at 60 μA and 80 μA, an angular modulation of the second harmonic signal is stillobserved but it is not described by a simple cosine function(see Supplemental Material [ 33]). An example of the negative R 2ω nlat low current is shown in Fig. 2(a) (top panel), and an example of the positive R2ω nlat high current is plotted inFig. 2(a) (bottom panel). The absolute amplitude |R2ω nl|in general decreases with increasing current at the heater, asplotted in Fig. 2(b). Its value for positive amplitude at high current is one to two orders of magnitude lower than itsvalue for negative amplitude at low current, depending on theapplied current. To get better insight of the role of the complex temperature distribution in our device for this nonlinear behavior, weemploy a two-dimensional finite element model (FEM) sim-ulating a geometry of the x-zplane. Indeed the full hBN en- capsulation of the CrBr 3flake in this device brings inevitable additional heat conduction paths resulting in strong current-dependent thermal gradients in both xandzdirections ( ∂ xT and∂zT, respectively). As κCrBr 3, the thermal conductivity of CrBr 3, is unknown, we ran the computation for different ther- mal conductance ratios ηKso that κCrBr 3(T)=ηKκhBN(T), withκhBNthe thermal conduction of hBN, and taking into account the highly temperature dependent thermal conductionof the materials (see Supplemental Material VII [ 33]). This modeling reveals a strong dependence of the temperatureprofile as a function of the heating current. It qualitativelysupports that the main contribution of the thermal gradientin the Pt detector is in xdirection ( ∂ xT). Yet there also is a non-negligible thermal gradient in zdirection ( ∂zT), in the CrBr 3as well as in the Pt detector, allowing for SSE and possible unintended effects occurring in the Pt detector thatwill be discussed below. The in-plane magnetic field dependence on the second order nlADMR amplitude R 2ω nlis plotted in Fig. 3(a).W e apply a range of fields from 0 T to 7 T for the in-planerotation measurements at 5 K. At low current (20 μA), we observe a linear increase of |R 2ω nl|from 0 T to 3 T. After 4 T, the magnitude tends to saturate showing only a slightdecay [Figs. 3(a) and3(c)]. At high current (160 μA), we also observe a linear increase of R 2ω nlfrom 0 T to 4 T, but (a) (b)(d) (c) -180 -90 0 90911911911911R2ω(V/A2) ϕ(deg)7T 4T 1T 0T20μΑ 160μΑ -180 -90 0 90-1000100 ϕ(deg)-1000100-1000100-1000100 0T1T4T7TR2ω(V/A2) 02 0 4 0 6 0050 Temperature (K)|R2ω nl(V/A2)|20μΑ 01 4T 160μΑ R2ω nl(V/A2)0246050 5K 20μΑ R2ω nl(V/A2) Magnetic field (T)|R2ω nl(V/A2)| 01 160μΑ FIG. 3. (a) Magnetic field dependence of R2ω nlwith both low current (20 μA) and high current (160 μA). The fitted cosine amplitude increases with magnetic field until 3 T in both cases. Examples of measured signals are shown in (c) for low bias and in (d) for high bias, with the fitted cosine curves in solid line. (b) The low bias and the high bias signals measured at three different temperatures: 5 K, 10 K, and 60 K. The thermal spin signal measured at 10 K is smaller than 5 K for both low bias and high bias cases. 205407-3TIAN LIU et al. PHYSICAL REVIEW B 101, 205407 (2020) (b) (d) (e) (f)zCrBr3 PtSSE CrBr3 PtCrBr3 PtANEz ANEx MPtMPt x(a) V +- + -V ++ -- Sum(c) FIG. 4. (a) Schematics of the main effects contributing to the detected signal in OOP-nlADMR, ϕ=0◦,θ∈[−180◦,180◦]. (b) Second harmonic nlADMR for the forward (blue) and the reverse (red) configurations measured with applied current of 20 μA. (c) Sum (R2ω NL,For+R2ω NL,Rev )/2 (green) and difference ( R2ω NL,For−R2ω NL,Rev )/2 (purple) of the traces in (b), highlighting contributions that are fitted with cos(θ)a n ds i n ( θ) functions, respectively. (d) Second harmonic nlADMR shown for 40, 100, and 280 μA for an external magnetic field of 4 T r o t a t i n gi nt h e x-zplane. (e) The current dependence of pANEx(red) and SSE +pANEzsignals (blue) for the forward configuration. In insets of (e) are given the ratio ξ=−(RSSE+RpANE z)/RpANE x(bottom inset). (f) Current dependence of the calculated SSE resistance for a range of γ=∂zT/∂xT. with magnitudes about 50 times smaller than |R2ω nl|for low current. After 4 T, the magnitude still increases but at a lower rate [Figs. 3(a) and3(d)]. The lower magnitude at high current is consistent with the reduction of the magnetizationexpected for a temperature increase due to Joule heating. Theorigin of the magnetic field dependence remains unclear. Asthe saturation of the magnetization of trilayer CrBr 3in its hard plane is reported to occur at 400 mT [ 2], the linear increases cannot be simply explained by the saturation of themagnetization as from an isolated CrBr 3layer and reveals the contribution of additional field dependent effects. The second order nlADMR is also measured at three different temperatures, 5 K, 10 K, and 60 K, and the fittedamplitudes of R 2ω nlare shown in Fig. 3(b) for low (20 μA) and high current (160 μA) measured under 4 T. Compared with the signal at 5 K, the fitted cosine amplitude at 10 K decreasesfor both low and high bias. Far above T cat 60 K, a very small but nonzero value of R2ω nlis observed in our measurements (0.08±0.03 V/A2at 160 μA and −3±2V/A2at 20μA). We attribute this small nonzero value to an artifact from themeasurement setup (see Supplemental Material [ 33]). We present hereafter a series of out-of-plane nlADMR (OOP-nlADMR) measurements, i.e., fixing ϕ=0 ◦and vary- ingθby rotating the magnetic field in the x-zplane, as defined on Fig. 1. Some examples and the current dependence of this OOP-nlADMR are summarized in Fig. 4. The first observation, with Figs. 4(b) and 4(d) as examples, is that all OOP-nlADMR signals exhibit a nonzero angular phaseshift varying with the heating current. We investigated the origin of this phase considering the various effects that couldadd to the SSE signal. Nernst, Seebeck, and spin Nernstmagnetoresistance (SNMR) [ 25,34] effects are discarded as major contributions, either due to the probing geometry ortheir angular dependence; a detailed description is given in theSupplemental Material [ 33]. However, the anomalous Nernst effect (ANE), which has already been reported as a possibleeffect, arising from a proximity induced ferromagnetism intothe Pt [ 24,32,35–38], cannot be ruled out. Considering a proximity ANE (pANE) in Pt, a transverse pANE voltage /Delta1V pANE reads /Delta1VpANE LPt=| ∇ V|y=|− SpANE(m×[−∇T])|y, (2) where SpANE is the pANE coefficient, mis the unit vector of direction of the magnetization, and LPtis the y-axis length of the contact area of Pt with CrBr 3. As the magnetization of CrBr 3is expected to saturate for fields beyond 1 T in the hard plane [ 2,21], we also assume the proximity induced magnetization parallel to the magnetic field at 4 T. Then, twocontributions of the pANE are distinguished [Fig. 4(a)]: the pANE signal caused by the IP gradient ∂ xT, pANEx, which varies as sin( θ), and the pANE signal caused by the OOP gradient ∂zT, pANEz, which varies as cos( θ). The pANE induced by the temperature gradient along x (pANEx) can be isolated from the other signals by changing the heat flow direction. By interchanging the heater and 205407-4SPIN CALORITRONICS IN A CrBr 3-BASED … PHYSICAL REVIEW B 101, 205407 (2020) detector contacts, the heat flow direction along the xaxis (∝∂xT) is reversed, but the heat flow direction along the z axis (∝∂zT) remains the same. Hence the pANEzcontribution will stay unchanged, while the pANExwill reverse its sign. In Fig. 4(b), we provide a normalized second order nlADMR R2ω N=R2ωAPt/LPt, with APtthe Pt electrode cross section, at 20μA and 4 T, for the configuration forward defined in Fig. 1, and the nlADMR from a reversed geometry where heater anddetector are interchanged. As the width and length of the twoelectrodes are different, as well as their interface with CrBr 3 possibly, the heating power injected will differ by a smallfactor. Therefore, our comparison remains only qualitative.Nevertheless, the amplitudes and offsets are alike and the twotraces differ mainly by the apparent opposite phase shift. If both pANE xand pANEzcontributions are significant in our system, the difference between the forward geometry[Fig. 4(b)] signal and the reverse geometry [Fig. 4(b)] signal will reveal the sin( θ) behavior, and the sum of these two signals will reveal the cos( θ) behavior. As a result, we obtain the respective traces shown in Fig. 4(c). The good agreement of the fittings on both curves is a confirmation that the pANEis present in the Pt detector. Based on this observation, we extracted the two contri- butions for every ADMR at different current and at a con-stant magnetic field of 4 T, by fitting the expression R 2ω= R2ω 0+RSSE+pANE zcosθ+RpANE xsinθ. The measurements at 40, 100, and 280 μAa r es h o w ni nF i g . 4(c), and the fitted sinusoidal curve presents the phase shift in each case. Thecurrent dependence of the extracted amplitudes is providedin Fig. 4(d).T h e R SSE+pANE zand RpANE xcontributions both follow a similar decreasing trend with applied current. WhileR SSE+pANE zdominates at 20 and 40 μA,RpANE xbecomes close to twice RSSE+pANE zat higher current. The variation of the amplitude of RSSE+pANE zat low currents follows the variation of the signal for IP field rotation in Fig. 2(b); however, the sign reversal for the derived RSSE+pANE zdoes not occur in the OOP configuration. To elucidate the contribution of the spin Seebeck, we introduce the ratio ξ=− RSSE+pANE z/RpANE x= −(RSSE+RpANE z)/RpANE xof the two contributions [inset of Fig. 4(e)], the ratio δ=Sz pANE/Sx pANE to account for any difference between the IP ( Sx pANE) and OOP ( Sz pANE) proximity anomalous Nernst coefficients, as well as the ratioγ=∂ zT/∂xTof the temperature gradients in Pt. As a result, the SSE contribution to the measured signal simply reads(demonstration in the Supplemental Material [ 33]) R SSE=RpANE x(δγ−ξ). (3) Based upon the fact that the saturated magnetization of CrBr 3 has been reported to be of the same magnitude when oriented IP or OOP, we assume δ≈1, i.e., Sz pANE≈Sx pANE. Following this assumption, the estimated ratio of the two contributionsγlays between −0.20 and 0.15, according to our FEM simulation based on thermal conduction properties of CrBr 3 and hBN layers (i.e., the ratio ηK=κCrBr 3/κhBN) (see details in the Supplemental Material [ 33]). Even using δγ=±0.5 accounting for the possible underestimation of ∂zTdue to the omission of a small heat leakage via the Pt /Au contacts leads on SiO 2, we extract a significant SSE contribution to(b) (a) FIG. 5. (a) SSE angular dependence shown for 1, 4, and 7 T, with current fixed to 20 μA at 5 K. (b) Magnetic field dependence of pANExand SSE +pANEzsignal amplitude for the forward con- figuration. In the inset of (b) the ratio ξ=−(RSSE+RpANE z)/RpANE x is given. See the Supplemental Material [ 33] for the data extraction in detail. the nlADMR signal at low heating current, as plotted in Fig. 4(f). We provide the magnetic field dependence of the OOP-nlADMR in Fig. 5. Figure 5(a) shows examples of the evolution of the OOP-nlADMR for 1, 4, and 7 T, fora current fixed to 20 μA. The same operation to separate pANE z+SSE from pANExis applied to this measurement set and the amplitude variation of each component is shown inFig.5(b) for magnetic fields from 0 to 7 T. The pANE z+SSE variation is comparable to the one measured in in-plane rota-tion configuration [Fig. 3(a), blue curve], except that we do not observe the high field saturation decrease. The dependenceof the pANE xtrace follows a similar increase until 2 T, but shows a slight decrease for 3 and 4 T and increases again toreach the same value as pANE z+SSE at 7 T. This behavior is captured into the ξratio that shows a peak above 1.5 for 3 and 4 T and a value remaining around 1 for other fieldstrengths. As the temperature profile is fixed, the differencebetween SSE +pANE zand pANExmust be strongly linked to the magnetic properties of the CrBr 3/Pt structure. III. DISCUSSION By analyzing the OOP-nlADMR, we show that pANEx presents a different angular dependence than SSE and pANEz allowing one to separate the two contributions. Despite the lack of insight on the mechanism inducing the magnetizationin Pt, this assumption is based on the fact that the magneticmoments emerging on the Pt atoms are imprinted by themoments of CrBr 3. Yet the saturated magnetization of CrBr 3 has been measured to differ by less than 20% between theorientation along the easy axis and the orientation in the hardplane. Therefore, the induced magnetization in Pt is expectedto behave accordingly, leading to a comparable anomalousNernst coefficient depending on the magnetization value butweakly on its orientation. A pANE contribution to the ADMR has been identified in Pt/YIG systems as well, but the pANE zrepresents at most 205407-5TIAN LIU et al. PHYSICAL REVIEW B 101, 205407 (2020) 5% of the voltage signal, the left 95% being attributed to SSE induced ISHE [ 32]. Because of the significant magnetic exchange field already noticed in CrBr 3[3,39]a sw e l la s the strong temperature gradients involved (beyond two ordersof magnitude higher than in [ 32]), in our CrBr 3/Pt system, the pANE cannot be neglected and the SSE signal is at bestcomparable with the pANE z. In the Pt /YIG system, the magnon SSE signal decreases with the magnetic field [ 40]. In Fig. 3(a), we notice that, after 3 T, the fitted amplitude of the low current curve doesnot change with magnetic field, but R 2ω nlof the high current curve increases linearly with magnetic field. In other words,R 2ω nlat low current tends to decrease where SSE contributes most, compared to the amplitude at high current where theSSE contributes less. Hence our measurements, with supportof a temperature distribution simulation, suggest that the highamplitude signal observed at low current is dominated by SSEfrom CrBr 3. According to the expected angular dependence of the SSE and ANE, the SSE +pANEzsignal should appear in both IP- nlADMR and OOP-nlADMR, while pANExshould be only detected in OOP-nlADMR. Therefore, the same current de-pendence of SSE +pANE zin both configurations is expected. According to the FEM simulation, a reversal of ∂zToccurs at sufficiently high current, simultaneously in CrBr 3and Pt at the detection interface (see the Supplemental Material [ 33]). This leads to a reversal of the SSE +pANEz, most likely dominated by ANE in the high current range. However, thesign reversal is only observed in the IP measurements [inFig. 2(b)], not showing in the OOP measurements after the separation [in Fig. 4(e)]. As the IP and OOP measurements were performed with different cool-down processes, the ther-mal conductivity is possibly changed at the interface. Thisimplies that the sign reversal current is possibly shifted toa much higher value and therefore not observed in the OOPmeasurements. Furthermore, we also suggest that a quantitative discrim- ination between pANE and SSE is possible. We provide anindicative estimation of the magnitude of the SSE based onthe assumption that the pANE coefficient is equal for ANE x and ANE z. By far, we are limited by the current knowledge on the material properties of the 2D magnet. However, if thethermal conduction profiles and the magnetization dynamicsare characterized concretely, a more accurate separation of thetwo spin-caloritronic effects can be realized. Nevertheless, the magnetic field dependence of pANE x and SSE +pANEzand the difference between them bring new questions. The ANE scales with the magnetization viathe coefficient S pANE. The nonmonotonic field dependence of pANExsuggests a complex evolution of the induced magne- tization in Pt, due to either the presence of magnetic domainsor any additional interaction at the interface.IV . CONCLUSIONS To conclude, we demonstrate the relevance of the full hBN encapsulation and the bottom contacting design to enablethe integration of air-reactive materials such as CrBr 3,f o r studying spin-caloritronic effects in 2D magnets. By usingsecond order nlADMR measurement on such an encapsulatedCrBr 3/Pt device, we reveal, by detecting the presence of a proximity ANE voltage, a significant proximity inducedmagnetism from CrBr 3into the adjacent Pt contacts. With reasonable assumptions, we conjecture about the presenceof a weak SSE, dominating the signal in the low currentregime, while the pANE prevails for currents above 60 μA. The nontrivial magnetic field dependence of the separatedeffects leaves open questions as for the current understand-ing of magnetic effects at the interface of heavy metal and2D magnets. The encapsulation shows itself as an eleganttechnique to address these questions in deeper investigationsof air-degradable 2D materials, and opens the way to futuremagnon transport studies. V . METHODS CrBr 3and hBN crystals are provided by a commercial company HQgraphene. CrBr 3is an air sensitive material. To study magnonics with CrBr 3in a nonlocal geometry, we encapsulate a 7-nm-thin chromium tribromide flake andplatinum (Pt) strips into two hexagonal boron nitride (hBN)layers (top layer and bottom layer). The stacking of van derWaals materials was performed in a glove box filled with inertgas argon by using standard PC /PDMS dry transfer method. Pt strips were first grown on bottom hBN. After that CrBr 3 with a top hBN thin layer was transferred on top of the Ptstrips. See the Supplemental Material [ 33] for more details in fabrication process. ACKNOWLEDGMENTS The authors thank Prof. J. Ye and P. Wan for granting us ac- cess to their transfer system in an Ar glove box. We thank J. G.Holstein, H. de Vries, H. Adema, and T. J. Schouten for tech-nical assistance. We acknowledge fruitful discussions with G.R. Hoogeboom, A. A. Kaverzin, and J. Liu. This project hasreceived funding from the Dutch Foundation for FundamentalResearch on Matter (FOM, now known as NWO-I) as a part ofthe Netherlands Organization for Scientific Research (NWO),the European Union’s Horizon 2020 research and innovationprogramme under Grants Agreement No. 696656 and No.785219 (Graphene Flagship Core 1 and Core 2), and ZernikeInstitute for Advanced Materials. M.H.D.G. acknowledgessupport from NWO VENI 15093. [1] B. Huang, G. Clark, E. Navarro-moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. Mcguire,D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-herrero, and X. Xu,Waals crystal down to the monolayer limit, Nature (London) 546,270(2017 ).[2] H. H. Kim, B. Yang, S. Li, S. Jiang, C. Jin, Z. Tao, G. Nichols, F. Sfigakis, S. Zhong, C. Li, S. Tian, D. 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PhysRevB.84.094401.pdf

PHYSICAL REVIEW B 84, 094401 (2011) Parametric excitation of eigenmodes in microscopic magnetic dots Henning Ulrichs,*Vladislav E. Demidov, and Sergej O. Demokritov Institute for Applied Physics and Center for Nonlinear Science, University of Muenster, Corrensstrasse 2-4, 48149 Muenster, Germany Sergei Urazhdin Department of Physics, West Virginia University, Morgantown, WV 26506, USA (Received 22 June 2011; published 2 September 2011) We utilize time- and space-resolved Brillouin light scattering spectroscopy to study the parametric excitation of spin-wave eigenmodes in microscopic Permalloy dots. We show that the fundamental center eigenmode hasthe smallest excitation threshold. With the increase of the pumping power above this threshold, higher-orderdipole-dominated eigenmodes with both even and odd spatial symmetry also become excited. At microwavepower levels far above the threshold, the multimode excitation regime is suppressed due to the parametricexcitation of short-wavelength exchange-dominated spin-wave modes. Our results provide important insight intothe physics of parametric processes in microscopic magnetic systems. DOI: 10.1103/PhysRevB.84.094401 PACS number(s): 75 .40.Gb, 85 .75.−d, 75.30.Ds, 75 .75.−c I. INTRODUCTION Parametric processes in magnetic systems were observed by Bloembergen and Damon1and theoretically explained by Anderson and Suhl2more than 50 years ago. Since then, they have been intensively studied in the context of appliedphysics as well as basic research. 3–15Parametric processes can be utilized for amplification and manipulation of spin-wavepulses, 7–9parametric stimulation and recovery of microwave signals,10and wave-front reversal.11They also can be used as a powerful experimental tool in studies of spin-wavesolitons and two-dimensional bullets, 7,8,12,13as well as magnon Bose-Einstein condensates.14,15 One of the main parameters governing the efficiency of parametric excitation is dynamic damping.16Monocrystalline yittrium iron garnet (YIG) films are characterized by extremelylow magnetic damping (Gilbert damping parameter α<10 −4) and thus have become the material of choice for studiesof parametric excitation and amplification of magnetizationoscillations and waves. As a consequence of the low dampingin YIG, moderate microwave pumping levels are sufficient forparametric excitation, enabling studies of strongly nonequi-librium states such as parametrically driven magnon gas instrongly nonlinear regimes. 17 As a material for technical applications, YIG has several drawbacks. High-quality YIG films can be grown only onspecial substrates, such as gallium gadolinium garnet, whichmakes the fabrication process incompatible with conventionalsilicon-based semiconductor technology. Additionally, thismaterial is difficult to structure, and it also exhibits astrong dependence of the magnetic properties on temper-ature due to the relatively low Curie point. In contrast,polycrystalline transition-metal ferromagnetic films can beeasily grown by sputtering or evaporation on a variety ofsubstrates, including silicon, and they can be structured ona submicrometer scale by standard lithography techniques.Among these materials, Ni 80Fe20=Permalloy (Py) is most widely used for basic research and applied studies dueto its low crystalline anisotropy and small damping ( α< 10 −2). Py has been utilized as a working medium in spin-torque nano-oscillators,18–21magnonic crystals,22–26domain wall motion–based memory devices,27and spin-wave logic circuits.28 Magnetic damping in Py is relatively small compared to other metallic ferromagnets, but it is still much larger thanin YIG, resulting in a significantly higher threshold powerrequired for parametric excitation. For instance, parametricexcitation of spin waves in Permalloy films has been achievedonly with microwave power levels of at least a few Watts. 29,30 This problem can be overcome by reducing the dimensions of the magnetic samples to nanometer scale and concentratingthe pumping energy into a smaller volume, thus producing alarge local microwave field sufficient for parametric excitationusing moderate driving power. 31,32 In this paper, we report an experimental investigation of the parametric excitation of spin-wave modes in an elliptical Py dot with submicrometer dimensions fabricated on top of a microscopic microwave transmission line. To analyze thespectral and spatial characteristics of the excited modes, weutilized time- and space-resolved microfocus Brillouin lightscattering (BLS) spectroscopy. 33The parametric excitation threshold power of about 1 mW was significantly smallerthan in the extended Py films, enabling us to investigate parametric excitation processes far above the threshold. We show that above the threshold, many different dipole-dominated eigenmodes can be excited. At large paramet-ric pumping power, short-wavelength exchange-dominatedspin-wave modes also become excited. The transfer of theparametric pumping energy into the short-wavelength part ofthe mode spectrum results in a decreased excitation efficiency of the dipole-dominated modes. This redistribution of energy does not significantly affect the excitation of the lowest-frequency modes of the dot. Consequently, the fundamentalcenter and the edge modes can be efficiently excited bythe parametric pumping both at small and at large pumpingpower levels. These results are important for the devel-opment of integrated magnetic devices utilizing parametric processes for excitation and amplification of magnetization oscillations. 094401-1 1098-0121/2011/84(9)/094401(6) ©2011 American Physical SocietyULRICHS, DEMIDOV , DEMOKRITOV , AND URAZHDIN PHYSICAL REVIEW B 84, 094401 (2011) H y Microwave pulses500 nm xM Microstrip linePy ellipse h FIG. 1. (Color online) Scanning electron micrograph of the sample. II. SAMPLE AND MEASUREMENT SETUP Figure 1shows a scanning electron micrograph of the studied sample, which consists of a 10-nm-thick Py filmpatterned by electron-beam lithography and ion milling intoan elliptical dot with lateral dimensions of 1000 by 500 nm.The dot is fabricated directly on top of a 1- μm-wide and 160-nm-thick Au microstrip transmission line. A static mag-netic field of H=400–700 Oe was applied along the short axis of the Py ellipse. The data presented here were obtained atH=700 Oe. To excite the magnetization dynamics, microwave pulses with duration of 100 ns and a repetition period of 2 μs were applied to the transmission line. The pulse power wasvaried between 0.1 mW and 50 mW. The microwave pulsescreated a dynamic magnetic field hparallel to the direction of the static magnetization. The detection of the magnetization dynamics was per- formed by microfocus BLS technique described in detail inRef. 33. This technique combines the spectral and temporal resolution of the conventional BLS 34with diffraction-limited spatial resolution of about 250 nm determined by the size ofthe probing laser spot. The intensity of the scattered light at agiven frequency is proportional to the square of the dynamicmagnetization amplitude at this frequency, at the position ofthe probing spot. The dynamic magnetic field hwas parallel to the static magnetization in our experimental geometry. In this configura-tion, the microwave field cannot linearly excite magnetizationdynamics, since the corresponding component of the dynamicmagnetic susceptibility tensor is equal to zero. 16Instead, the dynamics can be excited by a higher-order parametricexcitation process. 35In the quasiparticle picture, this process can be understood as splitting of a microwave photon withfrequency f Pand wave vector kP≈0 into two magnons with frequency fP/2 and wave vectors that are equal in magnitude and opposite in direction.16In accordance with this picture, in our experiments, we detected magnetization oscillations athalf of the applied microwave pumping frequency f P. In confined sample geometries, quantization of the spin- wave spectrum imposes limitations on the parametric excita-tion. Specifically, the dynamic magnetization response exhibitsresonant spectral behavior, with resonant frequencies equal tothose of the system’s eigenmodes. By utilizing the spectralsensitivity of the BLS technique to detect only the dynamicsignal at the frequency of a particular mode, one can selectivelymap out its spatial profile. Additionally, by synchronizingthe microwave pulses with the spectrometer clock, the timeFIG. 2. BLS spectra recorded at powers of parametric pumping varying from 1 to 50 mW, as labeled. The horizontal scale is the detection frequency, equal to half of the pumping frequency. dependence of the magnetization response to the excitation pulses can be recorded with resolution of 1 ns. III. EXPERIMENTAL RESULTS AND ANALYSIS A. Spectral characteristics of parametric excitation Figure 2shows the BLS spectra recorded at different values of the microwave pumping power Pbetween 1 and 50 mW, providing a survey of the spectroscopic properties and power-dependent dynamical regimes of the system. To record thespectra, the laser spot was positioned at the center of the Pydot. The pumping frequency f Pwas varied between 8 and 20 GHz, and the BLS intensity was simultaneously measuredatf P/2. Because of the threshold nature of the parametric excitation, no dynamic magnetization was detected atP<1m W .A t P=1 mW, the spectrum exhibits a single peak at f 0=7.1 GHz ( fp=14.2 GHz), corresponding to the spin-wave eigenmode with the lowest parametric threshold[Fig. 2(a)]. At P>2 mW, a second peak appears at f 2= 8.5 GHz, as illustrated in Fig. 2(b) forP=2.5 mW. At P> 3.2 mW, a third peak appears at f1=7.7 GHz, as illustrated in Fig. 2(c) forP=5 mW. At even larger power levels, several additional peaks appeared in the spectra. For example, fourclosely spaced large peaks and an additional small peak atfrequency f esignificantly below f0can be distinguished at P=10 mW [Fig. 2(d)]. There is also a bump on the declining slope of the peak at f0, suggesting that at least one additional mode with frequency close to f0may be excited. This simple trend is reversed at excitation powers above 10 mW. The BLS spectra now exhibit only two peaks atfrequencies f 0andfe[Figs. 2(e) and2(f)]. These two peaks exhibit a nonlinear frequency shift with increasing P.I n addition, they broaden and become noticeably asymmetric.The asymmetry is especially pronounced for the peak at f 0, which clearly has a significantly steeper rising slope than thedeclining slope, characteristic for a nonlinear resonance. 36–38 B. Spatial characteristics of the parametrically excited modes To identify the normal modes associated with the observed spectral peaks, we performed spatially resolved measurements 094401-2PARAMETRIC EXCITATION OF EIGENMODES IN ... PHYSICAL REVIEW B 84, 094401 (2011) FIG. 3. (Color online) Left: Pseudocolor-coded maps of the BLS intensity. Right: One-dimensional cross sections of the maps along the major axis of the Py ellipse, as marked by the dashed lines. Panels (a)–(c) were acquired at the labeled frequency values correspondingto three different spectral peaks. atP=10 mW, where the largest numbers of peaks are observed. For each of the observed peaks, the excitationfrequency was fixed at twice its central frequency, and two-dimensional mapping of the BLS intensity was performed. Theprobing spot was scanned in x- and y-directions with a step size of 50 nm across a 500 by 1000 nm rectangular area covering thePy dot. The left-side panels in Fig. 3show pseudocolor-coded maps of the recorded BLS intensity. The right-side panels showone-dimensional cross sections of these maps along the majoraxis of the Py ellipse. It is important to note that the measuredtwo-dimensional maps and one-dimensional profiles representa result of convolution of the local dynamic magnetizationamplitude with the instrumental resolution function, resultingin a significant blurring of submicrometer spatial features. Figure 3(a) shows that the mode at frequency f 0=7 GHz has a half-sine profile along the major axis, and it does notexhibit any nodal lines. These characteristics indicate that itis the fundamental center mode of the Py dot. 39The profile of the mode at frequency f1=7.7 GHz [Fig. 3(b)] has two maxima on the long axis and a minimum at the center of thedot. This minimum is likely associated with the nodal lineof the eigenmode. The BLS intensity does not vanish at theminimum, likely due to the limited spatial resolution of ourtechnique. The spatial profile of the mode at f 2=8.3 GHz [see Fig. 3(c)] has a maximum at the center, similar to the fundamental mode. In contrast to that mode, the profile issharper near the maximum, and it forms two broad shoulderswith small bumps near the edges. As mentioned previously,fine spatial features are blurred due to finite resolution ofthe setup. Therefore, based on our data, the mode with thefrequency f 2can be interpreted as a mode with two nodal lines separating a central maximum from two side maximalocated on the major axis of the Py ellipse. The limitations of the spatial resolution of our technique prevented us from identifying the mode corresponding to thepeak at f 3. This higher-order mode likely has three nodallines. We also performed spatially resolved measurements at fe=5.5 GHz, which revealed a typical spatial structure of the so-called edge mode, with maxima of intensity close tothe edges of the dot on the axis parallel to the direction of thestatic field, and a vanishing intensity at the center of the dot. We note that the mode observed at f 1=7.7 GHz [Fig. 3(b)] is expected to have odd spatial symmetry, i.e., its amplitudeprofile is antisymmetric with respect to the minor axis ofthe dot. By symmetry, this mode cannot be directly excitedby the usual linear excitation mechanism with a spatiallyuniform dynamic magnetic field. The symmetry of the modeatf 2does not prohibit its linear excitation by a uniform field, but the excitation efficiency would be significantlysmaller than for the fundamental mode at f 0.40In contrast, the efficiency of parametric excitation for all these modes issimilar, as indicated by the similar amplitudes of the peaksin Fig. 2(d). Therefore, the parametric excitation mechanism presents significant advantages compared to linear excitationfor the experimental studies of the eigenmode spectra in micro-and nanomagnets. C. Dependence of the mode intensities on pumping power We now analyze and interpret the dependencies of the parametrically excited mode intensities on the pumping power.Figure 4shows these dependencies for the fundamental mode (filled triangles), the higher-order mode at f 2(open triangles), and the edge mode at fe(circles). Both of the center modes exhibit similar nonmonotonic behavior above their excitationthresholds: The intensities first increase and then start todecrease with further increases in the pumping power. Theintensity of the fundamental mode reaches a minimum atP≈6 mW and then increases again. In contrast, the BLS peak atf 2becomes indistinguishable from the background at P= 12 mW and does not recover at larger pumping powers. Similarbehaviors were also observed for the higher-order modes atfrequencies f 1andf3. These data suggest the presence of a mechanism limiting the energy flow from the parametric pumping to the observed FIG. 4. (Color online) Dependence of the BLS peak intensity on pumping power for the fundamental mode at f0(filled triangles), for the higher-order mode at f2(open triangles), and for the edge mode atfe(open circles). 094401-3ULRICHS, DEMIDOV , DEMOKRITOV , AND URAZHDIN PHYSICAL REVIEW B 84, 094401 (2011) modes at large P. Although this mechanism influences all the observed modes, its effect on the higher-order modesis stronger than on the fundamental mode, leading to theircomplete suppression. To interpret these behaviors, we recall that because of the intrinsic anisotropy of the magnetic eigenmode spectrum, thefrequencies of the modes only weakly depend on the numberof nodal lines perpendicular to the magnetization. 41As a consequence, for each mode with nodal lines parallel to thestatic magnetization (see Fig. 3), there are, generally speaking, many nearly degenerate modes with a finite number of nodallines perpendicular to the magnetization. For example, atfrequencies f 1–f3, there are a number of exchange-dominated modes with very short effective wavelengths in the directionparallel to magnetization. These modes cannot be detected bythe BLS technique, which is sensitive predominantly to thelong-wavelength modes. The exchange-dominated modes are generally character- ized by stronger damping and weaker coupling to the pumpingfield, and consequently they have larger excitation thresholdscompared to the dipole-dominated modes. 41Therefore, only the dipole-dominated modes are excited at small P, and, in this regime, their intensity increases with P.A sPreaches the threshold value for the excitation of the exchange-dominatedmodes, additional scattering channels become effective thatredistribute the energy among the modes. While the details ofthese processes are presently unknown, one can generally ex-pect that the increase in the amplitudes of exchange-dominatedmagnetization oscillations results in nonlinear scattering of thedipole-dominated oscillations into the short-wavelength partof the mode spectrum, creating additional nonlinear dampingchannels for the dipole-dominated modes. As a result, theflow of energy from the pumping to the dipole-dominatedmodes decreases, leading to a decrease of their intensity and, atsufficiently large pumping power, to the complete suppressionof the dipole-dominated modes. This suppression mechanism is significantly less efficient for the fundamental mode, since it has the lowest frequencyamong the center modes, and consequently there are noexchange-dominated modes at the same frequency. Neverthe-less, there are a number of modes with no nodal lines parallel tothe magnetization and several nodal lines perpendicular to themagnetization whose frequency is only slightly different fromthat of the fundamental mode. The onset of their parametricexcitation can be the origin of the decrease in the fundamentalpeak intensity at P>3m W . These modes are dipole dominated and thus should be detectable by the BLS measurements. Indeed, the broadeningof the fundamental peak at P>3 mW [compare Fig. 2(a)and 2(c)] and a bump on its declining slope [see Fig. 2(d)] can be interpreted as a signature of their excitation. In addition,in the interval P=3–8 mW, spatially resolved measurements revealed deviations of the spatial profile of the mode at f 0 from that shown in Fig. 3(a) forP=10 mW, which can be associated with simultaneous excitation of several dipole-dominated modes with different spatial profiles. The largestdeviations were observed at P=6 mW, corresponding to the minimum of the fundamental mode intensity. These deviationsare dramatically reduced at P>10 mW. Based on these data, one can conclude that, in contrast to the competition betweenthe dipole-dominated and exchange-dominated modes, the competition among the dipole-dominated modes results in thepredominant energy flow into the fundamental mode of the dotat large P. Finally, the intensity of the edge mode (circles in Fig. 4) increases monotonically with increasing P. This behavior is consistent with the intensity-suppression mechanisms dis-cussed already: since the frequency of the edge mode liesfar below the frequencies of all the other modes of the system,its intensity is not affected by their parametric excitation. D. Temporal characteristics of parametric excitation In addition to the significance of parametric excitation as a spectroscopic tool, it can be used to determine otherimportant dynamical parameters of the magnetic system. Thetime dependence of the excited mode amplitude at differentpumping powers provides information about the magneticdamping constant, the strength of the microwave pumpingfield, and its coupling to the magnetic system (see Sec. IVfor details). We performed time-resolved measurements of the fun- damental mode intensity at pumping powers between thethreshold value of 1 mW and 50 mW, with temporal reso-lution of 1 ns. Figures 5(a) and5(b) show time traces for P=1mW and 3.2 mW. These data demonstrate that just above the parametric threshold, the rate of intensity growth is small,but it quickly increases with increasing P. Plotting the time-dependent intensity of the fundamental mode on the logarithmic scale, at P=1–4 mW, we observe a well-defined initial exponential rise followed by saturation, asillustrated in Fig. 5(c)forP=3.2 mW. Fitting this exponential dependence, we obtain a characteristic rise time constant τas a function of the pumping power for P<4 mW. At larger powers, P>4 mW, the intensity growth becomes too abrupt to make a reliable estimate of τdue to the limited temporal resolution FIG. 5. (Color online) (a) and (b) Time traces of the fundamental mode intensity at the labeled values of pumping power; t=0 corresponds to the start of the pumping pulse. (c) Time dependence of intensity on the logarithmic scale at P=3.2 mW. Line shows the result of a fit by an exponential function. (d) The inverse of amplitude rise time constant vs√ P∝h. Line is the best linear fit of the data. 094401-4PARAMETRIC EXCITATION OF EIGENMODES IN ... PHYSICAL REVIEW B 84, 094401 (2011) of our measurement. Analysis given in Sec. IVsuggests that the inverse of the time constant τshould depend linearly on h, which is proportional to the square root of the pumping power, h=A√ P. Here, Ais a calibration parameter determined by the sample geometry and the microwave losses in thetransmission line. As expected, the experimentally determined values of 1 /τfollow a linear dependence on√ P[Fig. 5(d)]. IV. THEORY Rigorous understanding of the nonlinear dynamical regimes in microscopic structures requires a self-consistent theoryof parametric excitation taking into account the effects ofthe inhomogeneity of the internal demagnetizing field andthe magnetization in the sample, as well as the boundaryconditions governing spin-wave quantization. Nevertheless,the theory developed for extended magnetic films 41can still be used to analyze the behavior of the studied system close tothe threshold of parametric excitation. According to Ref. 41, the threshold amplitude of the dynamic magnetic field for theonset of parametric excitation is given by h th=ωr/V, (1) where ωr=αω is the relaxation frequency, and V= γ24πMs[P(k)(1+sin2(ϕ))−1]/(4ω) is a coefficient char- acterizing the coupling of the pumping field to the planewave with frequency ωand wave vector koriented in the film plane at an angle ϕwith respect to the direction of the static magnetization: ϕ=tan −1(ky/kx). Here, 4 πM sis the saturation magnetization, and P(k)=1−(1−exp[−kd])/kd, where d is the film thickness. To account for the finite lateral dimensions of the dot, we applied the standard spin-wave quantization scheme.42Within this approach, the fundamental center mode of the dot isapproximated by a two-dimensional standing spin wave withthe components of the wave vector k x=π/aandky=π/b, where a=1000 nm and b=500 nm, which represent the lateral sizes of the dot in the xandydirections, respectively. In this approximation, the coupling coefficient is V=1.63× 107(Oe·s)−1.Above the threshold, the amplitude of the parametrically excited mode is expected to grow exponentially with acharacteristic time constant (see Ch. 5.3 in Ref. 43) τ=1/(hV−ω r). (2) In agreement with this result, the experimental values for 1/τscale linearly with h[Fig. 5(d)]. As follows from Eq. ( 2), 1/τis equal to the relaxation frequency ωrath=0 and vanishes at h=hth. Fitting the experimental data of Fig. 5(d) with a linear function and extrapolating to P=0, we obtain ωr=0.36×109s−1, corresponding to the Gilbert damping parameter α=0.008, which is in excellent agreement with the known value for Py.44From the same fit, we also obtain the threshold power Pth=0.6 mW, corresponding to exact compensation of the magnetic relaxation by the parametricpumping. Finally, from the slope of the linear dependence, weobtain the calibration factor A=29 Oe/(mW) 1/2. This value is in reasonable agreement with the estimate A=24 Oe/(mW)1/2 based on the nominal geometrical parameters of the microstrip line. V. CONCLUSIONS In conclusion, we have demonstrated parametric excitation of spin-wave modes in microscopic magnetic-film structuresat moderate microwave powers. Parametric processes canbe utilized for studies of the eigenmode spectra and otherdynamical characteristics in micro- and nanomagnets. 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PhysRevB.78.020404.pdf

Inhomogeneous Gilbert damping from impurities and electron-electron interactions E. M. Hankiewicz,1,2,*G. Vignale,2and Y. Tserkovnyak3 1Department of Physics, Fordham University, Bronx, New York 10458, USA 2Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA 3Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA /H20849Received 7 April 2008; revised manuscript received 3 June 2008; published 22 July 2008 /H20850 We present a unified theory of magnetic damping in itinerant electron ferromagnets at order q2including electron-electron interactions and disorder scattering. We show that the Gilbert damping coefficient can beexpressed in terms of the spin conductivity, leading to a Matthiessen-type formula in which disorder andinteraction contributions are additive. In a weak ferromagnet regime, electron-electron interactions lead to astrong enhancement of the Gilbert damping. DOI: 10.1103/PhysRevB.78.020404 PACS number /H20849s/H20850: 76.50. /H11001g, 75.45. /H11001j, 75.30.Ds I. INTRODUCTION In spite of much effort, a complete theoretical description of the damping of ferromagnetic spin waves in itinerant elec-tron ferromagnets is not yet available. 1Recent measurements of the dispersion and damping of spin-wave excitationsdriven by a direct spin-polarized current prove that the the-oretical picture is incomplete, particularly when it comes tocalculating the linewidth of these excitations. 2One of the most important parameters of the theory is the so-called Gil-bert damping parameter /H9251,3which controls the damping rate and thermal noise and is often assumed to be independent ofthe wave vector of the excitations. This assumption is justi-fied for excitations of very long wavelength /H20849e.g., a homoge- neous precession of the magnetization /H20850, where /H9251can origi- nate in a relatively weak spin-orbit /H20849SO/H20850interaction.4 However, it becomes dubious as the wave vector qof the excitations grows. Indeed, both electron-electron /H20849e-e/H20850and electron-impurity interactions can cause an inhomogeneous magnetization to decay into spin-flipped electron-hole pairs,giving rise to a q 2contribution to the Gilbert damping. In practice, the presence of this contribution means that theLandau-Lifshitz-Gilbert equation contains a term propor-tional to − m/H11003/H11612 2/H11509tm/H20849where mis the magnetization /H20850and requires neither spin-orbit nor magnetic disorder scattering.In contrast, the homogeneous damping term is of the formm/H11003 /H11509tmand vanishes in the absence of SO or magnetic dis- order scattering. The influence of disorder on the linewidth of spin waves in itinerant electron ferromagnets was discussed in Refs.5–7, and the role of e-einteractions in spin-wave damping was studied in Refs. 8and9for spin-polarized liquid He 3 and in Refs. 10and11for two- and three-dimensional /H208493D/H20850 electron liquids, respectively. In this Rapid Communication,we present a unified semiphenomenological approach, whichenables us to calculate on equal footing the contributions ofdisorder and e-einteractions to the Gilbert damping param- eter to order q 2. The main idea is to apply to the transverse spin fluctuations of a ferromagnet the method first introducedby Mermin 12for treating the effect of disorder on the dynam- ics of charge-density fluctuations in metals.13Following this approach, we will show that the q2contribution to the damp- ing in itinerant electron ferromagnets can be expressed interms of the transverse spin conductivity, which in turn sepa- rates into a sum of disorder and e-eterms. A major technical advantage of this approach is that the ladder vertex corrections to the transverse spin conductivityvanish in the absence of SO interactions, making the dia-grammatic calculation of this quantity a straightforward task.Thus we are able to provide explicit analytic expressions forthe disorder and interaction contribution to the q 2Gilbert damping to the lowest order in the strength of the interac-tions. This Rapid Communication connects and unifies dif-ferent approaches and gives a rather complete and simpletheory of q 2damping. In particular, we find that for weak metallic ferromagnets the q2damping can be strongly en- hanced by e-einteractions, resulting in a value comparable to or larger than typical in the case of homogeneous damp-ing. Therefore, we believe that the inclusion of a dampingterm proportional to q 2in the phenomenological Landau- Lifshitz equation of motion for the magnetization14is a po- tentially important modification of the theory in stronglyinhomogeneous situations, such as current-driven nano-magnets 2and the ferromagnetic domain-wall motion.15,16 II. PHENOMENOLOGICAL APPROACH In Ref. 12, Mermin constructed the density-density re- sponse function of an electron gas in the presence of impu-rities through the use of a local drift-diffusion equation,whereby the gradient of the external potential is cancelled, inequilibrium, by an opposite gradient of the local chemicalpotential. In diagrammatic language, the effect of the localchemical potential corresponds to the inclusion of the vertexcorrection in the calculation of the density-density responsefunction. Here, we use a similar approach to obtain the trans-verse spin susceptibility of an itinerant electron ferromagnet,modeled as an electron gas whose equilibrium magnetizationis along the zaxis. Before proceeding we need to clarify a delicate point. The homogeneous electron gas is not spontaneously ferromag-netic at the densities that are relevant for ordinary magneticsystems. 13In order to produce the desired equilibrium mag- netization, we must therefore impose a static fictitious fieldB 0. Physically, B0is the “exchange” field Bexplus any external/applied magnetic field B0appwhich may be addition-PHYSICAL REVIEW B 78, 020404 /H20849R/H20850/H208492008 /H20850RAPID COMMUNICATIONS 1098-0121/2008/78 /H208492/H20850/020404 /H208494/H20850 ©2008 The American Physical Society 020404-1ally present. Therefore, in order to calculate the transverse spin susceptibility we must take into account the fact that theexchange field associated with a uniform magnetization isparallel to the magnetization and changes direction when thelatter does. As a result, the actual susceptibility /H9273ab/H20849q,/H9275/H20850 differs from the susceptibility calculated at constant B0, which we denote by /H9273˜ab/H20849q,/H9275/H20850, according to the well-known relation11 /H9273ab−1/H20849q,/H9275/H20850=/H9273˜ab−1/H20849q,/H9275/H20850−/H9275ex M0/H9254ab. /H208491/H20850 Here, M0is the equilibrium magnetization /H20849assumed to point along the zaxis /H20850and/H9275ex=/H9253Bex/H20849where /H9253is the gyromagnetic ratio /H20850is the precession frequency associated with the ex- change field. /H9254abis the Kronecker delta. The indices aandb denote directions /H20849xory/H20850perpendicular to the equilibrium magnetization and qand/H9275are the wave vector and the fre- quency of the external perturbation. Here we focus solely on the calculation of the response function /H9273˜because term /H9275ex/H9254ab/M0does not contribute to Gilbert damping. We do not include the effects of exchange and external fields on theorbital motion of the electrons. The generalized continuity equation for the Fourier com- ponent of the transverse spin density M ain the direction a/H20849x ory/H20850at wave vector qand frequency /H9275is −i/H9275Ma/H20849q,/H9275/H20850=−i/H9253q·ja/H20849q,/H9275/H20850−/H92750/H9280abMb/H20849q,/H9275/H20850 +/H9253M0/H9280abBbapp/H20849q,/H9275/H20850, /H208492/H20850 where Baapp/H20849q,/H9275/H20850is the transverse external magnetic field driving the magnetization and /H92750is the precessional fre- quency associated with a static magnetic field B0/H20849including exchange contribution /H20850in the zdirection. jais the ath com- ponent of the transverse spin-current-density tensor and weset/H6036=1 throughout. The transverse Levi-Civita tensor /H9280ab has components /H9280xx=/H9280yy=0,/H9280xy=−/H9280yx=1, and the summation over repeated indices is always implied. The transverse spin current is proportional to the gradient of the effective magnetic field, which plays the role analo-gous to the electrochemical potential, and the equation thatexpresses this proportionality is the analog of the drift-diffusion equation of the ordinary charge transport theory, j a/H20849q,/H9275/H20850=iq/H9268/H11036/H20875/H9253Baapp/H20849q,/H9275/H20850−Ma/H20849q,/H9275/H20850 /H9273˜/H11036/H20876, /H208493/H20850 where /H9268/H11036/H20849=/H9268xxor/H9268yy/H20850is the transverse dc /H20849i.e.,/H9275=0/H20850spin conductivity and /H9273˜/H11036=M0//H92750is the static transverse spin sus- ceptibility in the q→0 limit.17Just as in the ordinary drift- diffusion theory, the first term on the right-hand side of Eq./H208493/H20850is a “drift current” and the second is a “diffusion current,” with the two canceling out exactly in the static limit /H20849forq →0/H20850, due to the relation M a/H208490,0 /H20850=/H9253/H9273˜/H11036Baapp/H208490,0 /H20850. Combin- ing Eqs. /H208492/H20850and /H208493/H20850gives the following equation for the transverse magnetization dynamics:/H20873−i/H9275/H9254ab+/H9253/H9268/H11036q2 /H9273˜/H11036/H9254ab+/H92750/H9280ab/H20874Mb =/H20849M0/H9280ab+/H9253/H9268/H11036q2/H9254ab/H20850/H9253Bbapp, /H208494/H20850 which is most easily solved by transforming to the circularly polarized components M/H11006=Mx/H11006iMy, in which the Levi- Civita tensor becomes diagonal, with eigenvalues /H11006i. Solv- ing in the “+” channel, we get M+=/H9253/H9273˜+−B+app=M0−i/H9253/H9268/H11036q2 /H92750−/H9275−i/H9253/H9268/H11036q2/H92750/M0/H9253B+app, /H208495/H20850 from which we obtain to the leading order in /H9275andq2 /H9273˜+−/H20849q,/H9275/H20850/H11229M0 /H92750/H208731+/H9275 /H92750/H20874+i/H9275/H9253/H9268/H11036q2 /H927502. /H208496/H20850 The higher-order terms in this expansion cannot be legiti- mately retained within the accuracy of the present approxi-mation. We also disregard the q 2correction to the static sus- ceptibility, since in making the Mermin ansatz /H208493/H20850we are omitting the equilibrium spin currents responsible for thelatter. Equation /H208496/H20850, however, is perfectly adequate for our purpose, since it allows us to identify the q 2contribution to the Gilbert damping, /H9251=/H927502 M0lim /H9275→0Im/H9273˜+−/H20849q,/H9275/H20850 /H9275=/H9253/H9268/H11036q2 M0. /H208497/H20850 Therefore, the Gilbert damping can be calculated from the dc transverse spin conductivity /H9268/H11036, which in turn can be com- puted from the zero-frequency limit of the transverse spin-current–spin-current response function, /H9268/H11036=−1 m/H115692Vlim /H9275→0Im/H20855/H20855/H20858i=1NSˆiapˆia;/H20858i=1NSˆiapˆia/H20856/H20856/H9275 /H9275, /H208498/H20850 where Sˆiais the xorycomponent of the spin operator for the ith electron, pˆiais the corresponding component of the mo- mentum operator, m/H11569is the effective electron mass, Vis the system volume, Nis the total electron number, and /H20855/H20855Aˆ;Bˆ/H20856/H20856/H9275 represents the retarded linear-response function for the ex- pectation value of an observable Aˆunder the action of a field that couples linearly to an observable Bˆ. Both disorder and e-einteraction contributions can be systematically included in the calculation of the spin-current–spin-current responsefunction. In the absence of spin-orbit and e-einteractions, the ladder vertex corrections to the conductivity are absentand calculation of /H9268/H11036reduces to the calculation of a single bubble with Green’s functions, G↑,↓/H20849p,/H9275/H20850=1 /H9275−/H9255p+/H9255F/H11006/H92750/2+i/2/H9270↑,↓, /H208499/H20850 where the scattering time /H9270sin general depends on the spin band index s=↑,↓. In the Born approximation, the scattering rate is proportional to the electron density of states, and wecan write /H9270↑,↓=/H9270/H9263//H9263↑,↓, where /H9263sis the spin- sdensity of states and /H9263=/H20849/H9263↑+/H9263↓/H20850/2./H9270parametrizes the strength of theHANKIEWICZ, VIGNALE, AND TSERKOVNYAK PHYSICAL REVIEW B 78, 020404 /H20849R/H20850/H208492008 /H20850RAPID COMMUNICATIONS 020404-2disorder scattering. A standard calculation then leads to the following result: /H9268/H11036dis=/H9271F↑2+/H9271F↓2 6/H20849/H9263↓−1+/H9263↑−1/H208501 /H927502/H9270. /H2084910/H20850 This, inserted in Eq. /H208497/H20850, gives a Gilbert damping param- eter in full agreement with what we have also calculatedfrom a direct diagrammatic evaluation of the transverse spinsusceptibility, i.e., spin-density–spin-density correlationfunction. From now on, we shall simplify the notation byintroducing a transverse spin-relaxation time, 1 /H9270/H11036dis=4EF↑+EF↓ 3n/H20849/H9263↓−1+/H9263↑−1/H208501 /H9270, /H2084911/H20850 where EFs=m/H11569/H9271Fs2/2 is the Fermi energy for spin- selectrons andnis the total electron density. In this notation, the dc transverse spin conductivity takes the form /H9268/H11036dis=n 4m/H11569/H9275021 /H9270/H11036dis. /H2084912/H20850 III. ELECTRON-ELECTRON INTERACTIONS One of the attractive features of the approach based on Eq. /H208498/H20850is the ease with which e-einteractions can be in- cluded. In the weak-coupling limit, the contributions of dis-order and e-einteractions to the transverse spin conductivity are simply additive. We can see this by using twice the equa-tion of motion for the spin-current–spin-current responsefunction. This leads to an expression for the transverse spinconductivity /H20851Eq. /H208498/H20850/H20852in terms of the low-frequency spin- force–spin-force response function, /H9268/H11036=−1 m/H115692/H927502Vlim /H9275→0Im/H20855/H20855/H20858iSˆiaFˆia;/H20858iSˆiaFˆia/H20856/H20856/H9275 /H9275./H2084913/H20850 Here, Fˆia=pˆ˙iais the time derivative of the momentum opera- tor, i.e., the operator of the force on the ith electron. The total force is the sum of electron-impurity and e-einteraction forces. Each of them, separately, gives a contribution of or-der /H20841 vei/H208412and /H20841vee/H208412, where veiandveeare matrix elements of the electron-impurity and e-einteractions, respectively, while cross terms are of higher order, e.g., vee/H20841vei/H208412. Thus, the two interactions give additive contributions to the conductivity.In Ref. 18, a phenomenological equation of motion was used to find the spin current in a system with disorder and longi-tudinal spin-Coulomb drag coefficient. We can use a similarapproach to obtain transverse spin currents with transverse spin-Coulomb drag coefficient 1 / /H9270/H11036ee. In the circularly polar- ized basis, i/H20849/H9275/H11007/H92750/H20850j/H11006=−nE/H11006 4m/H11569+j/H11006 /H9270/H11036dis+j/H11006 /H9270/H11036ee, /H2084914/H20850 and correspondingly the spin conductivities are /H9268/H11006=n 4m/H115691 −/H20849/H9275/H11007/H92750/H20850i+1 //H9270/H11036dis+1 //H9270/H11036ee. /H2084915/H20850 In the dc limit, this gives/H9268/H11036/H208490/H20850=/H9268++/H9268− 2=n 4m/H115691//H9270/H11036dis+1 //H9270/H11036ee /H927502+/H208491//H9270/H11036dis+1 //H9270/H11036ee/H208502. /H2084916/H20850 Using Eq. /H2084916/H20850, an identification of the e-econtribution is possible in a perturbative regime where 1 //H9270/H11036eeand 1 //H9270/H11036dis /H11270/H92750, leading to the following formula: /H9268/H11036=n 4m/H11569/H927502/H208731 /H9270/H11036dis+1 /H9270/H11036ee/H20874. /H2084917/H20850 Comparison with Eq. /H2084913/H20850enables us to immediately identify the microscopic expressions for the two scatteringrates. For the disorder contribution, we recover what we al-ready knew, i.e., Eq. /H2084911/H20850. For the e-einteraction contribu- tion, we obtain 1 /H9270/H11036ee=−4 nm/H11569Vlim /H9275→0Im/H20855/H20855/H20858iSˆiaFˆ iaC;/H20858iSˆiaFˆ iaC/H20856/H20856/H9275 /H9275,/H2084918/H20850 where FCis just the Coulomb force, and the force-force cor- relation function is evaluated in the absence of disorder. Thecorrelation function in Eq. /H2084918/H20850is proportional to the func- tionF +−/H20849/H9275/H20850which appeared in Ref. 11/H20851Eqs. /H2084918/H20850and /H2084919/H20850/H20852in a direct calculation of the transverse spin susceptibility. Mak-ing use of the analytic result for ImF +−/H20849/H9275/H20850presented in Eqs. /H2084921/H20850and /H2084924/H20850of that paper we obtain 1 /H9270/H11036ee=/H9003/H20849p/H208508/H92510 27T2rs4m/H11569a/H115692kB2 /H208491+p/H208501/3, /H2084919/H20850 where Tis the temperature, p=/H20849n↑−n↓/H20850/nis the degree of spin polarization, a/H11569is the effective Bohr radius, rsis the dimensionless Wigner-Seitz radius, /H92510=/H208494/9/H9266/H208501/3, and /H9003/H20849p/H20850—a dimensionless function of the polarization p—is de- fined by Eq. /H2084923/H20850of Ref. 11. This result is valid to second order in the Coulomb interaction. Collecting our results, wefinally obtain a full expression for the q 2Gilbert damping parameter, /H9251=/H9253nq2 4m/H11569M01//H9270/H11036dis+1 //H9270/H11036ee /H927502+/H208491//H9270/H11036dis+1 //H9270/H11036ee/H208502. /H2084920/H20850 One of the salient features of Eq. /H2084920/H20850is that it scales as the total scattering ratein the weak disorder and e-einteraction limit, while it scales as the scattering time in the opposite limit. The approximate formula for the Gilbert damping inthe more interesting weak-scattering/strong-ferromagnet re-gime is /H9251=/H9253nq2 4m/H11569/H927502M0/H208731 /H9270/H11036dis+1 /H9270/H11036ee/H20874, /H2084921/H20850 while in the opposite limit, i.e., for /H92750/H112701//H9270/H11036dis,1 //H9270/H11036ee, /H9251=/H9253nq2 4m/H11569M0/H208731 /H9270/H11036dis+1 /H9270/H11036ee/H20874−1 . /H2084922/H20850 Equation /H2084920/H20850agrees with the result of Singh and Tesanovic6 on the spin-wave linewidth as a function of the disorder strength and /H92750. However, Eq. /H2084920/H20850also describes the influ- ence of e-ecorrelations on the Gilbert damping. A compari- son of the scattering rates originating from disorder and e-eINHOMOGENEOUS GILBERT DAMPING FROM IMPURITIES … PHYSICAL REVIEW B 78, 020404 /H20849R/H20850/H208492008 /H20850RAPID COMMUNICATIONS 020404-3interactions shows that the latter is important and can be comparable or even greater than the disorder contribution forhigh-mobility and/or low-density 3D metallic samples. Fig-ure1shows the behavior of the Gilbert damping as a func- tion of the disorder scattering rate. One can see that the e-e scattering strongly enhances the Gilbert damping for smallpolarizations/weak ferromagnets /H20851see the red /H20849solid /H20850line /H20852. This stems from the fact that 1 / /H9270/H11036disis proportional to 1 //H9270and independent of polarization for small polarizations, while 1//H9270/H11036eeis enhanced by a large prefactor /H9003/H20849p/H20850=2/H9261//H208491−/H92612/H20850 +/H208491/2/H20850ln/H20851/H208491+/H9261/H20850//H208491−/H9261/H20850/H20852, where /H9261=/H208491−p/H208501/3//H208491+p/H208501/3.O n the other hand, for strong polarizations /H20849dotted and dash- dotted lines in Fig. 1/H20850, the disorder dominates in a broad range of 1 //H9270and the inhomogenous contribution to the Gil- bert damping is rather small. Finally, we note that our calcu-lation of the e-einteraction contribution to the Gilbert damp-ing is valid under the assumption of /H6036 /H9275/H11270kBT/H20849which is certainly the case if /H9275=0/H20850. More generally, as follows from Eqs. /H2084921/H20850and /H2084922/H20850of Ref. 11, a finite frequency /H9275can be included through the replacement /H208492/H9266kBT/H208502→/H208492/H9266kBT/H208502 +/H20849/H6036/H9275/H208502in Eq. /H2084919/H20850. Thus 1 //H9270/H11036eeis proportional to the scatter- ing rate of quasiparticles near the Fermi level, and our damp-ing constant in the clean limit becomes qualitatively similarto the damping parameter obtained by Mineev 9for/H9275corre- sponding to the spin-wave resonance condition in some ex-ternal magnetic field /H20849which in practice is much smaller than the ferromagnetic exchange splitting /H92750/H20850. IV. SUMMARY We have presented a unified theory of the Gilbert damp- ing in itinerant electron ferromagnets at the order q2, includ- inge-einteractions and disorder on equal footing. For the inhomogeneous dynamics /H20849q/HS110050/H20850, these processes add to a q=0 damping contribution that is governed by magnetic dis- order and/or spin-orbit interactions. We have shown that thecalculation of the Gilbert damping can be formulated in thelanguage of the spin conductivity, which takes an intuitiveMatthiessen form with the disorder and interaction contribu-tions being simply additive. It is still a common practice,e.g., in the micromagnetic calculations of spin-wave disper-sions and linewidths, to use a Gilbert damping parameterindependent of q. However, such calculations are often at odds with experiments on the quantitative side, particularlywhere the linewidth is concerned. 2We suggest that the inclu- sion of the q2damping /H20849as well as the associated magnetic noise /H20850may help in reconciling theoretical calculations with experiments. ACKNOWLEDGMENTS This work was supported in part by the NSF under Grants No. DMR-0313681 and No. DMR-0705460 as wellas Fordham Research Grant. Y.T. thanks A. Brataas andG. E. W. Bauer for useful discussions. *hankiewicz@fordham.edu 1Y. Tserkovnyak, A. Brataas, G. E. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 /H208492005 /H20850. 2I. N. Krivorotov, D. V. Berkov, N. L. Gorn, N. C. Emley, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Phys. Rev. B 76, 024418 /H208492007 /H20850. 3T. L. Gilbert, IEEE Trans. Magn. 40, 3443 /H208492004 /H20850. 4E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys. Rev. B75, 174434 /H208492007 /H20850. 5A. Singh, Phys. Rev. B 39, 505 /H208491989 /H20850. 6A. Singh and Z. Tesanovic, Phys. Rev. B 39, 7284 /H208491989 /H20850. 7V. L. Safonov and H. N. Bertram, Phys. Rev. B 61, R14893 /H208492000 /H20850. 8V. P. Silin, Sov. Phys. JETP 6, 945 /H208491958 /H20850. 9V. P. Mineev, Phys. Rev. B 69, 144429 /H208492004 /H20850. 10Y. Takahashi, K. Shizume, and N. Masuhara, Phys. Rev. B 60, 4856 /H208491999 /H20850. 11Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404 /H208492002 /H20850.12N. D. Mermin, Phys. Rev. B 1, 2362 /H208491970 /H20850. 13G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid /H20849Cambridge University Press, Cambridge, UK, 2005 /H20850. 14E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2 , Course of Theoretical Physics Vol. 9, 3rd ed. /H20849Pergamon, Ox- ford, 1980 /H20850. 15Y. Tserkovnyak, A. Brataas, and G. E. Bauer, J. Magn. Magn. Mater. 320, 1282 /H208492008 /H20850, and references therein. 16In ferromagnets which nonuniformities are beyond the linearized spin waves, there is a nonlinear q2contribution to damping /H20851see J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,arXiv:0803.2175 /H20849unpublished /H20850/H20852which has a different physical origin, related to the longitudinal spin-current fluctuations. 17Although both /H9268/H11036and/H9273˜/H11036are in principle tensors in transverse spin space, they are proportional to /H9254abin axially symmetric systems; hence we use scalar notation. 18I. D’Amico and G. Vignale, Phys. Rev. B 62, 4853 /H208492000 /H20850.FIG. 1. /H20849Color online /H20850The Gilbert damping /H9251as a function of the disorder scattering rate 1 //H9270. The red /H20849solid /H20850line shows the Gil- bert damping for polarization p=0.1 in the presence of the e-eand disorder scattering, while the dashed line does not include thee-escattering. The blue /H20849dotted /H20850and black /H20849dash-dotted /H20850lines show Gilbert damping for p=0.5 and p=0.99, respectively. We took q=0.1 k F,T=54 K, /H92750=EF/H20851/H208491+p/H208502/3−/H208491−p/H208502/3/H20852,M0=/H9253pn /2, m/H11569=me,n=1.4/H110031021cm−3,rs=5, and a/H11569=2a0.HANKIEWICZ, VIGNALE, AND TSERKOVNYAK PHYSICAL REVIEW B 78, 020404 /H20849R/H20850/H208492008 /H20850RAPID COMMUNICATIONS 020404-4

PhysRevB.87.174409.pdf

PHYSICAL REVIEW B 87, 174409 (2013) Spin-transfer torques in helimagnets Kjetil M. D. Hals and Arne Brataas Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (Received 24 March 2013; published 6 May 2013) We theoretically investigate current-induced magnetization dynamics in chiral-lattice helimagnets. Spin-orbit coupling in noncentrosymmetric crystals induces a reactive spin-transfer torque that has not been previouslyconsidered. We demonstrate how the torque is governed by the crystal symmetry and acts as an effectivemagnetic field along the current direction in cubic B20-type crystals. The effects of the new torque are computedfor current-induced dynamics of spin spirals and the Doppler shift of spin waves. In current-induced spin-spiralmotion, the new torque tilts the spiral structure. The spin waves of the spiral structure are initially displaced bythe new torque, while the dispersion relation is unaffected. DOI: 10.1103/PhysRevB.87.174409 PACS number(s): 75 .78.−n, 75.60.Jk, 75.70.Tj, 75.70.Kw I. INTRODUCTION Current-induced magnetization dynamics continue to be a very active research area due to potential applications in futureelectronic devices. In metallic ferromagnets, the magnetizationcan be manipulated via the spin-transfer torque (STT), whicharises due to a misalignment between the spin polarizationof the current and the local magnetization direction. 1,2Slon- czewski and Berger were the first to predict the existence ofthe STT effect, 3,4which was later demonstrated in several experiments.1,2The anticipated application potential of the STT effect lies in the development of electromagnetic devicesthat utilize a current-induced torque instead of externalmagnetic fields to manipulate the magnetization. 2 The magnetization dynamics of an itinerant ferromagnet is described by the Landau-Lifshitz-Gilbert (LLG) equationextended to include the current-induced torques: 2,5 ˙m=−γm×Heff+αm×˙m+τ. (1) Here, m=M/Ms(Ms=|M|) is the unit direction vector of the magnetization M,Heff=−δF/δ Mis the effective field found by varying the free energy F[M] with respect to the magnetization, αis the Gilbert damping coefficient, γis (minus) the gyromagnetic ratio, and τdescribes the current-induced torques. In the absence of intrinsic spin-orbitcoupling (SOC), the torque becomes τ=τ ex:2 τex=− (1−βm×)(vs·∇)m. (2) In Eq. (2), the first term is the adiabatic torque, while the second term (parametrized by β) is the nonadiabatic torque. The vector vsis proportional to the current density Jand its polarization P:vs=− ¯hPJ/2es0. Here, s0is the total equilibrium spin density along −m, and eis the electron charge. The torque in Eq. (2)treats the ferromagnet within the exchange approximation, which assumes that the exchangeforces only depend on the relative orientation of the spins. Thisassumption is believed to be valid in metallic ferromagnets,including disordered systems in which impurities couple to thespin degree of freedom through random magnetic moments orspin-orbit coupling. In this case, impurity averaging restoresthe spin-rotational symmetry of the system. Recently, insystems with a broken spatial inversion symmetry, the intrinsicSOC in combination with an external electric field havebeen observed to induce an additional torque, such thatτ=τ ex+τso.6–10In general, the SOC-induced torque τso can be written as6,9,10 τso=−γm×Hso, (3) where the SOC field Hsois proportional to the electric field and its orientation is determined by the symmetry of the underlyingcrystal lattice and the direction of the external electric field.Therefore, in contrast to τ ex, which vanishes in a homogeneous ferromagnet, τsois finite even in this case. Several experiments have demonstrated that the SOC torque plays an important rolein magnetization dynamics. 7,8The underlying physics of the torque is that the SOC effectively acts as a magnetic field onthe spins of the itinerant quasiparticles when an electric fieldis applied to the system. The effective magnetic field inducesan out-of-equilibrium spin density that yields a torque on themagnetization. 6,7 In chiral magnets, the exchange interaction also contains an anisotropic term known as the Dzyaloshinskii-Moriya(DM) interaction. 11,12The DM interaction arises due to the characteristic crystalline asymmetry of the chiral magnet incombination with the SOC, and in cubic B20-type crystals, itleads to the formation of a spin spiral in the magnetic groundstate. We refer to these systems as helimagnets. Helimagnetshave recently attracted substantial interest because topologicalnontrivial spin structures, known as skyrmions, have beenobserved in such systems under the application of weakexternal magnetic fields. 13–19Current-induced responses of the formed skyrmion lattice to current densities that areover five orders of magnitude smaller than those typicallyobserved in conventional ferromagnetic metals have recentlybeen observed experimentally. 20,21To understand this striking feature of helimagnets, numerical simulations and a col-lective coordinate description have been applied to studythe current-induced dynamics of spin spirals and skyrmionlattices. 22–24However, an important aspect of helimagnets is the absence of spatial inversion symmetry, which implies thatthe magnetization experiences a SOC-induced torque given byEq.(3). In the present paper, we derive the form of the torque in Eq.(3)for cubic noncentrosymmetric (B20-type) compounds. An important example of such a system is the chiral itinerant-electron magnet MnSi, which was the first system in whicha two-dimensional lattice of skyrmions was observed. The 174409-1 1098-0121/2013/87(17)/174409(5) ©2013 American Physical SocietyKJETIL M. D. HALS AND ARNE BRATAAS PHYSICAL REVIEW B 87, 174409 (2013) effects of the SOC torque are studied for two different cases: current-induced spin-spiral dynamics and the Doppler shift ofmagnons that propagates along the spiral structure. We observethat for current-induced spin-spiral motion, the new torqueyields an enhanced tilting of the spiral structure, while thetorque does not affect the Doppler shift of spin waves exceptto induce an initial translation of the spiral structure. We alsobriefly discuss the effect of the SOC torque on the skyrmionlattice dynamics. This paper is organized in the following manner. Section II provides a derivation of the SOC torque in Eq. (3)for cubic B20-type crystals. Section IIIdiscusses the effects of the SOC torque on current-induced spin-spiral motion and the Dopplershift of spin waves that propagates along the spin spiral. Weconclude and summarize our results in Sec. IV. II. DERIV ATION OF THE SOC TORQUE In deriving the explicit form of the torque in Eq. (3), we are guided by the Onsager reciprocity relations andNeumann’s principle. Consider a system described by theparameters {q i|i=1,..., N }for which the rate of change ˙qiis induced by the thermodynamic forces fi≡−∂F/∂q i, where F(q1,..., q N) is the free energy. Onsager’s theorem states that the response coefficients in the equations ˙qi=/summationtextN j=1Lijfj are related by Lij(H,m)=/epsilon1i/epsilon1jLji(−H,−m), where /epsilon1i=1 (/epsilon1i=− 1) ifqiis even (odd) under time reversal.25mand Hrepresent any possible equilibrium magnetic order and an external magnetic field, respectively. In the present paper, theresponses of the itinerant ferromagnet are described by thetime derivative of the unit vector along the magnetizationdirection, ˙m, and the charge current density J. The associated thermodynamic forces are the effective field scaled with themagnetization, f m=MsHeff, and the electric field, fq=E, respectively, and the equations describing the dynamics in thelinear response regime are determined by /parenleftbigg˙m J/parenrightbigg =/parenleftbiggL mm Lmq Lqm Lqq/parenrightbigg/parenleftbiggfm fq/parenrightbigg . (4) The Onsager reciprocity relations imply that Lmiqj(m)= −Lqjmi(−m). In addition to the symmetry requirements imposed by the reciprocity relations, the symmetry of theunderlying lattice structure also decreases the number ofindependent tensor components. This fact is expressed byNeumann’s principle, which states that a tensor representingany physical property should be invariant with respect to everysymmetry operation of the crystal’s point group. 25 According to Eq. (4), the effect reciprocal to the adiabatic and nonadiabatic torque in Eq. (2)is a charge current density induced by a time-dependent magnetic texture. To the lowestorder in the texture gradients and the precession frequency, theinduced charge current density in the exchange approximationis 26 Jex i=¯h 2eσP/parenleftbigg m×∂m ∂ri−β∂m ∂ri/parenrightbigg ·˙m. (5) Here, eis the electron charge, Pis the spin polarization of the current, σis the conductivity, and riis component i of the spatial vector. Because the exchange approximationneglects any coupling (via intrinsic SOC) of the spins to thecrystal structure, the above expression is fully spin-rotationally symmetric and a textured magnetization, i.e., ∂m/∂r i/negationslash=0, is required to have a coupling between the momentum of theitinerant quasiparticles and the magnetization. If the effects ofintrinsic SOC are considered, additional terms are allowed bysymmetry in the phenomenological expansion for the pumpedcurrent. In particular, for inversion symmetry-breaking SOC,a homogenous magnetization pumps a charge current. To thelowest order in SOC and precession frequency, the expressionfor the pumped current then becomes J pump i=ηij˙mj+Jex i. (6) The second-rank tensor ηijis an axial tensor because the current is a polar vector while the magnetization is anaxial vector. η ijis linear in the SOC coupling and vanishes in systems with spatial inversion symmetry. According toNeumann’s principle, the particular form of η ijis governed by the crystal structure and is determined by the following setof equations produced by the generating matrices [ R ij]o ft h e crystal’s point group:25 ηij=|R|RinRjmηnm. (7) Here,|R|is the determinant of the matrix [ Rij]. Let us now consider a cubic B20-type crystal. Its crystal structure belongs to the noncentrosymmetric space groupP2 13, which has the cubic point group T. Common examples of cubic B20-type chiral magnets are MnSi, FeGe, and(Fe,Co)Si. From the symmetry relations in Eq. (7), one then finds that the tensor η ijis proportional to the unit matrix:25 ηij=ηδij, (8) where δijis the Kronecker delta. The tensor is isotropic because the high symmetry of the cubic crystal reduces thenumber of independent tensor coefficients to the single param-eterη. Substituting this tensor into Eq. (6)and expressing the time derivative of the magnetization in terms of the effectivefield by applying the first term on the right-hand side of Eq. (1), one obtains the response matrix: 27 Lqimj=−γη Ms/epsilon1ikjmk+Lex qimj. (9) Here,Lex qimjare the response coefficients describing the process reciprocal to the STT in Eq. (2), which have been previously derived in Ref. 26. The term proportional to ηdescribes the process reciprocal to the SOC-induced torque in Eq. (3).U s i n g the Onsager reciprocity relations, we find that the SOC fieldtakes the following form: H so=ηsovs, (10) where ηso=(2ηes 0)/(¯hσPM s). Thus, the torque induced by the SOC in noncentrosymmetric cubic magnets acts as aneffective magnetic field along the current direction. Note thatthe torque is reactive because it does not break the time reversalsymmetry of the LLG equation. III. RESULTS AND DISCUSSION In the following, we investigate the effects of the SOC torque on current-driven spin-spiral motion and the Dopplershift of spin waves. Additionally, a brief discussion of how we 174409-2SPIN-TRANSFER TORQUES IN HELIMAGNETS PHYSICAL REVIEW B 87, 174409 (2013) expect the torque to affect the skyrmion crystal dynamics is presented. A. Spin-spiral motion To the lowest order in the magnetic texture gradients, the free energy density of a ferromagnet with broken spatialinversion symmetry can be written phenomenologically as: 28 F(m)=Jij 2∂m ∂ri·∂m ∂rj+Dijkmi∂mj ∂rk. (11) Here, Jijis the spin stiffness describing the exchange inter- action between neighboring magnetic moments, and the termproportional to D ijkis the DM interaction. In Eq. (11) (and in what follows), summation over repeated indices is implied.The explicit form of the tensors J ijandDijkis determined by the crystal symmetry. In cubic B20-type ferromagnets, the free energy density becomes F(m)=J 2∂m ∂ri·∂m ∂ri+Dm·(∇×m). (12) The free energy of the system, F[m]=/integraltext drF, is minimized by a helical magnetic order, where the wave vector of thespiral structure is determined by the ratio between the DMparameter and the spin stiffness, k=D/J .F o ra kvector that points along the zaxis, the magnetic order of the ground state is m 0(z)=cos(kz)ˆx+sin(kz)ˆy, (13) where ˆxand ˆyare the unit direction vectors along the xandy axes, respectively. The action functional S[m] and the dissipation functional R[˙m] of the system are written as29,30 S[m]=/integraldisplay dtdrAi(˙mi+vs·∇mi)+γ MsF(m)−γm·Hso, (14) R[˙m]=/integraldisplay dtdrα 2/parenleftbigg ˙m+β αvs·∇m/parenrightbigg2 . (15) Here, A(m) is the Berry phase vector potential of a magnetic monopole, which satisfies /epsilon1ijk∂Ak/∂m j=mi[/epsilon1ijkis the Levi- Civita tensor]. The LLG equation in Eq. (1), with τ=τex+ τso, is determined by δS δm=−δR δ˙m. (16) A previous study on spin-spiral motion demonstrated that the response of the structure to an applied current (along z) can be described by the tilting angle ξand drift velocity ˙ζof the spiral structure.22To find an approximate solution of Eq. (16), we therefore employ the following variational ansatz: m(z,t)=cos[ξ(t)]m0[z−ζ(t)]+sin[ξ(t)]ˆz. (17) Substitution of this ansatz into Eqs. (14) and (15) and integration over the spatial coordinates yield an effective actionand dissipation functional for the variational parameters ξ(t)andζ(t): S[ζ,ξ]=/integraldisplay dt(˙ζ−v s)ksinξ +γ Ms/parenleftbiggJ 2k2cos2ξ−Dkcos2ξ/parenrightbigg −γH sosinξ, (18) R[˙ζ,˙ξ]=/integraldisplay dtα 2/bracketleftBigg ˙ξ2+/parenleftbiggβ αvsk−k˙ζ/parenrightbigg2/bracketrightBigg . (19) The equations of motion for the variational parameters are δS[ζ,ξ] δζ=−δR[˙ζ,˙ξ] δ˙ζ,δS[ζ,ξ] δξ=−δR[˙ζ,˙ξ] δ˙ξ.(20) We are interested in the steady-state regime in which ξ approaches a constant value. In this regime, the drift velocityand the tilting angle are ˙ζ=β αvs, (21) sin(ξ)=Ms γ1 Jk−2D/bracketleftbigg/parenleftbiggβ α−1/parenrightbigg vs−γ kHso/bracketrightbigg .(22) The expression for the drift velocity ˙ζagrees with the expression derived in Ref. 22. The SOC torque does not affect the drift velocity because the SOC torque effectively actssimilarly to the adiabatic torque, as can be observed from theexpression for the action S[ζ,ξ]i nE q . (18). The adiabatic and SOC torques initiate a spiral motion when a current is applied.However, the motion is damped due to the intrinsic pinningeffect caused by Gilbert damping in combination with theDM interaction. Thus, similar to what is observed for domainwalls in conventional ferromagnets, a nonadiabatic torque isrequired to obtain a steady-state spiral motion. An observableeffect of the SOC torque is the modification of the tilting angleobserved in Ref. 22by an amount of −M sHso/(Jk2−2Dk). B. Doppler shift of spin waves In ferromagnets with a homogeneous magnetization, a Doppler shift in the spin-wave dispersion relation under theapplication of a current has been observed. 31The frequency ω of the spin wave is shifted by vs·q, where qis the wave vector of the magnon: ω=(γJ/M s)q2+vs·q. Theoretical works on Goldstone modes in helimagnets with a spin spiral predict that these modes are much morecomplicated than those in homogeneous ferromagnets. 32We refer to these Goldstone modes as helimagnons. The dispersionrelation of the helimagnons is highly anisotropic, with a linearwave-vector dependency parallel to the spin-spiral directionand a quadratic dependency in the transverse direction (in thelong wavelength limit). That is, the soft modes behave like an-tiferromagnetic magnons along the spiral, while ferromagneticbehavior is observed for modes propagating in the transverseplane. Thus far, no works have studied the effect of an appliedcurrent on the dispersion relation of helimagnons. To derive an effective action for the Goldstone modes, we describe the local fluctuations by ξandζin Eq. (17) by allowing the parameters to be both position and timedependent: ξ=ξ(r,t) andζ=ζ(r,t). A similar parametriza- tion was performed in Refs. 32and 33in the analysis of 174409-3KJETIL M. D. HALS AND ARNE BRATAAS PHYSICAL REVIEW B 87, 174409 (2013) helimagnons. The parameter ζdescribes a local twist (around thezaxis) of the spiral structure, while ξdescribes a local tilting along the zaxis. In the analysis of the Doppler shift, we neglect dissipation and disregard the dissipation function.Reference 33demonstrated that simple closed-form solutions for the variational parameters can only be obtained for modespropagating along the spin-spiral direction. For simplicity, wetherefore restrict our study to Goldstone modes that propagatealong the zaxis. Expanding Eq. (14) to second order in ξ(z,t) andζ(z,t), we obtain the effective action (the current is applied along the zaxis): S[ξ,ζ]=/integraldisplay dtdzkξ/parenleftbigg ˙ζ+v s∂ζ ∂z−vs/parenrightbigg +γJ 2Ms/bracketleftBigg/parenleftbigg∂ξ ∂z/parenrightbigg2 +k2/parenleftbigg∂ζ ∂z/parenrightbigg2 +k2ξ2/bracketrightBigg −γH soξ. (23) The equations of motion are obtained by varying the action with respect to ξandζ, i.e.,δS/δζ =δS/δξ =0, which results in two coupled equations for the variational parameters: ˙ζ(z,t)+vs∂ζ(z,t) ∂z =−γJ kMs/parenleftbigg k2−∂2 ∂z2/parenrightbigg ξ(z,t)+vs+γ kHso,(24) ˙ξ(z,t)+vs∂ξ(z,t) ∂z=−γJk Ms∂2ζ(z,t) ∂z2. (25) Let us first consider the homogenous part of the equations and neglect the two last terms on the right-hand side in Eq. (24). Substitution of a plane wave ansatz of the form [ζ0ξ0]Texp(i(qz−ωt))into the equations yields the follow- ing dispersion relation: ω=γJ Msq/radicalbig k2+q2+vsq. (26) We see that the STT results in a Doppler shift similar to what is observed for spin waves in conventional ferromagnets. Inthe long wavelength limit, q→0, a linear dispersion relation is obtained: ω=(γJ/M s)kq+vsq. The SOC-induced torque only appears as a source term in the nonhomogeneous equa-tions. The particular solution (PS) of the nonhomogeneousequations in Eqs. (24) and(25) is /parenleftbiggζ(z,t) ξ(z,t)/parenrightbigg PS=/parenleftBigg [vs+(γ/k)Hso]t 0/parenrightBigg . (27) This solution describes a displacement of the spiral structure induced by the adiabatic and SOC torques. However, thiscurrent-driven spin-spiral motion is damped when dissipationis considered due to the intrinsic pinning effect. Thus, theSOC torque (together with the adiabatic torque) only causesan initial translation of the spin spiral.C. Skyrmion crystal dynamics In helimagnetic thin-film systems, skyrmions have been observed under the application of a weak external magneticfield Bperpendicular to the thin film. Each skyrmion has a vortexlike magnetic configuration, where the magneticmoment at the core of the vortex is antiparallel to the appliedfield while the peripheral magnetic moments are parallel.From the peripheral moments to the core, the magneticmoments swirl up in a counterclockwise or clockwise manner.The formed skyrmions arrange themselves in a crystallinestructure, a two-dimensional skyrmion crystal. Recent experiments have revealed current-driven skyrmion crystal motion at ultralow current densities. 20T h em o t i o no fa skyrmion lattice is only weakly affected by pinning, which is instark contrast to observations for current-induced domain walldynamics in conventional ferromagnets. A theoretical workhas indicated that the pinning-free motion arises because theskyrmion lattice rotates and deforms to avoid the impurities. 24 However, all analyses of current-driven skyrmion crystalmotion have disregarded the SOC torque. Section IIshowed that the SOC torque acts as an effective field along the current direction. For a current applied alongany direction in the thin film, the expected consequence ofthe SOC torque is therefore that this torque leads to a smallcorrection to the external magnetic field that stabilizes the two-dimensional skyrmion lattice such that the total field becomesH T=B+Hso. The expected response of the skyrmion crystal to this perturbation is a rotation of the two-dimensional(2D) lattice structure that aligns the core magnetic momentsantiparallel to H T. To confirm our predictions, a more thorough numerical simulation of the magnetic system is required,which is beyond the scope of the present paper. IV . SUMMARY In this paper, we performed a theoretical study of current- induced torques in cubic noncentrosymmetric helimagnets. Wedemonstrated that due to the broken spatial inversion symme-try, the SOC induces a reactive magnetization torque. The spe-cific form of the SOC torque is determined by the symmetry ofthe underlying crystal lattice and acts as an effective magneticfield along the current direction in B20-type chiral magnets. The consequences of the SOC torque are studied for two different cases: current-induced spin-spiral motion andthe Doppler shift of helimagnons. During the current-drivenspin-spiral motion, the SOC torque yields an enhanced tiltingof the spin-spiral structure, while the velocity is not affected.The dispersion relation of a helimagnon that propagates alongthe axis of the spin spiral is not affected by the SOC torqueexcept to induce an initial translation of the spiral structure. ACKNOWLEDGMENTS This work was supported by EU-ICT-7 Contract No. 257159 “MACALO.” 1D. C. Ralph and M. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008). 2A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012).3L. Berger, Phys. Rev. B 54, 9353 (1996). 4J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 174409-4SPIN-TRANSFER TORQUES IN HELIMAGNETS PHYSICAL REVIEW B 87, 174409 (2013) 5Note that alternative phenomenologies for the magnetization dynamics exist; see, e.g., V . G. Bar’yakhtar, Zh. Eksp. Teor. Fiz.87, 1501 (1984) [Sov. Phys. JETP 60, 863 (1984)]. 6A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008). 7A. Chernyshov et al. ,Nat. Phys. 5, 656 (2009). 8I. M. Miron et al. , Nat. Mater. 9, 230 (2010). 9I. Garate and A. H. MacDonald, P h y s .R e v .B 80, 134403 (2009). 10K. M. D. Hals, A. Brataas, and Y . Tserkovnyak, Europhys. Lett. 90, 47002 (2010). 11I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958). 12T. Moriya, Phys. Rev. 120, 91 (1960); Phys. Rev. Lett. 4, 228 (1960). 13U. K. R ¨oßler, A. N. Bogdanov, and C. Pfleiderer, Nature (London) 442, 797 (2006). 14S. M ¨uhlbauer et al. ,Science 323, 915 (2009). 15X. Z. Yu et al. ,Nature (London) 465, 901 (2010). 16X. Z. Yu et al. ,Nat. Mater. 10, 106 (2011). 17A. Tonomura et al. ,Nano Lett. 12, 1673 (2012). 18S. Seki, X. Z. Yu, S. Ishiwata, and Y . Tokura, Science 336, 198 (2012). 19N. Kanazawa et al. ,Phys. Rev. B 86, 134425 (2012). 20F. Jonietz et al. ,Science 330, 1648 (2010). 21X. Z. Yu et al. ,Nat. Commun. 3, 988 (2012). 22K. Goto, H. Katsura, and N. Nagaosa, arXiv: 0807.2901 [cond- mat.str-el].23J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys. Rev. Lett. 107, 136804 (2011). 24J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Commun. 4, 1463 (2013). 25R. R. Birss, Symmetry and Magnetism (North-Holland, Amsterdam, 1966). 26Y . Tserkovnyak and M. Mecklenburg, P h y s .R e v .B 77, 134407 (2008). 27We have disregarded the magnetization damping in the derivationthe Onsager coefficients. If damping is included, the same resultsare obtained, but a transformation between the Landau-Lifshitzequation and the Landau-Lifshitz-Gilbert equation is required attwo intermediate steps in the derivation. See Refs. 10and 26for further details. 28L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media , Course of Theoretical Physics V ol. 8, (Pergamon, Oxford, 1984). 29A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, New York, 1994). 30T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 31V . Vlaminck and M. Bailleul, Science 322, 410 (2008). 32D. Belitz, T. R. Kirkpatrick, and A. Rosch, P h y s .R e v .B 73, 054431 (2006). 33O. Petrova and O. Tchernyshyov, P h y s .R e v .B 84, 214433 (2011). 174409-5

PhysRevB.78.064429.pdf

Calculation of current-induced torque from spin continuity equation Gen Tatara1and Peter Entel2 1Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan 2Physics Department, University of Duisburg-Essen, 47048 Duisburg, Germany /H20849Received 3 August 2008; published 29 August 2008 /H20850 Current-induced torque is formulated based on the spin continuity equation. The formulation does not rely on the assumption of separation of local spin and charge degrees of freedom, in contrast to approaches basedon the s-dmodel or mean-field approximation of itinerant ferromagnetism. This method would be thus useful for the estimation of torques in actual materials by first-principles calculations. As an example, the formalismis applied to the adiabatic limit of the s-dmodel in order to obtain the analytical expression for torques and the corresponding /H9252terms arising from spin relaxation due to spin-flip scattering and spin-orbit interaction. DOI: 10.1103/PhysRevB.78.064429 PACS number /H20849s/H20850: 85.75. /H11002d, 72.10. /H11002d, 72.25. /H11002b I. INTRODUCTION Spin-transfer torque is a torque acting on local spins as a result of an applied current. Such a torque has been discussedmostly based on s-dtype of exchange interaction 1–6after the pioneering works by Berger7,8and Slonczewski.9Ins-d models, the conduction electrons and localized spins are dis-criminated and therefore the transfer of spin angular momen-tum between those two degrees of freedom occurs. However,in reality, this separation of degrees of freedom is not alwaysso obvious; since in an itinerant picture, all electronic bandscontribute to both conduction and magnetism with differentweights. Thus, the formulation of spin torques based on thes-dpicture is an approximation and this is a serious problem when one tries to evaluate current-induced torques in actualmaterials. For trustful estimates, formulations beyond thesimple s-dseparation is certainly required. Such formalism can be combined with first-principles calculations withoutany artificial assumption and would be useful for realisticestimates of current-induced torques and of efficiency ofcurrent-induced switching. The aim of this paper is to de-velop a calculational scheme satisfying these requirementsbased on the spin continuity equation. Theoretical determination of current-induced torques is difficult even in the simplest case of s-dmodel when spin relaxation and nonadiabaticity is present. 4,5,10–15So far, very few studies on the effect of spin relaxation due to spin-flipscattering by magnetic impurities have been donemicroscopically. 4,5,12In the s-dformalism, the current- induced torque is represented as the effective field due to thespin polarization of the conduction electron s. The torque is therefore given as /H9270/H20849sd/H20850=−JsdS/H11003s, where Sis the localized /H20849d/H20850electron spin and Jsdis the exchange interaction constant. Microscopic calculation using linear-response theory4,5re- vealed, in agreement with phenomenolocigal result,2that spin-flip interaction of conduction electrons with random im-purities induces a torque perpendicular to the spin-transfertorque /H20849called /H9252terms16/H20850. The torque is written as /H9270/H20849/H9252/H20850=−/H9252P eS2/H20851S/H11003/H20849j·/H11612/H20850S/H20852, /H208491/H20850 where Pis the spin polarization of the current and jis the current density. The coefficient /H9252was calculated by sum-ming over not a few Feynman diagrams, representing self- energy and vertex corrections.4,5,15 The case of itinerant ferromagnetism was studied by Tserkovnyak et al.10and Duine et al.12They introduced the magnetization as a mean-field expectation value of itinerantelectron spin and, thus, the models considered were effec-tively the s-dmodel. Tserkovnyak et al. considered a kinetic equation for the spin density with a consistency condition forthe magnetization, but the spin dephasing term was intro-duced phenomenologically. Duine et al. estimated the torques by calculating the effective action for the magnetiza-tion fluctuation, which has been assumed to be of small am-plitude. Within the mean-field treatment, the torque in theitinerant case turned out to be exactly the same as that of thes-dmodel. 4,12 It has recently been noticed that the coefficient /H9252is very important for the realization of highly efficient magnetizationswitching by the current. 2,15–17First, it affects the threshold current and the intrinsic pinning threshold is replaced by anextrinsic one, which is usually lower than the intrinsic one. Second, it results in a terminal speed of the wall v/H11008/H9252 /H9251j, which can exceed the pure spin-transfer speed limit if/H9252 /H9251is large /H20849/H9251is Gilbert damping parameter /H20850. Third, the deforma- tion of the wall depends on /H9252. When /H9252/H11011/H9251, deformation is suppressed and weak dissipation may be expected.18Experi- mental studies of the value of /H9252have recently been carried out. Significant wall deformation observed in permalloy in-dicated that /H9252/HS11005/H9251.18Thomas et al.19found for permalloy that the observed wall speed corresponds to /H9252/H110118/H9251. Therefore, determination of /H9252is of particular importance for device applications. In this paper, we will present a microscopic calculation scheme different from the s-dformalism.7,8The idea is sim- ply to use the continuity equation of spin and thus the for-mulation is not necessarily based on the s-dinteraction pic- ture. The formalism turns out to be quite powerful, inparticular, for the determination of spin-relaxation effect /H9252. The continuity equation, which we consider, is essentiallythe kinetic equation discussed by Tserkovnyak et al. , 10but all observables have been microscopically defined and can becalculated using our formalism. For instance, spin dephasingtime introduced phenomenolocigally in Ref. 10is repre- sented by the spin source term /H20849T/H20850defined by Green’s func-PHYSICAL REVIEW B 78, 064429 /H208492008 /H20850 1098-0121/2008/78 /H208496/H20850/064429 /H208496/H20850 ©2008 The American Physical Society 064429-1tion in our formalism. Microscopic details of this term Tturn out to be essential in determining the spin-relaxation-inducedtorque. Our scheme is applicable also to the s-dmodel or mean- field approximation of itinerant ferromagnetism. We will useour formalism to obtain the analytical expression of thetorques arising from both spin-flip scattering and spin-orbitinteraction arising from the impurities in the s-dmodel in the adiabatic limit. In the present formalism, the number of con-tributing diagrams is less than the number of diagrams usedin the s-dexchange formalism 4,5and thus the calculation is easier. II. FORMALISM The spin density sof the total system is defined as the expectation value of conduction-electron spin, summed overall bands nas s /H9251/H20849x,t/H20850/H11013/H20858 n/H20855cn†/H20849x,t/H20850/H9268/H9251cn/H20849x,t/H20850/H20856. /H208492/H20850 It satisfies the equation /H6036s˙/H9251=i/H20858n/H20849/H20855/H20851H,cn†/H20852/H9268/H9251cn/H20856 +/H20855cn†/H9268/H9251/H20851H,cn/H20852/H20856/H20850where His total Hamiltonian. We assume that Hconsists of free and spin-relaxation parts HsrasH =/H20848d3x/H20858n/H60362 2m/H20841/H11612cn/H208412+Hsr. Then the continuity equation is ob- tained as /H6036s˙/H9251=−1 e/H11612·js/H9251+T/H9251, /H208493/H20850 where erepresents the electron charge. Here the spin current jsis defined by the free part as js/H9262/H9251/H11013−ie/H6036 2m/H20858 n/H20855cn†/H20849x,t/H20850/H11612J/H9262/H9268/H9251cn/H20849x/H11032,t/H20850/H20856, /H208494/H20850 and the spin source /H20849or sink /H20850Tis a contribution arising from spin relaxation and interaction, i.e., T/H9251/H11013i/H20858 n/H20849/H20855/H20851Hsr,cn†/H20852/H9268/H9251cn/H20856+/H20855cn†/H9268/H9251/H20851Hsr,cn/H20852/H20856/H20850. /H208495/H20850 The continuity /H20851Eq. /H208493/H20850/H20852is sufficient to calculate the torque acting on the spin. Actually, the equation is equivalent to the equation of motion of spin /H6036s˙=/H9270where /H9270represents the total torque acting on the spin. The torque is thus simply given by /H9270/H9251=−1 e/H11612·js/H9251+T/H9251. /H208496/H20850 Note that the continuity equation describes the time de- pendence of the spin density and, therefore, the right-handside of Eqs. /H208493/H20850and /H208496/H20850is uniquely defined even in the pres- ence of spin relaxation, where the spin current can be definedin several different ways /H20849see Ref. 20/H20850. In the context of spin Hall effect, the continuity /H20851Eq. /H208493/H20850/H20852was used to obtain proper definition of spin current and to explore transportproperties. 21–23Concerning the current-induced torques, Eq. /H208496/H20850has been so far applied only in the absence of spin- relaxation term, where the torque is given by the divergenceof the spin current. 24,25The main aim of this paper is to study the spin-relaxation contribution T.Let us look explicitly at the continuity equation, in case of spin relaxation, due to spin impurities and spin-orbit interac-tion H sr=Hsf+Hso. Spin-flip interaction is described by Hsf=vs/H20885d3x/H20858 nSimp/H20849x/H20850·/H20849cn†/H9268cn/H20850, /H208497/H20850 where vsis a constant, Simp/H20849x/H20850/H11013/H20858inimpSimpi/H9254/H20849x−Ri/H20850,Simpirep- resents the impurity spin at x=Ri, and nimpdenotes the num- ber of impurity spins. The spin-orbit interaction is written as Hso=−i 2/H9261so/H20885d3x/H20858 ijkl/H9280ijk/H11612jVso/H20849l/H20850/H20849x/H20850/H20849cn†/H9268l/H11612Jkcn/H20850, /H208498/H20850 where the potential Vso/H20849l/H20850is here assumed to arise from random impurities and depends on the spin direction /H20849l/H20850. The spin-relaxation torque is given by a sum of contribu- tions from spin-flip and spin-orbit interactions as T/H9251=Tsf/H9251 +Tso/H9251, where Tsf/H9251/H20849x/H20850=2vs/H20858 /H9252/H9253/H9280/H9251/H9252/H9253/H20855Simp/H9252s/H9253/H20856i, /H208499/H20850 Tso/H9251/H20849x/H20850=−2 m/H9261so/H20858 /H9252/H9253/H9262/H9263/H9280/H9251/H9252/H9253/H9280/H9262/H9263/H9252/H20855/H11612/H9262Vso/H20849/H9253/H20850/H20849x/H20850jsv/H9253/H20856i. /H2084910/H20850 The average over random impurity spins and spin-orbit po- tential is represented by /H20855/H20856i. All the terms in the right-hand side of torque /H20851Eqs. /H208496/H20850, /H208499/H20850, and /H2084910/H20850/H20852are written in terms of local spin density and local spin current and so the torque acting on the spin iscalculated by estimating the spin density and the spin cur-rent. This representation of the spin torque applies to anyspin-relaxation processes and interaction and is directly cal-culable without assuming separation of spin and charge de-grees of freedom. Equations /H208496/H20850,/H208499/H20850, and /H2084910/H20850are thus suit- able starting points for realistic estimates based on first-principles calculations. This is the essential point of thispaper /H20849although ab initio calculations, using the present for- malism, still need to be undertaken /H20850. III. APPLICATION TO THE s-dMODEL In Secs. III A and III B of the paper, we will apply this formulation to estimate the current-induced torques in theadiabatic limit /H20849i.e., slowly varying magnetization compared to conduction-electron motion /H20850to show the validity and use- fulness of our formalism. We will calculate the torque arisingfrom the spin relaxation due to both the spin-flip scatteringand the spin-orbit interaction. It is found that the torque isrepresented by the so-called /H9252term in both cases and values of corresponding /H9252are calculated. Our formulation is thus useful for both analytical and numerical studies. We will now consider the s-dmodel with only one con- duction band. Please note that the assumption of separationofsand delectrons here is simply for analytical demonstra- tion and is not a requirement for the present formulation. Thes-dinteraction between a localized spin Sand the conduction electrons is given byGEN TATARA AND PETER ENTEL PHYSICAL REVIEW B 78, 064429 /H208492008 /H20850 064429-2Hex/H11013−Jsd/H20885d3xS/H20849x,t/H20850·/H20849c†/H9268c/H20850. /H2084911/H20850 We describe the adiabatic limit by the standard local gauge transformation in the spin space, choosing the electron spin-quantization axis along S/H20849x,t/H20850at each point. A new electron operator a/H11013/H20849a +,a−/H20850t/H20849tdenotes transpose /H20850is defined as c/H20849x,t/H20850/H11013U/H20849x,t/H20850a/H20849x,t/H20850where Ui sa2 /H110032 matrix, which we further define as U/H20849x,t/H20850/H11013m·/H9268with mbeing a real three- component unit vector m=/H20849sin/H9258 2cos/H9278,sin/H9258 2sin/H9278,cos/H9258 2/H20850. The gauge field is written as A/H9262/H9251/H11013/H20849m/H11003/H11509/H9262m/H20850/H9251. Then the Hamil- tonian of aelectrons is given by the free part /H20858k/H9268/H9280k/H9268ak/H9268†ak/H9268 /H20849/H9280k/H9268/H11013/H9280k−/H9268M,/H9268=/H11006represents the spin /H20850,HAdescribing the interaction with the SU /H208492/H20850gauge field, and Hemas the inter- action with the external electric field, which drives thecurrent. 1,15Here, we consider static local spins in the adia- batic limit, where the momentum transferred by the gaugefield to conduction electrons is negligibly small /H20849compared to k F/H20850and take into account the gauge field only in linear order. Then, the gauge interaction is given by15 HA=/H60362 m/H20858 q/H20858 /H9262k/H9262A/H9262/H9251/H20849−q/H20850ak†/H9268/H9251ak. /H2084912/H20850 The applied electric field is represented by the interaction, Hem=/H20858 /H9262ie/H6036E/H9262 m/H90240ei/H90240t/H20858 k/H20851k/H9262ak†ak+/H20858 /H9251qA/H9262/H9251/H20849q/H20850ak†/H9268/H9251ak/H20852+O/H20849E2/H20850, /H2084913/H20850 where /H90240is the frequency of the field chosen as /H90240→0a t the end of calculation. The spin-current part of the torque is calculated in the adiabatic limit as −/H11612·js/H9251/H11229−/H20849/H11612/H9262n/H20850js/H9262. /H2084914/H20850 Here, n/H11013S/Sand, therefore, this contribution corresponds to the standard spin-transfer torque. A. Torque from spin-flip scattering Let us turn to the spin-relaxation part of the torque arising from spin impurities, i.e., Eq. /H208499/H20850. The effect of spin relax- ation on the spin-current part can be shown to be simply dueto modification of lifetime /H9270. Here, we assume that the im- purity spins are influenced by a strong s-dexchange field and write Simp/H9251/H20849x/H20850=R/H9251/H9252/H20849x/H20850S˜ imp/H9252/H20849x/H20850, where S˜ imp/H9251represents impurity spin in the rotated frame, and R/H9251/H9252/H110132m/H9251m/H9252−/H9254/H9251/H9252, /H2084915/H20850 is a rotation matrix. Then the averaging is given by /H20855S˜ imp/H9251/H20849x/H20850S˜ imp/H9252/H20849x/H11032/H20850/H20856i=1 3/H9254/H9251/H9252/H9254/H20849x-x/H11032/H20850nimpSimp2where nimpis the im- purity spin concentration. Averaging taken with respect toS impturns out to lead to—essentially—the same result as in the case of S˜imp. The spin source term is written as Tsf/H9251/H20849x/H20850=−2 ivs/H20858 /H9252/H9253F/H9251/H9252/H9253/H20849x/H20850/H20855S˜ imp/H9252/H20849x/H20850tr/H20851/H9268/H9253G˜ x,x/H11021/H20852/H20856i, /H2084916/H20850 whereF/H9251/H9252/H9253 /H11013/H20858 /H9262/H9263/H9280/H9251/H9262/H9263R/H9262/H9252R/H9263/H9253, /H2084917/H20850 and G˜ x,x/H11032/H11021/H11013i/H20855a†/H20849x/H11032/H20850a/H20849x/H20850/H20856is the lesser component of the Green’s function defined on Keldysh contour in the complex time. To the lowest /H20849second /H20850order in vs, we obtain after averaging over spin impurities, Tsf/H9251/H20849x/H20850=−i2 3nimpvs2Simp2/H20858 /H9252/H9253/H9254/H20858 /H9262/H9263F/H9251/H9252/H9253/H20849x/H20850tr/H20851/H9268/H9252G˜ x,x/H208490/H20850/H9268/H9253G˜ x,x/H208490/H20850/H20852/H11021 +O/H20849vs4/H20850, /H2084918/H20850 where G˜/H208490/H20850denotes Green’s functions without impurity spins but including the gauge field Aand external electric field E. Including these fields in linear order, we obtain Tsf/H9251/H20849x/H20850=−2e 3mnimpvs2Simp2/H20858 /H9252/H9253/H9262/H9263F/H9251/H9252/H9253/H20849x/H20850E/H9262A/H9263/H9254/H20849x/H20850D/H9262/H9263/H9252/H9253/H9254, /H2084919/H20850 where D/H9262/H9263/H9252/H9253/H9254/H11013lim /H90240→01 /H90240/H20885d/H9275 2/H9266/H20858 kk/H11032tr/H20875/H9268/H9252/H20873/H9254/H9262/H9263gk/H11032/H9275/H9268/H9253gk/H9275/H9268/H9254gk/H9275+/H90240 +k/H9262k/H9263 m/H20853gk/H11032/H9275/H9268/H9253gk/H9275gk/H9275+/H90240/H9268/H9254gk/H9275+/H90240 +gk/H11032/H9275/H9268/H9253gk/H9275/H9268/H9254gk/H9275gk/H9275+/H90240/H20854/H20874/H20876/H11021 + c.c. /H2084920/H20850 Here, the Green’s function gk/H9275is the Fourier representation of free Green’s function and /H20851/H20852/H11021denotes the lesser compo- nent. They are diagonal in spin space, being defined ingauge-transformed space. Complex conjugates are denoted by c.c. Figure 1shows the contributions to D /H9262/H9263/H9252/H9253/H9254diagram- matically. The lesser component is calculated in standardmanner in the limit of /H9024 0→0. The first two diagrams of Fig. 1are simplified by the use of partial integration over kusing k/H9262 m/H20849gka/H208502=/H11509 /H11509k/H9262gka, etc. These contributions are obtained as D/H9262/H9263/H9252/H9253/H9254/H208491−2 /H20850= lim /H90240→0/H20885d/H9275 2/H9266/H20858 kk/H11032tr/H20875f/H11032/H20849/H9275/H20850k/H9262k/H9263 m/H20853/H9268/H9252gk/H11032/H9275r/H9268/H9253 +/H9268/H9253gk/H11032/H9275a/H9268/H9252/H20854/H20849/H20841gk/H9275r/H208412/H9268/H9254gk/H9275a+gk/H9275r/H9268/H9254/H20841gk/H9275a/H208412/H20850 +/H9254/H9262/H9263/H20875f/H20849/H9275/H20850 2/H20853/H20851/H9268/H9252/H20849gk/H11032/H9275a/H208502/H9268/H9253 −/H9268/H9253/H20849gk/H11032/H9275a/H208502/H9268/H9252/H20852gk/H9275a/H9268/H9254gk/H9275a− c.c. /H20854σβσγ σδk/primeωkω kωk,ω+Ω0Aδν EµEµ Aδνk,ω+Ω0k,ω+Ω0 FIG. 1. Diagrammatic representation of D/H9262/H9263/H9252/H9253/H9254. Double-dashed, dotted, and wavy lines denote interaction with impurity spin, ap-plied electric field E, and gauge field A, respectively.CALCULATION OF CURRENT-INDUCED TORQUE FROM … PHYSICAL REVIEW B 78, 064429 /H208492008 /H20850 064429-3−1 /H90240/H20875f/H20873/H9275−/H90240 2/H20874/H20849/H9268/H9252gk/H11032/H9275a/H9268/H9253 +/H9268/H9253gk/H11032/H9275a/H9268/H9252/H20850gk/H9275a/H9268/H9254gk/H9275a−f/H20873/H9275+/H90240 2/H20874/H20849/H9268/H9252gk/H11032/H9275r/H9268/H9253 +/H9268/H9253gk/H11032/H9275r/H9268/H9252/H20850gk/H9275r/H9268/H9254gk/H9275r/H20876/H20876/H20876, /H2084921/H20850 where f/H20849/H9275/H20850/H11013/H20849 e/H9252/H9275+1/H20850−1. Similarly, the third contribution in Fig.1is obtained as D/H9262/H9263/H9252/H9253/H9254/H208493/H20850= lim /H90240→0/H20885d/H9275 2/H9266/H20858 kk/H11032/H9254/H9262/H9263tr/H20875f/H11032/H20849/H9275/H20850/H20853/H9268/H9252gk/H11032r/H9268/H9253 +/H9268/H9253gk/H11032a/H9268/H9252/H20854gkr/H9268/H9254gka+/H20875−f/H20849/H9275/H20850 2/H20853/H20851/H9268/H9252/H20849gk/H11032/H9275a/H208502/H9268/H9253 −/H9268/H9253/H20849gk/H11032/H9275a/H208502/H9268/H9252/H20852gk/H9275a/H9268/H9254gk/H9275a− c.c. /H20849/H9268/H9252gk/H11032/H9275a/H9268/H9253 +/H9268/H9253gk/H11032/H9275a/H9268/H9252/H20850/H20851gk/H9275a/H9268/H9254/H20849gk/H9275a/H208502−/H20849gk/H9275a/H208502/H9268/H9254gk/H9275a/H20852− c.c. /H20854 +1 /H90240/H20875f/H20873/H9275−/H90240 2/H20874/H20849/H9268/H9252gk/H11032/H9275a/H9268/H9253+/H9268/H9253gk/H11032/H9275a/H9268/H9252/H20850gk/H9275a/H9268/H9254gk/H9275a −f/H20873/H9275+/H90240 2/H20874/H20849/H9268/H9252gk/H11032/H9275r/H9268/H9253+/H9268/H9253gk/H11032/H9275r/H9268/H9252/H20850gk/H9275r/H9268/H9254gk/H9275r/H20876/H20876/H20876. /H2084922/H20850 Noting that only antisymmetric part with respect to /H9252and/H9253 contribute to the torque, these contributions are summed to be D/H9262/H9263/H9252/H9253/H9254=−i/H20885d/H9275 2/H9266/H20858 kk/H11032f/H11032/H20849/H9275/H20850tr/H20877/H20851/H9268/H9252Im/H20849gk/H11032a/H20850/H9268/H9253−/H9268/H9253Im/H20849gk/H11032a/H20850/H9268/H9252/H20852 /H11003/H20875k/H9262k/H9263 m/H20849/H20841gkr/H208412/H9268/H9254gka+gkr/H9268/H9254/H20841gkr/H208412/H20850+/H9254/H9262/H9263/H20849gkr/H9268/H9254gka/H20850/H20876/H20878,/H2084923/H20850 where gkr/H11013gk,/H9275=0r, etc. We see that spin-flip processes con- tribute as additional lifetimes as indicated by the imaginary part of spin-scattered electron Green’s function Im gk/H11032a. To estimate the trace in the spin space, we use general identities, which hold for 2 /H110032 diagonal matrices B,C, and D/H20849containing only /H9268zand the identity matrix /H20850, tr/H20851/H20849/H9268/H9252B/H9268/H9253−/H9268/H9253B/H9268/H9252/H20850/H20849C/H9268/H9254D+D/H9268/H9254C/H20850/H20852 =2i/H20851/H20849/H9280/H9252/H9253/H9254−/H9280/H9252/H9253z/H9254/H9254z/H20850/H20851/H20849BC/H20850+D−+/H20849BC/H20850−D++/H20849BD/H20850+C− +/H20849BD/H20850−C+/H20852+2/H9280/H9252/H9253z/H9254/H9254z/H20851B+/H20849CD/H20850−+B−/H20849CD/H20850+/H20852/H20852, tr/H20851/H20849/H9268/H9252B/H9268/H9253−/H9268/H9253B/H9268/H9252/H20850/H20849C/H9268/H9254D−D/H9268/H9254C/H20850/H20852 =2/H20849/H9254/H9253z/H9254/H9252/H9254−/H9254/H9252z/H9254/H9253/H9254/H20850/H20851/H20849BC/H20850+D−−/H20849BC/H20850−D+−/H20849BD/H20850+C− +/H20849BD/H20850−C+/H20852, /H2084924/H20850 where the components B/H11006are defined as B=/H20851B++B−+ /H20849B+−B−/H20850/H9268z/H20852/2, etc. The result for D/H9262/H9263/H9252/H9253/H9254is then obtained asD/H9262/H9263/H9252/H9253/H9254=/H9254/H9262/H9263/H20851a/H20849/H9280/H9252/H9253/H9254−/H9280/H9252/H9253z/H9254/H9254z/H20850+b/H20849/H9254/H9252/H9254/H9254/H9253z−/H9254/H9253/H9254/H9254/H9252z/H20850/H20852, /H2084925/H20850 where the coefficients are given by a=−1 2/H9266/H20858 kk/H11032/H20858 /H9268/H9268/H11032/H20875k2 3m/H20841gk/H9268r/H208412/H20849gk,−/H9268a+gk,−/H9268r/H20850 +/H20849gk/H9268rgk,−/H9268a+gk/H9268agk,−/H9268r/H20850/H20876/H20849Imgk/H11032/H9268/H11032a/H20850, b=−1 2/H9266/H20858 kk/H11032/H20858 /H9268/H9268/H11032/H20849i/H9268/H20850gk/H9268agk,−/H9268r/H20849Imgk/H11032/H9268/H11032a/H20850. /H2084926/H20850 Using F/H9251/H9252/H9253=−/H9280/H9251/H9252/H9253−2/H20858/H9254m/H9254/H20849/H9280/H9251/H9253/H9254m/H9252−/H9280/H9251/H9252/H9254m/H9253/H20850and A/H9262 =1 2/H20849n/H11003/H11509/H9262n/H20850−A/H9262zn,15the torque due to spin flip is obtained as Tsf=−2e 3mvs2Simp2/H20858 /H9262E/H9262/H20851a/H20849n/H11003/H11509/H9262n/H20850−b/H11509/H9262n/H20852. /H2084927/H20850 The coefficients aand bare calculated as a=/H9266/H20849m/e2M/H20850 /H20849/H9268+−/H9268−/H20850/H20849/H9263++/H9263−/H20850and b=O/H20851a/H11003/H20849/H9280F/H9270/H20850−1/H20852/H112290, where /H9263/H11006and /H9268/H9268=e2n/H9268/H9270/H9268/mare the spin-resolved conductivity and density of states, respectively. Coefficient bis treated as zero within the present approximation. Therefore, the torque induced bythe spin relaxation is simply a /H9252term given by Tsf=−/H9252sfP e/H20851n/H11003/H20849j·/H11612/H20850n/H20852, /H2084928/H20850 where P/H11013/H20849/H9268+−/H9268−/H20850//H20849/H9268++/H9268−/H20850is the spin polarization of the current and /H9252sf=2/H9266 3Mnimpvs2Simp2/H20849/H9263++/H9263−/H20850. /H2084929/H20850 Defining the spin-flip lifetime /H20849/H9270sof Ref. 4/H20850as/H20849note that Sz2+S/H110362of Ref. 4corresponds to2 3Simp2here /H20850/H9270sf−1 =/H208494/H9266/3/H20850nimpvs2Simp2/H20849/H9263++/H9263−/H20850,w efi n d /H9252sf=/H6036//H208492M/H9270sf/H20850, which agrees with the results obtained in Refs. 4and5. B. Torque from spin-orbit interaction The torque from spin-orbit interaction /H20851Eq. /H2084910/H20850/H20852is calcu- lated in a similar way. The spin-orbit interaction is written inthe rotated frame as H so=/H9261so/H20885d3x/H20858 ijkl/H9280ijk/H11612jVso/H20849i/H20850/H20849x/H20850Ril/H20849x/H20850/H20873−i 2a†/H9268l/H11612Jka+Akla†a/H20874. /H2084930/H20850 The spin-orbit contributions to the spin current and the elec- tron density in the rotated frame are obtained asGEN TATARA AND PETER ENTEL PHYSICAL REVIEW B 78, 064429 /H208492008 /H20850 064429-4js/H9263/H9267/H20849x/H20850=−i 2m/H9261so/H20858 ijkl/H9280ijk/H20849/H11612x−/H11612x/H11032/H20850/H9263/H20885d3x1/H11612jVso/H20849i/H20850/H20849x1/H20850Ril/H20849x1/H20850 /H11003tr/H20877/H9268/H9267G˜ x,x1/H11032/H208490/H20850/H20875−i 2/H20849/H11612/H6023x1−/H11612/H6024x1/H11032/H20850k/H9268l +Akl/H20849x1/H20850/H20876G˜ x1,x/H11032/H208490/H20850/H20878 x/H11032→x,x1/H11032→x1/H11021 , ne/H20849x/H20850=−i 2/H9261so/H20858 ijkl/H9280ijk,/H20885d3x1/H11612jVso/H20849i/H20850/H20849x1/H20850Ril/H20849x1/H20850/H20849/H11612/H6023x1 −/H11612/H6023x1/H11032/H20850ktr/H20851G˜ x,x1/H11032/H208490/H20850/H9268lG˜ x1,x/H11032/H208490/H20850/H20852x/H11032→x,x1/H11032→x1/H11021+O/H20849A/H20850./H2084931/H20850 The torque is then calculated as Tso/H9251/H20849x/H20850=−i/H9261so2/H20858 /H9252/H9262/H9263/H9270/H20858 jklm/H9280/H9251/H9262/H9270/H9280lm/H9270/H9280/H9263jk /H11003/H20885d3x1R/H9262/H9252/H20849x/H20850R/H9263/H9253/H20849x1/H20850/H20858 kk/H11032p/H20858 k1k1/H11032 /H11003plpje−ip·/H20849x−x1/H20850e−i/H20849k−k/H11032/H20850·xe−i/H20849k1−k1/H11032/H20850·x1/H20855Vso/H20849/H9252/H20850/H20849p/H20850Vso/H20849/H9270/H20850/H20849−p/H20850/H20856 /H11003/H208771 2/H20849k+k/H11032/H20850m/H20849k1+k1/H11032/H20850ktr/H20851/H9268/H9252G˜ k,k1/H208490/H20850/H9268/H9253G˜ k1/H11032,k/H11032/H208490/H20850/H20852/H11021 +/H20849k+k/H11032/H20850mAk/H9253/H20849x1/H20850tr/H20851/H9268/H9252G˜ k,k1/H208490/H20850G˜ k1/H11032,k/H11032/H208490/H20850/H20852/H11021 +/H20849k1+k1/H11032/H20850kAm/H9252/H20849x1/H20850tr/H20851G˜ k,k1/H208490/H20850/H9268/H9253G˜ k1/H11032,k/H11032/H208490/H20850/H20852/H11021/H20878. /H2084932/H20850 In the adiabatic limit, we consider Green’s functions are di- agonal in wave vectors G˜ k,k/H11032/H208490/H20850=/H9254k,k/H11032G˜ k/H208490/H20850and the integration over x1can be carried out, treating the slowly varying vari- ables R/H20849x1/H20850and A/H20849x1/H20850as constants, resulting in /H20848dx1e−i/H20849p−k+k/H11032/H20850·/H20849x−x1/H20850=V/H9254p,k−k/H11032. We therefore obtain Tso/H9251/H20849x/H20850=−i/H9261so2/H20858 /H9252/H9262/H9263/H9270/H20858 jklm/H9280/H9251/H9262/H9270/H9280lm/H9270/H9280/H9263jkR/H9262/H9252/H20849x/H20850R/H9263/H9253/H20849x/H20850/H20858 kk/H11032 /H11003/H20849k−k/H11032/H20850l/H20849k−k/H11032/H20850j/H20855Vso/H20849/H9263/H20850/H20849k−k/H11032/H20850Vso/H20849/H9270/H20850/H20849−k+k/H11032/H20850/H20856 /H11003/H208751 2/H20849k+k/H11032/H20850m/H20849k1+k1/H11032/H20850ktr/H20851/H9268/H9252G˜ k/H208490/H20850/H9268/H9253G˜ k/H11032/H208490/H20850/H20852/H11021 +/H20849k+k/H11032/H20850mAk/H9253/H20849x/H20850tr/H20851/H9268/H9252G˜ k/H208490/H20850G˜ k/H11032/H208490/H20850/H20852/H11021 +/H20849k+k/H11032/H20850kAm/H9252/H20849x/H20850tr/H20851G˜ k/H208490/H20850/H9268/H9253G˜ k/H11032/H208490/H20850/H20852/H11021/H20876. /H2084933/H20850 We average over spin-orbit impurities so that the average remains finite only when the spin polarizations are parallel.Impurity averaging is thus given as /H20855V so/H20849/H9263/H20850/H20849p/H20850Vso/H20849/H9270/H20850/H20849−p/H11032/H20850/H20856i=nso/H9254/H9263/H9270/H9254p,p/H11032. /H2084934/H20850 The result of the torque isTso/H9251/H20849x/H20850=−i1 2nso/H9261so2/H20858 /H9252/H9262/H9263/H9270/H9280/H9251/H9262/H9263R/H9262/H9252/H20849x/H20850R/H9263/H9253/H20849x/H20850/H11003/H20858 kk/H11032/H20851/H20849k/H11003k/H11032/H20850/H9270 /H11003/H20849k/H11003k/H11032/H20850/H9263tr/H20851/H9268/H9252G˜ k/H208490/H20850/H9268/H9253G˜ k/H11032/H208490/H20850/H20852/H11021+/H20849k/H11003k/H11032/H20850/H9270/H20851/H20849k−k/H11032/H20850 /H11003A/H9253/H20852/H9263tr/H20851/H9268/H9252G˜ k/H208490/H20850G˜ k/H11032/H208490/H20850/H20852/H11021+/H20849k/H11003k/H11032/H20850/H9263/H20851/H20849k−k/H11032/H20850 /H11003A/H9252/H20852/H9270tr/H20851G˜ k/H208490/H20850/H9268/H9253G˜ k/H11032/H208490/H20850/H20852/H11021/H20852/H9263=/H9270. /H2084935/H20850 The last two terms lead to vanishing contribution in the adia- batic limit. In fact, these are already linear in Aand so G˜/H208490/H20850 does not contain spin-flip components, and thus /H9268zand G˜/H208490/H20850 commute each other. We therefore obtain /H20851/H20849k−k/H11032/H20850A/H9253/H20852/H9263tr/H20851/H9268/H9252G˜ k/H208490/H20850G˜ k/H11032/H208490/H20850/H20852/H11021+/H20851/H20849k−k/H11032/H20850A/H9252/H20852/H9263 /H11003tr/H20851G˜ k/H208490/H20850/H9268/H9253G˜ k/H11032/H208490/H20850/H20852/H11021 =/H20853/H9254/H9252,z/H20851/H20849k−k/H11032/H20850/H11003A/H9253/H20852/H9263+/H9254/H9253,z/H20851/H20849k−k/H11032/H20850 /H11003A/H9252/H20852/H9263/H20854tr/H20851/H9268zG˜ k/H208490/H20850G˜ k/H11032/H208490/H20850/H20852/H11021. /H2084936/H20850 This contribution is symmetric with respect to /H9252and/H9253and results in zero when multiplied by F/H9262/H9263/H9251/H9252/H9253, which is asymmetric with respect to /H9252and/H9253. The first term of Eq. /H2084935/H20850can be simplified by using the rotational symmetry of electron, /H20855/H20849k/H11003k/H11032/H20850/H9270/H20849k/H11003k/H11032/H20850/H9270/H20856=1 3/H20855/H20849k /H11003k/H11032/H20850·/H20849k/H11003k/H11032/H20850/H20856=1 3/H20855/H20851k2k/H110322−/H20849k·k/H11032/H208502/H20852/H20856 /H20849/H20855 /H20856 denotes the angular average /H20850,a s Tso/H9251/H20849x/H20850=−i1 6nso/H9261so2/H20858 /H9252/H9262/H9263/H9270/H20858 kk/H11032F/H9262/H9263/H9251/H9252/H9253/H20851k2k/H110322 −/H20849k·k/H11032/H208502/H20852tr/H20851/H9268/H9252G˜ k/H208490/H20850/H9268/H9253G˜ k/H11032/H208490/H20850/H20852/H11021. /H2084937/H20850 We therefore see that the expression is similar to that of spin-flip impurity case /H20851Eq. /H2084918/H20850/H20852. Including the effect of electric and gauge fields to linear order in both similarly tothe spin-flip impurity case, we obtain the torque as T so=−e 6mnso/H9261so2a/H11032/H20858 /H9262E/H9262/H20849n/H11003/H11509/H9262n/H20850, /H2084938/H20850 where coefficient is given as a/H11032=−1 2/H9266/H20858 kk/H11032/H20858 /H9268/H9268/H11032/H20849k2k/H110322−/H20849k·k/H11032/H208502/H20850/H20875k2 3m/H20841gk/H9268r/H208412/H20849gk,−/H9268a+gk,−/H9268r/H20850 +/H20849gk/H9268rgk,−/H9268a+gk/H9268agk,−/H9268r/H20850/H20876/H20849Imgk/H11032/H9268/H11032a/H20850. /H2084939/H20850 The coefficient is calculated as a/H11032=/H92662m 3e2M/H20849/H9268+kF+2 −/H9268−kF−2/H20850/H20849/H9263+kF+2+/H9263−kF−2/H20850. Therefore, spin-orbit interaction yields the /H9252term with coefficient given by /H9252so=1 2M1 n+/H9270+−n−/H9270−/H20873n+/H9270+ /H9270+/H20849so/H20850−n−/H9270− /H9270−/H20849so/H20850/H20874, /H2084940/H20850 whereCALCULATION OF CURRENT-INDUCED TORQUE FROM … PHYSICAL REVIEW B 78, 064429 /H208492008 /H20850 064429-51 /H9270/H11006/H20849so/H20850/H110132/H9266 9nso/H9261so2kF/H110062/H20849/H9263+kF+2+/H9263−kF−2/H20850, /H2084941/H20850 with/H9270/H11006/H20849so/H20850as the lifetime due to spin-orbit interaction. The total current-induced torque in the adiabatic limit is therefore given by Eqs. /H2084914/H20850,/H2084928/H20850,/H2084929/H20850, and /H2084940/H20850as /H9270=−P 2e/H20849/H11612·j/H20850n−/H9252srP e/H20851n/H11003/H20849j·/H11612/H20850n/H20852, /H2084942/H20850 with/H9252sr/H11013/H9252sf+/H9252so. IV . SUMMARY In summary, we demonstrated that the spin continuity equation represents the current-induced torque acting on themagnetization and that it can be used for microscopic deter-mination of the torques. The present formalism does not as-sume separation of magnetization and conduction-electrondegrees of freedom and can directly be applied to itinerantelectron systems without mean-field approximation. In thispaper, the formalism was applied to the s-dmodel in the presence of spin relaxation caused due to spin-flip scatteringand spin-orbit interaction with impurities. Both relaxationprocesses were shown to induce the so-called /H9252torque term.Application of the formalism to realistic itinerant system using first-principles calculations would be very interestingsince it would allow for quantitative estimations of current-induced switching. Of particular interest are the systems withenhanced spin-orbit interaction near surfaces and multilay-ers. Our formulation can be easily extended to describe thesesystems. Further improvement of the present theory would be to include effects caused by electron-electron correlation. If thecorrelation is represented within the mean-field approxima-tion by a local spin-dependent potential, the torque isstraightforwardly calculated similarly to the estimate of spin-flip scattering. Treatment beyond mean field would be animportant future work. Note added in proof. Recently, we found that the spin- transfer torque in the presence of spin-orbit interaction inferromagnetic semiconductors was studied in Ref. 26. ACKNOWLEDGMENTS The authors thank H. Akai, M. Ogura, and H. Kohno for their valuable discussions. G.T. acknowledges Grant-in-Aidfor Scientific Research on Priority Areas for their financialsupport. P.E. thanks the SFB491 and the DFG for their finan-cial support. 1G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 /H208492004 /H20850. 2S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 /H208492004 /H20850. 3X. Waintal and M. Viret, Europhys. Lett. 65, 427 /H208492004 /H20850. 4H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn. 75, 113706 /H208492006 /H20850. 5H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710 /H208492007 /H20850. 6G. Tatara, H. Kohno, and J. Shibata, arXiv:0807.2894, Phys. Rep. /H20849to be published /H20850. 7L. Berger, J. Appl. Phys. 49, 2156 /H208491978 /H20850. 8L. Berger, J. Appl. Phys. 55, 1954 /H208491984 /H20850. 9J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 10Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 74, 144405 /H208492006 /H20850. 11G. Tatara, H. Kohno, J. Shibata, Y. Lemaho, and K.-J. Lee, J. Phys. Soc. Jpn. 76, 054707 /H208492007 /H20850. 12R. A. Duine, A. S. Nunez, J. Sinova, and A. H. MacDonald, Phys. Rev. B 75, 214420 /H208492007 /H20850. 13M. Thorwart and R. Egger, Phys. Rev. B 76, 214418 /H208492007 /H20850. 14F. Piechon and A. Thiaville, Phys. Rev. B 75, 174414 /H208492007 /H20850. 15G. Tatara, H. Kohno, and J. Shibata, J. Phys. Soc. Jpn. 77,031003 /H208492008 /H20850. 16A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 /H208492005 /H20850. 17G. Tatara, T. Takayama, H. Kohno, J. Shibata, Y. Nakatani, and H. Fukuyama, J. Phys. Soc. Jpn. 75, 064708 /H208492006 /H20850. 18L. Heyne et al. , Phys. Rev. Lett. 100, 066603 /H208492008 /H20850. 19L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, and S. S. P. Parkin, Nature /H20849London /H20850443, 197 /H208492006 /H20850. 20S. Murakami, N. Nagaosa, and S. C. Zhang, Phys. Rev. B 69, 235206 /H208492004 /H20850. 21D. Culcer, J. Sinova, N. A. Sinitsyn, T. Jungwirth, A. H. Mac- Donald, and Q. Niu, Phys. Rev. Lett. 93, 046602 /H208492004 /H20850. 22J. Shi, P. Zhang, D. Xiao, and Q. Niu, Phys. Rev. Lett. 96, 076604 /H208492006 /H20850. 23P. Zhang, Z. Wang, J. Shi, D. Xiao, and Q. Niu, Phys. Rev. B 77, 075304 /H208492008 /H20850. 24C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C 4, 916 /H208491971 /H20850. 25S. Wang, Y. Xu, and K. Xia, Phys. Rev. B 77, 184430 /H208492008 /H20850. 26D. Culcer and R. Winkler, arXiv:0802.3717 /H20849unpublished /H20850.GEN TATARA AND PETER ENTEL PHYSICAL REVIEW B 78, 064429 /H208492008 /H20850 064429-6

PhysRevB.77.174410.pdf

Shape effects on magnetization state transitions in individual 160-nm diameter Permalloy disks Zhigang Liu *and Richard D. Sydora Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2G7 Mark R. Freeman Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2G7 and National Institute for Nanotechnology, 11421 Saskatchewan Drive, Edmonton, Alberta, Canada T6G 2M9 /H20849Received 29 November 2007; revised manuscript received 4 March 2008; published 8 May 2008 /H20850 The spin dynamics in individual Permalloy nanodisks has been investigated by using time-resolved magneto-optical Kerr effect microscopy. Transitions between the vortex and quasisingle domain states havebeen observed by sweeping the applied bias field, and the critical bias fields for triggering vortex annihilationand nucleation have been determined by associated frequency shifts of 5–10 GHz in ferromagnetic resonance.The shape of the nanodisks has to be taken into account in three-dimensional micromagnetic simulations toobtain consistent results for the critical fields when compared to the experiment. DOI: 10.1103/PhysRevB.77.174410 PACS number /H20849s/H20850: 75.75. /H11001a, 75.40.Gb, 75.60.Ej, 76.50. /H11001g In recent years, remarkable progress has been made in exploring the equilibrium and dynamic properties of smallmagnetic elements, which is motivated by their applicationsin magnetic storage technology such as magnetoresistive ran-dom access memory. 1,2Many other applications in spintron- ics, which rely on the spin degree of freedom in electronicdevices, are also based on nanoscale magnetic systems. 3The demands for a high storage density, a fast processingspeed, and low energy consumption have brought togetherknowledge and techniques to spur the subfield ofnanomagnetism. 4,5 One of the most interesting and useful features of nano- magnets is their different ground states under certain condi-tions, which can form a technical basis for encoding infor-mation. In general, the ground states result from thecompetition of all terms contributing to the Hamiltonian ofthe system that leads to a minimization of the total energy.The vortex and quasisingle domain states are the two mainconfigurations in a ferromagnetic nanodisk. The demagneti-zation field plays the key role in favoring the vortex configu-ration in low external fields, with the size of the vortex coredetermined by competition with the exchange interaction. By using methods such as energy functional theory, variationaltheory, and micromagnetic simulation, a variety of analyticaland computational studies were reported to elucidate the dy-namics of the two states and the transitions between them. 6–8 However, a potential problem arises because most theories and calculations have modeled the nanodisk as a perfect cyl-inder /H20849with a rectangular cross section from the side view /H20850, while with current fabrication technologies, it is difficult toapproximate the samples in that way. In this paper, we showthat the shape of the nanodisks significantly affects thevortex-to-quasisingle domain state transitions. Many experimental studies on nanomagnets measure the total signals from a large number of elements in two-dimensional arrays, self-assembled composite, etc. 9–11Un- certainties due to size and shape variation, and in some casesdipolar coupling within arrays, can give unclear informationconcerning the behavior of state transitions. Experiments onindividual nanomagnets below 200 nm have recently begunto be reported, 12–14which focused on either the vortex stateor the quasisingle domain state. In this work, we report on observations for both of these states, which were measuredon individual Permalloy nanodisks with a diameter of about160 nm. Time-resolved magneto-optical Kerr effect micros-copy /H20849TRMOKE /H20850was used to monitor the small-angle per- turbation response of the magnetization /H20849spin wave dynam- ics/H20850in the nanodisks. A bias magnetic field was applied parallel to the disk plane, and a short magnetic pulse wasused to excite the spin waves in the nanodisks. The resonantfrequency is a sensitive fingerprint of the equilibrium mag-netic configuration, providing insight into the details of hys-teresis in small structures complimentary to that obtained bymagnetic force microscopy, Lorentz microscopy, or electronholography. The Permalloy disks were fabricated on sapphire sub- strates by standard electron beam lithography /H20849EBL /H20850, metal- lization, and lift-off procedures. Then, photolithography wasperformed to fabricate gold transmission lines near the Per-malloy disks for excitation. A micrograph of a sample isshown in Fig. 1/H20849a/H20850, wherein the pump-probe measurement scheme is also sketched. The experimental details, includingthe fast pulse generation using a GaAs photoconductive switch, are described in previous work 15–17and remain simi- lar here. The pulse field has both in-plane and out-of-planecomponents since the disks with good signals have to bevery close to the transmission lines; the peak amplitude is inthe range of 10 3A/m, as can be estimated by sampling the current with a fast oscilloscope and then calculating the fieldwith the Biot–Savart law. Scanning electron microscope/H20849SEM /H20850images of the nanodisks are shown in Figs. 1/H20849b/H20850and 1/H20849c/H20850. The tops of the disks are somewhat domed as a result of shadowing during deposition through the double-layer resistopenings. The deposited thickness was measured as 30 nm ina witness film. In addition, there is an edge roughness on the10-nm scale that arises from the granularity of the film. Thedistance between neighboring disks is about 900 nm /H20849/H110225 disk diameters /H20850, allowing dipolar interactions to be ne- glected, and the probing laser was focused on a single disk/H20849spot size of /H11021700 nm /H20850. The time traces in Fig. 2show the characteristic time- domain response of the out-of-plane component of magneti-PHYSICAL REVIEW B 77, 174410 /H208492008 /H20850 1098-0121/2008/77 /H2084917/H20850/174410 /H208495/H20850 ©2008 The American Physical Society 174410-1zation /H20849Mz/H20850to the pulsed excitation. The disk is in a single- domain state when sufficiently saturated by strong bias fieldssuch as H 0=95.5 kA /m in Fig. 2/H20849a/H20850. It should be noted that there is no extrinsic dephasing due to averaging the signalsfrom an ensemble of magnets. The decay time constant1.8/H110060.3 ns reasonably agrees with those measured in thin films or bulk Permalloy. At very small bias fields /H20851H 0 =0.24 kA /m in Fig. 2/H20849b/H20850/H20852, the disk is always in a vortexstate, exhibiting much higher modal frequencies than the uni- formly magnetized state. For intermediate bias fields, dis-torted versions of both the single domain and vortex statescan be stable, depending upon the magnetic history. Figures2/H20849c/H20850and2/H20849d/H20850show the measurements for H 0=39.8 kA /m. In this case, the magnetization is only partially saturated toform a quasisingle domain or Cstate, as illustrated by the schematic insets. The bending of the “ C” shape increases with decreasing bias field until at the nucleation fieldstrength H nu, the magnetization will close around a core and generate a vortex state. At low bias fields, the vortex corestays near the center of the disk /H20851inset of Fig. 2/H20849b/H20850/H20852. The core is pushed toward the edge by increasing H 0until it is ex- pelled when it reaches the disk edge at the annihilation fieldstrength H an. These transitions are identifiable from the measured wave forms and most conveniently by their frequencies, which areobtained by Fourier transformation. For the quasisingle do-main state, the relatively uniform oscillation is the only sig-nificant mode in the spectra /H20851see the insets of Figs. 2/H20849c/H20850and 2/H20849e/H20850/H20852, and the frequency f 0was given by Guslienko et al.18 by using a modified Kittel’s formula f0=/H92530 2/H9266/H20881H02+H0Ms/H208511−3 F/H20849/H9252/H20850/H20852, /H208491/H20850 where Msis the saturation magnetization, /H92530is the gyromag- netic ratio, and F/H20849/H9252/H20850is the effective demagnetizing factor for a cylinder with /H9252equal to the thickness-to-radius ratio. For the vortex state, the high-frequency dynamics19exhibits rela- tively complicated profiles in the frequency domain, and f0is chosen as the frequency of the primary mode /H20851with the high- est spectral density; see the insets of Figs. 2/H20849b/H20850and2/H20849d/H20850/H20852; the observed secondary modes will be discussed below. The variation in f0as a function of H0are plotted as squares in Fig. 3/H20849a/H20850. The quasisingle domain state frequen- cies closely follow the modified Kittel’s formula /H20849gray curve /H20850 except for the low- H0region in which the C-shape configu- ration leads to significantly lower frequencies than those pre-dicted by assuming an ideal uniform state. When H 0is low- ered below 29 kA/m, there is a frequency jump of /H1101110 GHz as a result of a single domain to vortex state transition /H20849vor- tex nucleation /H20850. If the applied field amplitude is increased while in the vortex state, the resonance frequency is main-tained at /H1101113 GHz, which is characteristic of Permalloy disks at this aspect ratio, 20–22until H0is larger than /H1101140 kA /m, whereupon f0drops to /H1101111 GHz. The lower frequency is characteristic of a thinner disk and suggests thatthe vortex core is affected by the sloping disk edge. Anabrupt frequency drop of /H110115 GHz is observed when H 0in- creases to /H1101168 kA /m and registers the vortex to quasisingle domain transition /H20849vortex annihilation /H20850. The measured nucle- ation and annihilation fields corresponded within 1 kA/m forrepeated bias field sweeps and had variations of /H110111.6 kA /m for different disks. In a single sweeping cycle, it was observed that a transi- tion can be spontaneously triggered when the bias field wasfixed in the critical region. Figure 4/H20849a/H20850shows an example when H 0was fixed at 67.4 kA/m. Before the moment re- corded for the annihilation /H20851marked by the arrow in Fig. FIG. 1. /H20849Color online /H20850/H20849a/H20850Micrograph of the sample showing the gold transmission lines /H20849yellow /H20850and the Permalloy patterns /H20849light green /H20850; the visible Permalloy structures are useful for alignment and for locating the nanomagnets. The circuit connections forTRMOKE measurement are also sketched, with the photoconduc-tive switch /H20849PCS/H20850on one end of the transmission lines and a bias voltage on the other end /H20849V=10–15 V /H20850./H20849b/H20850SEM image of the Permalloy disks captured at a 45° tilted angle. /H20849c/H20850SEM top view of one of the disks shown in /H20849b/H20850; its diameter is displayed by the SEM software, which reads “164.5 nm.” FIG. 2. /H20851/H20849a/H20850–/H20849d/H20850/H20852Evolutions of Mzas a function of the pump- probe optical delay time measured under different bias fields andground states /H20849quasisingle domain or vortex /H20850, as illustrated by the cartoons in each panel. The power spectral densities /H20849PSD/H20850of the time traces are shown by the insets, with the arrows marking thecharacteristic frequency f 0as the indicator for state transitions dis- cussed in the text.LIU, SYDORA, AND FREEMAN PHYSICAL REVIEW B 77, 174410 /H208492008 /H20850 174410-24/H20849a/H20850/H20852, the vortex state was sustained for more than 10 min /H20849each 12 ps delay time step in the pump-probe measurement took abou t4si n real time /H20850. A similar behavior was also observed for the vortex nucleation process when H0was fixed at 29.1 kA/m /H20851Fig.4/H20849b/H20850/H20852; the quasisingle domain state had survived for more than 10 min before the vortex ap-peared. These facts demonstrate that when measuring asingle nanomagnet, its magnetic properties /H20849not only f 0but also the magnetization M, the susceptibility /H9273, and so on /H20850in a single bias field sweep should show an effectively discon-tinuous change when a state transition occurs. In contrast, ifconventional hysteresis data /H20849Mas a function of H 0/H20850were obtained by averaging over an array,9,23,24these state transi- tions would appear to be gradually completed within a smallrange of H 0, mostly due to the variation of shape and size within the array, and hence cannot grasp the details revealedin the present work. Similarly, in time-resolved measure-ments, the spontaneous switching events can only be ob-served by measuring individual nanomagnets; the transitionevents are stochastic among different disks and would pro-duce incoherent temporal data if many disks were collec-tively measured.To better understand the observed state transitions and frequency data, micromagnetic simulations were performedbased on the Landau–Lifshitz–Gilbert equation. 25In the cal- culations, Ms=8.2/H11003105A/m,/H92530=1.854 /H110031011s−1T−1, and the exchange stiffness coefficient A=1.0/H1100310−11J/m. A large Gilbert damping constant /H9251=1.8 was used for fast ground state stabilization so that reasonable estimations forH anand Hnucan be made. The sweep step for the bias field was 0.8 kA/m near critical fields and was larger for unim-portant regions. Then, in accordance to our experimentalconditions, the excitation pulse was applied to the systemunder a finite temperature T=350 K /H20849an upper-limit estima- tion, considering the laser heating /H20850and the relaxation dynam- ics were simulated with the real damping constant /H9251=0.008 to test the stability of corresponding states and find moreaccurate ranges for H anand Hnu. The corrections due to these perturbations were below 0.8 kA/m. The finite-element dis-cretization was done on a 64 /H1100364/H110038 rectangular grid, 26and two shapes for the nanodisk were used /H20849see the inset of Fig. 3/H20850; a flat-topped cylinder similar to those adopted by the aforementioned theoretical and computational work and amore realistic “domed” cylinder to qualitatively model theactual samples are shown in Figs. 1/H20849b/H20850and1/H20849c/H20850. For both models, granular defects of 5–10 nm in size were randomlygenerated at the edges. Simulation results for the f 0-H0curves are plotted in Fig. 3/H20849a/H20850/H20849circles for the domed cylinder model and triangles for the flat cylinder model /H20850to compare to the measurements. Despite small discrepancies in the frequencies, the main fea-tures are reproduced reasonably well, such as the low- H 0 deviation from the modified Kittel’s formula due to the C state and the small frequency drop when H0/H1102247.8 kA /m. Concerning the state transitions, the domed model fits theexperiments much better than the flat model. A significantdifference occurs for the vortex annihilation field; the flatcylinder model gives H an/H1101587.5 kA /m, which reasonably agrees with the results in Ref. 6, while the domed cylinder (a) (b) FIG. 3. /H20849Color online /H20850Spin wave frequencies of the Permalloy disk. The measured data are plotted by squares, and the simulatedresults obtained by using the domed cylinder model and the flatcylinder model are plotted as circles and triangles, respectively. /H20849a/H20850 Hysteresis behavior of the primary modal frequency /H20849f 0/H20850as a func- tion of the bias field /H20849H0/H20850, with the arrows indicating the sweep direction of the bias field. The representative magnetization statesare also sketched. The inset 3D cartoon shows the shapes of the twomodels. The gray curve is calculated based on Eq. /H208491/H20850/H20849for our samples, /H9252=0.375 and F/H20849/H9252/H20850/H110150.1537, see Ref. 18for details /H20850; the gray dashed curve gives a reference calculation based on the un-modified Kittel’s formula, i.e., F/H20849 /H9252/H20850=0./H20849b/H20850Frequencies of the sec- ondary modes in the vortex states. The insets show the simulatedmagnitude and phase distributions of the primary mode /H2085113.1 GHz, see/H20849a/H20850/H20852and the secondary mode /H2084912.4 GHz /H20850when H 0=0. The color bars /H20849not shown /H20850are scaled by the maximum and minimum of individual maps. FIG. 4. /H20849a/H20850The solid curve shows the temporal scan of Mzwith an abrupt change in the precession behavior, indicating a vortex-to-quasisingle domain transition /H20849marked by the arrow /H20850; the dashed curve shows an immediately followed scan to confirm the disk wasalready in the quasisingle domain state. The bias field was fixed at67.4 kA/m during the scans. /H20849b/H20850Similar consecutive scans for de- tecting a quasisingle domain to vortex transition. The bias field wasfixed at 29.1 kA/m.SHAPE EFFECTS ON MAGNETIZATION STATE … PHYSICAL REVIEW B 77, 174410 /H208492008 /H20850 174410-3model leads to a much smaller Han/H1101566.9 kA /m, which is very close to the measured value. The process can be easilyvisualized through the simulated spatial images, in which thevortex core can reach the round-corner surface at a relativelylow bias field. The upper portion of the core will then begeometrically destabilized by the curved shape, resulting inthe earlier annihilation. It is also interesting to investigate the distributions of the vortex-state modes, which, under a specific H 0, exhibit cer- tain secondary peaks near the primary mode with compa-rable intensities. The measured and simulated results for thesecondary modes are plotted in Fig. 3/H20849b/H20850. These modes, to- gether with the primary modes shown in Fig. 3/H20849a/H20850, appear to fill in discrete frequency levels when the bias field increases.This type of mode distribution was intensively investigatedin larger disks wherein spatially resolved measurements werepossible. 20,21The high-frequency vortex state oscillations can be quantized in radial and azimuthal directions by the num- ber pair /H20849n,m/H20850, which indexes the orders of specific spin wave modes. This picture does not cleanly map onto oursmall radius samples with imperfect circular symmetry,where a symmetrical, “uniform” oscillation /H208490, 0/H20850is no longer an eigenmode of the disk. No radial nodes are foundin simulations, suggesting that n=0 under the present exci- tation conditions. The multiple peaks in the range of 10–13GHz exhibit stationary phase as well as left- and right-phasecirculation patterns that can only approximately be identifiedwith different azimuthal indices m. The insets of Fig. 3/H20849b/H20850 present the simulated magnitude and phase distributions ofthe 13.1 GHz /H20849primary /H20850and the 12.4 GHz /H20849secondary /H20850modes when H 0=0. The asymmetry of the structure selects particu- lar nonuniform modes. The azimuthal distribution is not uni-form for the primary mode, although no nodes are visible. Inaddition, this mode is stationary after being reconstructed inthe time domain by using the method introduced in Ref. 21. The secondary mode, however, shows a clear quantization/H20849m=4/H20850and is rotating counterclockwise. When the bias field becomes strong enough /H20849H 0/H1102245 kA /m/H20850, the primary mode shifts to lower frequen- cies. This trend indicates an increasing influence of the thin-ner edge of the disk as the vortex is driven far off center. Thedeclining primary mode frequency is more accurately de-scribed by the domed cylinder model than the flat cylindermodel /H20851see the data between 45 and 65 kA/m in Fig. 3/H20849a/H20850/H20852. For the uniform-to-vortex state transition with decreasing field /H20849vortex nucleation /H20850, the simulations show that the criti- cal bias field does nothave significant dependence on the models with different shapes. The H nuvalues determined by both models are consistent with the measurements /H20851Fig.3/H20849a/H20850/H20852 and also agree with the results from other simulation work.6 The nucleation process simulated by the two models can becompared by using the images presented in Fig. 5/H20849for faster execution, we used /H9251=1.8 and T=0 K in these calculations, so the evolution is not on real time scales /H20850. The bias field was set slightly above the critical range to fully stabilize the qua-sisingle domain state /H20851Figs. 5/H20849a1/H20850and5/H20849b1/H20850/H20852so that a subse- quent 0.8 kA/m step down can trigger the nucleation process.We found that in both models, the vortex core emerges fromthe bottom edge of the nanodisk /H20849although for the flat cylin- der model, the bottom and top sides are symmetric in geom-etry/H20850, while the spins near the top of the nanodisk still remain in quasisingle domain state /H20851Figs. 5/H20849a2/H20850and5/H20849b2/H20850/H20852. Note that in these two cases, the spins in the nucleation volume precessout of the disk plane before forming a vortex core penetrat-ing the entire thickness. The shape at the top of the disk doesnot affect the early stage of vortex nucleation near the bot-tom, and the nucleation fields produced by the two modelsdiffer by just 1.6 kA/m, which is much smaller than thediscrepancy for the annihilation fields. After the nucleation,the vortex core moves from the edge to its equilibrium loca-tion near the disk center, and its height increases when pass-ing across the edge region /H20851Figs. 5/H20849b3/H20850and5/H20849b5/H20850/H20852. To summarize, we have measured the time-resolved mag- netic dynamics in individual Permalloy disks of 160 nm di-ameter. The fundamental mode frequencies of the nanodiskexhibit a distinct hysteresis behavior as a function of thein-plane bias field, and the critical fields for triggering thevortex annihilation and nucleation processes have been (a1) (a3)(a2) (a5)(a4)(b1) (b3)(b2) (b5)(b4) τ= 5.1 nsτ=0 τ= 5.3 nsτ= 5.2 ns τ= 8.3 nsτ= 6.4 nsτ=0 τ= 6.7 nsτ= 6.6 ns τ= 9.7 ns min max FIG. 5. /H20849Color online /H20850The evolution of magnetization configu- ration in the vortex nucleation process as simulated by the flat cyl-inder model /H20851/H20849a1/H20850–/H20849a5/H20850/H20852and the domed cylinder model /H20851/H20849b1/H20850–/H20849b5/H20850/H20852. The disks are stabilized into equilibrium with H 0=29.44 kA /m /H20849a1/H20850and 27.85 kA/m /H20849b1/H20850, respectively. Then, the bias fields de- crease to 28.65 and 27.06 kA/m, respectively, to trigger the nucle-ation, representative snapshots are recorded in /H20849a2/H20850–/H20849a5/H20850and/H20849b2/H20850– /H20849b5/H20850. The /H9270values are the “effective” time in the simulations with a large damping constant /H9251=1.8, so these snapshots do not reflect the real time points /H20849in real time, the evolutions would be much slower /H20850. In each frame, the intensity of Mzat the top and bottom layers of the 3D models are shown by the colored surfaces; thecolor bar shows a fixed minimum value /H20849−1, assigned for cells outside the magnetic disk /H20850and different maximum values for dif- ferent frames. The small cones between the two surfaces representthe spins in the 3D models that are within the vortex core /H20849the criterion Mbeing at least 25° off the disk plane /H20850; the colors of these cones are also scaled with M z. For clarity, a zoom-in view of the vortex core is presented aside /H20849b3/H20850.LIU, SYDORA, AND FREEMAN PHYSICAL REVIEW B 77, 174410 /H208492008 /H20850 174410-4determined. The realistic shape of the nanodisk has to be considered in micromagnetic simulations to explain the mea-sured critical fields. By modifying the shape of the nanodisk,it would be possible to control the annihilation field over aconsiderable range while keeping the nucleation field un-changed, which could be a useful feature for applications. The methods described in the present work can be applied to more general nanomagnets, such as rings, squares, ormultilayer elements. Issues on the dynamic transition behav-ior and critical bias fields can be analogously addressed. These investigations may benefit a variety of nanoscale tech-nologies such as magnetic quantum cellular automata 27that utilize the magnetization state transitions to store and processinformation. We thank Hue Nguyen for help with EBL fabrication in the NanoFab of the University of Alberta. This work wassupported by NSERC, iCORE, CIFAR, and CRC. *zliu2@ucsc.edu 1Ultrathin Magnetic Structures IV , Applications of Nanomag- netism , edited by B. Heinrich and J. A. C. Bland /H20849Springer, New York, 2005 /H20850. 2J. Åkerman, Science 308, 508 /H208492005 /H20850. 3Magnetoelectronics , edited by M. Johnson /H20849Elsevier, Oxford, 2004 /H20850. 4S. D. Bader, Rev. Mod. Phys. 78,1/H208492006 /H20850. 5C. L. Chien, F. Q. Zhu, and J. G. Zhu, Phys. Today 60/H208496/H20850,4 0 /H208492007 /H20850. 6K. Yu. Guslienko, V. Novosad, Y. Otani, H. Shima, and K. Fuka- michi, Phys. Rev. B 65, 024414 /H208492001 /H20850. 7K. Yu. Guslienko, W. Scholz, R. W. Chantrell, and V. Novosad, Phys. Rev. B 71, 144407 /H208492005 /H20850. 8R. Zivieri and F. Nizzoli, Phys. Rev. B 71, 014411 /H208492005 /H20850. 9R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E. Welland, and D. M. Tricker, Phys. Rev. Lett. 83, 1042 /H208491999 /H20850. 10V. V. Kruglyak, A. Barman, R. J. Hicken, J. R. Childress, and J. A. Katine, J. Appl. Phys. 97, 10A706 /H208492005 /H20850. 11A. V. Jausovec, G. Xiong, and R. P. Cowburn, Appl. Phys. Lett. 88, 052501 /H208492006 /H20850. 12I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman, Science 307, 228 /H208492005 /H20850. 13A. Barman, S. Wang, J. D. Maas, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, Nano Lett. 6, 2939 /H208492006 /H20850. 14V. S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Nat.Phys. 3, 498 /H208492007 /H20850. 15K. S. Buchanan, X. Zhu, A. Meldrum, and M. R. Freeman, Nano Lett. 5, 383 /H208492005 /H20850.16X. Zhu, Z. Liu, V. Metlushko, P. Grütter, and M. R. Freeman, Phys. Rev. B 71, 180408 /H20849R/H20850/H208492005 /H20850. 17Z. Liu, F. Giesen, X. Zhu, R. D. Sydora, and M. R. Freeman, Phys. Rev. Lett. 98, 087201 /H208492007 /H20850. 18K. Yu. Guslienko and A. N. Slavin, J. Appl. Phys. 87, 6337 /H208492000 /H20850. 19The gyrotropic motion of the vortex core, with a low frequency of/H110111 GHz, is also visible in some of our measurements, such as, for example, in Fig. 2/H20849b/H20850. 20J. P. Park and P. A. Crowell, Phys. Rev. Lett. 95, 167201 /H208492005 /H20850. 21M. Buess, R. Höllinger, T. Haug, K. Perzlmaier, U. Krey, D. Pescia, M. R. Scheinfein, D. Weiss, and C. H. Back, Phys. Rev.Lett. 93, 077207 /H208492004 /H20850. 22C. E. Zaspel, B. A. Ivanov, J. P. Park, and P. A. Crowell, Phys. Rev. B 72, 024427 /H208492005 /H20850. 23R. P. Cowburn, J. Phys. D 33,R 1 /H208492000 /H20850. 24M. Grimsditch, P. Vavassori, V. Novosad, V. Metlushko, H. Shima, Y. Otani, and K. Fukamichi, Phys. Rev. B 65, 172419 /H208492002 /H20850. 25We used our own micromagnetics code /H20851Z. Liu, Ph.D. thesis, University of Alberta, 2008 /H20852.This was benchmarked against M. Scheinfein’s, LLG Micromagnetics Simulator™ /H20849http:// llgmicro.home.mindspring.com/ /H20850. The code has also been checked by using OOMMF’s standard problem No. 4 /H20849http:// www.ctcms.nist.gov/~rdm/mumag.org.html /H20850. 26Test simulations were also performed with 32 /H1100332/H110038 and 64 /H1100364/H1100316 grid dimensions, and the results did not show a sig- nificant difference. 27A. Imre, G. Csaba, L. Ji, A. Orlov, G. H. Bernstein, and W. Porod, Science 311, 205 /H208492006 /H20850.SHAPE EFFECTS ON MAGNETIZATION STATE … PHYSICAL REVIEW B 77, 174410 /H208492008 /H20850 174410-5

PhysRevLett.108.074501.pdf

Turbulence in Noninteger Dimensions by Fractal Fourier Decimation Uriel Frisch,1Anna Pomyalov,2Itamar Procaccia,2and Samriddhi Sankar Ray1 1UNS, CNRS, OCA, Laboratoire Lagrange, Boı ˆte Postale 4229, 06304 Nice Cedex 4, France 2Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel (Received 5 August 2011; published 13 February 2012) Fractal decimation reduces the effective dimensionality Dof a flow by keeping only a (randomly chosen) set of Fourier modes whose number in a ball of radius kis proportional to kDfor large k. At the critical dimension Dc¼4=3there is an equilibrium Gibbs state with a k/C05=3spectrum, as in V. L’vov et al. ,Phys. Rev. Lett. 89, 064501 (2002) . Spectral simulations of fractally decimated two-dimensional turbulence show that the inverse cascade persists below D¼2with a rapidly rising Kolmogorov constant, likely to diverge as ðD/C04=3Þ/C02=3. DOI: 10.1103/PhysRevLett.108.074501 PACS numbers: 47.27.Gs, 05.20.Jj In theoretical physics a number of results have been obtained by extending the dimension dof space from directly relevant values such as 1, 2, 3 to noninteger values.Dimensional regularization in field theory [ 1] and the 4/C0/C15expansion in critical phenomena [ 2] are well-known instances. For this, one usually expands the solution in terms of Feynman diagrams, each of which can be analyti- cally continued to real or complex values of d. The same kind of extension can be carried out for homogeneousisotropic turbulence but a severe difficulty appears thenford<2: the energy spectrum EðkÞcan become negative in some band of wave numbers k, so that this kind of extension lacks probabilistic realizability [ 3]. Nevertheless, in Ref. [ 4], henceforth cited as LPP, it is argued that, should there exist an alternative realizable way of doing the extension below dimension two in which thenonlinearity conserves energy and enstrophy, then an in-teresting phenomenon—to which we shall come back—should happen in dimension 4=3. For diffusion and phase transitions there is a very differ- ent way of switching to noninteger dimensions, namely, toreformulate the problem on a fractal of dimension D(here a capital Dwill always be a fractal dimension) [ 5]. Are we able to do this for hydrodynamics ? Implementing mass and momentum conservation on a fractal is quite a challenge[6]. We discovered a new way of fractal decimation in Fourier space, appropriate for hydrodynamics. Since,here, we are primarily interested in dimensions less thantwo, we shall do our decimation starting from the standardd¼2case. The forced incompressible Navier-Stokes equations for the velocity field can be written in abstract notation as @ tu¼Bðu;uÞþfþ/C3u; (1) Bðu;uÞ¼/C0 u/C1ruþrp; /C3¼/C23r2; (2) where ustands for the velocity field uðx1;x2;tÞ,ffor the force fðx1;x2;tÞ,pis the pressure and /C23the viscosity. The velocity uis taken in the space of divergenceless velocityfields which are 2/C25periodic in x1andx2, such that uðt¼0Þ¼u0. Now, we define a Fourier decimation op- erator PDon this space of velocity fields: Ifu¼X k2Z2eik/C1x^uk;thenPDu¼X k2Z2eik/C1x/C18k^uk:(3) Here, /C18kare random numbers such that /C18k¼/C261with probability hk 0with probability 1/C0hk;k/C17jkj:(4) To obtain D-dimensional dynamics we choose hk¼Cðk=k 0ÞD/C02; 0<D /C202; 0<C/C201;(5) where k0is a reference wave number; here C¼k0¼1. All the /C18kare chosen independently, except that /C18k¼/C18/C0k to preserve Hermitian symmetry. Our fractal decimation procedure removes at random—but in a time-frozen(quenched) way—many modes from the klattice, leaving on average NðkÞ/k Dactive modes in a disk of radius k. The randomness in the choice of the decimation will becalled the disorder. Observe that P Dis a projector, that it commutes with the viscous diffusion operator /C3and that it is self-adjoint for the energy ( L2) norm, defined as usual as kuk2/C17 ð1=ð2/C25Þ2ÞRjuðxÞj2d2x, where the integral is over a 2/C25/C2 2/C25periodicity square. The conservation of energy (by the nonlinear term) for sufficiently smooth solutions ofthe Navier-Stokes equation can be expressed asðu;Bðu;uÞÞ¼0, where ðu;wÞ/C17ð1=ð2/C25Þ 2ÞRuðxÞ/C1wðxÞd2x is the L2scalar product. The decimated Navier-Stokes equation , written for an incompressible field v, takes the following form @tv¼PDBðv;vÞþPDfþPD/C3v: (6) The initial condition is v0/C17vðt¼0Þ¼PDu0. Thus, at any later time PDv¼v. Energy is again conserved; indeed ðv;PDBðv;vÞÞ¼0, as is seen by moving the self-adjoint operator PDto the left hand side of the scalar product and using PDv¼v. For enstrophy conservation, take the curlPRL 108, 074501 (2012) PHYSICAL REVIEW LETTERSweek ending 17 FEBRUARY 2012 0031-9007 =12=108(7) =074501(4) 074501-1 /C2112012 American Physical Societyof (1); the quadratically nonlinear term in the vorticity equation is then Bvortð!; !Þ/C17/C0 u/C1r!, where uis ex- pressed in terms of !by Biot–Savart. The relation ð!; B vortð!; !ÞÞ¼0expresses enstrophy conservation. In the decimated case, the proof of enstrophy conservation isidentical to that for energy conservation with Breplaced by B vort. If, in addition to decimation, we apply a Galerkin trun- cation which kills all the modes having wave numbersbeyond a threshold K G, the surviving modes constitute a dynamical system having a finite number of degrees of freedom. Such truncated systems with no forcing and no viscosity have been studied by Lee, Kraichnan and others[8]. Using suitable variables related to the real and imagi- nary parts of the active modes, the dynamical equationsmay be written as _y /C11¼P /C12/C13A/C11/C12/C13y/C12y/C13. For the purely Galerkin-truncated (not decimated) case it is well known that the above dynamical system satisfies aLiouville theoremP /C11@_y/C11=@y/C11¼0and thus conserves volume in phase space. This in turn implies the existenceof (statistically) invariant Gibbs states for which the proba- bility is a Gaussian, proportional to e /C0ð/C11Eþ/C12/C10Þ, where E¼P kj^ukj2is the energy and /C10¼P kk2j^ukj2is the ens- trophy. Such Gibbs states , called by Kraichnan absolute equilibria , play an important role in his theory of the two- dimensional (2D) inverse energy cascade [ 9]. If we now combine inviscid, unforced Galerkin truncation and deci-mation, it is easily checked that the Liouville theorem still holds, provided the decimation preserves Hermitean sym- metry. For such Gibbs states, and any active mode ( /C18 k¼ 1), one easily checks that the mean square energy hjukj2i¼ C0=ð/C11þ/C12k2Þ, where C0>0does not depend on k. The corresponding energy spectrum is the mean energy EðkÞof modes having a wave number between kandkþ1.U pt o fluctuations of the disorder, the number of active modes insuch a shell is Oðk D/C01Þ. Thus, EðkÞ¼kD/C01 /C11þ/C12k2; /C12> 0;/C11 > /C0/C12; (7) where various positive constants have been absorbed into a new definition of /C11and/C12. An instance is enstrophy equipartition :/C11¼0(all the modes have the same ens- trophy), for which the energy spectrum is EðkÞ/kD/C03.A s observed in LPP, this equilibrium spectrum coincides with the Kolmogorov 1941 k/C05=3spectrum at the critical dimen- sionDc¼4=3. Note that such Gibbs states are only condi- tionally Gaussian, for a given disorder. Otherwise, they arehighly intermittent, since a given high- kmode will be active only in a small fraction of the disorder realizations.We also note that similar phenomena have been observedin shell models [ 10]. The form ( 7) of the D-dimensional absolute equilibria also allows for the kind of Bose condensation in the gravest modes (here, those with unit wave number) found by Kraichnan for 2D turbulence. For this the ‘‘inversetemperature’’ /C11must be taken negative, close to its mini- mum realizable value /C0/C12. The arguments used by Kraichnan to predict an inverse Kolmogorov k /C05=3energy cascade for high-Reynolds number 2D turbulence withforcing near an intermediate wave number k injcarry over to the decimated case with D< 2. In particular the con- servation of enstrophy blocks energy transfer to high wavenumbers. This in itself does not imply that the energy willcascade to wave numbers smaller than k inj, producing a k-independent energy flux: it might also linger around and accumulate near kinj. It is now our purpose to show that for 4=3<D /C202, when the energy spectrum is prescribed to be EðkÞ¼k/C05=3 over the inertial range, there is a negative energy flux /C5E, vanishing linearly with D/C04=3near the critical dimen- sionDc¼4=3. For this we shall assume that a key feature of the two-dimensional energy cascade carries over tolower dimensions, namely, the existence of scaling solu-tions with local (in Fourier space) dynamics, so that theenergy transfer is dominated by triads of wave numbers with comparable magnitudes. Let us now decompose the energy inertial range into bands of fixed relative width, sayone octave, delimited by the wave numbers 2 0,21,22, etc. Because of locality there is much intraband dynamics but,of course, interband interactions are needed to obtain anenergy flux. Pure intraband dynamics (with no forcingand dissipation) would lead to thermalization. For dimen-sional reasons, thermalization and interband transfer have the same time scale, namely, the eddy turnover time k /C03=2E/C01=2ðkÞ. To get a handle on the combined intraband and interband dynamics we perform a thermodynamic thought experi- ment in which we artificially separate them in time. In the first phase, starting from a k/C05=3spectrum we prevent the various bands from interacting by introducing (impene-trable) interband barriers at their edges. In each band, themodes will then thermalize and achieve a Gibbs state with a spectrum ( 7) in which /C11and/C12are determined by the constraints that the total band energy and enstrophy be thesame as for the /C05=3spectrum. For example, in the first band this gives the constraints ( n¼0for the energy and n¼2for the enstrophy) Z 2 1dkkn½kD/C01=ð/C11þ/C12k2Þ/C0k/C05=3/C138¼0; (8) a system of two transcendental equations for the parame- ters/C11and/C12, which we solved numerically. For D¼2, the corresponding absolute equilibrium spectrum, obtained bysubstituting these values in ( 7), is shown in Fig. 1, together with the /C05=3spectrum. The two spectra are very close to each other. Specifically, in 2D the absolute equilibriumspectrum exceeds the /C05=3spectrum by about 10% at any lower band edge and by about 5% at any upper band edge. Of course, as we approach the critical dimension D c¼4=3the discrepancy goes to zero and can easily bePRL 108, 074501 (2012) PHYSICAL REVIEW LETTERSweek ending 17 FEBRUARY 2012 074501-2calculated perturbatively in D/C04=3. In the second phase of our thought experiment, we consider two adjacentbands, e.g., [ 2 0,21] and [ 21,22] that have thermalized, starting from the same k/C05=3spectrum and we remove the interband barrier at 21. A new thermalization leads then to an absolute equilibrium in the band [ 20,22], which again, can be easily calculated. In 2D, before the removal, theenergy between 2 0and21was 0.555. After the new ther- malization, this energy is found to have increased by 0.005 51. Thus energy has been transferred from the upperband [ 2 1,22] to the lower band [ 20,21]. Close to Dc¼4=3, we can again apply elementary perturbation techniques andobtain for the upper-to-lower-band energy transfer0:009ðD/C04=3Þto leading order. Our thermodynamic thought experiment thus suggests that the energy fluxvanishes linearly with D/C04=3, being negative above the critical dimension, which implies an inverse cascade.Variants of this thought experiment involving more bandsgive similar results. In the K41 inertial range, the energy spectrum and the energy flux /C5 Eare related by EðkÞ¼ CKolj/C5Ej2=3k/C05=3, where CKolis the Kolmogorov constant; thus the Kolmogorov constant diverges as ðD/C04=3Þ/C02=3. A closure calculation of eddy-damped quasinormalMarkovian (EDQNM) type also predicts a divergence with a /C02=3exponent. Kraichnan’s ideas about the inverse cascade in 2D got growing support a few years later from direct numerical simulations, which eventually achieved the resolution of 32 768 2modes [ 11]. As to our idea about the robustness of the inverse cascade and the growth of the Kolmogorovconstant when lowering the dimension D, some support can be already provided, using a D-dimensional decimated variant of spectral direct numerical simulation: First onegenerates an instance of the disorder, that is the list ofactive and inactive Fourier modes; then, one applies stan-dard time marching algorithms and, at each time step, setsto zero all inactive modes. In addition to the well-knowndifficulties of simulating 2D turbulence (see, e.g., [ 11] and references therein), there are new difficulties.A few words about the numerical implementation. We integrate the decimated Navier-Stokes Eq. ( 6) in vorticity representation. Instead of using as damping the viscous operator /C3¼/C23/C1(where /C1/C17r 2is the Laplacian), we use /C3/C17/C0/C23/C1þ2/C0/C22/C1/C02;/C23 > 0;/C22 > 0;(9) whose Fourier symbol is /C0/C23k4/C0/C22k/C04. In other words, we use hyperviscosity to avoid wasting resolution on theenstrophy cascade and large-scale friction to prevent anaccumulation of energy on the gravest modes and thusallow eventual convergence to a statistical steady state.The results reported here have a resolution of N¼3072 collocation points in the two coordinates. Time marching is done by an Adams-Bashforth scheme combined with ex- ponential time difference (ETD) [ 12] with a time step between 5/C210 /C05and 10/C04, depending on dimension. Energy injection at the rate "is done in a band of width three around kinj¼319 by adding to the time rate of change of the Fourier amplitude of the vorticity a term proportional to the inverse of its complex conjugate [ 13]. This allows a k-independent and time-independent energy injection. As Dis decreased the amplitude of this forcing is increased to keep the total energy injection on active modes fixed at "¼0:01. The damping parameters are /C23¼10/C011and/C22¼0:1. Runs are done concurrently for different values of Don a high-performance cluster at the Weizmann Institute and take typically a few thousand hours of CPU per run to achieve a statistical steady state. Energy spectra are obtained by angular averages over Fourier-space shells of unit width EðKÞ/C171 2X K/C20k<Kþ1j^vðkÞj2; (10)1.0 1.2 1.4 1.6 1.8 2.0k0.40.60.81.0Ek FIG. 1 (color online). The k/C05=3spectrum (continuous) and the associated 2D absolute equilibrium with the same energy and enstrophy in the first octave (dashed). 0.5 1 1.5 2 2.5 3012 log10 kE(k)k5/3 1.6 1.8 200.511.5 DPlateauD=1.5 D=2 FIG. 2 (color online). Compensated steady-state spectra for D¼2:0, 1.9, 1.8, 1.7, 1.6, 1.5 from bottom to top with spikes at injection. The inset shows the dependence on Dof the plateau of the compensated spectra, as an average over the interval between vertical dashed lines (with standard deviation errorbars).PRL 108, 074501 (2012) PHYSICAL REVIEW LETTERSweek ending 17 FEBRUARY 2012 074501-3where the ^vðkÞare the Fourier coefficients of the solution of the decimated Navier-Stokes Eq. ( 6). We also need the energy flux /C5EðKÞthrough wave number Kdue to non- linear transfer, defined as /C5EðKÞ/C17X k/C20K^v/C3ðkÞ/C1cNLðkÞ; (11) where cNLðkÞdenotes the set of Fourier coefficients of the nonlinear term PDBðv;vÞin the decimated Navier-Stokes Eq. ( 6) and the asterisk denotes complex conjugation. EðkÞand /C5EðKÞare mostly insensitive to the disorder realization. Figures 2and3(inset) show the steady-state compen- sated energy spectra k5=3EðkÞand the energy fluxes /C5EðkÞ, for various values of Dbetween 2 and 1.5, respectively. Both are quite flat, over a significant range of kvalues, evidence that D-dimensional forced turbulence, Fourier decimated down from the two-dimensional case, preservesthe key feature of two-dimensional turbulence of having aninverse cascade that follows the /C05=3law. Note that the inertial range (the flat region of the compensated energyspectrum) shrinks as the dimension Ddecreases. The absolute value of the energy flux is about 80% of theenergy injection "forD¼2, but drops to less than 50% forD¼1:5. Indeed, as the dimension Dis lowered, there are fewer and fewer pairs of active modes in the forcing band, capable through their beating interaction of drainingthe energy into the infrared direction; thus the energyinjection will be more and more balanced by direct dis-sipation near injection. Preventing this would require asubstantial lowering of the dissipation which in turn re-quires a substantial increase in the resolution at the high- k end. Anyway, the fact that the flux j/C5 Ejbecomes substan- tially lower than injection does not prevent us from calcu-lating the Kolmogorov constant, given (in terms of plateau values) by C Kol¼k5=3EðkÞ=ðj/C5EðkÞj2=3Þ. Figure 3shows the variation of the Kolmogorov constant with dimension.When lowering the dimension from 2 to 1.5, a combined effect of a rise in the compensated spectrum and a drop influx yields a monotonic growth of about a factor ten in the Kolmogorov constant and a substantial growth of errors due to fluctuations within the averaging interval. Probingthe conjectured divergence by moving closer to the criticalpoint D c¼4=3would require much higher resolution. A state-of-the-art 16 3842simulation of sufficient length might shed light. We finally observe that the fractal Fourier decimation procedure—that allows numerical experimentation by spectral simulation—can be started from any integer di- mension and can be applied to a large class of problems incompressible and incompressible hydrodynamics andMHD. It is also applicable to other problems in nonlineardynamics and condensed matter physics, such as criticaldynamical phenomena [ 14], Kardar–Parisi–Zhang dynam- ics [15], and nonlinear wave interactions [ 16]. We are grateful to E. Aurell, H. Chen, H. Frisch, B. Khesin, V. L’vov, T. Matsumoto, S. Musacchio, S. Nazarenko, R. Pandit, and W. Pauls for useful discussions.U. F. and S. S. R.’s work was supported by ANR OTARIEBLAN07-2_183172. A. P. and I. P.’s work was supportedby the Minerva Foundation, Munich, Germany. [1] G.’t Hooft and M. Veltman, Nucl. Phys. B B44, 189 (1972) . [2] K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972) . [3] J.-D. Fournier and U. Frisch, Phys. Rev. A 17, 747 (1978) . [4] V. L’vov, A. Pomyalov, and I. Procaccia, Phys. Rev. Lett. 89, 064501 (2002) . [5] Y. Gefen, A. Aharony, and B. B. Mandelbrot, J. Phys. A 17, 1277 (1984) ; B. O’Shaughnessy and I. Procaccia, Phys. Rev. A 32, 3073 (1985) . [6] The lattice structure that is common to lattice Boltzmann models may be amenable to fractal decimation [ 7]. [7] H. Chen (private communication). [8] T. D. Lee, Q. Appl. Math. 10, 69 (1952); R. H. Kraichnan, J. Acoust. Soc. Am. 27, 438 (1955) . [9] R. H. Kraichnan, Phys. Fluids 10, 1417 (1967) . [10] E. Aurell, G. Boffetta, A. Crisanti, P. Frick, G. Paladin, and A. Vulpiani, Phys. Rev. E 50, 4705 (1994) ; T. Gilbert, V. S. L’vov, A. Pomyalov, and I. Procaccia, Phys. Rev. Lett. 89, 074501 (2002) . [11] G. Boffetta and S. Musacchio, Phys. Rev. E 82, 016307 (2010) . [12] S. M. Cox and P. C. Matthews, J. Comput. Phys. 176, 430 (2002) [13] Z. Xiao, M. Wan, S. Chen, and G. Eyink, J. Fluid Mech. 619, 1 (2009) . [14] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977) . [15] J. P. Bouchaud and M. E. Cates, Phys. Rev. E 47, R1455 (1993) , and references therein. [16] V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, Berlin, 1992).1.5 1.6 1.7 1.8 1.9 2103050 DCKol 11 . 522 . 5−1−0.50 log10 kΠE(K)/εD=1.5 D=2 FIG. 3 (color online). Dependence of the Kolmogorov con- stant on D. The lowest value, at D¼2, is about 5. The inset shows the energy flux normalized by the energy injection "for the same values of Das in Fig. 2.PRL 108, 074501 (2012) PHYSICAL REVIEW LETTERSweek ending 17 FEBRUARY 2012 074501-4

PhysRevB.99.014431.pdf

PHYSICAL REVIEW B 99, 014431 (2019) Ferromagnetic resonance and control of magnetic anisotropy by epitaxial strain in the ferromagnetic semiconductor (Ga 0.8,Fe0.2)Sb at room temperature Shobhit Goel,1,*Le Duc Anh,1,2,†Shinobu Ohya,1,2,3,‡and Masaaki Tanaka1,3,§ 1Department of Electrical Engineering and Information Systems, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2Institute of Engineering Innovation, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 3Center for Spintronics Research Network (CSRN), The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan (Received 2 July 2018; revised manuscript received 21 November 2018; published 25 January 2019) We study the strain dependence of the magnetic anisotropy of room-temperature ferromagnetic semiconductor (Ga 1-x,Fex)Sb (x=20%) thin films epitaxially grown on different buffer layers, using ferromagnetic resonance measurements. We show that the magnetocrystalline anisotropy ( Ki)i n( G a 0.8,Fe0.2)Sb exhibits a dependence on the epitaxial strain and changes its sign from negative (in-plane magnetization easy axis) to positive(perpendicular magnetization easy axis), when the strain is changed from tensile to compressive. Meanwhile,the shape anisotropy ( K sh) is negative and dominant over Ki. Therefore, the effective magnetic anisotropy (Keff=Ki+Ksh) is always negative, leading to the in-plane magnetic anisotropy in all the (Ga 0.8,Fe0.2)Sb samples. This work demonstrates ferromagnetic resonance and strong shape anisotropy at room temperature inIII-V ferromagnetic semiconductors. We also observed very high Curie temperature ( T C/greaterorsimilar400 K) in p-type (Ga,Fe)Sb, which is the highest TCreported so far in III-V based ferromagnetic semiconductors. DOI: 10.1103/PhysRevB.99.014431 I. INTRODUCTION Ferromagnetic semiconductors (FMSs) have attracted much attention since they exhibit both semiconducting andferromagnetic properties, which provide a straightforward ap-proach for integrating spin-dependent phenomena into semi-conductor devices. From FMS thin films, one can inject aspin-polarized current into a nonmagnetic semiconductor us-ing methods such as electrical spin injection [ 1–4] and spin pumping [ 5], without suffering from severe problems such as conductivity mismatch and interface roughness as seen in gen-eral ferromagnetic metal-semiconductor contacts [ 1,2]. This good compatibility with conventional semiconductor technol-ogy is very important for the realization of semiconductorspintronic devices with nonvolatile functions and low powerdissipation [ 6–11]. Thus far, the mainstream studies of FMSs are based on the Mn-doped III-V FMSs such as (Ga,Mn)As.These Mn-doped FMSs, however, maintain ferromagneticorder only at low temperature (the highest Curie temperatureT Cis 200 K in (Ga,Mn)As [ 12]) and they have strong magne- tocrystalline anisotropy, which results in a difficulty to controlthe magnetization by nanofabrication processing. Besides,Mn-doped FMSs are only p-type because Mn acts as an acceptor in III-V semiconductors. These are severe drawbacksthat hinder the use of FMSs in practical spintronic devices. Recently, we found that Fe-doped narrow-gap III-V FMSs can be promising alternatives to overcome the problems ofthe Mn-based FMSs. By using Fe as the magnetic dopants, *goel@cryst.t.u-tokyo.ac.jp †anh@cryst.t.u-tokyo.ac.jp ‡ohya@cryst.t.u-tokyo.ac.jp §masaaki@ee.t.u-tokyo.ac.jpone can grow both n-type FMSs ((In,Fe)As [ 13–15], (In,Fe)Sb [16]) and p-type FMSs ((Al,Fe)Sb [ 17], (Ga,Fe)Sb [ 18,19]) because Fe atoms are in the isoelectronic Fe3+state and do not supply carriers. The most notable feature in the Fe-doped FMSs is their very high T C: Intrinsic room-temperature ferromagnetism has been confirmed in (Ga 1-x,Fex)Sb thin films with the Fe density x/greaterorequalslant23% [ 18], and in (In 1-x,Fex)Sb thin films with x/greaterorequalslant16% [ 16]. Therefore, these new Fe-doped FMSs are expected to be useful materials for spintronic deviceapplications at room temperature. In this paper, we study the growth and magnetic anisotropy (MA) of (Ga 1-x,Fex)Sb (x=0.2) thin films epitaxially grown on different buffer layers and thus subjected to differentepitaxial strains. MA plays an important role in control-ling the magnetization of the ferromagnetic (FM) thin films,which is a fundamental operation of magnetic/spintronic de-vices. Understanding and controlling the MA, thus, are es-sential for device applications of (Ga,Fe)Sb. In the past, re-searchers successfully observed and controlled the MA of theMn-doped III-V FMSs, (Ga,Mn)As [ 20–28] and (In,Mn)As [29–31] ferromagnetic thin films, by epitaxial strain. These Mn-doped FMSs showed a perpendicular-magnetization easyaxis under tensile strain, and an in-plane-magnetization easyaxis under compressive strain. In this work, we have grown (Ga 0.8,Fe0.2)Sb thin films on four different buffer layers (AlSb, GaSb, In 0.5Ga0.5As, and GaAs) by molecular beam epitaxy (MBE) to induce different epitaxial strains rangingfrom tensile to compressive, and examined its effect on theMA of these (Ga 0.8,Fe0.2)Sb thin films. We performed ferromagnetic resonance (FMR) measure- ments to investigate the MA fields of the (Ga 0.8,Fe0.2)Sb thin films. FMR is not only one of the most efficient andpowerful techniques to observe the MA [ 32], but is also used for spin pumping to inject spin angular momentum (spin 2469-9950/2019/99(1)/014431(7) 014431-1 ©2019 American Physical SocietyGOEL, ANH, OHYA, AND TANAKA PHYSICAL REVIEW B 99, 014431 (2019) TABLE I. Epitaxial strain ( ε) estimated from XRD, saturation magnetization ( μ0Ms) measured by SQUID, effective magnetization (μ0Meff), magnetocrystalline anisotropy field ( Hi), and gfactor obtained by the fitting to the FMR spectra of (Ga 0.8,Fe0.2) S bi ns a m p l e s A–D with different buffer layers. Sample Buffer ε(%) μ0MS(mT) μ0Meff(mT) Hi(Oe) gfactor A AlSb −1.7 89.9 104 .3±0.5 −144±52 .08±0.03 B GaSb −0.1 77.9 90 .4±0.1 −125±12 .07±0.03 CI n 0.5Ga0.5As +0.23 59.8 37 .1±0.2 227 ±22 .1±0.03 D GaAs +3.84 66.3 32 .3±0.2 340 ±22 .11±0.03 current) from an FM material into nonmagnetic metals and semiconductors. Therefore, observing FMR in the (Ga,Fe)Sbthin films, particularly at room temperature, is a fundamentaland important step to examine this material as a spin injectorin practical spin devices. Here, we measured the dependenceof the FMR resonance field on the external magnetic fielddirection and fitted a theoretical curve to the data to obtainthe MA fields of the (Ga,Fe)Sb thin films. We performedcareful analyses of the MA to separate the shape anisotropy(K sh), which is due to the dipole-dipole interactions, and the magnetocrystalline anisotropy ( Ki), which is due to the spin-orbit interactions, and discussed the effect of epitaxialstrain on these two components. II. SAMPLE GROWTH AND CHARACTERIZATIONS We have grown a series of four samples A–D of p-type FMS (Ga,Fe)Sb thin films on semi-insulating GaAs (001)substrates by low-temperature molecular beam epitaxy (LT-MBE), whose buffer layers and properties are given in Table I. The schematic structure of our samples is shown in Fig. 1(a). In samples A, B, and C, on a semi-insulating GaAs (001)substrate, we first grew a 100-nm-thick GaAs layer at asubstrate temperature T S=550◦C to obtain an atomically flat surface; next we grew a 10-nm-thick AlAs layer at thesame T S. Then, we grew a thick buffer layer, which is a 300-nm-thick AlSb layer at TS=470◦C for sample A, a 300-nm-thick GaSb layer at TS=470◦C for sample B, and a 500-nm-thick In 0.5Ga0.5As layer at TS=550◦Cf o rs a m p l e C. For sample D, a 500-nm-thick GaAs buffer layer wasgrown directly on a semi-insulating GaAs (001) substrate atT S=550◦C. Finally, we grew a 15-nm-thick (Ga 1-x,Fex)Sb layer with a Fe concentration of x=20% at a growth rate of 0.5μm/h and an Sb beam equivalent pressure of 7 .8×10−5 Pa atTS=250◦C for all the samples. As shown in Fig. 1(b), in situ reflection high-energy electron diffraction (RHEED) patterns in the [ ¯110] direction of the (Ga,Fe)Sb thin films in all four samples are bright and streaky, thereby indicating goodtwo-dimensional growth of a zinc-blende crystal structure.In this way, we obtained high-quality (Ga,Fe)Sb thin films,whose quality is better than that of our previous reports[18,19] because we optimized the MBE growth conditions: The properties of (Ga,Fe)Sb depend on the Sb pressure duringthe MBE growth and we found that by keeping a higher Sb 4 pressure at 7 .8×10−5Pa in the MBE growth chamber before Ga and Fe fluxes were supplied, we obtained high TC>300 K in (Ga 1-x,Fex)Sb with a lower Fe concentration of x=20% (this is an improvement from our previous reports [ 18,19]).We characterized the crystal structures and lattice constants of all the (Ga,Fe)Sb thin films and buffer layers by x-raydiffraction (XRD). Figures 2(a)–2(d) show the XRD results of samples A–D, respectively. All the samples show a sharpGaAs (004) peak. In samples A, B, and C, there is a broaderpeak which can be deconvoluted into two Gaussian peakscorresponding to the buffer layer and (Ga,Fe)Sb (004). Insample D, the (Ga,Fe)Sb (004) peak can be clearly seen. Fromthe peak positions, we estimated the intrinsic lattice constantsof (Ga,Fe)Sb ( a GaFeSb ) and of the buffer layer ( abuffer) (see Supplemental Material [ 33]). We define the epitaxial strain ε asaGaFeSb−abuffer aGaFeSb×100 (%). As listed in Table I, the estimated FIG. 1. (a) Schematic illustration of the (001)-oriented sample structure composed of (Ga 1-x,Fex)Sb grown on different buffer lay- ers on a semi-insulating GaAs(001) substrate. (b) In situ reflection high-energy electron diffraction (RHEED) patterns observed along the [¯110] axis during the MBE growth of the 15-nm-thick (Ga,Fe)Sb thin films on AlSb (sample A), GaSb (sample B), In 0.5Ga0.5As (sam- ple C), and GaAs (sample D) buffer layers. (c) Sample alignment and coordinate system used in the ferromagnetic resonance (FMR) measurement system. A radio-frequency (rf) magnetic field hwas applied along the [ ¯110] direction of the sample. θHandθMare the angles of the magnetic field Hand the magnetization Mwith respect to the [001] direction, respectively. 014431-2FERROMAGNETIC RESONANCE AND CONTROL OF … PHYSICAL REVIEW B 99, 014431 (2019) FIG. 2. X-ray diffraction rocking curves of samples A–D. The broad peak in sample A–C was fitted by the Gaussian curves cor-responding to the peaks of the (Ga,Fe)Sb thin film (red dotted line) and of the buffer layers of (a) AlSb (yellow dotted line), (b) GaSb (pink dotted line), and (c) In 0.5Ga0.5As (green dotted line). The sum of the two curves is the fitting curve which is plotted by the violet dashed line. (d) In sample D, the (Ga,Fe)Sb (004) peak (blue-violet dashed line) can be clearly seen. For each sample, the epitaxial strain (ε%) was estimated (see Supplemental Material [ 33]) and shown in the figure. values of εindicate that the (Ga,Fe)Sb films can have both tensile and compressive strains when they are grown ondifferent buffer layers. Here, samples A ( ε=−1.7%, AlSb buffer layer) and B ( ε=−0.1%, GaSb buffer layer) have tensile strain, whereas samples C ( ε=0.23%, In 0.5Ga0.5As buffer layer) and D ( ε=3.84%, GaAs buffer layer) have compressive strain. These results demonstrate that we cansystematically vary the epitaxial strain of (Ga,Fe)Sb in awide range, from tensile to compressive, by growing it onappropriate buffer layers. Next, we characterized the magnetic properties of all the samples using magnetic circular dichroism (MCD) spec-troscopy and superconducting quantum interference device(SQUID) magnetometry. As shown in Figs. 3(a)–3(h),t h e magnetic field dependences of MCD (MCD– Hcurves) show clear hysteresis, and the Arrott plots indicate that T Cis higher than 320 K in all the samples. Here the MCD intensity wasmeasured at the E 1peak ( ∼2.1 eV). To estimate the exact value of TC, we measured remanent magnetization versus temperature ( M−T) curves up to 400 K (see Supplemental Material [ 33]). It is shown that the remanent magnetization is still present even at 400 K. We have also measured mag-netization hysteresis ( M−H) curves at 400 K, as shown in Figs. 4(a)–4(d), in which we can see clear remanent magneti- zation. Therefore, T Cis higher than 400 K. These results prove that the room-temperature ferromagnetism is obtained in allfour (Ga,Fe)Sb samples with the Fe concentration of 20%. Figures 5(a)–5(d) show the magnetic field dependence of the magnetization ( M−H)o f( G a 0.8,Fe0.2)Sb measured for samples A–D at 50 K, with a magnetic field Happlied along the in-plane [110] axis (solid circles) and the perpendicular[001] axis (open circles). In all the samples, Msaturates at FIG. 3. (a)–(d) MCD- Hcurves at different temperatures, (e)–(h) Arrott plots of (Ga 0.8,Fe0.2)Sb grown on different buffer layers. The (Ga,Fe)Sb thin films in all the samples exhibit clear ferromagnetismwithT C>320 K. smaller HwhenH//[110] than when H//[001]. These results show that the easy magnetization axes of the (Ga,Fe)Sb thinfilms lie in the in-plane direction in all four samples regardlessof the different epitaxial strains. We note that the same resultswere obtained from the M−Hcurves measured using SQUID at room temperature. Also, we observed a tendency that thesaturation magnetization decreases with increasing ε, which can be attributed to the degradation of the crystal quality ofthe films due to the buffer layer. The crystal-quality changeis observed in the linewidths in the ferromagnetic resonance(FMR) spectra, which is discussed in Sec. III. III. EXPERIMENTAL SETUP OF FERROMAGNETIC-RESONANCE (FMR) MEASUREMENTS AND THEORETICAL MODEL We used a Bruker electron paramagnetic resonance (EPR) spectrometer for performing FMR measurements at9.066 GHz. As shown in Fig. 1(c), in our FMR measurements, 014431-3GOEL, ANH, OHYA, AND TANAKA PHYSICAL REVIEW B 99, 014431 (2019) FIG. 4. (a)–(d) Magnetization hysteresis curves ( M-H) mea- sured at 400 K for (Ga 0.8,Fe0.2)Sb grown on the (a) AlSb, (b) GaSb, (c) In 0.5Ga0.5As, and (d) GaAs buffer layers when the magnetic field was applied in the film plane along the [110] axis (red open circles).These characteristics show that the T Cof these (Ga 0.8,Fe0.2)Sb is higher than 400 K. the microwave radio frequency (rf) magnetic field ( h)i s applied along the [ ¯110] axis in the film plane and the direct- current (dc) magnetic field His rotated from the [001] direc- tion (perpendicular to the film plane) to the [110] direction (inthe film plane). Initially, we cut the sample into a (3 ×1)-mm- size piece with edges along [ ¯110] (3 mm) and [110](1 mm). Then, we put it on the center of a quartz sample rod and placedit inside the center of the microwave cavity that resonates inthe TE 011mode, where hand rf electric field ( e) are largest and smallest, respectively. The FMR spectrum was thenmeasured by sweeping the magnitude of H. The magnetic field derivative of the microwave absorption was obtained by FIG. 5. Magnetization hysteresis curves ( M-H) measured at 50 K for (Ga 0.8,Fe0.2)Sb grown on the (a) AlSb, (b) GaSb, (c) In0.5Ga0.5As, and (d) GaAs buffer layers when the magnetic field was applied in the film plane along the [110] axis (red solid circles) and perpendicular to the plane along the [001] axis (black open circles).superimposing an alternating-current (ac) modulation field Hac(1 mT, 100 kHz) parallel to H. Figure 1(c) also shows the coordinate system used for the FMR measurements. θH andθMare the angles of HandMfrom the [001] direc- tion, respectively. All the samples were measured under amicrowave power P=200 mW at 300 K. We note that the raw FMR spectra of all the samples included backgroundsignals, which were separately detected by measuring theFMR spectra without samples and then subtracted from theraw data (see Supplemental Material [ 33]). In the FMR experiments, the total magnetic moment M precesses around the direction of the external magnetic fieldat the Larmor angular frequency ω. Microwave absorption occurs when the microwave angular frequency coincides withω. This precessional motion of the magnetization is described by the well-known Landau-Lifshitz-Gilbert (LLG) equationas shown in Eq. ( 1), 1 γ∂M ∂t=−[M×(H+Heff)]+α (γM S)/bracketleftbigg M×∂M ∂t/bracketrightbigg ,(1) where the first term on the right side shows the precessional motion of the magnetization and the second term representsdamping [ 37,38]. Here, γ=gμ B/¯his the gyromagnetic ra- tio, where g,μB, and ¯ hare the gfactor, Bohr magneton, and reduced Planck’s constant, respectively, and α=G γM Sis the damping coefficient, where GandMSare the Gilbert coefficient and saturation magnetization, respectively; Heff represents the effective magnetic field which includes the rf microwave magnetic field, the demagnetizing field (shapeanisotropy field), and the magnetocrystalline anisotropy field.To determine the FMR condition, we used the first term ofEq. ( 1). In our case, the free-energy density Eis expressed as the summation of the magnetocrystalline anisotropy energy (E i), the shape anisotropy energy ( Esh), and the Zeeman energy ( EZeeman ). In our model, we assumed that Eidepends only on the out-of-plane magnetic field angle ( θH) because the in-plane magnetic field angle ( φH) dependence of FMR was almost isotropic in all the (Ga,Fe)Sb samples (data notshown). The following Eq. ( 2) shows the modified expression forE: E=E eff+EZeeman =−Keffcos2θM−MSμ0Hcos(θH−θM), (2) where EiandEshare combined into the effective magnetic anisotropy energy Eeff(=Ei+Esh). The corresponding effec- tive magnetic anisotropy constants of Ei,Esh, andEeffare denoted as Ki,Ksh, andKeff(=Ki+Ksh), respectively. Kshis given in Eq. ( 3), Ksh=−1 2μ0M2 S. (3) From Eq. ( 2), the in-plane (perpendicular) magnetic anisotropy corresponds to negative (positive) signs of Keff [37,39]. The resonance field ( μ0HR) of the FMR spectrum is determined by the resonance condition given by the Smith-Beljers relation [ 40,41] expressed as /parenleftbiggω γ/parenrightbigg2 =1 (MSsinθM)2/bracketleftBigg ∂2E ∂θ2 M∂2E ∂φ2 M−/parenleftbigg∂2E ∂θM∂φM/parenrightbigg2/bracketrightBigg ,(4) 014431-4FERROMAGNETIC RESONANCE AND CONTROL OF … PHYSICAL REVIEW B 99, 014431 (2019) where φMis defined as the in-plane magnetization angle (see Supplemental Material [ 33]). Here, θMandφMat the resonance condition are determined by the two equationsof∂E/∂θ M=0 and ∂E/∂φ M=0. However, in our case, because the dependence of FMR on φMwas almost isotropic, we used only ∂E/∂θ M=0. Using Eq. ( 2), this condition is expressed as sin(2θM)=(2μ0HR/μ0Meff)sin(θM−θH). (5) Here, μ0Meffis the effective magnetic field which is expressed as μ0Meff=μ0MS−Hi, where Hi=2Ki MSis the magnetocrystalline anisotropy field. From Eqs. ( 2) and ( 4), we obtained the following fitting equation (see SupplementalMaterial [ 33]): /parenleftbiggω γ/parenrightbigg2 =[μ0HRcos(θH−θM)−μ0Meffcos2θM] ×[μ0HRcos(θH−θM)−μ0Meffcos2θM]. (6) Equations ( 5) and ( 6) were simultaneously solved nu- merically to obtain the theoretical value of μ0HRandθM, where γ(orgfactor) and μ0Meffare fitting parameters. Using the μ0MSvalues obtained from the SQUID measure- ments, we first estimated Ksh(=−1 2μ0M2 s), and then esti- mated Ki(=−MSHi 2,where Hi=μ0MS−μ0Meff).Finally, Keff(=Ki+Ksh) was estimated for all the samples. IV . RESULTS AND DISCUSSIONS The FMR spectra of the (Ga,Fe)Sb layers in samples A–D measured at room temperature (300 K) are shown inFigs. 6(a)–6(d), where the data obtained with H//[110] and H//[001] are represented by open red circles and open black squares, respectively. In all the samples for both magneticfield directions, we observed clear FMR signals from the(Ga,Fe)Sb thin films at room temperature. We note that theFMR signal at room temperature has never been reported forother III-V FMSs. The resonance field μ 0HRof the FMR spectra measured with H//[110] is smaller than that with H//[001] in all the samples, indicating that the easy magne- tization axis is always in the film plane (in-plane magneticanisotropy). This result is consistent with the SQUID resultsshown in Sec. II. We also note that the linewidth of the FMR spectra becomes broader from 31 mT (sample A) to 56 mT(sample D) for H//[001] as shown in Fig. 7(black solid circles) when the strain is changed from tensile (sample A)to compressive (sample D). This increase in FMR linewidthis attributed to the degradation of the crystal quality of thefilms due to the buffer layer, which also causes the decreasein saturation magnetization as shown in Fig. 7(blue solid squares). Next, we measured the FMR spectra for variousdirections of Hbetween the direction normal to the film plane (H//[001]) and the in-plane direction ( H//[110]). The detailed angular dependence of μ 0HRon the Hdirection (θH) of all the samples is represented as the black solid cir- cles in Figs. 6(e)–6(h).T h eμ0HRvalue decreased smoothly with increasing θHfrom 0◦(H//[001]) to 90◦(H//[110]). The change of μ0HRwhenHis rotated from [001] to [110] monotonously decreases when one goes from sampleA (0.14 T) to sample D (0.05 T). This result reflects the FIG. 6. (a)–(d) FMR spectra observed for (Ga 0.8,Fe0.2)Sb grown on the (a) AlSb, (b) GaSb, (c) In 0.5Ga0.5As, and (d) GaAs buffer layers at room temperature (300 K) when the magnetic field H was applied along [110] (“red” circles) and [001] (“black” squares).(e)–(h) Resonance field μ 0HRas a function of the direction angle θH ofμ0H. different MA in these samples, likely due to the different epitaxial strains. On the other hand, μ0HRremained almost unchanged when we rotated Hin the film plane (data not shown), indicating very weak in-plane magnetic anisotropyof the (Ga,Fe)Sb thin films. The fittings (black solid curves) FIG. 7. FMR linewidth when the magnetic field is applied along [001] (black solid circles) and the saturation magnetization (blue solid squares) as a function of strain ( ε). 014431-5GOEL, ANH, OHYA, AND TANAKA PHYSICAL REVIEW B 99, 014431 (2019) FIG. 8. Strain ( ε) dependence of the (a) magnetocrystalline anisotropy constant Ki, (b) shape anisotropy constant Ksh, and (c) effective magnetic anisotropy constant Keffof (Ga,Fe)Sb thin films grown on different buffer layers. reproduce the observed angular dependence of the FMR fields quite well for all the samples, as shown in Figs. 6(e)–6(h).T h e fitting parameters ( μ0Meffandgfactors) that were obtained from the fitting to the experimental μ0HRdata are listed in Table I. In Table I, one can see that μ0Mefftends to decrease when the strain is changed from tensile (sample A) to compressive(sample D). This means that μ 0Meffwhich carries information of the magnetocrystalline anisotropy depends strongly on theepitaxial strain of the (Ga,Fe)Sb thin film. Figures 8(a)– 8(c) summarize the estimated values of magnetocrystalline anisotropy constant K i, shape anisotropy constant Ksh, and effective magnetic anisotropy constant Keff(=Ki+Ksh), as a function of ε. In all the samples, the magnitude of Kshis one or two orders of magnitude larger than Ki, indicating the dominance of the shape anisotropy in the MA properties of(Ga,Fe)Sb. The strong shape anisotropy is due to the largeμ 0MSof (Ga,Fe)Sb even at room temperature. The magnetocrystalline anisotropy constant Ki, though small, shows a systematic dependence on the strain ε. As shown in Fig. 8(a), when the strain is changed from tensile ( ε=−1.7%) to compressive ( ε=+3.84%), the magnitude of Kiincreases and changes from negative (in-plane anisotropy) to positive (perpendicular anisotropy).These results indicate that it is feasible to control themagnetocrystalline anisotropy of (Ga,Fe)Sb thin films byusing epitaxial strain. Meanwhile, K shis always negative, as shown in Fig. 8(b), making Keffalways negative (in-plane magnetic anisotropy), as shown in Fig. 8(c). As a result, all the (Ga,Fe)Sb thin films examined here have in-plane magneticanisotropy. These results of (Ga,Fe)Sb are contrasting to thoseof (Ga,Mn)As in the following two points: (i) In (Ga,Mn)As,K iis large (magnetocrystalline field Hiis∼several 1000 Oe [20]) and dominates MA, but (Ga,Fe)Sb shows small Hi∼ 100−300 Oe (listed in Table I) and possesses a very large Ksh. (ii) In (Ga,Mn)As, compressive (tensile) strain leads toin-plane (perpendicular) magnetic anisotropy, but in (Ga,Fe)Sb tensile (compressive) strain leads to in-plane(perpendicular) magnetic anisotropy, and thus the strain effectis opposite. Therefore, in (Ga,Fe)Sb, the shape anisotropyshould be utilized to control the in-plane magnetic anisotropy. V . CONCLUSION We have successfully grown a series of (Ga,Fe)Sb thin films with the Fe concentration of 20% on different bufferlayers, AlSb, GaSb, In 0.5Ga0.5As, and GaAs, which all exhibit room-temperature ferromagnetism. The epitaxial strain εin the (Ga,Fe)Sb layers was gradually varied over a wide rangefrom−1.7% (tensile strain) to +3.84% (compressive strain). We observed clear FMR signals in (Ga,Fe)Sb at room temper-ature (FMR has never been observed in III-V based FMSs atroom temperature), and determined the magnetic anisotropyconstants. We found that the magnitude of K iis weak and shows a monotonous dependence on the strain. By changingthe strain from tensile to compressive, K ichanged from nega- tive (in-plane magnetic anisotropy) to positive (perpendicularmagnetic anisotropy). Meanwhile, K shwas always negative and is dominant over Ki, leading to negative Keff(in-plane magnetization) in all the samples. This study suggests that theeasy magnetization axis of (Ga,Fe)Sb can be controlled bychanging the shape anisotropy. ACKNOWLEDGMENTS This work was partly supported by Grants-in-Aid for Scientific Research by MEXT (Grants No. 18H03860, No.17H04922, and No. 16H02095), CREST of JST (Grant No.JPMJCR1777), the Spintronics Research Network of Japan(Spin-RNJ), Yazaki Memorial Foundation for Science &Technology, and the Murata Science Foundation. [1] M. Oestreich, J. Hubner, D. Hagele, P. J. Klar, W. Heimbrodt, W. W. Ruhle, D. E. Ashenford, and B. Lunn, Appl. Phys. Lett. 74,1251 (1999 ). [2] M. Oestreich, Nature (London) 402,735(1999 ). [3] S. Ghosh and P. Bhattacharya, Appl. Phys. Lett. 80,658 (2002 ).[4] Y . 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PhysRevB.99.184407.pdf

PHYSICAL REVIEW B 99, 184407 (2019) Local spin moments, valency, and long-range magnetic order in monocrystalline and ultrathin films of Y 3Fe5O12garnet Y. Y. C h i n ,1,2,*H.-J. Lin,2Y . -F. Liao,2W. C. Wang,2P. Wang,3D. Wu,3A. Singh,2H.-Y . Huang,2Y .-Y . Chu,2D. J. Huang,2 K.-D. Tsuei,2C. T. Chen,2A. Tanaka,4and A. Chainani2 1Department of Physics, National Chung Cheng University, Chiayi 62102, Taiwan 2National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan 3Department of Physics, National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China 4Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima 739-8530, Japan (Received 25 March 2019; published 9 May 2019) We investigate and compare the electronic structure of a bulk single crystal of Y 3Fe5O12garnet [YIG, a high- TC(=560 K) ferrimagnet] with that of an epitaxial ultrathin (3.3 nm) film of YIG with a reduced ferrimagnetic temperature TC=380 K, using bulk-sensitive hard x-ray photoelectron spectroscopy (HAXPES), x-ray absorption spectroscopy (XAS), and x-ray magnetic circular dichroism (XMCD). The Fe 2 pHAXPES spectrum of the bulk single crystal exhibits a purely trivalent Fe3+state for octahedral and tetrahedral sites. The Fe 3sspectrum shows a clear splitting which allows us to estimate the on-site Fe 3 s-3dexchange interaction energy. The valence band HAXPES spectrum shows Fe 3 d,O2 p,a n dF e4 sderived features and a band gap of ∼2.3 eV in the occupied density of states, consistent with the known optical band gap of ∼2.7e V . FeL-edge XAS identifies the octahedral Fe3+and tetrahedral Fe3+site features. XMCD spectra at the Fe L2,3edges show that bulk single-crystal YIG exhibits antiferromagnetic coupling between the octahedral- and tetrahedral-site spins. The calculated Fe 2 pHAXPES, Fe L-edge XAS, and XMCD spectra using full multiplet cluster calculations match well with the experimental results and confirm the full local spin moments. In contrast,HAXPES, XAS, and XMCD of the Pt /YIG (3.3 nm) ultrathin epitaxial film grown by a pulsed laser deposition method show a finite Fe 2+contribution and a reduced Fe3+local spin moment. The Fe2+state is attributed to a combination of oxygen deficiency and charge transfer effects from the Pt capping layer to the ultrathin film.However, the conserved XMCD spectral shape for the ultrathin film indicates that the 3.3-nm epitaxial film isgenuinely ferrimagnetic, in contrast to recent studies on films grown by radio-frequency magnetron sputteringwhich have shown a magnetic dead layer of ∼6 nm. The presence of Fe 2+and the reduced local spin moment in the epitaxial ultrathin film lead to a reduced Curie temperature, quantitatively consistent with well-knownmean-field theory. The results establish a coupling of the local Fe spin moments, valency, and long-rangemagnetic ordering temperature in bulk single crystal and epitaxial ultrathin-film YIG. DOI: 10.1103/PhysRevB.99.184407 I. INTRODUCTION Spintronics, or spin-based electronics, relies upon repro- ducible and robust transport of spin and charge for deviceoperation. However, recent studies have identified pure spincurrents which could efficiently transport spin angular mo-mentum without an accompanying charge current. This wouldlead to the absence of an Oersted field and lower Jouleheating losses [ 1–8] and promises new functionalities as well as energy savings. In order to generate and manipulatepure spin currents, bilayers composed of a normal metal(NM)/ferromagnetic material with a nonmagnetic layer have been extensively investigated, and fascinating phenomenasuch as spin pumping [ 3,4], the spin Seebeck effect (SSE) [5], the spin Hall effect (SHE) [ 6,7], and the inverse spin Hall effect [ 8] were recently reported. A pure spin current could be generated by a thermal gradient in the SSE, whilea nonmagnetic metal with strong spin-orbit coupling, such *yiyingchin@ccu.edu.twas Pt, could convert a charge current into a spin current inthe SHE. More interestingly, heterostructures with a ferro- orferrimagnetic insulator (FMI) layer have attracted significantattention because only magnetic excitations (spin currents) areexpected to propagate in the FMI layer, leading to a naturalseparation of spin current from charge current. The ferrimag-net Y 3Fe5O12with a TC=560 K is one such insulating oxide, and consequently, the bilayer Pt /Y3Fe5O12has become a prototype for investigating spin-current phenomena. Further-more, recent studies reported an unconventional Hall effectdepending on the magnetic field, implying the importance ofthe interface between Pt and Y 3Fe5O12[9–13]. Y3Fe5O12(YIG) is an extremely important material for ultrahigh-frequency optical modulators, femtosecond photo-magnetic switching devices, and microwave applications. Italso shows giant magnetoelectric and magnetocapacitanceeffects and exhibits Bose-Einstein condensation of magnons[14,15]. YIG crystallizes in a cubic structure ( Ia3d) with magnetically active Fe 3+ions in 16a octahedral (O h) sites and 24d tetrahedral (T d) sites. It exhibits ferrimagnetic or- der below TC=560 K with antiparallel Fe spins due to 2469-9950/2019/99(18)/184407(9) 184407-1 ©2019 American Physical SocietyY. Y. C H I N et al. PHYSICAL REVIEW B 99, 184407 (2019) superexchange on the octahedral:tetrahedral sites in a 2:3 ratio with the magnetic easy axis along the 111 direction.Moreover, because it exhibits low magnetic damping and isa very good insulator (band gap of ∼2.7e V ) [ 16], YIG is a favorite choice for generating pure spin currents via a thermalgradient. It was demonstrated that the dc magnetic momentcurrent in YIG could reach a value of 10 24μB/cm2[17]. Although bulk YIG shows only weak magnetic anisotropy, a recent study indicated the presence of perpendicular mag-netic anisotropy in Pt /YIG thin films [ 18]. Moreover, the deviation between the bulk magnetization and the longitudinalspin Seebeck effect was attributed to the near-surface uniaxialmagnetic anisotropy, which is intrinsic to YIG [ 18]. More significantly, the threshold current for exciting spin wavesin Pt/YIG bilayer films is 2–3 orders of magnitude lower than what is expected for bulk YIG. It was theoreticallyshown that the strong reduction in threshold current is dueto an easy-axis surface anisotropy, which also increases thepower of the spin wave excitation by at least 2 orders ofmagnitude [ 19]. However, in a recent study using polarized neutron reflectometry, the authors concluded that the interfaceof Pt/YIG films can become nonmagnetic, and this will have important repercussions for the inverse SHE [ 20]. It is also known that the Curie temperature of YIG films can getreduced even for high-crystalline-quality epitaxial films [ 21]. Most importantly, in a recent study of epitaxial films grown byradio-frequency magnetron sputtering, the YIG films grownon Gd 3Ga5O12(GGG) (111) substrates showed a magnetic dead layer of ∼6 nm at the interface [ 22]. Thus, it is extremely important to carry out a spectroscopic characterization of theelectronic structure and its relation to the magnetic propertiesof YIG films in bilayers. Further, it is necessary to compareit with the electronic structure of bulk single-crystal YIGusing the same techniques. This would help us to identify thebest conditions required for developing high-quality films fordevice applications. In this work, we study single-crystal YIG(111), Pt/YIG(111), and Cu /YIG(111) epitaxial thin films using hard x-ray photoelectron spectroscopy (HAXPES) andFeL 2,3x-ray absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD). HAXPES is ideallysuited for bulk sensitive core-level and occupied valenceband studies of electronic structure [ 23–25]. On the other hand, XAS and XMCD are well suited to studying the site-and orbital-selective unoccupied density of states and fordetermining element-specific orbital and spin moments [ 26]. Our results show the presence of Fe 2+and reduced Fe spin moments in the epitaxial ultrathin film compared to the pureFe 3+and the full spin moment seen in the bulk single crystal. This causes a reduced Curie temperature compared to the bulksingle crystal but is quantitatively consistent with well-knownmean-field theory. The results indicate a direct coupling ofthe local Fe spin moments, valency, and long-range magneticordering temperature in the bulk single crystal as well as inepitaxial ultrathin-film YIG. II. EXPERIMENTS The YIG bulk single crystal was obtained commer- cially. Pt /YIG(111) and Cu /YIG(111) ultrathin films wereFIG. 1. (a) The Fe 2 pHAXPES spectra of Y 3Fe5O12bulk single crystal at room temperature. (b) The theoretical simulation of theHAXPES spectrum of YIG single crystal by the configuration- interaction cluster calculations. epitaxially grown on GGG(111) substrates using pulsed laser deposition by applying a KrF excimer laser at a repetition of4 Hz and a laser fluence of 2.7 J /cm 2. The growth temperature and oxygen pressure were 740◦C and 0.07 Torr, respec- tively. Clear in situ reflection high-energy electron diffraction (RHEED) patterns were observed during deposition, indi-cating the single crystallinity of the YIG films. The YIGthickness was estimated to be 3.3 nm for Pt /YIG and for Cu/YIG epitaxial films from calibrated RHEED oscillations. The film quality was further confirmed by x-ray diffraction.The YIG films were transferred into another vacuum chamberto deposit Pt /Cu films by dc magnetron sputtering. The thick- ness of the Pt and Cu capping layer was 3 nm. The growthrates of the YIG films and the Pt /Cu capping layers were also determined by x-ray reflectivity measurements, and thecharacterization procedures were reported in a recent study[27]. The Fe L 2,3XAS and XMCD experiments were carried out at the BL11A beamline of the National SynchrotronRadiation Research Center in Taiwan. The Fe L 2,3XAS and XMCD spectra were collected at room temperature in thetotal-electron yield mode with an energy resolution of betterthan 0.3 eV . Fe 2O3and NiO single crystals were measured simultaneously in a separated chamber to calibrate the photonenergy with an accuracy better than 10 meV . HAXPES experi-ments ( hν=6500 eV) were performed at room temperature at the Taiwan beamline BL12XU of SPring-8 in Hyogo, Japan.The overall energy resolution was 0.35 eV , estimated from afit to the Fermi edge of silver, which was also used to calibratethe binding energy. III. RESULTS AND DISCUSSION A. HAXPES Fe 2 presults of bulk single crystal The Fe 2 pHAXPES spectrum of the YIG bulk single crystal is presented in Fig. 1(a). The spectrum consists of the 2p3/2and 2 p1/2spectral features due to spin-orbit splitting. The 2 p3/2main peak consists of two features, positioned at binding energies (BEs) of 710.5 and 711.5 eV and a satellite 184407-2LOCAL SPIN MOMENTS, V ALENCY , AND LONG-RANGE … PHYSICAL REVIEW B 99, 184407 (2019) TABLE I. The parameters (in eV) for simulating the Fe 2 p HAXPES spectrum of the YIG bulk single crystal. Udd Upd /Delta1 10 Dq Veg Vt2g TdFe3+6.0 7.5 2.0 −0.4 2.82 1.72 OhFe3+6.0 7.5 2.0 0.9 1.27 2.38 feature at about 720 eV . Similarly, the 2 p1/2main peak con- sists of two features, positioned at binding energies of 724.5and 725.5 eV and a satellite feature at about 733.5 eV . In orderto understand the origin of the spectral features, we carried outmodel configuration interaction cluster calculations [ 28,29] for the Fe 2 pspectrum, including full atomic multiplets for octahedral FeO 6and tetrahedral FeO 4clusters. The basis states used for the calculations consist of a linear combinationof the d 5,d6L1, and d7L2states for Fe3+. The electronic parameters for the calculations are the on-site Coulomb en-ergy U dd, the charge transfer energy /Delta1, the Fe 3 d-O 2 p hybridization strength V, the crystal field splitting 10 Dq, and the Coulomb interaction in the presence of a 2 pcore hole Upd. The parameters were optimized to give the best match with the experimental data, and the results are shown alongwith the experimental spectrum. The bulk single-crystal Fe2pspectrum can be simulated nicely using a combination of O hand T dFe3+in a 2:3 ratio, consistent with the chemical formula. The two features of the main peak are assigned to theoctahedral and tetrahedral Fe 3+sites, respectively. The elec- tronic structure parameters obtained from the cluster calcula-tions are listed in Table I. It is understood that YIG is a typical charge transfer insulator with a small charge transfer energy(/Delta1=2 eV), large on-site Coulomb energy ( U dd=6 eV), and strong hybridization ( Veg=2.82 eV and Vt2g=2.38 eV for T d and O hsites, respectively) between the Fe 3 dand O 2 pligand states. The charge transfer nature of YIG is consistent with FIG. 2. (a) The Fe 3 sHAXPES spectra of Y 3Fe5O12bulk single crystal at room temperature. (b) The simulation consists of four peaks obtained from a fit to the HAXPES spectrum for estimating the Fe 3s multiplet splitting, as explained in the text.FIG. 3. (a) The Y 3 pHAXPES spectra of Y 3Fe5O12single crys- tal at room temperature. (b) The Y 3 dHAXPES spectra of Y 3Fe5O12 bulk single crystal at room temperature. The weak plasmon features are marked by asterisks. the known results of other trivalent Fe3+oxides, hematite (α-Fe 2O3)[30] and maghemite ( γ-Fe 2O3)[31]. B. HAXPES Fe 3 sresults of bulk single crystal In Fig. 2(a) we plot the Fe 3 sHAXPES spectrum of bulk single-crystal YIG. The spectrum consists of two broadfeatures, a higher-intensity feature at about 94 eV and alower-intensity feature at 100 eV binding energy. The Fe3sspectrum thus exhibits the well-known multiplet splitting due to 3 s-3dexchange interaction [ 32]. A closer look at the higher-binding-energy feature at 100 eV shows that it consistsof two peaks, which can be assigned to the tetrahedral andoctahedral Fe sites. Hence, we have carried out a peak fittingto the Fe 3 sspectrum using four peaks (Td1, Oh1, Td2, and Oh2) to accurately estimate the binding energies of thefeatures. The fitting results are overlaid on the experimentalspectrum. We obtain a splitting of ∼6.0 eV for the tetrahedral Fe site and ∼7.0 eV for the octahedral site. While it is known that the Fe 3 smultiplet splitting energy /Delta1E 3s=(2S+1)Jeff, where Sis the net spin on the Fe site and Jeffis the effective exchange integral between the 3 sand 3 dstates [ 32], the role of the intrashell correlation effects [ 33], final-state screening 184407-3Y. Y. C H I N et al. PHYSICAL REVIEW B 99, 184407 (2019) FIG. 4. The valence band HAXPES spectra of Y 3Fe5O12bulk single crystal at room temperature for linear horizontal (H) and vertical (V) polarizations of the incident x rays. [34], and charge transfer screening [ 35] has been recognized. More recently, a systematic study on a series of Fe compoundsshowed that charge transfer screening leads to a modificationof/Delta1E 3s[36]. It was shown that /Delta1E3sfollows a linear relation versus (2 S+1) given by /Delta1E3s=A+(2S+1)Jeff, where A is a constant. From a fit to the experimentally observed data,it was found that A=0.94 and J eff=1.01 eV for a series of Fe compounds. Using this relation with A=0.94, as we will show later with the XMCD measurements and analysis, sinceS∼2μ Bfor the tetrahedral and octahedral Fe3+sites in bulk YIG, we could estimate that Jeff∼1.0 eV for tetrahedral Fe sites and Jeff∼1.2 eV for the octahedral Fe sites. C. HAXPES Y 3 pand 3 dresults of bulk single crystal Figures 3(a) and 3(b) show the Y 3 pand Y 3 dcore- level HAXPES spectra of bulk single-crystal YIG. The Y 3 p spectrum exhibits a spin-orbit split 3 p3/2and 3 p1/2doublet at binding energies of 301 and 313 eV , respectively, while theY3dspectrum exhibits a spin-orbit split doublet at binding energies of 158 and 160 eV , respectively. The clean singlepeaks and the binding energies of the Y 3 pand Y 3 dspectra are indicative of typical Y 3+states. We also note that the spectra exhibit weak satellites at about 12 eV higher bindingenergies compared to the main peaks in both the Y 3 pand 3dspectra, and these are indicative of weak plasmon features. In particular, since the splitting between the main peaks of the3p 3/2and 3 p1/2doublet is also 12 eV , the plasmon of the main 3p3/2feature is hidden in the main 3 p1/2feature, resulting in a small deviation of the relative spectral intensities comparedto the expected ratio of 2:1 due to their degeneracies. D. HAXPES valence band spectra of bulk single crystal In Fig. 4, we plot the valence band HAXPES spectra obtained for horizontal and vertical linearly polarized incidentx rays. The spectra show small differences for the horizontaland vertical polarization spectra. The spectra mainly consistFIG. 5. (a) The wide-scan Fe 2 pHAXPES data for bulk single- crystal YIG and Pt /YIG (3.3 nm) epitaxial film at room temperature, showing the Pt 4 score level overlapping the Fe 2 p1/2feature. (b) The narrow-range Fe 2 p3/2HAXPES data for YIG bulk single crystal, Cu /YIG (3.3 nm), and Pt /YIG (3.3 nm) epitaxial thin films at room temperature, showing a weak feature at low binding energy( ∼708 eV) attributed to Fe2+states. of three broad features: the first feature is from about 2.3 to about 4 eV BE, the second is between 4.0 and 7.0 eV , andthe third feature occurs between 7.0 and nearly 10.0 eV BE.The vertical polarization enhances the Fe 3 dstates, while the horizontal polarization enhances the Fe 4 sstates. In addition, based on known band structure calculations, the first featureis dominated by Fe 3 dstates, while the second feature is due to mainly O 2 pstates mixed with the Fe 3 dstates. The third feature consists of O 2 pstates mixed with Fe 4 sstates, a st h eF e4 sstates are enhanced in the horizontal incident polarization spectrum. The onset of the first feature is at2.3 eV BE and indicates that the band gap in the occupieddensity of states is close to the optical band gap of YIG, whichis approximately 2.7 eV [ 16]. This implies that the chemical potential of the bulk YIG single crystal is pinned near thebottom of its conduction band. E. Comparative HAXPES Fe 2 pspectra of bulk single crystal and epitaxial thin films Next, we discuss the comparison of the HAXPES Fe 2 p spectra of the bulk single-crystal YIG, Cu /YIG, and Pt /YIG 184407-4LOCAL SPIN MOMENTS, V ALENCY , AND LONG-RANGE … PHYSICAL REVIEW B 99, 184407 (2019) FIG. 6. (a) The Fe L2,3XAS data for YIG bulk single crystal at room temperature. (b) The theoretical simulation of the XASspectrum of YIG bulk single crystal by the configuration-interaction cluster calculations. films, as shown in Fig. 5. Since the YIG films have a capping layer of 3 nm Pt /Cu metal, we could use HAXPES to measure the valency of Fe in the YIG films. However, since the Pt 4 s core-level binding energy ( ∼722 eV) lies very close to the Fe 2p3/2feature (binding energy of 710–715 eV) and it actually overlaps the Fe 2 p1/2feature [see Fig. 5(a)], we measured and compared the HAXPES of Cu /YIG and Pt /YIG films to identify the changes in the Fe 2 p3/2signal with bulk single- crystal YIG. As shown in Fig. 5(b) on an expanded scale, the presence of Fe2+in Cu/YIG and Pt /YIG epitaxial thin films in the Fe 2 p3/2HAXPES spectra can be identified as a weak feature with a chemical shift to low binding energy [ 37]. The finite intensity observed between 708 and 710 eV indicatesthe existence of Fe 2+in the epitaxial thin films. Thus, as seen in Fig. 5(b), the Fe 2 p3/2HAXPES of the Pt /YIG 3.3-nm epitaxial film shows a higher Fe2+content compared to the Cu /YIG 3.3-nm epitaxial film. This is consistent with a recent study which reported a charge transfer from the Ptcapping layer compared to negligible charge transfer from aCu capping layer in ultrathin Pt /YIG (1.6 nm) and Cu /YIG (1.6 nm) bilayers [ 27]. We have also confirmed there is no observable angular dependence of the spectra, indicating theabsence of surface effects. This is inferred from the fact thatFe 2 pspectra (not shown), measured with horizontal and vertical polarization at grazing incidence as well as at a 45 ◦ incidence angle, all show very similar spectral shapes. Thepresence of Fe 2+is expected to have an influence on the magnetic properties of the Pt /YIG epitaxial thin films, and to investigate this, we performed XAS and XMCD experi-ments on YIG bulk single crystal and Pt /YIG epitaxial thin films. F. Comparative Fe L2,3XAS spectra of bulk single crystal and epitaxial thin films The Fe L2,3XAS spectrum of bulk single-crystal YIG is presented in Fig. 6(a). The Fe L2,3XAS spectra consist of two main sets of features at ∼707–711 and ∼720–724 eVFIG. 7. (a) The Fe L2,3XAS spectra of Y 3Fe5O12bulk single crystal and Pt /Y3Fe5O14epitaxial thin film at room temperature. (b) The simulation of the XAS spectrum of Pt /YIG epitaxial thin film by the configuration-interaction cluster calculations. photon energies, which are the L3andL2edges derived from Fe 2 pspin-orbit coupling. The energy positions of spectral features and their multiplet structures are characteristic of thevalence state and the local symmetry of the Fe ion. We thenused the same electronic parameters obtained for the Fe 2 p photoemission spectrum to also calculate the Fe L-edge XAS spectrum using configuration interaction cluster calculations.We obtain a good match between the calculated and exper-imental spectra, as shown in Fig. 6. The main peak of the L 3edge at 708.5 eV is dominated by tetrahedral Fe3+, while the octahedral Fe3+states dominate the prepeak at 707.5 eV and also contribute significantly to the main peak at the higherphoton energy of 709.5 eV . In Fig. 7(a), we plot the Fe L-edge XAS spectrum of the Pt/YIG epitaxial thin film compared with the YIG bulk single crystal’s Fe L-edge XAS shown in Fig. 6. As seen in Fig. 7(a), the Pt/YIG epitaxial thin-film spectrum shows higher spectral weight at the low-energy shoulder ( ∼707 eV) in the Pt /YIG epitaxial thin film compared to the YIG single crystal. Thisimplies the presence of Fe 2+ions in the YIG epitaxial thin film, consistent with the Fe 2 pHAXPES spectrum shown in Fig. 5(b). In order to confirm and determine the Fe2+ content in the Pt /YIG epitaxial thin film, we subtracted the spectrum of the YIG bulk single crystal from that of theepitaxial thin film. The difference spectrum (blue line) is alsoshown in Fig. 7(a). We note that this spectral shape is different from not only the spectrum of O hFe2+in Fe-doped MgO [38] but also that of T dFe2+in CaBaFe 4O7[39]. However, it can be simulated by their combination and indicates thepresence of both O hFe2+and T dFe2+in the Pt /YIG thin film. We then carried out configuration-interaction clustercalculations to simulate the Fe L-edge XAS spectrum. The best match to the experimental data is shown together withthe experimental spectrum. The calculations indicate that thePt/YIG 3.3-nm epitaxial film consists of ∼90% Fe 3+with O h and T dcontributions in a 2:3 ratio, ∼6.9% T dFe2+(cyan line) and∼3.1% O hFe2+(magenta line). The electronic structure 184407-5Y. Y. C H I N et al. PHYSICAL REVIEW B 99, 184407 (2019) TABLE II. The parameters (in eV) for simulating the Fe L2,3 XAS and XMCD data of the YIG bulk single crystal and the epitaxial thin film. Udd Upd /Delta1 10 Dq Veg Vt2g Hex YIG bulk single crystal TdFe3+6.0 7.5 2.0 −0.4 2.82 1.72 0.04 OhFe3+6.0 7.5 2.0 0.9 1.27 2.38 0.04 Pt/YIG (3.3 nm) TdFe3+6.0 7.5 2.0 −0.4 2.82 1.72 0.02 OhFe3+6.0 7.5 2.0 1.0 1.27 2.38 0.016 TdFe2+6.0 7.5 7.0 −0.4 2.82 1.72 0.007 OhFe2+6.0 7.5 7.0 0.6 1.27 2.38 0.007 parameters obtained from the cluster calculations are listed in Table II. G. Comparative HAXPES O 1 sspectra of bulk single crystal and epitaxial thin films Having confirmed the presence of Fe2+in the Pt /YIG films compared to bulk single-crystal YIG, we analyzed the O 1 s core-level HAXPES to check the origin of Fe2+in the film. As seen in Fig. 8,t h eO1 score-level HAXPES of the bulk crystal shows a narrow single peak at about 531 eV . In contrast, the O1sspectrum of the Pt /YIG film shows a broader main peak as well as a broad satellite feature extending up to a higher BEof nearly 536 eV . Since it is known that oxygen adsorption(physisorption and chemisorption) on Pt can lead to satellitefeatures at higher BE than the main peak [ 40], we carried out a fitting of the spectrum to accurately identify if the satelliteconsists of more than one feature. The best fit is obtained byusing two satellites at binding energies of 533.5 and 536.0 eV .It is known [ 41,42] that oxygen vacancies or defects result in a satellite typically 2 eV higher BE from the main peak. In FIG. 8. The O 1 sHAXPES spectrum of Y 3Fe5O12bulk single crystal and Pt /Y3Fe5O14(3.3 nm) epitaxial thin film at room temper- ature. The Pt 4 p3/2feature lies close to the O 1 sfeatures. We have fitted the O 1 sfeatures of the Pt /Y3Fe5O14(3.3 nm) epitaxial thin film using V oigt functions, as shown in order to estimate the energy positions and peak widths of the features.FIG. 9. (a) The Fe L2,3XMCD curves of Y 3Fe5O12bulk single crystal and Pt /Y3Fe5O12(3.3 nm) epitaxial thin film at T=300 K. The theoretical simulations for the T=300 K XMCD curves of (b) the YIG single crystal and (c) Pt /YIG (3.3 nm) epitaxial thin film. contrast, a satellite due to oxygen physisorbed on the Pt(111) surface occurs at about 5 eV higher BE than the main peak[40]. We thus attribute the satellite at nearly 536.0 eV to physisorbed oxygen and the 533.5 eV BE satellite to oxygenvacancies in the YIG film. H. Comparative Fe L2,3XMCD results of bulk single crystal and epitaxial thin films TheT=300 K Fe L2,3XMCD data of the YIG bulk single crystal and the Pt /YIG 3.3-nm epitaxial film are presented in Fig. 9(a). The XMCD experiments were carried out at T= 300 K under a 1-T magnetic field. We also tried experimentsatT=30 K, but the strong insulating behavior of the YIG bulk single crystal at T=30 K led to spectral distortions due to charging. We first discuss the XMCD data of thebulk single-crystal YIG which show three features, S1–S3, aslabeled in Fig. 9(b). By comparing our results with known XMCD curves of GaFeO 3with O hFe3+andγ-Fe 2O3with both T dFe3+and O hFe3+[43] ,t h em a i nF e L3XMCD 184407-6LOCAL SPIN MOMENTS, V ALENCY , AND LONG-RANGE … PHYSICAL REVIEW B 99, 184407 (2019) TABLE III. Fe 3 dspin moments (in units of μB) as determined by XMCD sum rules and XMCD simulations. Bulk single crystal Pt /YIG (3.3 nm) TdFe3+1.97 1.46 OhFe3+−2.03 −1.30 TdFe2+−0.47 OhFe2+−0.48 Per Fe 0.37 0.27 XMCD sum rules 0.35 0.27 feature in Fig. 9(b) (labeled S2) of the YIG bulk single crystal is attributed to the T dFe3+. In contrast, features S1 and S3 at lower and higher photon energies mainly comefrom O hFe3+and are in the direction opposite that of the TdFe3+contribution. Therefore, the XMCD signal indicates an antiparallel alignment, i.e., an antiferromagnetic couplingof the T dFe3+and O hFe3+magnetic moments, similar to what was observed in γ-Fe 2O3and Fe 3O4[43,44]. Although the orbital and spin moments could be obtained by employing the XMCD sum rules [ 45–47], theoretical cal- culations are also necessary for explaining and quantifying theobserved behavior, particularly for systems with more thanone valence state and /or local symmetries. For the Pt /YIG film, although it contains a finite amount of Fe 2+as under- stood from the Fe L2,3XAS and Fe 2 pHAXPES spectra discussed earlier, the line shape of the XMCD signal of theepitaxial thin film is quite similar to that of the YIG bulk singlecrystal, implying magnetic contributions from Fe 3+dominate the XMCD signal. We note that in a recent study on a Pt /YIG (1.6 nm) ultrathin film [ 27], it was shown that the spectral shape deviates a little from that of thick YIG films (and ourbulk single-crystal data). In particular, it was shown that onlythe T dFe3+site XMCD signal weakened, while the O hFe3+ site XMCD signal did not change. This was interpreted to represent a preferential charge transfer from the Pt cappinglayer to the T dFe site, resulting in T dFe2+. However, in the present case, we find that for the Pt /YIG (3.3 nm) epitaxial thin film, the spectral shape matches the bulk single-crystalXMCD signal but is uniformly weakened, and the reductionis larger than 10%, the concentration of Fe 2+in the film. This indicates reduced spin moments for both the T dFe3+and OhFe3+sites. In order to quantify the magnetic dichroism of the YIG bulk single crystal as well as the Pt /YIG 3.3-nm epitaxial thin film, cluster calculations using the same parameters asthose for the XAS spectra were performed, and the results arepresented in Figs. 9(b) and9(c). As shown in Fig. 9(b), there is nice agreement between the theoretical (magenta line) andexperimental XMCD (black line) spectra, and thus, we canquantify the magnetic moments of Fe ions for bulk single-crystal YIG. Moreover, the site-resolved calculations shownin the bottom part of Fig. 9(b) confirm that the magnetic moments of T dFe3+and O hFe3+are indeed aligned opposite to each other. The estimated magnetic moments (listed inTable III) match very nicely with the spin moments calcu- lated using the local spin-density approximation with on-siteCoulomb energy U[48,49].Further, as depicted in Fig. 9(c), our cluster calculations for the Pt /YIG 3.3-nm epitaxial thin film also match nicely with the experimental data. However, we needed to include theFe 2+contributions to get the best match, as was discussed for the XAS data of Fig. 7. While the contribution from Fe2+is quite small compared with that of Fe3+, it can be expected that the Fe2+ions will disturb the magnetic interactions between Fe3+ions. However, more surprisingly, we find that the spin moments for the Fe3+ions are also significantly reduced in the epitaxial thin film as listed in Table III. This not only explains the reduction of the XMCD signal but would alsoimply a reduced Curie temperature in the Pt /YIG epitaxial thin film. We measured the Curie temperature of the Pt /YIG (3.3 nm) epitaxial film, and as shown in the SupplementalMaterial [ 50], we could fit the magnetization as a function of temperature to a Bloch T 3/2law typical of ferrimagnets. We could estimate TC=380 K for the Pt /YIG (3.3 nm) epitaxial film. Thus, the magnetization results and the XMCD spectralshape of the ultrathin film indicate that the Pt /YIG (3.3 nm) epitaxial film is genuinely ferrimagnetic. This is in contrast torecent studies on films grown by radio-frequency magnetronsputtering which have shown a magnetic dead layer of ∼6n m [22]. In fact, as discussed above, even Pt /YIG (1.6 nm) bilayer films grown by pulsed laser deposition were reported to beferrimagnetic at room temperature [ 27]. Based on mean-field theory, it is known that T C= μeff(CACB)1/2, where μeffis the effective spin moment and CAandCBare Curie constants for the A and B sublattices in a ferrimagnet [ 51]. This equation indicates that T Cis directly proportional to μeff. Indeed, the ratio of TCfor the epitaxial thin film compared to the bulk single crystal RTc∼ 0.68 and is close to the ratio of the effective spin momentsestimated from the XMCD data: R μ1=0.74±0.05 for the TdFe3+site and Rμ2=0.64±0.05 for the O hFe3+site. The small deviations for the ratio of effective moments forthe T dFe3+and O hFe3+sites probably originates from the preferential charge transfer as reported for the Pt /YIG (1.6 nm) ultrathin film [ 27]. However, the nearly similar XMCD signal for the present case of the Pt /YIG (3.3 nm) epitaxial film suggests that the reduced spin moments on bothT dFe3+and O hFe3+sites is dominated by the presence of oxygen vacancies, leading to both T dFe2+and O hFe2+ sites. This can be expected to disturb and effectively weaken the exchange interaction between the T dFe3+and O hFe3+ sites. Thus, the reduced TCdue to the presence of Fe2+is attributed to a combination of oxygen deficiency and chargetransfer effects from the Pt capping layer to the ultrathinfilm. IV . CONCLUSION In conclusion, we have carried out HAXPES, XAS, and XMCD of bulk single-crystal YIG compared to an epitaxialPt/YIG thin-film bilayer. The Fe 2 pHAXPES spectrum of the bulk single crystal indicates a purely trivalent Fe 3+state. The valence band HAXPES spectrum shows Fe 3 d,O2 p, and Fe 4 sderived features and a band gap of ∼2.3e V i n t h e occupied density of states, close to the known optical bandgap of 2.7 eV . Fe L-edge XAS was used to characterize the octahedral Fe 3+and tetrahedral Fe3+site features. Fe L-edge 184407-7Y. Y. C H I N et al. PHYSICAL REVIEW B 99, 184407 (2019) XMCD spectra showed that bulk single-crystal YIG exhibits antiferromagnetic coupling between the octahedral and tetra-hedral sites. Moreover, the full multiplet cluster calculationsof the Fe 2 pHAXPES, Fe L-edge XAS, and XMCD spectra matched well with the experimental results and confirmedthe full local spin moments. In contrast, HAXPES, XAS,and XMCD of the Pt /YIG (3.3 nm) ultrathin epitaxial film grown by a pulsed laser deposition method showed a finiteFe 2+contribution and a reduced Fe3+local spin moment. The Fe2+state is attributed to a combination of oxygen deficiency and charge transfer effects from the Pt capping layer to theultrathin film. However, the conserved XMCD spectral shapefor the ultrathin film indicates that the 3.3-nm epitaxial filmis genuinely ferrimagnetic, in contrast to recent studies onfilms grown by radio-frequency magnetron sputtering whichconcluded a magnetic dead layer of ∼6 nm. The presence of Fe 2+and the reduced local spin moment in the epitaxial ultra- thin film lead to a reduced Curie temperature, quantitativelyconsistent with known mean-field theory. The results show a coupling of the local Fe spin moments, valency, and long-range magnetic ordering temperature in bulk single-crystaland epitaxial ultrathin-film YIG. ACKNOWLEDGMENTS We thank Dr. Y . Tanaka for providing the single-crystal YIG and for valuable discussions. The authors would liketo thank the Ministry of Science and Technology of theRepublic of China, for financially supporting this researchunder Contracts No. MOST 106-2112-M-213-003-MY3, No.106-2112-M-213-001-MY2, and No. 107-2112-M-194-001-MY3. The synchrotron radiation experiments were performedat the BL12XU of SPring-8 with the approval of the JapanSynchrotron Radiation Research Institute (JASRI) (ProposalsNo. 2016B4255 and No. 2017A4251). [1] M. Johnson and R. H. 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PhysRevLett.108.017601.pdf

Spin-Wave Modes and Their Intense Excitation Effects in Skyrmion Crystals Masahito Mochizuki1,2 1Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan 2Multiferroics Project, ERATO, Japan Science and Technology Agency (JST), Tokyo 113-8656, Japan (Received 31 August 2011; published 5 January 2012) We theoretically study spin-wave modes and their intense excitations activated by microwave magnetic fields in the Skyrmion-crystal phase of insulating magnets by numerically analyzing a two-dimensional spin model using the Landau-Lifshitz-Gilbert equation. Two peaks of spin-wave resonances with frequencies of /C241 GHz are found for in-plane ac magnetic field where distribution of the out-of-plane spin components circulates around each Skyrmion core. Directions of the circulations are oppositebetween these two modes, and hence the spectra exhibit a salient dependence on the circular polarizationof irradiating microwave. A breathing-type mode is also found for an out-of-plane ac magnetic field. Byintensively exciting these collective modes, melting of the Skyrmion crystal accompanied by a redshift ofthe resonant frequency is achieved within nanoseconds. DOI: 10.1103/PhysRevLett.108.017601 PACS numbers: 76.50.+g, 75.10.Hk, 75.70.Ak, 75.78. /C0n Competing interactions in magnets often cause nontri- vial spin textures such as ferromagnetic domains and mag-netic bubbles, which have attracted a great deal of interest from the viewpoints of both fundamental science and technical applications in the field of spintronics [ 1,2]. In particular, response dynamics of such magnetic structuresunder external fields is an issue of vital importance becauseits understanding is crucial for their manipulations. The skyrmion, a nontrivial swirling spin structure carry- ing a topological quantum number, is one of the interesting examples of such spin textures. It was originally proposedby Skyrme to account for baryons in nuclear physics in the1960s as a quasiparticle excitation with spins pointing inall directions to wrap a sphere [ 3,4], and was recently realized experimentally in two-dimensional condensed matter systems, e.g., quantum Hall ferromagnets [ 5,6], ferromagnetic monolayers [ 7], and doped layered antifer- romagnets [ 8]. The formation of Skyrmion crystal (SkX) was theoreti- cally predicted in Dzyaloshinskii-Moriya (DM) ferromag-nets without inversion symmetry [ 9,10], and was indeed observed in the Aphase of metallic chiral magnets MnSi [11,12] and Fe 1/C0xCoxSi[13] by neutron-scattering experi- ments as a triangular lattice of Skyrmions with spins anti-parallel to the applied magnetic field at the Skyrmioncenters and parallel at their peripheries. A recentMonte Carlo study found a greater stability of the SkXphase in thin films [ 14]. This prediction was confirmed by the real-space observation of the Skyrmion triangular lattice inFe 0:5Co0:5Sithin films using the Lorentz force micros- copy in a wide temperature and magnetic-field range [ 15]. Typically the Skyrmion is 10–100 nm in size, which is determined by the ratio of DM interaction and exchangecoupling and is much smaller than magnetic bubbles. Moreover, recent experiments found that the Skyrmion is stable even near or above room temperature [ 16], and canbe manipulated by much lower electric currents than fer- romagnetic domain walls [ 17,18]. These properties, i.e., small size, high operational temperature, and low thresholdfield, are advantageous for technical application to high- density data storage devices. Therefore, understanding of the dynamics of Skyrmions and SkX under external fieldsis an important issue [ 19]. In this Letter, we theoretically study collective spin dynamics in the SkX phase of insulating ferromagnetswith DM interaction by numerical simulations of theLandau-Lifshitz-Gilbert (LLG) equation under time-dependent ac magnetic fields. We find a couple of spin- wave resonances with frequencies /C241 GHz for in-plane ac magnetic field where the out-of-plane spin componentsrotate around each Skyrmion core. The directions of theserotations are opposite between the higher-lying and lower-lying modes, and their spectra show strong circular-polarization dependence. A breathing-type mode is alsofound for out-of-plane ac magnetic field. Furthermore, westudy intense excitation effects of these collective modes, and find a redshift of the resonant frequency and melting of the SkX within nanoseconds. These findings will lead to afast manipulation of Skyrmions in nanoscale using spin-wave resonances. We start with a classical Heisenberg model on a two- dimensional square lattice [ 14], which contains nearest- neighbor ferromagnetic exchange, Zeeman coupling, andDM interaction as [ 20], H¼/C0JX hi;jiSi/C1Sj/C0½HþH0ðtÞ/C138 /C1X iSi þDX iðSi/C2Siþ^x/C1^xþSi/C2Siþ^y/C1^yÞ;(1) where H¼ð0;0;HzÞis a constant external magnetic field normal to the plane, and H0ðtÞis an applied time-dependent magnetic field. The norm of the spin vector is set to be unity.PRL 108, 017601 (2012) PHYSICAL REVIEW LETTERSweek ending 6 JANUARY 2012 0031-9007 =12=108(1) =017601(5) 017601-1 /C2112012 American Physical SocietyWe adopt J¼1as the energy unit and take D¼0:09. The spin turn angle /C18in the helical structure is determined by the ratio D=J astan/C18¼D=ðffiffiffi 2p JÞ, which is derived from a saddle point equation of the energy as a function of /C18. Our parameter set gives /C18¼3:64/C14or the periodicity of /C2499 sites, which corresponds to the Skyrmion diameter of /C2450 nm if we consider a typical lattice parameter of 5 A ˚. We study collective spin excitations of this model by numerically solving the LLG equation using the fourth-order Runge-Kutta method. The equation is given by @Si @t¼/C01 1þ/C112 G½Si/C2Heff iþ/C11G SSi/C2ðSi/C2Heff iÞ/C138;(2) where /C11Gis the dimensionless Gilbert-damping coeffi- cient. We derive a local effective field Heff iacting on the ith spin Sifrom the Hamiltonian H asHeff i¼ /C0@H=@Si. All the calculations are performed for systems with N¼288/C2288 sites under the periodic boundary condition. We fix /C11G¼0:04for simulations of the spectra shown in Fig. 2, while /C11G¼0:004for others. We first study phase diagram of the model ( 1)a tT¼0 as a function of Hz. Starting with spin configurations ob- tained in the Monte Carlo thermalization at low T,w e further relax them by sufficient time evolution in the LLG equation, and compare their energies. As shown in Fig. 1(a), helical (HL), SkX, and ferromagnetic (FM) phases appearsuccessively as Hzincreases where critical fields are Hz¼ 1:875/C210/C03andHz¼6:3/C210/C03, respectively. Here Hz¼1/C210/C03corresponds to /C243:4m T if we adopt a typical value of J¼0:4 meV andS¼1spins (see also Table I). In Fig. 1(b), we display spin configuration of the SkX phase where the in-plane components of the spinvectors at sites ( i x,iy) are described by arrows when modðix;6Þ¼modðiy;6Þ¼0. Here distribution of the spin z-axis components, Szi, is shown by a color map. One Skyrmion is magnified in Fig. 1(c)with a color map of the local scalar spin chiralities given by Ci¼Si/C1ðSiþ^x/C2Siþ^yÞþSi/C1ðSi/C0^x/C2Si/C0^yÞ:(3) The finite spin chirality is a source of the topological Hall effect [ 21] observed in experiments [ 22–25]. We then study the microwave-absorption spectra due to spin-wave resonances in the SkX phase. We trace spindynamics after applying a /C14-function pulse of magnetic field at t¼0, which is given by H 0ðtÞ¼/C14ðtÞH!. The absorption spectrum or the imaginary part of the dynamicalsusceptibility, Im/C31ð!Þ, is calculated from the Fourier transformation of magnetization mðtÞ¼ð 1=NÞP iSiðtÞ. In Fig. 2(a), we show calculated spectra for several values of Hzwhen H!is parallel to the xyplane. We find two resonance peaks in the spectra, and both of theirfrequencies increase as H zincreases as shown in the inset of Fig. 2(a). Note that !¼0:01corresponds to /C241 GHz forJ¼0:4 meV (¼96:7 GHz ). Thus these spin-wave resonances are located in the frequency range 500 MHz–1.2 GHz or in the microwave regime. On the other hand,the calculated spectra for H !parallel to the zaxis are shown in Fig. 2(b), which have only one resonance peak. The resonant frequency !Rdecreases as Hzincreases as shown in the inset. Again these resonances are located inthe microwave frequency regime. To identify each spin-wave mode, we trace the spin dynamics by applying a stationary oscillating magnetic FIG. 1 (color). (a) Phase diagram of the Hamiltonian ( 1)a t T¼0where HL, SkX, and FM denote helical, Skyrmion- crystal, and ferromagnetic phases, respectively. (b) Spin con- figuration of the SkX phase with a color map of the spin z-axis components SziatHz¼3:75/C210/C03. Spin vectors at sites (ix,iy) projected onto the xyplane are shown by arrows for modðix;6Þ¼modðiy;6Þ¼0. (c) One Skyrmion is magnified with a color map of the scalar spin chiralities Ci. TABLE I. Unit conversion table when J¼0:4 meV . Magnetic field H 1/C210/C03J /C243:4m T Frequency ! 0.01 J /C241 GHz Time t 1000 J/C01/C2410 nsecFIG. 2 (color online). Imaginary parts of (a) in-plane and (b) out-of-plane dynamical susceptibilities, Im/C31ð!Þ, in the SkX phase for several values of Hz. The insets show resonant frequencies !Ras functions of Hz.PRL 108, 017601 (2012) PHYSICAL REVIEW LETTERSweek ending 6 JANUARY 2012 017601-2field with resonant frequency !R. We first study the modes activated by the in-plane ac magnetic field by setting H0ðtÞ¼ð 0;H!ysin!Rt;0Þwith H!y¼0:5/C210/C03. The frequency !Ris fixed at !R¼6:12/C210/C03for the lower-energy mode, while at !R¼1:135/C210/C02for the higher-energy mode. We find that for all of the modes,all the Skyrmions show uniformly the same motion so thatwe focus on one Skyrmion hereafter. In Figs. 3(a)and3(b), we display calculated time evolutions of the spins. Thespins at sites ( i x,iy) are represented by arrows when modðix;6Þ¼modðiy;6Þ¼0together with distributions of the Szicomponents in the left panels, while those of the spin chiralities Ciin the right panels. Interestingly the area of larger Szior that of larger jCijcirculates around each Skyrmion core even though the applied ac field H0ðtÞ is linearly polarized in the ydirection. We find that direc- tions of their rotations are opposite, i.e., counterclockwise(CCW) with respect to the magnetic field Hkzfor the lower-lying mode while clockwise (CW) for the higher-lying mode. These directions are independent of the sign ofDM constant Dor winding direction of the spins. Instead they are determined by a sign of the applied field or by the spin orientation at the Skyrmion core. Because of these habits, the spin-wave excitations acti- vated by the in-plane ac magnetic field strongly depend onthe circular polarization of the irradiating microwave. InFig.4, we show calculated time evolutions of the magne- tization parallel to the yaxis,m yðtÞ¼ð 1=NÞP iSyiðtÞ, when we irradiate linearly polarized, left-handed circularly po- larized (LHP), and right-handed circularly polarized(RHP) in-plane microwaves with resonant frequency ! R¼ 6:12/C210/C03, which corresponds to the lower-lying mode atHz¼3:75/C210/C03. More concretely, we apply a time- dependent magnetic field H0ðtÞ¼½H0xðtÞ;H0yðtÞ;0/C138where FIG. 3 (color). Spin dynamics of each collective mode in the SkX phase calculated at Hz¼3:75/C210/C03. Spins at sites ( ix,iy) are represented by arrows when modðix;6Þ¼modðiy;6Þ¼0with color maps of the Szicomponents in the left panels, while in the right panels, distributions of the local spin chiralities Ciare displayed. Temporal waveforms of the applied ac magnetic fields, H!ysin!Rt andH!zsin!Rt, are shown in the uppermost figures where inverted triangles indicate times at which we observe the spin configurations shown here. (a) [(b)] Lower-energy [Higher-energy] rotational mode with !R¼6:12/C210/C03(!R¼1:135/C210/C02) activated by the in-plane ac magnetic field. Distributions of the Szicomponents and the spin chiralities Cicirculate around the Skyrmion core in a counterclockwise (clockwise) fashion. (c) Breathing mode with !R¼7:76/C210/C03activated by the out-of-plane ac magnetic field.PRL 108, 017601 (2012) PHYSICAL REVIEW LETTERSweek ending 6 JANUARY 2012 017601-3H0xðtÞ¼/C11H!xycos!RtandH0yðtÞ¼H!xysin!Rtwith/C11¼0 for the linearly polarized microwave and /C11¼1(/C01) for the LHP (RHP) microwave. In the LHP (RHP) microwave,its magnetic-field component rotates in a CCW (CW) way. Here we fix H !xy¼0:5/C210/C03. We find that irradiation of the LHP microwave significantly enhances the magnetiza- tion oscillation as compared to the linearly polarized mi- crowave, whereas the RHP microwave cannot activate collective spin oscillations. Next we discuss a spin-wave mode activated by the out-of-plane ac magnetic field. We again trace spin dy- namics by applying H0ðtÞ¼ð 0;0;H!zsin!RtÞwith!R¼ 7:76/C210/C03andH!z¼0:5/C210/C03. We observe a breath- ing mode where the area of each Skyrmion extends and shrinks dynamically as shown in Fig. 3(c). We finally study effects of the intense spin-wave exci- tation. We apply in-plane LHP ( /C11¼1) and RHP ( /C11¼/C01) microwaves of H0xðtÞ¼/C11H!xycos!t and H0yðtÞ¼ H!xysin!tto the SkX phase at Hz¼6:3/C210/C03. The system is located on the phase boundary between the SkX and FM phases. Here we take H!xy¼0:5/C210/C03, which corresponds to /C241:7m T when J¼0:4 meV andS¼1. The frequency !is fixed at 7:4/C210/C03. This value is nearly equal to the resonant frequency !R¼ 7:8/C210/C03of the lower-energy mode, but slightly deviates from it in reality. Because the intense spin-wave excitations necessarily change the spin structure fromits equilibrium configuration, and it results in redshifts of the resonant frequencies, we chose !slightly smaller than ! Rof the nearly equilibrium case in advance. In fact, the redshift can be seen in Fig. 4. The magnetization dynamics under the LHP microwave becomes slow as compared to that under the linearly polarized microwave when the oscillation amplitude becomes larger. One can easily notice this fact from different maximum points between these two oscillations. Indeed the oscillationfrequency in Fig. 4under the LHP microwave is !/C246:1/C210 /C03for0<t< 2000 , while !/C245:7/C210/C03 for3000 <t< 5000 . In Figs. 5(a) and5(b), we show snapshots of the spin configurations at several times under the irradiating LHP microwave. We observe melting of the SkX due to the intensively excited rotational spin-wave modes. The melt- ing occurs within t/C245000 –6000 . Here t¼1000 corre- sponds to /C2410 nsec when J¼0:4 meV . Thus the melting occurs within 50–60 nsec. We also find that the SkX melting is difficult to achieve either by the RHP microwave or even by the LHP microwave if its frequency is off resonant. Note also that the spatial pattern in Fig. 5(c) loses a periodicity of the original SkX, suggestive of a chaotic aspect of the melting dynamics. We finally compare the modes found in the SkX phase with those in the vortex-state nanodisks clarified in Refs. [ 26–29]. The twofold rotational modes and the breathing mode found in the SkX resemble, respectively,the twofold translational modes expressed by the Bessel functions with m¼/C6 1and the radial mode with m¼0in the vortex-state nanodisks. In Ref. [ 28], Ivanov and Zaspel theoretically showed that degeneracy of the translationalmodes with m¼/C6 1in the nanodisk is lifted under an applied magnetic field normal to the disk. We consider thata similar mechanism works in the SkX case for the doublet CW and CCW modes. There are also several differences. The modes in nanodisks are mainly governed by the long-range dipolar interaction, resulting in their salient aspect-ratio dependence. Note that their frequencies go to zero inthe zero aspect-ratio limit. In contrast, the SkX and itsdynamics considered here are governed by the nearest-neighbor spin interactions described in the Hamiltonian (1). The essential relevance of the DM interaction to the SkX is indicated by several experimental findings [ 15] such as its emergence only in chiral magnets, unique spin-0.800.8 my time0 1000 2000 3000 4000circular (LHP)linear circular (RHP)xyH’ LHP (CCW) xyH’xyH’ linear RHP (CW) FIG. 4 (color online). Calculated time evolutions of magneti- zation ( ky),myðtÞ¼ð 1=NÞP iSyiðtÞ, in the SkX phase at Hz¼ 3:75/C210/C03under linearly polarized, left-handed circularly polarized (LHP), and right-handed circularly polarized (RHP) in-pane ac magnetic fields with resonant frequency !R¼ 6:12/C210/C03corresponding to the lower-lying mode. FIG. 5 (color online). Melting of the SkX within nanoseconds under irradiating LHP microwave, which excites the rotationalspin-wave modes intensively (see text). Color maps of the S zi components are displayed at (a) t¼0, (b) t¼4000 , and (c)t¼7200 . Figures magnify a partial area with 220/C2220 sites for clarity, while the calculations are done for 288/C2288 sites with the periodic boundary condition. Temporal waveform of the microwave is also shown where times corresponding tofigures (a), (b), and (c) are indicated by inverted triangles.PRL 108, 017601 (2012) PHYSICAL REVIEW LETTERSweek ending 6 JANUARY 2012 017601-4swirling directions of Skyrmions, and considerably small size (10–100 nm) of Skyrmions compared to dipolar-force-induced magnetic bubbles. Thus we expect negligible aspect-ratio dependence of the modes as well as weak influences of the dipolar interaction. Our study focuseson thin films whose thickness is much smaller than theSkyrmion diameter because a greater stability of the SkXin thinner films has been confirmed [ 14,15]. In such a case, the system can be regarded as ferromagnetically stackedtwo-dimensional layers, which guarantees the validity ofour results based on a two-dimensional model. In summary, we have theoretically studied spin-wave excitations in the SkX phase of insulating ferromagnetswith DM interaction. We have found a couple of rotationalmodes with /C241 GHz frequencies for in-plane ac magnetic field. The rotations are in a CCW fashion for the lower-lying mode, while in a CW fashion for the higher-lyingmode. These habits give rise to strong dependence of thesespin-wave excitations on the circular polarization of the irradiating microwave. A breathing mode has been found for out-of-plane ac magnetic field. We have also observedthe melting of the SkX under the irradiating LHP micro-wave. These findings will open a route to manipulation ofthe Skyrmion as a nanoscale spin texture using spin-waveresonances. The author is deeply grateful to N. Nagaosa for fruitful discussion and insightful suggestions. The author also thanks Y. Tokura, M. Kawasaki, X. Z. Yu, and S. Seki for stimulating discussions. This work was supported byGrant-in-Aid (No. 22740214) and G-COE Program‘‘Physical Sciences Frontier’’ from MEXT Japan, byFunding Program for World-Leading Innovative R&D onScience and Technology (FIRST Program) on ‘‘QuantumScience on Strong Correlation’’ from JSPS, and byStrategic International Cooperative Program (Joint Research Type) from JST. Note added in proof .—Recently Petrova and Tchernyshyov analytically derived rotational spin-wavemodes in the SkX phase [ 30]. Analysis of the neutron- scattering data [ 31] based on our finding is an issue of future interest. [1] A. P. Malozemoff and J. C. Slonczewski, Magnetic Domain Walls in Bubble Materials , edited by R. Wolfe (Academic Press, New York, 1979). [2] S. D. Bader, Rev. Mod. Phys. 78, 1 (2006) . [3] T. H. R. Skyrme, Proc. R. Soc. A 260, 127 (1961) . [4] T. H. R. Skyrme, Nucl. Phys. 31, 556 (1962) . [5] S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, Phys. Rev. B 47, 16 419 (1993) . [6] M. Abolfath, J. J. 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Franz, F. Jonietz, R. Georgii, P. Bo ¨ni, B. Pedersen, M. Schmidt, A. Rosch, and C. Pfleiderer, Phys. Rev. B 81, 041203(R) (2010) . [14] S. D. Yi, S. Onoda, N. Nagaosa, and J. H. Han, Phys. Rev. B80, 054416 (2009) . [15] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature (London) 465, 901 (2010) . [16] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Nature Mater. 10, 106 (2010) . [17] F. Jonietz, S. Mu ¨hlbauer, C. Pfleiderer, A. Neubauer, W. Mu¨nzer, A. Bauer, T. Adams, R. Georgii, P. Bo ¨ni, R. A. Duine, K. Everschor, M. Garst, and A. Rosch, Science 330, 1648 (2010) . [18] K. Everschor, M. Garst, R. A. Duine, and A. Rosch, Phys. Rev. B 84, 064401 (2011) . [19] Current-driven motions of Skyrmions have been recently studied theoretically; K. S. Kim and S. Onoda, arXiv:1012.0631 [Phys. Rev. B (to be published)]; J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys. Rev. 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PhysRevB.103.014433.pdf

PHYSICAL REVIEW B 103, 014433 (2021) Current-induced spin-wave Doppler shift and attenuation in compensated ferrimagnets Dong-Hyun Kim,1Se-Hyeok Oh,2Dong-Kyu Lee,3Se Kwon Kim,4and Kyung-Jin Lee3,4,5,* 1Department of Semiconductor Systems Engineering, Korea University, Seoul 02841, Korea 2Department of Nano-Semiconductor and Engineering, Korea University, Seoul 02841, Korea 3Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea 4Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea 5KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea (Received 17 July 2020; revised 25 November 2020; accepted 7 January 2021; published 20 January 2021) We theoretically and numerically study current-induced modification of ferrimagnetic spin-wave dynamics when an electrical current generates adiabatic and nonadiabatic spin-transfer torques. We find that the sign of theDoppler shift depends on the spin-wave handedness because the sign of spin polarization carried by spin wavesdepends on the spin-wave handedness. It also depends on the sign of the adiabatic-torque coefficient, originatingfrom unequal contributions from two sublattices. For a positive nonadiabaticity of spin current, the attenuationlengths of both right- and left-handed spin waves increase when electrons move in the same direction withspin-wave propagation. Our result establishes a way to simultaneously measure important material parametersof a ferrimagnet, such as angular momentum compensation point, spin polarization, and nonadiabaticity usingcurrent-induced control of ferrimagnetic spin-wave dynamics. DOI: 10.1103/PhysRevB.103.014433 I. INTRODUCTION Conventional semiconductor devices use the electron charge to compute and store information, which inevitablycauses Joule heating. In contrast, spin wave (SW) devices,where the SW is used as the information carrier, avoid theJoule heating as the SW is a collective low-energy magneticexcitation that does not involve moving charges [ 1–3]. Several concepts of SW devices implementing Boolean/non-Booleancomputing and multi-input/output operations have been re-ported [ 4–17]. Up until now, most SW studies have focused on ferromagnetic SWs. In comparison to ferromagnetic SWs, antiferromagnetic SWs have several distinct features. Unlike ferromagnetic SWswhose frequency is in gigahertz (GHz) ranges, the frequencyof antiferromagnetic SWs can reach terahertz (THz) ranges[18,19], which allows fast SW operation. In addition, both right-handed and left-handed modes are allowed in antiferro-magnets because of the antiferromagnetic coupling betweentwo sublattice moments [ 20,21]. This gives an additional de- gree of freedom for SW operations [ 22–26] as compared to the ferromagnetic SW that has only the right-handed mode. This intriguing antiferromagnetic dynamics is also realized in compensated ferrimagnets [ 27–40]. Antiferromagnetically coupled ferrimagnets composed of rare-earth (RE) and tran-sition metal (TM) elements have two compensation points.One is the magnetic moment compensation point where thenet magnetic moment is zero and the other is the angularmomentum compensation point where the net angular mo-mentum is zero. These two compensation points are different *kjlee@kaist.ac.krwhen the Landé gfactors of RE and TM elements are dif- ferent [ 27]. As the intrinsic dynamics of localized spins is governed by the commutation relation between angular mo-mentum (not magnetic moment) and the relevant Hamiltonian,antiferromagneticlike spin dynamics is realized at the angularmomentum compensation point of compensated ferrimagnets.Given that the net magnetic moment is nonzero at the angularmomentum compensation point, RE-TM ferrimagnets allowus to investigate antiferromagneticlike spin dynamics with afinite Zeeman coupling. For this reason, antiferromagneticlikespin dynamics of compensated ferrimagnets has been exten-sively studied in recent studies [ 27–40]. Another intriguing feature of compensated ferrimagnets is that spin transport is distinct from both ferromagnets andantiferromagnets. When a spin-polarized current is injectedinto a magnetic material, it exerts a torque on the local mag-netic moment by transferring spin angular momentum. Thisspin-transfer torque (STT) [ 41,42] consists of two mutually orthogonal vector components, adiabatic torque and nonadi-abatic torque [ 43–51], for continuously varying spin textures such as SW, domain wall, and skyrmion. For ferromagnets, itis well known that the adiabatic STT causes current-inducedSW Doppler shift [ 52,53] whereas the nonadiabatic STT con- trols SW attenuation [ 54–56]. It was predicted [ 57] that the current-induced SW Doppler shift by the adiabatic STT isalso present for antiferromagnets. Although there has beenno study on the current-controlled SW attenuation for antifer-romagnets, a recent numerical study found a non-negligiblenonadiabatic STT for antiferromagnetic domain walls [ 58], suggesting that electrical currents can control the attenuationof antiferromagnetic SWs. In addition, a recent experiment on GdFeCo ferrimagnets shows that the adiabatic torque in this material can be large 2469-9950/2021/103(1)/014433(7) 014433-1 ©2021 American Physical SocietyKIM, OH, LEE, KIM, AND LEE PHYSICAL REVIEW B 103, 014433 (2021) FIG. 1. (a) A schematic illustration of ferrimagnetic spin waves when the current is applied along the xaxis. For numerical simu- lations, an ac field Hac(10 mT) is applied to excite SWs. (b) The Doppler shift for right- (solid line) and left-handed (dashed line) SW [/Delta1ω±=ω±(J=1×109A/m2)−ω±(J=0)] as a function of net spin density δs. Here, we assume that the exchange constant A=3×10−12J/m, the easy-axis anisotropy constant along the zdi- rection K=104J/m3, spin polarization PRE=0.1,PTM=0.4, and the wave vector is 0 .02, 0.03, and 0 .04 nm−1. [59], which stems from a finite net spin polarization of RE and TM sublattices. Thus, the current-induced SW Dopplershift of compensated ferrimagnets is expected to be similarin magnitude to that of ferromagnets. Moreover, the sameexperiment [ 59] shows that the nonadiabatic torque in this material is large (i.e., equivalently, the ratio of nonadiabaticityβto damping αis large). This large nonadiabaticity of spin current in compensated ferrimagnets is attributed to the en-hanced spin mistracking [ 46,48,58,59], originating from the weakened spin dephasing in the antiferromagnetically alignedspin moments [ 60,61]. This unique STT characteristic of com- pensated ferrimagnets motivates us to investigate STT effectson ferrimagnetic SWs. In this work, we theoretically and numerically study the STT-induced control of ferrimagnetic SW dynamics nearthe angular momentum compensation point. To begin with,we derive the equations of motion for ferrimagnetic SWsin the presence of two torque components. From the equa-tions of motion, we obtain current-driven ferrimagnetic SWDoppler shift and attenuation. Then, we perform atomisticlattice model simulations to verify the obtained analytic so-lutions. We show that the ferrimagnetic SW Doppler shift dueto the adiabatic torque for right-handed SWs is opposite tothat for left-handed SWs since they carry opposite spin polar-izations. We also find that the SW attenuation is suppressed,and the SW amplitude is even amplified when a sufficientlylarge nonadiabatic torque is exerted. II. ANALYTIC THEORY We consider a ferrimagnet consisting of RE and TM mo- ments, which are antiferromagnetically coupled as shown inFig. 1(a). We introduce two unit vectors A kandBkdenoting localized spins located at two sublattices, the AandBsites. We define the total magnetization vector and the staggeredmagnetization vector as m=A k+Bkandn=(Ak−Bk)/2, respectively. The spin density is sA(B)=MA(B)/γA(B)where MA(B)is the saturation magnetization and γA(B)is the gyro- magnetic ratio. The Lagrangian density Lfor ferrimagnets isgiven by [ 30,62–65] L=−s˙n·(n×m)−δsa(n)·˙n−U, (1) where s=(sA+sB)/2 is the sum of spin densities of two sub- lattices, δs=sA−sBis the net spin density, a(n) is the vector potential, and the potential energy Ucontains the exchange energy and easy-axis anisotropy energy, given by U=A 2(∇n)2+a 2m2+Lm·∂xn−K 2(ˆz·n)2. (2) Here Ais the inhomogeneous exchange, ais the homogeneous exchange, Lis the parity-breaking exchange term [ 65,66], and Kis the effective easy-axis anisotropy including the demag- netization effect in the zdirection. We assume that the Gilbert damping constant αis the same regardless of site ( αA=αB), which simplifies the Rayleigh dissipation function as R= αs˙n2. In this theory, we neglect nonlocal dipolar interaction because net magnetization is an order of magnitude smallerthan the ferromagnets. By solving the above equations for nandm, we obtain two staggered equations of motion to linear order in thecurrent-induced STT effective field (i.e., by working withinlinear-response theory) and to the first order in the net mag-netization |m|by assuming that a change from a ground state (with m=0) is small, i.e., |m|/lessmuch 1, due to the strong anti- ferromagnetic exchange coupling by following the approachtaken in Ref. [ 67]: ˙n=−1 sfm×n+Tn STT, (3) ˙m=−1 sfn×n+2α˙n×n−δs s˙n+Tm STT, (4) where fm=−∂U ∂m,fn=−∂U ∂n,Tn(m) STTis the STT that affects n(m) dynamics, given as (see Appendix) Tn STT=−b+ j 2∂n ∂x−βb− j 2n×∂n ∂x, (5) Tm STT=−b− j∂n ∂x−βb+ jn×∂n ∂x, (6) where b± j=−μB 2e(PAgA MA±PBgB MB)Jeis the magnitude of adia- batic spin torque, PA(PB) is the spin polarization, gA(gB)i s the Landé gfactor, μBis the Bohr magneton, eis the electron charge, Jeis the current density, and βis the nonadiabaticity. Note that, when the two sublattices are equivalent and thusb − j=0, Eqs. ( 5) and ( 6) are same (except for numerical factors) as the two spin-transfer torque terms for antiferro-magnets shown in Eqs. ( 5) and ( 6)o fR e f .[ 67]. When deriving Eqs. ( 5) and ( 6), we retained the terms involving the gradient of the order parameter nwhile neglecting the terms involving the small net magnetization mby assuming mis strongly suppressed by the antiferromagnetic exchange coupling. Here,we define that all of spin polarization, Landé gfactor, Bohr magneton, and electron charge are positive and assume β A= βB=βfor simplicity. We note that b+ jcorresponds to a stag- gered torque exerting on two sublattice moments, whereasb − jcorresponds to a uniform torque. For antiferromagnets, b− jvanishes and b+ jis the only torque to drive dynamics of antiferromagnetic spin textures. In contrast, for ferrimagnets, 014433-2CURRENT-INDUCED SPIN-W A VE DOPPLER SHIFT AND … PHYSICAL REVIEW B 103, 014433 (2021) both b+ jandb− jare nonzero in general so that both torques affect the dynamics. We derive the equation of motion by inserting the STT [Eqs. ( 5) and ( 6)] into the staggered equations of motion [Eqs. ( 3) and ( 4)]. Then, we obtain the equation of motion fornas ρnרn+2αsn×˙n+δs˙n =A∗n×∂2 xn+Kn×nzˆz−sb− j∂xn−sβb+ jn×∂xn,(7) where A∗=A−L2/ais the renormalized exchange stiffness constant [ 65] andρ=s2/ais the inertia. It is worthwhile to note that the STT effect, i.e., the third and fourth terms on theright-hand side of Eq. ( 7), comes from T m STT[Eq. ( 6)], which is the STT acting on a ferromagnetic component m. On the other hand, the contribution of Tn STT[Eq. ( 5)], i.e., the STT acting on a stagger vector n, does not appear in Eq. ( 7) because it is of the third order in small parameters and thus negligible. By defining a complex field as ψ±=nx∓inyfor right- and left-handed SWs and linearizing the above equation for nxand ny, we obtain ±δS˙ψ±−i2αs˙ψ±−iρ¨ψ± =− iA∗∂2 xψ±+iKψ±∓sb− j∂xψ±+isβb+ j∂xψ±.(8) The upper (lower) sign corresponds to right- (left-) handed SW. By inserting the plane wave solution ψ±= exp[i( kx−ω±t)] exp[ −x//Lambda1±] into Eq. ( 8), we obtain the SW dispersion and SW attenuation length /Lambda1, given as ω±=±δs+/radicalBig δ2s+4ρ(A∗k2+K∓sb− jk) 2ρ, (9) /Lambda1±=2A∗k∓sb− j s(2αω±−βb+ jk). (10) Equations ( 9) and ( 10) are our central results. We first discuss the current-induced SW Doppler shift [Eq. ( 9)]. For antiferromagnets ( δs=0 and b− j=0), Eq. ( 9) shows no current-induced SW Doppler shift. This is causedby the fact that we derive the equations with the second-orderexpansion of small parameters. When we consider up to thethird-order terms, there is a finite SW Doppler shift evenfor antiferromagnets, which is consistent with Ref. [ 57]. For ferrimagnets, the last term in the square root of Eq. ( 9) (i.e., ∓sb − jk) signifies the current-induced SW Doppler shift. It originates from the uniform adiabatic torque b− jacting on a ferromagnetic component m. Figure 1(b) shows the current-induced SW Doppler shift /Delta1ω±as a function of the net spin density δs, computed from Eq. ( 9). Three observations are worth mentioning. First, the sign of Doppler shift depends on the SW handedness be-cause opposite spin polarizations are carried by right- andleft-handed SWs. Second, the Doppler shift is also related tothe sign of b − jbecause b− jcan be positive or negative depend- ing on the material parameters such as polarization, Landé g factor, and saturation magnetization. For a specific RE-TMferrimagnet, i.e., a GdCo ferrimagnet, the sign of b − jwould not change with temperature in the vicinity of TAbecause gGd≈gCoand MGdis not much different from MCo[59],while PGdis four times smaller than than PCo[68]. Third, the Doppler shift /Delta1ω±is maximized in the vicinity of the angular momentum compensation point TA(i.e.,δs=0). To get an insight into the second observation, we expand Eq. ( 9)i nt h e limit of small current density and obtain ω±≈ω0,±+/Delta1ω±, where the current-independent frequency ω0is given by ω0,±=±δs+/radicalbig δ2s+4ρ(A∗k2+K) 2ρ, (11) and, the current-induced Doppler shift /Delta1ω±is given by /Delta1ω±=∓sb− jk /radicalbig δ2s+4ρ(A∗k2+K). (12) Equation ( 12) shows that, in this limit, the current-induced Doppler shift of ferrimagnetic SW is linear in kand in current density as for ferromagnetic SWs [ 52–54]. It also shows that the current-induced Doppler shift /Delta1ω±is maximized in the vicinity of TAwhere δsvanishes. This result suggests that one can experimentally determine TAby measuring the current- induced SW Doppler shift. We next discuss the current-induced control of SW atten- uation [Eq. ( 10)]. For antiferromagnets ( δs=0 and b− j=0), Eq. ( 10) shows that the staggered nonadiabatic torque (i.e., βb+ jk) modifies the SW attenuation length. It means that the SW attenuation length in antiferromagnets is determinedby the denominator (2 αω ±−βb+ jk), which describes the competition between the damping torque and the staggerednonadiabatic torque. For ferrimagnets, in addition to the stag-gered nonadiabatic torque, the uniform adiabatic torque (i.e.,∓sb − j) in the numerator of Eq. ( 10) also controls the SW at- tenuation length, but its contribution is independent of k. With typical material parameters, however, this adiabatic-torquecontribution to the SW attenuation length is usually negligibleso that the main contribution is the competition between thedamping torque and the staggered nonadiabatic torque, evenfor ferrimagnets. Current-induced effect on the SW attenuation length de- pends on the relative flow direction of the electron and theSW ( k). Considering β> 0, when electrons flow in the same (opposite) direction as SWs, the attenuation length increases(decreases). When a large current is injected, i.e., b + j>2αω± βk, Eq. ( 10) becomes negative so that the SW solution is ψ= exp[i( kx−ωt)] exp[ +x/|/Lambda1|], meaning that SWs are amplified for both antiferromagnets and ferrimagnets. III. NUMERICAL ANALYSIS To verify the above analytic results, we perform atomistic lattice model simulations. We consider the one-dimensionalatomistic Hamiltonian as H=A sim/summationdisplay i,jSi·Sj−Ksim/summationdisplay i(Si·ˆz)2, (13) where Asimis the exchange constant, Ksimis the easy-axis anisotropy constant, Siis the spin moment vector at the isite, andjis the notation representing the nearest lattice of the site i. The atomistic Landau-Lifshitz-Gilbert equation including 014433-3KIM, OH, LEE, KIM, AND LEE PHYSICAL REVIEW B 103, 014433 (2021) TABLE I. The saturation magnetizations for RE and TM elements. The index T2corresponds to the angular momentum com- pensation temperature TA. Index T1 T2(=TA) T3 MRE(kA/m) 426 386 344 MTM(kA/m) 455 424.6 392 δs(×10−8Js/m3) 7.02 0 −7.02 STT terms is given as ∂Si ∂t=−γiμ0Si×Heff,i+αiSi×∂Si ∂t−bJ,iSi+1−Si−1 2d −βibJ,iSi×Si+1−Si−1 2d. (14) We solve the above equation by using the Runge-Kutta fourth-order method. Here, γi=giμB/¯his the gyromagnetic ratio, μ0is the permeability in vacuum, giis the Landé g factor, Heff,i=−1 μi∂H ∂Siis the effective field, μiis the magnetic moment, αiis the Gilbert damping constant, bJ,i=−giPiμB 2eMiJe is the magnitude of adiabatic STT, Piis the spin polarization, Miis the saturation magnetization, and βiis the nonadia- baticity. We locally apply a circularly polarized external field,μ 0Hac=μ0H0(cos 2πft,sin 2πft,0) with μ0H0=10 mT to excite SWs in a ferrimagnet. We also inject an in-planecurrent corresponding to the current density J eranging from −5×1012A/m2to+5×1012A/m2to induce STT. We use the following simulation parameters: the lattice constant d= 0.4 nm, the exchange constant Asim=7.5 meV, the easy-axis anisotropy constant Ksim=0.004 meV, the Gilbert damping constant α=0.003, the nonadiabaticity β=10αandβ= 100α, the Landé gfactor gRE=2,gTM=2.2, and the spin polarization PRE=0.1,PTM=0.4. In the continuum limit, the corresponding parameters in Eq. ( 7)a r eg i v e nb y A∗= 4Asim/dandK=4Ksim/d3. We assume that both damping constant and nonadiabaticity are the same regardless of thesublattice site as assumed for the analytic theory. We use thesaturation magnetization listed in Table I, which is measured in Ref. [ 59] for GdFeCo. We consider the system size of 3200×100×0.4n m 3with cell size 0 .4×100×0.4n m3 and perform simulations up to 4 ns, after which the system reaches a sufficiently steady state. Figures 2(a) and 2(b), respectively, show dispersions of the right- and left-handed SWs at zero applied current. T2 corresponds to the angular momentum compensation point TAwhere δs=SRE−STM=0 and T1<T2<T3. In all cases, numerical results (symbols) are in agreement with analyticalresults [lines, Eq. ( 9)]. The frequency of the right-handed SW is the highest at T 1[Fig. 2(a)], whereas the frequency of the left-handed SW is the lowest at T1[Fig. 2(b)]. This difference originates from different δs[Eq. ( 9)]. Figures 2(c) and 2(d), respectively, show the current- induced SW Doppler shifts of the right- and left-handedSWs at the angular momentum compensation temperatureT A(=T2) when the current density Je=± 5×1012A/m2is applied. Numerical results (symbols) are in reasonable agree-ment with Eq. ( 9) (solid lines). For the right-handed SW [Fig. 2(c)] and k>0, a positive (negative) current decreases FIG. 2. Ferrimagnetic SW dispersion for (a) right- and (b) left-handed SW in the vicinity of angular momentum compensa-tion temperature ( T A) when no current is applied. Current-induced Doppler shift of (c) right- and (d) left-handed SWs in ferrimagnet atTA. (increases) the SW frequency. On the other hand, for the left- handed SW [Fig. 2(d)] and k>0, a positive (negative) current increases (decreases) the SW frequency. Therefore, the sign ofthe Doppler shift of the right-handed SW is opposite to that ofthe left-handed SW, consistent with the analytic expression[Eq. ( 9)]. The same tendency of Doppler shift is obtained for other temperatures (not shown). Figures 3(a) [3(b)] shows the SW attenuation length as a function of the current density for β=10αand right- (left-)handed SWs. The SW frequency ( ω/2π)i s0 . 4T H z . Numerical results (symbols) are in reasonable agreement withEq. ( 10) (solid lines). For a positive β, we find that the SW at- tenuation length of both right- and left-handed SWs increaseswhen electrons move in the same direction with the SW prop-agation. When the nonadiabatic torque is sufficiently large[β=100α,F i g s . 3(c) and 3(d)], the antidamping effect of nonadiabatic torque overcomes the intrinsic damping torqueand, as a result, the SW attenuation length becomes negative,meaning that the SW is amplified. IV . SUMMARY To summarize, we theoretically study the STT effects on ferrimagnetic SWs. Unlike antiferromagnetic SWs for whichcurrent-induced Doppler shift is small, ferrimagnetic SWsexhibit non-negligible Doppler shift because of a finite spinpolarization. The current-induced Doppler shift is maximizedin the vicinity of the angular momentum compensation pointT A, providing a way to measure TA. The sign of the Doppler shift depends on the SW handedness, because the spin polar-ization carried by SWs also depends on the SW handedness.A recent experiment has identified the SW handedness in 014433-4CURRENT-INDUCED SPIN-W A VE DOPPLER SHIFT AND … PHYSICAL REVIEW B 103, 014433 (2021) FIG. 3. The SW attenuation length as a function of current den- sity at β=10αfor (a), (b) and β=100αfor (c), (d). (a), (c) are for right-handed SWs and (b), (d) are for left-handed SWs. The solidlines are analytic results and symbols are numerical results. ferrimagnets by measuring the relative magnitudes of Stokes and anti-Stokes peak in the Brillouin light scattering [ 39]. Our work suggests an alternative way to identify the SW handed-ness by measuring the sign of current-induced SW Dopplershift. It is found that the attenuation length of ferrimagnetic SWs is modified by nonadiabatic staggered torque, which can be used to experimentally determine the nonadiabaticity βof a ferrimagnet. Combined with the current-induced SW Dopplershift, our work provides a way to simultaneously determineimportant material parameters of ferrimagnets, namely, theangular momentum compensation point T A, the spin polar- ization P, and the nonadiabaticity β, by performing a single series of time-domain measurements of current-induced SWdynamics in a ferrimagnet. However, the determination ofthe handedness or the unknown parameters is experimentallychallenging and may need to be combined with other indepen-dent measurements of the spin polarization [ 68] and T A[29]. ACKNOWLEDGMENTS K.-J.L. was supported by the National Research Founda- tion (NRF) of Korea (Grant No. NRF-2020R1A2C3013302).S.K.K. was supported by Brain Pool Plus Program throughthe National Research Foundation of Korea funded bythe Ministry of Science and ICT (Grant No. NRF-2020H1D3A2A03099291).APPENDIX: DERIV ATION OF EXPRESSION FOR STT ON FERRIMAGNETS In this part, we derive Eqs. ( 5) and ( 6) from the STT exerting on each sublattice as ∂Ak ∂t=−bj,A∂Ak ∂x−βAbj,AAk×∂Ak ∂x, (A1) ∂Bk ∂t=−bj,B∂Bk ∂x−βBbj,BBk×∂Bk ∂x, (A2) where bj,i=−giμBPi 2eMiJeand the first (second) term represents the adiabatic (nonadiabatic) torque. Using Ak=m 2+nand Bk=m 2−n, we obtain ∂ ∂t/parenleftBigm 2+n/parenrightBig =−bj,A∂ ∂x/parenleftBigm 2+n/parenrightBig −βAbj,A/parenleftBigm 2+n/parenrightBig ×∂ ∂x/parenleftBigm 2+n/parenrightBig , (A3) ∂ ∂t/parenleftBigm 2−n/parenrightBig =−bj,B∂ ∂x/parenleftBigm 2−n/parenrightBig −βBbj,B/parenleftBigm 2−n/parenrightBig ×∂ ∂x/parenleftBigm 2−n/parenrightBig . (A4) Combining Eqs. ( A3) and ( A4) and assuming a uniform β,w e obtain ∂n ∂t=−b− j 4∂m ∂x−b+ j 2∂n ∂x−βb− j 8m×∂m ∂x−βb+ j 4m ×∂n ∂x−βb+ j 4n×∂m ∂x−βb− j 2n×∂n ∂x, (A5) ∂m ∂t=−b+ j 2∂m ∂x−b− j∂n ∂x−βb+ j 4m×∂m ∂x−βb− j 2m ×∂n ∂x−βb− j 2n×∂m ∂x−βb+ jn×∂n ∂x, (A6) where b± j=−μB 2e(PAgA MA±PBgB MB)Je. 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PhysRevA.95.022327.pdf

PHYSICAL REVIEW A 95, 022327 (2017) Hybrid quantum systems with trapped charged particles Shlomi Kotler,*Raymond W. Simmonds, Dietrich Leibfried, and David J. Wineland National Institute of Standards of Technology, 325 Broadway St., Boulder, Colorado 80305, USA (Received 9 August 2016; published 21 February 2017) Trapped charged particles have been at the forefront of quantum information processing (QIP) for a few decades now, with deterministic two-qubit logic gates reaching record fidelities of 99 .9% and single-qubit operations of much higher fidelity. In a hybrid system involving trapped charges, quantum degrees of freedom of macroscopicobjects such as bulk acoustic resonators, superconducting circuits, or nanomechanical membranes, couple tothe trapped charges and ideally inherit the coherent properties of the charges. The hybrid system thereforeimplements a “quantum transducer,” where the quantum reality (i.e., superpositions and entanglement) of smallobjects is extended to include the larger object. Although a hybrid quantum system with trapped charges couldbe valuable both for fundamental research and for QIP applications, no such system exists today. Here we studytheoretically the possibilities of coupling the quantum-mechanical motion of a trapped charged particle (e.g.,an ion or electron) to the quantum degrees of freedom of superconducting devices, nanomechanical resonators,and quartz bulk acoustic wave resonators. For each case, we estimate the coupling rate between the chargedparticle and its macroscopic counterpart and compare it to the decoherence rate, i.e., the rate at which quantumsuperposition decays. A hybrid system can only be considered quantum if the coupling rate significantly exceedsall decoherence rates. Our approach is to examine specific examples by using parameters that are experimentallyattainable in the foreseeable future. We conclude that hybrid quantum systems involving a single atomic ionare unfavorable compared with the use of a single electron because the coupling rates between the ion and itscounterpart are slower than the expected decoherence rates. A system based on trapped electrons, on the otherhand, might have coupling rates that significantly exceed decoherence rates. Moreover, it might have appealingproperties such as fast entangling gates, long coherence, and flexible topology that is fully electronic in nature.Realizing such a system, however, is technologically challenging because it requires accommodating both atrapping technology and superconducting circuitry in a compatible manner. We review some of the challengesinvolved, such as the required trap parameters, electron sources, electrical circuitry, and cooling schemes in orderto promote further investigations towards the realization of such a hybrid system. DOI: 10.1103/PhysRevA.95.022327 I. INTRODUCTION Trapping of charged particles [ 1,2] has enabled long interrogation times of their external and internal states,enabling precision metrology, such as atomic clocks. Applyingthese tools to atomic ions, paired with laser-enabled statemanipulation, can also turn ions into a quantum informationprocessing (QIP) platform [ 3–7]. Ions have demonstrated record fidelities for initialization, readout, individual spinmanipulation [ 8], and entanglement [ 9,10]. Other quantum-coherent systems might therefore benefit, by coupling to trapped ions, potentially inheriting aspectsof their high controllability and coherence. For example,as described below, we might be able to use a single 9Be+ion coupled to a ∼10 mg quartz resonator to cool the latter close to its ground state. By placing the ion ina superposition state of motion and transferring it to amacroscopic resonator, we could explore bounds on quantummechanics for massive objects. The ion therefore could providea “quantum transducer” that enables the manipulation of amuch larger object in a coherent way at the single-phononlevel. For the purpose of QIP, ions might be used as excellentmemory units, e.g., for superconducting devices, as long asquantum information can be exchanged between the twosystems on timescales that are sufficiently short compared *shlomi.kotler@nist.govwith the decoherence time of the superconducting circuit. Theinternal degrees of freedom of an ion can remain coherentfor tens to hundreds of seconds [ 8,11–13], significantly exceeding the lifetime of coherent excitation in currentsuperconducting devices, typically limited to less than 100 μs (see, e.g., Ref. [ 14]), setting the timescale for useful quantum exchange. The resonant interaction of ions with radio frequency electrical resonators was studied in Ref. [ 15]. Complementary parametric interaction schemes for the nonresonant case werestudied in Refs. [ 16–20]. Other suggestions include interfacing nanomechanical resonators [ 21–25], electrical wires [ 26], and superconducting qubits [ 25]. These reports analyzed the basic physics involved in each of the different couplingmechanisms as well as the prospects of using such hybridsystems. Here, we focus on a few specific examples of hybrid systems rather than presenting a general treatment. For theseexamples we take into account available materials, achievablequality factors. and practical limitations. Nevertheless, ouranalysis is based on a unified framework (Sec. II) that allows for direct comparison of relevant figures of merit associatedwith the different systems. We hope these examples arerepresentative of the different opportunities available and canilluminate some of the issues of hybrid QIP with chargedparticles. A charged particle moving in a harmonic trap gives rise to an oscillating electric dipole. This dipole in turn can couple 2469-9926/2017/95(2)/022327(29) 022327-1 ©2017 American Physical SocietyKOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) to nearby charged objects [ 21,27,28], generate image currents in a nearby conductor [ 15], polarize a dielectric material, or induce motion in a piezoelectric crystal. If the coupled systemalso has a harmonic mode resonant with the ion motion, energyexchange will occur between the ion harmonic motion and thecoupled system. The analysis that follows below is guided by the realization that coupling two quantum systems is a double-edged sword.Ideally, we would like to benefit from the useful propertiesof both systems. In reality, the hybrid system often inheritsthe disadvantages of both constituents. Therefore, to retainany useful quantum characteristics, we require that thecoupling rate between the two systems exceeds the fastestrelevant decoherence rate in both systems. Additionally, wefocus on specific architectures where the two technologiesinvolved could be compatible and not preclude either ofthe coupled systems from being close to a pure quantumstate. Although we cannot completely rule out all mechanisms considered here that involve an atomic ion, the analysisemphasizes how challenging it would be to incorporate ionsinto a hybrid system at the quantum level. The coupling rateswe calculate, based on experimentally attainable parameters,are either well below the decoherence rates or marginally closeto them. This conclusion changes when considering couplinga charged particle to a superconducting resonator, assumingan electron rather than an ion (e.g., see Refs. [ 15,18,29,30] as well as Sec. VIfor a more extensive reference list). This follows from the fact that for a particle of mass mthe coupling rate is proportional to m −1/2(see Sec. IV), rendering coupling rates for an electron on the order of ∼0.1 to 1 MHz, where we expect to exceed decoherence rates. The shift from using an atomic ion to using an electron has significant practical implications, as detailed in Sec. VI. Laser-enabled state manipulation, specifically laser cooling,plays an important role in trapped atomic ion QIP experiments.Without these tools, electrical-circuit-based alternatives needto be considered along with their implications on the systemas a whole. We therefore embark on a feasibility study thattakes these implications into account, considering amongother factors trap stability, trap depth, maintaining super-conductivity, the requirements from a low-energy electron source, electrical wiring, and the superconducting resonator involved. A previous report [ 18] has already suggested a specific electron trap that would support a parametric couplingscheme. The different aspects considered in the feasibil-ity study bear significance on the trap design, suggestingthat a larger trap be used for an electron-based hybridsystem. Although technically challenging, these issues do not seem to preclude a hybrid system based on a trapped electron. Asdetailed in Sec. VI, such a platform might offer appealing qualities such as fast entangling gates ( ∼10 ns) and long coherence times (seconds), rendering a coherence time togate time ratio of /greaterorequalslant10 8, far exceeding any current QIP system. Moreover, the platform could offer a flexible couplingtopology enabled by interfacing engineered electrical circuits,potentially enabling high-fidelity electron spin readout. This,in turn, could open new avenues of basic research, interestingin their own right.II. ELECTRICAL EQUIVALENT OF MECHANICAL MOTION There are various systems that could, in principle, couple to a trapped charged particle. Those systems differ from thecharged particle and from one another in frequency, mass,length scale, and coupling mechanism, as highlighted inFig. 1. With the exception of the electrical LC resonator, all other systems considered here are mechanical resonatorsactuated by an electromagnetic field. To place all of themon an equal footing we associate an electrical equivalent foreach of these mechanical systems. This reduces the analysisof any of the hybrid systems into an all-electrical circuitproblem. Our discussion extends the treatment in Ref. [ 31] where the electrical equivalent circuit of a trapped ion wasderived. This could also be derived by using the generalframework developed by Butterworth and Van Dyke [ 32–34] that associates a circuit equivalent for electrically actuatedmechanical systems. We refer to the resulting electricalnetwork as the BVD equivalent circuit. Suppose a mechanical system of mass mis placed near an electrode that is biased with voltage V, resulting in a forceF=βV acting on it. For simplicity, we assume the geometry in Fig. 2(a), where two electrodes form the two plates of a parallel plate capacitor, separated by a distanced. An important example (analyzed in Refs. [ 31,35]) is that of a single charged particle with charge qresulting in F=qV/d , i.e., β=q/d. In general, electrical actuation could also result from dipolar interaction, electrostriction,piezoelectricity, etc. Since microscopically these mechanismsoriginate from having nonzero local charge densities within themechanical system, we lump the overall effect of the voltagewith a single effective parameter β. When the mass mmoves at velocity v[see Fig. 2(b)], it will induce a current I=βvat the electrode. This is an immediate generalization of the single-charged-particle case:if it is at a distance xfrom an electrode it induces an image charge of q image=qx/d . Therefore, within the electrostatic approximation, a velocity v=˙xwould translate into a current I=qv/d . The induced charges will back-act on the mass mwith an additional force /Delta1F. This force, however, will be independent of Vand will not contribute to the induced current I. The effect of /Delta1F can therefore be lumped into a (usually but not necessarily) small change of the system’s mechanicalproperties, e.g., its spring constant in the case of a harmonicoscillator (for a rigorous derivation see Refs. [ 31,35]). Now assume that the mechanical system is harmonic, i.e., that it has a resonant frequency ω 0and a friction coefficient γ. If the voltage is time varying, V(t), the equation of motion for the harmonic-oscillator position xis m¨x+γ˙x+mω2 0x=βV(t). (1) By using the relation I(t)=β˙xthis can be rewritten as m β2dI dt+γ β2I+mω2 0 β2/integraldisplayt dt/primeI(t/prime)=V(t). (2) Therefore, from the perspective of the electrical circuit, the mechanical system is equivalent to a series combination of 022327-2HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) Electrostatic Ind. currentsPiezoelectric Piezoelectric198Hg+ FIG. 1. Examples of different platform candidates for a hybrid architecture considered in this paper. Clockwise from the top-middle: 198Hg+ion trap, quartz bulk acoustic wave resonator, gallium nitride nanobeams, superconducting LC circuit, and nanomechanical silicon nitride (SiN) membrane. The ion (green shading) is coupled via piezoelectricity (red shading), induced image currents (purple shading) orelectrostatics (blue shading). Ion trap photograph courtesy of J. Bergquist, NIST, Boulder, Colorado 80305, USA. Gallium nitride nanobeams photographs courtesy of K. Bertness, NIST, Boulder, Colorado 80305, USA. Quartz resonator device courtesy of S. Galliou, FEMTO-ST institute, 25000 Besanc ¸on, France. SiN membrane photograph courtesy of K. Cicak, NIST, Boulder, Colorado 80305, USA. mFV d C0mI vxV C0 RLC C0(a) (b) (c) FIG. 2. Simplified geometry for an electrically actuated mechan- ical system. (a) A mechanical system of mass mis placed inside a capacitor C0that is biased at a voltage V. The force acting on mis assumed to be proportional to the capacitor bias voltage F=βV.( b ) If the mechanical system velocity is v/negationslash=0 an image current I=βvis induced. (c) BVD equivalent circuit. The mechanical system electrical response is identical to that of a series RLC circuit connected inparallel with the capacitor C 0.resistance, inductance, and capacitance; namely, LdI dt+RI+1 C/integraldisplayt dt/primeI(t/prime)=V(t), (3) where L↔m β2,R↔γ β2,C↔β2 mω2 0, (4) and their series combination is added in parallel to the capacitance of the drive electrode C0[see Fig. 2(c)]. Throughout this paper, we refer to the mechanical system and its electrical equivalent interchangeably, in order tosimplify the coupling analysis. III. COUPLING IN STRONG QUANTUM REGIME Our general problem is concerned with two resonantly coupled harmonic oscillators (mechanical or electrical). Weassume that the coupling rate gis much smaller than the frequencies of the harmonic oscillators so that the couplingHamiltonian can be treated perturbatively. The Hamiltoniancoupling term for two mechanical harmonic oscillators ofmasses m 1,m 2and bare frequencies ω/prime 1,ω/prime 2by a spring of constant k[Fig. 3(a)]i s Hc=kx1x2, (5) 022327-3KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) k1 m1k m2k2 System A System B C1L1 CL2 C2(a) (b) FIG. 3. (a) Coupled mechanical harmonic oscillators. (b) Cou- pled electrical harmonic oscillators. where x1andx2are the displacements of the oscillators from equilibrium. The resonant frequency for each of the harmonicoscillators in the presence of the coupling spring is ω i= (ω/prime2 i+k/m i)1/2.I fω1=ω2=ω0the coupling Hamiltonian can be rewritten in terms of a coupling rate gby expressing xiin terms of their respective harmonic-oscillator ladder operatorsx i=√¯h/(2miωi)(ˆai+ˆai†) so that Hc=¯hg(ˆa1+ˆa1†)(ˆa2+ˆa2†), (6) where g=k 2ω0√m1m2, (7) and ¯his the Planck constant divided by 2 π. It will be useful later to express gin terms of an analog electrical system [Fig. 3(b)] of two LC resonators coupled by a shunt capacitor C. In this case, the coupling Hamiltonian is Hc=1 Cq1q2, (8) where q1andq2are the charges on the capacitors C1and C2, respectively. The resonant frequency for each of the LC resonators is ωi=1//radicalbig LiC/prime iwhere C/prime i=CiC/(Ci+C)i st h e series capacitance of CiandC. Assuming ω1=ω2=ω0,w e can rewrite Eq. ( 8) in terms of the ladder operators, qi=/radicalbig ¯hω0C/prime i/2(a+a†), so that Hctakes the form of Eq. ( 6) with g=ω0 2/radicalBigg C1C2 (C1+C)(C2+C). (9) We stress that both Eqs. ( 7) and ( 9) are valid only if the coupling rategis smaller than the harmonic oscillator’s frequency, i.e., g/lessmuchω0. We will be particularly interested in the strong-coupling quantum regime , i.e., when a large number of complete energy oscillations occur between the two oscillators beforethey significantly lose coherence: N osc≈τcoh/τosc/greatermuch1. Here τosc=π/g is the time required for a complete energy os- cillation (back and forth) between the two oscillators. Fora system of two harmonic oscillators, τ cohis the average exchange period of a single energy quantum with any of thethermal baths of the oscillators. We assume that coherence islimited by energy relaxation. In reality, there are additional decoherence mechanisms that could decrease N oscfurther and the values calculated here should be considered as an upperbound. An important case is motional dephasing of a trappedcharged particle [ 21,36]. Although the motional heating rate for trapped ions could be as low as a few quanta per second(see Appendix B), trap-frequency drifts, for example, could cause motional dephasing at a higher rate. Another well-knownsource of motional decoherence is the nonlinear couplingbetween trap axes due to trap imperfections [ 21]. Although these mechanisms could be reduced by technical means, itwould be highly favorable from a practical standpoint thatthe coupling strength g/greaterorequalslant2π×1 kHz, posing an additional constraint in what follows. When expressing the above condition in terms of the lower of the two quality factors Qassociated with the two oscillators and the temperature Tof their environment, we observe two regimes. At “high” temperatures ( k BT/greaterorequalslant¯hω0), the thermal equilibration time constant τthermal=Q/ω 0of the oscillators can be thought of as the 1 /etime required to heat the mechan- ical oscillator from 0 K to the surrounding temperature T, i.e., the time it takes to acquire an average of (1 −1/e)nthermal phonons where nthermal=[exp (¯hω0 kBT)−1]−1≈kBT ¯hω0energy quanta and kBis the Boltzmann constant. Any quantum coherent phenomena will therefore be restricted to timesshorter than τ coh=τthermal/nthermal≈¯hQ/k BT, roughly the time required to absorb one phonon at the rate of thermalequilibration. At “low” temperatures ( k BT/lessorequalslant¯hω0) the equili- brated oscillator contains one phonon or less on average andtherefore τ coh=Q/ω 0. The strong quantum regime condition therefore translates to Nosc≈gQ π(nthermal+1)ω0/greatermuch1. (10) At typical liquid-helium temperatures of ∼4K ,kBT/¯h= 2π×83 GHz, so for frequencies below 83 GHz we require Nosc≈gQ 2π×262 GHz/greatermuch1. (11) For dilution-refrigerator temperatures of /lessorequalslant50 mK for example, kBT/¯h=2π×1 GHz, so for frequencies below 1 GHz we require Nosc≈gQ 2π×3.3 GHz/greatermuch1. (12) The inequalities in Eqs. ( 10)–(12) introduce stringent constraints both on the coupling strength gand the Qfactors involved. The need for high Qfactors accounts for why superconducting circuits, which often have high Qfactors, naturally arise in the context of hybrid systems, as will be seenin the next section. If the two oscillators have different eigenfrequencies ( ω 1/negationslash= ω2) their weak off-resonant coupling could be brought into a strong effective resonant coupling by modulating one ormore of the system parameters by a fraction 0 <η< 1, at the difference frequency, ω 1−ω2, usually at the expense of a lower coupling rate. For example, if the two mechanicaloscillators in Fig. 3(a) have different resonant frequencies, 022327-4HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) they can still be coupled by modulating the spring constant kat the difference frequency. The expression for the coupling ratein Eq. ( 6) generalizes to g=ηk/(4√ ω1ω2m1m2). Therefore, the coupling strength is reduced by η/2, where ηis typically at the 0 .05 to 0 .2 range to avoid nonlinear behavior of the coupling spring. We note that parametric schemes can havecertain advantages. For example, by coupling a low-frequencyresonator to a high-frequency resonator, a low number ofphonon or photon occupancy for the low-frequency resonatorcan be achieved which could be useful for experimentinitialization, for example. This, however, does not improvethe coherence time of either system unless they are coupledto different thermal baths with different temperatures (e.g.,see Ref. [ 37]). Since large coupling rates compared with decoherence rates are critical, we concentrated on resonantoscillators in the above discussion and in what follows. Fordetails of parametric coupling schemes in the context of hybridsystems involving ions, see Refs. [ 17–20]. IV . TRAPPED CHARGED PARTICLE COUPLED TO AN ELECTRICAL RESONATOR The first hybrid system we consider is that of a trapped charged particle coupled to an electrical resonator, followingRef. [ 15]( s e ea l s oR e f .[ 25]). Schematically, a point particle of mass mand charge qis elastically bound by a trap, here modeled by a spring (see Fig. 4), with a resonant radial frequency ω 0. If the particle is placed between the two plates of a capacitor, any voltage difference Vbetween the plates would result in a force F=αqV/d acting on it, where dis the distance between the plates and αis a unitless geometric factor ( α=1 for a parallel plate capacitor with infinite plate areas). The equivalent electrical circuit [Eq. ( 4)] is composed of an effective inductance Lpand capacitance Cp, where Lp=md2 α2q2,C p=1 Lpω2 0. (13) Therefore, the hybrid system composed of a harmonically confined charged particle and resonator is equivalent to alumped element LC circuit ( L p,Cp) shunted by the trap capacitance Ctrapand coupled to the electrical resonator, as shown in Fig. 4(b).F r o mE q .( 9) and assuming C/greatermuchCtrapfor m, qL CdCpLp CtrapL C(a) (b) FIG. 4. (a) A simplified picture of a trapped particle coupled to an LC resonator. (b) The corresponding electrical BVD equivalent circuit. The trap capacitance Ctrapin panel (b) is formed by the two equivalent parallel plates, which are a distance dfrom one another in panel (a).maximal coupling, we get g=ω0 2/radicalBigg Cp Ctrap=αq 2d1/radicalbigmC trap. (14) Notice that this is an upper bound on the coupling rate g.I n any realistic implementation, the two trap electrodes need tobe dc biased independently and therefore a finite value of C should be taken into account. This coupling can be increased by trapping more than one charged particle. If N pparticles are trapped and form a Wigner crystal, their common mode motion can be treated as that ofa single particle with a charge of N pqand a mass of Npm. From Eq. ( 14) it follows that g∝/radicalbigNp. For very small traps however, Npwill be limited by the Coulomb repulsion between the charges. B a s e do nE q .( 10), Table Isummarizes the constraints on the Qfactor of the electrical resonator required to be in the strong- coupling quantum regime for various charged particles. Theseshould be compared with experimentally attainable values forlumped-element superconducting resonators that are typicallyin the range of Q∼10 4–105and in some cases up to 106, mostly limited by dielectric losses [ 14,38]. Since the required Qis greater than these values, achieving strong coupling of an ion to a superconducting resonator at 4 K does not seemfeasible. In fact, the only two candidates from Table Ithat stand out in terms of reasonable Qfactors are 9Be+(Q/greatermuch7×105at 50 mK) and electrons ( Q/greatermuch4×105at 4 K and Q/greatermuch7×103 at 50 mK). For9Be+it would require incorporating atomic ion trapping technology into a dilution refrigerator, the discussionof which is beyond the scope of this paper and can be foundelsewhere [ 20]. We discuss the prospects of electron coupling in the last part of the paper. Our estimates are compatible withprevious results [ 18,26]. In the above discussion we considered only lumped- element electrical resonators. A different approach would be touse low frequency transmission line resonators. Those can besimpler to fabricate and could potentially have higher qualityfactors. As an example, Fig. 5(a) shows a simple geometry where an ion is trapped close to the voltage antinode of aquarter-wave resonator. Near resonance, the transmission lineresonator is equivalent to a parallel LC circuit [see Fig. 5(b)] with effective capacitance C=π/(4ω 0Z0) and inductance L=1/(ω2 0C) where ω0is the resonance frequency and Z0 is the characteristic impedance of the transmission line [ 39]. The coupling strength is calculated, as before, by using theelectrical equivalent circuit g=ω 0 2/radicalBigg Cp C+Ctrap. (15) The main concern is that the effective capacitance Cof these resonator modes is very large. For a typical Z0=50 ohm transmission line and ω0=2π×10 MHz, C∼250 pF. The coupling strength gwill therefore degrade by a factor of ∼70 as compared with the numbers in Table I, requiring, for example, a quality factor satisfying Q/greatermuch4×109for9Be+at 4 K. This number exceeds the best quality factors for such resonators,having Q∼10 7at 10 MHz [ 40]. Moreover, our estimate for gis an upper bound since, in a real geometry, the field lines at 022327-5KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) TABLE I. Coupling strengths of different trapped charged particles coupled to an electrical resonator. The mass of the proton and the electron are mpandme, respectively. We assume the geometry in Fig. 4, withd=50μm,Ctrap=50 fF, and α=1, and use Eq. ( 14) to calculate g. The table states a lower bound for the required Qfactors; namely, Qcorresponding to Nosc=1. Actual Qfactors should be at least an order of a magnitude greater to comfortably satisfy inequality ( 10). These estimates are consistent with Ref. [ 26], where 600 Hz coupling strength was estimated for40Ca+in a 1 MHz trap with 2 .5 pF trap capacitance, d=50μm, and α=1. Our trap-capacitance estimate of Ctrap=50 fF can only be achieved in small trap geometries through careful design (see, for example, Sec. VI A ). Moreover, additional capacitors required for the trap circuit operation may add to the total capacitance resulting in a lower coupling strength (see Sec. VI E ). The values for ghere, therefore, should be considered as an upper bound estimate. Particle Mass, m Trap frequency, ω0 Coupling strength, gQ min(4 K) Qmin(50 mK) Electron me 1.3 GHz 1.2 MHz 4 ×1057×103 9Be+9mp 10 MHz 9 kHz 56 ×1067×105 24Mg+24mp 6M H z 6k H z 9 2 ×1061.1×106 40Ca+40mp 4.7 MHz 4 kHz 119 ×1061.5×106 88Sr+88mp 3.2 MHz 3 kHz 176 ×1062×106 the voltage antinode of the resonator will differ from those of an ideal parallel plate capacitor. For these reasons, ouranalysis has focused on coupling the charged particle to alumped-element electrical oscillator, where the same resonantfrequency can usually be achieved with significantly lessoverall capacitance. V . COUPLING TO MACROSCOPIC MECHANICAL RESONATORS To circumvent the limitations of attainable Qfactors of superconducting devices, it has been suggested to try andcouple an ion directly to a high- Qmacroscopic mechanical object by using electrostatic coupling [ 15,21,23–25] or piezo- electricity [ 15,41]. m, qd xV CpLp Ctrap C L(a) (b) FIG. 5. (a) A simplified picture of a trapped ion coupled to a trans- mission line resonator. The ion is trapped close to the voltage antinodeof a short-circuited quarter-wave resonator. (b) The corresponding electrical equivalent circuit. The ion is replaced with its equivalent series capacitance C pand inductance Lpwhile the resonator is replaced with its equivalent lumped element representation formed by a parallel LC resonator. Additional capacitance due to trap electrodes is represented by Ctrap.A. Electrostatic coupling to a nanomechanical membrane Commercial nanomechanical membrane resonators can have high quality factors, over 107at 300 mK [ 42]. Recent advances in membrane fabrication [ 43–47] have resulted in quality factors as high as 108, even at room temperature. If such a membrane is metalized on one side, and biased with avoltage U, it could electrostatically couple to an ion trapped near its surface. To estimate this coupling, we assume thesimple geometries shown in Fig. 6. In both cases, the coupling Hamiltonian is H=αqUz izm d2 0, (16) where zi,zmare the displacements of the ion zmotion and the membrane, respectively, d0is the distance between the membrane and the bottom electrode of the ion trap, and αis a geometric factor as in Sec. IV. For the geometries considered here, 0 .5/lessorequalslantα/lessorequalslant1 and we assume α=1 to get an upper bound forg.A si nE q .( 6), we can derive the coupling strength g=αqU 2d2 0ω0√mionM, (17) where Mis the membrane mode mass and ω0its resonant frequency. These masses are significantly larger than the ion q,m ionU d0q,m ionU(a) (b) FIG. 6. Electrostatic coupling of a trapped ion (charge qand mass mion) to a nearby rectangular nanomechanical membrane biased by a voltage U. The ion is assumed to be trapped at a height d0/2 above a surface trap, which is dc grounded with respect to the membrane, suspended above the ion (for simplicity the trap rf electrodes areomitted). (a) A membrane (blue) is clamped at its rim, allowing for a sinusoidal fundamental mode as in Ref. [ 43]. (b) A membrane (blue) is attached by thin wires (red), allowing for a center-of-massfundamental mode as in Refs. [ 45,46]. 022327-6HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) mass, thereby lowering the coupling strength, with a mass ratio on the order of M/m ion∼1014for9Be+. We assume thatd0=100μm and the ion is trapped midway between the membrane and the trap. For a SiN membrane [ 43] with dimensions 500 μm×500μm, coupled to a9Be+ion, we get a mode mass of M∼2×10−11kg, a resonant frequency ω∼2π×1 MHz, and a coupling strength of g/2π∼0.24 Hz atU=1 V bias. Combined with an assumed quality factor of 2×108, such a device does not satisfy the strong quan- tum criteria at T=50 mK since gQ/ 2π∼0.048 GHz [see Eq. ( 12)]. For a suspended trampoline membrane [ 45,46] with dimensions 100 μm×100μm, coupled to a9Be+ion, we get a mode mass of M∼10−12kg, a resonant frequency of ω∼2π×140 kHz, and a coupling strength of g/2π∼12 Hz atU=1 V, leading to gQ/ 2π∼1.2 GHz. The latter nearly enters the strong quantum regime for T=50 mK. However, taking into account ion heating rates still makes this schemeunfavorable, because ion motional heating rate and motionaldephasing would typically exceed g. The coupling can be made stronger by increasing the bias voltage Uat the expense of changing the trapping potential, the ion position, and possibly the trapping stability. Evenwith the U=1 V assumed above, the equilibrium position of, say a 9Be+ion in a 10 MHz harmonic trap, would move by∼7μm. This might be mitigated by adding additional electrodes that compensate for the static voltage bias effectof the membrane (e.g., see Ref. [ 25]). Those electrodes, however, might shield some of the trapping field and needto be taken into account when estimating the ion-trappingpotential. In addition, a more careful estimation of gwould take the membrane mode shape and finite size into account.Finally, adding an electrode to a membrane might decrease itsQfactor. Previous experiments [ 48] with lower quality factors (Q∼10 6) showed that metallization of the membrane was not the limiting factor. Whether this is also true for the case ofQ∼10 8would need to be tested experimentally. B. Piezoelectric coupling to acoustic resonator A piezoelectric resonator is an acoustic resonator made from piezoelectric materials and can therefore be excited byusing external electric fields [ 49]. Quartz resonators have been optimized for stable frequency operation and are therefore natural candidates for ion coupling, despite being relativelymassive. A different plausible candidate is GaN-nanobeamsthat have low masses. To estimate the coupling strength, we start by considering the geometry shown in Fig. 7. An ion is trapped at a distance h above a GaN nanobeam. Such an arrangement can be achieved,for example, by bringing a surface ion trap [ 50,51]o ra stylus ion trap [ 52,53] close to the beam. The main challenge would seem to be to compensate for electric fields from straycharges on the dielectric beam due to its close proximity.We assume throughout that those are compensated for. Whensuch a beam undergoes small oscillations, the position ofeach point in the beam can be written as /vectorr+/vectoru(/vectorr,t) where /vectorr=(r 1,r2,r3) is the equilibrium position and /vectoru=(u1,u2,u3) is the time-dependent displacement from equilibrium. In aflexure acoustic mode, /vectoruis along the ˆr 3direction and its spatial dependence is restricted to the first component of /vectorrˆr1ˆr3 lq,m hr1,opt (r1,t) displacement FIG. 7. Piezo coupling between an ion of mass mand charge q to a nanobeam. The ion is held at a height habove a beam of length lby a Paul trap (not shown). The geometry shown is not to scale sinceh/greatermuchl(see Sec. VC). Harmonic motion about the trap center generates an alternating electric field which drives the mechanicalflexure mode of the beam (light blue) via the piezoelectric effect. The ion position r 1,optmaximizes the coupling and is close to but smaller than the beam length l, due to edge effects. (see Fig. 7). Moreover, the dependence on time and spatial coordinates can be separated, i.e., /vectoru(r1,t)=a(t)/vectors(r1), where /vectors(r1)=(0,0,s3(r1))is the mode shape (unitless) and a(t) is its amplitude. The acoustic oscillation can therefore be reducedto a one-dimensional harmonic oscillator a(t) with frequency ω 0, effective mode mass M, and effective spring constant K as M¨a=−Ka, (18a) M=ρ/integraldisplay Vd3r|s|2, (18b) K=E/integraldisplay Vd3r/vextendsingle/vextendsingle/vextendsingle/vextendsingleds dr1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 , (18c) where ρis the material density, Vis the volume of the beam, andEis its Young’s modulus. The harmonic motion of the ion can couple to the beam acoustic mode via piezoelectricity. A simplified model of thebeam piezoelectric material is that of an ionic lattice. When thebeam is at rest, the electric fields generated by the positive andnegative charges inside it ideally cancel each other. If, howeverthe ions are displaced from equilibrium nonuniformly, 1the beam will exhibit a bulk polarization Pthat can interact with the electric field of the ion. Such a polarization therefore,depends linearly on the strain tensor composed of all the partialderivatives of the displacement components ∂ iuj≡∂uj/∂ri fori,j∈{1,2,3}. Since the strain tensor is symmetric, this linear relation can be written as /vectorP=eu/primewhere eis the 3 ×6 matrix of piezo coefficients (in units of C /m2) and u/primerep- resents strain in V oigt notation u/prime=(∂1u1,∂2u2,∂3u3,∂2u3+ ∂3u2,∂3u1+∂1u3,∂1u2+∂2u1). This bulk polarization will in turn be influenced by the ion electric field /vectorEion. The coupling constant between the ion motion along the ith axis and the 1A uniform displacement of all the ions cannot generate bulk polarization. 022327-7KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) piezoelectric beam is gi=/integraltext Vd3r∂i/vectorEiones/prime 2ω0√Mm ion,i=1,2,3. (19) Here we used the assumption that /vectoru=a(t)/vectors(r1) and s/primeis defined in the same manner as u/prime. The expression in Eq. ( 19) is general and not particular to any specific beam geometry. While the denominator is thestandard term we encountered for two coupled mechanicaloscillators [see Eq. ( 6)], the numerator is a rather involved overlap integral. To appreciate its complexity, we write itsintegrand in explicit matrix form as ∂ i(Eion,1,Eion,2,Eion,3)⎛ ⎜⎝e1,1···e1,6 ... e3,1···e3,6⎞ ⎟⎠⎛ ⎜⎜⎜⎜⎜⎝∂ 1s1 ∂2s2 ∂3s3 ∂2s3+∂3s2 ∂3s1+∂1s3 ∂1s2+∂2s1⎞ ⎟⎟⎟⎟⎟⎠. (20) This integrand can be understood as a dipole-dipole energydensity. To see this, notice that since the field of the ion is that of a monopole, its spatial derivative ∂ i/vectorEionis equivalent to a dipole field aligned along the ith axis ˆi. We may therefore rewrite Eqs. ( 19) and ( 20) in terms of an integral over an effective dipole-dipole interaction as gi=1 4π¯h¯/epsilon1/integraldisplay Vd3r3(/vectorpion·ˆr)(/vectorP·ˆr)−/vectorpion·/vectorP r3, (21) where /vectorpion=q/radicalBigg ¯h 2mionω0ˆi, (22a) /vectorP=es/prime/radicalBigg ¯h 2Mω 0, (22b) and we use ¯ /epsilon1=(/epsilon10+/epsilon1dielectric )/2 since the field of the ion inside the piezoelectric material can be approximated as thatof an ion in vacuum, with the dielectric constant of vacuum/epsilon1 0replaced by ¯ /epsilon1, the average of the vacuum and dielectric constants [ 54]. Ap r i o r i , the overlap integral in the numerator of Eq. ( 19) should not be expected to be large. The piezoelectric coefficientmatrix eis a material property, while the mode shape /vectorsis a result of both geometry and material constraints. Those impose a polarization density /vectorPwhich need not necessarily align with /vectorp ion. We next perform a calculation for two specific piezoelectric resonators in order to demonstrate this difficulty.We use Eqs. ( 19) and ( 21) interchangeably. C. Ion coupled to GaN nanobeam Figure 8shows an image of gallium nitride (GaN) nanobeams. A single beam, clamped at one end, can resonatein a flexure mode [ 55] with a resonance frequency of ω 0= (βa2E/ρl4)1/2. Here, ais the cross-section radius, lis the beam length, Eis its Young’s modulus, ρis its density, and β 100 nm 1μm FIG. 8. SEM microscopy of GaN nanobeams with hexagonal cross section. Gallium nitride nanobeam photographs courtesy ofK. Bertness, NIST, Boulder, Colorado 80305, USA. is a numerical factor (3 .09 for a circular cross section, 2 .57 for a hexagonal cross section2). We can estimate an upper limit on the coupling rate based on Eq. ( 19) and by using the simplified geometry in Fig. 7: g=q˜eA 4π¯/epsilon1h3ω0√Mm ionf(h/l), (23) where fis a unitless geometric factor depending on the h/l aspect ratio, Ais the cross-section area, ˜eis the largest element of the 3 ×6 GaN piezo-coefficient matrix, and ¯ /epsilon1is the average of its dielectric constant and that of vacuum. The ion positionalong the beam r 1,optis chosen so as to maximize the coupling. It turns out that r1,opt∼0.6ldue to edge effects. Figure 9shows the coupling coefficient as a function of ion height h. At an experimentally attainable height of h=50μm, beam length l=15μm, and frequency ω0= 2π×868 kHz, the coupling strength is g=2π×235 Hz. Even for a beam with a relatively high quality factor of Q= 6×104[56], the product gQ/ 2π=1.4×107Hz whereas the strong quantum regime requires gQ/ 2π/greatermuch2.6×1011Hz at 4 K and gQ/ 2π/greatermuch3.3×109Hz at 50 mK [Eq. ( 10)]. Based on Eq. ( 23), the coupling to materials other than GaN can be estimated. Another notable material is lithium niobate 2For a hexagon, the radius is defined to be that of the smallest circle enclosing it. 25 30 35 40 45 50012 h[μm]g/2π(kHz) FIG. 9. Ion to GaN nanobeam piezoelectric coupling strength gvs ion height habove the beam. The beam cross section is as in Fig. 8. The geometry is as in Fig. 7withl=15μm, E=3×1011kg m−1s−2,ρ=6.15×104kg/m3,˜e=0.375 Cm−2 (the strongest piezo coefficient of GaN), ¯ /epsilon1=5/epsilon10,w i t h /epsilon10being the vacuum permittivity. The beam flexure mode frequency is 868 kHz. 022327-8HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) where the strongest of the piezoelectric coefficients is an order of a magnitude larger than for GaN, with the other parametersreasonably close to those of GaN [ 57]. That, however, would still have a gQfactor which is below our criteria ( N osc∼10−4 at 4 K), and even that estimate assumes a high- Qlithium niobate resonator, which has yet to be demonstrated. Anotherapproach would be to use beams with higher quality factorsthat are close to 10 6; for example, silicon nitride [ 58] doubly clamped beams or other resonators (see Tables 1 and 2 inRef. [ 59]). However, since these resonators are not made from piezoelectric material, it would require incorporatingpiezoelectric material into the beam while maintaining thehigh quality factors. D. Ion coupled to quartz resonator Recent work with quartz bulk acoustic resonators at both 4 K and at temperatures of tens of millikelvin demonstratedquality factors of up to 7 .8×10 9and might therefore be useful as part of a hybrid quantum system [ 60–64]. Conveniently, the resonance frequencies of these devices are compatible withthose of trapped ions, i.e., in the 5 to 15 MHz range. A BV A resonator (Bo ˆıtier ´a Vieillissement Am ´elior ´e, enclosure with improved aging) is a quartz resonator designedfor high- Qclock oscillators [ 65]. The resonator described here is formed from a disk of L=6.5 mm radius and t=1m m thickness mechanically clamped at its rim (see Fig. 10). The mechanical motion of the disk is actuated by placing the diskbetween the two plates of a capacitor. The origin of the high- Q factors becomes apparent when considering the mechanicaldisplacement profiles of one family of its acoustic modes [ 66]: /vectors(x,y,z )=e −(x2+z2)/2σ2sin(kny)ˆs. (24) Here an acoustic standing wave is formed along the unit vector ˆs=(0.226,0.968,0.111) which is approximately along the ˆyaxis (see Fig. 11). The mode kvector satisfies knt= nπ, n =3,5,... and has a radial Gaussian profile, with σ∼1m m <L . This is very similar to the standing wave formed in a Fabry–P ´erot optical cavity. The acoustic mode is therefore well protected from dissipation through contactsat the rim, where the disk is clamped. Other acoustic-modefamilies are not considered here since they exhibit lowerquality factors [ 62]. This is also the reason why we do not consider the fundamental n=1 mode of Eq. ( 24). An ion can be coupled to the quartz resonator by trapping it a distance h=50μm from the surface, as shown in Fig. 11. Calculating the coupling strength can be accomplished byusing Eq. ( 19) and considering the acoustic-mode shape [see Eq. ( 24)]. An upper bound, which does not take into account the relative angle between the derivative of the field of the ionand the polarization of the bulk, yields g∼2π×1k H z .T h i si s calculated by applying the Cauchy–Schwarz inequality to theintegrand in Eq. ( 20) of the overlap integral in Eq. ( 19). When combined with the high quality factors involved ( Q∼10 9), this yields gQ/ 2π∼1012Hz. This bound, however, cannot be saturated when using the actual integrand in Eq. ( 20). To see this, recall Eq. ( 21) where gis expressed as an integral over the dipole-dipole interaction between the dipole defined by the ion motion, /vectorpion, and the piezoelectrically induced polarization density /vectorP. Figure 12 t Lxy(a) (b) FIG. 10. High- Qquartz bulk acoustic resonator. (a) Photograph of a resonator. Device courtesy of Serge Galliou, FEMTO-ST institute, 25000 Besanc ¸on, France. (b) Schematic cross section. Quartz resonator of thickness tis shown by the light blue fill. Quartz holders (dark blue fill) clamp the resonator at its rim. The resonator is sandwiched between two metallic electrodes forming the actuating capacitor (yellow fill). Thickness of the electrodes as well as the gapsbetween the quartz resonator and the quartz holders are exaggerated for clarity. The modes with highest Qfactor can be described by standing waves, approximately along the yaxis, with resonant frequencies of f n≈nvs 2t=n×3.38 MHz where vs=6757 m /si s the speed of sound and nis the mode number. illustrates the structure of /vectorP. Naturally its magnitude follows that of the acoustic mode, having a Gaussian radial profile andforming a standing wave along the ˆyaxis. The polarization direction of each standing-wave antinode is approximatelyconstant and opposite to that of its neighboring antinodes.Based on this structure, we can refine our upper bound for g xy t Lhq,m ion FIG. 11. Basic geometry for ion-to-quartz resonator coupling. An ion of mass mionand charge qis hovering at a distance h=50μm (exaggerated) above a disk of radius L=6.5 mm and thickness t= 1 mm. The Gaussian radial profile of the acoustic mode is shown in gray. The ion motion generates an oscillating electric field thatactuates the acoustic modes via piezoelectricity. 022327-9KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) -6 -4 -2 0 2 4 6-0.500.5 z (mm)y (mm) 01|P| |P|maxion FIG. 12. Piezoelectrically induced polarization density /vectorPfor third-overtone acoustic mode [Eq. ( 24)w i t h n=3]. Magnitude (relative) is shown by the color plot. Direction is shown by the unit-vector arrows (darker arrows indicate stronger field strengths). Inset shows modeoverlap between the electric-dipole field due to a fixed dipole /vectorp ionat the ion position, which is associated with its motion along ˆy, and the quartz resonator polarization density /vectorP. The ion is assumed to be trapped 50 μm above the resonator surface. The integral over the dipole-dipole interaction between /vectorpionand/vectorP[Eq. ( 21)] yields a coupling strength g/2π/lessorequalslant1 Hz (see Appendix A). by using g/lessorequalslant2|/vectorpion||/vectorPmax| 4π¯h¯/epsilon1/integraldisplay Vd3r r3≈3.2|/vectorpion||/vectorPmax| 4π¯h¯/epsilon1, (25) where we utilized the fact that the interaction energy between two dipoles obtains a maximum when they are aligned withthe vector /vectorrconnecting them. For the mode configuration in Fig. 12, we get g/lessorequalslant2π×1.7 Hz. This bound is confirmed in Appendix A, where we numerically calculate the coupling strengths for various ion motion axes according to Eq. ( 19) and get g/2πin the range of 0 .49 to 1 .46 Hz. To increase the coupling strength, we could reshape the dipole field associated with the trapped ion to better matchthe acoustic-mode polarization density. A simple and practicalway to do this is to use a capacitor to mediate the electric fieldsbetween the ion and the quartz resonator (see Ref. [ 15], Ap- pendix C), as in Fig. 13. Here, the ion motion generates image currents on the trap electrodes that generate a time-varying,but uniform, electric field near the center of the crystal. The coupling gcan be calculated directly as done in Eq. ( A3). However, since the BVD equivalent capacitance C quartz and inductance Lquartz of the quartz resonator have been measured for various acoustic modes, we present herea simpler analysis based on the BVD equivalent circuit ofboth the ion and the quartz resonator, shown in Fig. 13(b) .W e rewrite Eq. ( 9) for this case as g=ω 0 2/radicalbigCionCquartz Ctrap+Cshunt, (26) where we utilized the fact that the trap and shunt capacitances are much larger than the mechanical equivalent capacitancesC ionandCquartz . In fact, Cion<0.2 aF [see Eq. ( 13)] and typical values for Cquartz are in the 1 to 200 aF range [ 67,68]. Therefore, it is imperative that the sum of the trap and shuntcapacitance C total≡Ctrap+Cshunt are kept to a minimum. On the other hand, the quartz capacitor must be large enoughto have considerable overlap with the quartz acoustic mode.Because the mode radius is on the order of σ∼1m m , t h ecapacitor plate area should have a comparable radius, leading toC shunt∼0.13 pF, given the dielectric constant of these crystals /epsilon1=4×10−11F/m. The trap capacitance, therefore, should be comparable or lower than that value. Figure 13(c) shows an ion-trap design where these low capacitances canbe realized. The crux of the design is that, instead of forminga trap capacitor separate from the quartz resonator capacitorand connecting them with wires, the top capacitor plate of theBV A also serves as the trap bottom dc plate. This arrangementis therefore able to minimize the effect of additional straycapacitances. By using an electrostatic simulation, we estimateC total=0.18 pF. The capacitor reshaping of the ion electric field indeed improves the coupling to 10 to 20 Hz for known parametersofC quartz . With Nions we get gQ/ 2π∼√ N×1010Hz, requiring a Wigner crystal of more than 100 ions in orderto satisfy the strong-coupling regime constraint at 4 K.Maintaining such a crystal in the trap might not be trivial due tothe anharmonicities and finite size of the trap. In Appendix A, we show that the coupling dependence on different deviceparameters and mode overtone number does not allow forsubstantial increases in g. It has been shown that high-overtone modes, e.g., n=65, can exhibit quality factors of almost Q∼10 10[62]. That high Qis counteracted by the n−0.5 dependence of gin the mode number (see Appendix A). Nonetheless, it is worth noting the outstanding properties of such a device. The mechanical mode, which is resonantlycoupled to the ion motion, can potentially be cooled to nearits ground state by laser cooling the ion. Since laser coolingcan be done much faster than the coupling rate, the quartzcooling rate is close to 2 g/2π. The thermal heating rate is (1−e −1)nthermalτ−1 thermal≈(1−e−1)kBT/(¯hQ) (see Sec. III). The steady-state number of quanta of the quartz acoustic modewould therefore be ¯n≈π(1−e −1)kBT ¯hgQ. (27) If operated at 4 K, the 5 to 15 MHz mechanical modes of the quartz resonator could be cooled to ¯n∼16 quanta by 022327-10HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) Quartz resonatorRF DC2FL 1m m RF DC1 FL9Be+FL DC 2 DC 1q,m ion CionLion CFLLquartz Cquartz(b) (c)(a) FIG. 13. Coupling an ion to a quartz resonator mediated by a shunt capacitor. (a) The ion is trapped between two endcap electrodes forming a capacitor between FL and DC 1. Ion motion generates image currents in the wires connecting the trap endcap DC1 and thequartz shunt capacitor (formed between FL and DC 2), which in turn generate an oscillating electric field at the quartz resonator, actuating its acoustic modes through piezoelectricity. (b) BVD equivalentcircuit of the two coupled systems. The capacitance C FLis the total capacitance between FL and ground. (c) A Paul trap design minimizing CFLfor maximal coupling of a9Be+ion to the quartz resonator. The trap is formed from a circular inner dc electrode (DC 1), surrounded by an outer cylindrical shell rf electrode (RF). Two disks of 1 mm radius placed at the top (FL) and bottom (DC 2) of the quartz resonator form the quartz shunt capacitance. Ideally, the top plate should be kept floating (FL) or connected to ground by alarge ( >G/Omega1) resistor. The trap drive circuity that connects to the RF electrode and the RF ground connection between DC 1 and DC 2 is omitted. laser cooling the coupled ion. Starting at dilution-refrigerator temperatures ( <50 mK) would result in ¯n∼0.2 quanta. The mechanical coherence times τcoh=¯hQ/k BTcould reach ∼2 ms in a 4 K environment and up to 150 ms in a 50 mK environment. Due to its very large mode mass (1 to 10 mg),such a device, if placed in a superposition state of motion, couldbe used to restrict certain decoherence theories of massiveobjects (see Sec. VII).VI. PRACTICAL CONSIDERATIONS FOR COUPLING AN ELECTRON TO A SUPERCONDUCTING RESONATOR In Sec. IV, we concluded that, based on its small mass, the electron is potentially the most favorable candidate fora strongly coupled hybrid system composed of a chargedparticle and a superconducting resonator. Coupling strengthson the order of 0 .1 to 1 MHz can be expected for an electron trapped 50 to 100 μm away from the trap electrodes, requiring a very moderate quality factor of Q/greaterorequalslant10 4for the electrical resonator, at dilution-refrigerator temperatures. To estimateelectron motional decoherence, we take the measured heatingrates for trapped ions and extrapolate them to an electron witha secular oscillation frequency of 1 GHz. We find a heatingrate of ˙n∼100 quanta /s, well below the coupling rate (see Appendix B). An electron-based hybrid system might enable a fast and coherent quantum information processing technology. A plat-form of trapped electrons could be realized where the electronspin serves as the quantum bit (qubit). Unitary single andtwo-qubit gates can be implemented using rf gradients [ 69,70]. In the presence of magnetic gradients, the electron spin couplesto its motion, which in turn is coupled to the underlying LCresonator. Spin initialization and readout could therefore beimplemented with the superconducting resonator acting bothas a reservoir and as an interface for readout circuitry (e.g., seeRef. [ 71]). The proposed architecture may be more scalable compared with trapped ion QIP since the interconnectingelements are chip based, requiring only rf or microwavecontrol and no optical elements or laser beams. The absenceof optical design constraints could allow for smaller traps,which translates into stronger coupling between electrons andsuperconducting elements, enabling faster two-qubit gates.Moreover, recent advances in QIP with trapped ions havereached gate speeds that are only an order of magnitudeslower than the trap frequency [ 9]. If that scaling holds for electrons, that would correspond to tens of nanosecond gatetimes, making them on par with superconducting qubit gatetimes (see, for example, Ref. [ 72]). Qubit (i.e., spin) coherence could extend to seconds [ 73]. Therefore, an electron-based QIP platform could allow for a coherence time to gate time ratioof/greaterorequalslant10 8, far exceeding any other QIP technology. Moreover, if the motional heating rates estimated in Appendix Bare experimentally verified, such fast gates would correspond to aBell-state generation fidelity error of ∼10 −6(see Ref. [ 74]). The hybrid nature of such a system might offer an additional way, albeit slower, to entangle electrons, usingthe coupling to the underlying circuitry. This would enrichthe QIP toolbox available for electron spins, for example, byentangling electrons in different traps that are far apart. Here,by using magnetic-field gradients, the spin of the electroncan be entangled with its motion. Since the motion of eachelectron is strongly coupled ( g/2π∼0.1t o1M H z )t oa corresponding LC resonator, entanglement can be achievedby electrical coupling (either inductive or capacitive) ofthe two LC circuits. Moreover, the inclusion of Josephson-junction–based (JJ-based) devices could play an importantrole within the rf circuitry, allowing for greater flexibilityin addressing and connecting electrons, e.g., by enablingtunable and/or parametric coupling [ 75,76] between electrons. 022327-11KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) In addition, the electron could couple to an on-chip JJ-based qubit, a nonlinear resource with a high speed of operation.For example, swapping information to a JJ qubit could enablehigh-fidelity state readout (e.g., see Ref. [ 77]). The idea of using trapped electrons as part of a hybrid quantum system was first suggested for Penning traps [ 15,29]. To that end, novel planar Penning traps have been developedand demonstrated [ 78,79]. Moreover, electrons were trapped with cryogenic planar Penning traps [ 80]. Although single electrons have already been detected in three-dimensional Pen-ning traps by driving their motion [ 16,81], the anharmonicity of planar traps makes single-electron detection challenging.An optimization of the design of the planar trap electrodes [ 82] led to the detection of one or two electrons [ 83]. The outlook for planar Penning traps is discussed elsewhere [ 83–85]. Recently, an ensemble of ∼10 5electrons trapped on super- fluid Helium with normal-mode frequencies in the tens of giga-hertz range were nonresonantly coupled to a superconductingresonator at ∼5 GHz [ 86]. By measuring dispersive shifts in the resonator frequency in the presence of the electrons,the authors could deduce a coupling strength of ∼1 MHz per electron. Further studies of that technology could determine ifthe single-electron regime can be achieved, establishing a newand interesting route for quantum information processing withelectrons, as proposed in Refs. [ 30,87–90]. The potential advantages and prospects of using rf Paul traps for electron-based quantum information processing weresuggested and analyzed [ 18]. Clearly, since a Paul trap does not involve the strong magnetic fields required in a Penning trap, itnaturally avoids exceeding the typical critical magnetic fieldsof superconducting circuitry. Strontium ions, for example,have been trapped with a superconducting niobium planar-chiptrap [ 91]. Two-dimensional trapping of electrons with rf fields was recently demonstrated, resulting in guiding electronsalong a given trajectory [ 92]. To date, however, electrons have been almost exclusively trapped in three-dimensional Penningtraps, with the exception of Ref. [ 93]. There, a macroscopic combined Penning and Paul trap was used to simultaneouslytrap tens of ions and electrons. In Ref. [ 18], a ring Paul trap design for electrons is analyzed, where a parametric coupling scheme is suggested, based ongeometric nonlinearities of the potential. The coupling rates and decoherence rates reported here are consistent with those results. The trap volume used in Ref. [ 18] was relatively small [5μm×π×(15μm) 2] with a trap depth of 1 meV, placing the electron 5 μm away from the nearest electrode, rendering a strong coupling of g=2π×1.1M H z . Here, we analyze the experimental conditions of two trap geometries, aimed at achieving the strong-coupling regime, fora larger trapping volume and a deeper trap. As will be apparentin what follows, the design of these traps involves a delicateinterplay between the trap stability and depth, its ability tomaintain superconductivity, the energy range of the electronsource, and the strong-coupling requirement. In broad strokes,it is easier to build a large trap that is stable and deep so thatcurrently available electron sources could be used. Large trapdimensions, however, would prevent satisfying the couplingcriteria in Eq. ( 10). On the other hand, a small trap is optimal for strong coupling, but it can only support a shallow trappingpotential and therefore requires a low-energy electron source toensure trapping. Because these problems are intertwined, our presentation includes a discussion of each of these aspects, aswell as their compatibility. A. Stable trapping of electrons A Paul trap [ 1] is formed when a time-varying voltage Vrfcos(/Omega1rft) is applied to an electrode arrangement that gives a quadratic spatial dependence for the electric potential in theneighborhood of its electric-field null point. For simplicity,we assume cylindrical symmetry and write the time-varyingpotential in terms of the standard ( ρ,z) cylindrical coordinates as φ=qV rfcos(/Omega1rft)/Phi1(ρ,z), /Phi1(ρ,z)=βρ2−2z2 d2forρ,z/lessmuchd, (28) where qis the electron charge, βis a unitless geometry prefactor ( β=1 for an ideal quadrupole), and dis the trap electrodes’ length scale (e.g., distance from the trap center tothe nearest point of an electrode surface). The time-varyingfield generates a confining potential provided that the Mathieucriterion for stability is satisfied [ 1]: q mathieu ≡8βqV rf md2/Omega12 rf<1. (29) The confinement can then be described, to lowest order, by a time-independent pseudopotential: φpseudo=q2V2 rf 4m/Omega12 rf|∇/Phi1|2, (30) where mis the electron mass. It follows that the pseudopoten- tial trap depth can be expressed as D=qVrfqmathieu/ζ, where ζis a unitless factor dependent only on the trap geometry. For a perfect quadrupole trap D=qVrfqmathieu/6, whereas, for example, for a planar “five-wire” surface electrode trap [ 94], D=qVrfqmathieu/404. The first constraint we consider is trap stability [Eq. ( 29)]. Since the electron mass is small compared with ions, either the trap voltage should be lowered or the trap scale dand/or frequency /Omega1rfshould be increased, as compared with ion traps, to maintain stability. Lowering the voltage would reducethe trap depth and increasing dwould diminish the coupling strength. Therefore, it appears to be advantageous to increasethe trap frequency to the gigahertz regime. The second parameter we consider is trap depth. Naturally, it is easier to trap electrons in a deeper trap. For that purpose,increasing V rfis beneficial. Other constraints; namely, the need to maintain superconductivity in the trap electrodes andcircuitry, limit the maximal rf voltage to a few tens of volts (seeSec. VI B ). Thus far, the shallowest Penning trap that was able to maintain trapped electrons had a trap depth of D∼1e V , the electrons being loaded first into a 5-eV-deep trap whosevoltages were subsequently lowered to form the 1 eV trap [ 83]. We therefore require the trap depth to be at least D∼1e V . Figure 14shows two different three-dimensional geome- tries of traps satisfying the above constraints. Table II 022327-12HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) TABLE II. Trap parameters for the designs shown in Fig. 14.T h e pseudopotential secular frequencies are ωx,ωy,a n dωzwhere x,y are in the plane of the rf ring and zis perpendicular to it, Dis the trap depth, Ctrap[see Eq. ( 14)] is the inherent total capacitance between the dc endcaps and gis the electron-superconducting resonator coupling rate [see Eq. ( 14) as well as Fig. 17for circuit schematics]. With the above choices of d, the geometric parameter in Eq. ( 14)i sα∼1 for both traps. Parameters are estimated by using an electrostatic simulation (this is a reasonable approximation since in both traps the rf wavelength is >10 cm, i.e., much larger than trap dimensions). The maximal rf current Irfis estimated based on Irf=/Omega1rfCtrapVrf. Additional capacitance would result in higher values for the rf current. Parameter Fig. 14(a) Fig. 14(b) Vrf 50 V 50 V Irf 42 mA 243 mA /Omega1rf/2π 9 GHz 7 .15 GHz ωx/2π 0.6 GHz 0 .75 GHz ωy/2π 0.6 GHz 0 .75 GHz ωz/2π 1.2 GHz 1 .5 GHz D 1e V 0 .9e V qmathieu 0.40 .6 Ctrap 15 fF 108 fF d 100μm 200 μm g 2π×1.2M H z 2 π×203 kHz summarizes the resulting trap parameters. Figure 14(a) de- scribes a three-dimensional configuration of electrodes similarto that of Ref. [ 95]. Here, the trap endcap-to-endcap distance is set tod=100μm in order to yield reasonable coupling, while keeping a minimum distance of 50 μm between the ion and the nearest electrode to avoid large heating rates. The coupling also DC U DC Lrf 100µme−DC U DC Lrf(a) (b) FIG. 14. Two Paul trap designs for electron trapping. (a) An rf ring with 300 μm inner diameter and 500 μm outer diameter forms a quadrupole field at its center with respect to two dc endcaps. Theflat-ended endcaps have a diameter of 200 μm and are 100 μma p a r t . (b) A two-dimensional cut through a stacked chip version of (a). The blue region is a silicon substrate. The electron is trapped at the centerof the middle rf ring electrode. The upper and lower endcap disks are 200 μm apart. The center ring inner diameter is 240 μma n d the silicon-free region diameter is 500 μm. Table IIsummarizes the resulting trap parameters.benefits from having no nearby dielectrics, thereby minimizing the trap capacitance. The challenge in constructing sucha trap, however, is the tolerance required for holding andaligning the electrodes. One way to solve this is shown inFig. 14(b) where a trap is constructed from stacked chips, with lithographically patterned metal electrodes, pressed andaligned together [ 96,97]. Because convenient wafer thickness is/greaterorequalslant100μm,d=200μm and the trap capacitance is increased (due to the additional dielectrics), lowering the coupling rates. B. Maintaining superconductivity An immediate concern with the above designs is that the relatively high rf currents involved will generate dissipationand magnetic fields that could potentially lead to breakdownof the superconductivity in the trap electrodes. Usually, theelectrodes of Paul traps form part of the capacitance Cof a parallel rf LC resonator (e.g., in Fig. 17, it would be the total capacitance between the two leads of L rf). We can estimate the on-resonance peak current Imaxfrom the rf voltage amplitude Vrfby using1 2LI2 max=1 2CV2 rf. We find Imaxin the range of 200 to 400 mA for the conditions described below. For simplicity, we restrict our analysis to thin film wires on chip, where an analytic treatment is available. The criticalcurrent I c, above which a thin film wire is no longer superconducting, is Ic=/Lambda1√ wb 0.74Jc, (31) where bis the film thickness, wis its width, /Lambda1is the London penetration depth of the superconducting material, and Jcis its critical current density [ 98]. Of the two commonly used materials for superconducting circuits; namely, aluminum (Al) and niobium (Nb), aluminumis disadvantageous due to its lower values for J cand/Lambda1 and, with a critical temperature of Tc=1.2 K, it requires operation at dilution-refrigerator temperatures. For example,a 100 nm ×10μm aluminum wire has a critical current of I c=11.3 mA. A niobium wire with the same dimensions would have a critical current of Ic=221 mA and would be fairly strongly superconducting even at 4 K ( Tc=9.2K ) . To maintain superconductivity in the chip-based design in Fig. 14(b) with niobium films, we require thicknesses and widths that satisfy bw > 16μm2. Here, the features of the narrowest electrode or wire would serve as the bottleneckdetermining the critical current for the entire circuit. Forexample, a 50 μm×500 nm film cross section would be convenient to fabricate and could provide I c=1.105 A. These numbers are compatible with those measured in asuperconducting niobium trap for strontium ions [ 91]. Equation ( 31) actually constrains the dc critical current through a wire; however, the rf critical current for a su-perconducting resonator has similar values [ 99], at least for the case of a half-wavelength stripline resonator. Whether asimilar result holds for a lumped element resonator, wherethe current distribution is significantly different, has yet to bedemonstrated. 022327-13KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) C. Low-energy electron source In principle, one method to load electrons into the trap would be to target the trapping volume with slow electronsand capture them by turning the trap on when they reachthe trap center. In this case, the challenge lies in the fastelectronics required. A slow electron source could be, forexample, an ultracold GaAs photocathode [ 100,101], which has demonstrated beams with less than 1 eV average energyand less than 50 meV energy spread [ 102]. Such slow 0 .1 to 1 eV electrons traversing a trap with a typical length of100 to 200 μm requires turning the trap on faster than 0 .1t o 1 ns. In Sec. VI D , however, we show that the trap resonator quality factor should exceed 10 4in order to comply with the typical cooling power of a cryogenic refrigerator. Thiswould realistically limit the switching time of such a trapto the microsecond regime. We could mitigate this problemby constructing even slower electron sources. For example,using electron tunneling from bound states on the surfaceof liquid helium [ 103] could potentially generate <1m e V electrons, thereby relaxing the trap-switching-time constraint.The analysis of such a source is beyond the scope of thispaper. A second type of electron source, which is commonly used in Penning traps, is based on secondary electrons [ 104,105]. For example, in Ref. [ 83], a sharp tungsten tip was used to field emit high-energy ( /greaterorequalslant200 eV) electrons that collided with the trap surfaces, liberating gas molecules. During this process,some of these molecules reach the trapping region wherethey have a probability of being ionized by the incoming fastelectrons. The relatively slow “secondary” electrons generatedin the ionization process can then be trapped. This approach seems to be effective with deep ( /greaterorequalslant5e V ) and large ( d=0.1 to 2 cm) traps [ 83]. Trap depth U depth is defined as the maximum minus the minimum of the trap pseudopotential within the trap volume. It is not obviousthat this technique would be efficient enough for a U depth= 1 eV trap with a typical length scale of 100 μm. As an alternative, photo-ionization of a cold atomic gas could bemore compatible with a shallow trap (e.g., see Ref. [ 106]), albeit at the expense of requiring optical access to the cryogenicchamber of the electron trap. One would also have to considerwhether the cold atoms would immediately stick to the trapsurfaces thereby creating a possible charging effect that wouldchange the trapping potential. Here, we consider a refinementof the secondary electron technique that might be less violent tothe trap electrodes, as well as increase the trapping probability,while not requiring optical access. Rather than directing the incoming beam of electrons at the trap electrodes, we consider focusing the beam into thecenter of the trapping region and away from any surfaces. As asource of secondary electron emitters, a cold charcoal adsorbercontaining helium might be used. Primarily used for pumpingresidual helium gas, a charcoal adsorber can be heated witha resistor in order to liberate some helium and increase itsvapor pressure in the chamber [ 107]. Incoming electrons will ionize the helium gas and generate secondary electrons thatcould then be trapped. In Sec. VI D we show that, in order to accommodate for the heat load generated by the trap, itshould be operated at temperatures in the range of 1 to 4 Kand not dilution-refrigerator temperatures. That would also leave enough cooling power to remove the heat generated bythe charcoal heating resistor. We henceforth assume that therefrigerator is operated at 4 K. The total cross section for helium ionization is maximal when the incoming electrons have a kinetic energy of E p∼ 120 eV [ 108]. Here, however, we are interested in maximizing the cross section for generating low-energy secondary elec-trons rather than the total ionization cross section. In fact,since the threshold ionization for helium is ∼24.58 eV, it is not surprising that the low-energy cross-section peaks atE p∼30 eV [ 109,110]. The incoming electron energy should therefore be set to around 30 eV, resulting in an optimal cross section of σion∼0.05˚A2for secondary electrons with energy below 1 eV [ 109]. The resulting ionizing rate of helium atoms within the trapping volume is /Gamma1ion/similarequalJπr2 0 qenHelσion, (32) where Jis the incoming current density of electrons, qeis the electron charge, r0is the incoming electron beam radius, lis the radius of the spherical trapping volume, and nHeis the vapor density of helium atoms. We restrict the discussionto secondary electron generation due to the interaction ofhelium with the primary incoming electron beam. Additionalionization events due to, for example, elastically scatteredelectrons, could only increase /Gamma1 ion. In the presence of the rf trap, the incoming electron energy Epwill be spread by less than ±15 eV around 30 eV, as shown in Appendix C. This, in turn, could reduce the average value of σionby <18% to σion>0.041 ˚A2(see Ref. [ 109]). Equation ( 32) can therefore be considered as an average estimate for /Gamma1ion.I n addition, trap rf voltage can deflect the incoming electrons, causing the average beam radius to expand to r1=ξr0. Since the rf trap voltages Vrfconsidered in this paper have the same order of magnitude as Ep/qe(see Table II),ξ/lessorequalslant4a s shown in Appendix C. We can still use r0in Eq. ( 32) since it depends on the total current of electrons traversing thetrapping region. As long as r 1<l, electrons are not lost due to collisions with the trap walls and this total current should bepreserved. The steady-state number of trapped electrons is determined by the ratio between the low-energy secondary electrongeneration rate /Gamma1 ionand the total electron loss rate. Electrons that have already been trapped may collide with incomingelectrons or with the surrounding helium atoms. The averageenergy of the electrons gradually increases due to thesecollisions (heating) until eventually it exceeds the trap depthand they are lost (boiling). In Appendix C, we derive analytically an upper bound on the contribution to the heating rate due to collisionswith incoming electrons. Briefly, since each collision is aRutherford-type scattering problem, it cannot be attributed afinite cross section. Its geometric scale is therefore dictated bythe incoming electron beam finite radius rwhere r 0/lessorequalslantr/lessorequalslantr1. Therefore, the average energy a single trapped electron gainsin a single collision is <q 2 e/(4π/epsilon10r0). Since the rate of collisions is Jπr2 0/qethe resulting heating rate is ( dE/dt )|e< 022327-14HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) Jr0qe/(4/epsilon10). This translates to an electron loss rate of /Gamma1e=1 Udepth/parenleftbiggdE dt/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle e<Jr0qe 4/epsilon10Udepth. (33) The contribution to the heating rate due to collisions with the helium gas is known as “rf-heating.” This follows fromthe helium atom playing the role of a hard immovable ballin the collision process, being much heavier than the electron.Therefore, when an electron collides with it, its instantaneousmicromotion kinetic energy before the collision transformsinto the secular motion energy after the collision [ 111,112]. During the harmonic secular motion of the ion, kinetic energyis exchanged between rf and secular motion, the rf fractionbeing maximal farthest from the trap center and ideally zeroat the center. Therefore, collisions that occur farther from thecenter will potentially transfer more energy into the secularmotion. If the secular energy of the trapped electron priorto collision is E in, the energy gain after a single collision is/lessorequalslantEin/2, when averaging over the secular motion period. Assuming that the trapped electrons have a uniform energydistribution between 0 and U depth, the average energy gain per collision with a single helium atom is less than Udepth/4. The rate of collisions in this case is ∼σelasticnHe/angbracketleft|v|/angbracketrightwhere σelastic∼ 6˚A2is the electron-helium elastic cross section for low-energy (/lessorequalslant2 eV) electrons [ 113] and /angbracketleft|v|/angbracketright ∼4√ 2 3π/radicalbigUdepth/meis the average velocity of the trapped electrons, with mebeing the electron mass. The resulting heating rate is ( dE/dt )|He< σelasticnHe/angbracketleft|v|/angbracketrightUdepth/4. We translate it to an electron loss rate of /Gamma1He<1 Udepth/parenleftbiggdE dt/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle He=σelasticnHe 3π/radicalBigg 2Udepth me. (34) Combining Eqs. ( 32)–(34), the steady-state number of electrons in the trap, Ne, is dictated by setting dNe/dt=0 in the rate equation dNe dt=/Gamma1ion−Ne(/Gamma1e+/Gamma1He). (35) For trapping, we require the steady-state number of elec- tronsNebe greater than a threshold value Nthreshold ,a sw e discuss below. This can always be satisfied if the current density Jand the density of helium, nHe, are large enough [Eq. ( 35)]. To see this quantitatively, in Fig. 15(a) ,w ep l o tt h e number of steady-state electrons for different current densitiesand helium-pressure values. The value for N threshold depends on the dynamics of the electron loading process; specifically, on the cooling rate ofthe electron motion /Gamma1 coolduring loading. Without cooling, once the incoming electron source is turned off ( J→0), any trapped electrons would rapidly boil out of the trap due tocollisions with the helium background gas. Indeed, the heliumpressure can be decreased significantly to avoid this process byallowing the charcoal adsorber to cool to its 4 K surroundings.However, the timescale for removing the helium is likely to belong compared with 1 //Gamma1 He. The latter is inversely proportional to the helium pressure and, for example, equals 1 .3μsa t a helium pressure of 10−2Pa. Collisions with other atoms are neglected in our discussion because we expect the trap10−410−310−210−1100100101102 Pressure (Pa)J(A/m2) 10−310−210−1100Ne 10−410−310−210−1100100101102 Pressure (Pa)J (A/m2) 10−210−1100101τsteady (μs)(a) (b) FIG. 15. Effect of loading parameters. (a) Estimated steady-state number electrons, Ne, in a 1-eV-deep trap having a trapping volume of∼(95μm)3when the electron gun is on. Incoming electron beam radius is assumed to be r0=10μm. (b) 1 /etime to reach steady-state number of electrons. chamber to be in an ultrahigh vacuum cryogenic environment with pressures of 10−10Pa or less. In the design we consider below, we assume the zmotion of the trapped electrons is strongly coupled to an LC resonatorto experience damping. In Sec. VI E , we show that a ∼1 GHz LC resonator with a quality factor Q det∼1000 should suffice for single-electron detection. Therefore, the LC resonatorequilibrates with its 4 K surroundings at a ∼1 MHz rate, i.e., much faster than the coupling rate gbetween the LC resonator and the electron motion. The resulting z-motion damping rate is dictated by the slower of the rates, /Gamma1 cooling∼g/2π/greaterorequalslant 100 kHz. To cool the xandymotion, these modes could be parametrically coupled to the zmotion [ 114] as discussed in Sec. VI F . We will henceforth assume a similar damping rate for all axes. Once the incoming electron beam is turned off, the trapped- electron energy Eis dictated by the cooling rate and the helium collision-induced heating rate: dE dt=−/Gamma1coolE+1 π(σelasticnHe)/radicalBigg 2E meE. (36) 022327-15KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) For this equation to be correct, the initial energy of the electron must be below a value Einitdetermined by trap anharmonicity, which manifests as an amplitude dependence of the resonantfrequency. Since damping is based on resonant coupling to theLC resonator, large-amplitude motion will not cool effectively.Based on Sec. VI E , we can estimate E init/lessorsimilar0.3 meV. This threshold can be increased in a few ways. One techniquecould be to detune the trap in order to match the resonantfrequency of higher-energy electrons, then adiabatically followtheir frequency as they cool down. Another way is to designa trap with lower anharmonicity (see references in Sec. VI E ). A third way could be to design an LC resonator with a tunablequality factor by using a tunable coupler [ 75,115] where the Qfactor is first lowered for cooling purposes and increased once the electrons are cold. For the sake of the discussionhere, we will adopt the more conservative estimate for E initof /lessorequalslant0.3m e V . To achieve net cooling, the right-hand side of Eq. ( 36) should be negative, i.e., E/lessorequalslantEcapture≡me 2/parenleftbiggπ/Gamma1 cool σelasticnHe/parenrightbigg2 . (37) Therefore, if the electron zmotion satisfies E<E thresh≡ min(Ecapture,Einit), it will remain trapped. For helium pressures below 0 .027 Pa, Einitis the smaller of the two and determines Ethresh=0.3 meV. For a pressure Pgreater than that, Ethresh= Ecapture=0.3m e V ×(0.027 Pa /P). Equation ( 36) was based on the assumption that excess micromotion can be neglected. Excess micromotion occurswhen the ion experiences rf fields even at its equilibriumposition that is usually shifted from the rf null due to strayfields. This would lead to a constant heating term in Eq. ( 36), thereby limiting both E capture as well as the steady-state energy. By using dc compensation fields, the electron positioncan be adjusted back to the rf null. We require the heatingrate due to excess micromotion to be much lower than theheating rate for electrons with E capture energy. If the electron is at a position xaway from the rf null, this constraint can be written as mev2 mm(x)/lessmuchEcapture where vmm(x)i st h e micromotion velocity amplitude at x. For a 1 GHz trap and Ecapture=0.3 meV this constrains x/lessmuch1μm. From Figs. 15(a) and15(b) we can extract the time needed to trap a single electron. Within the parameters explored, thesteady-state number of trapped electrons, N e, is less than one and the threshold energy is Ethresh∼0.3 meV or smaller. Therefore, the loading process should be operated in pulsedmode, with ∼(U depth/Ethresh )/Nepulses required on average to trap a single electron (provided that the electron energydistribution is uniform between zero and U depth). Combined with the 1 /etime required to reach the steady state [Fig. 15(b) ], we extract the average total time required for trapping asingle electron, shown in Fig. 16(a) . As long as E thresh is not dominated by the helium pressure P, i.e., by Ecapture , increasing Pis beneficial since Neincreases. An optimal helium pressure of ∼0.027 Pa is reached, beyond which Ethresh=Ecapture∝1/P2. These estimates assume that, once a single electron is trapped, it is immediately detected. Realistically, some sortof detection procedure needs to be applied in order to verify10−410−310−210−1100100101102 Pressure (Pa)J(A/m2) 10−310−210−1100101102Ttot(s) 10−410−310−210−1100100101102 Pressure (Pa)J(A/m2) 100101102103104Ttot(s)(a) (b) FIG. 16. Estimated average total time Ttotfor trapping and detecting a single electron, based on the same parameters used for Fig. 15. The incoming electron beam gun is operated in pulse mode, the duration of each pulse [Fig. 15(b) ] allows a steady-state number of electrons [Fig. 15(a) ]. This translates into a probability of trapping a single electron after a single pulse. The process must be repeated a number of times, which is inversely proportional to that probability.After the electron loading pulse, a detection procedure needs to be applied for T det. (a) Assuming Tdet=0, i.e., negligible. (b) Assuming Tdet=10μs based on the conservative detection-time estimates from Sec. VI E . that indeed an electron is present. In Sec. VI E , we analyze the detection scheme of Ref. [ 31]. We estimate that the time to detect a single electron Tdetis in the 1 to 10 μs range. In Fig. 16(b) , we plot the total time required to trap and detect a single electron for the more conservative estimateofT det=10μs. Based on the plot, working in the helium pressure range of 10−4to 10−1Pa and a current-density range of 1 to 100 A/m2, the range of times we get is similar to that of Paul trap loading times for ions. The current-density range in Figs. 15and 16is chosen such that the total current of incoming electrons is in thenano-ampere regime for a beam radius of r 0=10μm. The beam radius was chosen so that, even after expansion to r1 due to the trap rf fields, it would avoid the trap walls. These parameters can be easily obtained with commercial electronsources. Smaller beam radii with the same total current wouldreduce the total time required to trap an electron even further. 022327-16HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) That would require a design of electron optics combined with either a commercial or homemade cold field emission source,the details of which are beyond the scope of this paper. D. Electrical circuitry Stable trapping requires applying large voltages and cur- rents in a cryogenic environment, next to a sensitive detectionresonator. This has implications on the heat load experiencedby the refrigerator and the circuit design of the trap. Achieving a trap drive amplitude of V rf=100 V at fre- quencies in the 7 to 9 GHz range requires resonating the trapcapacitance C rfwith an inductor. The resulting dissipation rate would be Pdis=/Omega1rfCrfV2 rf/Q, where Qis the rf resonator quality factor. With Crf/lessorequalslant150 fF (based on simulations of the traps in Fig. 14) andQin the 104to 105range, implies /lessorequalslant0.2 to 2 mW of dissipated power for frequencies in the 7 to 9 GHz range. With the cooling power of a dilution refrigeratortypically being in the 100 to 400 μW range at T=100 mK, too low to survive such heat loads, it seems that working at4 K would be required, where 2 mW of power dissipation iseasily absorbed, even with a lower ( Q∼10 4) quality factor. In fact, even 1–2 K cryostats with ∼60 to 200 mW of cooling power would suffice. To understand the implications of the trap drive on the electron detection circuit, we model the traps in Figs. 14(a) and 14(b) with the lumped-element circuit shown in Fig. 17. Detecting the presence of electrons would be accomplished byusing a tank circuit technique [ 31,81]. The electron thermal motion generates image currents that couple to the resonatorformed from the trap capacitance and the inductor L det, chosen to be resonant with the ∼1 GHz secular motion. The trap is driven by a different resonator, formed from the ring-to-end-caps capacitance and another inductor, L rf, chosen to resonate at the 7 to 9 GHz drive frequency. The possible cross talk between the drive and detection resonators could deteriorate their respective quality factors.If the trap is electrically symmetric, i.e., C rf,1=Crf,2and Ciso,1=Ciso,2, the two circuits are essentially orthogonal. The detection circuit is connected to equipotential points inthe trap drive circuit and is therefore not influenced by thehigh currents flowing there. Moreover, due to the Wheatstonebridge topology, the detection circuit is not sensitive to the rfinductor L rfand its coupling port. It is only influenced by the additional capacitances Ciso,jforj=1,2 that add to the total trap capacitance. Similarly, the rf resonator is indifferent tothe added impedance of the detection resonator. The impact oftrap asymmetry on the quality factor of the two resonators canbe estimated by /Delta1Q rf Qrf∼Qrfω0 Qdet/Omega1rfCcap Ciso,1+Crf,1+2Ccap/epsilon1, (38a) /Delta1Q det Qdet∼Qdetω0 Qrf/Omega1rfCiso,1+Crf,1 Ciso,1+Crf,1+2Ccap/epsilon1, (38b) /epsilon1=|Crf,1−Crf,2|+|Ciso,1−Ciso,2| Crf,1+Ciso,1, (38c) where QrfandQdetare the rf and detection resonator quality factors, respectively, when the trap is completely symmetric,/Delta1Q rfand/Delta1Q detis their respective change due to asymmetry,ω0∼2π×1 GHz is the secular frequency, /Omega1rf/(2π)∼7t o 9 GHz is the trap drive frequency and /epsilon1is the asymmetry parameter. Clearly, if QrfandQdetare comparable, and the capacitances involved are of the same order of magnitude,then keeping /epsilon1below a few percent should suffice. E. Nonlinearity and detection of a single electron One of the main concerns with detecting a single electron in Penning trap experiments is the trap anharmonicity [ 80,82,84]. In these traps, the signal of a single electron has a few-hertzlinewidth due to damping resulting from its coupling to thedetection circuit, whereas the effect of anharmonicity in theseplanar traps is to broaden the electron detection signal to10 kHz to 1 MHz. However, in Ref. [ 82], it was shown that, by adding compensation electrodes and carefully adjusting their relative voltages, we could avoid the dominant anharmonicterms of the potential. Similarly, careful consideration forelectrode shape and geometry allow for higher degree ofharmonicity in three-dimensional traps [ 116,117]. In the designs considered here, the electron is strongly coupled to the detection circuit, giving a relatively broadsignal linewidth, which in turn relaxes the constraints onthe trap harmonicity. By assuming a moderate quality factorfor the detection circuit Q det∼1000, the detection-circuit linewidth is on the order of ∼1 MHz and therefore larger than anharmonicity-induced broadening of the electron signal, aswe show below. To reach the strong quantum regime, however,we require Q det/greatermuch7000 (see Table I). However, with a tunable coupler [ 75,115], we could potentially tune the quality factor of the detection circuit to accommodate for both Q-factor regimes. Detailed analysis of such a coupler is beyond thescope of this paper. Therefore, in this section and in Sec. VI F , we use the lower Q det∼1000 value. Figure 17shows the schematics of a typical tank detection circuit and Fig. 18shows a simplified equivalent circuit. The simplification follows first from replacing the trappedelectron with its BVD equivalent network L e,Ceand a current source Iecorresponding to the induced currents due to electron motion. Further simplification is achieved byreplacing the entire network connected to the two ends of thedetection inductor L detwith its total equivalent capacitance Ctotal. This will define the tank circuit resonant frequency ω0=1/√LdetCtotal, which we assume to be resonant with the electron trap frequency. Finally, the amplification networkthat couples to L detvia mutual inductance to the coupling inductor Lcplis replaced by an equivalent resistor Rdet.T h e coupling inductor Lcpltransduces the input impedance of the amplifier, the real part of which presents an effective resistanceR extin parallel with the internal resistance Rintof the LC tank circuit. The total resistance of the detection circuit is thereforeR det=RextRint/(Rext+Rint). The width of the electron signal can be estimated to be Rdet/Le∼2π×100 kHz, expressed in terms of the trap parameters Rdet Le=Qdetq2 eα2 ω0Ctotalmed2, (39) where d∼200μm is the endcap-to-endcap distance, ω0= 2π×1 GHz is the trap secular motion frequency, and Ctotal∼ 180 fF for the trap in Fig. 14(b) . The capacitance Ctotalis 022327-17KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) CcapRVcap,1 DCU R Vcap,2DCLCiso,3 Ldet Ciso,4e− Crf,2Crf,1 Ciso,2Ciso,1 Lrfrf VrfLcplAMP FIG. 17. Schematic of trap and detection resonators for the traps illustrated in Fig. 14. The electrodes DC U,D C L, and rf of Fig. 14are indicated here in the schematic. The trap capacitances are shown in blue where Crf,1andCrf,2are the capacitances between the center ring and each endcap and Ccapis the endcap-to-endcap capacitance. For the trap in Fig. 14(a) , these equal 21.3, 21.3, and 4.6 fF, respectively. For the trap in Fig. 14(b) , these equal 146, 146, and 35 fF, respectively. The Lrfinductor forms a resonator with the total capacitance between its ends generating the quadrupole trapping field. The Ldetinductor along with the capacitance shown forms the detection resonator that monitors the electron motion (double red arrow). The four isolation capacitors enable independent dc biasing of the upper and lower endcaps ( Vcap,j,j=1,2) with bias resistors R/greaterorequalslant10 M/Omega1to avoid loading the detection circuit, assumed to have a quality factor of ∼1000 [see Sec. VI E ]. The leftmost isolation capacitors Ciso,1andCiso,2are chosen to equal Crf,1=Crf,2. The rightmost isolation capacitors Ciso,3andCiso,4are chosen to be much greater than the total capacitance between DC Land DC U, e.g., on the order of 1 pF. The mutual inductance of LdetandLcplallows for electron detection using an amplifier. calculated by expressing it in terms of the other capacitances in Fig. 17as Ctotal=Ccap+Crf,1Crf,2 Crf,1+Crf,2+Ciso,1Ciso,2 Ciso,1+Ciso,2, (40) assuming that Ciso,k(k=3,4) are much larger than Ctrap. While Crf,k,k=1,2 and Ccapare dictated by the trap electrodes, Ciso,k,k=1,2 can be chosen independently. There is an inherent tradeoff in this choice, however. On the one hand, IeCeLe Ctotal Ldet Rdet FIG. 18. Simplified electron-detection circuit, based on the cir- cuit in Fig. 17. Here, Ctotalis the total capacitance between the two ends of the detection inductor Ldet. The trapped electron is replaced by its electrical equivalent of a series LC resonator with inductance Leand capacitance Ce. Currents generated by electron motion are represented by Ie. The coupling inductor Lcplin Fig. 17transduces the input impedance of the amplifier to an effective resistance, which, combined with LC internal dissipation, are represented byan equivalent shunt resistor R det.these should be much larger than Crf,kin order to maximize the trap drive voltage. On the other hand, these should be as smallas possible so as to minimize C trapand increase the coupling rateg. For simplicity, here we choose Ciso,1=Ciso,2=Crf,1= Crf,2but other choices could be explored. For the trap in Fig. 17(a) ,Ctotal∼26 fF, so Rdet/Lion∼2π×0.7 MHz. See the caption of Fig. 17for the capacitance values for both traps. The relatively large difference between the signal bandwidthscalculated above and the typical signal bandwidth in a Penningtrap experiment follows from the small dimensions and smallcapacitance of the designs considered here. The width of the electron signal should be compared with the frequency spread resulting from the trap anharmonicity.By using first-order perturbation theory, we can estimate theeffects of the r 4,r2z2,z4terms in the trap potential (see, for example, Ref. [ 80]) resulting in /lessorequalslant0.5 MHz dispersion in the signal for both traps in Fig. 14, assuming the electron thermal motion equilibrates to a 4 K bath. This should contribute verylittle to the broadening of a single-electron signal, therebysimplifying its detection without the need for a more elaborateelectrode design. Notice also that the /lessorequalslant0.5 MHz dispersion falls within the bandwidth of the detection circuit describedabove, rendering the cooling induced by coupling to thedetection circuit to be effective for electrons with temperatures/lessorequalslant4 K (energies /lessorequalslant0.34 meV). Even in the presence of nonlin- earities, a single electron could be detected by parametricallydriving its motion and coherently detecting the resulting imagecurrents in the detection circuit [ 16]. The bandwidths calculated above fall in between 0 .1 and 1 MHz, and therefore correspond to a single-electron detection 022327-18HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) time of 1 to 10 μs. By integrating the thermal power spectral density at Rdetover a bandwidth of B≡Rdet/(2πLe) centered atω0, the total detected power will vary from Pdet∼4kbTB when no electron is trapped to Pdet∼0 when an electron is trapped [ 31]. This is a result of the fact that, on resonance, the electron equivalent circuit is effectively a short which shuntsR det, as seen in Fig. 18. To avoid a large noise background, an amplifier with an effective noise temperature that is /lessorequalslantT is required. As an example, for the T=4 K experiments explored here, using an amplifier with a noise temperatureof 2 K at ω 0/2π∼1 GHz such as in Ref. [ 118] could suffice, giving an estimated signal-to-noise ratio of unity or larger indetermining the variation in P detbefore and after trapping. F. Parametric cooling The low-energy electron source described in Sec. VI C relies on the ability to cool the motion in all three spatialaxes. As described there, adequate z-motion cooling can be achieved when the detection circuit is resonant with the z motion. By parametrically coupling the radial xandymotion to the zmotion, cooling on all axes can be achieved [ 114]. Such a scheme has the benefit of not needing an extra radialelectrode for damping or additional resonant circuitry on theexisting ring electrode. The coupling scheme in Ref. [ 114] was based on xyandxz terms in the pseudopotential, which were proportional to a volt-ageU. Time modulating U(t)=U 0cos(/Delta1ωt ) at the difference frequency /Delta1ω=ωi−ωjcauses energy exchange between the motion along the iandjaxes. The traps considered in Fig. 14, however, are axially symmetric and therefore should have negligibly small cross terms of that type. We could alsoconsider this approach by modifying the electrodes to be ableto induce couplings of this form. Alternatively, a variationon this coupling scheme could be used that incorporates thesymmetry of the simpler electrode structures. To see this, weapproximate the trap pseudopotential around its minimum as φ pseudo=1 2me/parenleftbig ω2 xx2+ω2 yy2+ω2 zz2/parenrightbig +βx2z2+γy2z2, (41) where the x2y2anharmonic term is also negligible for the axially symmetric traps considered and β≈γ. In terms of the harmonic ladder operators, the x2z2cross term, for example, contains the following summands: ¯hξ(a2b†2+b2a†2), (42) where a, a†are the z-motion operators and b, b†are the x- motion counterparts. Coherently driving the zmotion at ωd= 2ωx−ωzcan be described mathematically by replacing a/mapsto→ αe−iωdt+a. Rewriting Eq. ( 42) and neglecting fast rotating terms introduces terms of the form 2¯hξα(ab†2+b2a†). (43) As an example, consider the trap design in Fig. 14(a) . There, in order to achieve x-zcoupling, ωdshould be ∼2π× 90 MHz. By expressing βin terms of the pseudopotential parameters, β=ζ2q2 eV2 rf me/Omega12rfd6, (44)where ζ=0.166 is a geometric prefactor, we can express the x-zcoupling frequency as 2ξα=ζ√ 2¯hq3 eV2 rfVd m3.5e/Omega12 rfωxω2.5zd7, (45) where Vdis the drive voltage applied to the trap endcaps. For the trap in Fig. 14(a) , we get a rate of 2 π×0.92 MHz /V×Vd. Therefore, a Vd∼109 mV drive, corresponding to ∼3.36μm of motion amplitude, would render an x-zcoupling rate of 2π×100 kHz. This would enable cooling of the xmotion on the order of that rate. With a Qfactor of 1000 for the detection circuit, a 109 mV drive at ωd∼2π×90 MHz would dissipate less than 10 nW of power, well within thecryogenic capabilities of the refrigerator. G. Planar arrangements Planar chip traps have some advantages over the three- dimensional traps analyzed above. They can be easier to fabri-cate, require no alignment, and are more suited for scalability.Such traps, however, have a much shallower trapping potentialfor the same applied voltages and frequencies, as comparedwith three-dimensional traps. This can be mitigated by addinga cover electrode a few millimeters away from the trap chip,and applying a negative voltage [ 82,119,120]. 0 50 100 150 200 250 300 350100150200250300350400 r(μm)z(μm) 012eV(a) (b)r DC RF GNDz e FIG. 19. Planar point Paul trap for electrons. (a) Inner DC disk radius is 100 μm. Outer RF ring radius is 250 μm. The electron is trapped at a height of ∼100μm above the surface. (b) Pseudopotential trap depth of the trap in panel (a), with 100 V trap drive at 7 .1 GHz and a capping electrode, here represented by adding a uniform field of 58.5 V/cm along −z. The trap minimum is at r=0,z∼100μm. Resulting secular frequency along zisωz=2π×1.46 GHz. 022327-19KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) Figure 19shows an example of a planar electrode Paul trap [ 119,121], here chosen to be cylindrically symmetric for simplicity. With the addition of a cover electrode gen-erating a uniform field of 58 .5V/cm, the trap depth is D=0.02qV rfqmathieu . When applying a trap drive voltage ofVrf=100 V to the RF annulus electrode (DC and GND electrodes are rf grounded) and assuming a Mathieu parameterofq mathieu ∼0.5, we expect a 1 eV trap depth, as in the three-dimensional designs shown earlier. The relevant trapcapacitance that dictates the value of the coupling rate g is formed between the central dc electrode and ground.Due to the trap geometric aspect ratio, the coupling ratedecreases to g=2π×180 kHz. As a side effect of using a cover electrode, the electron equilibrium position should shifttowards the trap chip by 7 .7μm. This would result in ∼2μm micromotion amplitude (corresponding to a pseudopotentialenergy of ∼15 meV) that should be compatible with a stable trap operation. This, however, would compromise electronloading into the trap due to the additional rf heating resultingfrom excess micromotion (see Sec. VI C ). One remedy could be to compensate for micromotion by applying dc voltages onthe center DC electrode. In the example considered here, 1 .5V of dc bias would restore the ion position to the rf-null point,while still rendering a 1.6-eV-deep trap. Although planar traps seem promising, separating the detection circuit from the drive circuit would be more difficultdue to the lack of symmetry assumed in Sec. VI D . Also, since planar traps tend to be more anharmonic compared with three-dimensional ones, additional compensation electrodes may berequired in order to enable single-electron detection [ 82]. VII. CONCLUDING REMARKS We first consider coupling the motion of a confined charged particle to a superconducting resonator. Limited bythe currently achieved quality factors of such resonators(Q/lessorequalslant10 6), we conclude that, for the systems considered, it will be very difficult to reach the strong-coupling regimeby using a single trapped charged particle, with perhaps theexception of 9Be+at dilution-refrigerator temperatures or trapped electrons. We explored coupling a trapped ion to a nanomechanical resonator, either through electrostatics or piezoelectricity. Based on recent advances in the fabrication of membranes(Q/greaterorequalslant10 8), we considered their electrostatic coupling to a trapped ion. By coating such a membrane with a thin metallicfilm and applying a voltage bias to it, the coupling could beon the order of 10 Hz for a 1 V bias, within reach of thestrong-quantum regime at T=50 mK. We analyzed the possibility of direct piezoelectric coupling of ion motion to a mechanical resonator. An interestingcandidate was a quartz acoustic resonator with a very highquality factor ( Q> 10 9). However, due to the relatively small overlap between the ion electric field and the acoustic-modeshape, the coupling strength is found to be on the order of1 Hz. Reshaping the ion field with the aid of a capacitor led toan increase in the coupling, to 10 Hz, approaching the strongquantum regime. By laser cooling a single 9Be+ion that interacts with the quartz resonator, the acoustic mode with an effective mass of/greaterorequalslant1 mg (!) could be cooled close to its ground state of motion. If such a massive object is placed in a superposition state, it couldbe used to restrict various macroscopic decoherence theories.For example, quantum gravity has been suggested to result ina motional decoherence rate that is proportional to M 2for an object of mass M[122]. If a few-milligram mechanical oscil- lator is placed in a superposition of position states differing bytwice its zero-point motion, that superposition would decoherein∼10 ps. This effect should be testable since the expected coherence time of the quartz resonator is much longer, evenat 4 K. To be well within the strong quantum regime, wecould engineer a different resonator, perhaps with strongerpiezoelectric coefficients, that maintains a high Qfactor and where the acoustic-mode shape has a large overlap with theion electric field. Such a task, however, is not straightforwardbecause these different demands may not be compatible. Lastly, we considered coupling an electron to a supercon- ducting electrical resonator. We examined two specific trapdesigns with a 1 eV trap depth, a depth we view as crucialfor initial trapping where laser cooling is not available. Therelatively high voltages and currents required to create sucha trap depth suggest the need for thick niobium conductorsto form the trap, in order to maintain superconductivity.Additionally, a 1 eV trap requires a low-energy source ofelectrons, and damping to combat heating. We examined athree-dimensional trap arrangement, which can separate thehigh voltage, high current rf trapping circuitry from the lowvoltage, low currents flowing in the electron detection circuitby using trap symmetry. Obtaining a similar effect for aplanar-chip trap geometry would be more complicated dueto the lack of symmetry. It is worth noting the appealing properties that a hybrid system based on a trapped electron might have. Such anarchitecture might be more scalable compared with trappedion QIP since the interconnecting elements are chip based,requiring only rf control and no optical elements or laserbeams. The absence of optical elements could allow forsmaller traps, enabling stronger coupling between electronsand superconducting elements. Moreover, as the speed ofentangling gates based on the Coulomb interaction of twocharged particles scales with the trapping frequency, and as atrap for electrons would typically have secular frequencies that are two orders of magnitude larger than for ions, we expect shorter electron gate times as compared with trappedions [ 69]. Recent advances in entangling trapped ions have reached gate speeds that are only an order of magnitudeslower than the trap frequency [ 9]. If that were to scale for a trapped electron, it would correspond to a ∼10–100 ns gate time, comparable to superconducting qubit gate times [ 72]. Electron spin-coherence times can exceed a second [ 73] and therefore be orders of magnitude larger than coherence timesfor superconducting qubits, where the best values to dateare close to a millisecond [ 123]. Therefore, a hybrid QIP platform based on trapped electrons might have a much largerqubit coherence time to gate time ratio. The platform mightoffer an additional way to entangle electrons, mediated bythe underlying circuitry. This would enrich the QIP toolboxavailable for electron spins. For this second method, gate speedis limited to the exchange rate between the electron and itsaccompanying superconducting resonator, which we estimate 022327-20HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) to be on the order of g∼2π×1M H z f o r 5 0 μm distance between electrons and superconducting circuitry and faster forsmaller traps. ACKNOWLEDGMENTS The authors would like to thank K. Cicak for her help in estimating the coupling of an ion to a membrane and forher help with electron trap design and resulting fabricationconstraints. We thank K. Bertness for discussions regardingGaN nanobeams. We thank M. Goryachev, S. Galliou, andM. E. Tobar for introducing us to the physics of BV A resonatorsas well as lending us devices to measure. We thank A. Sandersfor introducing us to electron source and electron opticstechnology and for his help in assessing their relevance. Wethank F. Lecocq, J. D. Teufel, and J. Aumentado for discussionsregarding the superconducting and rf measurement aspectsof this manuscript. We thank A. Sirois and D. Allcock forcarefully reading this manuscript and their helpful comments.We appreciate support of the NIST Quantum Informationprogram. APPENDIX A: CALCULATING QUARTZ RESONATOR TO ION COUPLING Coupling calculations require knowing the quartz resonator mode shape /vectors, the orientation of the crystallographic axes of the resonator, the corresponding 3 ×6 piezoelectric coefficient matrix of quartz e, and the ion electric field. We focus on the high-Qmodes [Eq. ( 24)] that are quasilongitudinal, i.e., along the ˆn=(0.226,0.968,0.111) unit vector, in the coordinate system described in Fig. 20. The BV A quartz resonators are made from doubly rotated SC (stress-compensated) cutquartz [ 62]. The coefficient matrix efor this cut is taken from Table 7 in the IEEE standard of piezoelectricity [ 124]. Denote the overlap integral in the numerator of Eq. ( 24)b y g c, i.e., gi=/integraltext Vd3r∂iEiones/prime 2ω0√Mm ion≡gc 2ω0√Mm ion,i=x,y,z. (A1) The mode mass is calculated by the integral M=/integraldisplay vd3rρquartz|s|2 =ρquartzπσ2t 2(1−e−L2/σ2), (A2) Rt Lry FIG. 20. BV A geometry. Cylindrically symmetric about the yaxis with a maximal thickness t. The BV A lower surface is a flat disk of radius L. The BV A upper surface can be described by a curved surface y=t(1−r2 2Rt) with a radius of curvature R. We consider a resonator (not to scale) with R=300 mm, L=6.5m m , t=1.08 mm.where σis the Gaussian profile radial scale of the mode shape /vectors.F r o mR e f .[ 66], σ=/parenleftbiggRt3 3n2π2/parenrightbigg1/4 , (A3) where R=300 mm is the radius of curvature of the upper surface of the resonator, tis its thickness, and nis the mode number (see Fig. 20). An approximate formula for the resonance frequency is ω0=csoundnπ t, (A4) where csound=6750 m/s is the speed of sound for the quasi- longitudinal modes. An exact calculation of gccan be found in Appendix A2. Before doing so, we first estimate in Appendix A1an upper bound on gcand correspondingly g, by avoiding the vector nature of the overlap integrand. 1. Upper bound on direct ion-quartz coupling An upper bound can be obtained by using the Cauchy– Schwarz inequality applied to gc: gc=/integraldisplay d3r∂iEioneu/prime /lessorequalslant/radicalBigg/integraldisplay d3r(∂iEion)2/integraldisplay d3r(eu/prime)2 /lessorequalslant/radicalBigg/integraldisplay d3r(∂iEion)2emax/integraldisplay d3r(u/prime)2, (A5) where emax≈0.234 C/m2is the square root of the maximal eigenvalue of e†e. The electric field of an ion hovering at a height halong the ˆyaxis is Eion(/vectorr)≈q/vectorR/4π/epsilon1R3where /vectorR=/vectorr−hˆyand/epsilon1is the average dielectric constant of vacuum and quartz. We can therefore write gc/lessorequalslantγemaxq 4π/epsilon1√ h3/radicalBigg/integraldisplay d3r(u/prime)2, (A6) where γis a numerical factor of order unity for all i=x,y,z directions. To estimate the last integral of the strain ( u/prime)2, recall that the mode mass M=/integraltext d3rρquartzu2, where ρquartz=2.6× 103kg/m3is the quartz density. Due to the mode shape [Eq. ( 24)] we may approximate u/prime∼ku, where kis the wave number of the longitudinal oscillations within the BV A, i.e.,kt=nπ fortthe resonator thickness and n=1,3,5,.... Therefore,/integraltext d 3r(u/prime)2∼k2/integraltext d3ru2and we may estimate an upper bound, g≡gc 2ω0√Mm ion /lessorequalslantγemaxq 4π/epsilon1cs/radicalBig mionρquartzh3 0∼2π×1k H z, (A7) where cs=6757 m /s is the speed of sound for the quasilon- gitudinal mode. 022327-21KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) TABLE III. Direct coupling of a9Be+i o nt oaB V Aq u a r t z resonator. The ion is assumed to be trapped 50 μm above the quartz. The quartz thickness is assumed to be 1 .08 mm. Coupling strength gifori=x,y,z corresponds to an ion motion along the iaxis. The longitudinal mode number is n. n Frequency gy gx gz 3 9.4 MHz 2 π×1.46 Hz 2 π×1.09 Hz 2 π×0.49 Hz 5 15.6 MHz 2 π×1.39 Hz 2 π×1.02 Hz 2 π×0.47 Hz 7 21.9 MHz 2 π×1.33 Hz 2 π×0.97 Hz 2 π×0.44 Hz 9 28.1 MHz 2 π×1.28 Hz 2 π×0.94 Hz 2 π×0.43 Hz 2. Direct ion-quartz-coupling calculation Now that the upper bound has been established, we numerically calculate the integral in Eq. ( A1)f o rt h el o w - frequency modes of the quartz resonator (Table III). We see that all coupling strengths are below 1 .5H z . It is interesting to notice the weak dependence of the coupling strengths on the mode number n. Due to the frequency and mode mass scaling, the denominator of Eq. ( A1) scales like√n. On the other hand, because the derivative of the ion field is equivalent to a dipole field, the integrand of gc scales as 1 /r3whereas its Jacobian scales as rdrso overall we should expect a 1 /r∼1/σ∼√ndependence, which nearly cancels the similar dependence in the denominator for theexpression in g. Although, for very high frequency modes, g should deteriorate due to high spatial frequency averaging ofthe ion field. 3. Ion-quartz coupling via shunt capacitor In the body of the paper, we estimate the coupling of the ion to the quartz resonator via a shunt capacitor by usinga BVD equivalent electrical circuit. The main advantage ofthat approach, other than its simplicity, is that the effectivecapacitance of a BV A quartz resonator is a rather easilymeasured quantity [ 67]. H e r e ,w eu s eE q .( A1) to directly calculate the coupling strength, in order to infer its dependence on mode parameters.To this end, we have to introduce parameters that describe thegeometry involved. The ion is assumed to be trapped at thecenter of a parallel plate capacitor whose plates are a distanced Tfrom one another (see Fig. 21). The quartz resonator is dQA Cshunt Leq,m ion dT Ctrap FIG. 21. Coupling an ion to a quartz resonator mediated by a shunt capacitor. An ion is elastically trapped (trap electrodes notshown) at the center of a parallel plate capacitor. The ion motion generates image currents that in turn generate an electric field between the parallel plate capacitor (each plate with area A=πL 2 e) encapsulating the quartz resonator.assumed to be enclosed in another parallel plate capacitor, with a distance dQbetween the plates and a plate area of A. If the ion is displaced by /Delta1yfrom equilibrium towards one of the plates, it will generate an image charge q∗=/Delta1yq/d T. A portion of these image charges spread uniformly on theBV A shunt capacitor plates, creating a charge density σ= q ∗/A(1+Ctrap/Cshunt) and exerting a field inside the BV A volume E=σ//epsilon1. We get dE d/Delta1y=q /epsilon1AdT(1+Ctrap/Cshunt), (A8) where Ctrapis the trap capacitance, and Cshunt is the BV A shunt capacitance and the field is perpendicular to the plates.As before, we focus on the quasilongitudinal mode shapes[Eq. ( 24)]. Performing the overlap integral in this case results in g c=4q¯e /epsilon1dTσ2 L2e/parenleftbig 1−e−L2 e/2σ2/parenrightbig 1 1+Ctrap Cshunt, (A9) where Leis the electrode radius, ¯eis the mode- shape weighted average of e22,e2,4,e26, i.e., ¯e=nye22+ nze24+nxe26=7.43×10−2C/m−2and ˆn=(nx,ny,nz)= (−0.23,−0.97,0.1) is the quasilongitudinal mode direction vector. By maximizing gcas a function of Leand for Ctrap=50 fF trap capacitance, we estimate Le=1.05σ,s o the coupling rate is g=0.58qe /epsilon1dTω0√Mm ion=2π×10 Hz, (A10) where we assumed coupling to a9Be+ion, trapped between capacitor plates a distance dT=200μm away from one another. To see the geometric scaling of this, recall that σ=(t3R 3π2n2)1 4 andω0≈csnπ/t [66]. We get g≈0.3qe /epsilon1cs√mionρquartz1 dT(tR/2)1 4√n. (A11) From Eq. ( A11), we expect the coupling to diminish for higher modes (increasing n). The dependence in the geometrical parameters t,R is also very weak (1 /4 exponent) with values limited to thicknesses in the range of 0 .5t o1m m and radii of curvature in the R∼300 mm range. APPENDIX B: ESTIMATING ELECTRON “ANOMALOUS” MOTIONAL HEATING RATE FROM AMBIENT NOISE We estimate the “anomalous” heating rate of the electron motion by extrapolating from known ion heating rates [ 125– 127]. Ifndenotes the average number of motional quanta in a trap with frequency fthen ˙n∝q2 m1 d4f1+α, (B1) where q,m are the particle charge and mass, respectively, and αvaries between 0 .5 and 2 in various experiments. In this expression, dis the distance of the charge from the nearest electrode and we assume the electric-field noise is generatedby independent fluctuating patch potentials of extent <d[125]. 022327-22HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) TABLE IV . Selected measured heating rates ˙nfor ion traps. Ion- to-surface distance is d,fis the trap frequency. Trap material T ion df ˙n Ref. Au on sapphire 5 K88Sr+50μm1.32 MHz 4 quanta /s[128] Au on quartz 300 K9Be+40μm3.6 MHz 58 quanta /s[129] Nb on sapphire 6 K88Sr+100μm 1 MHz 2 quanta /s[91] From Table IVand Eq. ( B1) we can estimate the electron heating rate to be between 30 to 160 quanta/s for a trap-to-electron distance of ∼50μm and an electron motional frequency of ∼1 GHz, assuming α=0.5. For α=2, all of the extrapolated heating rates are below 0 .02 quanta/s. These rates are at least three orders of magnitude smaller than the couplingrates we expect between the electron and the superconductingresonator. Specifically, with the traps considered in this paper,the coupling rates were estimated to be in the range of 0 .18 to 1.06 MHz. APPENDIX C: ELECTRON HEATING RATE DUE TO INCOMING ELECTRONS DURING THE LOADING PROCESS We estimate an upper bound for the heating rate of trapped electrons due to collisions with incoming electrons during traploading. We assume that a single electron is trapped in athree-dimensional harmonic potential with ∼1 GHz secular frequency in all axes with a trap depth of U depth=1e V . Incoming electrons, each having Ep=30 eV of kinetic energy, collide with the trapped electron, causing heating. We focus on a single trapped electron collision process since we are aiming at a steady-state number of just one to afew trapped electrons. Moreover, we assume that the trappedelectron interacts with just one incoming electron at a time.This is consistent with the incoming electron current valueswe considered in Sec. VI C and the timescale for the collision process (see below). We ignore the trap dynamics during any single collision since the former is relatively slow compared with the latter. To see this, first note that the timescale for a collision process isb/v p,0where bis the impact parameter and vp,0is the incoming electron initial velocity. The impact parameter is limited by theoverall incoming electron beam radius r 0, which we assume is <100μm. The incident electron speed is vp,0=/radicalbig2Ep/me= 3.2×106m/s where meis the electron mass. Therefore, the collision duration times are /lessorequalslant3×10−11s, i.e., shorter than the trap drive period ( ∼10−10s) and much shorter than the trap harmonic period ( ∼10−9s). Based on our trap parameters, we can estimate that, during a collision, trap forces will changethe positions of the two electrons by no more than ∼20% as compared with a collision where no trap is involved. Sincewe are interested only in an order-of-magnitude estimate, weignore these deviations from a trap-free calculation. For our purposes, however, the trap still plays a role in determining the initial conditions of the collision process.Trapped electrons have an initial energy below U depth.F o r simplicity we assume that the initial energy distribution isuniform in the range 0 /lessorequalslantE s,0/lessorequalslantUdepth (see, for example,Fig. 5 in Ref. [ 109]). The incoming electron, at the moment of entrance into the trapping region, either accelerates ordecelerates prior to the collision, depending on the phaseof the trap drive. For concreteness we use the geometry inFig. 14(b) , the trap parameters of Table II, and assume that the incoming electron velocity is initially along the trap symmetryaxisz. The incoming electron’s initial kinetic energy prior to collision will be spread by ±15 eV around E p=30 eV, as we show later. Since the primary electron beam is initiallyaligned parallel to the rf electric field, the rf-trap-inducedspread in E pis maximal. If, for example, the electrons come at an angle of ∼54.7◦with respect to z, the energy spread inEpreduces to ±2.5 eV. At this angle, to first order, the rf-trap field lines are perpendicular to the incoming electrons’ initial velocity. Our choice of geometry and electron direction therefore accentuates the spread in Epdue to the rf in order to fully appreciate its influence on the heating rate. Another effectof the trap is electron deflection in the transverse directionresulting in a rastering of the incoming beam. It can be shownby using elementary electrostatic consideration that the beamradius will expand by /lessorequalslantexp{2a r c s i n[ q eVrf/(Ep+eVrf)]}< 4. Therefore, we must make sure that the initial beam diameter is small enough such that the beam does not strike the trap electrodes from rastering. We assume that the process can be reasonably captured by classical mechanics. We therefore ignore the spins of theelectrons, as well as scattering interference effects. The ratiobetween the quantum-mechanical differential cross sectionfor electron-electron Coulomb scattering ( dσ/d/Omega1 ) quantum and its classical counterpart ( dσ/d/Omega1 )classical can be bounded by 0.5<|(dσ/d/Omega1 )quantum /(dσ/d/Omega1 )classical|<1.03, based on our parameters; see, e.g., Ref. [ 130]. The quantity of interest is the energy gain per collision, /Delta1E, which is the average of the energy gained per scattering direction over an appropriaterange of solid angle. Therefore, our classical estimation of/Delta1E will also not deviate from a full quantum-mechanical estimation by more than the above bounds. The geometry of a collision process is shown in Fig. 22(a) . An incoming electron with velocity /vectorv p,0and position /vectorrp collides with a relatively slow trapped electron (the target electron) with velocity /vectorvs,0and position /vectorrs. Our subscripts follow the convention of electron scattering terminology where the incoming electrons are called “primary” whereas the (pos- sibly) scattered electrons are called “scattered.” The scatteringproblem can be described in the center-of-mass and reduced-mass coordinates: R c.m.≡(/vectorrp+/vectorrs)/2, and /vectorr≡/vectorrp−/vectorrs,r e - spectively. Ignoring the trapping potential as mentioned above,we can assume that the center of mass will move at a constantvelocity of /vectorV c.m.=(/vectorvp,0+/vectorvs,0)/2. The relative motion is described in the primed coordinate system shown in Fig. 22(b) . It is subsequently reduced to a Rutherford scattering problemof a particle of one electron charge and a reduced mass ofμ=m e/2, moving with an initial velocity /vectorv=/vectorvp,0−/vectorvs,0and an impact parameter b, in the Coulomb potential of a fixed elec- tron at the origin [see Fig. 22(b) ]. The relative velocity vector will therefore be deflected with respect to its initial direction by θR=2a r c t a n/parenleftbiggq2 e/4π/epsilon10b μv2/parenrightbigg , (C1) where v≡| /vectorv|. 022327-23KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) x yz s,0 e−,me s p,0p e−,mexz be−,μθR(b) (a) FIG. 22. Geometry of electron-electron scattering. (a) Laboratory frame. An incoming fast electron with velocity /vectorvp,0collides with a slow (trapped) electron with velocity /vectorvs,0. (b) Reduced-mass frame of reference. Here, /vectorr≡/vectorrp−/vectorrs,/vectorv≡/vectorvp,0−/vectorvs,0,a n dμ=me/2i st h e reduced mass. The angle θRis the deflection angle of /vectorvwith respect to its initial direction, after the collision. Returning to the laboratory frame, the target electron final velocity is /vectorvs=/vectorvp,0+/vectorvs,0 2−/vectorvcosθR+vˆusinθR 2, (C2) where ˆu=/vectorr−(/vectorr·ˆv)ˆv |/vectorr−(/vectorr·ˆv)ˆv|,ˆv=/vectorv v. (C3) By using Eq. ( C2) and the triangle inequality, we can find an upper bound for |/vectorvs|as |/vectorvs|/lessorequalslant|/vectorvp,0−/vectorvs,0|/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinθ R 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|/vectorv s,0| /lessorequalslant/parenleftBigg 1+/radicalBigg Ethresh Ep/parenrightBigg |/vectorvp,0|/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinθ R 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|/vectorv s,0|,(C4) where Ethresh is the maximal energy of an initially trapped electron (see Sec. VI C ). This translates into a bound on the change in the kinetic energy of the targetelectron |/Delta1E|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 2me|/vectorvs|2−1 2me|/vectorvs,0|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle /lessorequalslantγE p/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinθ R 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (C5) where γ=/parenleftBigg 1+/radicalBigg Ethresh Ep/parenrightBigg/parenleftBigg 1+3/radicalBigg Ethresh Ep/parenrightBigg . (C6) If we use Udepth as a bound for Ethresh , we get γ≈1.83. However, in Sec. VI C we showed that only electrons with Ethresh=0.3 meV are expected to be trapped, correspondingtoγ≈1.01. The average change in the absolute value of the target electron kinetic energy is therefore /angbracketleft|/Delta1E|/angbracketright/lessorequalslantγq2 e 4π/epsilon10r0, (C7) where r0is the incoming electron beam radius. Here, we averaged over all possible impact parameters b, assuming that the incoming electrons are uniformly distributed in an electronbeam having a radius of r 0: /angbracketleftbigg γEp/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinθ R 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketrightbigg =γE p 2r2 0/integraldisplay2r0 0dbb1/radicalBig 1+/parenleftbig2π/epsilon10bmev2 q2e/parenrightbig2 ≈γEp 2r2 0/integraldisplay2r0 0dbb1/radicalBig 1+/parenleftbig4π/epsilon10bEp q2e/parenrightbig2 ≈γq2 e 4π/epsilon10r0, (C8) where the approximation v∼vpwas used. A subtle point in the calculation of the average in Eq. ( C8) is the assumption of a uniformly distributed (spatial) incidentelectron beam. While this assumption is reasonable in thelaboratory frame, it is not immediately clear that it is adequatefor the center-of-mass frame. For trapped electrons withan initial energy /lessorequalslantE thresh=0.3 meV; that is, significantly smaller than Ep=30 eV, the assumption of uniformity is a good approximation since the laboratory frame and center-of-mass frame are nearly identical. The value of E thresh might be larger if measures are taken to decrease trap anharmonicity.The ultimate bound for E thresh is therefore Udepth. In that case, we can see numerically that going to the center-of-mass frame 0 10−702004006008001000120014001600 |ΔE|/EpEvent count FIG. 23. Histogram of the absolute value of the change in total energy of a trapped electron, |/Delta1E|, due to collisions with anEp=30 eV incoming electrons. Shaded pink region shows the analytic bound in Eq. ( C7). We numerically integrate the equations of motion for an electron, trapped initially at x=y=z=0, with initial energy Esinteracting with the incoming electron. We assume a pseudopotential harmonic trap with 1 GHz frequencies in all axes in a trapping volume of l3=(95μm)3. Here we do not account for micromotion dynamics. The energy Esis assumed to be uniformly distributed between 0 /lessorequalslantEs/lessorequalslantUdepth=1.01 eV and the incoming electron position is assumed to be at z=− 1 mm with x,y uniformly distributed in the beam cross section, ( x2+y2)1/2/lessorequalslantr0=100μm. 022327-24HYBRID QUANTUM SYSTEMS WITH TRAPPED CHARGED . . . PHYSICAL REVIEW A 95, 022327 (2017) 00 . 511 . 522 . 530100200300400 t/τrf||(μm) 05101520253035 Ep(eV) 012345050100150200250300 t/τrf||(μm) 253035404550 Ep(eV) 10−410−310−210−110010110210−1210−1010−810−610−410−2100102 b(μm)Es(eV) 10010110210−510−410−310−2 r0(μm)ΔEbound (eV)(a) (b) (c) (d) FIG. 24. The effect of micromotion on electron-electron scattering for a harmonic trap with dimensions and frequencies similar to those described in Fig. 14(b) . (a) Example of a simulated trajectory of a trapped electron (dotted blue curve) colliding with an incoming electron (dashed red curve) at an impact parameter b=10˚A, as a function of time in units of the trap rf period τrf. Trap center is assumed at the origin x=y=z=0a n d/vectorris the particle position. The instantaneous kinetic energy of the incoming electron Ep(solid green curve) decreases prior to the collision due to the varying rf potential. When the incoming electron is at a distance of the order of ∼bfrom the trap center (dashed black line), the incoming electron looses 1 .39 eV giving the trapped electron enough energy to escape the trap. (b) Same as panel (a), but for an initial rf phase shifted by πradians as compared to panel (a). In this case, the trapped electron gains 0 .27 eV due to the collision, resulting in confined oscillations. (c) Blue vertical lines show the spread in the final target electron energy Esvs impact parameter b, resulting from different initial trap rf phases. Analytic theory of Eq. ( C10) is shown by the solid red line. (d) Bound on the average energy gain per collision vs incoming electron beam radius r0. Target electron initial kinetic energy is assumed to be uniformly distributed from 0 eV to 1 eV. Analytic theory of Eq. ( C8) (solid blue line) is compared with a numerical integration of Eq. ( C11) that includes the spread in impact parameters and incoming electron kinetic energies due to micromotion (blue circles). The spread in these values is calculated by numerical integration of the equations of motion for the two electrons, for various initial conditions, and under the influence of the trap rf field as well as their Coulombrepulsion. Initial conditions are assumed uniform as in Fig. 23. redistributes the impact parameters to include a larger range of distances and consequentially a lower average impact energy.The calculation in Eq. ( C8) can therefore be regarded as an upper bound on the actual average value of |sin(θ R/2)|.A sa n example, we compare this bound to a histogram of |/Delta1E|de- rived from a numerical integration of the collision equation ofmotion for a random set of initial conditions, as seen in Fig. 23. The target electron energy before collision E s,0is assumed to be uniformly distributed 0 /lessorequalslantEs,0/lessorequalslantUdepth. The incoming electron beam is assumed to be uniformly distributed. From thehistogram, the average absolute value of the energy imparted tothe target electron per collision is ∼0.74×10 −7Ep. Assuming 0/lessorequalslantEs,0/lessorequalslantUthresh , this average decreases to ∼10−9Ep.B o t h values are consistent with the analytic expression in Eq. ( C7) which yields a bound of 4 .8×10−7Ep. The simulation is set up to account for the effect of the trapping pseudopotentialduring the collision process, thereby serving as an independentvalidation of the omission of trap dynamics in our analytic derivation. Since the bound in Eq. ( C7) does not depend on the target electron initial velocity, it can be translated to a correspondingaverage heating rate bound by multiplying it by the incomingrate of electrons. A current density of Jincoming electrons results in Jπr 2 0/qecollisions per second, which in turn results in a heating rate bound of /parenleftbiggdE dt/parenrightbigg e/lessorequalslantqeJr0 4/epsilon10, (C9) where we approximated γ≈1. The above discussion did not include the effect of micro- motion on the collisions. The effect of the trap drive is tospread the kinetic energy of the incoming electron as well asthe impact parameter of the collision. The bound in Eq. ( C9) changes only by a factor of order unity due to micromotion. To 022327-25KOTLER, SIMMONDS, LEIBFRIED, AND WINELAND PHYSICAL REVIEW A 95, 022327 (2017) see this, we first consider the simple case of a target electron initially at rest in the absence of rf fields. By using Eq. ( C2) we can write the target electron exact final kinetic energy dueto a single collision: E s=Epx2 1+x2,x≡q2 e/4π/epsilon10b Ep. (C10) Forx/lessmuch1 (equivalently b/greatermuch1˚A), faster (slower) incoming electrons result in a smaller (larger) increase of the targetelectron energy, E s∝1/Ep. In the presence of a rf trap, the incoming electron can either accelerate or decelerate before the collision, dependingon the initial phase of the trap drive when it entered thetrapping region. An accelerated (decelerated) electron willtherefore transfer less (more) energy to the target electron ascompared with the no-trap collision. This is exactly the casefor the two examples shown in Figs. 24(a) and 24(b) . These simulate collision processes for initial rf phases that differbyπradians. For concreteness, we assumed an rf trap with dimensions and frequencies as in Fig. 14(b) and Table II.W e simplified the calculation by assuming the trap is harmonic inthe entire cylindrical volume bounded by the electrodes. Theincoming electron initial velocity is assumed to be parallel tothe trap zaxis. Figure 24(a) shows a collision process where the incoming electron is maximally decelerated to a kinetic energyofE p=15 eV at the beginning of the collision. This results in the ejection of the target electron from the trap. Figure 24(b) shows the other extreme case where the incoming electronexperiences maximal acceleration resulting in E p=46 eV so the target electron remains trapped. Although this may seemparadoxical, it follows immediately from Eq. ( C10) for impact parameters which satisfy b/greatermuch1˚A.To see how well this explanation encapsulates the effect of micromotion for the general case, we compare the theoryin Eq. ( C10) to the values of E sextracted from numerical simulations as a function the impact parameter b.W ev a r yt h e values of bfrom 1 ˚A, below which collisions are essentially head on [equivalently x∼1i nE q .( C10)], to 100 μm, i.e., the electron beam radius. For a given value of b, the different values of the trap initial rf phase result in the spread in Es values shown in Fig. 24(c) (blue markers). The center of these distributions, however, follows the theory in Eq. ( C10), which assumes no trap drive [solid red line in Fig. 24(c) ]. Overall, the effect of micromotion is a ∼60% spread in the value of Es, centered at the value given by Eq. ( C10). Finally, we extend our treatment to include nonzero initial velocity for the target electron. To this end, we repeat thecalculation in Eq. 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PhysRevApplied.10.024031.pdf

PHYSICAL REVIEW APPLIED 10,024031 (2018) Effect of (Co xFe1−x)80B20Composition on the Magnetic Properties of the Free Layer in Double-Barrier Magnetic Tunnel Junctions Shalabh Srivastava,1Andy Paul Chen,2,5Tanmay Dutta,1,3Rajagopalan Ramaswamy,1Jaesung Son,1,4 Mohammad S. M. Saifullah,3Kazutaka Yamane,4Kangho Lee,4Kie-Leong Teo,1Yuan Ping Feng,2,5 and Hyunsoo Yang1,* 1Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 2Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542 3Institute of Materials Research and Engineering, A*STAR (Agency for Science, Technology, and Research), 2 Fusionopolis Way, #08-03 Innovis, Singapore 138634 4GLOBALFOUNDARIES Singapore Pte. Ltd., 60 Woodlands Industrial Park D, Street 2, Singapore 738406 5NUS Graduate School of Integrative Sciences and Engineering, National University of Singapore, 28 Medical Drive, Singapore 117456, Singapore (Received 14 December 2017; revised manuscript received 31 May 2018; published 22 August 2018) The alloy Co-Fe-B finds extensive application in spintronics, and in particular the perpendicular mag- netic anisotropy characteristic of Co-Fe-B-MgO systems is of great interest. While some efforts have been made to examine the effect of composition on magnetic properties of Co-Fe-B materials, the magnetic- property–composition relationship for the Co-Fe-B-MgO system is still not fully understood. Therefore,it is fundamentally and practically important to understand the Co-Fe-B composition dependence of the magnetic properties of Co-Fe-B-MgO systems. This work focuses on the data-storing free layer of double-barrier magnetic tunnel junctions with perpendicular magnetic anisotropy (PMTJs), which include (Co xFe1−x)80B20ultrathin films sandwiched between two MgO layers and a W insert layer. We study magnetic properties of various (Co xFe1−x)80B20compositions at different annealing conditions and find x∼35% to have the highest anisotropy energy to ensure high thermal stability, while maintaining the lowest Gilbert damping parameter which is essential to achieve a low critical switching current. This composition also shows the highest thermal stability at elevated temperatures. In addition, we performfirst-principle calculations to explain the anomalous composition trends of the magnetization and Gilbert damping parameter. Moreover, we find that the conventional Slater-Pauling curve is not applicable, and it is necessary to consider the magnetization’s dependence on the magnetic anisotropy which in turn dependson (Co xFe1−x)80B20composition and the oxide interface. Our results provide a perspective for a better understanding of metal-oxide systems with desirable properties for DBL PMTJ applications. DOI: 10.1103/PhysRevApplied.10.024031 I. INTRODUCTION Spin-transfer-torque magnetic random-access memory (STT MRAM) based on a magnetic tunnel junction (MTJ) with a MgO tunnel barrier and Co-Fe-B magnetic elec- trodes provides a promising solution for universal memory. STT MRAM has the potential to achieve a low power consumption, high density, fast read-and-write speed, very high endurance, and excellent scalability [ 1,2]. MTJs with perpendicular magnetic anisotropy (PMA), called PMTJs, are currently the subject of extensive research and com- mercialization work. PMA has a magnetic anisotropy easy-axis orientation in the orthogonal direction of the magnetic film, while, on the other hand, in-plane mag- netic anisotropy has an easy-axis orientation in the plane *eleyang@nus.edu.sgof the film, which is a preferred orientation because of the demagnetization field. Compared to materials with in-plane magnetic anisotropy, materials with PMA show a higher STT switching efficiency, lower power consump- tion, and better scalability with improved thermal stabil- ity, which are the essential properties for long-term data storage [ 3–5]. The switching between the memory states in a MTJ is governed by two mechanisms, namely, coherent uniform switching and domain-nucleation-based switching. In the case of coherent uniform switching, the thermal stability of a magnetic thin film mainly depends on the factor /Delta1= KeffV/kBT, where Keffis the effective anisotropy energy, Vis the volume of the magnetic layer, kBis the Boltz- mann constant, and Tis the operation temperature [ 3].Keff is the effective anisotropy energy, which is given by the intrinsic uniaxial anisotropy ( KU) and the demagnetization 2331-7019/18/10(2)/024031(11) 024031-1 © 2018 American Physical SocietyS. SRIVASTAVA et al. PHYS. REV. APPLIED 10,024031 (2018) energy according to the relation Keff=KU−2πMS2. The coherent-uniform-switching mechanism is valid for small magnetic dimensions, whereas for larger magneticdimensions (approximately 4 δ w, where δwis the domain- wall width) the switching proceeds predominantly via a domain-nucleation mechanism. The dimension limit of the transition between the above two mechanisms is deter- mined by the exchange stiffness ( AS) of magnetic mate- rials [ 6–9]. A double-barrier layer (DBL) is a PMTJ structure that has a free layer sandwiched between two MgO layers [ 10]. A DBL is effective in increasing /Delta1of the free layer by increasing Vand Keff. The perpendicular magnetic ori- entation of the free layer is stabilized with a relatively strong uniaxial magnetic anisotropy, which is character- ized by twofold symmetry and assisted by the interfa- cial anisotropy originating from the interface of Co-Fe-B with MgO and the heavy metal (e.g. W). The uniaxial anisotropy causing PMA can be attributed to the spin-orbit interaction (SOI) between the ferromagnet and the heavy metal, as well as overlap between the O porbital and SOI-induced hybridization of the dorbital of the ferro- magnetic material [ 11]. For reliable memory applications, PMA materials should have a high value of Keffto ensure /Delta1> 60 for high thermal stability [ 12]. At the same time, a small Gilbert damping parameter ( α) is required to achieve a low switching current ( Ic) while maintaining a high /Delta1. Maximizing thermal stability at higher temperatures and maintaining annealing robustness are crucial factors in the development of a PMTJ. Therefore, it is important to study the annealing and temperature dependence of the magnetic properties of PMTJ stacks. For example, it has been shown that the temperature dependence of magnetocrystalline anisotropy for (Fe 1−xCo x)2B ingot systems arises from the changes in the electronic structure induced by spin fluctua- tions [ 13,14]. Moreover, temperature-dependent measure- ments of Keffand the saturation magnetization ( MS) can also provide insight into the nature of anisotropy by a rela- tionship given by the Callen-Callen model [ 15,16]. Apart from the temperature dependence of the magnetic proper- ties, there are few experimental and theoretical studies in the literature on the composition dependence of magnetic properties such as the Curie temperature ( TC),MS[17,18], Keff[17–20], lattice parameter ( a0), coercivity ( HC)[21], gfactor [ 17],α[22–24], and tunneling magnetoresis- tance (TMR) [ 25]. Since it is hard to give a semi-classical description for most of these above properties, some stud- ies have attempted to provide quantum-mechanical expla- nations [ 26]. However, it is noted that most of these studies are limited only to bulk magnetic materials. In contrast, the current STT MRAM involves ultra- thin films with the MgO tunnel barrier (approximately 1 nm) and metals as adjacent layers, whose properties are very different from the bulk magnetic materials. In ultrathin films, the break in symmetry, strains, interfaces,oxygen hybridization, and change in crystallization are the determining factors for the change in magnetic properties (MS,TC,Keff,a n dα)[27–30]. Therefore, there is a need to calculate the band structure of thin-film systems with proper consideration given to oxide interfaces and film thickness since the Slater-Pauling model, which predicts the dependence of magnetic moment on the composition in the bulk materials [ 31–33], may not be applicable for thin- film systems. Furthermore, it is observed that the tempera- ture dependence of αis independent of intrinsic factors of the bulk and instead depends on the interfacial anisotropy [16]. Thus, the role of the MgO-Co-Fe-B interface must be considered in engineering the αof the materials. In addi- tion, increases in ASand decreases in αwith the annealing temperature are found to be related to the Fe-O interface, Co-Fe-B crystallization, and the differential modification of the thin films compared to the bulk [ 34–37]. In this work, we present a comprehensive analysis of the factors that are affected by changing the composition of (Co xFe1−x)80B20in the free layer of DBL PMTJs. The effects of annealing on MS, the saturation field in the hard- axis direction Heff K[5],Keff,a n dαare studied to identify a suitable (Co xFe1−x)80B20composition for CMOS integra- tion (at least 400 °C annealing temperature). The results based on density-functional-theory calculations are used to explain the anomalies in trends of αand MSwith respect to (Co xFe1−x)80B20composition. We also identify com- positions that show the lowest effects of temperature on magnetic properties to ensure data retention during the solder-reflow process (260 °C for 90 s). The results of temperature-dependent measurements are used to compute ASas a function of (Co xFe1−x)80B20composition. The tem- perature dependences of MSand Keffare also analyzed using the Callen-Callen model to give an insight into the natures of the magnetization and anisotropy for different (Co xFe1−x)80B20compositions. II. EXPERIMENTAL DETAILS The films with the stack structure of substrate/Ta(27)/ Co20Fe60B20(4)/MgO(13)/(Co xFe1−x)80B20(13)/W(4)/(Co x Fe1−x)80B20(9)/MgO(9)/Ta(9)/Ru(27) (thicknesses in Å) are sputter deposited onto thermally oxidized silicon wafers in a magnetron-sputtering chamber with a base pressure of 5 ×10−9Torr. A schematic diagram of the deposited structure is shown in Fig. 1(a). The refer- ence layer is taken to be 0.4 nm of Co 20Fe60B20so that it is magnetically dead, and we can independently study the properties of the free layer of the DBL PMTJ. The W is chosen as the insertion material between two (Co xFe1−x)80B20layers, because it has a better tolerance for 400 °C annealing conditions, as compared to a Ta insertion layer [ 38–40]. In order to achieve the different compositions of (Co xFe1−x)80B20, the two composite tar- gets of Co 60Fe20B20(target 1) and Co 20Fe60B20(target 2) 024031-2EFFECT OF (Co xFe1−x)80B20COMPOSITION ON MTJ FREE LAYER. . . PHYS. REV. APPLIED 10,024031 (2018) are co-sputtered, keeping the boron constant in terms of mole fraction. Since the deposition rates of both targets are similar (close to 0.029 nm/s), the sputtering power of tar- get 1 is fixed at 60 W (P1), while the sputtering power of target 2 (P2) is varied to achieve an intended composi- tion x. TOF SIMS coupled with plasma etching is utilized to determine the composition of Co and Fe relative to B (which is assumed to be constant). The deposited compo- sitions of (Co xFe1−x)80B20are very close to the intended compositions as shown in Fig. 1(b). The deposited films are annealed at 300, 400, and 450 °C for 30 min. A VSM is used to characterize the magnetic properties of the films, such as, MS,Heff K,a n d HC. TEM is performed on selected samples to check the crystallinity and the film thickness. The TEM image and hysteresis loop obtained from VSM for the sample with a composition of x=25% are shown in Figs. 1(c) and1(d), respectively. The W insertion layer is indistinguishable in the middle of two (Co xFe1−x)80B20 free layers because of a very small thickness and simi- larly the 0.4-nm-thick (Co xFe1−x)80B20reference layer is indistinguishable from the Ta layer in Fig. 1(c).Heff Kis determined by the saturation magnetic field in the hard- axis direction (in-plane) in Fig. 1(d).MSis determined by the saturation magnetic moment in the easy-axis (out-of- plane) hysteresis and the coercivity HCis shown in the inset of Fig. 1(d). Ferromagnetic resonance (FMR) measurements are car- ried out to determine αand the gfactor. A radiofrequency signal with an amplitude of 4 dBm is applied to the waveguide to excite FMR. In order to determine the res- onance magnetic field Hresat each frequency fres,a n external magnetic field perpendicular to the waveguide is swept at different rf signal frequencies, in stepwise changes. The value of αis determined by the relationship /Delta1H=/Delta1H0+4πα fres/μ0γ, where /Delta1His the resonance field linewidth and γis the gyromagnetic ratio calcu- lated using the Kittel equation, fres=γμ 0Hres/2π, for the out-of-plane magnetic field configuration [ 41]. In order to determine the exchange stiffness AS, the temperature dependence of the magnetization is analyzed using the Kuz’min model [ 42,43], MS(T) MS(0)=/bracketleftBigg 1−s/parenleftbiggT TC/parenrightbigg3/2 −(1−s)/parenleftbiggT TC/parenrightbigg5/2/bracketrightBiggβ ,( 1 ) where s=0.0586 (gμB/βM0)(kT C/D)3/2,TCis the Curie temperature, βis a constant based on anisotropy, and is typically taken as 1/3, Dis the spin-wave stiffness, and MS(0) is the saturation magnetization at 0 K. Dis related to exchange stiffness according to the equation [ 44]D= 2gμBAS(T)/MS(T). In order to provide a qualitative understanding of the observed magnetic properties, we carry out first-principles calculations using the Vienna ab initio simulation package(VASP ). The Co xFe100−x/MgO structure is modeled using a superlattice model that comprises three layers of (001)- oriented MgO and five layers of (001)-oriented bcc Co xFe100−xper unit cell. The in-plane lattice constant of the Co xFe100−xlayer is fixed at 2.84 Å, producing an in- plane compressive strain of 5.2% and a corresponding volume-conserving tetragonal distortion perpendicular to the plane of the interface by approximately 4% on the MgO layer. To achieve a reasonable resolution of Co composi- tion in our study, we use an enlarged supercell containing 16 metal atoms per atomic layer of Co xFe100−x(4×4 units) and the [110] direction of MgO aligned with the [100] direction in bcc Co xFe100−x[45]. The Co compo- sition xin Co xFe100−xis varied from 0 to 50%, where bcc Co xFe100−xis known to be stable [ 23]. We assume an ordered alloy structure of Co xFe100−xconstructed by replacing N(N=0, 2, 4, 5, 6, 7, or 8) Fe atoms per atomic layer with Co atoms. The site of each subsequent replacement of Fe with Co is chosen to form an arrange- ment approximating an even distribution of Co and Fe atoms in the Co xFe100−xmonolayers. The atomic arrange- ment is maintained for each Co xFe100−xmonolayer in our construction of the model. Structural relaxations and total energy calculations are performed using the pseudopotential plane-wave method with the generalized gradient approximation [ 46] of the exchange-correlation energy implemented in VASP [47,48]. A plane-wave basis set with a kinetic energy cutoff of 500 eV is used to expand the Kohn-Sham orbitals to obtain reliable atomic coordinates. For the reciprocal space sam- pling, a Monkhorst-Pack mesh of 3 ×3×3kpoints is used for the Co xFe100−x/MgO supercell to maintain a k-point spacing under 0.03 Å−1. The method of Methfessel-Paxton is used to treat partial occupancies, with a smearing width of 0.2 eV. The energy convergence threshold is 10−4eV per unit cell in structural relaxation steps or 10−5eV per unit cell for electronic structure optimization. After structural relaxation, the stable Fe—O or Co—O bond lengths are found to range from 2.14 to 2.19 Å, similar to the values found in earlier calculations [ 49]. During the analysis of superlattice density-of-states (DOS) pro- files, the minimum DOS n(EF) at the Fermi surface and magnetic moment for each atom is obtained. While the two interfaces (MgO/Co xFe100−xand Co xFe100−x/MgO) in the supercell can be magnetically inequivalent, their differ- ences in the quantities that interest us, i.e., the calculated magnetic moments, DOS, etc., are negligible. To simplify the calculations we do not consider the W insert layer, and we assume that no diffusion of W has taken place in the (Co xFe1−x)80B20layer. III. RESULTS AND DISCUSSION Figures 2(a)–2(d) show the variations of MS,Heff K,Keff, and HC, respectively, as a function of (Co xFe1−x)80B20 024031-3S. SRIVASTAVA et al. PHYS. REV. APPLIED 10,024031 (2018) 0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.8)eF+oC(/oC Power ratio P2/(P1+P2) SIMS calculated composition Nominal deposition composition P1 – Power of Co20Fe60B20 P2 – Power of Co60Fe20B20 -30 -20 -10 0 10 20 30-1000-50005001000Magnetization(emu/cm3) Magnetic field (kOe) Out-of-plane field In-plane field Slope fit-200 0 200-100001000MS cm3)/ume( Magnetic field (Oe)Saturation field(a) (c)(b) (d)Co20Fe60B20(4 Å)MgO (13 Å) Si/SiO2substrateTa (27 Å) 2.2 ÅRu (27 Å) Ta (9 Å) MgO (9 Å) (CoxFe1-x)80B20(9Å) (CoxFe1-x)80B20W buffer (4 Å) (CoxFe1-x)80B20(13Å) FIG. 1. (a) Schematic diagram for the film deposited. (b) Composition of deposited samples versus the sputterpower. (c) TEM image of the cross section of a film with composition x=25% annealed at 400 °C for 30 min. (d) In-plane and out-of-plane magnetic hysteresis loops for a film with x=25% annealed at 400 °C for 30 min. The insetshows a hysteresis loop in the low field range. composition for the samples annealed at 300, 400, and 450 °C, where xis the percentage mole fraction of Co/(Co +Fe) in (Co xFe1−x)80B20. First, we discuss the variation of magnetic properties with respect to annealing temperature. In order to ensure annealing robustness, the change in the magnetic properties of the free layer with respect to annealing temperature should be minimum near the annealing temperature of 400 °C. For all the composi- tions, MSis stable for the annealing range of 300–400 °C without significant deviation [Fig. 2(a)]. However, as we increase the annealing temperature to 450 °C, MSincreases significantly for all compositions. This increased value of MSat a higher annealing temperature could arise from the degradation of uniaxial anisotropy and an increase in the contribution of cubic anisotropy [ 50–52]. From Fig. 2(b), it is observed that the effect of annealing on Heff Kis mini- mal for x=35–42% in the annealing range of 300–400 °C, and then Heff Kdecreases drastically at 450 °C for all com- positions. This decrease in Heff Kat 450 °C annealing tem- perature is also observed as a reduction in Keffat the 450 °C annealing temperature as shown in Fig. 2(c).T h e decrease in Heff Kand Keffat higher annealing temperatures is consistent with earlier studies [ 53,54] and can again be attributed to the degradation of interfacial (uniaxial) anisotropy, as the higher annealing temperatures do not affect bulk (cubic) anisotropy appreciably. Some of the reasons for this degradation in uniaxial anisotropy, which is responsible for PMA in our sys- tem, at a high annealing temperature of 450 °C, include the crystallization of (Co xFe1−x)80B20and overoxidation.A good interface is necessary to enable the hybridiza- tion of 3 delectrons of Fe and 4 dor 5 delectrons of the heavy-metal underlayer to obtain a high PMA [ 55–57]. However, a high temperature of 450 °C also causes an enhancement of (Co xFe1−x)80B20crystallization and crys- tal relaxation, which results in a detrimental effect on PMA due to the decrease in (Co xFe1−x)80B20-MgO interfacial strain [ 58,59]. Weakening of PMA is also attributed to overoxidation of (Co xFe1−x)80B20[11]. Figure 2(d) shows an improvement in the out-of-plane coercivity for the sam- ples annealed at 450 °C, compared to the 300 and 400 °C annealing cases. This increase in the out-of-plane coerciv- ity at 450 °C can result from the enhanced (Co xFe1−x)80B20 crystallization at higher annealing temperature as reported before [ 58,59], leading to a significant reduction in crys- tal defects and dislocations. In our DBL PMTJ structure,when the annealing temperature is changed from 300 to 400 °C, there is limited change in M S,Heff K,and Kefffor x=35–37.5% compared to other compositions. This indi- cates that the compositions x=35–37.5% are more robust in terms of annealing. Next, we discuss the effect of variation of Co composi- tion on the magnetic parameters MS,Heff K,a n d Keff.F r o m Fig. 2(a), it is observed that the MStrend as a function of Co composition has a sharp minimum at x=37.5% for all the annealing temperatures. This contradicts an earlier report [ 17], in which the MSprofile shows a broad max- imum at x∼40% in accordance with the Slater-Pauling curve which is valid for bulk (cubic anisotropy) mag- netic materials. Hence, our observation of a minimum at 024031-4EFFECT OF (Co xFe1−x)80B20COMPOSITION ON MTJ FREE LAYER. . . PHYS. REV. APPLIED 10,024031 (2018) 25 30 35 40 45 5003006009001200MS (emu/cm3) Co/(Co+Fe) (%)300 °C 400 °C 450 °C 25 30 35 40 45 50-30369Heff K )eOk( Co/(Co+Fe) (%)300 °C 400 °C 450 °C 25 30 35 40 45 50-101234Keff cm3)/greM( Co/(Co+Fe) (%)300 °C 400 °C 450 °C 25 30 35 40 45 50020406080100HC )eO( Co/(Co+Fe) (%)300 °C 400 °C 450 °C(a) (c)(b) (d)FIG. 2. Composition dependence of (a) magnetization ( MS), (b) saturation field(Heff K), (c) effective anisotropy energy ( Keff), and (d) coercivity ( HC) for the DBL PMTJ samples annealed at 300, 400, and 450 °C for 30 min andmeasured at room temperature. x=37.5% indicates that, for our thin films, the Slater- Pauling curve is not suitable for explaining the MStrend and more careful considerations of uniaxial anisotropy are required, as we show later using VASP simulations. Further, for the case of 400 and 450 °C annealing temperatures, both Heff Kand Keff, shown in Figs. 2(b) and2(c), respec- tively, remain almost constant with a change in the Co concentration up to x=37.5%, and then decrease with a further increase in the Co concentration beyond x=37.5%. The decrease in Heff Kand Kefffor higher Co concentra- tions arises due to the decrease in the out-of-plane orbital moment because of the hybridization of Co xFe100−xand W [19,57], as we discuss later in this section. The effect of annealing on the Gilbert damping param- eterαis studied using FMR. Figure 3(a) shows the extracted αas a function of (Co xFe1−x)80B20composi- tion for different annealing temperatures. For the sam- ples annealed at 300 °C, αis almost independent with respect to (Co xFe1−x)80B20composition, while the αpro- file shows a minimum at x=35% for the annealing temperatures of 400 °C and above. A similar profile of αis also observed in the composition studies of the metal/Co xFe100−x/metal system [ 22,23], where a minimum inαis found at x∼25%. Similarly, in another study [ 60], for the Ta/(Co xFe1−x)80B20/MgO system, a minimum in αis found at x∼50% when studying only three com- positions of x=25, 50 and 75%. In our case of DBL MgO/(Co xFe1−x)80B20/W/(Co xFe1−x)80B20/MgO systems, the minimum value of αoccurs at x=35%. The rea- sons for this occurrence of minimum αatx=35% willbe discussed later in this section. Nevertheless, the find- ing of this Co composition for the minimum value of αis important for achieving low switching currents in (Co xFe1−x)80B20/MgO-based DBL PMTJ devices. Further- more, αshows a decreased value for all compositions for the 450 °C annealing case, compared to the 300 and 400 °C cases, which suggests that the enhanced crys- tallinity at a high annealing temperature of 450 °C reduces the contribution of extrinsic factors to α. A similar depen- dence of damping on the Co xFe100−xcrystallinity is also observed in previous studies [ 39,61], where epitaxially grown Co xFe100−xshows a smaller αthan polycrystalline Co xFe100−x. Figure 3(b) shows a local minimum in the product of Keffandαat x=35% (for 400 °C annealing tempera- ture), which is critical for achieving a low STT switching current density ( JC∝ Keffα), while maintaining a high value of Keff[62]. Even though x=46% shows a lower value of the Keffαproduct compared to that of x=35%, a very low Keffatx=46% [see Fig. 2(c)] is not ben- eficial for thermal stability of the device. Figure 3(c) shows the composition trend of the gfactor, calculated using the equation [ 17],γ=gμB//planckover2pi1. As we increase the annealing temperature, the gfactor decreases, but this decrease becomes very prominent for Co compositions above 37.5%. The gfactor can be used to calculate the orbital and spin components of average magnetic moment using the relations μL/μ S=(g−2)/2a n dμ=μS+μL, where μLandμSare, respectively, the orbital and spin components of the magnetic moment μ[63].μLis one 024031-5S. SRIVASTAVA et al. PHYS. REV. APPLIED 10,024031 (2018) 25 30 35 40 45 5001234×104(tcudorP Keff) Co/(Co+Fe) (%)300 °C 400 °C 450 °C 25 30 35 40 45 500.0000.0050.0100.015(gnipmaD) Co/(Co+Fe) (%)300 °C 400 °C 450 °C 25 30 35 40 45 502.002.052.102.152.202.25rotcafg Co/(Co+Fe) (%)300 °C 400 °C 450 °C 25 30 35 40 45 500.000.020.040.060.08 Co/(Co+Fe) (%)(tnemomlatibrO) 02468 KU (Merg/ cm3) Orbital moment (B) Uniaxial anisotropy energy ( KU)(a) (c)(b) (d)FIG. 3. FMR results showing com- position dependence of (a) damping α for samples annealed at 300, 400, and450 °C for 30 min, (b) the product of effective anisotropy energy K effandα for the samples annealed at 300, 400,and 450 °C for 30 min, and (c) gfac- tor calculated using gyromagnetic ratio for samples annealed at 300, 400, and450 °C for 30 min. (d) Comparison of orbital moment and uniaxial anisotropy. of the main factors in determining the spin-orbit coupling (SOC) of the Co xFe100−x-W system [ 19,57], which, in turn, is correlated with the uniaxial magnetic anisotropy, KU[64,65]. Figure 3(d) shows the comparison of μLand KU, where KU=Keff+2πMS2. It is noted that the dras- tic decrease of μLafter 42% leads to a weakening of SOC and thus, can be correlated to the decrease in KU atx=46%. Figure 4(a) shows the results of DOS calculations for different compositions of Co xFe100−x/MgO superlattices, and Fig. 4(b) shows the DOS calculated at the Fermi level, n(EF), and its comparison with αfor the samples annealed at 450 °C. We find that there is a minimum for n(EF) between x=31.25 and x=37.5% [Fig. 4(b)], which arises due to the shift of the DOS to the lower energies [as shown in Fig. 4(a)] as Co concentration increases. In the literature, for a metal/Co xFe100−x/metal system [ 23], this minimum inn(EF) is observed at a Co composition of 25%. How- ever, in our case, the introduction of an MgO interface causes the minimum in n(EF) to shift to a Co compo- sition in the range of 31.25–37.5%. This shift is due to the presence of metal-oxygen bonds across the interface, which divert a portion of the metallic electrons to localized bonding states and decrease the conduction band-filling in Co xFe100−x. In contrast, increasing the Co concentration increases the filling of the conduction band in Co xFe100−x. Therefore, a larger concentration of Co is needed to achieve the minimum DOS for the interface model with metal-oxygen bonds in comparison with bulk Co xFe100−x. In the limit of intraband scattering, the damping param- eter has been demonstrated to be largely proportional ton(EF)[23,66,67]. Therefore, the profile for n(EF) in the MgO/Co xFe100−x/MgO structure is in close agreement to our experimental observation of a minimum αatx=35%. In order to ensure the program-retention capability of the devices, it is important that the samples can maintaingood magnetic properties, that is, limited change in M Sand high/Delta1, at an elevated temperature up to 260 °C. Figures 5(a)–5(c) show the temperature dependence (measured at 60, 85, 125, 150, 175, 225, and 260 °C) of magnetic prop- erties for each composition ( x)o f( C o xFe1−x)80B20samples annealed at 400 °C. Figure 5(a) shows the temperature variation of MSfor different compositions. It is observed that x=42 and 46% show the highest MSvalues and small- est rate of decrease in MSwith respect to temperature. However, from Fig. 5(b), it is observed that x=35% main- tains the highest Heff Kvalues and the lowest rate of decrease ofHeff Kwith respect to temperature .The trend of Heff Kis also reflected in the trend of Keffas shown in Fig. 5(c), in which it is observed that x=35% has the highest Keff values for higher temperatures. Since a high Keffvalue is important for a high /Delta1,x=35% is a good candidate from a thermal stability perspective for sustaining solder- reflow processes at 260 °C. Furthermore, for our DBL PMTJ structure with the free-layer thickness of 2.6 nm, the calculated thermal-stability factor for x=35% at 260 °C is approximately 60 (assuming a 50 nm diameter), which is enough for embedded MRAM applications [ 12]. Subse- quently, we fit the temperature-dependent magnetization, using Kuz’min’s model given by Eq. (1), to calculate the value of ASfor different Co compositions as shown in Fig. 5(d). The value of ASshows a maximum at x=37.5%. As 024031-6EFFECT OF (Co xFe1−x)80B20COMPOSITION ON MTJ FREE LAYER. . . PHYS. REV. APPLIED 10,024031 (2018) -0.2 -0.1 0.0 0.1 0.205101520 37.5%n(E) (eV-1) 50% 31.25% 25% 43.75% 25 30 35 40 45 509101112n(EF) (Simulation) Damping (Experimental) Co/(Co+Fe) (%)n(EF) (eV-1) 0.0040.0060.0080.0100.012 Damping ( )(a) (b) FIG. 4. DOS calculation and com- parison with measured damping val- ues. (a) DOS calculation results forCo xFe100−x/MgO for different Co compositions. (b) Majority DOS at the Fermi level n(EF) and damping αobtained from FMR plotted as a function of xfor 450 °C annealed samples. the domain-wall width δwis proportional to√AS, this com- position can be beneficial for uniform coherent switching for larger diameter devices. We analyze the temperature-dependent MSand Keff data using the Callen-Callen law, MS(T)/MS(0)= (Keff(T)/Keff(0))m, in which mshould be 3 for uniaxial anisotropy and 10 for cubic anisotropy [ 68,69]. Figure 6(a) shows the composition dependence of mfor the 400 °C annealed samples. The obtained values of mvary in the range of 3–10, which arises due to the combined contributions of both uniaxial and cubic anisotropies. Since there is no direct way to find the ratio of the sites with uniaxial and cubic anisotropy, we assume wto be the weighting of the uniaxial contribution to the anisotropy and the factor mto be weighted between 3 and 10 (m= w×3+(1−w)×10). We assume that this weight wis the ratio of the sites with uniaxial anisotropy and use thisfactor to calculate the average moment per atom, ma, using the equation ma=mu×w+mc×(1−w),( 2 ) where muis the moment per atom at the sites contributing to the uniaxial anisotropy and mcis the moment per atom at the sites contributing to the cubic anisotropy. For different Co compositions, the cubic contribution mccan be extracted from the Slater-Pauling curve for Co xFe100−x[70]. In order to obtain the uniaxial con- tribution mu, we performed VASP simulations for the Co xFe100−x/MgO system at different Co concentrations. For the uniaxial Co xFe100−x/MgO system, the magnetic moments of Co and Fe delectrons are found to be in par- allel directions and are the prime contributors to the net magnetic moment per atom for both species. Switching 0 50 100 150 200 250 30002468Heff K (kOe) Temperature (°C)25% 32% 35% 37.5% 42% 46% 25 30 35 40 45 500123 25 °C 60 °C 85 °C 105 °C 125 °C 150 °C 175 °C 225 °C 260 °CKeff )cm3/greM( Co/(Co+Fe) (%)25 30 35 40 45 500246810)m/Jp(ssenffitsegnahcxE Co/(Co+Fe) (%)(a) (c)(b) (d)0 50 100 150 200 250 30002004006008001000 MS /cm3) ume( Temperature (°C)25% 32% 35% 37.5% 42% 46%FIG. 5. Temperature dependence of (a) magnetization and (b) anisotropy field for different Co compositions. (c) Anisotropy energy dependence on Cocomposition at different temperatures. (d) Exchange stiffness plotted as a function of Co composition. Samplesannealed at 400 °C for 30 min. 024031-7S. SRIVASTAVA et al. PHYS. REV. APPLIED 10,024031 (2018) 25 30 35 40 45 50246810m )walnellaC-nellaC( Co/(Co+Fe) (%)Cubic anisotropy Uniaxial anisotropy 0 1 02 03 04 05 01.21.62.02.42.8(mota/tnemoM) Co/(Co+Fe) (%)Fe – Interface Fe – 1 away Fe – 2 away Co – Interface Co – 1 away Co – 2 away 0 1 02 03 04 05 02.02.12.22.32.42.5(mota/tnemoM) Co/(Co+Fe) (%) Slater-Pauling ( mc) Simulation ( mu) 25 30 35 40 45 50700800900ma (Simulation) MS (Experimental) Co/(Co+Fe) (%)MS cm3)/ume( 2.22.32.4 Moment/atom ( )(a) (c)(b) (d)FIG. 6. (a) Dependence of mon com- position for 400 °C annealed samples. (b)VASP simulation results for magnetic moment per atom of Co and Fe lay- ers at different positions with respect to MgO. (c) Comparison of the moment peratom from the Slater-Pauling curve and the average moment per atom calculated for the PMA system. (d) Comparison ofmagnetization from VSM measurement and average magnetic moment per atom calculated using Eq. (2). The simulation curves in (b), (c), and (d) are fitted using splines. the magnetization direction out of or into the MgO layer is found to have a negligible effect on the absolute value of the magnetization per atom. Figure 6(b) shows the contri- butions of individual Co and Fe atoms to the magnetization obtained from the simulations at different positions with respect to the MgO interface. The contribution of moment from each Co atom is less than that from each Fe atom, which leads to a net decrease in mu[blue line in Fig. 6(c)] as the composition of Co increases from 0 to 50%. This agrees with Madelung’s rule of the ordering of spins in atomic subshells [ 71]. Interestingly, as Co is introduced into the system, the magnetic moment per Co atom rises, showing a peak at x∼24%, then decreases sharply again in Fig. 6(b). Increasing the Co content higher than x=30% does not affect the atomic magnetic moment of Co further. From Fig. 6(b), it is also observed that the presence of the MgO interface enhances the magnetic moment per atom in the case of Fe and suppresses it in the case of Co [ 11]. Figure 6(c) shows the calculated values of mufor the PMA (interfacial) system that are fitted with spline curves to interpolate the intermediate values from the simulation. Figure 6(c)also shows mcobtained from the Slater-Pauling (cubic) curve [ 70]. Subsequently, the values of muand mcare used to calculate the average magnetic moment per atom, mausing Eq. (2). Figure 6(d) shows the calcu- lated values of mafor different Co compositions. Figure 6(d) also shows the trend of the saturation magnetization (MS) obtained from VSM for 400 °C annealed samples. It is observed that both maand MSshow a very close relationship in terms of Co concentrations, which con- firms that the observed trend of MSwith Co composition is due to the transition from uniaxial anisotropy to cubicanisotropy with increasing Co concentration. Moreover, this close relationship of maand MSsuggests that, for the case of ultrathin films with a MgO interface, the tradi- tionally used Slater-Pauling curve is not suitable, and it is necessary to consider the contributions of both uniaxial and cubic anisotropies as well as their dependence on com- position for magnetization calculations. Furthermore, the uniaxial anisotropy can be indicative of a better hybridiza- tion between Fe and O, resulting in higher /Delta11symmetry, which can be useful in determining materials with a high TMR in DBL PMTJs [ 72]. IV. CONCLUSIONS We study the (Co xFe1−x)80B20composition dependence of magnetic properties of the free layer in a double-barrier PMTJ. We investigate the annealing robustness of our system, with the range of Co composition x=35–37.5% showing the least deviation in the magnetization and anisotropy energy with respect to annealing temperature. We also extract the orbital angular moment that explains the profile for Keffwith respect to Co composition. The composition dependence of the damping parameter reveals that a low critical current density for STT switching is achievable for x=35%. The analysis of the magnetic prop- erties at elevated temperatures demonstrates that a high thermal stability at elevated temperature can be achieved atx=35%. Further, we identify that the exchange stiff- ness maximizes at x=37.5%, which is indicative that this composition supports a larger diameter size for coherent uniform switching. Using the Callen-Callen law, we iden- tify that the anisotropy of our system transits from uniaxial 024031-8EFFECT OF (Co xFe1−x)80B20COMPOSITION ON MTJ FREE LAYER. . . PHYS. REV. APPLIED 10,024031 (2018) to cubic as Co concentration increases. This transition from uniaxial to cubic anisotropy is used to explain the anoma- lous dip in the magnetization. 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PhysRevApplied.13.064050.pdf

PHYSICAL REVIEW APPLIED 13,064050 (2020) Laser-Induced Abnormal Cryogenic Magnetoresistance Effect in a Corbino Disk Xinyuan Dong,1,2Diyuan Zheng,1,2Meng Yuan,1,2Yiru Niu,1,2Binbin Liu,1,2and Hui Wang1,2,* 1State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, People’s Republic of China 2Key Laboratory for Thin Film and Microfabrication Technology of the Ministry of Education, Research Institute of Micro/Nano Science and Technology, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, People’s Republic of China (Received 7 September 2019; revised manuscript received 27 April 2020; accepted 1 June 2020; published 19 June 2020) The geometric magnetoresistance effect in semiconductors has remained a heated discussion for many years. However, there are few reports on laser-triggered geometric magnetoresistance in traditional struc-tures. In this work, we use a laser to change the carrier concentration to obtain a large magnetoresistance (212.6%) under a low magnetic field (1 T) at 150 K in a Corbino disk with Co-Ag films. One unantici- pated finding is that the large positive magnetoresistance does not change monotonously with temperature,which is different from previous research. Theoretical calculation reveals that the interaction among a pho- togenerated carrier, bending of the current path, and magnetic nanoparticles in low temperature improves the magnetoresistance in a Corbino disk. These findings reveal an important strategy for creating laser-trigged nanoscale magnetoresistance devices, while presenting a wide range of possibilities for exploring the dependence of photogenerated carriers on temperature under magnetic field. DOI: 10.1103/PhysRevApplied.13.064050 I. INTRODUCTION The geometric magnetoresistance (MR) effect [ 1,2]i n nanoscale films has attracted increasing research attention since its discovery [ 3,4], which is widely used in accurate measurement of carrier concentration [ 5–7], evaluation of transferred electron device performance [ 8], high magnetic field sensors [ 9], current sensors [ 10], and other fields [11]. A Corbino disk is a quintessential structure to inves- tigate the geometric magnetoresistance effect thoroughly since it can eliminate Hall voltage. Considerable research efforts have been devoted to explain the underlying mech- anism of the geometric magnetoresistance effect, which can be attributed to the bending of the current path and uneven carrier distribution [ 12–14]. However, using a laser to change the carrier concentration and mobility is rarely involved in these studies. Moreover, relatively little atten- tion has been paid to the temperature dependence of the effect, despite its importance for the selection of mech- anisms and the development of a microscopic theory of geometric magnetoresistance. Previously, our group reported a large laser-triggered positive magnetoresistance in a Corbino disk of Cu/SiO 2/ Si [15]. With the combined application of laser and mag- netic field, the magnetoresistance is significantly improved by more than 60 times compared with other research *huiwang@sjtu.edu.cn[16,17] at the same magnetic field (1 T). On the basis of our previous research, this work investigates the laser-triggered geometric magnetoresistance effect in Co- Ag films, especially to further explore the temperature response of this effect. Thin films consisting of cobalt nanoparticles embedded in a silver matrix are attractive for magnetoresistive research. The phase diagram indi- cates there is very limited mutual solubility of Co with Ag, which offers the possibility of heterogeneity [ 18,19]. Experimental results show the laser-triggered magnetore- sistance of Co-Ag films can reach 212.6% and 8.2% at 150 K and room temperature under a magnetic field of 1 T, respectively. The magnetoresistance is significantly enhanced compared with previous studies under the same circumstances (1 T magnetic field) [ 12,15,18,20]. We also notice, most surprisingly, that the magnetoresistance does not change monotonously with temperature, unlike the monotonous rise in magnetoresistance of granular films caused by scattering with a decreasing temperature [ 19]. This research extends the knowledge into the geometric magnetoresistance mechanisms induced by laser and tem- perature, while opening the door to the possibilities in temperature sensors and magnetoresistance devices. II. FABRICATION AND METHODS We fabricate Co-Ag composite films on the doped n-type Si (111) wafers (approximately 0.3 mm thickness, 20–50 /Omega1cm resistivity) with a native ultrathin oxide layer 2331-7019/20/13(6)/064050(8) 064050-1 © 2020 American Physical SocietyXINYUAN DONG et al. PHYS. REV. APPLIED 13,064050 (2020) approximately 1.2 nm thick on one side. Co-Ag films are deposited by co-sputtering of Co and Ag targets (purity better than 99.9%) using dc magnetron sputtering from two confocal sputter magnetron guns at room tempera- ture. The whole sputtering process is in an argon pressure of 0.78 Pa, and the base pressure of the vacuum system is better than 3.8 ×10−4Pa. The dc power of cobalt and silver is fixed at 10 and 20 W, respectively. The depo- sition rate is 3.2 Å s−1, which is determined by the step profiler on thick calibration samples fabricated under the same condition. A ring-shape mask is utilized to deposit the film. First, the ring groove is covered by the mask and deposited for 145 s. Then, the mask is removed, and the whole part is deposited for 5 s. Counting with the depo- sition rate, the electrode layer of 48 nm Co-Ag alloy is deposited on center and peripheral areas of the Si substrate, as annotated in Fig. 1(a). The annular region between two electrodes is 1.6 nm (nominal thickness) superthin Co-Ag composite films. We switch the dc power of the Co target to regulate the content of cobalt in the samples. The compo- sitional distribution of the alloy film is further investigated by energy dispersive spectroscopy (EDS). Figure 1(a) shows a representative schematic diagram of a Corbino disk, where rais the radius of center region and rbis the outer radius of the ring. The width of thering groove is defined as rb−ra. During the experiment, the ring width of all samples is maintained at 1.5 mm. Electrodes A and B of alloying indium (less than 1 mm in diameter) are pressed on the central and peripheral bulk Co-Ag areas, respectively. Figures 1(b) and 1(c) show SEM images of the two different regions, and the inset shows the corresponding EDS spectra. SEM images recorded at high magnification clearly show that the as- deposited films consist of a bulk region with uniform shapes [Fig. 1(b)] and nanoparticles [Fig. 1(c)]. In the experimental process, electrode A is irradiated by a 635-nm, 3-mW laser focused on a roughly 50- µm diam- eter spot continuously, and without any background light illumination. We measure the magnetoresistance using the two-probe method in a vacuum cavity with a pressure of 1.0×10−4Pa, and the chamber temperature ranged from 20 to 300 K in the presence of a magnetic field in the z- axis direction varying up to 1 T. The chamber temperature is regulated by helium compressors and a temperature con- troller, and the resistance is determined using the Keithley 4200-SCS Semiconductor Characterization System. The magnetoresistance is defined as /Delta1R/R0=[(RH−R0)/R0]×100%. (1) (3 mm)(1.5 mm)(a) (b) (c)FIG. 1. (a) Schematic diagram of Corbino structure with Co-Ag/SiO 2/Si and the experimental measurement method. (b) SEM image of the bulk Co-Ag layer. Inset shows the corresponding EDS spectra. (c) SEM image of the annular groove region where superthinCo-Ag films are deposited. 064050-2LASER-INDUCED ABNORMAL . . . PHYS. REV. APPLIED 13,064050 (2020) Here, RHrepresents the resistance with the external mag- netic field applied, and R0is the resistance in the absence of magnetic field. III. RESULTS AND DISCUSSION Figure 2(a) shows the I-Vcurves of the Co-Ag/SiO 2/Si sample between electrode A and B under different condi- tions at 20 K (equipment limit). We define the forward- sweep voltage to represent the case where electrode A is the anode and B is the cathode. The original I-Vcurve (i.e., without laser irradiated and magnetic field applied) is symmetrical, and the sample is in a high-resistance state. However, when a fixed 635-nm laser is applied perpendicularly to electrode A, the resistance is signifi- cantly reduced. Most surprisingly, the I-Vcurve exhibited extreme asymmetry, which suggested a laser-induced polar resistance effect. This bipolar resistance effect has been reported in our previous research, which can be attributed to diffusion and scattering of carriers based on Schottkybarriers [ 15]. On this basis, we apply a 1-T magnetic field perpendicular to the sample, the I-Vcharacteristic as indi- cated by the blue line in Fig. 2(a). The resistance of the sample increases observably under the combined effect of laser and magnetic field. Nevertheless, when the laser is removed, the I-Vcharacteristic is almost the same as the original curve. From the results we can conclude that the laser plays an indispensable role in the effect. In order to further investigate the effect of temperature and magnetic field on magnetoresistance, we fix the laser position and change the ambient temperature, and mea- sure the magnetoresistance under different magnetic fields. The MR values are calculated by Eq. (1). Figure 2(b) shows MR data of the Co-Ag/SiO 2/Si sample versus mag- netic field for various temperatures. As the magnetic field increases, the MR value is drastically promoted, which is in consonance with the work we have reported before [ 15]. Besides, the magnetoresistance in a Corbino disk struc- ture presents excellent symmetry to the direction of applied magnetic field. As for temperature dependence, the MR (a) (b) (c) (d) FIG. 2. (a) I-Vcurves of the Co-Ag/SiO 2/Si sample with different laser and magnetic field conditions. The ambient temperature is set to 20 K, and measurement details are shown in the inset. A fixed 635-nm, 3-mW laser on a roughly 50- µm diameter spot perpendicular to electrode A. (b) MR ratio of the Co-Ag/SiO 2/Si sample as a function of magnetic field at different temperatures. (c) The Co-Ag/SiO 2/Si sample’s MR ratio versus cobalt content as a function of temperature. The applied magnetic field is fixed at 1 T and the measurement condition is identical to before. (d) The dependence of the Co-Ag/SiO 2/Si sample’s MR on temperature with different laser conditions. The measurement condition is the same compared to before (with a 1-T magnetic field and 635-nm laser applied). 064050-3XINYUAN DONG et al. PHYS. REV. APPLIED 13,064050 (2020) values increase with the increasing temperature and grow up to the maximum when the temperature is 150 K, and then decrease sharply above 170 K. We obtain a large magnetoresistance of 212.6% at 150 K, only 1-T magnetic field, which is comparable to other research. To gain deeper insight related to the laser-triggered MR effect in Co-Ag films, a systematic study has been carried out in Co-Ag/SiO 2/Si samples with cobalt content varying from 3.01% to 26.53%. The cobalt content is governed by sputtering time and power, and determined by energy dis- persive spectroscopy. We prepare samples with the same nominal Co-Ag thickness but different cobalt contents. During the experiment, the applied magnetic field is fixed at 1 T and the samples are irradiated by a 635-nm laser con- tinuously. A clear dependence of MR on the cobalt content is observed [Fig. 2(c)]. The same general trend is found at different temperatures: the MR value increases with the cobalt content up to a maximum and then drops off with the higher cobalt content. As is widely known, a laser may cause the temperature change in the place of irradiation, and further leading to the apparition of temperature gradients. In order to eliminate the possibility that the local temperature change caused by a laser contributes to the magnetoresistance, we mea- sure the dependence of MR on temperature with different laser conditions in a 1-T magnetic field. As annotated in Fig.2(d), MR values present the same tendency with tem- perature changes under different illumination conditions. When the temperature is below 150 K, the magnetoresis- tance increases slowly with temperature. With temperature increasing from 150 to 200 K, the magnetoresistance value is drastically reduced. Once the temperature is greater than 200 K, the magnetoresistance decreases very slowly with increasing temperature. Besides, the MR effect with 1550- nm laser irradiation almost has no change compared with the case with no laser irradiation, but it is greatly enhanced with 635-nm laser irradiation, indicating the local temper- ature change caused by the laser does not contribute to the magnetoresistance effect. The influence of magnetic materials on magnetoresis- tance is non-negligible. To further investigate the mag- netism of samples, magnetic hysteresis, the field cooled (FC), and the zero field cooled (ZFC) are employed. Figure 3(a) shows the full hysteresis loops of the mag- netization measured at room temperature on the Co-Ag (nominal thickness, 1.6 nm, and 13.27% Co) sample for parallel ( ||) and perpendicular ( ⊥) to film plane orienta- tions of the applied magnetic field. The inset shows the enlarged image of the M-Hcurve corresponding to H perpendicular ( ⊥) to the film, and the sample exhibits negligible coercivity force (16.8 Oe). It is found that thetwo M-Hcurves could not reach saturation even at the maximum magnetic field of 1.5 T, which clearly indicates the presence of superparamagnetic particles [ 21]. And the easy axis of the nanoparticles is out of plane. To confirmthe superparamagnetic behavior, we measure the ZFC and FC data in the temperature range 2–400 K, as shown in Fig.3(b). For samples with 13.27% Co, the M ZFC(T) curve exhibits the maximum at TB∼170 K, while the MFC(T) curve decreases monotonously with the increasing temper- ature. Such magnetic properties indicate the superparam- agnetic character of the Co nanoparticles in our samples [22]. Besides, the blocking temperature in the sample with 13.27% Co is larger than the sample with 8.22% Co, which suggests the increase in particle size. Moreover, the FC curves suggest the absence of cobalt oxide. Because CoO is antiferromagnetic, the magnetization of FC curve will exhibit a sharp drop above the Neel temperature of 290 K. However, this is not observed in our samples. Magnetic measurements of the samples are carried out to determine whether the Co-Ag materials contribute greatly to magnetoresistance. Figure 3(c) shows the MR ratio as a function of temperature in Co-Ag/SiO 2/Si with ordinary structure (without Corbino geometry). The nominal thick- ness of Co-Ag film is fixed at 1.6 nm, which is the same as the thickness in a Corbino disk. And the content of cobalt is 13.27%. As can be seen in Fig. 3(c), the magne- toresistance values are quite small in the absence of laser, and the temperature dependence of MR is consistent with previous researches [ 19,23]. However, the magnetoresis- tance is considerably improved when the laser is applied. And the temperature dependence is completely different from that without laser. With the temperature increasing from 20 to 170 K, the laser-triggered MR declines slowly. Once the temperature is higher than 170 K, the laser- triggered MR ratio decreases sharply until 230 K. Obvi- ously, the magnetic material has a certain contribution to the magnetoresistance below TB(170 K), but it makes lit- tle contribution once the temperature is higher than TB.W e think these results can be attributed to the transition from ferromagnetism to superparamagnetism, which involves a transition from ordered to disordered orientations of the electron spins. The neighboring islands tend to be parallel aligned by the external field and reduce the resistance in the absence of laser, thereby, the negative magnetoresistance. But when the laser is applied, the photogenerated carriers play a vital role in the magnetoresistance. In the diffusion process of carriers, electrons in Co-Ag also may recombine with holes in silicon. And the carrier recombination rate is affected by scattering. When the temperature is below TB, the particles exhibit ferromagnetism. Under the applied magnetic field, the spin-dependent scattering of conducting electrons contribute to the increase of photogenerated car- rier recombination rate, thereby, the increased resistance and positive magnetoresistance. But when the temperature is higher than TB, the thermal energy can disrupt the mag- netic moment, thereby weakening the magnetism, which contributes little to the magnetoresistance. For comparison, we also measure the temperature dependence of laser-triggered MR in Corbino disks with 064050-4LASER-INDUCED ABNORMAL . . . PHYS. REV. APPLIED 13,064050 (2020) (a) (b) (c) (d) FIG. 3. (a) M-Hhysteresis loops corresponding to Hparallel ( ||) and perpendicular ( ⊥) to film measured at room temperature for the Co-Ag (nominal thickness, 1.6 nm, and 13.27% Co) sample. (b) Temperature dependence of magnetization in Co-Ag samples with different Co contents, in the ZFC and FC protocols in the presence of magnetic field of 100 Oe. (c) The MR ratio as a function of temperature in Co-Ag/SiO 2/Si with ordinary structure (without Corbino geometry). The nominal thickness of the Co-Ag film is 1.6 nm and the content of cobalt is 13.27%. The inset shows the measurement method. (d) Dependence of laser-triggered MR ratio on temperature with nonmagnetic materials in Corbino disks. The width of the ring groove is fixed at 1.5 mm, and the thickness of metalkept consistent. nonmagnetic materials, as shown in Fig. 3(d). Both of the samples show similar nonmonotonic temperature depen- dence. Compared with the magnetoresistance in magnetic materials shown in Fig. 2(d), there is a striking difference when the temperature is quite low. The magnetoresis- tance in magnetic materials is much larger than that in nonmagnetic materials. To explain these phenomena, we propose a model based on the Schottky barrier. The samples contain bulk Co- Ag layers and superthin films, and the Schottky barrier is much lower in the annular groove region, which covered by superthin films [ 15]. Therefore, the equivalent circuit can be considered as two reverse diodes and a pure resis- tor, as indicated in Fig. 4(left). Obviously, the system is in a high-resistance state without laser irradiation. When the laser is applied, a large amount of photogenerated carriers are generated in the silicon substrate. There was a high car- rier concentration at the laser spot, so carriers diffuse to the surrounding. [ 24–26] And photogenerated electrons havethe opportunity to tunnel into the alloy layer. The applica- tion of magnetic field brought on the deflection of a carrier motion path under Lorentz force, thereby, the increased resistance (Fig. 4, right). If we suppose the initial resistivity without a laser and magnetic field is ρ0, the resistance at position x(i.e., the distance from the laser spot) can be written as [ 27,28] ρ(x)≈ρ0/parenleftbigg 1−n0 N0+n0 N0λx/parenrightbigg .( 2 ) Hereλis the diffusion length, n0and N0represent the den- sity of laser-induced electrons and drift carriers at the laser spot, respectively. In previous research, we have derived the current path formula under the applied magnetic field, which can be written as [ 15,29] s=αx,( 3 ) 064050-5XINYUAN DONG et al. PHYS. REV. APPLIED 13,064050 (2020) FIG. 4. Schematic diagram of photogenerated carriers’ move- ment only laser irradiated (left).The carriers’ motion path with the combined effect of laser and mag- netic field (right). here, α=/radicalbigg 1+μ2H2 c2,( 4 ) andμis the carrier mobility, His the density of magnetic field, and cis a constant in the Gauss unit system. We find that the new parameter αcharacterizes the current path change caused by magnetic field in the Corbino disk. More simply, αcharacterizes the extent of motion-path bending. Note that a laser is a prerequisite in the experiment. R0rep- resents the resistance with laser irradiation only, and it can be written as a path integral of ρ(x): R0=/integraldisplayrb raρ(x)dx.( 5 ) However, when the magnetic field is applied, the current path changed, thereby, the resistivity at different positions. RH=/integraldisplayrb raρ(s)ds=α/integraldisplayrb raρ(α x)dx.( 6 ) Finally, the laser-triggered magnetoresistance can be writ- ten as /Delta1R/R0≈α+(α2−α) 1+kλ/parenleftBig N0 n0−1/parenrightBig.( 7 ) Here k=1/(ra+rb)is a constant. From Eq. (7), we can see the temperature Thas an influence on three parameters: λ,N0/n0,a n dα. The diffusion length λcan be written as [26,30] λ=√ Dτ∝T−1,( 8 ) where Dis diffusion coefficient affected by temperature. And according to the Boltzmann distribution function, thecarrier concentration satisfies N0 n0∝e−EC−EF k0T∝e−1 T.( 9 ) The parameter α, which measures the current path change caused by magnetic field described in Eq. (4), can be sim- plified as α∝μ. Here, μis the mobility, which can be written as [ 17] μ=qτ m∗. (10) Here m∗andτare the effective mass and lifetime of the carrier, respectively. Hence, the MR can be simplified as /Delta1R/R0∝μ(T)/braceleftbigg 1+[μ(T)−1]T e−1/T/bracerightbigg . (11) Here, e−1/Tapproaches a constant as temperature increases. Therefore, we mainly take the mobility μinto consideration. The scattering processes influence the life- time, thereby, limiting the mobility. Due to the large carrier density, the thermal vibration of the lattice has a non- negligible influence on the mobility even at low tempera- ture. Therefore, the ionized impurity scattering dominates at low temperatures, where μ∝T3/2. Apparently, the mag- netoresistance increases with the increasing temperature. But as the temperature further rises, the lattice vibration scattering dominates, which satisfies μ∝T−3/2[17]. As a result, the magnetoresistance decreases. Similar tem- perature dependence of magnetoresistance (without laser irradiation) in a Corbino disk has been reported in previous research [ 17]. According to the above analysis, we con- clude that the nonmonotonic temperature dependence of magnetoresistance is mainly due to the change of mobility in Corbino structure. And the laser plays a significant role in generating photogenerated carriers and amplifying mag- netoresistance. Under the influence of these factors, themagnetoresistance grows monotonously up to maximum, above which it precipitously decreases. Besides, taking into account the contribution of mag- netic nanoparticles below the blocking temperature T B, 064050-6LASER-INDUCED ABNORMAL . . . PHYS. REV. APPLIED 13,064050 (2020) the recombination rate of photogenerated carriers increases due to the spin-dependent scattering. As a result, the diffu- sion length λdecreases. According to Eq. (7), the magne- toresistance is enhanced below TB, which also explains the reason why the MR effect in magnetic particles is better than nonmagnetic at low temperatures. As for the cobalt content dependence of MR shown in Fig. 2(c), it can be attributed to changes in parti- cle size. When the concentration of magnetic particles is small, there is less scattering and larger particle spac- ing, which leads to the small magnetoresistance. There- fore, as the cobalt content is increased, the MR effect is improved. However, as indicated in Fig. 3(b),TB increases with increasing cobalt content, which suggests the increase in particle size. As the particles grow larger, the surface:volume ratio decreases, which weaken the spin-dependent scattering of conducting electrons [ 19,31]. As a result, the photogenerated carrier recombination rate decreases, thereby, the magnetoresistance is reduced. IV. CONCLUSION In conclusion, we obtain a colossal magnetoresistance effect using a simple laser-triggered method in Corbino disks with Co-Ag films. The temperature dependence of the laser-triggered magnetoresistance effect is investigated in the temperature range from 20 to 300 K. What is surpris- ing is that the dependence of the MR ratio on temperature is nonmonotonic. Moreover, the MR effect is closely asso- ciated with the elemental component of samples. We show that the Corbino geometry, diffusion length, and magnetic nanoparticles contribute to the magnetoresistance. This work expands the possibility of design for laser-trigged and temperature-regulated magnetoresistance devices. ACKNOWLEDGMENTS We acknowledge the financial support of the National Natural Science Foundation of China under Grants No. 11874041, No. 61574090, No. 11374214, and No. 10974135. 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PhysRevB.92.104430.pdf

PHYSICAL REVIEW B 92, 104430 (2015) Respective influence of in-plane and out-of-plane spin-transfer torques in magnetization switching of perpendicular magnetic tunnel junctions A. A. Timopheev, R. Sousa, M. Chshiev, L. D. Buda-Prejbeanu, and B. Dieny Univ. Grenoble Alpes, INAC-SPINTEC, F-38000 Grenoble, France; CEA, INAC-SPINTEC, F-38000 Grenoble, France; CNRS, SPINTEC, F-38000 Grenoble, France (Received 3 June 2015; revised manuscript received 26 August 2015; published 28 September 2015) The relative contributions of in-plane (damping-like) and out-of-plane (field-like) spin-transfer torques (STT) in the magnetization switching of out-of-plane magnetized magnetic tunnel junctions (pMTJ) has been theoreticallyanalyzed using the transformed Landau-Lifshitz-Gilbert (LLG) equation with the STT terms. It is demonstratedthat in a pMTJ structure obeying macrospin dynamics, the out-of-plane torque influences the precession frequency,but it does not contribute significantly to the STT switching process (in particular to the switching time andswitching current density), which is mostly determined by the in-plane STT contribution. This conclusion isconfirmed by finite temperature and finite writing pulse macrospin simulations of the current field switchingdiagrams. It contrasts with the case of STT switching in in-plane magnetized magnetic tunnel junction (MTJ) inwhich the field-like term also influences the switching critical current. This theoretical analysis was successfullyapplied to the interpretation of voltage field STT switching diagrams experimentally measured on pMTJ pillars36 nm in diameter, which exhibit macrospin behavior. The physical nonequivalence of Landau and Gilbertdissipation terms in the presence of STT-induced dynamics is also discussed. DOI: 10.1103/PhysRevB.92.104430 PACS number(s): 85 .75.−d,75.78.−n,85.70.Ay I. INTRODUCTION Fully perpendicular magnetic tunnel junctions (pMTJ) constitute the storage element of spin-transfer torque magne-toresistive random access memory (STT-MRAM) [ 1–6]. STT- MRAMs are very promising emerging nonvolatile memoriessince they combine nonvolatility, low energy consumption,high thermal stability, and almost unlimited endurance. Thestrongest research and development efforts are nowadaysfocused on out-of-plane magnetized MgO-based MTJs. In-deed, the latter combine several advantages. They exhibit ahigh tunnel magnetoresistance effect [ 7] amplitude due to a very efficient spin-filtering phenomenon associated with thesymmetry of the tunneling electron wave function [ 8,9]. Fur- thermore, they present a very large perpendicular anisotropyat the interface between the magnetic electrode and the MgOoxide barrier (Ks ∼1.4 erg /cm²)[10], which allows the storage layer magnetization to achieve a quite high thermalstability and therefore long memory retention. Additionally,a remarkable property of this interfacial anisotropy is thatit exists in materials having weak spin-orbit coupling andtherefore relatively low Gilbert damping α(α< 0.01). This is very important in STT-MRAM since the critical current forSTT-induced switching [ 11,12] of storage layer magnetization is directly proportional to the Gilbert damping. The advantageof using out-of-plane rather than in-plane magnetized MTJsin STT-MRAM is twofold: first, the interfacial perpendicularanisotropy at the CoFeB/MgO interface provides higherthermal stability at smaller dimensions (sub-60 nm) thandoes the usual-shaped anisotropy by giving elliptical shapeto in-plane magnetized MTJs. Second, for a given retention,i.e., a given thermal stability factor, the critical current forSTT-induced switching is lower with an out-of-plane than withan in-plane magnetized storage layer [ 13,14]. From a theoretical point of view, a first approach to STT-induced switching can be developed by solving theLandau-Lifshitz-Gilbert (LLG) equation under the assump- tions of 0 K macrospin approximation under stationary applied spin-polarized current. The equilibrium configurations of thesystem can thus be calculated, and the precessional dynamicsof the system submitted to a small perturbation from thestatic equilibrium can be studied. This allows derivation ofthe threshold current required to achieve STT switching,as was done in Refs. [ 13–15]. Thermal fluctuations can be taken into account in several limiting cases using the Fokker-Planck equation. Thermal activation mainly decreasesthe threshold current value and the switching time, introducingan undesirable stochastic magnitude effect in both parameters[16,17]. The influence of the writing pulse duration was also theoretically studied [ 16,18–21]. Despite the numerous experimental results [ 22,23] and micromagnetic simulations [ 24–26] generally pointing to quantitative disagreements with the macrospin-based esti-mations, usage of the macrospin approach is still justifiedfor at least two reasons. First, it gives a simple but solidpicture of the physical processes involved in STT switchingthat creates a common basis for qualitative analysis ofthe different magnetic multilayered systems, while most of the conclusions derived from micromagnetic approaches are rather of particular character. Micromagnetic behavior canbe mimicked, for example, by introduction of an effectiveactivation volume instead of Stoner-Wohlfarth behavior, butstill using a thermal activation model for the subvolume [ 22]. Second, considering the general trend to reduce the volume ofthe storage element (and, consequently, the energy needed per write/read cycle), magnetic memory elements will eventually behave in a macrospin manner. Based on these viewpoints, we investigated STT switching in fully perpendicular magnetic tunnel junction systems, wherein addition to the Slonczewski STT term (sometimes calledparallel or in-plane torque since it lies in the plane defined 1098-0121/2015/92(10)/104430(9) 104430-1 ©2015 American Physical SocietyA. A. TIMOPHEEV et al. PHYSICAL REVIEW B 92, 104430 (2015) by the local magnetization and that of the spin polarization usually defined by the magnetization direction of the referencepinned layer), having a damping-like structure, an out-of-plane(also called field-like or perpendicular) term exists. Severaltheoretical papers predicted that the torque produced by theout-of-plane STT term could reach an amplitude comparableto that of in-plane torque [ 27–29]. Several experimental papers carried out on in-plane MTJ structures have alreadyestimated it to be in the range of 30–40% of the in-planetorque [ 30–33]. It was mentioned [ 34] that its presence may lead to a backswitching process, a very undesirable effect inmagnetic memory applications causing write errors. In this paper, after having analyzed the Landau-Lifshitz- Gilbert-Slonczewski equation mathematically transformedinto Landau-Lifshitz (LL) form, we show that in fullyperpendicular MTJ structures, the field-like torque plays anegligible role in the switching process. In contrast to in-planeMTJ systems [ 30–34], it only influences the precessional frequency preceding the switching, but the switching currentdensity is primarily determined by the in-plane STT term.The experiment carried out on 36-nm-diameter pMTJ pillarsupports our conclusions. II. PHASE BOUNDARIES FROM LLG EQUATION TRANSFORMED INTO LL EQUATION The most accepted form of the LLG equation describing dynamics of a macrospin under constant spin-polarized currentcan be presented as dˆm dt=−γ(ˆm×/vectorHeff)+α/parenleftbigg ˆm×dˆm dt/parenrightbigg −γˆm×(ˆm×a/bardblˆp)+γˆm×a⊥ˆp, (1) where ˆm=/vectorM MSis the unit vector along the free layer magnetization direction (in which M Sis the free layer’s volume magnetization saturation parameter), /vectorHeffis the effective field (comprising applied field, anisotropy field, and demagnetizingfield), ˆpis the unit vector along the magnetization direction of the polarized layer, αis Gilbert damping, γis the gyromagnetic ratio, and a /bardblanda⊥are in-plane (damping-like) and out-of- plane (field-like) STT prefactors, respectively. Both prefactorscan be phenomenologically represented as functions of spinpolarization in the magnetic electrodes, current density, orvoltage bias applied to the tunneling barrier, as will be donelater in the text. In-plane and out-of-plane STT terms as written in Eq. ( 1) are geometrically equivalent to the precession and dampingterms of the LL equation. One can therefore transform Eq. ( 1) into LL form using the standard technique, i.e., by making an ˆm×product on both sides of the equation, ˆm×dˆm dt=−γˆm×(ˆm×/vectorHeff)+αˆm×/parenleftbigg ˆm×dˆm dt/parenrightbigg −γa/bardblˆm×[ˆm×(ˆm׈p)]+γa⊥ˆm×(ˆm׈p) FIG. 1. (Color online) Geometry of the fully perpendicular MTJ system. and replacing the damping term in Eq. ( 1) with the result. This yields, (1+α2) γdˆm dt=− ˆm×[/vectorHeff−(a⊥+αa/bardbl)ˆp] −ˆm×{ˆm×[α/vectorHeff−(αa⊥−a/bardbl)ˆp]}.(2) To this moment, all the transformations born only a character of mathematical identities, and Eq. ( 2)i sv a l i df o r any system with any configuration of /vectorHeffand ˆp. Rewritten in such a way, it acquires a more suitable form for furtheranalytical treatment because dynamics in this system are fully determined by two vectors, namely [ /vectorH eff−(a⊥+αa/bardbl)ˆp] and [α/vectorHeff−(αa⊥−a/bardbl)ˆp], which have many similarities and whose form can be significantly simplified as soon as the geometry of /vectorHeffand ˆphas been set. Also, the use of Eq. ( 2) is more convenient in numerical integration schemes. Furtheranalysis will focus on the case of the pMTJ structure assumingmacrospin dynamics of the storage layer described by Eq. ( 2). We consider fully perpendicular magnetic tunnel junctions submitted to an out-of-plane external magnetic field /vectorH ext and, therefore, applied parallel to the symmetry axis. This situation allows analytical analysis wherein the quantities /vectorHeff,ˆp,/vectorHext,ˆzremain collinear independently of the in- stantaneous direction of ˆm.The magnetic free energy density functional U of such a system depends only on one variable,θ,the angle between magnetization vector ˆmand quantization axis ˆz(see Fig. 1) and is written: U=/parenleftbig K ⊥−2πM2 S/parenrightbig sin2θ−MSHextcosθ. (3) When |Hext|<H ⊥,H⊥=2K⊥ MS−4πM S, and Hext= /vectorHext·ˆz, there are two stable magnetic moment orientations 104430-2RESPECTIVE INFLUENCE OF IN-PLANE AND OUT-OF- . . . PHYSICAL REVIEW B 92, 104430 (2015) independent of Hextand always collinear with ˆz: ∂U ∂θ=0,∂2U ∂θ2>0, −H⊥<H ext<H ⊥, H⊥>0.→θ0=0,θ 0=π. (4) The collinearity of the four vectors /vectorHeff,ˆp,/vectorHext,ˆzgreatly simplifies Eq. ( 2), allowing it to work only with the magnitudes a⊥,a/bardbl,andHeff: (1+α2) γdˆm dt=− ˆm×Aˆz−ˆm×(ˆm×Bˆz), A=Heff−(a⊥+αa/bardbl), B=αH eff−(αa⊥−a/bardbl), Heff=/vectorHeff·ˆz=−∂U ∂/vectorM·ˆz=H⊥/parenleftbig cosθ0+Hext/H⊥/parenrightbig .(5) Here, two scalar parameters AandBare introduced, which represent the direction and magnitude of the perpendicular andin-plane (the plane is formed by ˆmand ˆp) effective torques (see Fig. 1) acting on magnetization when the latter departs from its equilibrium position θ 0(0 orπ) because of thermal fluctuations. An important specific of the considered system is that the Aparameter cannot change orbit (i.e., the angle θ); rather, it only influences the frequency of the precession. One can derivethe ferromagnetic resonance (FMR) condition, which is just amodified “easy axis” Kittel formula for this case: ω/γ=H eff−(a⊥+αa/bardbl), (6) where ωis the angular frequency of the resonance precession. One can see, that if a⊥>H eff−αa/bardbl,the precession direction will be changed, while an increase or decrease of θis exclusively determined by the sign of the Bparameter, wherein the damping-like STT term is dominating since α is usually small (typically in the range 0.007–0.02). Theprecessional response of the system before the switching couldbe measured—for instance, by measuring ωversus the dc applied voltage bias V biason a single pMTJ pillar either by rf voltage frequency detection, noise measurements [ 35], STT experiments, or microfocused Brillouin light scattering (BLS)FMR technique. The excitation frequency would give accesstoa ⊥(Vbias) dependence, while the FMR linewidth parameter change versus Vbiaswould reflect mostly a/bardbl(Vbias) dependence. Turning back to the analysis of Eq. ( 5) and Fig. 1, one can note that only the damping term, ˆm×(ˆm×Bˆz), can change the precession angle θ. It is therefore possible to derive the boundary conditions for a current-magnetic field stabilityphase diagram. The magnetization switching process startswhen the Bparameter changes sign. This condition yields the threshold criterion for the STT-induced magnetizationswitching: αH eff+a/bardbl−αa⊥=0. (7) One can see from Eq. ( 7) that the contribution from the in-plane STT term ( a/bardbl) is largely dominating the switching process. Indeed, the in-plane torque is on the order of αH eff while the contribution of the perpendicular torque is weightedby the Gilbert damping, resulting in a much weaker influence in the switching process. Here one can note again that thebest method to determine a ⊥experimentally is through FMR measurements, and not from the influence of a⊥on the (current, field) phase diagram boundaries since the latter isvery weak. Indeed, from the above discussion, being able tosee an influence of a ⊥on the phase diagram boundaries would require a⊥≈a/bardbl/αwhich seems to be physically unachievable in standard pMTJ systems [ 27–34]. Also, as will be shown in Sec. VI,t h eαa⊥term in Eq. ( 7) disappears if one chooses the dissipation term in the LL formulation. In any case, Eqs. ( 6,7) are quite useful for the analysis of STT switching experimentsperformed on pMTJ systems. III. STABILITY PHASE DIAGRAM BOUNDARIES Having set the relations between electric current flowing through pMTJ and the STT prefactor magnitudes, one canconstruct the stability phase diagram explicitly from Eq. ( 7) assuming that the spin-polarized current pulse is long enoughto complete any STT-induced switching while influence ofthe thermal fluctuations is limited to setting a small initialmisalignment angle θ 0so that |cosθ0|≈1.Modification of the phase boundaries due to thermal fluctuations andunder a short-pulse writing regime, which are essential inreal magnetic memory applications, will be analyzed in thefollowing sections, while in this section, the conditions oflong-pulse and low-temperature regime are assumed. In most investigated pMTJs, one can expect the condition a ⊥<a/bardblandαa⊥/lessmucha/bardblto be fulfilled. In this case, one can set a⊥=0 and build up the boundaries of the (current, field) sta- bility phase diagram. In absence of the spin-polarized current(a /bardbl=0,a⊥=0), switching occurs when αH effchanges sign, i.e., when Hext=−H⊥forθ0=0 andHext=H⊥forθ0=π. This defines the vertical boundaries of the diagram shown inFig. 2(a), depicted by dashed vertical lines. For H ext=0 and by setting a/bardbl=st/bardblGpVbias[where st/bardbl=/planckover2pi1 2e·η tFMS=STT is the conversion efficiency factor in units of Oe/(A cm−2);ηis the effective spin polarization parameter; and Gpis the tunneling conductance factor, generally dependent on θandVbias, in units of/Omega1−1cm−2, representing in the simplest interpretation the inverse of the R ×A product], one can obtain that the switching current density Iswis proportional to αH⊥: Isw0=GpVsw0=αH⊥ st/bardbl=2e /planckover2pi1·tFαH⊥Ms η. (8) In the case of Hext/negationslash=0,relation ( 8) leads to a linear depen- dence between the switching current and external magneticfield, yielding a linear slope on the switching phase diagramgiven by dI sw dH ext=α st/bardbl=2e /planckover2pi1·tFαM s η. (9) One can conclude that if the effective spin polarization parameter ηis constant (i.e., weakly dependent on the bias voltage Vbias), then the STT-driven parts of the switching dia- gram are linearly dependent on the applied field, with the slope 104430-3A. A. TIMOPHEEV et al. PHYSICAL REVIEW B 92, 104430 (2015) FIG. 2. (Color online) (a) Stability phase diagram constructed from Eq. ( 7) assuming a/bardbl=st/bardblGpVbiasanda⊥=0; (b) modification of the phase boundaries for the same a/bardblprefactor ( a/bardbl=st/bardblGpVbias,st/bardbl=67G−1 pOe/V) and different forms of a⊥prefactor: solid line a⊥=0; circles a⊥=st⊥2(GpVbias)2withst⊥2=154G−2 pOe/V2;d a s h e dl i n e a⊥=st⊥1GpVbias+st⊥2(GpVbias)2withst⊥1=500G−1 pOe/Va n d st⊥2= 10 000 G−2 pOe/V2.Other system parameters are α=0.05 and H⊥=200 Oe. proportional to the intrinsic damping parameter αand inversely proportional to the STT efficiency prefactor st/bardbl, and with the zero-field switching current magnitude being proportional tothe effective perpendicular anisotropy H ⊥. One should also note that Eq. ( 8) is in full agreement with previously obtained expressions [ 13–15,36] for the zero-field-threshold switching current derived from analysis of the precessional response ofthe system, assuming linear dependence of the damping-likeSTT prefactor versus the applied current. In our case, Eq. ( 7) allows one to calculate I-Hstability phase diagram boundaries for any a /bardbl,a⊥prefactors with arbitrary bias current (voltage) dependence, or by choosing it from the theoretical estimationsmade for the concrete MTJ system [ 28,29]. Simultaneous influence of both in-plane and out-of-plane STT terms on the phase boundaries is shown in Fig. 2(b). We have chosen realistic values for the magnetic system (seethe figure caption), letting the in-plane prefactor be linearlydependent on bias voltage with s t/bardbl=67G−1 pOe/V. A s f o r the out-of-plane prefactor a⊥,we show three different cases: zero, quadratic dependence with st⊥2=154G−2 pOe/V2, and quadratic +linear dependence (which mimics features ofan asymmetric MTJ structure; see the expression in the caption of Fig. 2) with unreasonably large STT conversion coefficients. One can see that within the difference betweenthe phase boundaries in all three cases is negligible. Thesecond case uses exactly the same parameters as those inRef. [ 15]i nF i g . 3. We can see that the boundaries calculated and simulated there are identical to our three cases, nomatter what prefactor dependence is introduced −H ⊥< Hext<H ⊥,for the out-of-plane STT term. This confirms that the out-of-plane STT term has a negligible influence onthe STT switching diagram. Parabolic shape of boundariesstarts being observed only in the third case, and it becomesnoticeably different only for current magnitudes several timeslarger than the threshold switching current. Thus, one canconclude that under long-pulse/low-temperature conditions,STT switching in fully perpendicular MTJ structures obeyingmacrospin dynamics is almost not influenced by the out-of-plane STT term and by its prefactor bias voltage or currentdependence. Below, we will show that this statement isstill valid at finite temperature and reasonably short writingpulses. FIG. 3. (Color online) Finite writing pulse phase diagrams for different in-plane and out-of-plane STT prefactor magnitudes: (a) T=0K ; (b)T=300 K .The model parameters are H⊥=200 Oe ,g=2.20 (g-factor), α=0.01. Integration time was 1 μs in each field point, and the writing pulse width is 40 ns. Each diagram is an average of 10 identical simulations. 104430-4RESPECTIVE INFLUENCE OF IN-PLANE AND OUT-OF- . . . PHYSICAL REVIEW B 92, 104430 (2015) IV . MACROSPIN SIMULATIONS Aiming at extending the conclusions made in the previous sections to the case of finite temperatures and finite writ-ing pulse regime, a series of macrospin simulations wereperformed using Eq. ( 2) (i.e., with Gilbert damping). The simulations were carried out with a fixed writing pulse duration of 40 ns and a cumulative integration time of 1 µs for each field point. The following assumptions of bias voltage dependencesfor the STT prefactors were used: a ⊥=st⊥2G2 pV2 biasanda/bardbl= st/bardblGpVbias, which is the case of symmetrical MTJ systems with a high spin polarization parameter. For convenience, theparameter G pwas set constant and equal to 1 /Omega1−1cm−2.The temperature was included in the form of a stochastic thermalfield H thwith Gaussian distribution [ 37], added directly to the effective field Heff.Statistical properties of these thermal fluctuations are given by the following relations: /angbracketleftHth,i(t)/angbracketright=0 and /angbracketleftHth,i(t)Hth,j(t/prime)/angbracketright=2αkBT γM SVpδijδ(t−t/prime) where kBis the Boltzmann constant, and Vpis the free layer volume. The chosen LLG equation is inte-grated with a (predictor-corrector) Heun scheme [ 38]. Here we used V p=2.07×10−17cm3,H⊥=200 Oe, Ms= 1000 emu /cm3, which gives the effective stability factor at T=300 K: /Delta1=H⊥MSVp 2kBT=50. This set of the parameters was chosen to mimic working conditions of an actual STT-MRAM device. Two sets ofmacrospin simulations, at T=0 K and T=300 K, respec- tively, presented in Fig. 3show how the phase boundaries are changed for the different combinations of in-plane andout-of-plane STT prefactor magnitudes. We will discuss firstthe results shown in Fig. 3(a) corresponding to the case with finite pulse duration and no thermal fluctuations ( T=0K ) . The finite duration of the writing pulse brings two main ef- fects. First, the STT-driven boundaries are shifted toward muchhigher voltages (currents). Evidently, to achieve switchingwithin the considered finite time period, one has to apply higheramplitudes for the writing pulses. In the initial stage, when ˆm is almost collinear with the symmetry axis ˆz, the torque is very weak, which results in very slow STT-induced dynamics in the system. It is evident that in absence of thermal fluctuations, theswitching time from ˆm/bardblˆzinitially would be infinite for any spin-polarized current magnitude [ 13,14]. To avoid this in the T=0 K simulations, a small misorientation (0 .1 ◦) between ˆp andHextwas introduced in the system. The second effect is nonlinearities of the phase boundaries, which are seen even onthe diagrams with the in-plane STT term only. This effect islinked with a nonlinear dependence of time necessary for STTswitching versus the applied magnetic field. Both effects areentirely of dynamical nature, and their influence on the phaseboundaries can be theoretically described using the formalismdeveloped in Ref. [ 16]. Renormalization of the effective dynamic time allows one to link dependence between thecritical current, pulse width, and finite temperature. This alsowill be done in the next section, while here the discussion will be focused on a qualitative analysis of the relative contributionsof the in-plane and out-of-plane STT terms to phase boundariesshapes. One can see from Fig. 3(a) that the general behavior of the phase boundary modifications on the simulated phasediagrams under finite writing pulse regime is in agreementwith the conclusions made in the previous sections for the dc regime. For the case of s t⊥2=400G−2 pOe/V2andst/bardbl= 0G−1 pOe/V, the simulated phase diagram demonstrates a unidirectional STT switching due to quadratic dependence ofa⊥versus applied voltage. In other words, switching to the antiparallel configuration is possible only for st⊥2>0, st/bardbl=0.Zero-field ( Hext=0) STT switching voltage for this diagram is ±1.6 V. This voltage induces an effective STT field in the damping term of Eq. (2) of ∼1000 Oe, which is five times higher than the effective perpendicularanisotropy field H ⊥=200 Oe. At the same time, if one adds a relatively small damping-like prefactor st/bardbl=30G−1 pOe/V, it completely removes any apparent influence of the field-like STT term from the phase diagram, despite the huge valuechosen for its prefactor. When the effective contributions fromboth prefactors are comparable, the phase diagram acquires anoticeable asymmetry, as can be seen for the last two diagramsin the middle column. However, such a combination of s t/bardbland st⊥2already can be physically unrealistic. Figure 3(b) shows the same set of simulations made under T=300 K. Several temperature-induced effects are observed: (i) decrease of the coercive field showing thatthermally activated magnetization reversal takes place whenthe external magnetic field substantially lowers the effectivebarrier height in the system; (ii) shift of the voltage-driven partsof the boundaries toward lower switching voltages. Thermalfluctuations of the magnetic moment direction increases theprobability of launching STT switching thanks to a thermallyinduced misorientation between ˆmand ˆp. This increases the initial STT amplitude and substantially decreases theswitching time for a given writing pulse amplitude. Thisis consistent with earlier observations in STT-MRAM cellsand with theoretical expectation of a I c=Ic0{1−kBT /Delta1Eln(τ τ0)} dependence of switching current on the pulse duration underfinite temperature [ 39]. Therefore, Fig. 3(b) Indicates that the general features observed in the switching phase diagram at0 K [i.e., Fig. 3(a)] are conserved at finite temperature and illustrates again the negligible role of the out-of-plane STTterm in the switching process [see in particular the last columnin Fig. 3(b)]. V . EXPERIMENTAL MEASUREMENTS OF THE ( I-H) SWITCHING DIAGRAM In this section, the STT efficiency and other magnetic parameters of pMTJ pillars are directly extracted from themeasured diagram. Nominal 50-nm-diameter pMTJ pillarswere fabricated from an MTJ stack grown by magnetronsputtering. The stack contains a 1.7-nm-thick Co 20Fe60B20 free layer sandwiched between two MgO barriers. Magneti-zation saturation parameter of the free layer was measuredto be 1030 emu /cm 3.Current in-plane magnetotransport measurements (CIPTMR) yielded R ×A=5.7/Omega1μm2and 104430-5A. A. TIMOPHEEV et al. PHYSICAL REVIEW B 92, 104430 (2015) FIG. 4. (Color online) Experiment carried out on pMTJ pillar at room temperature applying 100-ns writing pulses. (a) Examples of magnetoresistance loops measured with zero writing pulses; (b) stability phase diagram; (c) extracted phase boundaries and their linear fittings. TMR=126%. The second MgO barrier was introduced to increase the perpendicular anisotropy of the free layer. It hasa negligible resistance-area (R ×A) product compared with the main tunnel barrier. The bottom fixed layer is a syntheticantiferromagnetic-based, perpendicularly magnetized multi-layer, and the polarizer material has the same compositionas the free layer. The metallic electrode above the secondMgO barrier is nonmagnetic. Experimentally, it was foundthat the pillar diameter slightly differs from its nominal valuedue to the nanofabrication technology (36 nm instead of 50 nmnominal). This was recalculated using the values of the lowresistive state ( R pp=5.6k/Omega1) of the magnetoresistance curve [Fig. 4(a)] and assuming that the R ×A value is preserved after the nanofabrication. Knowing the volume of the free layer inthe pillar V p,its room temperature coercivity, measurement time (∼1s ),and attempt frequency f0=1010s−1,one can recalculate the perpendicular magnetic anisotropy from theN´eel-Brown formula [ 37,40], H C(T)=H⊥/parenleftBigg 1−/radicalBigg 2kBTln(tmf0) MSH⊥Vp/parenrightBigg , (10) which gives H⊥=2.6 kOe and /Delta1=56. The phase diagram measured at room temperature is shown in Fig. 4(b). At each magnetic field point, a 100-ns writing pulse with fixed amplitude was applied to the pMTJ pillar.Subsequently, the resistance was measured under small dcbias current, and the next magnetic field point was set. Toreduce the stochasticity in the switching field values, themagnetoresistance loop was measured 15 times, and theiraverage was used for switching field determination. The sameprocedure was used for all writing pulse amplitudes, and thefinal phase diagram was constructed from these averaged mag- netoresistance loops. Magnetic field loop repetition frequency was 2 Hz. The extracted phase boundaries are shown in Fig. 4(c).T h e coercive field of the free layer is 940 Oe, and the coupling fieldwith the reference layer is only 11 Oe and is ferromagnetic. Thevoltage driven parts are linear and almost parallel to each other.To reduce the influence of small nonlinearities at the edgesof the boundaries, only the central parts (within the ±500 Oe region) were used in the fitting. The extracted slopes are 1 .27× 10 −4and 1.23×10−4V/Oe; their difference is within the fitting error. The zero-field switching voltages are 0.359 and0.385 V , respectively. The difference is most probably due to the small dc bias current used for the resistance measurements. The phase diagram shape is similar to those obtained from the theoretical analysis (Sec. III) and the simulations (Sec. IV) where the out-of-plane STT term is not dominating. For thissystem, we can choose the STT prefactor model a ⊥=0,a/bardbl= st/bardblGpVbias. It corresponds to the dc diagram shown in Fig. 2 whose boundaries are described by Eqs. ( 8,9). To recalculate thest/bardblparameter from the extracted diagram slopes, one first needs to remap the experimental finite temperature–finite writ-ing pulse diagram to that of the long pulse–low temperaturemodel case. Here, we will follow the formalism described inRef. [ 16]. Thermal effects in our case can be reduced to the regime of thermally assisted ballistic STT switching. In thisregime, the main role of thermal fluctuations is to increasethe probability of STT switching thanks to an increased initialmisorientation angle θ 0,|cos(θ0)|/negationslash=1. As already mentioned, STT switching dynamics starting from a tilted state reducesthe switching time τ,in agreement with [ 13,14]. The cone angle 2 θ 0, for which the equilibrium probability for the magnetic moment orientation distribution is 0.5, is determinedby thermal stability parameter /Delta1and applied magnetic field θ 0=(ln 2//Delta1)1/2(1+Hext/H⊥)−1/2,while the final angle, the extremum on the energy barrier θτ=arccos( −Hext/H⊥)( f o r θ0<π / 2), is determined by magnetic field (see Eq. (77) in Ref. [ 16]). Having defined the initial θ0and final θτangles of the STT-induced dynamics, one can calculate analytically theswitching time τ(see Eq. (58) in Ref. [ 16]): (i−1)τ τD=ln/parenleftbiggxτ x0/parenrightbigg −1 i+1ln/parenleftBiggi−1 i+1+x2 τ i−1 i+1+x2 0/parenrightBigg , (11) Here,x0=tanθ0,xτ=tanθτ,τD=(1+α2) αμ 0γH⊥,and, according to our formalism, i=Iτ sw/Isw0−Hext H⊥.Having calculated θ0=6◦andτD=9.9 ns and assuming α=0.02 [ 41] and writing pulse duration τ=100 ns, we recalculated Iτ sw(Hext) dependence from Eq. ( 11) (Fig. 5, blue line) and compared it with the Isw(Hext) dc diagram case (Fig. 5, circles) derived from Eqs. ( 8,9). One can conclude that 100-ns writing pulses are long enough to remove the effect of dynamical distortion of thephase boundaries. For the measured device of Fig. 5, we find τ τD=100.6,which is quite high. This makes it possible to work directly with the phase boundaries [Eqs. ( 8,9)] derived from 104430-6RESPECTIVE INFLUENCE OF IN-PLANE AND OUT-OF- . . . PHYSICAL REVIEW B 92, 104430 (2015) dc FIG. 5. (Color online) Finite pulse–finite temperature diagram boundary forτ τD=100.6 (blue for experiment) andτ τD=1.5 (red for simulations). The dots are the respective boundary obtained fromEqs. ( 8,9). Eq. ( 7). However, ifτ τD<10 (the writing pulse width in the experiment would be <10 ns) and/or θ0is too small, the phase boundary remapping procedure is necessary before furtheranalysis of the phase boundaries can be made. Indeed, inthe simulations shown in the previous sections, the respectivevalue of τ τDis 1.54. Therefore, the switching currents are much higher and the linear slope is different from that expected fromthe model. One also should notice that this formalism worksonly in high- /Delta1approximation. Therefore, the parts of the phase boundaries close to the regions where H extapproaches H⊥ should be removed from the analysis. From extrapolation of the voltage -driven boundaries to V=0 one can estimate H⊥∼2.8−3.1 kOe, which is slightly higher than the corresponding value extracted from Eq. ( 10) (2.6 kOe). Nevertheless, the obtained H⊥values are in quite good agreement considering these two values are derivedfrom very different physical phenomena (superparamagnetismvs STT switching). The spin-torque efficiency prefactor s t/bardbl can be directly determined from the experimental slope using Eq. ( 9):st/bardbl=162G−1 pOe/V.From this, assuming that Gp=1/R×A, the effective spin polarization parameter in the system can be derived as η=0.49. If one uses the measured TMR value to estimate the polarization factor,assuming that η=√ TMR (TMR +2)/[2(TMR +1)] [ 42] and TMR =1.26, this would yield η=0.44, which is close to the value extracted from the diagram boundary slope. Thezero-field switching current, recalculated using Eq. ( 8)f o r obtained values of H ⊥,st/bardbl,and known parameter α,gives Isw0=0.35GpV. Therefore, one can conclude that the experiments carried out in the 36-nm pMTJ system can be well described withinthe macrospin approximation and thermally activated ballisticregime of STT switching. The H ⊥,st/bardblparameters extracted from the phase boundaries of the Vbias-Hstability diagram are in good agreement with those extracted independently from themagnetoresistance loop and N ´eel-Brown model. It is worth noting that a macrospin behavior is not specific only to themeasured device but is a generally observed feature for pillarswith a nominal diameter <80 nm.VI. LANDAU vs GILBERT In this section, we emphasize an important issue naturally arising from the analysis carried out in the previous sections.If the STT terms are added directly to the LL equation [ 43], then instead of Eq. ( 2) (obtained with the Gilbert dissipation term [ 44]), the following modified equation is obtained: 1 γμ 0dˆm dt=− ˆm×/parenleftbigg1 1+α2/vectorHeff−a⊥ˆp/parenrightbigg −ˆm ×/bracketleftbigg ˆm×/parenleftbiggα 1+α2/vectorHeff+a/bardblˆp/parenrightbigg/bracketrightbigg . (12) Still preserving the main features and general behavior of STT switching in fully perpendicular structures, Eq. ( 10) forbids switching only by the out-of-plane STT term, incontrast to Eq. ( 2), where the [ αa ⊥ˆm×(ˆm׈p)] component allows the system to change its energy even if a/bardbl=0. That turns us to the still open discussion [ 45–52] of physical validity of Gilbert damping and Landau damping formulation in themagnetization dynamics equation. Although it is generallyclaimed that LL and LLG equations are mathematicallyequivalent, we can see a significant difference when the STTterms are added: the field-like STT term written in the LLequation is fully conservative , and it cannot change the system energy if Eq. ( 12) is chosen to describe the STT-induced dynamics. Leaving this fact “as is,” one should notice thatin numerical simulations, it is more common to use the LLform instead of the LLG form, and different ways to introduceSTT terms [i.e., explicitly into the LL equation (Eq. 12)o rv i a transformation of LLG +STT (Eq. 2)] can lead to significantly different results. Figure 6demonstrates this important issue by comparing examples of macrospin simulations using either LL or Gilbertdamping terms to describe dissipation during STT-inducedswitching. Here, we adjusted the relative magnitudes of thefield-like and damping-like STT prefactors to have comparablecontributions in the second part of Eq. ( 2), which is the LLG + STT case. As soon as the field-like STT prefactor is set tohave only a quadratic-bias voltage dependence (the case of asymmetrical tunnel junction), the produced torque always pullsthe free layer magnetization in the antiparallel configurationwith the fixed layer. The damping-like STT prefactor is set to belinear on the bias voltage, and therefore the torque direction is FIG. 6. (Color online) Two identical macrospin simulations of a stability phase diagram carried out at T=0 K: (a) using Eq. ( 2), LLG+STT; (b) using Eq. ( 12), LL+STT. STT prefactors: st/bardbl= 12G−1 pOe/Va n d st⊥2=400G−2 pOe/V2. Other parameters are the same as used for the simulations in Sec. IV. 104430-7A. A. TIMOPHEEV et al. PHYSICAL REVIEW B 92, 104430 (2015) determined by the current polarity. When a negative voltage is applied to the system, field-like torque helps the damping-liketorque switch magnetization in the antiparallel state. It shiftsthe phase boundary toward lower switching voltages. Howeverthe expected boundary shift is too small to be visible in oursimulations considering the chosen step for the voltage writingpulse amplitude. Also, a quadratic dependence of the field-likeSTT prefactor allows it to compete with the damping-liketorque only at relatively high writing pulse voltages. At thesame time, for positive pulses, field-like torque works againstthe damping-like torque, which shifts the phase boundaryto higher voltages. The higher the switching voltage, thehigher the relative contribution from the field-like torque.Finally, when the writing pulse is about 1.6 V , field-like torquecompensates the damping-like one, and further increase ofthe writing pulse amplitude starts shifting the phase boundaryback toward negative fields, decreasing the field window of thebipolar STT switching. The same effect is observed at finitetemperatures in Fig. 3(b) for the bottom middle diagram. This competition between the STT terms, however, is impossiblein simulations with the Landau damping term because theαa ⊥ˆm×(ˆm׈p) term is absent in Eq. ( 12). Finally, it is traditionally accepted that the LLG and LL equations are geometrically equivalent, and the mathematicaltransformation from one to another ends up with 1 1+α2 rescaling of the gyromagnetic ratio. This1 1+α2correction in real physical systems is very small and experimentallyundetectable. However, this is not the case anymore if theSTT terms are added to the LLG equation. The equationsare now different .The same transformation (i.e., LLG + STT→LL) leads to the appearance of two additional STT pseudo-torques [ αa ⊥ˆm×(ˆm׈p),α a /bardblˆm׈p], which are linearly proportional to the damping constant αand in principle can be detected experimentally. Experimentally, it should be possible to assess which formulation of damping is correct by measuring the variation ofthe precession frequency in the subswitching threshold regimein samples having various damping constants. Such samplescould be produced, for instance, by depositing a wedge ofPt above the storage layer before the patterning of the wafer.For this experiment, it would be preferable to use symmetricMTJs so that the field-like torque has a quadratic dependenceon bias voltage. If the LLG formulation is correct, we expecta linear variation of the frequency with damping constantunder fixed bias voltage, whereas if the LL formulation isvalid, no dependence of the frequency on damping should beobserved. VII. CONCLUSIONS It has been shown that the LLG equation with the field-like and damping-like STT terms transformed into the LL formconsiderably simplifies the analysis of the STT switchingprocess. In the case of a fully perpendicular MTJ system, theboundaries of the I-Hstability phase diagram can be obtained directly from the transformed Eq. ( 2). It was shown that the field-like term has negligible influence on the STT switchingprocess in pMTJs with low damping, influencing mainly theFMR precession frequency for the small oscillations near theequilibrium. Considering that in standard pMTJ structures its effective magnitude cannot be much higher than the magnitudeof the in-plane torque, it would be hard to track its bias voltage(current) dependence from experimentally measured stabilityphase diagrams. Measuring the bias voltage dependence ofthe frequency in the precessional regime would certainlybetter reveal the influence of the field-like STT term, butthe contribution of the field-like term still would have tobe separated from the nonlinear influence of the oscillationamplitude on the frequency. Finite temperature macrospin simulations in LLG-STT formalism under finite writing pulse duration have confirmedthe negligible role of the field-like term in the STT switchingprocess of pMTJ structure. Limitations of the macrospinmodel are not expected to be important in the case of pMTJpillars with diameters comparable to or below the exchangelength. This is confirmed by the experiments carried out on36-nm-diameter pMTJ pillars. One should note that the method developed for the phase boundaries construction gives the same results as thoseobtained from the analysis of dynamical response of thesystem carried out by different groups supposing the lineardependence of the damping-like STT prefactor versus appliedbias voltage. However, we believe that it will be more useful inthe interpretation of the experiment and simulations, becauseit is much more flexible, and it allows the introduction of anydesirable current (voltage) dependences for the in-plane andout-of-plane STT prefactors. Using the developed formalism, the spin-torque efficiency and effective spin polarization parameters have been derivedfrom the current field stability diagram boundaries experi-mentally measured on a 36-nm pMTJ pillar. The obtainedparameters have been cross-checked by estimations frommagnetoresistance curves and from the thermally activatedmagnetization reversal regime. Good agreement betweenthe values derived from the analysis of different physicalprinciples strongly supports the assumption of macrospinbehavior in the measured sample. We also showed that the different dissipation terms (i.e., LL or Gilbert) give rise to different analytical expressionsdescribing the phase boundaries of I-Hswitching diagrams, which can be important in heavily damped systems. If theLandau damping term is physically correct, the action of thefield-like and the damping-like torques in the pMTJ system iscompletely separated in precession and dissipation terms in theequation of dynamics. If the Gilbert damping term is correct,then two additional torques [ αa ⊥ˆm×(ˆm׈p) andαa/bardblˆm׈p] are mixed in with the main STT contributors [ a/bardblˆm׈pand a⊥ˆm×(ˆm׈p),respectively]. An experimental way to assess which damping formulation is correct in combination with STT was proposed. 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PhysRevB.93.224410.pdf

PHYSICAL REVIEW B 93, 224410 (2016) Phase locking of spin-torque nano-oscillator pairs with magnetic dipolar coupling Hao-Hsuan Chen,1Ching-Ming Lee,2,*Zongzhi Zhang,1,†Yaowen Liu,3,‡Jong-Ching Wu,4 Lance Horng,4and Ching-Ray Chang5 1Shanghai Ultra-Precision Optical Engineering Center, Department of Optical Science and Engineering, Fudan University, Shanghai 200433, People’s Republic of China 2Graduate School of Materials Science, National Yunlin University of Science and Technology, Douliou, 64002, Taiwan, 3School of Physics Science and Engineering, Tongji University, Shanghai 200092, People’s Republic of China 4Department of Physics, National Changhua University of Education, Changhua 500, Taiwan 5Department of Physics, National Taiwan University, Taipei 10617, Taiwan (Received 1 December 2015; revised manuscript received 17 May 2016; published 8 June 2016) A spin-torque nanopillar oscillator (STNO) that combines a perpendicular-to-plane polarizer (PERP) with an in-plane magnetized free layer is a good candidate for phase locking, which opens a potential approach toenhancement of the output power of STNOs. In this paper, the magnetic dipolar coupling effect is used as thedriving force to synchronize two STNOs. We develop an approximation theory for synchronizing two identicaland nonidentical pairs of PERP STNOs, by which the critical current of synchronization, dipolar couplingstrength, phase-locking transient time, and frequency can be analytically predicted. These predictions are furtherconfirmed by macrospin and micromagnetic simulations. Finally, we show the phase diagrams of the phaselocking as a function of applied current and separation between two STNOs. DOI: 10.1103/PhysRevB.93.224410 I. INTRODUCTION A spin-polarized current can be used to excite persistent magnetization oscillations in a nanomagnet through the spin-transfer torque (STT) effect [ 1,2]. Such STT-driven magnetic precession has attracted considerable attention because ofboth the fundamental interest for studying nanoscale magneticdynamics and the applications in the frequency tunablemicrowave oscillators [ 3], which can be used in telecom- munications, microwave signal processors, and microwavefield detectors [ 4–9]. The frequency of STT oscillators can be tuned by the strength of magnetic fields or current. However,the present output power of a single spin-valve spin-torquenanopillar oscillator (STNO) is typically in the range ofpicowatts to nanowatts [ 10,11], which is still too weak for any practical applications. Increasing output power of a STNO is essential for suc- cessful adaptation of the STT excitation scheme for advancedmicrowave oscillators. Several ways to enhance the outputsignals have been reported. For example, using a magnetictunnel junction (MTJ) to replace the spin valve element canincrease the output power to microwatt level [ 12–14]; using the perpendicular-to-plane magnetized layer as the spin polarizerof STNOs can excite large angle out-of-pane (OP) precessionfor enhancement of the power output [ 15–21]. Up to now, a single STT device has been significantly optimized, but theoutput power still cannot reach the required milliwatts. Another promising approach to increasing the emitted power has been suggested by using the phase-locking modeof an array of STNOs through the synchronization technique.This is a very challenging issue due to the strongly nonlinearproperty of STNOs [ 22,23]. A phase-locking experiment has *cmlee@yuntech.edu.tw †zzzhang@fudan.edu.cn ‡yaowen@tongji.edu.cnbeen reported in spin-torque devices with multiple nanocon- tacts [ 24–31], in which the magnetization in all the nanocontact regions can be locked in the same phase via propagatingspin-waves [ 32–34]. Alternatively, the phase locking by using the coupled electrical circuits has also been proposed in anarray of STNO nanopillars electrically connected in series or inparallel [ 35–37]. Recently, a third scheme using the magnetic dipolar coupling effect as the driving source of synchronizationhas been demonstrated in nanopillars that combines the out-of-plane magnetized polarizer and the in-plane magnetizedfree layer namely, perpendicular-to-plane polarizer (PERP)STNO [ 19,38,39]. Additionally, another kind of oscillator based on the spin Hall effect (SHE), i.e. spin Hall oscillators(SHOs), have been also reported recently [ 40–45]. Moreover, the synchronization of vortex-based nonuniformly magnetizedSTNOs in a horizontal array has also been reported [ 46–49]. Among these synchronization schemes, the scheme using themagnetic dipolar effect displays special features [ 38,47,48]: First, the dipolar coupling among STNOs with nanopillarstructure is an intrinsic property, so that it does not need anyother external sources such as external microwave field or aspecial design of resistor-inductor-capacitor (RLC) circuit toassist synchronization. Second, unlike the scheme employinga propagating spin wave [ 31], in which the phase-locking mode can be either in phase or antiphase, depending on theintercontact distance and current strength, the magnetizationphase-locking state induced by the dipolar interaction is verystable, and the antiphase mode is independent of the currentand separation between neighboring STNOs [ 38]. In this paper, we present a phase-locking scheme to synchronize two horizontally arranged PERP STNOs throughthe dipolar coupling effect of the free layers. The paper isorganized as follows: In Sec. II, we develop a theory for synchronizing two identical and nonidentical pairs of PERPSTNOs. The pairs are horizontally arranged. The sufficient andnecessary parametric conditions for the synchronization areaddressed, based on the assumption of strong demagnetization 2469-9950/2016/93(22)/224410(12) 224410-1 ©2016 American Physical SocietyCHEN, LEE, ZHANG, LIU, WU, HORNG, AND CHANG PHYSICAL REVIEW B 93, 224410 (2016) energy and the dipolar coupling approximation with a single domain model. We analytically predict the critical current,critical dipolar coupling strength, as well as the phase-lockingfrequency and phase-locking transient time. In Sec. III,w e perform both macrospin and micromagnetic simulations. Thesimulation results are consistent with our analytical predic-tions, by showing the time evolution of the phase difference,spectrum analysis of the synchronization oscillations, thecurrent range, and the separation distance between the twosynchronized STNOs. We also show parameter diagramsof phase locking. Finally, a brief summary and discussionsare given in Sec. IV. Appendix Aprovides details of the calculation of magnetic dipolar interaction between twocircular, uniformly magnetized discs, and Appendix Bpresents an approximation theory using a low-energy orbit to derive theNewton-like Eq. ( 5). II. MODEL AND THEORETICAL FRAMEWORK As a model system, we consider here a pair of STNOs shown in Fig. 1. The bottom layer is the spin polarizer layer (P) whose magnetization is fixed along the perpendicular-to-planedirection. The top layer is supposed to be etched down tothe nonmagnetic metal layer. The free layers (F1 and F2) ofthe two nanopillars are separated by an edge-to-edge distanced ee. We assume that the free layer has a quasiuniform in- plane magnetization due to its small size. A dc electric currentseparately flows from the bottom layer to the two free layersF1 and F2. We assume that the two pillars have the sameamount of injected current (each one has −I). When the current strength is larger than a critical value, the current-inducedSTT effect will drive the two free-layer magnetizations into a precessional state [ 15,21]. Owing to the magnetic dipolar interaction between the two free layers, the two STNOs canoscillate synchronously under certain conditions. The magnetization dynamics of the two free layers can be described by the Landau-Lifshitz-Gilbert (LLG) equationincluding the STT term [ 19,50–52] dm i dτ=−/parenleftbig ∇miG/parenrightbig ×mi+α/parenleftbigg mi×dmi dτ/parenrightbigg −aJ(mi)[mi×(mi×p)], (1) FIG. 1. (a) Sketch of a horizontal array containing two PERP STNOs. P denotes the spin polarizer layer (i.e. the fixed layer), and F denotes the free layer. (b) The unit vector mof free layer magnetization is illustrated in the polar coordinate representation(θ,φ).where the subscript i(=1,2) is used to distinguish the two nanopillars. Here, m=M/Msis the unit vector of the free-layer magnetization, Msis the saturation magne- tization, and τ=(4πMsγ)tis the scaled time, with γ= 1.76×107Oe−1·s−1being the gyromagnetic ratio. Also, G(m) is the total energy density of the free layer which has been normalized by 4 πM2 s. Further, αis the Gilbert damping constant. The third term on the right side ofEq. ( 1) is the STT term, in which pis a unit vector of the polarizer magnetization along the zdirection, and a J(mi)=AJ(mi)(4πMsγ)−1=aJ0ε(θi,Pi,/Lambda1i) is the scaled- down STT strength in which AJ(mi)=AJ0ε(θi,Pi,/Lambda1i)= (γ/planckover2pi1J/2eMsd)ε(θi,Pi,/Lambda1i). Here, Jis the injected current density, Pis the spin polarization, dis the free-layer thickness, andθis the angle between the magnetization vectors of the free layer and the polarizer layer. Also, ε(θi,Pi,/Lambda1i)= Pi/Lambda1i2/[(/Lambda1i2+1)+(/Lambda1i2−1) cos θi](i=1,2) is the angular dependence factor of the Slonczewski STT [ 1], in which Pand/Lambda1are dimensionless quantities which determine the spin-polarization efficiency. In the spherical coordinates ( θ,φ)[ s e eF i g . 1(b)], the total energy density G(m) is given by the sum of the demagnetiza- tion energy, uniaxial anisotropic energy, and magnetic dipolarinteraction energy G dem(θ1,θ2)=1 22/summationdisplay i=1m2 zi=1 22/summationdisplay i=1cos2θi, (2) Gu(θ1,φ1,θ2,φ2)=k 22/summationdisplay i=1m2 xi=k 22/summationdisplay i=1sin2θisin2φi, (3) Gdip(θ1,φ1,θ2,φ2)=Adisc(dee)[3(m1·r)(m2·r)−m1·m2] =Adisc(dee)[sinθ1sinθ2(sinφ1sinφ2 −2 cosφ1cosφ2) +cosθ1cosθ2], (4) where kis the uniaxial anisotropy constant, either a magnetic anisotropy or a shape anisotropy. The easy-axis of theanisotropy is along the x-axis direction. Here, A disc(dee)i s the strength coefficient of the dipolar field that describes themagnetostatic interaction effect between the two nanopillars,andd eeis the edge-to-edge separation distance. The vector ris a unit vector of the displacement between two magnetic dipoles. In order to improve the accuracy of our approximation,the dipolar interaction in Eq. ( 4) is treated as the interaction between two circular uniformly magnetized discs (for detailssee Appendix A). The strength A disc(dee) is more realistic than the point dipoles strength Apoint(dee) due to the finite size effect [ 53]. We find that, when the distance deeis smaller than 30 nm, then Adisc(dee) is significantly larger than Apoint(dee), see Appendix A. In order to get proper parameters and gain insight into the phase-locking behavior, an approximation theory is intro-duced here. We assume that the free-layer magnetization isapproximately suppressed in the easy plane with θ i∼π/2 due to the strong demagnetizing field. In this case, thesystem executes low-energy orbits, and the total magneticenergy density of the system can be approximately writtenas|G|∼k>|G dip|∼Adisc(dee). These low-energy orbits 224410-2PHASE LOCKING OF SPIN-TORQUE NANO-OSCILLATOR . . . PHYSICAL REVIEW B 93, 224410 (2016) FIG. 2. Effective potential energy Geff(φ+,φ−) for the (a) and (b) identical STNO pair and (c) and (d) nonidentical STNO pair at different uniaxial anisotropy kand different current I. Three cross-sections taken at φ+=0,π/2,πare shown in the corresponding bottom panels of each figure. Here, dee=20 nm, and Adisc(dee)=0.002. All parameters are marked in the figures. satisfy [ θi(τ),φi(τ)]=[π/2+δθi(τ),φi(τ)](i=1,2), where |δθi|∼√ k. If the damping constant and the STT strength satisfy Eq. ( B13), then Eq. ( 1) can be rewritten as a pair of Newton-like equations (see Appendix Bfor details) ¨φ1+α˙φ1=Adisc(dee)[−sin(φ1+φ2)−sinφ1cosφ2] −k 2sin 2φ1+aJ1/parenleftbigg θ1=π 2/parenrightbigg , (5a) ¨φ2+α˙φ2=Adisc(dee)[−sin(φ1+φ2)−cosφ1sinφ2] −k 2sin 2φ2+a21/parenleftbigg θ2=π 2/parenrightbigg , (5b) where the effective force is dominated by the dipolar interac- tion term (the first term of the right-hand side), the uniaxialanisotropy (the second term), and the STT term (the thirdterm). For simplicity, by using a new set of variables formedby the phase sum φ +=φ1+φ2and the phase difference φ−=φ1−φ2, we rewrite Eqs. ( 5a) and ( 5b)a s d2φ+ dτ2+αdφ+ dτ=−∂G eff(φ+,φ−) ∂φ+, (6a) d2φ− dτ2+αdφ− dτ=−∂G eff(φ+,φ−) ∂φ−, (6b) where the effective potential energy is now given by Geff(φ+,φ−)=Adisc(dee)(−3 cosφ+−cosφ−) −kcosφ−cosφ++aJ+φ++aJ−φ−.(7) Here,aJ+≡aJ1+aJ2andaJ−=aJ1−aJ2.A. An identical STNO pair According to the design shown in Fig. 1, the STT strength is the same for two identical PERP STNOs, thatis,a J−=0 and aJ+=2aJ.F r o mE q .( 6a) with Eq. ( 7), by setting |∂G eff/∂ φ+|>0, we obtain |aJ+|>3Adisc(dee)+k, and under this condition, all equilibria of Geffalong the φ+-axis direction are eliminated (Fig. 2). This condition indicates that there exists a critical STT strength (or critical current) to drivethe two STNOs into a steady OP precessional state |a J1,c|=|aJ2,c|=(1/2)|aJ+,c|=(1/2)[3Adisc(dee)+k]. (8a) Atdee=20 nm, we have Adisc(dee)=0.002 (Fig. 10in Appendix A). In the absence of uniaxial anisotropy (i.e. k=0), we further get the critical STT strength aJ+,c=0.006 and the current Ic=0.29 mA. Similarly, in the presence of uniaxial anisotropy ( k=0.008), we have aJ+,c=0.014 and Ic=0.68 mA. Note that, when the current is larger than the critical value given by Eq. ( 8a), the two STNOs can be driven into a precessional state, but the precession may not be synchronous.Therefore, the synchronization or phase-locking state requiresadditional conditions. From Eqs. ( 6b) and ( 7), by setting ∂G eff/∂(φ−)=0 and ∂2Geff/∂(φ−)2>0, the condition for φ−=0 as the only stable equilibrium point in the range of φ+∈[0,2π] can be derived Adisc(dee)>k . (8b) We would like to emphasize that Eq. ( 8b) guarantees that the two free layers of the coupled system always evolve intoa final state with a stable phase difference beginning with anarbitrary initial state. If Eqs. ( 8a) and ( 8b) are simultaneously satisfied, Eqs. ( 6a) and ( 6b) can be reduced to a single equation 224410-3CHEN, LEE, ZHANG, LIU, WU, HORNG, AND CHANG PHYSICAL REVIEW B 93, 224410 (2016) of motion ¨φ+α˙φ=− (1/2)[3Adisc(dee)+k]s i n2φ−aJ, (9) where φ≡(1/2)φ+=φ1=φ2andaJ≡(1/2)aJ+=aJ1= aJ2. It should be noticed that Eq. ( 9) has the same form as Eq. ( 5) for a single oscillator, but the anisotropy energy is enhanced by including the dipolar coupling term 3 Adisc(dee). Therefore, we conclude that Eqs. ( 8a) and ( 8b)a r et h e necessary and sufficient conditions for the phase locking ofmagnetization precession of two nano-oscillators. On the otherhand, note from Eqs. (8) and ( 8b) that the anisotropy kcan raise the threshold values of the dipolar strength A disc, and of the critical spin-transfer strength aJcas well. This analytical result suggests that the reduction of anisotropy is a possible way toreduce the critical current as well as to enhance the stability ofa phase-locked array of STNOs. In order to obtain a qualitative insight into Eq. (8), we regard the dipolar coupled STNOs pair as an effective Newton-likeparticle moving on the energy surface G eff.A ss h o w ni n Fig. 2(a), in the absence of uniaxial anisotropy when the current I=0.34 mA (larger than Ic=0.29 mA at k=0), the energy surface will be tilted along −φ+direction by the sum of the STT strengths aJ+. Because there are no stable equilibrium points along the φ+axis, the particle will move downward along the −φ+direction with an average terminal velocity |/angbracketleft˙φ/angbracketright|τ=|aJ+|/α. Furthermore, the dipolar coupling creates stable equilibrium points at φ−=0 on the energy surface. The barrier height between the local equilibria alongtheφ −axis is Adisc(dee), as is shown in the lower panel of Fig.2(a). Due to energy dissipation, the particle will eventually move downward along the ditch from any initial state withthe average terminal velocity |a J+|/α, indicating that the two STNOs precess in phase. In the presence of uniaxialanisotropy, as shown in Fig. 2(b), besides the elevation of the critical current, also the cross-section shape of the ditchon the energy surface is changed with φ +[see the lower panel of Fig. 2(b)], meaning that uniaxial anisotropy is certainly detrimental to phase locking. B. A nonidentical STNO pair Now we consider two nonidentical PERP STNOs. The non- identical property may be caused by asymmetric STT strengths(that is, a J−/negationslash=0) or by other parameters (for example, shape difference). Similar to the identical case, by analytically setting|∂G eff/∂ φ+|>0,∂G eff/∂(φ−)=0, and ∂2Geff/∂(φ−)2>0, one can obtain the phase-locking conditions as the followingform: |a J+|>3Adisc(dee)+k, (10a) Adisc(dee)>k , (10b) and |aJ−|<A disc(dee)−k. (10c) Here, Eq. ( 10c) guarantees the difference of STT strengths is not so strong as to destroy the phase-locking state.Additionally, from the Eq. ( 10a), we can estimate that the critical currents I cin the absence and presence of the energy kare 0.26 and 0.62 mA, respectively. Assuming that the edge-to-edge distance deeis approxi- mately 30 nm or less, the corresponding dipolar interactionstrength Adisc(dee) does not easily satisfy the condition of Adisc(dee)>k. This circumstance is due to the fact that, for the given value k=0.008, the value of Adisc(dee) is, according to Fig. 10, smaller than kunless the separation deeis decreased down to 5 nm. By inserting Eq. ( 7) into Eq. ( 6b), we now obtain d2φ− dτ2+αdφ− dτ=− sin(φ−)[Adisc(dee)+kcos(φ+)]−aJ−, (11) The first term on the right-hand side of Eq. ( 11)i st h e restoring force, in which the uniaxial anisotropy kactually contains a prefactor rapidly varying in time cos( φ+). This is because the φ+varies faster than the growth of the phase difference φ−when|aJ−|is smaller than the dipolar interaction strength Adisc(dee). Thus, the terminal velocity of the phase difference |˙φ−|<A disc(dee)/αmust be smaller than that of the phase sum |˙φ+|>(3Adisc(dee)+k)/α. Here, the perturbation of anisotropy koscillates very fast compared to the phase difference change, so that the perturbation can be omitted bythe approach presented below [ 54]. Taking a time average of φ −in Eq. ( 6b) over a period of /Delta1T=2π/|˙φ−|=2πA disc(dee)/α, one can easily find that the contribution of the time-varying part in φ−becomes close to zero, i.e. /angbracketleftcos(φ+)/angbracketright/Delta1T≈0. Therefore, the right-hand side of Eq. ( 6b) takes on the form −Adisc(dee)s i n (φ−)−aJ−. As a consequence, a soft phase-locking condition for thenonidentical pair of PERP STNOs is obtained |a J+|>3Adisc(dee)+k, (12a) |aJ−|<A disc(dee). (12b) These two Eqs. ( 12a) and ( 12b) are supported by numerical solutions of Eqs. ( 6a) and ( 6b). When the above conditions are satisfied, Eq. ( 6a) for phase sum φ+is rewritten as d2φ+ dτ2+αdφ+ dτ=− [3Adisc(dee)+kcosφ+0]s i nφ+−aJ+, (13) in which φ−0is the stable, nonzero phase difference. Accord- ingly, the phase-locked angular velocity is given by |˙φ1|=|˙φ2|=1 2/vextendsingle/vextendsingle/vextendsingle/vextendsingled(φ +) dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle=1 2α|aJ+|. From the viewpoint of the Newton-like particle, in the absence of uniaxial anisotropy k=0, when the current I= 0.69 mA is larger than the critical value 0.26 mA, the energy surface will not only be tilted along the −φ+direction by the sum of the STT strengths aJ+, but will also be tilted along the +φ−direction by the STT strength difference aJ−,a ss h o w n in Fig. 2(c). Thus, the position of the ditch created by dipolar coupling is shifted by aJ−slightly away from φ−=0, and the barrier height between the local equilibriums along the φ− axis is smaller than Adisc(dee), implying that the phase-locking ability of dipolar coupling is weakened by negative aJ−values [lower panel of Fig. 2(c)]. Due to the energy dissipation, the particle will eventually move downward along the new ditchfrom any initial state, meaning that the two STNOs precesswith a small phase difference. However, similar to the identicalcomponents case, the uniaxial anisotropy still changes thestability of the local equilibrium points. 224410-4PHASE LOCKING OF SPIN-TORQUE NANO-OSCILLATOR . . . PHYSICAL REVIEW B 93, 224410 (2016) III. NUMERICAL SIMULATIONS: RESULTS AND DISCUSSION In order to verify the analytical model, both macrospin and full micromagnetic (FMM) simulations have been performedfor a coupled PERP STNO pair with dipolar magneticinteraction. In this section, we will show the time dependenceof the phase difference φ −and the inclination angle θ,t h ex andzcomponents of the precessional magnetization, and the spectrum analysis of magnetization oscillation. Furthermore,the critical conditions for triggering phase locking of magneti-zation with a minimum current Iand a maximum edge-to-edge distance d eewill be discussed. The simulated parameter ranges for phase locking will be compared with the results from theapproximate theory. In this paper, both macrospin and FMM simulations are conducted. The macrosopin code is developed in our groupindependently, and the micromagnetic simulations are carriedout by using two open micromagnetic codes, the finiteelement package magpar [ 55] and the finite difference package MuMax3 [ 56]. In these simulations, we assume that the STNOs have an elliptical shape with size 70 ×60 nm, and that the free-layer thickness d=3 nm. For simplicity, we only focus on the magnetization dynamics of the free layers. The initialmagnetization state is aligned along the xaxis (long axis of the sample) direction. The thickness of the free layer is 3 nm.Typical material parameters are used for the Co free layer[16]: 4πM s=1.09×104Oe (saturation magnetization), k= 0.008 (in-plane uniaxial anisotropy), A=2.5×10−11Jm−1 (exchange stiffness constant), α=0.02 (Gilbert damping constant). The discretization cell size for MuMax3 is set at1×1n m×3 nm, while the magpar average size of tetrahedron mesh is 2 nm. The spin polarization of the left STNO is setto be P 1=0.38 and /Lambda11=1.8. The right STNO is given byP2=0.44,/Lambda1 2=2. Without dipolar interaction, the two STNOs have different oscillation frequencies (we will showthis later) due to different spin polarizations. In this paper, the current-induced Oersted field [ 57]i s ignored. Our calculations indicate that the maximum valueof the Oersted field created by a current of 0.7 mA is ∼40 Oe located in the perimeter zone of an isolated nanopillar (notshown). This is a reasonable estimation for considering theOersted fields created by current as an infinite wire. TheOersted field is therefore much smaller than other fieldssuch as the in-plane uniaxial anisotropy field ( ∼170 Oe) and the demagnetizing field ( ∼1.09×10 4Oe). For a pair of nanopillars horizontally arranged with an edge-to-edgeseparation changing from 5 to 20 nm, the calculated Oerstedfield is further reduced down to ∼25 Oe due to the cancellation between the two STNOs. For this reason, the Oersted field isignored in this paper. A. Synchronization of an STNO pair: Phase-locking state First, a phase-locking state is obtained both from the analytical theory and simulations: The injected current is setto be I=0.8 mA. Using the above parameter values and θ 1=θ2=π/2, we analytically obtained the STT difference |aJ−|=|aJ1−aJ2|=0.0017, which is smaller than the value ofA(dee)=0.002, meaning that the analytical condition forphase locking shown in Eq. ( 12b) is satisfied. Numerically, both the macrospin and micromagnetic simulations with theseparameters indicate that the phase-locking magnetization statecan be achieved within several nanoseconds, as shown in theupper panels of Figs. 3(a) and 3(c). In this paper, the total simulation time is 50 ns. In order to show clearly the transientbehavior, the time scales in Fig. 3are confined to the initial several nanoseconds. The phase-locking state has a small phasedifference φ −=0.23 rad/πin the macrospin simulation and 0.34 rad/ πin the micromagnetic simulation. This nonzero φ− corresponds to the position shift of equilibrium points [see Fig. 2(b)], caused by aJ−. In addition, the phase-locking state of the inclination angle θof the two STNOs has also been achieved in both simulations, as shown in the lower panel ofFig.3(a). This phase locking of θcan also be inferred from the locking of ˙φ, according to the conjugacy between the variables ofφandθ[see Appendix B,E q .( B5)]. In other words, when the locking of φoccurs (that is ˙φ 1=˙φ2),θ1must be equal to θ2. Similarly, the phase locking can also be clearly seen from the magnetization plot in Cartesian coordinates, as shown inFig. 3(b). From Figs. 3(a)–3(c), one can see that there exists a transient state before the STNO pair synchronizes into a stable phase-locked state. The typical time order of the transient state can betheoretically estimated from the Newton-like motion equationof Eq. ( 6b). As mentioned before, the uniaxial anisotropy kin Eq. ( 6b) contains a fast time-varying prefactor cos( φ +) which makes it possible to neglect k.Therefore, Eq. ( 6b) can be linearized close to the equilibrium point φ−∼0. For a small angleδ(φ−), we have d2δ(φ−) dτ2+αdδ(φ−) dτ=− {/radicalBig [Adisc(dee)]2−(aJ−)2}δ(φ−). The general solution δ(φ−)(τ) of this equation has a decay factor e−(α/2)τ, in which τcan be defined as the time order of the transient state. For example, if τ=460 for α=0.02 then e−(α/2)τ∼1%. Furthermore, the real physical time tcan be easily derived from the relation of τ=(4πMsγ)t. This yields the transient time t=2.4 ns which is in good agreement with our simulation results shown in Fig. 3. Figure 3(d) shows the phase difference of the final state φ−=φ1−φ2as a function of the injected current I,including both the prediction curve of the approximate theory (blackline) and the simulation curves from macrospin (blue solidsquares) and micromagnetic (red open squares) calculations.From the approximate theory, the stable phase differencefor the softer (without k) phase-locking condition satis- fies−A disc(dee)s i n (φ−)−aJ−=0 and |aJ−/Adisc(dee)|<1. Therefore, the relationship between φ−andI, and the max- imum injected current for phase locking can be analyticallyobtained φ −=sin−1/bracketleftbigg −/parenleftbigg/planckover2pi1 8πeM2 SV/parenrightbigg/Delta1ε Adisc(dee)I/bracketrightbigg , |I|</parenleftbigg8πeM2 SV /planckover2pi1/parenrightbiggAdisc(dee) |/Delta1ε|. (14) Here, /Delta1ε=ε1−ε2with P1=0.38,P2=0.44,/Lambda1 1= 1.8,/Lambda1 2=2, and θ1=θ2=π/2. Inserting all the pa- rameters into Eq. ( 14), we obtain φ−=sin−1[1.087×I], 224410-5CHEN, LEE, ZHANG, LIU, WU, HORNG, AND CHANG PHYSICAL REVIEW B 93, 224410 (2016) FIG. 3. The phase difference φ−=φ1−φ2and of the inclination angles θifor two nonidentical PERP STNOs. Macrospin simulations for (a) the time evolution of θiand the phase difference φ−and (b) time evolution of the xandzcomponents of the free-layer magnetization. The currents flowing through the STNO-1 and STNO-2 are 0.8 mA. The dipolar interaction strength is Adisc(dee)=0.002. (c) Micromagnetic simulation for time evolution of φ−ave. Here, φ−aveis the spatial averaged phase difference between the two free layers. (d) The current dependence of the phase difference φ−calculated from different models: The black curve for the approximate theory, the blue solid squares for the macrospin simulation, and the red open squares for the micromagnetic simulations. Ic=0.618 mA denotes the threshold current predicted by the approximation and ( Ic)M=0.5 mA by the macrospin simulation. |I|<0.92 mA. It should be noted that the theoretical curve predicted by Eq. ( 14) is quite close to the macrospin and micromagnetic simulation results for the low-current case. Inthe high-current case, the Idependence of φ −predicted by the approximate theory is still quite close to the micromagneticresult, but a little different from the macrospin result. Inter-estingly, these results confirm that the angular profile of thedisc dipolar coupling used in the approximation theory (seeAppendix A) is quite reasonable. We would like to point out that the approximation theory is in principle valid only forprecessions close to the thin film plane, i.e. for θ i=π/2+δθi and|δθi|∼√ k/lessmuch1, which corresponds to the case where the STT reaches its maximum magnitude when the free-layermagnetization lies in the plane of the film [ 19], but our calculations indicate that the dipolar coupling coefficient A disc can still be used if θ=0.34π=61.54◦forI=0.8m A . Another interesting point is that the dependence of the phase difference φ−=φ1−φ2on current in Fig. 3(d) shows that the analytical curve is much closer to the micromagneticsimulation curve. We assume that unexpected behavior iscaused by the fact that the analytical theory is in principleonly valid for the case of magnetization precession close to the film plane. At large currents, the phase difference can beenhanced in the analytical approximation by the fact that thehigh order terms of δθin the expansion of the STT torque in Eqs. ( B7) and ( B8) have been eliminated. By contrast, in the micromagnetic simulation, due to the nonuniformity ofthe local magnetization configuration, the calculated dipolarcoupling is actually smaller than that of the macrospin modelin the high-current region. As a result, the phase difference inthe micromagnetic simulation is enhanced at a large currentwhen compared with the macrospin simulation. The critical current to excite magnetization oscillation can be derived from Eq. ( 12a) I c=/parenleftbigg8πeM2 SV /planckover2pi1/parenrightbigg/bracketleftbigg3Adisc(dee)+k ε1+ε2/bracketrightbigg . (15) Note that, in Fig. 3(d), the theoretical critical current is Ic= 0.618 mA, which is slightly larger than that of the macrospin (0.5 mA) and micromagnetic results (0.4 mA). 224410-6PHASE LOCKING OF SPIN-TORQUE NANO-OSCILLATOR . . . PHYSICAL REVIEW B 93, 224410 (2016) FIG. 4. Frequency spectra of the two nonidentical STNOs calcu- lated by the FFT technique from the time evolution of xcomponents of the free-layer magnetization. The current is fixed to I=0.8m A for each STNO. (a) Macrospin simulations: The blue curves show therespective frequency of the two STNOs in the case without dipolar coupling; the red curve shows the frequency of the phase-locking state in the case with dipolar coupling. (b) Micromagnetic simulations:The blue curves show the respective frequency of the two STNOs without dipolar coupling, and the red curve shows the phase-locking frequency by the dipolar coupling. B. Frequency spectra of magnetization oscillations The oscillation frequency can be calculated from the time evaluation of magnetization. Figures 4(a) and 4(b) show the oscillation frequency spectra for the STNO pair with(red curves) and without (blue curves) dipolar interactioneffect simulated by the macrospin and the micromagneticmodel, respectively. Here, the frequency spectra are calculatedfrom the xcomponents through the fast Fourier transform (FFT) technique. The applied current for each STNO is0.8 mA. Note that both the macrospin and micromagneticsimulations display two separate oscillation frequencies (bluecurves) for the case without dipolar interaction betweenthe two STNOs. This corresponds to the case where theseparation d ee=∞ orAdisc(dee)=0. The left STNO has a low frequency (10.58 GHz in macrospin and 9.34 GHz inmicromagnetics) due to its relatively small spin-polarizationefficiency ( P 1=0.38,/Lambda1 1=1.8), while the right STNO has a higher frequency (12.2 and 9.76 GHz for macrospin andmicromagnetic simulations, respectively). When the separation is decreased to 20 nm [i.e. A disc(dee)= 0.002], the frequency of the STNO pair is locked at a medium FIG. 5. Precession frequency of two nonidentical STNOs as a function of the current Icalculated from (a) approximation theory, (b) macrospin simulation, and (c) micromagnetic simulation. The redcurves show the frequency for the STNO-1, the blue curves show the frequency for the STNO-2, and the black curves show the phase- locking frequency of the two STNOs through the dipolar coupling.The yellow background color regions show the current tunable range to achieve the phase-locking state. The threshold currents to excite precession states of STNO-1 and STNO-2 are indicated by I c1and Ic2, respectively. The threshold current for the phase-locking state of the two STNOs is marked by Ic. value, 11.39 GHz in the macrospin simulation and 9.7 GHz in the micromagnetic simulations. This is shown by the redcurves in Figs. 4(a) and4(b). The synchronized frequency in the macrospin simulation is exactly located at the center ofthe two separated blue peaks, while in the micromagneticsimulation, there is a little shift to that of STNO-2. Thisresult clearly confirms that, for a synchronized STNO pair, themagnetization of the two free layers precesses with the sameangular velocity ˙φas described by Eq. ( 13). The phase-locking angular velocity ˙φis an average of the two original angular velocities ˙φ 1,2. Figure 5shows the current tunable range of the phase- locking frequency in two nonidentical PERP STNOs. Themacrospin simulations for the two STNOs with P 1= 0.38,/Lambda11=1.8 and P2=0.44,/Lambda12=2a r es h o w nb yt h er e d and blue branches in Fig. 5(b). The critical driving current Ic is around 0.3 ∼0.4 mA. This critical value Iccan be estimated 224410-7CHEN, LEE, ZHANG, LIU, WU, HORNG, AND CHANG PHYSICAL REVIEW B 93, 224410 (2016) from Eq. ( 5). From |aJ1,2|/greaterorequalslantk/2, one obtains aJ01ε1(θ1=π/2,P1=0.38,/Lambda11=1.8)/greaterorequalslant0.008/2 aJ02ε2(θ2=π/2,P2=0.44,/Lambda12=2)/greaterorequalslant0.008/2. The calculated critical current for the left STNO is thus I1c=0.39 mA, and for the right one I2c=0.323 mA. We attribute the lower critical current in the right STNO toits relatively larger P. The right STNO therefore requires a relatively smaller current which can generate a strong enoughSTT to overcome the system barrier and then lead to amagnetization precession state. Note that, for current rangingfrom 0.4 to 1.2 mA, the macrospin simulation shows that thecurrent dependence of the precessional frequency is linear[Fig. 5(b)], which is consistent with the prediction of the approximate theory [Fig. 5(a)]. Theoretically, an approximate relationship between current and frequency can be derivedfrom Eq. ( 5) for a steady precession angular velocity |˙φ 1,2|= |aJ1,2|/α, f1,2(GHz) =|aJ1| 2πα(4πMsγ) =/planckover2pi1(4πMsγ) 8πeM2sV(2πα)ε1,2(θ,P,/Lambda1 )I1,2 =/braceleftbigg 15.6×I1(mA) 18.9×I2(mA). (16) As we have mentioned before, the synchronization fre- quency of the two STNOs is an average value between theirindividual natural frequencies. From Eq. ( 14), the phase- locking frequency as a function of current is given by theblack curve shown in Fig. 5(a). This result has been confirmed by both macrospin and micromagnetic simulations, as shownin Figs. 5(b) and 5(c). On the other hand, compared with individual STNO, it should be noticed that the critical currentfor the phase locking of the STNO pair increases due to thedipolar effect [Figs. 5(a) and5(b)], as indicated in Eq. ( 12a). Note that, not only no synchronization is observed at a smallcurrent, but that the dipolar coupling effect will also failto achieve the phase-locking state for a very large current.This is caused by the enhanced frequency difference betweenthe two STNOs at an increased current [see Eq. ( 16)]. Our simulations indicate that the effective current of phase lockingis 0.5–1.1 mA for the macrospin model and 0.4–0.8 mA forthe micromagnetic model. The phase-locking state of the two nonidentical STNOs precession strongly depends on the edge-to-edge distance d ee between the two nanopillars. This is due to the fact that the dipolar coupling decreases with increasing distance. Figure 6 shows the onset of phase locking as a function of theseparation distance d eefor a given current I=0.8m Afl o w i n g through each nanopillar. Clearly, both the macrospin and themicromagnetic simulations show almost the same parameterrange of the phase-locking state. The maximum edge-to-edgedistance ( d ee)Mis∼20 nm. Below this critical value, the two STNOs have the same precessional frequency, implying thatthe dipolar coupling is strong enough to drive them into aphase-locked state. On the contrary, when the distance d eeis larger than this value, the two nonidentical STNOs lose phase, FIG. 6. Precession frequencies of two nonidentical STNOs as a function of the edge-to-edge distance dee. The injected current for each STNO is I=0.8 mA. The yellow background color shows the parameter region of the phase-locking synchronization state. ( dee)M denotes the maximum edge-to-edge distance of the phase-locking state. (a) Macrospin simulation results. The dipolar coupling strength as a function of the edge-to-edge distance is taken from Fig. 10in Appendix A. (b) Micromagnetic simulation results, in which the red and blue curves are results from the MuMax3 and magpar simulation codes, respectively. and the frequency difference between them increases gradually with increasing distance dee, showing the decreased frequency in the left STNO, and the increased frequency in the right one. Compared with the macrospin model, the locked frequency in the micromagnetic simulations increases gradually withthe decrease of the separation d ee. This interesting result can be attributed to the following: For a small separation (e.g.d ee=4 nm), the stray fields generated by the neighboring STNO slightly reduce the nonuniformity of the magnetizationconfiguration. The increase of the uniform magnetization willin turn enhance the demagnetization fields. Therefore, theenhanced demagnetization field will increase the oscillationfrequency. C. Phase-locking diagram Finally, the phase diagrams of the two nonidentical STNO pairs as a function of distance and current are summarizedin Fig. 7. The phase diagram is divided into three regions, 224410-8PHASE LOCKING OF SPIN-TORQUE NANO-OSCILLATOR . . . PHYSICAL REVIEW B 93, 224410 (2016) FIG. 7. Phase diagrams as a function of the edge-to-edge distance deeand of the injected current I. (a) Prediction of the approximate theory. (b) Macrospin simulation. (c) Micromagnetic simulation. The central blue region represents the phase-locking (PL) mode. The yellow region is the steady (S) state without magnetization precession. The yellow region denotes the asynchronous (AS) precession mode. The border between the PL and AS states is separated by ( dee)M. including the steady state without magnetization precession (S state), the phase-locking precession state (PL state), andasynchronous state (AS state). The boundary between S andPL is the threshold current I c, defined in Fig. 5. The boundary between the PL and AS states is the maximum edge-to-edgedistance ( d ee)M, defined in Fig. 6. In the S state region, the current is too small to trigger the free-layer magnetizationoscillation. In contrast, in the AS state region, the dipolarcoupling between the two STNOs is not strong enough to drivea phase-locking state. From Fig. 7, one can see that the phase region of the approximation theory gives a good qualitativeprediction with the numerical simulations, indicating thatthe dipolar coupling strength A disc(dee) estimated from the assumption of uniformly magnetized thin film disc is quitereasonable for study of the phase-locking precession. IV . SUMMARY We show that the magnetic dipolar coupling between PERP STNOs can be used as a driving force to synchronize a seriesof horizontally aligned nanopillar oscillators. In this paper, wehave developed an approximate theory for two identical ornonidentical STNOs to predict their stable phase-locking stateand the requisite parametric conditions. The theoretical pre-dictions have been well confirmed qualitatively by macrospinand micromagnetic simulations. We calculated the relationshipbetween the critical current of synchronization, the criticaldipolar coupling strength, the phase-locking frequency, andthe transient time as well. These results may open a startingpoint for the design of a reliable horizontal array of PERPSTNOs phase locked through the dipolar coupling effect. Thiswould represent an effective way to raise the output power ofSTNOs. ACKNOWLEDGMENTS This paper is supported by the National Basic Research Program of China (Grants No. 2015CB921501 and No.2014CB921104). Z. Zhang thanks for the support fromNational Natural Science Foundation of China (Grants No.51222103, No. 51171047, and No. 11474067). Y . Liu thanksfor the support from NSFC of China (Grants No. 11274241and No. 51471118).APPENDIX A: APPROXIMATION THEORY FOR CALCULATION OF MAGNETIC DIPOLAR INTERACTION BETWEEN TWO CIRCULAR UNIFORMLY MAGNETIZED DISCS As shown in Fig. 8, for two uniformly magnetized circular thin-film discs separated by an edge-to-edge distance dee,t h e magnetic dipolar interaction can be calculated via integratingthe magnetostatic energy due to magnetic surface charges A disc/parenleftbiggπ 2,φ1,π 2,φ2/parenrightbigg =1/parenleftbig 4πM2s/parenrightbig V/contintegraldisplay S1/contintegraldisplay S2(σ1dS1)(σ2dS2) η, (A1) Here, we assume that the magnetizations of the two discs are aligned in the film plane, i.e. θ1=θ2=π/2. The surface charge densities accumulated on the edges of two discs are written as σ1=Mscos(φ1−φ/prime) and σ2=Mscos(φ2−φ/prime/prime), where φ1andφ2are the φcoordinates of magnetizations, andφ/primeandφ/prime/primeare the surface charge densities. The area elements dS1=(Rd)dφ/primedz/primeanddS2=(Rd)dφ/prime/primedz/prime/prime, where Ris the radius of the discs. The distance between any pair of surface charges on the two discs is written as η=/radicalbig [2R+dee+R(cosφ/prime/prime−cosφ/prime)]2+[R(sinφ/prime/prime−sinφ/prime)]2+(z/prime/prime−z/prime)2. In fact, when the edge-to-edge distance deeis much larger than FIG. 8. A top view of two uniformly magnetized circular discs. The plus and minus signs represent the magnetic surface charge distributions σ1(φ/prime)a n dσ2(φ/prime/prime).φ/primeandφ/prime/primeare the coordinates used to designate the locations of σ1andσ2.ηis the distance between any pair of magnetic surface charges between the two discs. φ1andφ2 are the magnetization directions, Ris the common radius of the two discs, and deeis the edge-to-edge distance between them. 224410-9CHEN, LEE, ZHANG, LIU, WU, HORNG, AND CHANG PHYSICAL REVIEW B 93, 224410 (2016) the disc thickness d,E q .( A1) can be well approximated by calculating the magnetic interaction between two uniformly magnetized circle discs modeled by the inscribed regular npolygons Adisc/parenleftbiggπ 2,φ1,π 2,φ2/parenrightbigg ≈1/parenleftbig 4πM2s/parenrightbig Vlim s,l→∞s−1/summationdisplay n=0l−1/summationdisplay m=0q1nq2m η12. (A2) Here, the magnetic surface charges accumulated on the nth and mth edges of the two discs are given by q1n=Mscos(φ1− φn)×2Rsin(π l)×dandq2m=Mscos(φ2−φm)×2Rsin(π s)×d. In these expressions, landsdenote the edge numbers of two regular npolygons. The distance between the two charges has the form η12=/radicalBigg/bracketleftbigg 2R+dee+Rcos/parenleftbiggπ s/parenrightbigg cos(φm)−Rcos/parenleftbiggπ l/parenrightbigg cos(φn)/bracketrightbigg2 +/bracketleftbigg Rcos/parenleftbiggπ s/parenrightbigg sin(φm)−Rcos/parenleftbiggπ l/parenrightbigg sin(φn)/bracketrightbigg2 . In order to analyze the phase-locking behavior from the point of view of dipolar interactions, a new set of variables(φ +,φ−) are introduced to replace ( φ1,φ2). Note that, the form of the angular profile of Adisc(φ+,φ−) is very similar to that of point dipolar interaction Apoint (φ+,φ−), especially in the locations of local energy maxima and minima, asshown in Fig. 9. However, when the distance d eeis smaller than 30 nm, the energy difference between local maximaand local minima of A disc(φ+,φ−) along φ−direction is obviously larger than that of Apoint(φ+,φ−). This means that the dipolar coupling strength of two uniformly magnetizationdiscsA disc(dee) is larger than that of two point magnetic dipoles Apoint(dee). This is because, when the magnetizations of two discs are placed in head-to-head or tail-to-tail configuration,they correspond to the local maximum of A disc(φ+,φ−), i.e.|φ−|=π. The magnetic energy comes mainly from the surface charges with the same sign, which are accumulated onthe face-to-face edges of the two discs. The actual distancebetween any pair of magnetic charges is much smaller thanthe center-to-center distance 2 R+d ee. Hence, for a smaller dee, the maximum dipolar interaction significantly grows with decreasing dee. Conversely, for configurations with local minimum of Adisc(φ+,φ−), corresponding to the head-to-tail or tail-to-head configuration, the minimum dipolar interactionbecomes significantly lower. The energy difference between the local maxima and minima of A point (φ+,φ−) along φ−isApoint(dee)= V/4π(2R+dee)3. Since Adisc(φ+,φ−)i ss i m i l a rt o Apoint (φ+,φ−), the disc dipolar field strength Adisc(dee) can be approximately estimated as the energy difference along φ− between the local maxima and minima of Adisc(φ+,φ−). Figure 10shows the comparison of Adisc(dee) andApoint(dee). Obviously, the growth rate of Adisc(dee) is faster than that of Apoint(dee) when deedecreases. APPENDIX B: NEWTON-LIKE EQUATIONS For the low-energy orbits with the total magnetic energy density |G|∼k>|Gdip|∼Adisc(dee), the orbits can be written as [ θi(τ),φi(τ)]=[π/2+δθi(τ),φi(τ)](i= 1,2), where |δθi|/lessmuch 1. In the absence of damping and the STT effect, these orbits obey the energy con-servation law G 0(π/2,π/2,φ01,φ02)=G1(π/2+δθ1,π/2+ δθ2,φ11,φ12), where G0andG1denote the energies of the initial and final states. By substituting Eqs. ( 3) and ( 4)i n t o the energy conservation equation and expanding ( δθi)2on theright-hand side, one can easily obtain 1 22/summationdisplay i=1(δθi)2≈k 22/summationdisplay i=1(sin2φ0i−sin2φ1i)+Adisc(dee) ×[(sinφ01sinφ02−2 cosφ01cosφ02) −(sinφ11sinφ12−2 cosφ11cosφ12)].(B1) Thus, the order of magnitude of |δθi|∼√ k. In the absence of Gilbert damping and the STT effect, by substituting θi=π/2+δθiinto Eq. ( 2) and expanding it to the first order of δθi, we obtain/braceleftBigg δ˙θi=−∂G ∂φi ˙φi=∂G ∂δθi,(i=1,2). (B2) Here, the total energy density Gis G(δθ1,δθ2,φ1,φ2)=1 22/summationdisplay i=1δθi2+k 22/summationdisplay i=1sin2φi+Adisc(dee) ×(sinφ1sinφ2−2 cosφ1cosφ2).(B3) Note that δθiandφiin Eq. (B2) form a set of conjugate variables in the Hamiltonian formulation. Accordingly, aneffective Hamiltonian can be defined as H(δθ 1,δθ2,φ1,φ2)=1 22/summationdisplay i=1δθi2+k 22/summationdisplay i=1sin2φi+Adisc(dee) ×(sinφ1sinφ2−2 cosφ1cosφ2),(B4) and Eq. (B2) becomes/braceleftBigg δ˙θi=−∂H ∂φi ˙φi=∂H ∂δθi,(i=1,2). (B5) We can obtain an effective Lagrangian by introducing the Legendre transformation L(φ1,φ2,˙φ1,˙φ2)=/summationtext2 i=1˙φiδθi− H(δθ1,δθ2,φ1,φ2). Thus, the effective Lagrangian is given by L(φ1,φ2,˙φ1,˙φ2)=1 22/summationdisplay i=1˙φ2 i−k 22/summationdisplay i=1sin2φi−Adisc(dee) ×(sinφ1sinφ2−2 cosφ1cosφ2). (B6) The STT and the Gilbert damping torques are nonconser- vative effects, and we therefore need to construct it from the 224410-10PHASE LOCKING OF SPIN-TORQUE NANO-OSCILLATOR . . . PHYSICAL REVIEW B 93, 224410 (2016) FIG. 9. Magnetic dipolar coupling as a function of the phase sum and the phase difference ( φ+,φ−). The edge-to-edge distance deeis 20 nm. The dipolar coupling profile is produced by two magnetic dipoles which are arranged (a) horizontally and (b) by two uniformlymagnetized circular discs. exact energy balance equation dG dτ=2/summationdisplay i=1−/bracketleftbigg α/vextendsingle/vextendsingle/vextendsingle/vextendsingledm i dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +aJi(mi)(mi×p)·dmi dτ/bracketrightbigg .(B7) Under the low-energy approximation, θi=π/2+δθi, |δθi|∼√ k/lessmuch1, the energy Eq. ( B7) can be approximated as dG dτ∼=−α2/summationdisplay i=1/parenleftbig δ˙θ2 i+˙φ2 i/parenrightbig −2/summationdisplay i=1aJi/parenleftbigg θi=π 2/parenrightbigg ˙φi.(B8) For a low-energy orbit, the order of magnitudes of |˙φi|and |δ˙θi|in Eq. (B2) can be estimated as /braceleftbiggδ˙θi∼−k ˙φ∼√ k,(i=1,2) (B9) FIG. 10. The dependence of the dipolar coupling strength on the edge-to-edge distance dee. The red circles represent the strength produced by two magnetic dipoles in a horizontal array. The black circles represent the strength produced by two uniformly magnetizedcircular discs in a horizontal array. Here,|˙φi|/greatermuch|δ˙θi|. Therefore, the energy balance equation can be further approximated in the form dG dτ∼=−α2/summationdisplay i=1˙φ2 i−2/summationdisplay i=1aJi/parenleftbigg θi=π 2/parenrightbigg ˙φi. (B10) Besides the damping effect, the contribution from the STT is also taken rigorously into account. We then can easily definean effective dissipation function in the Lagrangian dynamics F dis≡1 2α2/summationdisplay i=1˙φ2 i+2/summationdisplay i=1aJi/parenleftbigg θi=π 2/parenrightbigg ˙φi. (B11) From the Euler-Lagrangian equations with dissipation, Eq. ( 5) is obtained. Equation ( 5) is formally equivalent to /braceleftBiggδ˙θi+α˙φi+aJi/parenleftbig θi=π 2/parenrightbig =−∂G ∂φi ˙φi=∂G ∂δθi,(i=1,2), (B12) and if the magnitudes of |δ˙θi|,|˙φi|,|∂G/∂φ i|, and|∂G/∂δθ i| are on the same order as those in Eq. 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PhysRevB.100.054506.pdf

PHYSICAL REVIEW B 100, 054506 (2019) Electrical control of magnetization in superconductor /ferromagnet /superconductor junctions on a three-dimensional topological insulator M. Nashaat,1,2I. V . Bobkova ,3,4A. M. Bobkov,3Yu. M. Shukrinov,1,5I. R. Rahmonov,1,6and K. Sengupta7 1BLTP , Joint Institute for Nuclear Research, Dubna, Moscow Region, 141980, Russia 2Department of Physics, Cairo University, Cairo, 12613, Egypt 3Institute of Solid State Physics, Chernogolovka, Moscow reg., 142432, Russia 4Moscow Institute of Physics and Technology, Dolgoprudny, 141700, Russia 5Dubna State University, Dubna, 141980, Russia 6Umarov Physical Technical Institute, TAS, Dushanbe, 734063, Tajikistan 7School of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur, Kolkata-700032, India (Received 29 April 2019; published 6 August 2019) Strong dependence of the Josephson energy on the magnetization orientation in Josephson junc- tions with ferromagnetic interlayers and spin-orbit coupling opens a way to control magnetization byJosephson current or Josephson phase. Here we investigate the perspectives of magnetization control insuperconductor /ferromagnet /superconductor (S /F/S) Josephson junctions on the surface of a 3D topological insulator hosting Dirac quasiparticles. Due to the spin-momentum locking of these Dirac quasiparticles astrong dependence of the Josephson current-phase relation on the magnetization orientation is realized. It isdemonstrated that this can lead to splitting of the ferromagnet’s easy axis in the voltage driven regime. We showthat such a splitting can lead to stabilization of an unconventional fourfold degenerate ferromagnetic state. DOI: 10.1103/PhysRevB.100.054506 I. INTRODUCTION By now it is well known that current-phase relation (CPR) in Josephson junctions with multilayered ferromagnetic in-terlayers is strongly sensitive to the mutual orientation of the magnetizations in the layers [ 1–12]. CPRs of Joseph- son junctions with ferromagnetic interlayers in the presenceof spin-orbit coupling also depends on the magnetizationorientation. This occurs primarily via the appearance ofthe magnetization-dependent anomalous phase shift [ 13–26]. This coupling between the Josephson and magnetic subsys-tems leads to the supercurrent-induced magnetization dynam- ics [ 1,6,27–32]. In particular, the reversal of the magnetic moment by the supercurrent pulse [ 33] was predicted. A unique possibility of controlling the magnetization dynamicsvia external bias current and series of specific magnetizationtrajectories has been reported [ 34]. In Refs. [ 27,35]i tw a s also reported that in the presence of spin-orbit coupling thesupercurrent can cause reorientation of the magnetization easy axis. Assuming the initial position of the easy axis along the zdirection these works demonstrate that under the applied supercurrent stable position of the magnetization becomesbetween zandyaxes depending on parameters of the system. Here we investigate prospects of superconductor / ferromagnet /superconductor (S /F/S) Josephson junctions constructed atop a three-dimensional topological insulator (3D TI) surface, which hosts Dirac quasiparticles, in the field of supercurrent-induced magnetization control. Ourmotivation is that these Dirac quasiparticles on the surface ofthe 3D TI exhibit full spin-momentum locking: An electronspin always makes a right angle with its momentum. Thisgives rise to a very pronounced dependence of the CPRon the magnetization direction [ 17,36,37]. In particular, the anomalous ground state phase shift proportional to thein-plane magnetization component perpendicular to thesupercurrent direction was reported. The second reason to study magnetization dynamics in such a system is that at present there is a great progressin experimental realization of F /TI hybrid structures. In particular, to introduce the ferromagnetic order into the TI,random doping of transition metal elements, e.g., Cr or V , hasbeen employed [ 38–41]. The second option, which has been successfully realized experimentally, is a coupling of the non-magnetic TI to a high T cmagnetic insulator to induce strong exchange interaction in the surface states via the proximityeffect [ 42–46]. Here we demonstrate that the anomalous phase shift causes the magnetization dynamics analogously to the case of aspin-orbit coupled system. However, in contrast to the spin-orbit coupled systems, where the magnetization dynamicswas studied before, for the system under consideration theabsolute value of the critical current also depends stronglyon the magnetization orientation. It only depends on the in-plane magnetization component along the current direction.We demonstrate that such dependence, in a suitably cho-sen voltage-driven regime, can lead to supercurrent inducedsplitting of the magnetic easy axis of the ferromagnet. Weshow that this effect may lead to stabilization of a fourfolddegenerate ferromagnetic state, which is in sharp contrast tothe conventional twofold degenerate easy-axis ferromagneticstate. The paper is organized as follows. In Sec. IIwe derive a CPR for the S /F/S junction atop a topological insula- tor surface starting from the quasiclassical Green function 2469-9950/2019/100(5)/054506(7) 054506-1 ©2019 American Physical SocietyM. NASHAAT et al. PHYSICAL REVIEW B 100, 054506 (2019) FIG. 1. Sketch of the system under consideration. Superconduct- ing leads and a ferromagnetic interlayer are deposited on top of the TI insulator. formalism. This is followed by a discussion of the mag- netization dynamics of such systems in Sec. III. Next, in Sec. IV, we discuss the stabilization of the fourfold degenerate ferromagnetic state. Finally, we conclude in Sec. V. II. CURRENT-PHASE RELATION IN A BALLISTIC S /F/S J U N C T I O NO NA3 DT I The sketch of the system under consideration is presented in Fig. 1. Two conventional s-wave superconductors and a ferromagnet are deposited on top of a 3D TI insulator to forma Josephson junction. First of all, we consider a current-phase relation of a Josephson junction. The interlayer of the junction consists ofthe TI conducting surface states with a ferromagnetic layer ontop of it. It is assumed that the magnetization M(r)o ft h ef e r - romagnet induces an effective exchange field h eff(r)∼M(r) in the underlying conductive surface layer. The Hamiltonianthat describes the TI surface states in the presence of anin-plane exchange field h eff(r) reads: ˆH=/integraldisplay d2r/primeˆ/Psi1†(r/prime)ˆH(r/prime)ˆ/Psi1(r/prime), (1) ˆH(r)=−ivF(∇×ez)ˆσ+heff(r)ˆσ−μ, (2) where ˆ/Psi1=(/Psi1↑,/Psi1↓)T,vFis the Fermi velocity, ezis a unit vector normal to the surface of TI, μis the chemical potential, and ˆσ=(σx,σy,σz) is a vector of Pauli matrices in the spin space. It was shown [ 37,47] that in the quasiclassical approx- imation ( heff,ε,/Delta1 )/lessmuchμthe Green’s function has the fol- lowing spin structure: ˇ g(nF,r,ε)=ˆg(nF,r,ε)(1+n⊥σ)/2, where n⊥=(nF,y,−nF,x,0) is the unit vector perpendicular to the direction of the quasiparticle trajectory nF=pF/pF and ˆgis the spinless 4×4 matrix in the particle-hole and Keldysh spaces containing normal and anomalous quasiclas-sical Green’s functions. The spin structure above reflectsthe fact that the spin and momentum of a quasiparticle atthe surface of the 3D TI are strictly locked and make aright angle. Following standard procedures [ 48,49]i tw a s demonstrated [ 37,47,50] that the spinless retarded Green’s function ˆ g(n F,r,ε) obeys the following transport equations in the ballistic limit: −ivFnFˆ∇ˆg=[ετz−ˆ/Delta1,ˆg]⊗, (3) where [ A,B]⊗=A⊗B−B⊗Aand A⊗B=exp[( i/2) (∂ε1∂t2−∂ε2∂t1)]A(ε1,t1)B(ε2,t2)|ε1=ε2=ε;t1=t2=t.τx,y,zare Pauli matrices in particle-hole space with τ±=(τx±iτy)/2.ˆ/Delta1=/Delta1(x)τ+−/Delta1∗(x)τ−is the matrix structure of the superconducting order parameter /Delta1(x) in the particle-hole space. We assume /Delta1(x)=/Delta1e−iχ/2/Theta1(−x−d/2)+ /Delta1eiχ/2/Theta1(x−d/2). The spin-momentum locking allows for including heffinto the gauge-covariant gradient ˆ∇ˆA=∇ˆA+(i/vF)[(hxey−hyex)τz,ˆA]⊗. Equation ( 3) should be supplemented by the normaliza- tion condition ˆ g⊗ˆg=1 and the boundary conditions at x= ∓d/2. As we assume that the Josephson junction is formed at the surface of the TI, the superconducting order parameter /Delta1 andheffare effective quantities induced in the surface states of TI by proximity to the superconductors and a ferromagnet.In this case there are no reasons to assume existence ofpotential barriers at the x=∓d/2 interfaces and we consider these interfaces as fully transparent. In this case the boundaryconditions are extremely simple and are reduced to continuityof ˆgfor a given quasiparticle trajectory at the interfaces. To obtain the simplest sinusoidal form of the current-phase relation we linearize Eq. ( 3) with respect to the anomalous Green’s function. In this case the retarded component ofthe Green’s function ˆ g R=τz+fRτ++˜fRτ−. The anomalous Green’s function obeys the following equation: −1 2ivF,x∂xfR+heffn⊥fR=εfR−/Delta1(x). (4) Equation for ˜fRis obtained from Eq. ( 4)b yvF→−vF,/Delta1→ −/Delta1, andχ→−χ. The solution of Eq. ( 4) satisfying asymptotic conditions fR→(/Delta1/ε)e±iχ/2atx→± ∞ and continuity conditions at x=∓d/2 takes the form [the solution is written for x∈ (−d/2,d/2), the solution in the superconducting leads is also found, but it is not required for finding the Josephson current]: fR ±=/Delta1e∓iχ/2 εexp/bracketleftbigg∓2i(heffn⊥−ε)(d/2±x) vx/bracketrightbigg , (5) ˜fR ±=−/Delta1e∓iχ/2 εexp/bracketleftbigg∓2i(heffn⊥−ε)(d/2∓x) vx/bracketrightbigg , where the subscript ±corresponds to the trajectories sgnvx=±1. The density of electric current along the xaxis is jx=−eNFvF 4/integraldisplay∞ −∞dε/integraldisplayπ/2 −π/2dφ 2πcosφ ×[(gR +⊗ϕ+−ϕ+⊗gA +)−(gR −⊗ϕ−−ϕ−⊗gA −)], (6) where φis the angle, which the quasiparticle trajectory makes with the xaxis.ϕ±is the distribution function corresponding to the trajectories sgn vx=±1. Here we consider the voltage-biased junction. In principle, in this case the electric current through the junction consistsof two parts: the Josephson current j sand the normal current jn. The Josephson current is connected to the presence of the nonzero anomalous Green’s functions in the interlayerand takes place even in equilibrium. Here we assume thelow applied voltage regime eV /(k BTc)/lessmuch1. In this case the deviation of the distribution function from equilibrium is weakand can be disregarded in the calculation of the Josephsoncurrent: ϕ +=ϕ−=tanh(ε/2T). Exploiting the normaliza- 054506-2ELECTRICAL CONTROL OF MAGNETIZATION IN … PHYSICAL REVIEW B 100, 054506 (2019) tion condition one can obtain gR ±≈1−fR ±˜fR ±/2. Taking into account that gA ±=−gR∗ ±we find the following final expression for the Josephson current: js=jcsin(χ−χ0), (7) jc=evFNFT/summationdisplay εn>0/integraldisplayπ/2 −π/2dφcosφ/Delta12 ε2n ×exp/bracketleftbigg −2εnd vFcosφ/bracketrightbigg cos/bracketleftbigg2hxdtanφ vF/bracketrightbigg , (8) χ0=2hyd/vF, (9) where εn=πT(2n+1). At high temperatures T≈Tc/greatermuch/Delta1 the main contribution to the current comes from the lowestMatsubara frequency and Eq. ( 8) can be simplified further j c=jb/integraldisplayπ/2 −π/2dφcosφ ×exp/bracketleftbigg −2πTd vFcosφ/bracketrightbigg cos/bracketleftbigg2hxdtanφ vF/bracketrightbigg , (10) where jb=evFNF/Delta12/(π2T). A similar expression has al- ready been obtained for Dirac materials [ 50]. The normal current is due to deviation of the distribution function fromthe equilibrium. However, for the system under consideration,where we assume the ferromagnet to be metallic, practicallyall the normal current flows through the ferromagnet becausein real experimental setups the TI resistance should be muchlarger as compared to the resistance of the ferromagnet. Asfor the Josephson current, it is carried by Cooper pairs andis strongly suppressed inside the ferromagnetic layer becausethe exchange field there is typically much larger as comparedto the induced exchange field h effin the TI surface layer. Therefore, it flows through the TI surface states and we canassume that it is equal to the total electric current flowing viatheTI surface states . III. MAGNETIZATION DYNAMICS INDUCED BY A COUPLING TO A JOSEPHSON JUNCTION The dynamics of the ferromagnet magnetization can be described in the framework of the Landau-Lifshitz-Gilbert(LLG) equation ∂M ∂t=−γM×Heff+α MsM×∂M ∂t, (11) where Msis the saturation magnetization, γis the gyromag- netic ratio, and Heffis the local effective field. The electric current flowing through the TI surface states causes spin-orbital torque [ 51–54] due to the presence of a strong coupling between a quasiparticle spin and momentum. In principle, ifthe ferromagnetism and spin-orbit coupling spatially coexist,this torque is determined by the total electric current flowingthrough the system. However, for the case under considerationonly the supercurrent flows via the TI surface states, wherethe spin-momentum locking takes place. Therefore, only thissupercurrent generates a torque acting on the magnetiza-tion. The normal current flows through the homogeneousferromagnet, where we assume no spin-orbit coupling. Con- sequently, it does not contribute to the torque. The torque caused by the supercurrent can be accounted for as an additional contribution to the effective field. In orderto find this contribution we can consider the energy of thejunction as a sum of the magnetic and the Josephson energies: E tot=EM+EJ, (12) where the Josephson energy EJ=Ec[1−cos(χ−χ0)] with Ec=/Phi10Ic/2π,Ic=jcS(Sis the junction area) and χ= 2e Vtin the presence of the applied voltage. EM= −KVFM2 y/2M2 sis the uniaxial anisotropy energy with the easy axis assumed to be along the yaxis. VFis the volume of the ferromagnet. The effective field Heff=−(1/VF)(δEtot/δM) and takes the form: Heff,x HF=/Gamma1/bracketleftbigg/integraldisplayπ/2 −π/2e−˜d/cosφsinφsin(rmxtanφ)dφ/bracketrightbigg ×[1−cos(/Omega1Jt−rmy)], (13) Heff,y HF=/Gamma1/bracketleftbigg/integraldisplayπ/2 −π/2e−˜d/cosφcosφcos(rmxtanφ)dφ/bracketrightbigg ×sin(/Omega1Jt−rmy)+my, (14) Heff,z=0, (15) where we have introduced the unit vector m=M/Ms, ˜d=2πTd/vFis the dimensionless junction length, /Gamma1= /Phi10jbSr/2πKVFis proportional to the ratio of the Joseph- son and magnetic energies, r=2dheff/vF,/Omega1J=2e V i s t h e Josephson frequency, and HF=/Omega1F/γ=K/Ms. The effective field consists of two contributions: The anisotropy field, which is directed along the easy axis, isrepresented by the last term in Eq. ( 14). The other terms are generated by the supercurrent. The same approach tostudy magnetization dynamics in voltage biased junctionshas already been applied to systems with spin-orbit couplingin the interlayer [ 27,35]. The qualitative difference of our system based on the TI surface states from these works is thatthe critical current demonstrates strong dependence on the x component of magnetization in our case, while it has beenconsidered as independent on the magnetization directionearlier. This dependence leads to nonzero H eff,x∼mxat small mx. This means that the easy yaxis can become unstable in a voltage-driven or current-driven junction, while this axisis always stable if the critical current does not depend onmagnetization direction. Moreover, there is no difference forthe system between ±m xcomponents of the magnetization. This leads to the remarkable fact that in a driven system theeasy axis is not reoriented keeping two stable magnetizationdirections, as has already been obtained before, but is splitdemonstrating four stable magnetization directions. In the following section we study this effect in detail. IV . MAGNETIZATION EASY AXIS SPLITTING It is obvious that mx=mz=0 is an equilibrium point of Eq. ( 11) with Heffdetermined by Eqs. ( 13)–(15). Now we investigate stability of this point. In the linear order with 054506-3M. NASHAAT et al. PHYSICAL REVIEW B 100, 054506 (2019) respect to mxthe effective field can be written as follows: Heff,x=AHFmx[1−cos(/Omega1Jt−r)], Heff,y=HF[1+Bsin(/Omega1Jt−r)], (16) where A>0 and B>0 are constants. By comparing Eqs. ( 16) and ( 13) it is seen that A=r/Gamma1/integraldisplayπ/2 −π/2e−˜d/cosφsinφtanφdφ. (17) The LLG equation ( 11) in the linear order with respect to mx andmztakes the form ˙mx=γ 1+α2[Heff,y(mz−αmx)+αHeff,x], (18) ˙mz=γ 1+α2[−Heff,y(mx+αmz)+Heff,x], while ˙ my=0. One can estimate the parameter /Omega1F//Omega1J∼γHF/(eIcRn) for 3D TI Josephson junctions. Taking IcRn∼10−3V, as has been reported for Nb /Bi2Te3/Nb Josephson junctions [ 55], and the easy-axis anisotropy field HF∼500 Oe, what was reported for Py [ 56,57], we obtain /Omega1F//Omega1J∼3×10−3.I n the regime /Omega1F//Omega1J/lessmuch1 the magnetization varies slowly at t∼/Omega1−1 J, therefore we can average Eqs. ( 18) over a Josephson period thus obtaining the following system: ˙mx=/Omega1F 1+α2[mz−α(1−A)mx], (19) ˙mz=/Omega1F 1+α2[−(1−A)mx−αmz]. The general solution of this system takes the form mx(z)=/summationtext k=1,2Ck,x(z)exp[λkt]. The equilibrium point mx=mz=0 becomes unstable under the condition Re λ1>0o rR e λ2>0. One can obtain that it is realized at A>1. It is rather difficult to make accurate estimates of the nu- merical value of Afor realistic parameters. The main problem is the absence of experimental data on the value of heff.H o w - ever, if we take K=500 J/m3from the measurements [ 58] on permalloy with very weak anisotropy, Ic=10μA,vF∼ 105m/c from Ref. [ 55] and the permalloy volume d× l×w∼100 nm ×10 nm ×50 nm, then we can obtain A∼ r/Gamma1∼Icheff/(vFeKlw)∼0.4–8 for heff∼10–200 K. There- fore, we can conclude that the range of Avalues discussed in our work should be experimentally accessible. Now we turn to study the stationary points of the magne- tization and their stability. Beyond the linear approximation(with respect to m xandmz) it is convenient to parametrize the magnetization as m=(sin/Theta1cos/Phi1,cos/Theta1,sin/Theta1sin/Phi1). Then from the LLG equation one obtains ˙/Theta1=γ 1+α2[−αHeff,ysin/Theta1 +Heff,x(sin/Phi1+αcos/Theta1cos/Phi1)], ˙/Phi1sin/Theta1=γ 1+α2[−Heff,ysin/Theta1 +Heff,x(−αsin/Phi1+cos/Theta1cos/Phi1)].(20) At/Omega1F//Omega1J→0 effective fields Heff,x,ydetermined by Eqs. ( 13), (14) should be averaged over a Josephson period0. 0. 1.0 1.5 2.0.0.0.0.0.1.00. 0. 1.0 1.5 2.0.0.0.0.0.1.00. 0. 1.0 1.5 2.0.0.0.0.0.1.00. 0. 1.0 1.5 2.0.0.0.0.0.1.0 ΦΘΘΘΘ 0π 2π0π 2π0π 2π0π 2π 0π 2π3π 22π(a) (b) (c) (d) FIG. 2. Vector fields according to Eq. ( 20). (a) A=0.90 (/Gamma1= 1.26), (b) A=1.05 (/Gamma1=1.46), (c) A=1.25 (/Gamma1=1.84), (d) A= 1.50 (/Gamma1=2.10).r=0.5,˜d=0.3,α=0.25 for all the panels. Blue points indicate unstable stationary solutions, and the stable solutions are marked by red points. Heff,x,y→/angbracketleftHeff,x,y/angbracketright. The stationary points are to be found as solutions of Eqs. ( 20) corresponding to ˙/Theta1=˙/Phi1=0. Figure 2shows vector fields in the plane 0 /lessorequalslant/Phi1< 2π, 0/lessorequalslant/Theta1<π according to Eq. ( 20) at four different values of A. The stationary solutions are indicated by color points. The blue points correspond to unstable stationary solutions, whilethe red points indicate the stable magnetization directions.The Gilbert damping constant α=0.25. We have chosen such a large unrealistic value of the Gilbert constant in order toclearly show the stability /instability of the stationary points because for α=0.01, which is more appropriate for a realistic situation, stability /instability of a solution is not clearly seen in the figure [compare Figs. 3(a) and3(b)], although in fact the topology of the vector field is not changed. Figure 2(a) 054506-4ELECTRICAL CONTROL OF MAGNETIZATION IN … PHYSICAL REVIEW B 100, 054506 (2019) 0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.00. 0. 1.0 1.5 2.0.0.0.0.0.1.0 ΦΘΘ 0π 2π0π 2π 0π 2π3π 22π(a) (b) FIG. 3. (a) Vector field corresponding to the parameters of Fig. 4, but for /Omega1F//Omega1J→0. (b) The same as in panel (a), but for α=0.25 in order to demonstrate stability /instability of the stationary points. represents the regime A<1, when the only stable solutions mstaremst y=±1, which corresponds to upper and bottom horizontal lines in the figure. Panels (b) and (c) demonstratethe vector fields in the regime of not very large A>1. Four stable red points are clearly seen. Upon further increase ofAthe stable points get closer to /Theta1=π/2 and finally merge into two stable points at some A crit, as is shown in Fig. 2(d). Therefore, there exists a finite range of 1 <A<Acrit, where the ferromagnet has four stable magnetization directions inthe voltage-biased regime considered here. From Fig. 2it is seen that all the stationary points correspond to m z=±1o r mz=0. The stationary points mz=±1 are always unstable. Let us consider the stationary points corresponding to mz=0, that is /Phi1=0,π. It is obvious that in order to have four stable points |mst x|and|mst y|should be nonzero simultaneously. Substituting mz=0 into Eq. ( 20) and taking into account that/angbracketleftHeff,y/angbracketright=HFmy, we obtain that mst xcan be determined from the simple nonlinear equation mx=/angbracketleftHeff,x/angbracketright/HF.T h i s equation always has the solution mx=0, but at 1 <A<Acrit it also has the second nonzero solution mst x. In this situation there are four stable points mst=(±|mst x|,±|mst y|,0). Further in Fig. 4we demonstrate the full time evolution of the magnetization mobtained from the numerical solution of the LLG equation. It is seen that starting from different initialconditions it is possible to reach all four stable magnetizationsolutions. The results are obtained at /Omega1 F//Omega1J=0.2, but the averaged values of magnetization at large times are in goodagreement with the results for stable points obtained in thelimit/Omega1 F//Omega1J/lessmuch1, which are demonstrated in Fig. 3(a) for the same parameters /Gamma1,r,α, and d. Figure 3(b) only differs from (a) by the value of α=0.25. While the topology of the vector fields presented in panels (a) and (b) is the same,the stability /instability of all the stationary points is more clearly seen for larger values of the damping constant α.A t FIG. 4. Time evolution of the magnetization starting from different initial conditions. (a) mx(t=0)=−0.6,my(t=0)= 0.8, (b) mx(t=0)=0.6,my(t=0)=0.8, (c) mx(t=0)=−0.6, my(t=0)=−0.8, and (d) mx(t=0)=0.6,my(t=0)=−0.8. For all the panels we take mz(t=0)=0. The four panels correspond to four possible stable states, which are reached by the system atlarge t./Gamma1=1.57,r=0.5,˜d=0.3,α=0.01,/Omega1 F//Omega1J=0.2; time is measured in units of /Omega1−1 J. finite values of /Omega1F//Omega1Jthe magnetization oscillates around the vector trajectory presented in Fig. 3and the amplitude of the oscillations is suppressed at /Omega1F//Omega1J→0. In order to show that the system under consideration can demonstrate spontaneous behavior we investigate the system FIG. 5. Time evolution of the magnetization starting from an unstable point with the initial condition mx=my=0a n d mz=1 in the presence of noise. The four panels correspond to four possible stable states, which are reached by the system at large t./Gamma1=1.57, r=0.5,˜d=0.3,α=0.01,/Omega1F//Omega1J=0.2; time is measured in units of/Omega1−1 J. 054506-5M. NASHAAT et al. PHYSICAL REVIEW B 100, 054506 (2019) FIG. 6. (a) Averaged values of magnetization components at large times as functions of /Omega1J//Omega1F.˜d=0.2,/Gamma1=1.62. (b) The same as functions of /Gamma1.˜d=0.2,/Omega1J//Omega1F=5. (c) The same as functions of˜d./Omega1F//Omega1J=0.2,/Gamma1=1.62. For all the panels r=0.5,α=0.01.evolution starting from one of the unstable points mz=±1. A small noise is introduced to the system in order to allow forleaving the unstable equilibrium point. From the vector fieldsrepresented in Fig. 3(a) it is seen that at small values of α the system finally comes to one of the four stable states withpractically equal probabilities. It is shown in Fig. 5, where different panels correspond to all the possible final states. Figure 6demonstrates the behavior of the absolute values of averaged magnetization at t→∞ depending on essential parameters of the system. The dependence on /Omega1 J//Omega1Fis represented in Fig. 6(a). It is seen that at /Omega1J//Omega1F/greatermuch1|/angbracketleftmi/angbracketright| tend to constant values and, in particular, |/angbracketleftmz/angbracketright| → 0, as it follows from our analysis of stationary points of Eqs. ( 20). The dependence on /Gamma1is plotted in Fig. 6(b)./Gamma1is linearly proportional to A. For this reason one can explicitly see in this panel the range of Awhere four stable limiting magnetization directions exist: it corresponds to the regions where |/angbracketleftmx/angbracketright|and |/angbracketleftmy/angbracketright|are nonzero simultaneously. Panel (c) of Fig. 6represents the dependence of |/angbracketleftmi/angbracketright|on the junction length. Analogously to the previous panel, therange of existence of four stable limiting magnetization direc-tions is also clearly seen. The dependence on ris qualitatively very similar to the dependence on /Gamma1, therefore we do not represent it. Figures 6(b) and6(c) also provide the optimal range of parameters /Gamma1and dfor experimental realization of the easy axis splitting. The effect can be experimentallyinvestigated, for example, by measuring the magnetic fieldpattern away from the nanomagnet. V . CONCLUSIONS In this work we study a S /F/S Josephson junction atop a topological insulator and discuss the possibility of electricalcontrol of magnetization in such a device. Our analysis, whichcombines microscopic Keldysh Green function techniquefor obtaining the Josephson current with phenomenologicalLandau-Lifshitz-Gilbert equations for studying magnetizationdynamics, shows that the property of full spin momentumlocking can lead to destabilization of a transverse easy magne-tization axis m yin such systems in the presence of appropriate voltage or current bias. Such an instability, in turn, resultsin a ferromagnet with two easy axes allowing for four sta- blemagnetization directions instead of usual two. Switching between these states by means of voltage or current impulsesis of interest from the applied point of view. ACKNOWLEDGMENTS The work of I.V .B. and A.M.B. was carried out within the state task of ISSP RAS. 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PhysRevLett.109.067203.pdf

Atomistic Molecular Dynamic Simulations of Multiferroics Dawei Wang,1,2,*Jeevaka Weerasinghe,2and L. Bellaiche2,3 1Electronic Materials Research Laboratory—Key Laboratory of the Ministry of Education, and International Center for Dielectric Research, Xi’an Jiaotong University, Xi’an 710049, China 2Physics Department, University of Arkansas, Fayetteville, Arkansas 72701, USA 3Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA (Received 18 October 2011; revised manuscript received 9 February 2012; published 8 August 2012) A first-principles-based approach is developed to simulate dynamical properties, including complex permittivity and permeability in the GHz–THz range, of multiferroics at finite temperatures. It includesboth structural degrees of freedom and magnetic moments as dynamic variables in Newtonian andLandau-Lifshitz-Gilbert (LLG) equations within molecular dynamics, respectively, with the couplingsbetween these variables being incorporated. The use of a damping coefficient and of the fluctuation fieldin the LLG equations is required to obtain equilibrated magnetic properties at any temperature. Noelectromagnon is found in the spin-canted structure of BiFeO 3. On the other hand, two magnons with very different frequencies are predicted via the use of this method. The smallest-in-frequency magnon corresponds to oscillations of the weak ferromagnetic vector in the basal plane being perpendicular tothe polarization while the second magnon corresponds to magnetic dipoles going in and out of this basalplane. The large value of the frequency of this second magnon is caused by static couplings betweenmagnetic dipoles with electric dipoles and oxygen octahedra tiltings. DOI: 10.1103/PhysRevLett.109.067203 PACS numbers: 75.85.+t, 31.15.xv, 75.30.Gw, 76.50.+g Multiferroics form a promising class of materials exhib- iting a rare coexistence between ferroelectricity and mag-netism. They are experiencing a huge resurgence ininterest, partly for designing novel advanced technologies(see, e.g., Refs. [ 1,2] and references therein). The develop- ment and use of ab initio atomistic schemes recently had helped in gaining a better knowledge of these complexmaterials. For instance, first-principles-based simulationsexplained why so few compounds are multiferroics [ 3] and how their magnetic ordering can be controlled by theapplication of electric fields along specific directions[4–7]. They also provided a deep insight into the strain- driven phase transition towards states with giant axial ratio and large out-of-plane polarization in BiFeO 3 (BFO) multiferroics [ 8–11]. Similarly, ab initio techniques revealed that a dramatic enhancement of magnetoelectriccoefficients can be achieved near this latter phase transition[12,13]. Another example is the prediction of array of ferroelectric (FE) vortices in BFO films [ 14], that was then experimentally confirmed [ 15]. Interestingly, all these latter breakthroughs from first principles concerned static properties of multiferroics. On the other hand, one particu- larly challenging issue that remains to be tackled byatomistic methods, in general, and by first-principles tech-niques, in particular, is the prediction of dynamical prop-erties of multiferroics at finite temperature. One particularreason behind such lack of numerical tool is that ionicvariables (e.g., polarization and/or tilting of oxygen octa-hedra) obey Newton’s equations of motion while the spin degrees of freedom do not [such latter degrees of freedom follow Landau-Lifshitz [ 16] or even more complicatedequations such as Landau-Lifshitz-Gilbert (LLG) [ 17]]. In order to realistically mimic dynamical properties ofmultiferroics, one thus needs to develop a tool wheredifferent equations are simultaneously obeyed, and thatalso includes the coupling between ionic and magnetic variables. Moreover and in order to predict finite- temperature dynamics of multiferroics, this hypotheticaltool should also be able to control (at the same time) thetemperature associated with ionic motions and the tem-perature associated with magnetic degrees of freedom.Such simultaneous control is not an easy task to accom-plish. On the other hand, developing such code can havelarge benefits. For instance, it may help in understanding what is the nature of the excitation, having a frequency that is larger than typical magnon frequencies but smaller thanphonon frequencies, which has been recently observed inBFO systems (see Ref. [ 18] and Fig. 2 of Ref. [ 19] for the ‘‘mysterious’’ excitation having a frequency of the orderof30–60 cm /C01). Can it be an electromagnon [ 20–22], as suggested in Ref. [ 18]? The purpose of this Letter is to demonstrate that it is possible to develop such an ab initio scheme and to apply it to the study of BFO. It is found that LLG equations, inwhich a realistic damping coefficient is used and in which afluctuation field is incorporated, coupled with classicalNewtonian equations allows us to reach the equilibriumstates of both the structural and magnetic variables at anytemperature. The use of this tool also yields the computa-tion of the complex electric and magnetic susceptibilities for any frequency in the GHz–THz range. In particular, it predicts that the aforementioned mysterious excitation is inPRL 109, 067203 (2012) PHYSICAL REVIEW LETTERSweek ending 10 AUGUST 2012 0031-9007 =12=109(6) =067203(6) 067203-1 /C2112012 American Physical Societyfact a magnon rather than an electromagnon, and that its large frequency originates from static (rather thandynamic) couplings between the magnetic dipoles with electric dipoles and oxygen octahedra tilting. Here, we first take advantage of the first-principles- based effective Hamiltonian developed for BFO systems[23,24]. Its total internal energy E totis written as a sum of two main terms: Etot¼EFE/C0AFDðfuig;f/C17g;f!igÞ þEMAGðfmig;fuig;f/C17g;f!igÞ; (1) where uiis the local soft mode in unit cell i, which is directly proportional to the electrical dipole centered on that cell. The f/C17gis the strain tensor and contains both homogeneous and inhomogeneous parts [ 25,26]. The !i pseudo-vector characterizes the oxygen octahedra tilt, which is also termed the antiferrodistortive (AFD) motion,in unit cell i[23]. The m iis the magnetic dipole moment centered on the Fe-site iand has a fixed magnitude of 4/C22B [27]. The EFE/C0AFDis given in Ref. [ 28] and involves terms associated with ferroelectricity, strain, and AFD motions, and their mutual couplings. The EMAG gathers magnetic degrees of freedom and their couplings and is given inRefs. [ 23,24]. Note that the use of this effective Hamiltonian approach within Monte Carlo (MC) simula-tions was shown to (i) correctly yield the R3cground state that exhibits a coexistence of a spontaneous polarizationwith antiphase oxygen octahedra tilt in BFO bulks [ 23], (ii) provide accurate Ne ´el and Curie temperatures and intrinsic magnetoelectric coefficients in BFO bulks andthin films [ 13,23,29], and also (iii) reproduces the spin- canted magnetic structure that is characterized by a weakmagnetization superimposed on a large G-type antiferro- magnetic (AFM) vector in BFO films (note that this spin-canted structure originated from the AFD motions ratherthan the polarization) [ 24]. Note, however, that the current version of this effective Hamiltonian approach does not yield a spin cycloid structure in BFO bulks, unlike inexperiments [ 20]. The probable reason for that is either the lack of an additional energetic term that generates suchcycloid or that the period of the cycloid [ 20] is too large to be mimicked by atomistic simulations. The present resultsshould thus be relevant to BFO thin films (for which nocycloid exists) [ 30,31]. Here, we decided to combine the effective Hamiltonian scheme within an original molecular dynamics (MD)scheme in order to be able to predict dynamical properties.Technically, and as done in Refs. [ 32–34], Newtonian equations are implemented for the fu ig;f/C17g;f!igvariables, with the corresponding forces appearing in these equationshaving been obtained by taking partial differential of theE totenergy of Eq. ( 1) with respect to each variable. As also previously implemented [ 32,33], the temperatures of these lattice variables are controlled by Evans-Hoover thermo-stats [ 35]. The novelty here is to also include the dynamics of the magnetic moments on the same footing than thedynamics of the structural variables at a given temperature (note that we are not aware of any previous study address-ing such simultaneous ‘‘double’’ dynamics and that con- trolling temperature for the magnetic sublattice is a challenging problem [ 17,36,37]). For that, we imple- mented the stochastic LLG equation [ 17] for the m i’s degrees of freedom: dmi dt¼/C0/C13mi/C2½Bi effðtÞþbi flðtÞ/C138 /C0/C13/C21 jmijð1þ/C212Þmi /C2fmi/C2½Bi effðtÞþbi flðtÞ/C138g; (2) where Bi eff¼/C0@Etot=@miis the effective magnetic field acting on the ith magnetic moment, /C13is the gyromagnetic ratio, /C21is the damping coefficient, and bi flis a fluctuation field that also acts on the ith magnetic moment. As we will see below, the introduction of this latter fluctuation field iscrucial to obtain correct magnetic properties in a multi-ferroic at finite temperature, as consistent with previousstudies done on magnetic systems [ 38,39]. Technically, we use the Box–Muller method (that generates Gaussian dis- tributed numbers for each magnetic moment) to simulate b i fland to enforce the following conditions to be obeyed by this fluctuation field at the finite temperature T[17,38]: hbi fli¼0; (3) hbi fl;/C11ðt1Þbi fl;/C12ðt2Þi ¼ 2/C21kBT /C13jmij/C14/C11;/C12/C14ðt1/C0t2Þ; (4) where /C11and/C12denote Cartesian coordinates and t1andt2 are two different times. The hiindicates an average over possible realizations of the fluctuating field [ 17],/C14/C11;/C12is Kronecker delta function, and /C14ðt1/C0t2Þis a Dirac delta function. A semi-implicit method devised by Mentink et al. [40] is adopted here to (i) properly integrate the LLG equation, which is a Stratonovich stochastic differentialequation [ 17] (the need to properly integrate LLG equation is a pivotal point that has been discussed in several studies [17,40–44]), and (ii) to enforce the conservation of the magnetic moments’ magnitude. The Mentink algorithmis efficient by limiting the matrix inversion procedure—which is needed by an implicit integrator—for each mag-netic moment at each time step. We have checked that thisalgorithm indeed conserves the magnitude of the individ-ual magnetic moments very well and satisfies our need for efficiency and stability. Simulations on a periodic 12/C212/C212supercell (8640 atoms) are performed within the presently developedMD scheme to obtain finite-temperature properties ofBFO. The system is first equilibrated at a chosen tempera-ture and pressure ( NPT ensemble), and then, depending on the purpose of the simulation, we either continue havingNPT steps to extract static properties or adopt NVE steps to obtain time-resolved properties, such as autocorrelation functions of electric or magnetic dipoles, to predict dy-namical properties. A time step of 0.5 fs is used in allsimulations.PRL 109, 067203 (2012) PHYSICAL REVIEW LETTERSweek ending 10 AUGUST 2012 067203-2One important problem to address when dealing with dynamics of magnetic degrees of freedom and the LLGequation is to determine the realistic value, or range ofvalues, of the damping coefficient for a given system. Oneway to solve such problem is to realize that MC and MDshould give identical results for static properties at anytemperature. As a result, MC can be used as a way ofgauging MD simulations and extracting the proper damp-ing constant /C21[45]. We numerically found that, at any temperature, /C21has little effect on the spontaneous polar- ization and oxygen octahedra tilting, therefore yieldingMD results being similar to the MC predictions for thesestructural properties for a wide range of damping coeffi- cients. In fact, the effect of /C21can be clearly seen when investigating magnetic properties in the multiferroicBFO—as consistent with the fact that /C21‘‘only’’ appears in the spin equations of motions. Consequently, Fig. 1 shows the temperature evolution of the magnitude of theG-type AFM vector ( L) for different /C21values within the MD scheme, as well as the MC prediction for such quan-tity. Moreover, parts (a) and (b) of this figure display theresults when the fluctuation field is neglected and ac-counted for, respectively, in the MD simulations, in orderto also reveal the importance of b i flon finite-temperature magnetism. One can see that, without the fluctuation field, (i) MD simulations with /C21*1:0/C210/C04give an AFM vector that is significantly larger than that from the useof the MC technique for any temperature ranging between 10 and 800 K and therefore also generates a larger Ne ´el temperature, while (ii) for damping coefficients smallerthan 1:0/C210 /C04(including the case of /C21¼0), the MD results are consistent with the MC calculations for tem-peratures larger than 250 K but yield too small AFM vectors for lower temperatures [ 46]. Therefore, not a single proper /C21value allowing the MD simulations of the AFM vector to match the MC results across all temperaturescan be found without a fluctuation field. On the otherhand, Fig. 1(b) demonstrates that a wide range of /C21 (namely, 1:0/C210 /C04/C20/C21/C201:0/C210/C01) leads to a satis- factory agreement (i.e., a difference of less than 3%)between the MD and MC results at any temperature, when the fluctuation field is included. Such results thus prove the crucial importance of a fluctuation field foraccurately modelling finite-temperature spin dynamics inmultiferroics. Note also that a large range of /C21can be adopted to obtain equilibrated static properties, whichmakes the MD approach suitable to model different multi-ferroic or ferromagnetic bulks or nanostructures that mayhave very different damping constants due to different damping mechanisms [ 47]. Let us now use the proposed MD scheme, incorporating the fluctuation field and choosing /C21¼1:0/C210 /C04to com- pute the complex electric and magnetic susceptibilities ofBFO, to be denoted by /C31 eand/C31m, respectively. Such quantities can be calculated as follows [ 32,48,49]: ½/C31eð/C23Þ/C138/C11/C12¼1 "0VkBT/C20 hd/C11ðtÞd/C12ðtÞi þi2/C25/C23Z1 0dtei2/C25/C23thd/C11ðtÞd/C12ð0Þi/C21 ;(5) ½/C31mð/C23Þ/C138/C11/C12¼/C220 VkBT/C20 hM/C11ðtÞM/C12ðtÞi þi2/C25/C23Z1 0dtei2/C25/C23thM/C11ðtÞM/C12ð0Þi/C21 ;(6) where /C23is the frequency while /C11and/C12define Cartesian components, with the x,y, and zaxes being along the pseudocubic [100], [010], and [001] directions, respec- tively. The dðtÞandMðtÞare the electric and magnetic dipole moments at time t, respectively. Here, we focus on a fixed temperature of 20 K, for which the crystallographicequilibrium state is R3c. Figure 2(a) shows the isotropic value of the ½/C31 eð/C23Þ/C138/C11;/C12 dielectric response, that is f½/C31eð/C23Þ/C138xxþ½/C31eð/C23Þ/C138yyþ ½/C31eð/C23Þ/C138zzg=3. Four peaks can be distinguished, having resonant frequencies of 151 cm/C01,176 cm/C01,240 cm/C01, and263 cm/C01. They correspond to E,A1,E, and A1 symmetries, respectively [ 50]. Not all the modes appearing in measured Raman or infrared spectra [ 19,54–61] can be reproduced by our simulations because of the limitednumber of degrees of freedom included in the effectiveHamiltonian. In particular, the modes observed around 74 and81 cm /C01, and that are E(TO) and E(LO) modes, re- spectively, according to Ref. [ 61], are missing in our computations. Moreover, we numerically found that thefirst two (lowest-in-frequency) peaks of Fig. 2(a) are0.00.51.01.52.02.53.03.54.0Magnetic moment (Bohr magneton)(a)λ = 2.0 ×10-4 λ = 1.0 ×10-4 λ = 5.0 ×10-5 λ = 1.0 ×10-5 λ = 0.0 MC 0.00.51.01.52.02.53.03.5 0 200 400 600 800 1000 1200 1400 Temperature (K)(b)λ = 1.0λ = 1.0 ×10-1 λ = 1.0 ×10-4 λ = 1.0 ×10-5 MC FIG. 1 (color online). Temperature dependency of the magni- tude of the antiferromagnetic vector within the proposed MDscheme and for different damping coefficients, when the fluc-tuation field is neglected (a) and incorporated (b). For compari- son, the MC results are also indicated by the red line with red solid circles.PRL 109, 067203 (2012) PHYSICAL REVIEW LETTERSweek ending 10 AUGUST 2012 067203-3mostly related to the sole FE degree of freedom incorpo- rated in the effective Hamiltonian scheme, while the lasttwo peaks have also a significant contribution from AFDdistortions, as consistent with Ref. [ 51]. As revealed in Refs. [ 33,62], bilinear couplings between the FE and AFD modes in the R3cphase allow the AFD mode to acquire some polarity, which explains why these last two peaksemerge in the dielectric spectra. Regarding the permeability, two peaks can be seen in Fig. 2(b). Their predicted resonant frequencies are /C247c m /C01and/C2485 cm/C01, respectively [ 63]. Since none of the frequencies coincides with the dielectric resonantfrequencies shown in Fig. 2(a), we can safely conclude that they are not electromagnons. They are rather ‘‘solely’’ magnons. Interestingly, we further numerically found thatthe lowest-in-frequency magnon entirely disappears whenwe switch off in our simulations the parameter responsible for the spin-canted structure of BFO. In other words, the purely AFM G-type structure does not possess such mag- non. Moreover, the video shown in the SupplementalMaterial S1 [ 64] demonstrates that this magnon is associ- ated with the rotation of magnetic dipoles inside the (111) plane (which contains the polarization). In other words,this magnon is the low-in-frequency excitation (possessinga gap) that has been predicted in Refs. [ 65,66] and that corresponds to the oscillation of the weak ferromagnetic moment about its equilibrium position in the basal plane[67]. Let us now try to understand the origin of the second magnetic peak, for which the frequency is much larger than those of typical magnons (which are usually lower than 20 cm /C01) but is smaller than the phonon fre- quencies shown in Fig. 2(a) (this second peak is thusconsistent with the ‘‘mysterious’’ excitations observed in Refs. [ 18,19]. This second magnetic peak is associated with fast oscillations of the magnetic dipoles going in and out of the (111) plane, as well as, a change in length of the weak FM vector (see Fig. 3and video in Supplemental Material S1 [ 64]). This second peak there- fore corresponds to the so-called optic antiferromagnetic mode of Ref. [ 68] and to the high-frequency gapped mode of Ref. [ 66]. Interestingly, we also numerically found that this second peak (i) has a resonant frequency that is in- sensitive to the effective masses associated with the FE and AFD motions (in other words, the frequency of this second magnetic peak is insensitive to a change of FE or AFDresonances), and (ii) becomes a broad peak ranging from 0c m /C01to’16 cm/C01when switching off the coupling parameters between magnetic moments with FE and AFD motions in our simulations (in that case, the corre- sponding motions of the magnetic dipoles are not only inand out of the (111) plane but also are within the (111) plane). As a result, we can safely conclude that the abnor- mally large frequency of the second peak results from static (rather than dynamic) couplings between the m i’s and structural variables, with these couplings generating alarge magnetic anisotropy. Furthermore, this second peak has a resonant frequency of around 60 cm /C01rather than /C2485 cm/C01, if one ‘‘only’’ switches off the static coupling between magnetic degrees of freedom and AFD motions. In other words, AFD distortions (that have not been ex- plicitly incorporated in phenomenological models so far to study dynamics of BFO systems) do significantly affect the resonant frequency of this second peak. Analytical expres-sions derived in the Supplemental Material S2 [ 69] from energetic terms included in the effective Hamiltonian con- firm and even shed more light on such features, such as revealing that the resonant frequency of this second mag- netic peak also depends on the values of the spontaneouspolarization and angle of oxygen octahedra tilting [ 70]. We thus hope that our proposed atomistic MD method is, and will be, of large benefit to gain a deeper knowledge of the-3-2-10123456 100 150 200 250 300χe(ν) (103) Frequency (cm-1)(a)Re(χe) Im(χe) -3-2-101234567 0 20 40 60 80 100 120χm(ν) (10-3) Frequency (cm-1)(b)Re(χm) Im(χm) FIG. 2 (color online). Complex electric (a) and magnetic (b) susceptibilities as a function of frequency in BFO at T¼20 K .FIG. 3 (color online). Sketch of the FM vector (the short green vector) and the Lvector (the long red vector) at one instance. We note, at this instance, these two vectors slightly deviate from the (111) plane due to couplings with AFD and FE (see Supplemental Material S2 [ 69]). The (weak) FM vector is enhanced by /C2459times to be seen in this figure.PRL 109, 067203 (2012) PHYSICAL REVIEW LETTERSweek ending 10 AUGUST 2012 067203-4fascinating multiferroic materials [ 71]. Note that it can also open the door to many exciting studies, such as the com- putation and understanding of the cross-coupled electro- magnetic susceptibility defined in Ref. [ 68]. Discussions with J. In ˜iguez, Dr. Kamba, M. Cazayous, and M. Bibes are greatly acknowledged. We mostly thank Office of Basic Energy Sciences, under contract ER-46612 for personnel support. NSF Grants No. DMR-0701558 and No. DMR-1066158, ARO Grant No. W911NF-12-1-0085, and ONR Grants No. N00014-11-1-0384 and No. N00014- 08-1-0915 are also acknowledged for discussions with scientists sponsored by these grants. D. W. acknowledges support from the National Natural Science Foundation of China under Grant No. 10904122. Some computationswere also made possible thanks to the MRI grant 0959124 from NSF, and N00014-07-1-0825 (DURIP) from ONR. *dawei.wang@mail.xjtu.edu.cn [1] T. Choi, S. Lee, Y. J. Choi, V. Kiryukhin, and S.-W. Cheong, Science 324, 63 (2009) . [2] S. Y. Yang, J. Seidel, S. J. Byrnes, P. 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[46] Item (ii) thus reveals that, at higher temperatures, the equilibrated lattice degrees of freedom in the system— local modes, AFD variables, strain tensor—act as a very good heat reservoir for the magnetic degrees of freedomeven when no damping is included, but a too small damp- ing prevents the magnetic sublattice from reaching its ground state at low temperatures. [47] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004) . [48] J. Caillol, D. Levesque, and J. Weis, J. Chem. Phys. 85, 6645 (1986) . [49] J. Hlinka, T. Ostapchuk, D. Nuzhnyy, J. Petzelt, P. Kuzel, C. Kadlec, P. Vanek, I. Ponomareva, and L. Bellaiche, Phys. Rev. Lett. 101, 167402 (2008) . [50] The phonon frequencies of BFO have been investigated using ab initio computations [ 51–53] and Raman/infrared spectroscopy [ 19,54–61]. Surprisingly, rather different results were obtained between these different studies,even for the modes’ symmetry—in addition to the quanti- tative value of the resonant frequencies. As a result, Tutuncu et al. [52] had assigned a margin of 40–50 cm /C01 when comparing different results. Here, we have matched our MD results for the resonant frequencies and symmetry of the peaks to LDA þUphonon calculations of BFO in itsR3cphase, by tuning the effective masses of the FE and AFD modes. [51] P. Hermet, M. Goffinet, J. Kreisel, and P. Ghosez, Phys. Rev. B 75, 220102(R) (2007) . [52] H. Tu ¨tu¨ncu¨,J. Appl. Phys. 103, 083712 (2008) . [53] I. Apostolova, A. T. Apostolov, and J. M. Wesselinowa, J. Phys. Condens. Matter 21, 036002 (2009) . [54] R. Haumont, J. Kreisel, P. Bouvier, and F. Hippert, Phys. Rev. B 73, 132101 (2006) . [55] H. Fukumura, S. Matsui, H. Harima, T. Takahashi, T. Itoh, K. Kisoda, M. Tamada, Y. Noguchi, and M. Miyayama, J. Phys. Condens. Matter 19, 365224 (2007) . [56] R. P. S. M. Lobo, R. L. Moreira, D. Lebeugle, and D. Colson, Phys. Rev. B 76, 172105 (2007) . [57] D. Rout, K.-S. Moon, and S.-J. L. Kang, J. Raman Spectrosc. 40, 618 (2009) . [58] J. Lu, M. Schmidt, P. Lunkenheimer, A. Pimenov, A. A. Mukhin, V. D. Travkin, and A. Loidl, J. Phys. Conf. Ser. 200, 012106 (2010) . [59] R. Palai, H. Schmid, J. F. Scott, and R. S. Katiyar, Phys. Rev. B 81, 064110 (2010) . [60] A. A. Porporati, K. Tsuji, M. Valant, A.-K. Axelsson, and G. Pezzotti, J. Raman Spectrosc. 41, 84 (2010) . [61] J. Hlinka, J. Pokorny, S. Karimi, and I. M. Reaney, Phys. Rev. B 83, 020101(R) (2011) . [62] J. Weerasinghe, D. Wang, and L. Bellaiche, Phys. Rev. B 85, 014301 (2012) . [63] Note that we also computed the magnetic response asso- ciated with the G-type AFM vector by replacing the magnetic dipole moments by the AFM ones in Eq. ( 6). We found that such AFM response has also two peaks, with their resonant frequencies coinciding with those of the permeability. [64] See Supplemental Material S1 at http://link.aps.org/ supplemental/10.1103/PhysRevLett.109.067203 for avideo showing the time evolution of the ferromagnetic and antiferromagnetic vectors. [65] P. Pincus, Phys. Rev. Lett. 5, 13 (1960) . [66] R. de Sousa and J. E. Moore, Appl. Phys. Lett. 92, 022514 (2008) . [67] Note that switching off the parameter responsible for the spin canting should shift down our lowest-in-frequency magnetic peak to 0c m/C01[66]. We cannot observe such resulting zero-frequency peak because our numerical tech-nique cannot efficiently probe frequencies lower than’3c m /C01due to the picosecond time scale inherent to MD simulations. [68] K. L. Livesey and R. L. Stamps, Phys. Rev. B 81, 094405 (2010) . [69] See Supplemental Material S2 at http://link.aps.org/ supplemental/10.1103/PhysRevLett.109.067203 for de- tailed analysis using our effective Hamiltonian. [70] Since the spontaneous polarization, angle of oxygen octa- hedra tiltings and, especially, magnetic-structural couplingparameters are rather difficult to be precisely determinedfrom first principles, it is possible that the second magnonpeak will be experimentally found at a different resonant frequency in BFO films (we are not aware of any pub- lished data reporting magnetic peaks in BFO films that donot possess magnetic cycloid). However, if future Ramanor infrared measurements do confirm the existence of theelectromagnon peak around /C2485 cm /C01, one has to realize that it will nearly overlap with the E(LO) dielectric peak[61]. We also note that our MD simulations indicate that increasing the damping coefficient results in a decrease ofthe magnitude of the second magnetic peak. [71] It is also important to realize that two main limitations are currently associated with the proposed method: (1) onecan not study magnetic excitations associated with verylowkvectors because of the relatively small size of the supercell [ 66,72]; and (2) excitations lower than ’3c m /C01 cannot be investigated because of the time scale of usual MD simulations. On the other hand, in addition to provid-ing insightful atomistic details, our proposed scheme hasalso the advantage (with respect to phenomenologicalworks) to extract its parameters from first principles. Forinstance, it provides an effective magnetic field associated with the parameter leading to spin canting [which is related to the coefficient K ijin Eq. ( 1) of the Supplemental Material S2 [ 69]] of about 1 Tesla at low temperature. This value is about 7 times larger than thephenomenological value used in Ref. [ 68], which there- fore questions the accuracy of this latter value since ourK ijparameter was shown to provide a weak ferromagnetic vector that agrees very well with experiment [ 24]. Our numerical tool also gives a Ne ´el temperature of only ’150 K if one neglects the static couplings between magnetic degrees of freedom and structural variables, to be compared with the value of ’660 K when these couplings are included. Note that these latter couplingsare those related to the E ijandGijparameters of Eq. ( 1)o f the Supplemental Material S2 [ 69], since Kijwas found to have a negligible effect on the Ne ´el temperature [ 24]. [72] A. K. Zvezdin and A. A. Mukhin, JETP Lett. 89, 328 (2009) .PRL 109, 067203 (2012) PHYSICAL REVIEW LETTERSweek ending 10 AUGUST 2012 067203-6

PhysRevB.101.140404.pdf

PHYSICAL REVIEW B 101, 140404(R) (2020) Rapid Communications Deterministic approach to skyrmionic dynamics at nonzero temperatures: Pinning sites and racetracks Josep Castell-Queralt , Leonardo González-Gómez , Nuria Del-Valle , and Carles Navau* Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain (Received 6 February 2020; revised manuscript received 27 March 2020; accepted 30 March 2020; published 16 April 2020) The discovery of room-temperature skyrmions in some magnetic materials has boosted the investigation of their dynamics in view of future applications. We study the dynamics of skyrmions in the presence of defectsor borders using a deterministic (finding and solving a deterministic Fokker-Planck differential equation) whileprobabilistic (the solution is the probability density for the presence of a skyrmion) approach. The probability thata skyrmion becomes trapped in a pinning center or the probability of the survival of a skyrmion along a racetrackare obtained as a function of temperature. The present work can be relevant in the design of skyrmionic deviceswhere the probability of finding skyrmions at a given position and time is crucial for their feasibility. DOI: 10.1103/PhysRevB.101.140404 Magnetic skyrmions are whirling magnetic structures that can be found on certain magnetic materials [ 1]. Their small size and high mobility have promoted them as promisinginformation carriers as well as basic elements in ultradensemagnetic memories, logic devices, or computational sys-tems [ 2–6]. In ferromagnetic ultrathin films, it has been found that skyrmions can be stabilized with the aid of interfacialDzyaloshinskii-Moriya (iDM) interactions with a heavy-metalsubstrate [ 7–10]. The same mechanism allows the formation of skyrmions in multilayers with alternate ferromagnets andheavy metals [ 11,12]. The experimental discovery of room- temperature skyrmions [ 13] has boosted the potentiality of skyrmions for applications and, as a result, the study of theircurrent-driving dynamics at nonzero temperatures. Skyrmionic racetracks were proposed to transport skyrmions using the spin-orbit torque produced by aspin-polarized current fed into a heavy-metal substrate [ 2,14]. In such systems, defects or granularity result in a thresholdcurrent density for the activation of the movement [ 15–20] and the borders of the track create a confining potential that sets adriving velocity threshold above which the skyrmions wouldescape [ 21–24]. At increasing temperature, the stochastic effects on the skyrmions’ position [ 25–27] could compromise their existence when approaching the borders or de-fects [ 22,28]. Also, their topological protection is weakened, which can lead to their collapse [ 22,29–32]. This stochastic motion sets different conditions (or restrictions) on the appli-cability of racetracks that should be addressed. In particular,some questions arise: What is the probability for the skyrmionto escape from a pinning center? What is the probability fora skyrmion to overcome the racetrack border’s confiningpotential? What is the probability of finding a skyrmion at agiven distance from the initial position after a given time? At temperature T=0 the movement of the skyrmion is not probabilistic [ 21,23,33,34] and the previous questions do *Corresponding author: carles.navau@uab.catnot apply. The inclusion of thermal effects in micromagnetic simulations can be done either by using a stochastic Landau-Lifshitz-Gilbert equation, Landau-Lifshitz-Bloch equation, orby stochastic atomic spin dynamics [ 35–38]. Thiele’s equa- tion [ 39], originally introduced for magnetic bubble domain motion, with the inclusion of extra stochastic terms [stochas-tic Thiele’s equation (STE)], is also used to study the dy-namics of skyrmions [ 17,25,27,28,40,41]. In this case, it is assumed that the skyrmion maintains its shape during themovement (rigid approximation). Even at room temperature,this assumption can be a valid approximation as shown forPt/Co/Ir, Pt/CoFeB /MgO, or Pt /Co/Ta multilayers [ 13,20]. In all these cases, the simulations for a single skyrmion have tobe repeated a large number of times and the average quantitiesevaluated with the corresponding statistical dispersion. Here, we use a deterministic, while probabilistic, approach for studying the dynamics of skyrmions including tempera-ture. We evaluate the effects of pinning potentials as well asracetrack borders. Instead of solving the STE many times, wesolve the corresponding deterministic Fokker-Planck equa-tion [ 42] (FPE) for the probability density of the presence of a skyrmion as a function of time. The main advantagesof this approach are as follows: (i) One needs to solve theFPE only once; (ii) some approximations can be analyticallyworked out; (iii) FPE is a partial differential equation thatcan be solved using well-known numerical techniques (evencommercial software); and (iv) complex potentials can beincluded without a significant increase in the computationtime. Some previous works have also used the same de- terministic approach for studying the dynamics of rigidskyrmions [ 26,43] (and vortices [ 44] ) .I nR e f .[ 26] the mobil- ity of skyrmions in a periodic potential was evaluated in thestationary limit. Also, in Ref. [ 43] the steady-state velocity (of antiferromagnetic skyrmions in this case) for the lowest-order traveling wave solutions of the probability density wasobtained. In contrast to these works, here we obtain the fulltime and position dependence of the solution of the FPE for 2469-9950/2020/101(14)/140404(5) 140404-1 ©2020 American Physical SocietyJOSEP CASTELL-QUERALT et al. PHYSICAL REVIEW B 101, 140404(R) (2020) the case in which rigid skyrmions travel in the presence of a pinning site or along a racetrack. From these solutions, all theprobabilistic properties of the dynamics can be obtained. Our starting point is the stochastic Thiele’s equation for a rigid skyrmion moving on a magnetic ultrathin film (thicknessd, volume V) with background magnetization pointing in the −ˆzdirection and located on the z=0 plane. The movement of the skyrmion is described by the position of its center of masswhose time derivative is the velocity of the skyrmion V s.W e consider here that the skyrmion is driven through dampingliketorques produced by spin-polarized currents coming from thespin-Hall effect after feeding an in-plane current J Hin a heavy-metal substrate [ 45]. The STE can be written as [ 46] (G−MsαD)Vs+MsNVH+γM2 s(Fext+Fst)=0,(1) whereGandDare the gyrocoupling and dissipation ma- trix, respectively. Ncomes from the integration of the spin- transfer torque term in the Landau-Lifshitz-Gilbert equation.They all are 2 ×2 matrices whose elements are ( u,v=x,y) G uv=/integraltext VM0·(∂M0 ∂u×∂M0 ∂v)dV,Duv=/integraltext V(∂M0 ∂u·∂M0 ∂v)dV, and Nuv=(1/d)/integraltext V(∂M0 ∂u×M0)vdV.VH=−μBθH eMs(z×JH), with μBthe Bohr magneton, θHthe Hall angle, and e(>0) the charge of the electron. In the present case, consideringaxisymmetric skyrmions, G xy=−Gyx≡G,Dxx=Dyy≡D, Nxy=−Nyx≡−N, where G>0,D>0, and N>0. All other elements of the matrices are zero. γis the gyromag- netic constant ( γ=2.21×105mA−1s−1),αis the Gilbert damping constant, and Msthe saturation magnetization. The force terms FextandFstcome, respectively, from the exter- nal and stochastic forces.1Fstis considered a white noise with/angbracketleftFst,j/angbracketright=0 and/angbracketleftFst,iFst,j/angbracketright=2αDkBT γμ 0Msδijδ(t−t/prime)[25], with i,j=x,y,z,μ0the vacuum permeability, kBthe Boltzmann constant, δijthe Kronecker delta, and δ(t−t/prime) the temporal Dirac’s delta. For a given initial position of the skyrmion, the stochastic nature of Eq. ( 1) results in different trajectories for each simu- lated solution. However, the probability density of finding thecenter of mass of the skyrmion at position r=(x,y) and time t,p(r,t), can be directly evaluated from its corresponding FPE (see Supplemental Material for the derivation [ 47] ) that can be written as ∂ ∂tp(r,t)=−∇·[p(r,t)(Vdrv+Vext)]+Dd∇2p(r,t), (2) where we have used the definitions Dd=γM3 sαDkBT μ0(G2+D2α2M2s), (3) Vdrv=− (G−αMsD)−1MsNVH, (4) Vext=− (G−αMsD)−1γM2 sFext. (5) Equation ( 2) is a convection-diffusion equation. The first term on the right-hand side indicates that the probability 1Strictly, FextandFstdo not have units of force, but we follow the usual nomenclature.density is transported at a velocity Vdrv+Vext, whereas the second term is a linear, homogeneous, and isotropic diffusionterm with constant D d. Actually, the FPE is also a continuity equation ∂ ∂tp(r,t)=−∇·Jp(r,t), (6) where one can define the current of probability density Jp(r,t)=(Vdrv+Vext)p(r,t)−Dd∇p(r,t). Equation ( 2) can be analytically solved in some cases. If one considers a free skyrmion [no driving ( VH=0), no exter- nal forces ( Fext=0)], whose initial position probability den- sity is described by a Gaussian function centered at the originof coordinates with a given variance σ 2,p0(x,y)=N(0,σ), the solution of Eq. ( 2)i s p(x,y,t)=N(0,/radicalbig σ2+2Ddt), which indicates that the initial Gaussian distribution diffuseswithout translation, maintaining the Gaussian shape, wherethe variance is a linear function of the time and proportional tothe temperature. These results are in complete agreement withRef. [ 25] which used a completely different approach (thermal agitation of classical spins on a triangular lattice) to monitorthe Brownian motion of skyrmions. Another interesting analytical solution of the FPE is the stationary ( t→∞ ) solution of the probability density when a harmonic pinning center is present. In this case, if the pinningcenter is at the origin, the force felt by the skyrmion canbe described by F ext=−k(xˆx+yˆy)(kindicates the restor- ing coefficient of the force, assumed constant). No drivingcurrent is considered. The stationary solution of Eq. ( 2)i s found to be, independently of the initial probability density, p(x,y,∞)=N(0,/radicalbigg MskBT μ0Ddk). It is also a Gaussian distribution whose variance increases linearly with temperature and is inversely proportional to k. This indicates that the skyrmion, regardless of the initial position, will go to the pinning site,jiggling around it with a variance that represents a competitionbetween the thermal diffusive effect and the attractive pinningforce. Consider a more realistic case of a skyrmion driven by current density J Hwhich finds an attractive pinning center whose force is described by [ 46] Fext=−F0pr λexp/parenleftbigg −|r|2 λ2/parenrightbigg , (7) where λandF0pcontrol the scope and the strength of the pinning potential, respectively. Although only the position ofthe center of mass of the skyrmion is evaluated, the rigidmodel assumes a fixed shape of the skyrmions. However, itsradius can change in an order of magnitude from T=0t o room temperature [ 20,31]. Since λin Eq. ( 7) is related to the radius of the skyrmion [ 21], we set λ=R sand we use the results of Ref. [ 31] to find its dependence on T(see Supplemental Material for the details [ 47]).λ(T) increases with temperature in a nonlinear way, with a larger slope atlarger temperatures. We assume that GandDare independent of temperature (models predict that Gis basically independent of the radius and that Ddepends on the diameter /domain wall width ratio [ 40,48]; we are thus assuming that the change in temperature maintains this ratio constant). 140404-2DETERMINISTIC APPROACH TO SKYRMIONIC DYNAMICS … PHYSICAL REVIEW B 101, 140404(R) (2020) FIG. 1. Snapshots for the probability density p(r,t) (color bar in units of 10−3nm−2), evaluated at different times, indicated in each figure in ns. The central green cross indicates the position of the pinning center and the solid blue square indicates the initial position. The open blue squares indicate the initial position of the skyrmions considered in Fig. 2. The parameters used in this simulation are T=150 K, α=0.3,Ms=580 kA m−1,D=G=4π,VH=277.4ms−1,λ(T)=38.3 nm, and F0p=5.8×10−14m2A−1. See Supplemental Material for numerical details and video [ 47]. We want to evaluate the probability that a skyrmion is trapped by this pinning center, as a function of the temper- ature. The resulting FPE [Eqs. ( 2)–(5) with Eq. ( 7)] has to be solved numerically (see Supplemental Material [ 47] and Ref. [ 49] for the numerical details). Results do depend on the initial position of the skyrmion. As described in Ref. [ 46]f o r T=0, for a given VHbelow a threshold velocity, the solution of Thiele’s equation yields two differentiated regimes depend-ing on the initial position of the skyrmion: (i) The skyrmionis trapped and (ii) the skyrmion escapes. Actually, there is asaddle point in the T=0 phase portrait of the trajectories that determines if the skyrmion is or is not trapped, depending onfrom which side of the saddle point the skyrmion is approach-ing. However, at T>0 these two regimes are not so clearly differentiated. Due to the thermal diffusion, it is possible thata skyrmion that would escape at T=0 does not at T>0. The opposite is also true: A skyrmion that would be trappedatT=0 has some “thermal chance” of escaping. We show in Fig. 1some snapshots, at different times, of the calculated probability density p(x,y,t) in a region close to a pinning center (indicated as a green cross) for a given initial position(blue solid square) and a given temperature ( T=150 K). We observe that some probability density escapes from thepinning center. After a long time the situation stabilizes andthe probability of being trapped, P t=/integraldisplay Scp(x/prime,y/prime,∞)dx/primedy/prime, (8) can be evaluated. To evaluate Pt, we have considered a calcula- tion window Scof dimensions much larger than λ(T)×λ(T). Note that the persistence of the driving currents makes thatthe final probability density distribution is not centered on thepinning site but at a nearby position [ 46,50]. In Fig. 2we show the calculated P tas a function of the tem- perature, for different initial positions. At T=0 the situation is binary: Ptis either one or zero, depending on the initial position. Increasing T, the potential well becomes broader and shallower [due to the λ(T) dependence], increasing the probability for those skyrmions that would escape at T=0 (red lines) to fall into the well at relatively low T.A tt h e same time, for those skyrmions that would be trapped atT=0 (blue lines) the probability of trapping decreases when the potential becomes shallower, although the broadeningacts as a counteracting effect. Finally, at large T, regardlessof the initial position of the skyrmions, the thermal energy overcomes the potential well and P tgoes to zero in all cases. An increase in the F0pfactor would shift the temperature at which Ptgoes to zero to higher values. One of the important issues in the skyrmionic roadmap is the transport along racetracks [ 2]. The feasibility of such systems is based on the survival of skyrmions when theyare driven along the racetrack. We now want to evaluate theprobability of such a survival. Consider a long racetrack alongthexaxis, with a width 2 Win the yaxis (the center line of the racetrack is y=0). The force over the skyrmion due to the borders can be modeled as [ 21] F ext=F0t/bracketleftbig −e−W−y λ+e−W+y λ/bracketrightbig ˆy. (9) The corresponding FPE [Eqs. ( 2)–(5) with Eq. ( 9)] is also solved numerically (see Supplemental Material for numer-ical details [ 47]). In Fig. 3we show the probability den- sity evolving with time for two temperatures [ T=100 K in Fig. 3(a), and T=300 K in Fig. 3(b)] at a fixed driving FIG. 2. Probability Ptthat a skyrmion is trapped in a Gaussian pinning center when it is driven by currents as a function of the temperature, for different initial positions of the skyrmion, indicatedwith the xcoordinate with respect the green cross in Fig. 1(the initial positions are shown as open blue squares in Fig. 1). The blue (red) dots correspond to cases where the skyrmion would be trapped(escape) at T=0. The rest of the parameters are the same as in Fig. 1. The encircled point corresponds to the case of the snapshots in Fig. 1. 140404-3JOSEP CASTELL-QUERALT et al. PHYSICAL REVIEW B 101, 140404(R) (2020) FIG. 3. Snapshots [(a) T=100 K; (b) T=300 K] of the prob- ability density (color bar in units of 10−5nm−2) for the central position of a rigid skyrmion driven along a track, p(x,y,t). In each track, the blue dot (at left) indicates the initial position and p(x,y,t) at several times t(indicated in the figure in ns) are plotted. The gray regions correspond to y’s such that W>|y|> W−λ(T). The parameters used are W=150 nm, VH=325 m s−1, F0t=5.325×10−14m2A−1,λ(T=100 K) =25.05 nm, λ(T= 300 K) =70.33 nm. The rest of the parameters are the same as in Fig. 1. See Supplemental Material for numerical details and videos [ 47]. velocity VHbelow the threshold (thus, at T=0, the skyrmion would not escape from the track). When T>0 there is some probability of escaping through the borders due to thermaldiffusion. One interesting figure of merit is the probability that a skyrmion reaches a certain position along the track, which isevaluated from the flux of current density through a sectionof the track at position x[initially, the skyrmion is located at (x,y)=(0,0)], P s(x)=/integraldisplayW−λ(T) −W+λ(T)dy/prime/integraldisplay∞ 0dt/primeˆx·Jp(x,y/prime,t/prime). (10) Note that, in order to consider the radius of the skyrmion at different temperatures, apart from considering λ(T)i nE q .( 9), we have considered that the skyrmion escapes from the trackwhen|y|>W−λ(T) (that is, when the “skyrmion border”— not its center—reaches the track border). In Fig. 4(a) we show P s(x) at different temperatures. The region of Ps/similarequal1 corresponds to small distances where the skyrmion has not yetreached the borders. The subsequent decay in the probabilityindicates that, when the skyrmion is moving along the border,the probability of escaping increases as it goes further away[thus, the P s(x) track decreases with increasing x]. In Fig. 4(b) we show Psat a fixed value of x=xL,a s a function of temperature, for several values of the drivingvelocity. In general, the probability of survival up to a givendistance decreases with increasing temperature. We observea plateau, even a slight increase in P s(xL) at intermediate temperatures: The confining potential from the borders cancompensate the thermal diffusion (and the increase in theradius of the skyrmion) up to a certain temperature and theskyrmion can reach the desired x L. For large temperatures, the borders are not able to compensate the diffusion and itbecomes less likely to reach x L. FIG. 4. (a) Probability of surviving a certain distance xalong the track as a function of x,Ps(x), for different temperatures (indicated in the figure). (b) Ps(xL=1115 nm) as a function of temperature for different driving currents (indicated in the figure). The parameters not shown are as in Fig. 3. The encircled points in (b) correspond to the simulations of Fig. 3. Averages over infinite runs of the STE could give the same information as the FPE. Comparing similar computationaltimes and /or precision, the presented calculations are a better approach since the numerical errors in the solution of FPEare orders of magnitude lower that the statistical errors inthe averages of the solutions of the STE (see SupplementalMaterial for a comparison of both methods [ 47]). Some of the most promising applications of skyrmions are those based on the one-skyrmion one-bit principle. Knowingthe probability of finding a skyrmion at a given position andtime is crucial for assessing the viability of such systems atroom temperature. Since borders and defects are unavoidablein realistic samples, the present results may help in the futuredesign of skyrmionic devices. We thank Dr. S. Serna for her advice in the nu- merical techniques in solving the FPE. We also ac-knowledge Catalan Project No. 2017-SGR-105 and Span-ish Project No. MAT2016-79426-P of Agencia Estatalde Investigación /Fondo Europeo de Desarrollo Regional (UE) for financial support. J.C.-Q. acknowledges a Grant(FPU17 /01970) from Ministerio de Ciencia, Innovación y Universidades (Spanish Government). 140404-4DETERMINISTIC APPROACH TO SKYRMIONIC DYNAMICS … PHYSICAL REVIEW B 101, 140404(R) (2020) [1] K. Everschor-Sitte, J. Masell, R. M. Reeve, and M. Kläui, J. Appl. Phys. 124,240901 (2018 ). [2] A. Fert, V . Cros, and J. Sampaio, Nat. Nanotechnol. 8,152 (2013 ). [3] G. Bourianoff, D. Pinna, M. Sitte, and K. 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PhysRevB.98.220410.pdf

PHYSICAL REVIEW B 98, 220410(R) (2018) Rapid Communications Damping and antidamping phenomena in metallic antiferromagnets: An ab initio study Farzad Mahfouzi*and Nicholas Kioussis† Department of Physics and Astronomy, California State University, Northridge, California 91330, USA (Received 13 November 2018; revised manuscript received 10 December 2018; published 26 December 2018) We report on a first-principles study of antiferromagnetic resonance (AFMR) phenomena in metallic systems [MnX(X=Ir, Pt, Pd, Rh) and FeRh] under an external electric field. We demonstrate that the AFMR linewidth can be separated into a relativistic component originating from the angular momentum transfer between thecollinear AFM subsystem and the crystal through spin-orbit coupling, and an exchange component that originatesfrom the spin exchange between the two sublattices. The calculations reveal that the latter component becomessignificant in the low-temperature regime. Furthermore, we present results for the current-induced intersublatticetorque which can be separated into fieldlike and dampinglike components, affecting the intersublattice exchangecoupling and AFMR linewidth, respectively. DOI: 10.1103/PhysRevB.98.220410 Spintronics is a field of research exploiting the mutual influence between the electrical field/current and the magneticordering. To date, the realization of conventional spintronicdevices has relied primarily on ferromagnetic (FM) basedheterostructures [ 1–6]. On the other hand, antiferromagnetic (AFM) materials have been recently revisited as potentialalternative candidates for active elements in spintronic devices[7,8]. In contrast to their FM counterparts, AFM systems have a weak sensitivity to magnetic field perturbations, produceno perturbing stray fields, and can offer ultrafast writingschemes in the terahertz (THz) frequency range. The THzspin dynamics due to AFM ordering has been experimentallydemonstrated using all-optical [ 9,10], and Néel spin-orbit torque (SOT) [ 11,12] mechanisms. One of the most important parameters in describing the dynamics of the magnetic materials is the Gilbert dampingconstant α. Intrinsic damping in metallic bulk FMs [ 13,14] is associated with the coupling between the conduction elec-trons and the time-dependent magnetization /vectorm(t), where the latter in the presence of spin-orbit coupling (SOC) leads to amodulation (breathing) of the Fermi surface [ 13] and hence excitation of electrons near the Fermi energy. The excitedconduction electrons in turn relax to the ground state throughinteractions with the environment (e.g., phonons, photons,etc.), leading to a net loss of the energy/angular momentum inthe system. While the damping in FMs has been extensivelystudied both experimentally and theoretically, the damp-ing in metallic AFM has not received much attention thusfar. Manipulation of the damping constant in magnetic devices is one of the prime focuses in the field of spintronics. Con-ventional approaches to manipulate the damping rate of aFM rely on the injection of a spin-polarized current into theFM. The spin current is often generated either through thespin Hall effect (SHE) [ 15,16] by a charge current passing *Farzad.Mahfouzi@gmail.com †Nick.Kioussis@csun.eduthrough a heavy metal (HM) adjacent to the FM in a lateral structure, or spin filtering in a magnetic tunnel junction (MTJ)in a vertical heterostructure [ 17]. Similar mechanisms have also been proposed [ 8,18–22] for AFM materials, where the goal is often to cause spontaneous THz-frequency oscilla-tions or reorientation [ 11,23–25] of the AFM Néel ordering, /vectorn(t)=(/vectorm 1−/vectorm2)/2. Here, /vectormsis a unit vector along the magnetization orientation of the sublattice s. In contrast to the aforementioned studies that require breaking of inversionsymmetry to induce Néel ordering switching, in this RapidCommunication we focus on the current-induced excitation ofthe sublattice spin dynamics of bulk metallic AFM materialswith inversion symmetry intact, and hence no Néel SOT[11,12,26,27]. The objective of this work is to (1) provide a general analytical expression for the antiferromagnetic resonance(AFMR) [ 28] frequency and linewidth in the presence of current-induced sublattice torque, and (2) employ the Kubo-like formalism with first-principles calculations to calculatethe Gilbert damping tensor α ss/prime(s,s/prime=↑,↓), and the field- like,/vectorτFL, and dampinglike, /vectorτDL, components of the sublat- tice torque for a family of metallic AFM materials includ-ing Mn X(X=Ir,Pt,Pd,Rh) and FeRh, shown in Fig. 1. We demonstrate that the zero-bias AFMR linewidth can beseparated into the relativistic, /Gamma1 r=λα0/2M, and exchange, /Gamma1ex=Kαd/2Mcomponents [ 29], where αd≡/summationtext sαss,α0≡ αd−/summationtext sαs¯s,Mis the magnetic moment of each sublat- tice,λis the intersublattice exchange interaction, and Kis the magnetocrystalline anisotropy energy. In agreement withrecent first-principles calculations [ 30], we find that α dis about three orders of magnitude larger than α0, indicating the crucial role of the exchange component to the AMFRlinewidth. Our calculations reveal that at high temperaturesdue to the interband contribution, the relativistic componentbecomes the dominant term in the AFMR linewidth, whileat low temperatures both exchange and relativistic compo-nents contribute to the AFMR linewidth on an equal footing.We further demonstrate that the current-induced antidamp-inglike (fieldlike) torque changes the AFMR linewidth 2469-9950/2018/98(22)/220410(6) 220410-1 ©2018 American Physical SocietyFARZAD MAHFOUZI AND NICHOLAS KIOUSSIS PHYSICAL REVIEW B 98, 220410(R) (2018) FIG. 1. Crystal structure of (left) MnX with (X=Ir, Pt, Pd, Rh) and (right) FeRh used for the first-principles calculations, where the corresponding spin configuration and primitive cells are shown withsolid lines. (intersublattice exchange interaction), thereby allowing the manipulation of the damping constant (Néel temperature) inbulk AFM materials. Precessional magnetization dynamics of AFMs is often described by a system of coupled equations for each spinsublattice [ 20,22,31], where a local damping constant αis assigned to each of the two sublattices ignoring the effects ofthe rapid (atomic scale) spatial variation of the magnetizationon the damping constant due to the AFM ordering. Takinginto account the Gilbert damping tensor α ss/prime, the coupled Landau-Lifshitz-Gilbert (LLG) equations of motion for thetwo sublattices can be written as d/vectorm s(t) dt=−γ/vectorms(t)×/vectorHeff s+/summationdisplay s/primeαss/prime/vectorms(t)×d/vectorms/prime(t) dt, (1) where the local effective field in the presence of the external electric ( /vectorEext) and magnetic ( /vectorBext) fields is given by /vectorHeff s=/vectorBext+/summationdisplay i=xyz/parenleftbig K(2) i;s+K(4) s/bracketleftbig 1−m2 i;s(t)/bracketrightbig/parenrightbigmi;s(t) Msˆei +e/vectorτ0 DL·/vectorEext/vectorms(t)×/vectorm¯s(t) +/parenleftbiggλ Ms+e/vectorτ0 FL·/vectorEext/parenrightbigg /vectorm¯s(t). (2) Here,λis the exchange coupling between the two sublattices, /vectorτ0 DL(/vectorτ0 FL) is the current-induced intersublattice dampinglike (fieldlike) torque component, and K(2) i;s(K(4) s) is the second- order (fourth-order) magnetocrystalline anisotropy energy(MCAE). Equation ( 2) shows that the effect of /vectorτ 0 FLis to renormalize the intersublattice exchange coupling, λ/prime=λ+ Mse/vectorτ0 FL·/vectorEext. In the following, without loss of generality, we assume Kz 2=0 and Kx,y 2/greaterorequalslant0, where in the absence of an external magnetic field the magnetization relaxes towards the ˆezaxis which can be either in or out of plane. Consequently, we con-sider/vectorm s(t)=mz sˆez+δ/vectorms(t), where mz s=±1 and δ/vectorms(t)i s a small deviation of the magnetic moment normal to the easy(ˆe z) axis. Solving the resulting linearized LLG equations of motions, the poles of the transverse dynamical susceptibilityyield two oscillating modes with resonance frequencies ω j,given by /parenleftbiggωj γ−i/vectorτ0 DL·/vectorEext/parenrightbigg2 =/parenleftbig ω0 j/parenrightbig2+2i/Gamma1jωj γ,j=x,y, (3a) ω0 j=/radicalbig KxKy+2λ/primeKj M, (3b) where M=|Ms|,Kj=K(2) j+K(4), and the AFMR linewidth /Gamma1j≡/Gamma1r+/Gamma1ex j=1 2M(λ/primeα0+Kjαd)( 4 ) can be separated into a relativistic component originating from the angular momentum transfer between the collinearAFM orientation and the crystal through the SOC, and anexchange component that originates from the spin currentexchange between the two AFM sublattices. For a system withuniaxial MCAE, Eq. ( 3a) can be used in both cases of out-of- and in-plane precessions with K (2) x,y=|K(2) ⊥|andK(2) y=0, K(2) x=|K(2) ⊥|, respectively, where |K(2) ⊥|is the amplitude of the out-of-plane MCAE. Equation ( 3a) is the central result of this Rapid Communication which is used to calculatethe AFMR frequency and linewidth and their correspondingcurrent-induced effects. A more general form of Eq. ( 3a)i n the presence of an external magnetic field along the precessionaxis is presented in the Supplemental Material [ 32]. Equation ( 3a) also yields the effective Gilbert damping α eff j≡δIm(ωj) δRe(ωj)=λα0+Kjαd 2M/radicalbig KxKy+2λKj,j=x,y. (5) Similarly to the linewidth, αeff jcan be separated into the relativistic, αr j=/Gamma1r j/ω0 jand exchange, αex j=/Gamma1ex j/ω0 j, contri- butions. To understand the origin of the relativistic componentof the AFMR linewidth, one can use a unitary transformationinto the rotating frame of the AFM direction, where α 0can be written in terms of the matrix elements of ˆHSOCusing the spin-orbital torque correlation (SOTC) expression [ 14], also often referred to as Kambersky’s formula [ 13], α0=¯h πNkM/summationdisplay /vectorkTr(ˆA/vectork[ˆHSOC,σ+]ˆA/vectork[ˆHSOC,σ−]).(6) Here, ˆA/vectork=Im(Gr /vectork) is the spectral function, ˆGr /vectorkis the retarded Green’s function calculated at the Fermi energy, 2 σ±=σx± iσyare the spin ladder operators, and Nkis the number of k-point sampling in the first Brillouin zone. A similar approach applied to the intersublattice elements of the damping tensor leads to a relationship between differentelements of α ss/prime, rather than an explicit expression for each element. This is due to the fact that in the rotating frameof one sublattice, the other sublattices precess. Therefore,to calculate α dwe employ the original torque correlation expression [ 14], αd=/summationdisplay s¯h πNkM/summationdisplay /vectorkTr/parenleftbigˆA/vectorkˆ/Delta1s /vectorkˆσ+ˆA/vectorkˆ/Delta1s /vectorkˆσ−/parenrightbig , (7) 220410-2DAMPING AND ANTIDAMPING PHENOMENA IN METALLIC … PHYSICAL REVIEW B 98, 220410(R) (2018) TABLE I. Calculated sublattice magnetic moment ( Ms), magnetocrystalline anisotropy energy per unit cell ( K⊥ 2), intersublattice exchange coupling per unit cell ( λ), ratio of the resistivity ( ρxx) to the broadening parameter η, and the experimental values of ρxx. We also list values ofαd,α/vectorms/bardbl/vectora(/vectorc) 0 for sublattice magnetization parallel to the /vectora(/vectorc) axis, the relativistic ( αr ⊥) and exchange ( αex ⊥) damping parameter for the out-of-plane oscillation mode, for ηandη/10 corresponding to the high- and low-temperature regimes, respectively. Finally, we list values of the sublattice current-induced fieldlike ( τ0,/vectorE/bardbl/vectora(/vectorc) FL ) and antidampinglike ( τ0,/vectorE/bardbl/vectora(/vectorc) DL ) components of the spin-orbit torques under an external electric field along the /vectora(/vectorc) axis for room-temperature broadening. |Ms|c/a K⊥ 2 λρ xx/η ρexpt xx ηα dα/vectorms/bardbl/vectorc 0 α/vectorms/bardbl/vectora 0 αr ⊥ αex ⊥ τ0,/vectorE/bardbl/vectora(/vectorc) FL τ0,/vectorE/bardbl/vectora(/vectorc) DL (μB) (meV) (eV) (μ/Omega1cm meV)(μ/Omega1cm) (meV) (10−3)( 1 0−3)( 1 0−3)( 1 0−3)( 1 0−3Å) (10−3Å) FeRh 3.1 1 +x−1.2x0.44 3.4 ≈100a29 0.25 0.8 0.8 1 .7/√|x|1.5√|x|33 (33) −14 (−14) 2.9 2.5 0.27 0.27 0 .6/√|x|15√|x| MnRh 3.1 0.94 −0.7 0.42 0.57 95b166 0.13 3.3 3.9 10 0.6 10 (7) 6 ( −3) 16.6 0.45 1.5 1.7 5 2 MnPd 3.9 0.93 −0.6 0.5 2.6 223c103 0.3 0.5 0.6 1.6 0.9 −2(−5) 93 (4) 10.3 1.8 0.1 0.5 1.3 5.6 MnPt 3.8 0.93 0.45 0.48 2.7 119,d164c48 0.43 2.2 7.1 6.7 1.2 −15 (17) 1 (11) 4.8 3.5 1.5 21 4.6 10 MnIr 2.6 0.97 −5.9 0.4 0.5 176–269e350 0.22 36 35 39 3.6 7 (13) 18 ( −7) 35 0.36 14 11 12 6 aReference [ 33]. bReference [ 34]. cReference [ 35]. dReference [ 36]. eReferences [ 35,37–39]. where ˆ/Delta1s /vectorkis the exchange spitting of the conduction electrons for sublattice s[32]. Since for AFMs with Néel temperatures above room tem- perature λ/greatermuchKj, one might conclude that αr/greatermuchαexand the effects of the intersublattice spin exchange on the AFMRlinewidth become negligible. However, since |α ss|is propor- tional to the intersublattice hopping strength [ 32], one can ex- pect to have /bardblαss/prime/bardbl/greatermuchα0. Therefore, the interplay between the relativistic and exchange terms is material dependent, wherefor systems with λ/greatermuchK j, the effect of the intersublattice spin exchange on the AFMR linewidth may dominate. The crystal structure, conventional and primitive cell, and the AFM ordering of the MnX (X=Pt, Pd, Ir, Rh) family of metallic bulk AFMs and the biaxially strained AFM bulkFeRh is shown in Fig. 1. The details of the electronic structure calculations of the various damping and antidamping proper-ties are described in detail in the Supplemental Material [ 32]. Table Ilists the ab initio results of the sublattice magnetic moment M s,c/aratio, magnetocrystalline anisotropy energy K(2) ⊥, intersublattice exchange interaction λ, and ratio of the longitudinal conductivity to the broadening parameter ρxx/η for the FeRh and Mn Xsystems, respectively. We also list experimental values of the room-temperature ρxxwhich were used to determine the broadening parameter. For FeRh we provide the linear dependence of K(2) ⊥as a function of biaxial strain, x≡c/a−1, which shows that under compressive (tensile) biaxial strain the magnetization is along the c(a) axis [40]. For the Mn Xfamily the magnetization is along the aaxis except for MnPt. The MCA values for both Mn Xand FeRh are in good agreement with previous ab initio calculations [40–43].We also list in Table Ivalues of αdandα/vectorms/bardbl/vectora(/vectorc) 0 for sublat- tice magnetization parallel to the /vectora(/vectorc) axis, and the relativistic (αr ⊥) and exchange ( αex ⊥) damping components of the effective Gilbert damping for ηat room temperature and η/10 corre- sponding to low temperature. The decrease (increase) of thedamping constants with decreasing temperature is associatedwith the conductivitylike (resistivitylike) regime where theinterband (intraband) scattering contribution is dominant. Wefind that for the ηvalue corresponding to room temperature, the AFMR linewidth is mostly dominated by the relativisticcomponent, while at low temperatures the two components arecomparable in magnitude. For FeRh a relatively large strain(i.e.,x≈0.1) is required to render the exchange component to have a significant contribution to the AFMR linewidth atlow temperature. In Fig. 2(a) we show the variation of α dandα/vectorms/bardbl/vectora 0 withηfor cubic FeRh as a representative example. We find that in the experimentally relevant range of η(≈10– 100 meV), α0is in the resistivity regime where the in- terband component is dominant. On the other hand, αd decreases monotonically with η, suggesting that the intra- band component is dominant. Unlike α0which may depend on the orientation of the Néel ordering, αdis relatively isotropic. Finally, Table Ilists the values for the current-induced FL and DL intersublattice torque coefficients τ0,i FL/DLunder an external electric field along the i(aorc) direction. The sublattice torques are determined by fixing the orien-tation of the ¯ssublattice magnetization and calculating the torque for different magnetization orientations of the ssub- lattice, using the symmetric and antisymmetric correlation 220410-3FARZAD MAHFOUZI AND NICHOLAS KIOUSSIS PHYSICAL REVIEW B 98, 220410(R) (2018) FIG. 2. (a) Sublattice Gilbert damping αd(dashed blue curve) andα0(solid red curve) components for bulk cubic FeRh vs broad- ening parameter η. (b) Sublattice current-induced FL (dashed blue) and DL (red solid) torque coefficients for FeRh under an external electric field along the aaxis. The coefficients were calculated by fitting the vector dependence of the DL [ ∝/vectorms×(/vectorms×/vectorm¯s)] and FL (∝/vectorms×/vectorm¯s) expressions for the symmetric and antisymmetric components in Eq. ( 8a), respectively. The insets display the top view of the vector field of the FL and DL torques for cone angles /lessorequalslant30◦. expressions [ 44], /vectorτS s;i=¯h πNkMs/vectorms×/summationdisplay /vectorkTr/parenleftBigg ˆA/vectorkˆ/Delta1s /vectork/vectorˆσˆA/vectork∂ˆH/vectork ∂ki/parenrightBigg , (8a) /vectorτAS s;i=2 MsNk/vectorms×/summationdisplay nm/vectorkRe⎡ ⎣Im/bracketleftbig/parenleftbigˆ/Delta1s /vectork/vectorˆσ/parenrightbig nm/parenleftbig∂ˆH/vectork ∂ki/parenrightbig mn/bracketrightbig (εn/vectork−εm/vectork−iη)2⎤ ⎦fn/vectork. (8b) Here,fn/vectorkis the Fermi-Dirac distribution function and εm/vectorkare the eigenvalues of the Hamiltonian ˆH/vectork. Having determined the torques, we fit the results to the expected τ0,i FL/vectorms×/vectorm¯sand τ0,i DL/vectorms×(/vectorms×/vectorm¯s) expressions and find the values for the FL and DL torque coefficients. The calculations reveal thatthe symmetric (antisymmetric) torque expression leads to theDL (FL) component, in contrast to the SOT results in HM/FM bilayers [ 44]. Figure 2(b) displays the current-induced FL and DL inter- sublattice torques under an external electric field along theadirection for FeRh, as a representative example, versus the broadening parameter η. Note the FL component that originates from the antisymmetric torque term [Eq. ( 8b)] is relatively insensitive to η(or temperature). On the other hand, the DL intersublattice torque varies almost linearly with η(for η< 0.1 eV) and is of extrinsic origin. Thus, in the ballistic regime where the electronic spin diffusion length is infinite,there is no current-induced transfer of angular momentum be-tween the two sublattices, as it would violate the conservationlaw of total angular momentum. In the extreme opposite limit,where the spin diffusion length is much smaller than the latticeconstant, each sublattice can be viewed as a magnetic leadin a spin valve system where the intersublattice DL torque isanalogous to the DL spin transfer torque. In summary, we have employed ab initio based calcula- tions to investigate the AFMR phenomena in Mn X(X=Ir, Pt, Pd, Rh) and biaxially strained FeRh metallic AFMs inthe presence or absence of an external electric field. Wedemonstrate that both the AFMR linewidth and effectiveGilbert damping parameter can be separated into relativisticand exchange contributions, where the former dominates atroom temperature while the latter becomes significant at lowtemperatures. We find that both the AFMR linewidth and theintersublattice exchange interaction (and hence the AFMRfrequency and Néel temperature) can be tuned by the externalelectric field. 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PhysRevB.96.054444.pdf

PHYSICAL REVIEW B 96, 054444 (2017) Gate-controlled magnon-assisted switching of magnetization in ferroelectric/ferromagnetic junctions Yaojin Li,1Min Chen,1Jamal Berakdar,2and Chenglong Jia1,2 1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, China 2Institut für Physik, Martin-Luther Universität Halle-Wittenberg, Halle (Saale) 06099, Germany (Received 21 June 2017; revised manuscript received 23 July 2017; published 30 August 2017) Interfacing a ferromagnet with a polarized ferroelectric gate generates a nonuniform, interfacial spin density coupled to the ferroelectric polarization. This coupling allows for an electric field control of the effective fieldacting on the magnetization. To unravel the usefulness of this interfacial magnetoelectric coupling we investigatethe magnetization dynamics of a ferroelectric/ferromagnetic multilayer structure using the Landau-Lifshitz-Baryakhtar equation. The results demonstrate that the interfacial magnetoelectric coupling is utilizable as ahighly localized and efficient tool for manipulating magnetism by electrical means. Ways of enhancing thestrength of the interfacial coupling and/or its effects are discussed. DOI: 10.1103/PhysRevB.96.054444 I. INTRODUCTION Electrical control of magnetism has the potential to boost spintronic devices with a number of novel functionalities[1–5]. To mention an example, magnetization switching can be achieved via a spin-polarized electric current by virtue of thespin-transfer torque or the spin-orbital torque in the presence ofa spin orbital interaction [ 6–15]. One may also use an electric field to manipulate the magnetization dynamics [ 16–22]i n which case the electric field may lead to modulations in the charge-carrier density or may affect the magnetic propertiessuch as the magnetic moment, the exchange interaction,and/or the magnetic anisotropy [ 16–19]. Compared to driving magnetization via a spin-polarized current, an electric fieldgoverning the magnetization has a clear advantage as it allowsfor nonvolatile device concepts with significantly reduced energy dissipation. On the other hand, an external electric field applied to an itinerant ferromagnet (FM) is shielded by chargeaccumulation or depletion caused by the spin-dependentscreening charge that extends on a length scale of only a fewangstroms into the FM [ 23]. This extreme surface confinement of screening hinders its utilization to steer the magneticdynamics of bulk or a relatively thick nanometer-sized FM [24,25]. Experimentally, ultrathin metallic FM films were thus necessary to observe an electric-field influence on the dynamicof an FM [ 16,17,26]. In this work we show that while the spin-polarized screening charge is surface confined, in the spin channel a local nonuniform spiral spin density builds up at the interface and goes over into the initial uniform (bulk) magnetization awayfrom the interface. Hence, this interfacial spin spiral acts as a topological defect in the initial uniform magnetization vector field. The range of the spiral defect is set by the spin diffusion length λ m[27] which is much larger than the charge screening length. This spin-spiral constitutes a magnetoelectric coupling with a substantial influence on the transversal magnetization dynamics of the FM layer with a thickness as large as tens of nanometers [ 28]. The interfacial spiral spin density can be viewed as a magnonic accumulation stabilized by the interfacial, spin-dependent charge rearrangement at the contact region between the FM and the ferroelectrics (FE) (with aFE polarization P) and by the uniform (bulk) magnetization of FM far away from the interface [ 30].Presponds to an external electric field and so does the magnetic dynamics. As shown below, this magnonic-assisted magnetoelectric coupling arising when using a dielectric FE gate allows a (ferro)electric-field control of the effective driving field that governs the magnetization switching of a FM layer with a thickness on the range of the spin diffusion length λm, which is clearly of an advantage for designing spin-based, nonvolatile nanoelectronic devices. In Sec. IIwe discuss the mathematical details of the spin-spiral magnetoelectric coupling, followed by its imple-mentation into the equations of motion for the magnetiza-tion dynamics in Sec. III. In Sec. IVresults of numerical simulations are presented and discussed showing to whichextent the spin-spiral magnetoelectric coupling can allowfor the electric-field control of the magnetization in FE/FMcomposites. Ways to enhance the effects are discussed andbrief conclusions are made in Sec. V. II. INTERFACIAL MAGNETOELECTRIC COUPLING Theoretically, the above magnon accumulation scenario maybe viewed as follows: When a FE layer with remanentelectric polarization Pand surface charges σ FEis brought in contact with an itinerant (charge-neutral) FM, bond rearrange-ments occur within a few atomic layers in the interface vicinity[31]. On the FM side, the rearranged spin polarized charge density implies a spin configuration different from the bulkone. The modifications of the magnitude of the interfaciallocal magnetic moments are dictated by hybridization andcharge transfer and were studied thoroughly both theoreticallyand experimentally (e.g., Ref. [ 31]). Here we are interested in the consequence on the long-range magnetic order extendingto the asymptotic bulk magnetization. In the mean-fieldformulations, the induced spin density sis exchange coupled with the localized magnetic moments S, which can be treated classically as an effective magnetization M=− gμ B a3Swith μB,g, andabeing the Bohr magneton, gfactor, and lattice constant, respectively. The associated sdexchange coupling 2469-9950/2017/96(5)/054444(6) 054444-1 ©2017 American Physical SocietyY AOJIN LI, MIN CHEN, JAMAL BERAKDAR, AND CHENGLONG JIA PHYSICAL REVIEW B 96, 054444 (2017) energy at the FM interface is Fsd=JsdM Mss·m, (1) where mis a unit vector in the direction of M.Msis the intrinsic saturation magnetization. Within the Stoner mean-field theory[32] the spin polarization ηof the electron density in transition FM metals is usually less than 1; we can decompose theinduced spin density sas [30] s=s /bardbl+s⊥, (2) where s/bardblrepresents the spin density whose direction follows adiabatically the intrinsic magnetization Mat an instantaneous timet.s⊥describes the transverse deviation from M.G i v e n that the steady-state charge accumulation entails much higherenergy processes than spin excitations, in the absence of acharge current across the FE/FM interface, the spin diffusionnormal to the FM/FE interface (hereafter refereed to as the z direction with its origin at the interface) follows the dynamicequation (see Refs. [ 30,33] for details) ∂s /bardbl ∂tm+s/bardbl∂m ∂t+∂s⊥ ∂t−D0∇2 zs/bardbl−D0∇2 zs⊥ =−s/bardbl τsf−s⊥ τsf−s⊥×m τex, (3) where D0is the diffusion constant and τex≈¯h/(2Jsd).τsf is the spin-flip relaxation time due to scattering from impu- rities, electrons, and phonons; τsf∼10−12–10−14s[34] and τex/τsf∼10−2for typical FM metals [ 27]. The time-derivative terms∂s/bardbl ∂t,∂m ∂t, and∂s⊥ ∂tbelow THz are negligible compared withs/τsfands/τex. Thus the steady state is set by [ 30] D0∇2 zs/bardbl=s/bardbl τsfandD0∇2 zs⊥=s⊥×m τex, (4) implying an exponentially decaying spiral spin density [ 30], s/bardbl=ησFM λmee−z/λ m, (5) s⊥=(1−η)QmσFM ee−(1−i)Qm·r. (6) HereσFM=σFE≈/epsilon1FEEis the surface charge density due to the electric neutrality constraint at the interface, /epsilon1FEandEare the dielectric permittivity of FE and an applied normal electricfield, respectively. λ m=√D0τsfis the effective spin-diffusion length and the normal spin spiral wave vector Qm=1√2D0τexˆez. Clearly, in the presence of the exchange interaction with long- range FM ordering, the accumulated (magnonic) spin densityextends in the FM system over a nanometer characteristiclength ( ∼λ mbeing 38 ±12 nm in Co [ 27]) which is much larger than the electrostatic screening length (a few angstroms),albeit both are associated by largely different energy scales. As we are interested in the effect of the low-energy accumulated magnonic density on the spin dynamic in FM wecan safely assume that the spin-dependent charge excitationsare frozen (because of the higher energy scale) during the(GHz-THz) spin dynamics in the FM. Treating the magneticdynamics, we consider the additional effective magnetoelectricfieldH meacting on the magnetization dynamics M(t) due to the interfacial spin order. To leading terms, from the sdinteraction FIG. 1. Schematics of the plane of variation for the magnetization M=M{cosφsinθ,sinφsinθ,cosθ}. The FM/FE interface is re- ferred to as the xyplane. Hme θandHme φare the transversal components of the interface magnetoelectric field. energy [Eq. ( 1)] we derive Hme=−δFsd/δM=−Jsd Mss. (7) We choose nanometer-thick layers Co and BaTiO 3as prototyp- ical FM and FE layers for estimating the characteristics of Hme. The density of surface charges [ 35] reads σFE=0.27 C/m2and the parameters of Co are [ 36]Ms=1.44×106A/m,K1= 4.1×105J/m3,λm=40 nm [ 27], and η=0.45 [ 32]. We find thus |Hme|≈0.2 T with Jsd≈0.1e V/atom and the FM thickness dFM=40 nm. Such a strong magnetoelectric field is comparable with the uniaxial anisotropic fieldK1 Ms≈0.3To f Co. More importantly, note that the nonadiabatical componentH me ⊥is always perpendicular to the direction of magnetization M, acting as a fieldlike torque and a dampinglike torque at all time (cf. Fig. 1), which would play a key role for electric-field assisted magnetization switching. III. MAGNETIZATION DYNAMICS We start from the Landau-Lifshitz-Baryakhtar equation (LLBar) [ 37–39] for the magnetization dynamics at the FM interface, ∂M ∂t=−γM×Heff+ˆ/Lambda1r·Heff−ˆ/Lambda1e,ij∂2Heff ∂xi∂xj, (8) where γis the gyromagnetic ratio. The last two terms describe the local and nonlocal relaxations. ˆ/Lambda1rand ˆ/Lambda1eare generally the relaxation tensors of relativistic and exchangenatures, respectively. The anisotropy of relaxations decreaseswith increasing temperature. Experimentally, the isotropy ofrelaxations were discussed in Ref. [ 40]. We can represent the relaxation tensors as ˆ/Lambda1 r=λrand ˆ/Lambda1e=λewhere λr=γαM s andλe=γgμ B¯hG 0/(8e2) with αandG0being the Gilbert damping coefficient and the conductivity of FM system,respectively. eis the electron charge. In contrast to the Landau-Lifshitz-Gilbert equation, the LLBar equation doesnot conserve the magnitude of the magnetization capturingthe magnetic relaxations in metals, especially the case for FM 054444-2GATE-CONTROLLED MAGNON-ASSISTED SWITCHING OF . . . PHYSICAL REVIEW B 96, 054444 (2017) metal interfaces. This is necessary in our case to ensure that the local magnetic order which is in equilibrium with the interfaceregion relaxes to the asymptotic bulk magnetization. By introducing M=Mminto the LLBar equation, we infer the following equation for the direction of magnetization [ 39]: ∂m ∂t=−γm×Heff+1 MsR⊥ (9) withR=λrHeff−λe∇2 zHeffandR⊥=−m×(m×R).Here Heff=H0 eff+Hme(10) is the effective magnetic field, in which H0 efffollows from the functional derivative of the free-energy density via [ 41] H0 eff=−δF0/δM, F0=−K1(sin2θcos2φsin2θu+cos2θcos2θu) −K1 2sin 2θsin 2θucosφ −/parenleftbig Ks/dFM−μ0M2 s/2/parenrightbig cos2θ−M·B. (11) K1is the uniaxial magnetocrystalline anisotropy energy, Ks is the magnetic surface anisotropy contribution which is significant for relatively thin magnetic films and favors amagnetization out of the xyplane. μ 0M2 sdenotes the demag- netizing field contribution, which favors a magnetization inplane. M·Bis the Zeemann interaction and θ uis the tilted angle of the easy axis from the zdirection. Clearly, the nonuniform effective field Hmedue to the s-d interaction with the exponentially decaying spiral spins wouldgive rise to a nonlocal damping of the magnetization dynamics.Considering that the contribution of the induced spin densityto the spatial distribution of local ferromagnetic moments issmall, we have /angbracketleftbig ∇ 2 zs⊥/angbracketrightbig =2Q2 m(/angbracketleftsφ ⊥/angbracketrightˆeθ−/angbracketleftsθ ⊥/angbracketrightˆeφ). (12) Without loss of generality one can take /angbracketleftsφ ⊥/angbracketright=/angbracketleftsθ ⊥/angbracketright=1√ 2/angbracketlefts⊥/angbracketright. It is also convenient to redefine some dimensionless pa- rameters which are ˜dFM=dFM λm,˜t=tγT≈28tGHz, and ˜Jsd=Jsd eVσFM Psλm dFMwith the FE spontaneous polarization Ps.I n the following ˜Jsdis taken as an adjustable parameter in view of ferroelectric tuning of magnetoelectric field Hme. IV . NUMERICAL RESULTS AND DISCUSSIONS For the surface anisotropy Ks≈10−3J/m2andμ0M2 s/2≈ 1.3×106J/m3of Co sample [ 36], the dominant contribution of the anisotropic term ( Ks/dFM−μ0M2 s/2) in Eq. ( 11) has the form either Ks/dFMor−μ0M2 s/2 depending on the thickness dFM, i.e., the magnetization will be either normal to the FM interface ( θu=0) or in the interface plane ( θu=π/2). Case I . Normal FM magnetization with θu=0: The free energy density is F0=−Keffcos2θ−M·B,K eff=K1+Ks dFM−μ0M2 s 2, (13)which leads to H0 eff=2Keff Mscosθˆez (14) without an applied magnetic field B. The LLBar equation reads then ∂θ ∂˜t=γe +√ 2Hme ⊥−αKeff Mssin 2θ, (15) sinθ∂φ ∂˜t=−γe −√ 2Hme ⊥+Keff Mssin 2θ (16) withγe ±=1−2Q2 mλe γM s±λr γM s=γe±α. Clearly, under a weak interfacial ME field, the condition Hme ⊥=√ 2α γe +Keff Mssin 2θ (17) can be satisfied; the polar angle θends up processionally in the equilibrium state [cf. Fig. 2(a) with∂θ/∂ ˜t=0]. Otherwise, the strong transversal field Hme ⊥results in a magnetization flip over the normal ˆezdirection [Fig. 2(b)]. Considering that the ME field depends linearly on the applied electric field andthe reciprocal of FM thickness, one would expect a transitionfrom the magnetization procession around the zaxis (for a small electric field Eand/or relatively thick FM layers) to the magnetization flip over the normal direction (for a strongelectric field and/or ultrathin FM film) at the critical points, asdemonstrated in Figs. 2(c) and2(d). Case II . In-plane magnetization with θ u=π/2: Disregard- ing the surface anisotropy ( Ks/dFM/lessmuchμ0M2 s/2) for a thick FM film, the effective magnetic field reads H0 eff=2K1/M ssinθcosφˆex−μ0Mscosθˆez+B (18) and the magnetization favors an in-plane ˆexaxis, which means φ(0)=0 with the external magnetic field B=0. Upon some simplifications the LLBar equation reads ∂θ ∂˜t=γe +√ 2Hme ⊥+αμ0Ms 2sin 2θ+αK1 Mssin 2θcos2φ −K1 Mssinθsin 2φ+αBcosθcosφ−Bsinφ, (19) sinθ∂φ ∂˜t=−γe −√ 2Hme ⊥−μ0Ms 2sin 2θ−αK1 Mssinθsin 2φ −K1 Mssin 2θcos2φ−Bcosθcosφ−αBsinφ. (20) In the absence of an external magnetic field B, the magnetiza- tion dynamics is determined by three parameters: α,Hme ⊥, and K1/M s. First, let us ignore the damping terms for small Gilbert damping coefficient α; the weak ME field Hme ⊥satisfies∂θ ∂˜t=0 and∂φ ∂˜t=0, resulting in a relocation of the magnetization with an equilibrium tilted angle in the vicinity of xaxis, as shown in Fig. 3(a). However, when Hme ⊥is stronger than the anisotropic fieldK1/M sand the demagnetization field μ0Ms, no solutions exist for ∂θ/∂ ˜t=0 at all time; the magnetization possesses az-axial flip mode in the whole spin space [cf. Fig. 3(b)] similar to the case of normal FM magnetization. On the otherhand, after accounting for terms containing αin the LLBar 054444-3Y AOJIN LI, MIN CHEN, JAMAL BERAKDAR, AND CHENGLONG JIA PHYSICAL REVIEW B 96, 054444 (2017) FIG. 2. Dynamics of the normal magnetization. The polar angle θvs dimensionless time ˜tfor different ME field (a) ˜Jsd=0.005 and (b) ˜Jsd=0.03, respectively. Panels (c) and (d) demonstrate the thickness and electric-field dependence of δθmax=θmax−θ(0), where θ(0) andθmaxare respectively the initial value [ θ(0)=0] and the maximum value of the polar angle during the time evolution of magnetization. δθmax=πindicates a magnetization flip over the normal ˆezdirection. Here, α=0.1,/epsilon1FE=1000, and Keff∼K1. equations, we would have additional magnetization rotation around the zaxis [Fig. 3(c)]. Further insight into the detailed characterization of magnetization dynamics is delivered bynumerics for a varying strength of the ME field ˜J sdand the uniaxial anisotropy K1/M sin Fig. 4withα=0.1. There are two new phases, the z-axial flip mode and the z-axial rotational mode, which were unobserved in the FM systems inthe absence of interface ME interaction. With decreasing thedamping α, the area of the z-axial rotational mode shrinks and vanishes eventually. By applying an external magnetic field B along the xdirection, only slight modifications are found in the phase diagram. However, the initial azimuthal angle φ(0) deviates from the easy axis with a rotating magnetic field Bin thexyinterface plane. Considering the LLBar equations with the initial condition θ(t=0)=π/2, we have ∂θ ∂˜t|˜t=0≈γe +√ 2Hs−d ⊥−K1 Mssin 2φ(0) (21)with a small damping α. As the dynamic equation is sensitive to the initial azimuthal angle φ(0), the calculations show that the magnetization dynamics may change between the processionalmode around the xaxis and the z-axial flip or z-axial rotational mode, depending on the initial value of φ(0). Phenomenologically, such z-axial flip mode and z-axial rotational mode are exhibited as a precessional motion of themagnetization with a negative damping, as shown in the exper-imental observation for polycrystalline CoZr/plumbum mag-nesium niobate–plumbum titanate (PMN-PT) heterostructures[5], where an emergence of positive-to-negative transition of magnetic permeability was observed by applying external elec-tric field. There is also some analogy between these nonequi-librium switching behaviors in FM/FE heterostructures andthe negative damping phenomenon in trilayer FM/normal-metal/FM structures, in which the supplying energy is thoughtto be provided by injecting spin polarized electrons from FIG. 3. Dynamics of the in-plane magnetization for different interface ME field and anisotropic field: (a) ˜Jsd=0.03 and 2 K1/M s=0.6T , (b)˜Jsd=0.03 and 2 K1/M s=0.3T ,a n d( c ) ˜Jsd=0.015 and 2 K1/M s=0.1 T, respectively. Here ˜dFM=1a n dα=0.1. 054444-4GATE-CONTROLLED MAGNON-ASSISTED SWITCHING OF . . . PHYSICAL REVIEW B 96, 054444 (2017) FIG. 4. Phase diagrams of the in-plane magnetization dynamics withα=0.1,˜dFM=1, and B=0: (a) the localized precessional mode, (b) the z-axial flip mode, and (c) the z-axial rotational mode, respectively. The characterization of the dynamic behavior of the magnetization in three different phases is illustrated in Fig. 3.I n s e t s show the corresponding time evolution of the magnetization in eachphase. an adjacent FM layer, magnetized in the opposite direction compared to the FM layer under consideration [ 42,43]. V . CONCLUSION AND OUTLOOK The above theoretical considerations along with numerical simulations for specific FE/FM composites endorse that themagnetization dynamics can be controlled by an electricfield of moderate strength. The excitations triggered by theelectric field are transferred to the spin system via theinterface spiral-mediated magnetoelectric coupling and mayresult in a magnetization switching. This direct electric-field control of the magnetization switching offers a qualitatively different way to manipulate magnetic devices swiftly andwith low-power write capability. On the other hand, even FIG. 5. Schematic structure diagram of the FE/FM multilayer system with enhanced magnetoelectric effect. The arrows mark the directions of the FE polarization Pand the FM magnetization M, respectively. though the spin-mediated magnetoelectric coupling has a much longer range than the surface localized charge-mediatedFE/FM coupling, its range is still limited by the spin-diffusionlength which is material dependent but yet is in the rangeof several tens of nanometers. Hence, the full power of thepredicted effect is expected for multilayer systems such asthose schematically shown in Fig. 5: Starting from a bilayer structure with a thick FE interfaced with a FM layer, whichhas a thickness in the range of the spin-diffusion length, wesuggest to cap this structure with a spacer layer, for instance an(oxide) insulator. Repeating the whole structure as proposedin Fig. 5allows for a simple serial extension from a double to multilayer structure while enhancing the influence of themagnetoelectric coupling. ACKNOWLEDGMENTS This work is supported the National Natural Science Foun- dation of China (Grant No. 11474138), the German ResearchFoundation (Grant No. SFB 762), the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT-16R35), and the Fundamental Research Funds for theCentral Universities. [1] W. Eerenstein, N. D. Mathur, and J. F. Scott, Multiferroic and magnetoelectric materials, Nature (London) 442,759 (2006 ). 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PhysRevB.98.165444.pdf

PHYSICAL REVIEW B 98, 165444 (2018) Magnetization nutation induced by surface effects in nanomagnets R. Bastardis,*F. Vernay,†and H. Kachkachi‡ Laboratoire PROMES CNRS (UPR-8521) & Université de Perpignan Via Domitia, Rambla de la thermodynamique, Tecnosud, 66100 Perpignan, France (Received 19 July 2018; revised manuscript received 4 October 2018; published 30 October 2018) We investigate the magnetization dynamics of ferromagnetic nanoparticles in the atomistic approach taking account of surface anisotropy and the spin misalignment it causes. We demonstrate that such inhomogeneousspin configurations induce nutation in the dynamics of the particle’s magnetization. More precisely, in additionto the ordinary precessional motion with frequency f p∼10 GHz, we find that the dynamics of the net magnetic moment exhibits two more resonance peaks with frequencies fcandfnwhich are higher than the frequency fp: fc=4×fp∼40 GHz is related with the oscillations of the particle’s magnetic moment between the minima of the effective potential induced by weak surface anisotropy. On the other hand, the much higher frequency fn∼ 1 THz is attributed to the magnetization fluctuations at the atomic level driven by exchange interaction. We havecompared our results on nutation induced by surface effects with those rendered by the macroscopic approachbased on the Landau-Lifshitz-Gilbert equation augmented by an inertial term (proportional to the second-ordertime derivative of the macroscopic moment) with a phenomenological coefficient. The good agreement betweenthe two models has allowed us to estimate the latter coefficient in terms of the atomistic parameters such as thesurface anisotropy constant. We have thus proposed a different origin for the magnetization nutations as beinginduced by surface effects and have interpreted the corresponding resonance peaks and their frequencies. DOI: 10.1103/PhysRevB.98.165444 I. INTRODUCTION Research on nanoscale magnetic materials benefits from a continuing impetus owing to an increasing demand of ourmodern societies for ever smaller devices with ever higherstorage densities and faster access times. These devices arethe upshot of spintronics or magnonic applications with ma-terials exhibiting thermally stable magnetic properties, energyefficient magnetization dynamics, and controlled fast magne-tization switching. In the macroscopic approach, the magneti-zation dynamics on time scales ranging from microsecondsto femtoseconds can be described by the Landau-Lifshitz-Gilbert (LLG) equation [ 1–3] dm dt=m×/parenleftbigg γHeff−α mdm dt/parenrightbigg , (1) where Heffis the effective field acting on the macroscopic magnetic moment mcarried by the nanomagnet, γthe gy- romagnetic factor, and αthe phenomenological damping pa- rameter. Equation ( 1) describes the relaxation of mtowards Heffwhile maintaining a constant magnitude, i.e., /bardblm/bardbl=m, assuming that the nanomagnet is not coupled to any heatbath or other time-dependent external perturbation. The firstterm on the right hand of Eq. ( 1) describes the precessional motion of the magnetic moment maround the effective field H eff. This is well known from the classical mechanics of a gyroscope. Indeed, if an external force tilts the rotation axis of *roland.bastardis@univ-perp.fr †francois.vernay@univ-perp.fr ‡hamid.kachkachi@univ-perp.frthe gyroscope away from the direction of the gravity field, therotation axis no longer coincides with the angular-momentumdirection. The consequence is an additional movement of thegyroscope around the axis of the angular momentum. Thismotion is called nutation . In the case of the magnetic moment m, this additional motion (nutation) can occur if the effective field H effbecomes time dependent. Indeed, in the presence of a time-dependent magnetic field (rf or microwave field),there appears the fundamental effect of transient nutationswhich has been widely investigated in NMR [ 4], EPR [ 5,6], and optical resonance [ 7]; see also the review by Fedoruk [8]. Magnetic or spin nutation was first predicted in Joseph- son junctions [ 9–13] and was later developed using various approaches based on first principles [ 14], electronic structure theory [ 15–19], or in a macrospin approach where the LLG equation ( 1) is extended by a second-order time derivative [20–23]. Magnetic nutation may also occur at the level of atomic magnetic moments on ultrashort time scales. For instance, inRef. [ 24] it is argued that nutation is enhanced for atomic spins with low coordination numbers and that it occurs ona time scale of the magnetic exchange energy, i.e., a fewtens of femtoseconds. More generally this spin nutation iscaused by a nonuniform spin configuration which leads toan inhomogeneous effective field H effwhose magnitude and orientation are different for different lattice sites. These spatialinhomogeneities are a typical result of surface effects thatbecome very acute in nanoscale magnetic systems such asmagnetic nanoparticles. In this work we adopt this atomisticapproach and show that, for a magnetic nanoparticle regardedas a many-spin system, a model henceforth referred to asthemany-spin problem (MSP), surface effects do induce 2469-9950/2018/98(16)/165444(9) 165444-1 ©2018 American Physical SocietyR. BASTARDIS, F. VERNAY , AND H. KACHKACHI PHYSICAL REVIEW B 98, 165444 (2018) nutations of the net magnetic moment of the nanoparticle. More precisely, this approach involves at least three energyscales, namely the core (magnetocrystalline) anisotropy, thesurface anisotropy, and exchange coupling. Consequently,there appear at least three different frequencies: the lowestcorresponds to the ordinary precession around a fixed axiswith a constant projection of the net magnetic moment on thelatter and the other two frequencies correspond to nutationswith a time-dependent projection of m. In the limiting case of weak surface effects, inasmuch as the spin configurationinside of the nanomagnet can be regarded as quasicollinear,the dynamics of the nanomagnet can be described with thehelp of an effective macroscopic model for the net magneticmoment of the nanomagnet. This model will be referred toin the sequel as the effective one-spin problem (EOSP). More precisely, it has been shown that a many-spin nanomagnet of agiven lattice structure and energy parameters (on-site core andsurface anisotropy, local exchange interactions) can approx-imately be modeled by a macroscopic magnetic moment m evolving in an effective potential [ 25] that comprises second and fourth powers of the components m α,α=x,y,z . Within this approach we find two precession frequencies fpandfc: fpcorresponds to the precession of maround the reference z axis with constant mzandfcto the frequency of oscillations of mbetween the four minima of the effective potential produced by its quartic term. When surface or boundary effects are toostrong, the spin configuration can no longer be considered asquasicollinear, and thereby the effective model is no longer agood approximation, one has to take account of higher-orderfluctuations of the atomic spins. Doing so numerically, wefind an additional nutation frequency f nwhich is much higher thanfpandfcas it corresponds to a movement of the atomic spins that occurs at the time scale of the magnetic exchangeinteraction. Observation of nutation in magnetization dynamics is dif- ficult because the effect is rather small and the correspondingfrequency is beyond the detection capabilities of standardtechniques using the magnetization resonance such as thestandard FMR or a network analyzer with varying frequency.Nevertheless, from the high-frequency FMR (115–345 GHz)spectra obtained for ultrafine cobalt particles, the authors ofRef. [ 26] inferred low values for the transverse relaxation time τ ⊥(two orders of magnitude smaller than the bulk value) and suggested that this should be due to inhomogeneousprecession which possibly originates from surface spin dis-order. Likewise, in Ref. [ 24] it was shown that nutation in magnetization dynamics of nanostructures occurs at edgesand corners, with a much smaller amplitude than the usualprecession. More recently, Li et al. [27] performed HF-FMR measurements of the effective magnetic field and showed thatthere was an additional contribution which is quadratic infrequency as obtained from the additional term d 2m/dt2in the LLG equation [ 20,21]. To sum up, in this work we first demonstrate that surface effects or, more generally, noncollinear atomic spin orderinginduce nutation in the magnetization dynamics of a nano-magnet. Second, it establishes a clear connection betweennutation within our atomistic approach and that describedby the quadratic frequency dependence of the effective fieldas described within the macroscopic approach includingmagnetization inertia. If we cannot provide an analytical con- nection between the corresponding parameters, we do providea numerical correspondence between the phenomenologicalparameter of the macroscopic approach and our atomisticparameters, such as the surface anisotropy constant. We alsopropose an intermediate macroscopic model which accountsfor all three resonance frequencies. Finally, we speculate thatthe resonance peak at f c, induced by surface effects, provides a route for observing nutation in well prepared assembliesof nanomagnets. All in all, the main objective of the presentwork is to show that magnetic nutation in a nanoparticle origi-nates from surface effects which lead to spin noncollinearitieswithin the nanoparticle and the latter affect the high-frequencydynamics. The paper is organized as follows. In Sec. IIwe present our model of many-spin nanomagnets, discuss the effects of sur-face anisotropy on the magnetization dynamics, and presentour main results showing two resonance peaks which weattribute to two kinds of magnetization nutation. In Sec. II A we also discuss a particular situation where it is possible toanalytically derive the equation of motion of the net magneticmoment of the (many-spin) nanomagnet, which makes it clearthat nutation is related with the spin fluctuations at the atomiclevel. In Sec. II B we compare our results with other works in the literature mostly based on the macroscopic approachusing the Landau-Lifshitz-Gilbert equation augmented by aninertial term, and establish a quantitative relationship betweenthe corresponding sets of parameters. Finally, in Sec. IIIwe summarize the main results of this work and then discussthe possibility to observe the magnetization nutations inresonance experiments. II. MODEL AND HYPOTHESIS We consider a nanomagnet with Natomic spins sion a simple cubic lattice described by the (classical) Hamiltonian(/bardbls i/bardbl= 1) H=−1 2/summationdisplay i,jJijsi·sj−h·N/summationdisplay i=1si−N/summationdisplay i=1Han,i, (2) where h=μaH,μais the magnetic moment associated with the atomic spin, His the magnetic field, Jijis the exchange interaction (that may be different for core-surface, surface-surface, and core-core links), and H an,iis the anisotropy energy at site i,a function of sisatisfying the symmetry of the problem. More precisely, Han,iis the energy of on-site anisotropy which is here taken as uniaxial for core spins andof Néel’s type for surface spins [ 28], i.e., H an,i=/braceleftbigg−Kc(si·ez)2,i ∈core, +1 2Ks/summationtext j∈n.n.(si·eij)2,i∈surface ,(3) where eijis the unit vector connecting the nearest neighbors at sites iandjandKc>0 and Ks>0 are respectively the core and surface anisotropy constants. The spin dynamics is described by the Landau-Lifshitz equation (LLE) for the atomic spin si dsi dτ=si×heff,i−αsi×(si×heff,i), (4) 165444-2MAGNETIZATION NUTATION INDUCED BY SURFACE … PHYSICAL REVIEW B 98, 165444 (2018) with the (normalized) local effective field heff,iacting on si being defined by heff,i=−δH/δsi;τis the reduced time defined by τ≡t τs, (5) where τs=μa/(γJ) is a characteristic time of the system’s dynamics. By way of example, for cobalt J=8 meV leading toτs=70 fs. Henceforth, we will only use the dimensionless timeτ. In these units, heff,i=μaHeff,i/J. Equation ( 4)i sas y s t e mo f2 Ncoupled equations for the spins si,i={1,..., N}. In this work, it is solved using itera- tive optimized second-order methods using Heun’s algorithm. The particle’s net magnetic moment is defined as s0=1 NN/summationdisplay i=1si. (6) Next, we introduce the unit vector of s0 m≡1 s0s0,s 0=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1 N/summationdisplay isi/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble. (7) As discussed in the Introduction, because of surface effects or spatial inhomogeneities of the effective field (mainly due tothe fact that the anisotropy constant and the easy axis dependon the lattice site), the spin configuration is not uniform foran arbitrary set of energy parameters. As a consequence, thevectors s iare not all parallel to each other and as such we may define their deviation from the direction mas [29] si=(m·si)m+ψi, where we have introduced the vector ψi≡si−(m·si)m. It can be easily checked that ψiis perpendicular to si, i.e., ψi·si=0=ψi·mand satisfies/summationtextN i=1ψi=0. This means that the transverse vector ψicontains the Fourier components with k/negationslash=0and describes spin waves in the nanomagnet. Whereas in the standard spin wave theory s0is a constant corresponding to the ground-state orientation, here it is treatedas a time-dependent variable. Note that using the condition /bardbls i/bardbl= 1, we may write si=m√ 1−ψ2 i+ψi. Now, in the realistic case, Ks/lessmuchJ, the deviations of sifrom the homogeneous state mare small and one can adopt the following approximation: si/similarequalm/parenleftbig 1−1 2ψ2 i/parenrightbig +ψi≡m+δsi, where δsi≡−1 2ψ2 im+ψi. (8) Then, we define the magnetization deficit due to surface anisotropy as follows: /Delta1m≡−1 N/summationdisplay im·δsi. (9) Using Eq. ( 8) and/summationtextN i=1ψi=0we obtain /Delta1m=1 2N/summationdisplay iψ2 i=1−1 N/summationdisplay i(m·si)=1−s0.(10)In what follows, we will show that the magnetization nutations are a consequence of the magnetization deficit /Delta1m (which is due to the transverse spin fluctuations ψi) with respect to s0. In order to study nutation, we compute /Delta1m(τ) or the components mα(τ), with α=x,y,z . In the sequel, we will mainly study the latter as their behavior clearly illustratesthe precession and nutation phenomena. In the next sectionwe present a sample of our results obtained for a cube-shapednanomagnet described by the Hamiltonian ( 2) together with the anisotropy model in Eq. ( 3). A. Magnetization nutation induced by surface anisotropy In order to clearly illustrate the central result of this work, namely that spin noncollinearities, induced by surfaceanisotropy, lead to nutation in the magnetization dynamicsof a nanomagnet, we consider a simple shape, e.g., a cube.Today, nanocubes (of iron or cobalt) are routinely investigatedin experiments since their synthesis has become fairly wellcontrolled [ 30–35]. Here we consider a nanocube of N=729 spins located on the vertices of a simple cubic lattice (i.e.,N x=Ny=Nz=9). This choice has the main advantage that the number of core spins ( Nc=343) is comparable to that of surface spins ( Ns=386), a configuration suitable for study- ing the role of surface effects versus core properties. Then, wecompute the time evolution of the net magnetic moment mby solving the system of equations ( 4), using Eqs. ( 6) and ( 7). We start from the initial state s i(t=0)=(1/2,1/2,1/√ 2), which corresponds to all spins tilted to the same angle withrespect to the zaxis of the laboratory frame. Let us first consider the case of a nanocube with anisotropy energy defined in Eq. ( 3), i.e., uniaxial for core spins and of Néel’s type for surface ones. A surface spin is defined as thespin whose coordination number is smaller than in the core(here six on a simple cubic lattice). For simplicity, we set allexchange couplings equal to a reference value Jeverywhere in the core, on the surface and at the interface betweenthem, i.e., J cc=Jcs=Jss=J. All energy constants are then measured in units of J, so that J=1 andkc≡Kc/J=0.01, ks≡Kc/J=0.1. These are typical values extracted from experiments on cobalt and iron nanoparticles [ 36–38]. In this calculation, the external magnetic field and damping are bothset to zero. Solving the LLE ( 4) renders the components of m(τ) de- fined in Eq. ( 7). These are shown in Fig. 1. In the lower panel, m x(τ) andmy(τ) show the usual precessional movement of m(τ) around the zaxis. The corresponding frequency for the parameters given above is fp=14 GHz. If m(τ)w e r et o exhibit only this precession, its component mz(τ) would be a constant with a constant tilt angle between m(τ) and the z axis. However, as can be seen in the upper panel, it is clearlynot the case. Indeed, we see a double modulation of m z(τ) in time; there are two oscillations: (i) one with frequencyf c=4×fp=56 GHz and an amplitude that is an order of magnitude smaller than precession and (ii) another oscillationwith the much higher frequency f n=1.1 THz and an ampli- tude two orders of magnitude smaller than precession. Theseoscillations are further illustrated in Fig. 2. Let us now discuss the origin of these oscillations. As discussed in the Introduction, in the case of not-too-strong 165444-3R. BASTARDIS, F. VERNAY , AND H. KACHKACHI PHYSICAL REVIEW B 98, 165444 (2018) 0 1 02 03 04 0 50 60 70-0.6-0.4-0.200.20.40.60 1 02 03 04 0 50 60 700.70.710.720.73 mx mymz t(ps)Reduced magnetic moment FIG. 1. Time evolution of the average magnetization components (mx,my,mz) for a nanomagnet of N=9×9×9=729 spins with uniaxial anisotropy in the core and Néel surface anisotropy onthe surface. surface effects, the MSP may be mapped onto an EOSP [25,39–41] for the net magnetic moment mof the particle evolving in an effective potential containing a quadratic anda quartic term in the components of m. This work has recently been extended to cube-shaped magnets [ 42]. So for a nano- magnet within the EOSP approach the equation of motionreads dm dτ=m×/bracketleftbig 2k2mzez−4k4/parenleftbig m3 zez+m3 yey+m3 xex/parenrightbig/bracketrightbig .(11) Herez=6 is the coordination number and k2=kcNc/N. For a sphere k4=κk2 s/zJ, where κis a surface integral [ 25], and for a cube we have k4=(1−0.7/N1/3)4k2 s/zJ [42]. The components mα(τ) rendered by Eq. ( 11) exhibit two resonance peaks corresponding to (i) the ordinary precessionwith frequency f pand (ii) the oscillation with frequency fc between the minima of the effective potential induced by the term in k4. The latter is due to the fact that the effective magnetic moment has now to explore a potential-energy sur-face that comprises four saddle points because of the cubicanisotropy (with constant k 4). Therefore, mzvisits a minimum each time mpasses over one of these saddle points, and this occurs with the frequency fc=4×fp=56 GHz. Thus FIG. 2. Illustration of the nutation of the macrospin s0in the presence of damping ( α/negationslash=0). We have used nonzero damping for later reference.01234 56 7-0.8-0.6-0.4-0.200.20.40.60.8 0 0.5 11.5 20.7050.710.715 mxmy mz mz(0)(1+ Δm) t (ps)ΔmReduced magnetic moment FIG. 3. Time evolution of the net magnetic moment compared with that of the magnetization deficit. The exchange parametersare homogeneous ( J=J cs=Js=1); both surface and core spins have a uniaxial anisotropy along the zaxis with surface anisotropy ks=0.1 and core anisotropy kc=0.01. fcis a consequence of the first correction stemming from (relatively weak) surface effects. In the case of larger values of ksand thereby stronger spin noncollinearities, it is no longer possible to map themany-spin particle onto an effective particle. One then has tofully deal with the spin fluctuations. As a consequence it is nolonger an easy matter to derive an equation of motion similarto Eq. ( 11) in the general case. Nevertheless, in Ref. [ 29] two relatively simpler configurations of anisotropy were stud-ied, namely a uniform uniaxial anisotropy (with the sameconstant and orientation) or a random anisotropy (with thesame constant and random orientation). It was then possibleto derive a system of (coupled) equations for m(t) andψ i(t) containing higher-order terms in ψi(t); see Eqs. (21) and (26) in Ref. [ 29]. In the present situation with a nonuniform anisotropy configuration, these higher-order contributions areresponsible for the nutation movement with frequency f n, as they lead to a net magnetization deficit; see Eq. ( 10) and Fig. 3where the plot of /Delta1mshows such a movement. More precisely, these fluctuations of the atomic spins lead to a pre-cession of the latter around their local effective field h eff,ithat evolves in time due to exchange interaction. Unfortunately, inthis complex situation it is a rather difficult task to derive anexplicit expression for h eff,iand thereby an analog of Eq. ( 11). However, we may consider a simpler model of a nanomagnetwith a uniaxial anisotropy having the easy axis along e zfor all sites, but with a constant that is different in the core from thaton the surface, i.e., e i/bardblez,kc/negationslash=ks. Therefore, instead of the model in Eq. ( 3) we consider the following one: Han,i=/braceleftbigg −kc(si·ez)2,i ∈core, −ks(si·ez)2,i∈surface .(12) This configuration is quite plausible especially in elon- gated nanomagnets such as nanorods [ 43] and nanowires [ 44] where the magnetostatic energy is strong enough to induce 165444-4MAGNETIZATION NUTATION INDUCED BY SURFACE … PHYSICAL REVIEW B 98, 165444 (2018) an effective uniaxial anisotropy along the major axis of the nanomagnet. Then, it is possible to derive a system of equations for mandψi(to second order in ψi). The equation for ψiis cumbersome and thus omitted here as it is not necessary tothe discussion that follows. That of mreads dm dτ/similarequalm×2 N/summationdisplay iki/parenleftbig mz+ψz,i−mzψ2 i/parenrightbig ez +m×2 N/summationdisplay iki⎡ ⎣1 N/summationdisplay j/parenleftBigg mzψ2 j 2/parenrightBigg⎤ ⎦ez −m×2 N/summationdisplay iki[(mz)2+mzψz,i]ψi. (13) First, setting ψi=0above we obtain dm/dτ=m× 1 N/summationtext i(2ki)mzez=m×2keffmzez, which describes the pre- cession of maround the effective field heffwith heff=2keffmzez,k eff=Nckc+Nsks N. (14) This clearly shows that nutation disappears in the absence of the spin fluctuations ψi. Furthermore, projection on the z axis of Eq. ( 13) yields the relation dmz/dτ/similarequalmzd(/Delta1m)/dτ, where/Delta1mis the magnetization deficit defined in Eqs. ( 9) and (10). Upon integrating over time we obtain (to lowest order in ψi) mz(τ)/similarequalmz(0)[1+/Delta1m(τ)]. (15) This expression shows that mz(τ) and/Delta1m(τ)h a v et h es a m e frequency, as confirmed by the green dots in the inset of Fig. 3. Therefore, this simplified model emphasizes the appear- ance of two relevant frequencies: the low-frequency of theordinary precession and the higher frequency of nutationrelated with spin fluctuations at the atomic level driven by theexchange coupling. These two frequencies clearly show up inFig. 1(blue wiggles in m z). Furthermore, in Eq. ( 13)w ea l s o see that the spin fluctuations ψiare directly coupled to the anisotropy parameters ki, and this implies that the nutation’s magnitude is not only related to the ratio of surface-to-corespin number, but also to the value of the anisotropy constants.Note, however, that the connection between Eq. ( 13) and Eq. ( 11) is not a direct one, and one has to eliminate the fast variables ψ i, e.g., by integration or by making use of their equations of motion in a perturbative way. Finally, we have systematically varied the physical pa- rameters ( Jij,ki) and studied the effect on nutation and the frequencies fp,fc, and fn. First, we confirm that in the absence of surface anisotropy (e.g., the same uniaxialanisotropy k cfor all spins), no nutation has been observed. This is a direct consequence of the fact that, in this specificcase, there is no magnetic inhomogeneity in the particle thatcan lead to a nonuniform effective field. Second, we findthat the precession frequency f pmainly depends on kcsince all spins are parallel to each other forming a macrospin thatprecesses in the effective uniform field. In general, this wouldalso include the shape anisotropy and the dc magnetic field.On the other hand, the frequency f nstrongly depends on the exchange coupling as can be seen in Table I.TABLE I. Precession and nutation frequencies for fixed values of the exchange couplings J=Jcs=Js=1 (top) and for fixed values of core and surface anisotropies kc=0.005 and ks=0.01 (bottom). Precession frequency Nutation frequencykc ksfp(GHz) fn(THz) 0.001 0.001 3.2 0 0.001 0.01 19 ≈1 0.001 0.05 86 ≈1 0.001 0.1 170 ≈1 0.005 0.01 25 ≈1 0.005 0.05 93 ≈1 0.005 0.1 180 ≈1 0.01 0.1 185 ≈1 Precession frequency Nutation frequencyJcs Jsfp(GHz) fn(THz) 2 2 25 1.5 1 2 25 1.25 11 2 5 1 1 0.5 25 0.75 1 0.1 25 0.25 0.5 0.5 25 0.7 We have checked that the observed magnetic nutation features also occur in cube-shaped particles of different size(20×20×20 or 30 ×30×30). We have obtained qualita- tively the same oscillating behavior. A more detailed system-atic and quantitative analysis of this data is being carried outand will be published later as it is beyond the scope of thepresent paper. We have also performed these calculations for a spherical nanomagnet which has a different distribution of coordinationnumbers than in a cube. The results are qualitatively the samebut the nutation frequency f nis higher. B. Comparison with the macroscopic approach to magnetization nutation As discussed in the Introduction, magnetization nutation has been studied by many authors within the macroscopicapproach based on Eq. ( 1) augmented by an inertial term pro- portional to the second time derivative of the (macroscopic)magnetic moment m: dm dτ=m×/bracketleftbigg heff−αm×heff−β τsd2m dτ2/bracketrightbigg , (16) where the coefficient βis often taken proportional to the damping parameter αand to a phenomenological relaxation timeτ1related with, e.g., the dynamics of the angular mo- mentum, which is on the order of a femtosecond. In Ref. [ 14], it was shown that the inertial damping results from high-ordercontributions to the spin-orbit coupling effect and is relatedto the Gilbert damping through the magnetic susceptibilitytensor. In the sequel, we shall use the notation ˜β≡β/τ sand this macroscopic model, with the equation of motion ( 16) and phenomenological parameter ˜β, will be referred to as the inertial one-spin problem (IOSP). Solving the equation above, in the presence of dc and ac magnetic fields, i.e., heff=hdc+hac,O l i v e et al. [21] observed two resonance peaks, the first of which corresponds 165444-5R. BASTARDIS, F. VERNAY , AND H. KACHKACHI PHYSICAL REVIEW B 98, 165444 (2018) to the ordinary large-amplitude precession at frequency fp and a second resonance peak, at a much higher frequency fnwith smaller amplitude, that was attributed to the nutation dynamics. A number of other authors made similar observa-tions by also investigating the IOSP model [ 16,20,24,27]. In Ref. [ 21] it was suggested that ω nutation =2πfn=1/β. Let us summarize the situation. On one hand, we have the EOSP model (applicable when surface effects are nottoo strong) in which the dynamics of the net magneticmoment is described by the equation of motion ( 11). The solution to the latter only exhibits two resonance peaks withfrequencies f pandfc. On the other hand, we have the IOSP model where the equation of motion is given by ( 16) (with the phenomenological parameter ˜β) whose solution only provides the two resonance peaks with frequencies fpandfn.N o w ,t h e MSP approach, when treated in its full generality, provides uswith a self-consistent scheme in which all three frequenciesappear in a natural manner. In particular, it shows hownutation with the high-frequency f nsets in, in the presence of surface effects which induce noncollinear spin configurationsand generate high-frequency and small-amplitude spin-waveexcitations. See, for example, a thorough study of spin-waveexcitations in a nanocube in Ref. [ 45]. However, within the MSP approach, the derivation of the equation of motionfor the net magnetic moment m(and the spin-wave vectors ψ i) is too cumbersome, if not intractable. This issue will be investigated in the future. Nevertheless, in the case of aspherical nanomagnet, a Helmholtz equation was derivedfor the vectors ψ iin Ref. [ 41], see Eq. ( 8) therein, which is nothing other than the propagation equation for the spin wavesdescribed by ψ i. Now, using the expansion si/similarequalm+ψi,w e may infer that the exchange contribution J/Delta1siis proportional to the second time derivative of mand, as such, the coefficient β∝1/Jand thereby ωnutation ∝J. The exact relation will be investigated in a future work. Nevertheless, there is a specific situation in which we can establish a clear connection between the MSP approachand the IOSP model. This is the case of weak surfaceeffects or, equivalently, a quasicollinear spin configuration. Indeed, under this condition, we may combine the EOSPand IOSP models and write an equation of motion whosesolution renders all three frequencies, f p,fc, andfn.M o r e precisely, we start from Eq. ( 11) with the effective field heff=2k2mzez−4k4(m3 zez+m3 yey+m3 xex) and add a term similar to that in Eq. ( 16) with coefficient ˜β, leading to the following equation of motion: dm dτ=m×/bracketleftbig 2k2mzez−4k4/parenleftbig m3 zez+m3 yey+m3 xex/parenrightbig/bracketrightbig −˜βm×d2m dτ2, (17) where again we have k2=kcNc/Nand for a cube k4=(1−0.7/N1/3)4k2 s/zJ, and ˜β=β/τs. Henceforth, this model will be referred to as the inertial effective one-spin problem (IEOSP). Compared with Eq. ( 16), the field heffhas been replaced in Eq. ( 17) by the effective field produced by the combined uniaxial and cubic anisotropies, induced by relatively weaksurface effects. Of course, we could also include an externalmagnetic field and a demagnetizing field in the EOSP equa-tion. The advantage of the IEOSP model is twofold: (i) itrenders the three resonance peaks at the frequencies f p,fc, andfnand (ii) it allows us to establish a clear connection between the phenomenological parameter ˜βand the atomistic physical parameters of the MSP approach, such as the surfaceanisotropy constant k s. For solving Eq. ( 17) one needs to set the initial velocity form. For Néel’s anisotropy, the system exhibits several different velocities, depending on the spin position in thestructure (edge, corner, face, or core). In this case, one wouldhave to set up a global constraint by imposing an initialvelocity for the net magnetic moment ( 6). In practice, we have found it sufficient to use the average velocity ˙m(t=0)=/summationtext i˙si(t=0)/N. The solution of Eq. ( 17) is plotted (in dots) in Fig. 4(left). 01 0 2 0 3 0 4 0 50 60 70-0.6-0.4-0.200.20.40.601 0 2 0 3 0 4 0 50 60 700.70.710.720.73 sxsymx mxsxmz t(ps)Reduced magnetic moment 0 5 10 15 20 25 30 35 40 45 50 55-0.8-0.400.40.80 5 10 15 20 25 30 35 40 45 50 550.70710.707120.707140.7071600 . 5 1 1.5 2 t (ps)mxmymzMSP MSP MSPIOSP IOSP IOSPReduced magnetic moment FIG. 4. Time evolution of the components of the macroscopic magnetic moment m(dots) and the net magnetic moment (lines) for MSP. On the left, for Néel surface anisotropy, the MSP results are compared to the IEOSP model ( 17) and on the right, for uniaxial anisotropy, they are compared to the IOSP model ( 16). The inset shows a magnification of the mz(t) component with a typical period ∼0.9p s(ωnutation /similarequal7T H z ) . 165444-6MAGNETIZATION NUTATION INDUCED BY SURFACE … PHYSICAL REVIEW B 98, 165444 (2018) In Fig. 4we show the results from the MSP, IOSP, and IEOSP models. The parameters for the MSP calculations aret h es a m ea si nF i g . 1, i.e.,k c=0.001,ks=0.01. On the left, we compare the MSP approach to the IEOSP model Eq. ( 17) withk2=0.00475 ,k4=0.0011,˜β=2.25. On the right, the MSP approach is compared to the IOSP model ( 16) with the effective field given in Eq. ( 14) and parameters keff= 0.00576 ,˜β=2.2. Note that instead of using the expression forkeffin Eq. ( 14) one might perform a fitting to the MSP curves. Doing so, we find a slight discrepancy in keff(here 0.00585) as well as in the initial velocities ˙mα(t=0),α= x,y,z . This is most likely due to the fact that the velocity average does not exactly account for the spin noncollinear-ities. All in all, the results from the MSP approach are invery good agreement with those rendered by the macroscopicmodel, either IOSP or IEOSP, upon using the correspondingeffective field for the given anisotropy configuration in MSP,namely ( 12)o r( 3), respectively. In Fig. 4(left), the MSP approach with the anisotropy model ( 3) is in good agreement with the IEOSP model with a given parameter ˜β. Both models exhibit the three frequencies f p,fc, andfn. Regarding the nutation with frequency fn, there is a slight discrepancy in amplitude between the two models. As mentioned above,this is attributed to the average over the initial velocities.In Fig. 4(right) we see that, for MSP with the anisotropy model ( 12), the IOSP model ( 16) with the effective field ( 14) recovers the two resonance peaks with f pandfn.W ed r a w the attention of the reader to the difference in time scale andamplitude for the zcomponent. Indeed, the oscillations of the zcomponent on the right are to be identified with the wiggles of the same component on the left panel. In Ref. [ 21]t h e authors argued that ω nutation =1/β. Here, from Fig. 4(right) we extract β/similarequal1.43×10−13s, which should be compared to ˜βτs/similarequal1.5×10−13s, showing a good agreement. Finally, the major difference between the results on the left and right panels is related with the frequency fc. This implies that the model with uniaxial anisotropy, same easy axis butdifferent constants in the core and the surface, cannot accountfor this frequency. This confirms the fact that the latter isrelated with the inhomogeneity of the on-site anisotropy easydirection and thereby with the cubic effective anisotropy as a first correction to surface effects. In general, the relation between ˜βand the frequency f n, within the MSP approach, is difficult to derive analyticallysince ˜βdepends on the atomic parameters. Nevertheless, we have tried to establish a quantitative correspondence betweenthe phenomenological parameter ˜βand the microscopic pa- rameters such as k s,kcor the effective parameters k2,k4that appear in Eq. ( 11). Accordingly, in Fig. 5we plot 1 /˜βas the result of the best fit between the MSP and IEOSP models. Onthe right panel of Fig. 5, this is done for the uniform uniaxial anisotropy model ( 12) and on the left panel for the anisotropy model in Eq. ( 3). These results show that 1 /˜βis nearly linear ink sand that the value of the phenomenological parameter ˜βinvolved in the IEOSP model can be estimated for a given value of the surface anisotropy constant ks, which is an input parameter of the MSP approach. Finally, we have investigated the effect of damping with parameter α[see Eq. ( 4)] within the MSP approach. The results are shown in Fig. 6for the magnetization deficit. Together with the 3D picture in Fig. 2, this indicates how the spin fluctuations and thereby /Delta1mdecays in time towards zero. This result is obviously in agreement with those of Fig. 1(b) inRef. [ 21]. We would like to emphasize, though, that the IOSP approach in its actual formulation cannot account for the mag-netization nutation in the absence of damping because thecoefficient βappearing in Eq. ( 16) before the inertial term d 2m/dt2is proportional to damping and thus vanishes when the latter does. This is one of the major discrepancies withthe MSP approach since the latter does produce magnetizationnutation even in the absence of such a damping ( α=0). How- ever, within the MSP approach the surface-induced nutationis due to local spin fluctuations and is thus affected by thespin-spin correlations or multimagnon processes which causedamping effects and relaxation of the magnetization deficit.But in the absence of a coupling of the spin subsystem tothe lattice, referred to in Ref. [ 46] as the direction relaxation, these damping effects are not dealt with in this work and thisis why when we set α=0 the time evolution of m αor/Delta1mis undamped, but does exhibit nutation. 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.0500.050.10.150.20.250.30.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050.45750.460.46250.465 k4 k2 ksks1 β~ 0.005 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.20.2750.30.3250.350.3750.40.4250.450.475 0.005 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.200.0250.050.0750.10.1250.15β1 keff ksks~ FIG. 5. Left: k4/k2and 1/˜βagainst ks. Right: keffand 1/˜βagainst ks. 165444-7R. BASTARDIS, F. VERNAY , AND H. KACHKACHI PHYSICAL REVIEW B 98, 165444 (2018) 0 1 02 03 04 0 50 60 7000.00050.0010.0015Δm t (ps) FIG. 6. Time evolution of the magnetization deficit, showing the damping effect. Same parameters as in Fig. 1with a damping value ofα=0.01. III. CONCLUSION AND PERSPECTIVES We have proposed an atomistic approach for studying the effects of surface anisotropy and investigating nutation inthe magnetization dynamics in ferromagnetic nanoparticles.We have then shown that because of these effects, whichinduce spin noncolinearities leading to nonuniform local ef-fective fields, the magnetization dynamics exhibits severalresonance peaks. In addition to the ordinary precessionalmotion with frequency f p∼10 GHz, we have shown that the dynamics of the net magnetic moment exhibits two moreresonance peaks with frequencies f candfn, which are higher than the FMR frequency. Indeed, fc=4×fp∼40 GHz is related with the oscillations of the particle’s magnetic momentbetween the minima of the effective potential induced byweak surface anisotropy. On the other hand, the much higherfrequency f n∼1 THz is attributed to the magnetization fluc- tuations at the atomic level driven by exchange couplingwhich becomes relevant in the presence of strong nonuniformspin configurations. We have compared our results on nutation induced by sur- face effects with those rendered by the macroscopic approachbased on the Landau-Lifshitz-Gilbert equation augmented byan inertial term (proportional to the second-order time deriva-tive of the macroscopic moment) with a phenomenologicalcoefficient. The good agreement between the two modelsmakes it possible to estimate this coefficient in terms of theatomistic parameters such as the surface anisotropy constant.In brief, the atomistic approach provides an origin for themagnetization nutations and a global and a self-consistentpicture that renders all three frequencies.In the case of not-too-strong surface effects, an effective model renders two frequencies f pandfc. On the other hand, the Landau-Lifshitz-Gilbert equation with an inertial termonly renders the frequencies f pandfn. Now, in the case of arbitrary surface effects, it is a rather difficult task to derive aneffective equation of motion for the magnetization dynamics.As such, we have proposed an intermediate model that startsfrom the effective model established for weak surface effectsand added magnetization inertia through the term proportionalto the second-order time derivative of the magnetization.Then, we have shown that this macroscopic model is in verygood agreement with the atomistic approach and renders allresonance peaks and their frequencies. This establishes a clearquantitative connection between the phenomenological pa-rameters of the macroscopic approach to the atomistic energyparameters. Our final word is devoted to the possibility of experimen- tal observation of nutation in magnetization dynamics. Firstof all, establishing the fact that surface effects do inducemagnetization nutation may provide us with an additionalmeans for observing the latter. Indeed, surface effects onferromagnetic resonance in nanoparticles have been studiedfor a few decades now. For example, the authors of Ref. [ 26] reported on high-frequency FMR (115–345 GHz) spectra forultrafine cobalt particles and inferred rather small values of thetransverse relaxation time τ ⊥, which suggests that this should be due to an inhomogeneous precession caused by (relativelyweak) surface spin disorder. There are several other pub-lications on FMR measurements on magnetic nanoparticles[47–52]. However, these measurements can only capture the two frequencies f pandfc. Nevertheless, the observation of the frequency fc, which is on the order of tens of GHz, should be an easy matter using a network analyzer with variablefrequency covering this range. Doing so would clearly provethe existence of the first nutation motion induced by spindisorder as a consequence of surface anisotropy. A variantof the FMR spectroscopy, called magnetic resonance forcemicroscopy [ 53–55], yields a highly sensitive local probe of the magnetization dynamics and consists in mechanicallydetecting the change in the longitudinal fluctuations of themagnetization, i.e., /Delta1m z. This would be particularly suited for detecting the fluctuations in mzseen in Figs. 1and4, if not for the mismatch in the frequency range. Now, the frequencyf nis rather in the optical range and we wonder whether the corresponding oscillations could be detected by coupling themagnetization of the nanoparticle to a plasmonic nanoparticleof gold or silver, thus making use of the magnetoplasmoniccoupling evidenced in many hybrid nanostructures [ 56–58]. Graphene plasmons is another promising route for detectionof THz radiation [ 59]. ACKNOWLEDGMENT We would like to acknowledge useful discussions with J.-E. Wegrowe on his early work on magnetization nutation. [1] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935).[2] T. L. Gilbert, Ph.D. thesis, Illinois Institute of Technology, Chicago, 1956. 165444-8MAGNETIZATION NUTATION INDUCED BY SURFACE … PHYSICAL REVIEW B 98, 165444 (2018) [3] T. L. Gilbert, IEEE Trans. Magn. 40,3443 (2004 ). [4] H. C. Torrey, Phys. Rev. 76,1059 (1949 ). [5] N. C. Verma and R. W. Fessenden, J. Chem. Phys. 58,2501 (1973 ). [6] P. W. Atkins, A. J. Dobbs, and K. A. McLauchlan, Chem. Phys. Lett.25,105(1974 ). [7] G. B. Hocker and C. L. Tang, P h y s .R e v .L e t t . 21,591(1968 ). [8] G. G. Fedoruk, J. Appl. Spectrosc. 69,161(2002 ). [9] J.-X. Zhu and J. Fransson, J. Phys.: Condens. Matter 18,9929 (2006 ). [10] J. Fransson and J.-X. Zhu, New J. Phys. 10,013017 (2008 ). [11] J. Fransson, Nanotechnology 19,285714 (2008 ). [12] Z. Nussinov, A. Shnirman, D. P. Arovas, A. V . 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PhysRevB.97.134405.pdf

PHYSICAL REVIEW B 97, 134405 (2018) Magnetism of a Co monolayer on Pt(111) capped by overlayers of 5 delements: A spin-model study E. Simon,1,*L. Rózsa,2K. Palotás,3,4and L. Szunyogh1,5 1Department of Theoretical Physics, Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary 2Department of Physics, University of Hamburg, D-20355 Hamburg, Germany 3Department of Complex Physical Systems, Institute of Physics, Slovak Academy of Sciences, SK-84511 Bratislava, Slovakia 4MTA-SZTE Reaction Kinetics and Surface Chemistry Research Group, University of Szeged, H-6720 Szeged, Hungary 5MTA-BME Condensed Matter Research Group, Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary (Received 21 January 2018; published 9 April 2018) Using first-principles calculations, we study the magnetic properties of a Co monolayer on a Pt(111) surface with a capping monolayer of selected 5 delements (Re, Os, Ir, Pt, and Au). First we determine the tensorial exchange interactions and magnetic anisotropies characterizing the Co monolayer for all considered systems.We find a close relationship between the magnetic moment of the Co atoms and the nearest-neighbor isotropicexchange interaction, which is attributed to the electronic hybridization between the Co and the capping layers,in the spirit of the Stoner picture of ferromagnetism. The Dzyaloshinskii-Moriya interaction is decreased forall overlayers compared to the uncapped Co/Pt(111) system, while even the sign of the Dzyaloshinskii-Moriyainteraction changes in the case of the Ir overlayer. We conclude that the variation of the Dzyaloshinskii-Moriyainteraction is well correlated with the change of the magnetic anisotropy energy and of the orbital momentanisotropy. The unique influence of the Ir overlayer on the Dzyaloshinskii-Moriya interaction is traced by scalingthe strength of the spin-orbit coupling of the Ir atoms in Ir/Co/Pt(111) and by changing the Ir concentrationin the Au 1−xIrx/Co/Pt(111) system. Our spin dynamics simulations indicate that the magnetic ground state of Re/Co/Pt(111) thin film is a spin spiral with a tilted normal vector, while the other systems are ferromagnetic. DOI: 10.1103/PhysRevB.97.134405 I. INTRODUCTION Owing to promising technological applications, chiral mag- netic structures have become the focus of current experimentaland theoretical research activities [ 1,2]. Chiral magnetism is essentially related to the breaking of space-inversion symme-try, since in this case spin-orbit coupling (SOC) leads to theappearance of the Dzyaloshinskii-Moriya interaction (DMI)[3,4] that lifts the energy degeneracy between noncollinear magnetic states rotating in opposite directions. Noncollinearchiral magnetic structures stabilized by the DMI, such as spinspirals and magnetic skyrmion lattices, have been explored incrystals with bulk inversion asymmetry such as MnSi [ 5–7]. Magnetic thin films and multilayers with broken interfacialinversion symmetry represent another class of systems inwhich chiral magnetic structures can emerge. In these systems,magnetic transition-metal thin films are placed on heavy metal(e.g., Pt, Ir, W) substrates supplying strong spin-orbit inter-action. For instance, spin spiral ground states were reportedfor Mn monolayers on W(110) [ 8,9] and on W(001) [ 10], spin spirals and skyrmions were detected in the Pd/Fe/Ir(111)bilayer system [ 11–13], while in the case of an Fe monolayer on Ir(111) the formation of a spontaneous magnetic nanoskyrmionlattice has been observed [ 14]. Competing ferromagnetic (FM) and antiferromagnetic (AFM) isotropic exchange couplingsare also capable of stabilizing noncollinear spin structures *esimon@phy.bme.huin magnetic thin films and nanoislands [ 15–18], while the chirality of these structures is still determined by the DMI. Understanding and controlling the sign and strength of the DMI at metallic interfaces is one of the key tasks inexploring and designing chiral magnetic nanostructures. A large number of experiments has been devoted to the study of the influence of different nonmagnetic elements on theDMI at magnetic/nonmagnetic metal interfaces [ 19–21], also supported by first-principles calculations [ 22]. Recently, it was shown that at 3 d/5dinterfaces the trend for the DMI follows Hund’s first rule as the number of valence electrons in themagnetic layer is varied [ 23], while for a Co/Pt bilayer it was studied how the DMI depends on the number of occupied states close to the Fermi energy by resolving the DMI in reciprocalspace [ 24]. It was also demonstrated that the magnetic ground state of an Fe monolayer on 5 dmetal surfaces is strongly influenced by the electronic properties of the substrate [ 25,26]. Because of the interplay between large spin-orbit coupling andhigh spin polarizability, particular attention has been paid to the influence of the heavy metal Ir on the DMI. This includes the formation of noncollinear spin structures in ultrathinmagnetic films on Ir substrates [ 12–14] and the insertion of Ir into multilayer structures [ 27–30]. It was demonstrated that the insertion of Ir leads to a sign change of the DMI in the Pt/Co/Ir/Pt system [ 20,31], and it was suggested that the Ir/Co/Pt stacking order in magnetic multilayers can lead to an enhancement of the DMI [ 22,27]. Motivated by previous experimental and theoretical inves- tigations, in the present paper we explore the role of selected 2469-9950/2018/97(13)/134405(11) 134405-1 ©2018 American Physical SocietySIMON, RÓZSA, PALOTÁS, AND SZUNYOGH PHYSICAL REVIEW B 97, 134405 (2018) monatomic 5 d(Re, Os, Ir, Pt, Au) overlayers in influencing the magnetic properties of a Co monolayer deposited onPt(111). We focus on the investigation of how the electronichybridization with heavy metal capping layers possessingdifferent numbers of valence electrons and different strengthsof the spin-orbit interaction influences the magnetic propertiesof the Co layer. In Sec. II, the parameters of an extended classical Heisen- berg model are discussed, where the coupling between the spinsis described by tensorial exchange interactions using first-principles electronic structure calculations. It is also explainedhow these interactions can be converted to effective or micro-magnetic parameters. In Sec. III A, the modifications of the Co magnetic moments and of the nearest-neighbor (NN) isotropicexchange coupling between the Co atoms are found to correlatewith the change of the electronic states in the Co and the 5 d overlayers. In Sec. III B, the correlations between the DMI, the magnetic anisotropy energy (MAE), and the orbital momentanisotropy are highlighted. In the case of the Ir/Co/Pt(111)system, we find that the DMI in the Co monolayer changes signcompared to Co/Pt(111) and the systems with the other cappinglayers, and we scale the spin-orbit coupling of the Ir layer in order to get a more profound insight into this phenomenon. This investigation is supplemented by investigating the DMIand the MAE in Au 1−xIrx/Co/Pt(111) thin films with an alloy overlayer. Finally, in Sec. III C we determine the magnetic ground state of the Co monolayer on the Pt(111) substrate withdifferent capping layers using spin dynamics simulations. Theresults are summarized in Sec. IV. II. COMPUTATIONAL METHODS A. Details of ab initio calculations We performed self-consistent electronic structure calcula- tions for X/Co/Pt(111) ( X=Re, Os, Ir, Pt, Au) ultrathin films in terms of the relativistic screened Korringa-Kohn-Rostoker(SKKR) method [ 32,33]. For the case of chemically disordered overlayers, we employed the single-site coherent-potentialapproximation (CPA). We used the local spin-density approx-imation as parametrized by V osko et al. [34] and the atomic sphere approximation with an angular momentum cutoff of/lscript max=2. The energy integrals were performed by sampling 16 points on a semicircle contour in the upper complexenergy semiplane. The layered system treated self-consistentlyconsisted of nine Pt atomic layers, one Co monolayer, one X monolayer, and four layers of vacuum (empty spheres) betweenthe semi-infinite Pt substrate and semi-infinite vacuum. Formodeling the geometry of the thin films we used the value a 2D=2.774˚A for the in-plane lattice constant of the Pt(111) surface and fcc growth was assumed for both the Co andthe different overlayers. The distances between the atomiclayers were optimized in terms of V ASP calculations [ 35–37]. Relative to the interlayer distance in bulk Pt, we found aninward relaxation between 5 and 10% for the Co monolayerand between 8 and 15% for the different overlayers. In order to study the magnetic structure in the Co layer we use the generalized classical Heisenberg model H=−1 2/summationdisplay i,j/vectorsiJij/vectorsj+/summationdisplay i/vectorsiK/vectorsi, (1)where /vectorsidenotes the spin vector of unit length at site i,Jijis the exchange coupling tensor [ 38], and Kis the on-site anisotropy matrix. The tensorial exchange coupling can be decomposedinto an isotropic, an antisymmetric, and a traceless symmetriccomponent [ 31]: J ij=JijI+1 2/parenleftbig Jij−JT ij/parenrightbig +/bracketleftbig1 2/parenleftbig Jij+JT ij/parenrightbig −JijI/bracketrightbig . (2) The isotropic part Jij=1 3TrJijrepresents the Heisenberg couplings between the magnetic moments. The antisymmetricpart of the exchange tensor can be identified with the DMvector: /vectors i1 2/parenleftbig Jij−JT ij/parenrightbig /vectorsj=/vectorDij(/vectorsi×/vectorsj). (3) From the diagonal elements of the traceless symmetric part of the exchange tensor the two-site anisotropy may becalculated. The second term of Eq. ( 1) comprises the on-site anisotropy with the anisotropy matrix K. Note that for the case of C 3v symmetry the studied systems exhibit, the on-site anisotropy matrix can be described by a single parameter, /vectorsiK/vectorsi=K(sz i)2. The effective MAE of the system can be obtained as a sum ofthe two-site and on-site anisotropy contributions as will bediscussed in Sec. II C. Note that the sign convention for J ij, /vectorDij, andKis opposite to Ref. [ 31], from which we include the values for the Co/Pt(111) system without a capping layer forcomparison with the present results. The exchange coupling tensors were determined in terms of the relativistic torque method [ 38,39], based on calculating the energy costs due to infinitesimal rotations of the spins atselected sites with respect to the ferromagnetic state orientedalong different crystallographic directions. For these orienta-tions we considered the out-of-plane ( z) direction and three different in-plane nearest-neighbor directions, being sufficientto produce interaction matrices that respect the C 3vsymmetry of the system. The interaction tensors were determined for allpairs of atoms up to a maximal distance of 5 a 2D, for a total of 90 neighbors including symmetrically equivalent ones. B. Determining the ground state of the system To find the magnetic ground state of the Co monolayer, we calculated the energies of flat harmonic spin spiral configura-tions: /vectors i=/vectore1cos/vectork/vectorRi+/vectore2sin/vectork/vectorRi, (4) where /vectorkdenotes the spin spiral wave vector, /vectore1and/vectore2are unit vectors perpendicular to each other, and /vectorRiis the lattice position of spin /vectorsi. Substituting Eq. ( 4) into Eq. ( 1) yields 1 NESS(/vectork,/vectorn)=−1 2/summationdisplay /vectorRij1 2/parenleftbig TrJij−/vectornJsymm ij/vectorn/parenrightbig cos/vectork/vectorRij −1 2/summationdisplay /vectorRij/vectorDij/vectornsin/vectork/vectorRij+1 2(TrK−/vectornK/vectorn),(5) with/vectorn=/vectore1×/vectore2the normal vector of the spiral, /vectorRij=/vectorRj− /vectorRi, and Jsymm ij=1 2(Jij+JT ij). The ground state configuration was approximated by optimizing Eq. ( 5) with respect to /vectorkand 134405-2MAGNETISM OF A Co MONOLAYER ON Pt(111) CAPPED … PHYSICAL REVIEW B 97, 134405 (2018) /vectorn, and comparing it to the energy of the ferromagnetic state: 1 NEFM(/vectoreFM)=−1 2/summationdisplay /vectorRij/vectoreFMJij/vectoreFM+/vectoreFMK/vectoreFM, (6) which was minimized with respect to the ferromagnetic direc- tion/vectoreFM. Due to the magnetic anisotropy, actual spin spiral config- urations become distorted compared to the harmonic shapedefined in Eq. ( 4). In order to take this effect into account, we further relaxed the configurations obtained above usingzero-temperature spin dynamics simulations by numericallysolving the Landau-Lifshitz-Gilbert equation [ 40,41]: ∂/vectors i ∂t=−γ 1+α2/vectorsi×/vectorBeff i−αγ 1+α2/vectorsi×/parenleftbig /vectorsi×/vectorBeff i/parenrightbig ,(7) where αis the Gilbert damping parameter and γ=2μB/¯h is the gyromagnetic ratio. The effective field /vectorBeff iis obtained from the generalized Hamiltonian Eq. ( 1)a s /vectorBeff i=−1 m∂H ∂/vectorsi=1 m/summationdisplay j(/negationslash=i)Jij/vectorsj−2 mK/vectorsi. (8) The spin magnetic moment of the Co atom mwas determined from the electronic structure calculations. We used a two-dimensional lattice of 128 ×128 sites populated by classical spins with periodic boundary conditions and considered the fulltensorial exchange interactions and the on-site anisotropy termwhen calculating the effective field. In all considered cases wefound that the harmonic model provided a good approximationfor the wave vector and normal vector of the spin spiral orcorrectly determined the ferromagnetic ground state. We alsoperformed simulations initialized in random initial configura-tions to investigate whether noncoplanar configurations canemerge in the systems, but found no indication for such abehavior in the absence of external magnetic field. C. Effective interaction parameters In order to allow for a comparison between different ab initio calculation methods and experimental results, here we discuss how one can transform between the atomic interactionparameters calculated for many different neighbors used inthis paper, and effective nearest-neighbor interactions andparameters in the micromagnetic model. Complex magnetic textures are often studied in terms of micromagnetic models, where it is assumed that the magneti-zation direction is varying on a length scale much larger thanthe lattice constant, and the spins may be characterized by thecontinuous vector field /vectors(/vectorr), the length of which is normalized to 1. In order to describe chiral magnetism, for a magneticmonolayer with C 3vpoint-group symmetry, the energy density is usually expressed as e(/vectors)=J/summationdisplay α=x,y,z(/vector∇sα)2+DwD(/vectors)−K(sz)2, (9) with the linear Lifshitz invariant: wD(/vectors)=sz∂xsx−sx∂xsz+sz∂ysy−sy∂ysz. (10) The relationship between the micromagnetic parameters J,D, and Kand the atomic parameters in Eq. ( 1) may beobtained by calculating the energy of the same type of spin configurations. Here we will consider spin spiral states withwave vectors along the ydirection: /vectors(/vectorr)=/vectore zcosky+/vectoreysinky, (11) where the plane of the spiral is spanned by the wave-vector direction /vectoreyand the out-of-plane direction /vectorez, corresponding to cycloidal spin spirals. In the micromagnetic model, the averageenergy over the spin spiral reads E micromagnetic =JVak2+DVak−1 2KVa, (12) if it is calculated for the atomic volume Va. For the atomic model one obtains [cf. Eq. ( 5)] Eatomic=−1 2/summationdisplay /vectorRij1 2/parenleftbig Jyy ij+Jzz ij/parenrightbig coskRy ij +1 2/summationdisplay /vectorRijDx ijsinkRy ij+1 2K. (13) Expanding Eq. ( 13) up to second-order terms in kyields Eatomic≈Jeffk2+Deffk+1 2Keff (14) apart from a constant shift in energy, with the effective spin- model parameters defined as Jeff=1 4/summationdisplay jJij/parenleftbig Ry ij/parenrightbig2, (15) Deff=/summationdisplay jDx ijRy ij, (16) Keff=K+1 2/summationdisplay /vectorRij/parenleftbig Jyy ij−Jzz ij/parenrightbig . (17) The effective parameters JeffandDeffare also known as spin stiffness and spiralization, respectively [ 42,43]. The relationship between the micromagnetic and the effectiveparameters is given by J=1 VaJeff,D=1 VaDeff,K=−1 VaKeff. (18) Note that it is possible to define the atomic volume as Va=√ 3 2a2 2Dtwhere√ 3 2a2 2Dis the area of the in-plane unit cell and tis the film thickness. In Ref. [ 22]t h ev a l u eo f t=nlayer√2 3a2Dwas used with nlayerthe number of magnetic atomic layers, corresponding to the ideal interlayer distancein an fcc lattice along the (111) direction. However, thisapproximation becomes problematic when lattice relaxationsare taken into account at the surface, since in this descriptionthe positions of the centers of the atoms are defined instead ofthe thickness of the layers. Therefore, we used the expressionV a=4π 3R3 WS, where RWSis the radius of the atomic spheres used in the SKKR calculations, with RWS≈1.49˚Af o rt h e considered X/Co/Pt(111) systems. The cycloidal spin spiral defined in Eq. ( 11) is called clock- wise or right-handed for k>0, meaning that when looking at the system from the side with the out-of-plane directiontowards the top the spins are rotating clockwise when movingto the right along the modulation direction of the spiral [ 44]. For 134405-3SIMON, RÓZSA, PALOTÁS, AND SZUNYOGH PHYSICAL REVIEW B 97, 134405 (2018) k<0, the spin spiral is called counterclockwise or left-handed. According to Eq. ( 12), the micromagnetic DMI creates an energy difference between the two rotational senses, withD>0 preferring a counterclockwise rotation. Equation ( 16) demonstrates that the micromagnetic parameter is connected tothexcomponent of the atomic DM vector for spin spirals with wave vectors along the ydirection, or the in-plane component D /bardbl ijof the vector for general propagation directions. Note that the magnitude of D/bardbl ijis the same for all neighbors that can be transformed into each other via the C3vsymmetry of the system, while the sign can be defined based on whether thevectors prefer clockwise or counterclockwise rotation of thespins. Note that in the case of C 3vsymmetry /vectorDijalso has a nonvanishing zcomponent, the effect of which on domain walls was investigated in Ref. [ 31]. Finally, we also define nearest-neighbor atomic interaction parameters JandD, which reproduce the effective parameters in Eqs. ( 15) and ( 16): Jeff=3 4a2 2DJ, D eff=3 2a2DD, (19) where Dis the in-plane component of the nearest-neighbor DM vector with the sign convention discussed above. Instead of performing the direct summations in Eqs. ( 15) and ( 16), we fitted the spin spiral dispersion relation in Eq. ( 13) calculated from all interaction parameters in Eq. ( 1) with an effective nearest-neighbor model containing J,D, andKeff. The fitting was performed in a range that is sufficiently largeto avoid numerical problems, but sufficiently small that themicromagnetic approximations may still be considered valid,corresponding to |k|a 2D/2π/lessorequalslant0.1. We note that this procedure is similar to how the atomic interaction parameters are deter-mined from spin spiral dispersion relations directly obtainedfrom total-energy calculations (see, e.g., Refs. [ 8,12,45]), but we used the spin model containing interaction parametersbetween many neighbors to determine the dispersion relationin the first place. We confirmed with spin dynamics simula-tions that in ferromagnetic systems the domain-wall profilescalculated with the full model Hamiltonian ( 1) agree well with the profiles that can be calculated analytically from amicromagnetic model with the interaction parameters obtainedusing the above procedure. Nevertheless, we found that not allsystems can be sufficiently described by the three parametersused in the micromagnetic model, and this discrepancy canbe attributed to the competition between ferromagnetic andantiferromagnetic isotropic Heisenberg interactions (see Sec.III C for details). In order to support the comparison of our calculated param- eters with corresponding values obtained from experimentsor other theoretical approaches we shall present the micro-magnetic, effective, and nearest-neighbor atomic parameters asdefined above for all considered systems. As an example, in Ta-bleIwe present the comparison between DMI values obtained for the Co monolayer on Pt(111) without a capping layer usingdifferent ab initio calculation methods in Refs. [ 12,22,31,43], similarly to the summary given in Ref. [ 47]. Using the above definitions, we find reasonable agreement between the differenttheoretical descriptions, and all parameters fall into the rangewhere a ferromagnetic ground state is expected based on theexperimental investigations in Ref. [ 47].TABLE I. Nearest-neighbor atomic ( D), effective ( Deff), and mi- cromagnetic ( D) DM coupling obtained in several earlier publications for the Co monolayer on Pt(111). Positive values indicate that the counterclockwise (left-handed) chirality is preferred in the system.For a consistent transformation between the different parameters we used the values a 2D=2.774˚Aa n d RWS=1.44˚A. For Ref. [ 12]w e took into account the different definition of the atomic interactionparameters compared to Eq. ( 1). For Ref. [ 22] we considered the DMI value for the Co(3)/Pt(3) structure and the correction in Ref. [ 46]. D(meV) Deff(meV ˚A) D(mJ/m2) Ref. [ 31] 2.86 11.90 15.11 Ref. [ 43] 2.72 11.30 14.35 Ref. [ 12] 3.60 14.98 19.02 Ref. [ 22] 3.12 12.98 16.48 III. RESULTS A. Isotropic exchange interactions Figure 1shows the calculated isotropic exchange constants Jijbetween the Co atoms as a function of interatomic distance for the different capping layers (CL) and for the uncapped sys-tem (no CL). According to Eq. ( 1), positive and negative signs of the isotropic exchange parameters refer to FM and AFMcouplings, respectively. For all overlayers the ferromagneticNN interaction is dominating: it is the largest in magnitude forthe Au overlayer, for Pt and Ir a small decrease can be seen,while for Os and Re overlayers it is dramatically reduced. Thesecond- and third-nearest-neighbor couplings are considerablysmaller in magnitude than the NN couplings and the trendfor the different overlayers is also less systematic; e.g., in thecase of Au, Pt, and Os overlayers the second-NN couplingis ferromagnetic, while for Ir and Re it is AFM. Overall, themagnitude of the isotropic interactions decays rapidly with thedistance, becoming negligible beyond the third-NN shell. In Table II, the NN exchange couplings ( J 1) and the spin magnetic moments of Co atoms ( mCo) are summarized for the different overlayers. FIG. 1. Calculated Co-Co isotropic exchange parameters Jijas a function of the interatomic distance and different overlayers, and forthe Co/Pt(111) system without the capping layer (no CL) [ 31]. 134405-4MAGNETISM OF A Co MONOLAYER ON Pt(111) CAPPED … PHYSICAL REVIEW B 97, 134405 (2018) TABLE II. Calculated nearest-neighbor exchange interactions J1 between the Co atoms and the spin magnetic moment of Co mCofor all considered capping layers and for the Co/Pt(111) system without the capping layer (no CL) [ 31]. J1(meV) mCo(μB) Re 5.03 1.04 Os 9.66 1.55Ir 31.73 2.11 Pt 31.55 2.17 Au 37.54 2.10No CL 42.46 2.10 We find that capping by 5 doverlayers systematically reduces J1compared to the uncapped case, which can be at- tributed to the hybridization between the Co and the overlayer.The magnetic moment of Co is almost constant for the Au, Pt,and Ir overlayers, while it shows an apparent decrease for Osand Re, similarly to the NN isotropic exchange. This decreaseis, however, much less drastic than for J 1:mCoin the case of the Re overlayer is about half of mCoin the case of the Au layer, while this ratio is about 1 /7f o rJ1. According to the Stoner model of ferromagnetism, the density of states (DOS) of the delectrons of Co at the Fermi level, n(/epsilon1F), in the nonmagnetic phase plays the crucial role in stabilizing spontaneous magnetization: in the case ofIn(/epsilon1 F)>1 (with Ibeing the Stoner parameter) the system becomes ferromagnetic. Hence the observed trends in mCoand J1are governed by the filling of the 5 dband of the overlayer that influences the 3 dband of Co via hybridization. In order to trace this effect, in Fig. 2we plot the density of states of thedelectrons in the Co layer and in the overlayer in the nonmagnetic phase, meaning that the exchange-correlation magnetic field was set to zero during the density functional theory calculations. Since all the dstates of Au are occupied, the corresponding 5 dband lies well below the Fermi level, 024Re 024Os 024Ir 024Pt −6 −4 −2 0 2 /epsilon1−/epsilon1F(eV)024AuDOS (states/eV) FIG. 2. DOS of delectrons in the Co layer (solid red line) and in the overlayer (dashed blue line) in nonmagnetic X/Co/Pt(111) (X=Re, Os, Ir, Pt, Au) systems.TABLE III. Nearest-neighbor atomic ( J), effective ( Jeff), and micromagnetic ( J) parameters of Co for the isotropic exchange interaction of X/Co/Pt(111) ( X=Re, Os, Ir, Pt, Au) and Co/Pt(111) thin films (no CL) [ 31] obtained from the calculated spin-model parameters by the fitting procedure in Sec. II C. J(meV) Jeff(meV ˚A2) J(pJ/m) Re 0.82 4.73 0.56 Os 22.58 130.32 15.48 Ir 6.94 40.05 4.71 Pt 41.89 241.76 27.99Au 49.23 284.12 31.98 No CL 54.40 313.96 39.86 leaving the Co 3 dband localized around the Fermi level, with a large n(/epsilon1F) that explains the strong magnetic moment of Co in this case. Although the 5 dband of Pt is shifted upwards due to the decrease of the band filling and the hybridization withthe Co dband increases, the large peak in the Co DOS at the Fermi level still pertains, keeping m Coat a high value. This trend remains also in the case of the Ir overlayer, where the3d-5dhybridization further increases and n(/epsilon1 F) of Co clearly decreases, but the magnetic moment of Co is of similar valueas for the Au overlayer. For the cases of Os and Re overlayersthe Co 3 dband gets rather delocalized due to hybridization with the wider 5 dbands and n(/epsilon1 F) is further reduced leading to the observed drop in mCo. Note that a similar dependence of the Co moments on the overlayer was obtained for othersystems [ 48–51]. From the calculated isotropic exchange interactions we obtained the spin stiffness constant ( J eff), the corresponding micromagnetic parameter ( J), and the NN atomic value ( J) for all considered overlayers as described in Sec. II C, and presented them in Table III. Apparently, these values follow the variation of mCoorJ1for Os, Pt, and Au capping layers; however, in the case of Ir and Re they are considerably reduced.The reason for this behavior is the amplification of the role ofexchange interactions between farther atoms in J effas follows from Eq. ( 15). From Fig. 1one can see that in the case of the Ir overlayer both the second- and third-NN couplings arenegative (AFM), which drastically reduces the value of J eff. The decrease of the NN coupling is apparently insufficientin itself to explain the very small value of J effin the case of the Re overlayer. However, a detailed investigation ofFig.1shows that the seventh-NN interaction, J 7=−0.39 meV , gives a dominating negative contribution to Jeffdue to the large distance ( d=3.606a2D) and the large number (12) of neighbors in this shell. B. Relativistic spin-model parameters 1. Different capping layers Next, we investigate the in-plane components of Dzyaloshinskii-Moriya interactions between the Co atomswhich are shown in Fig. 3for all capping layers as a function of the distance between the Co atoms, compared to the valuesin the absence of a capping layer [ 31]. The sign changes of the DMI indicate switchings in the preferred rotational sense from 134405-5SIMON, RÓZSA, PALOTÁS, AND SZUNYOGH PHYSICAL REVIEW B 97, 134405 (2018) 4 6 8 10 12 14−1.5−1.0−0.50.00.51.01.52.0D/bardbl ij(meV) clockwisecounterclockwiseRe Os Ir Pt Au no CL FIG. 3. In-plane component of the DM vectors D/bardbl ijas a function of the distance between the Co atoms for different overlayers, and for the Co/Pt(111) system without the capping layer (no CL) [ 31]. shell to shell, analogously to the oscillation between ferro- magnetic and antiferromagnetic isotropic exchange interactioncoefficients. Except for the case of the Au capping layer, the NNDMI is the largest in magnitude; however, the DM vectors formore distant pairs also play an important role. This is somewhatdifferent for the isotropic couplings, J ij,i nF i g . 1, where the NN interaction is much larger in magnitude than the interactionsfor farther shells; therefore, the slow decay with the Co-Codistance is less visible than for the DMI in Fig. 3. To illustrate the overall effect of the overlayers on the DMI, we calculated the NN atomic, effective, and micromagneticDMI coefficients of Co from the ab initio spin-model parame- ters as discussed in Sec. II C. These values are summarized in Table IVfor different capping layers. For comparison, we also included the corresponding values for Co/Pt(111).It is worthwhile to mention that the effective parameters inTable IVfollow exactly the same order for the different capping layers as the in-plane NN DM vectors in Fig. 3, unlike in the case of the isotropic exchange interactions. Regardless ofthe choice of the capping layer, the DMI is shifted towardsthe direction of clockwise rotational sense compared to theuncapped system. For the Pt/Co/Pt(111) system, the DMI isexceptionally weak, which is to be expected since inversionsymmetry is almost restored in this system if we consider that TABLE IV . Nearest-neighbor atomic ( D), effective ( Deff), and micromagnetic ( D) DM coupling of Co obtained from the spin-model parameters for X/Co/Pt(111) thin films ( X=Re, Os, Ir, Pt, Au) and for Co/Pt(111) without any capping layer (no CL). D(meV) Deff(meV ˚A) D(mJ/m2) Re 1.82 7.57 8.94 Os 2.58 10.74 12.75 Ir −1.75 −7.28 −8.56 Pt 0.20 0.83 0.96Au 1.50 6.24 7.02 No CL 2.86 11.90 15.11−0.15−0.10−0.05-Δmorb(μB) Co/Pt(111) 05Keff(meV) in-plane out-of-plane Re Os Ir Pt Au010Deff(meV ·) counterclockwise clockwise FIG. 4. Calculated values of orbital moment anisotropy in the Co layer with negative sign −/Delta1m orb,M A E Keff, and effective DMIDeffforX/Co/Pt(111) thin films ( X=Re, Os, Ir, Pt, Au). The corresponding parameters for Co/Pt(111) are also illustrated bydashed green lines. generally the interfacial DMI is dominated by the magnetic and nonmagnetic heavy metal layers directly next to each other. We would also like to point out that the Ir capping layer is the only one that switches the sign of the DMI preferring clockwiserotation. This is somewhat unexpected since the Ir layer alsochanged the preferred rotational sense to clockwise when itwas introduced between the Co monolayer and the Pt(111)substrate [ 31], so it should prefer a counterclockwise rotation for the opposite stacking order according to the three-sitemodel of the DMI [ 52]. A possible reason for this effect is that the reduced coordination number of the Ir atoms in the cappinglayer as well as the electrostatic potential barrier at the surfacesignificantly modify the electronic structure of the cappinglayer compared to the bulk case or when the Ir is inserted belowthe Co layer. This sign change of the DMI in Ir/Co/Pt(111)indicates that ultrathin-film systems can display qualitativelydifferent features compared to magnetic multilayers, where theIr/Co/Pt stacking was suggested as a way of enhancing the DMI[27]. The different behavior of Ir as a capping layer and as an inserted layer was recently investigated in Ref. [ 53]. In order to study the dependence of the DMI on the capping layer, we calculated additional quantities determinedby the strength of the spin-orbit coupling, namely, the totalMAE K effand the anisotropy of the orbital moment of Co atoms, /Delta1m orb=m⊥ orb−m/bardbl orb, where the superscripts ⊥and /bardblrefer to calculations performed for a normal-to-plane and an in-plane orientation of the magnetization in the Co layer,respectively. Figure 4shows /Delta1m orbwith a negative sign (top panel), Keff(middle panel), and Deff(bottom panel) for the Co monolayer depending on capping layer. Note that negative andpositive signs of K effrefer to easy-axis and easy-plane types of magnetic anisotropy, respectively. For 3dtransition metals, where the spin-orbit coupling is small compared to the bandwidth, second-order perturbationtheory describes the uniaxial magnetic anisotropy well [ 54]. According to Bruno’s theory, neglecting spin-flop coupling andfor a filled spin-majority dband, a negative proportionality between the MAE and /Delta1m orbapplies, that was confirmed 134405-6MAGNETISM OF A Co MONOLAYER ON Pt(111) CAPPED … PHYSICAL REVIEW B 97, 134405 (2018) 4 6 8 10 12 14 d−1012D/bardbl ij(meV)counterclockwise clockwiseλ=0.0 λ=0.2 λ=0.4 λ=0.6 λ=0.8 λ=1.0 FIG. 5. In-plane DMI as a function of the distance between the Co atoms for various values of the SOC scaling parameter λin the Ir capping layer of the Ir/Co/Pt(111) system. theoretically and experimentally for Co layers [ 54–58]. From Fig. 4a good qualitative correlation can be inferred between Keffand−/Delta1m orbwith the exception of the Re overlayer. Indeed, due to the large 3 d-5dhybridization, the delocalization of the spin-majority band of Co is increased in the case of theRe overlayer such that the above-mentioned conditions for thesimple proportionality do not apply. From Fig. 4it turns out that the variations of K effand Deffalso correlate well with each other. This is somewhat surprising since, as mentioned above, the MAE is of secondorder in the SOC, while the DM term appears in the firstorder of the perturbative expansion [ 4]. Compared to the Co/Pt(111) system, the Os capping layer does not modify theDMI significantly, but we observe a strong easy-plane MAE.The Re and the Au capping layers decrease the magnitudeofD eff, and the preferred magnetization direction is also in plane. An out-of-plane magnetization was obtained for Ir andPt capping layers, and as discussed above the Ir cappinglayer prefers a clockwise rotation, while in the case of thePt overlayer the DMI is close to zero. 2. Scaling of the spin-orbit coupling in the Ir overlayer To gain further insight into the the sign change of the DMI in the Co monolayer with the Ir capping layer, we artificiallymanipulated the strength of SOC at the Ir atoms. Ebert et al. introduced a continuous scaling of the SOC via the parameter λ within the relativistic KKR formalism [ 59]: calculation without scaling ( λ=1) corresponds to the fully relativistic case, while λ=0 can be identified with the so-called scalar-relativistic description. Importantly, in the above formalism the scaling ofthe SOC can be used selectively for arbitrary atomic cells. Wethus applied it to the Ir monolayer, while the SOC at all othersites of the system remained unaffected. Figure 5shows D /bardbl ijas a function of the distance between the Co atoms for different scaling parameters. Varying λ has a strong influence on the NN in-plane DMI: it changescontinuously from preferring counterclockwise ( λ=0) to preferring clockwise ( λ=1) rotational direction, while the−3−2−10Keff(meV) 0.0 0.2 0.4 0.6 0.8 1.0 λ−505Deff(meV ·) FIG. 6. Calculated MAE Keff, and effective DMI Deffas a function of the SOC scaling parameter λin the Ir overlayer of the Ir/Co/Pt(111) system. changes in the other shells are smaller in relative and in absolute terms. In the case of λ=0, the NN in-plane DMI takes a value of 2.32 meV , which means that the NN DMI of the Co/Pt(111)system (1.98 meV) [ 31] is nearly restored in this case. In accordance with the results of first-order perturbation theory, Fig. 6illustrates that the variation of the effective DMI is rather linear with λ.F o rλ=0,K effis close to the value of the uncapped Co/Pt(111) system ( −0.20 meV [ 31]) and it increases in magnitude to −3m e Vf o r λ=1. Following the change in the NN in-plane DMI interaction in Fig. 5, the sign of the effective DMI turns from preferring counterclockwiseto preferring clockwise rotation when increasing the strengthof the SOC in the Ir overlayer. On the other hand, at λ=0D eff is somewhat smaller in magnitude than in the case of the uncapped Co/Pt(111) (11 .90 meV ˚A). This indicates that the Ir overlayer influences the DMI of the system not just due toits strong SOC but also by modifying the electronic states inthe Co monolayer via hybridization. 3. Changing the capping layer composition in Au1−xIrx/Co/Pt(111) Controlling the Ir concentration xin the alloy capping layer Au 1−xIrx(0/lessorequalslantx/lessorequalslant1) represents a transition where the effect of increasing hybridization between the 3 dband of Co and the 5 dband of the capping metal can be traced, as shown in Fig. 2. On the other hand, the strength of the SOC, defined by the operator ξ/vectorL/vectorS, in Au and Ir is roughly the same ( ξ≈600 meV), meaning that the alloying is expected to have a different effect than the scaling of the SOC discussedin the previous section. Thus, we performed calculations ofthe spin-model parameters for x=0.1,0.2,..., 0.9b yu s i n g the CPA for the chemically disordered overlayer. The layerrelaxation was varied as a function of xaccording to Vegard’s law using the calculated layer relaxation of the Au/Co/Pt(111)and Ir/Co/Pt(111) systems. The in-plane components of the DM vectors in the Co monolayer from the first to the fourth shell are shown in Fig. 7 as a function of the Ir concentration. When increasing the Ir 134405-7SIMON, RÓZSA, PALOTÁS, AND SZUNYOGH PHYSICAL REVIEW B 97, 134405 (2018) 0.0 0.2 0.4 0.6 0.8 1.0 x(Ir concentration)−1.0−0.50.00.5D/bardbl ij(meV)counterclockwise clockwise D1 D2 D3 D4 FIG. 7. In-plane components of the DM vectors D/bardbl ijof Co from the first ( D1) to the fourth shell ( D4), as a function of the Ir concentration ( x)i nt h eA u 1−xIrx/Co/Pt(111) system. concentration, the sign of the first-NN and the second-NN D/bardbl ijchanges from positive to negative. The third-NN in-plane DM for the Au/Co/Pt(111) system is negative, it turns positivearound x≈0.1, and it has approximately the same magnitude around 20% Ir concentration as for the pure Ir/Co/Pt(111)layer, with a maximal amplitude at about x=0.5. The sign of the fourth-NN D /bardbl ijis not changed by the alloying, and the magnitude remains nearly constant. The changes of the effective DMI and MAE are shown in Fig. 8as a function of the Ir concentration. Unlike the case where the SOC was scaled (Fig. 6), the variation of KeffandDeffwithxis nonmonotonous, with a maximum of Keffat around 10% and a minimum of Deffat around 90% Ir concentration. The effect of alloying the nonmagnetic heavy metals on the DMI was also investigated recently in Ref. [ 60], where 4-ML Ir xPt1−x/1-ML Co /4-ML Pt and 4-ML Pt xAu1−x/1- FIG. 8. Calculated total MAE Keffand effective DMI Deffin the Co monolayer as a function of Ir concentration ( x)i nt h e Au1−xIrx/Co/Pt(111) system.TABLE V . Obtained magnetic ground states for X/Co/Pt(111) thin films ( X=Re, Os, Ir, Pt, Au). KJ/D2Ground state Re −0.20 Tilted SS Os −6.77 In-plane FM Ir 2.27 Out-of-plane FM Pt 436.13 Out-of-plane FMAu −1.46 In-plane FM ML Co /4-ML Pt trilayers were considered along the (111) stacking direction. Similarly to the results presented here,a nonmonotonous dependence on the concentration was re-ported, together with a switching to negative DMI due to thepresence of Ir in the capping layer, although only at smallerIr concentration. In Ref. [ 60], for pure Au or pure Ir capping layers similar values of the DMI were obtained to the uncappedvalues summarized in Table I. As discussed above, in the present calculations the decrease of the DMI due to the Auoverlayer and the sign change due to the Ir overlayer canprobably be attributed to the reduced coordination numberof the atoms if the capping layer is only 1 ML thick. As apossible alternative for obtaining a microscopic understanding,an interesting perturbative model for the DMI in zig-zag chainscan be found in Ref. [ 61], where the dependence of the sign and strength of the DMI on different parameters is reported. C. Magnetic ground states The ground states of the systems were determined by com- bining harmonic spin spiral calculations with spin dynamicssimulations as described in Sec. II C. After scaling out the energy and length scales, the micromagnetic energy density inEq. ( 9) can be described by a single dimensionless parameter KJ/D 2, which governs the formation of the magnetic ground state. As already discussed in earlier publications [ 44,62,63], noncollinear ground states are expected to be formed for −1< KJ/D2<π2 16≈0.62 in this model; the upper limit denotes where magnetic domain walls become energetically favorablein out-of-plane oriented ferromagnets, while the lower limitindicates the instability of the in-plane oriented ferromagneticstate towards the formation of an elliptic conical state. The calculated values are summarized in Table Vfor these systems. For most considered capping layers the parameterKJ/D 2is outside the range where the formation of non- collinear states is expected, and in the simulations we indeedobserved FM ground states. This can be explained eitherby the strong easy-plane (Os) or easy-axis (Ir) anisotropies,the weakness of the DMI for the Pt/Co/Pt(111) system, orthe combination of the above for the Au capping layer. Forthe Re/Co/Pt(111) system, the micromagnetic model predicts[62,63] a cycloidal spin spiral ground state with the normal vector in the plane, just as it was assumed in Eq. ( 11). However, by minimizing Eq. ( 4) with respect to the wave vector /vectorkand the normal vector /vectorn, we obtained a tilted spin spiral state of the form /vectors i=/vectorexcoskRy isin/Phi10−/vectoreysinkRy i+/vectorezcoskRy icos/Phi10. (20) 134405-8MAGNETISM OF A Co MONOLAYER ON Pt(111) CAPPED … PHYSICAL REVIEW B 97, 134405 (2018) xy z xz yΦ0=3 8◦allJij(a) xy z xz yΦ0=0◦NNJ(b) xy z xz yΦ0=5 8◦NNNJ1,J2(c) FIG. 9. The tilted spin spiral ground state found in the Re/Co/Pt(111) system in spin dynamics simulations. The tilting angle /Phi10is defined in Eq. ( 20). (a) Ground state obtained using the full Jij exchange interaction tensors. (b) Ground state obtained with only nearest-neighbor (NN) atomic interaction parameters, J=0.82 meV from Table III, NN DMI, and effective on-site anisotropy. (c) Ground state obtained by performing the fitting procedure discussedin Sec. II C for NN and next-nearest-neighbor (NNN) exchange interactions, J 1=53.46 meV and J2=−18.20 meV, NN DMI, and effective on-site anisotropy. Red and blue colors correspond topositive and negative out-of-plane spin components, respectively. The ground state obtained from the spin dynamics simu- lations is displayed in Fig. 9(a). Although the spiral became slightly distorted due to the anisotropy, we found that it couldstill be relatively well described by Eq. ( 20) using a wavelength ofλ=2π/k≈3.5 nm and a tilting angle of /Phi1 0≈38◦.T h eenergy gain due to the tilting is approximately 0 .04 meV /atom. Note that the tilted spin spiral state is still a cycloidal spiral inthe sense that the wave vector is located in the rotational planeof the spirals, but the normal vector is no longer confined tothe surface plane. This is different from the case of weak DMIin out-of-plane magnetized films, where the normal vector ofdomain walls gradually rotates in the surface plane from Néel-type to Bloch-type rotation due to the presence of the magne-tostatic dipolar interaction (see, e.g., Ref. [ 64] ) .I ta l s od i f f e r s from the elliptic conical spin spirals discussed in Refs. [ 62,63] because the tilted spin spiral state has no net magnetization. The formation of such a ground state can be explained by the easy-plane anisotropy preferring an in-plane orientationof the spiral, the DMI preferring a spiral plane perpendicularto the surface, and the simultaneous presence of competingferromagnetic and antiferromagnetic isotropic exchange inter-actions in the system, the latter also leading to the reducedvalue of the effective J effparameter for the Re capping layer in Table II. This is illustrated in Figs. 9(b) and9(c):i nt h e nearest-neighbor atomic model, a vertical cycloidal spin spiralground state is obtained, in agreement with the prediction of themicromagnetic description [ 62,63]. The spin spiral wavelength is also significantly shorter, λ≈1.4 nm, due to the inaccuracy of the nearest-neighbor fitting procedure. On the other hand, ifthe fitting is performed with taking nearest- and next-nearest-neighbor isotropic exchange interactions into account, thetilted spin spiral ground state is recovered with λ≈3.8n m and/Phi1 0≈58◦, in reasonable agreement with the full model. IV . SUMMARY AND CONCLUSIONS In conclusion, we examined the X/Co/Pt(111) ( X=Re, Os, Ir, Pt, Au) ultrathin films using first-principles and spin-modelcalculations. We determined the Co-Co magnetic exchangeinteraction tensors between different pairs of neighbors andthe magnetic anisotropies. From the results of the ab initio calculations we also determined effective and micromagneticspin-model parameters for the Co layers. For the isotropicexchange couplings we found dominant ferromagnetic nearest-neighbor interactions for all systems, which decrease withthed-band filling of the capping layer. This effect due to the hybridization between the 3 dstates of the Co layer and the 5 dstates of the capping layer can be qualitatively explained within a Stoner picture, which also accounts forthe similarly decreasing magnetic moment. Considering theeffective isotropic couplings of Co, we found significantlylower values for Re and Ir overlayers than what would beexpected simply based on the decrease of the nearest-neighborinteractions; this we attributed to competing antiferromagneticcouplings with further neighbors. We also investigated the in-plane Dzyaloshinskii-Moriya interactions of Co, and found it to be weaker for all capping layers compared to the uncapped Co/Pt(111) system. For the Ircapping layer we found a switching from counterclockwise toclockwise rotation, which is unexpected since the same switch- ing can also be observed if the Ir is inserted between the mag- netic layer and the substrate [ 31]. We attributed this effect to the reduced coordination number of Ir atoms and the electrostaticpotential barrier at the surface. We also found a correlation between the effective Dzyaloshinskii-Moriya interactions D eff, 134405-9SIMON, RÓZSA, PALOTÁS, AND SZUNYOGH PHYSICAL REVIEW B 97, 134405 (2018) the effective magnetic anisotropies Keff, and the anisotropy of the orbital moment /Delta1m orb. We further investigated the sign change of the effective Dzyaloshinskii-Moriya interaction ofCo for the Ir capping layer by scaling the strength of thespin-orbit coupling at the Ir sites and by tuning the filling of the 5dband in a Au 1−xIrx/Co/Pt(111) system. We found a linear dependence of the effective Dzyaloshinskii-Moriya interactionon the spin-orbit coupling strength in agreement with theperturbative description, and a nonmonotonic dependence on the band filling. Using the spin-model parameters we determined the mag- netic ground state for all considered systems. For Os, Ir, Pt,and Au capping layers we found a ferromagnetic ground state,in agreement with the analytical prediction based on the calcu-lated micromagnetic parameters. For the Re/Co/Pt(111) sys-tem we found a tilted spin spiral ground state, the appearanceof which can only be explained if competing ferromagnetic andantiferromagnetic isotropic exchange interactions are takeninto account alongside the Dzyaloshinskii-Moriya interactionand the easy-plane anisotropy.Our results highlight the importance of ab initio calculations and atomic spin-model simulations in cases where simplermodel descriptions might lead to incomplete conclusions. 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PhysRevB.101.134423.pdf

PHYSICAL REVIEW B 101, 134423 (2020) Realization of Su-Schrieffer-Heeger states based on metamaterials of magnetic solitons Gyungchoon Go,1,*Ik-Sun Hong,2Seo-Won Lee,1Se Kwon Kim ,3and Kyung-Jin Lee1,2 1Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea 2KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea 3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA (Received 10 October 2019; revised manuscript received 23 March 2020; accepted 1 April 2020; published 21 April 2020) We theoretically investigate coupled gyration modes of magnetic solitons whose distances to the nearest neighbors are staggered. In a one-dimensional bipartite lattice, we analytically and numerically find that thereis a midgap gyration mode bounded at the domain wall connecting topologically distinct two phases which isanalogous to the Su-Schrieffer-Heeger model. As a technological application, we show that a one-dimensionaldomain-wall string in a two-dimensional soliton lattice can serve as a waveguide of magnetic excitations, whichoffers functionalities of a signal localization and selective propagation of the frequency modes. Our resultoffers an alternative way to control the magnetic excitation modes by using a magnetic metamaterial for futurespintronic devices. DOI: 10.1103/PhysRevB.101.134423 I. INTRODUCTION Topological properties embedded in band structures are one of the central themes in modern condensed-matterphysics. In two-dimensional (2D) electron systems, represen-tative examples supporting topologically protected edge states[1] are the Haldane model [ 2] and the Kane-Mele model [3], which exhibit the quantum Hall and quantum spin Hall phases, respectively. A classical example in one-dimensional(1D) topological systems is the Su-Schrieffer-Heeger (SSH)model supporting a midgap bound state with fermion number1/2[4,5]. Inspired by the topological effects in electronic systems, numerous studies have been devoted to investigatingtopological properties in bosonic systems such as magnons[6–8], phonons [ 9,10], and their hybridized states [ 11–14]. Such topological effects of band structures can also be realized in artificially structured composites, called meta-materials, whose functionalities arise as the collective dy-namics of local resonators [ 15]. Analogs of topologically protected edge states in 2D systems have been proposed andexperimentally observed in acoustic [ 15,16], optical [ 17–21], magnetic [ 22–24], mechanical [ 25,26], and electric circuit [27–29] systems. Moreover, the 1D SSH model has been realized in optical waveguides [ 30], electric circuits [ 31,32], and magnetic spheres [ 33]. An intriguing feature of the meta- materials is that the band structures and their topologicalproperties can be manipulated by changing the crystal param-eters. This tunability of metamaterials is of crucial importancefor widespread applications of topological properties in, forinstance, reconfigurable logic devices [ 15,19]. Magnetic solitons such as magnetic vortices and skyrmions are resonators whose dynamics exhibit gyroscopic mo-tion [ 34–36]. Theoretical [ 37–39] and experimental [ 40–42] *gyungchoon@gmail.comresults on the dynamics of coupled gyration modes of the magnetic solitons provide a potential application for a dif-ferent type of information device [ 39]. Moreover, internal degrees of freedom of magnetic solitons such as polarity andchirality can offer efficient control of the functionalities ofsoliton-based metamaterials [ 43]. One of us has shown that the collective excitation of the magnetic solitons supports achiral edge mode in a honeycomb lattice [ 23], which was later confirmed by micromagnetic simulation [ 24]. Recently, the topological corner states have been realized in breathingkagome [ 44] and honeycomb lattice [ 45]. However, the SSH state in the 1D system has not been realized for collectivegyration modes of magnetic solitons. In this paper, we first study a metamaterial composed of the magnetic soliton disks structured in a one-dimensionalbipartite chain. By using both analytic calculation and micro-magnetic simulation, we show the existence of a midgap statebounded at a domain wall connecting topologically distincttwo configurations, which is analogous to the electronic SSHmodel. Then we derive a 2D extension of our 1D magneticSSH model, which is shown to be able to support a magneticwaveguide with selective propagation of frequency modes. II. REALIZATION OF THE SSH MODEL WITH AN ARRAY OF SOLITON DISKS We consider a quasi-one-dimensional array of nanodisks containing magnetic vortices or skyrmions. In general, thesteady-state motion of topological solitons can be describedby the dynamics of the center-of-mass position R(t) and m=m[r−R(t)], where mis a unit vector along the direc- tion of local magnetization. The dissipationless magnetizationdynamics of the coupled vortices /skyrmions is described by Thiele’s equation [ 46]: Gˆz×dU j dt+Fj=0, (1) 2469-9950/2020/101(13)/134423(7) 134423-1 ©2020 American Physical SocietyGO, HONG, LEE, KIM, AND LEE PHYSICAL REVIEW B 101, 134423 (2020) where Uj=Rj−R0 jis the displacement of the soliton from the equilibrium position R0 j,G=− 4πMstDQ/γis the gy- rotropic coefficient, Msis the saturation magnetization, tD is the thickness of the disk, and γis the gyromagnetic ratio. Here, Q=1 4π/integraltext dxdym·(∂xm×∂ym) is the topologi- cal charge which characterizes the topological solitons. Thetopological charge of the magnetic vortices and skyrmionsareQ=± 1/2 and ±1, respectively. F j=−∂W/∂Ujis the conservative force from the potential energy W=/summationdisplay jK 2U2 j+/summationdisplay j/negationslash=kUjk 2, (2) where K>0 is the spring constant and Uj≡(uj,vj)i s the displacement vector. Here, Ujkis the interaction energy between two solitons: Ujk(djk)=Ix(djk)ujuk−Iy(djk)vjvk, (3) where djk(=|R0 j−R0 k|) is the distance between centers of two neighboring disks, and Ix(djk) and Iy(djk) are interac- tion parameters between two disks. This system of coupledmagnetic solitons has been studied both theoretically andexperimentally [ 37,38,41,42]. In particular, the values of the parameters in Eqs. ( 2) and ( 3) have been experimentally mea- sured and theoretically calculated for certain sizes of solitondisks. Let us first consider the situation where the nearest- neighbor disk pairs are separated by a uniform distance. Usingthe complex variable ψ j≡uj+ivj, we write Eq. ( 1)i na simplified form [ 23,24]: i˙ψj=ωKψj+/summationdisplay k∈/angbracketleftj/angbracketright(ζψk+ξψ∗ k), (4) where ωK=K/Gis the gyration frequency of an isolated soliton and ζ=(Ix−Iy)/Gandξ=(Ix+Iy)/Gare the reparametrized interactions. In order to eliminate ψ∗ k,w e expand the complex variable as ψj=χjexp(−iω0t)+ηjexp(iω0t), (5) where χj(ηj) is a counterclockwise (clockwise) gyration am- plitude. Substituting Eq. ( 5) into Eq. ( 4) and applying |χj|/greatermuch |ηj|(|χj|/lessmuch|ηj|) for counterclockwise (clockwise) soliton gyrations, we have i˙ψj=/parenleftbigg ωK−ξ2 ωK/parenrightbigg ψj+ζ/summationdisplay k∈/angbracketleftj/angbracketrightψk−ξ2 2ωK/summationdisplay l∈/angbracketleft/angbracketleftj/angbracketright/angbracketrightψl,(6) where /angbracketleft/angbracketleftj/angbracketright/angbracketrightrepresents second-neighbor sites of j. The right- hand side of Eq. ( 6) contains zeroth-order ( ωK), first-order ( ζ), and second-order ( ξ2) terms of the interdisk interactions. For 1D chain systems, we have i˙ψj=/parenleftbigg ωK−ξ2 ωK/parenrightbigg ψj+ζ(ψj+1+ψj−1) −ξ2 2ωK(ψj+2+ψj−2). (7) Taking the Fourier transformation, we obtain an eigenvalue equation, i˙/Psi1(kx,t)=Hk/Psi1(kx,t) with a momentum space FIG. 1. A schematic illustration of the staggered 1D chain of magnetic nanodisks without the domain-wall defect (a), and witha pair of domain-wall and anti-domain-wall defects (b). A single (double) bond represents the longer (shorter) interdisk distance. Band structure of the system without the domain-wall defect (c), and with a pair of domain-wall and anti-domain-wall defects (d). A pair of states at ω=ω Kis induced by the defects (red). Hamiltonian Hk=ωK+2ζcoskx−2ξ2 ωKcos2kx, (8) describing a single-band Hamiltonian of magnetic excitations. Now, let us consider a staggered 1D chain of magnetic nanodisks [Fig. 1(a)] with periodic boundary condition which mimics the SSH system [ 4]. Because of the staggered lattice structure, the interdisk interactions ( ζandξ) are divided into two different types: ζ→/braceleftbigg ζ(1+/Delta1) ζ(1−/Delta1),ξ →/braceleftbigg ξ(1+/Delta1/prime) ξ(1−/Delta1/prime). (9) Here,/Delta1and/Delta1/prime, which can be either positive or negative, rep- resent the staggeredness of the SSH system. By substitutingEq. ( 9) into Eq. ( 6) and introducing sublattice indices Aand B,w eh a v e i˙ψ A 2m=/parenleftBigg ωK−ξ2(1+/Delta1/prime2) ωK/parenrightBigg ψA 2m +ζ(1+/Delta1)ψB 2m+1+ζ(1−/Delta1)ψB 2m−1 −ξ2(1−/Delta1/prime2) 2ωK/parenleftbig ψA 2m−2+ψA 2m+2/parenrightbig , (10) i˙ψB 2m+1=/parenleftBigg ωK−ξ2(1+/Delta1/prime2) ωK/parenrightBigg ψB 2m+1 +ζ(1−/Delta1)ψA 2m+2+ζ(1+/Delta1)ψA 2m −ξ2(1−/Delta1/prime2) 2ωK/parenleftbig ψB 2m−1+ψB 2m+3/parenrightbig . (11) We note that ξ2/Delta1/prime2terms, which are induced from the stag- geredness of ξ, appear in the identity matrix part of the momentum space Hamiltonian. Because these terms cannot 134423-2REALIZATION OF SU-SCHRIEFFER-HEEGER STATES … PHYSICAL REVIEW B 101, 134423 (2020) change the topology of the Hamiltonian and are negligible in the small /Delta1/primelimit, we discard ξ2/Delta1/prime2terms in this paper. Taking the Fourier transformation, we obtain Hk=/parenleftBigg ωK−2ξ2 ωKcos2kx 2ζ(coskx−i/Delta1sinkx) 2ζ(coskx+i/Delta1sinkx) ωK−2ξ2 ωKcos2kx/parenrightBigg =/parenleftbigg ωK−2ξ2 ωKcos2kx/parenrightbigg I2×2+n(kx)·σ, (12) where the basis of the Hamiltonian is /Psi1(kx)= (ψA(kx),ψB(kx))Tandσ=(σx,σy) are the Pauli matrices. The eigenvalues of Eq. ( 12)a r eg i v e nb y ω±=ωK−2ξ2 ωKcos2kx±2ζ/radicalBig cos2kx+/Delta12sin2kx.(13) In Fig. 1(c), we show the dispersion relation of Eq. ( 13). For calculation, we take the model parameters ωK/2π= 0.955 GHz, ζ/2π=− 0.04 GHz, ξ/2π=0.13 GHz, and /Delta1=0.3 in accordance with micromagnetic simulation results in next section. The staggeredness /Delta1induces a finite gap. We note that the particle-hole symmetry is broken because of themomentum dependent diagonal component in Eq. ( 12)f r o m the second-order interactions ( − 2ξ2 ωKcos2kxI2×2). The second- order interaction term can be treated as a smooth perturbationof the Hamiltonian and does not change the topological char-acter of the system. The topological number of the Hamil-tonian ( 12) is the winding number of the two-component unit vector ˆn(k x)=n(kx)/|n(kx)|≡(cosθk,sinθk) which is expressed by the integral [ 47–49] N=1 2π/integraldisplay BZdkx/parenleftbiggdθk dkx/parenrightbigg =sgn(/Delta1), (14) where θk=tan−1(ny/nx)=tan−1(/Delta1tankx) is a polar angle of the unit vector in momentum space. The winding number iscorresponding to the homotopy map π 1(S1)=Z. Equation (14) implies that there are two topologically distinct phases which are represented by the sign of /Delta1. Expanding Eq. ( 12) around kx=π/2, which minimizes the band gap, we obtain an effective Dirac Hamiltonian Hk=/parenleftBigg ωK −2ζ/bracketleftbig/parenleftbig kx−π 2/parenrightbig +i/Delta1/bracketrightbig −2ζ/bracketleftbig/parenleftbig kx−π 2/parenrightbig −i/Delta1/bracketrightbig ωK/parenrightBigg .(15) Diagonalizing Eq. ( 15), we obtain the eigenfrequencies with a band gap /Delta1, ω±=ωK±2ζ/radicalbigg/parenleftBig kx−π 2/parenrightBig2 +/Delta12. (16) Because a topological bound state exists at the interface between the two topologically distinct phases, we considera situation where the staggeredness /Delta1is reversed its sign at x=0:/Delta1(x)=/Delta1 0sgn(x). In this case, a midgap bound state appears at ω=ωK, without changing the bulk dispersions of upper and lower bands [see Fig. 1(d)]. From Eq. ( 15), we read that the midgap bound state satisfies /parenleftbigg 0 i∂x−i/Delta1(x) i∂x+i/Delta1(x)0/parenrightbigg /Psi1bound=0, (17) FIG. 2. Band structures of the Dirac Hamiltonian (a) and lo- calization of the bound state (b) for /Delta10=0.2 (dotted), /Delta10=0.3 (dashed), and /Delta10=0.5 (solid). which results in /Psi1bound (x)∼/parenleftbigg0 e−/Delta10|x|/parenrightbigg (/Delta10>0), /Psi1bound (x)∼/parenleftbigg e/Delta10|x| 0/parenrightbigg (/Delta10<0). (18) Equation ( 18) shows that the bound state is exponentially localized at the domain wall. This is a magnetic analogof the SSH system which possesses a soliton with half-electric charge [ 4]. Creation of the bound state is compensated by one-half of a state missing from the two bulk bandscorresponding to ω=ω ±.I nF i g . 2, we show the band structures of the effective Dirac Hamiltonian in Eq. ( 15) and localization of the bound state for different valuesof/Delta1 0. III. MICROMAGNETIC SIMULATION We perform micromagnetic simulations to visualize the collective dynamics of the magnetic vortex lattice. Here,we use following parameters of typical permalloy [ 37]: the saturation magnetization M s=800 erg /cm3, the exchange stiffness A=1.3×10−6erg/cm. In order to obtain the clear fast Fourier transform (FFT) image, we choose asmall Gilbert damping constant α=0.001. The diameter and thickness of magnetic nanodisk are chosen to be 80and 20 nm, respectively. The unit-cell size is chosen to be4×4×20 nm 3. We consider a 1D bipartite chain of 40 identical magnetic nanodisks with a periodic boundary condition as shown inFig. 3(a). Each disk has a magnetic vortex with the same polarity ( p=1) and chirality ( C=− 1) [see Fig. 3(b)]. We simulate the collective dynamics of the vortex gyration inthe bipartite lattice with a domain wall (11th disk) and ananti-domain wall (31st disk) which are separated by 20 disks.To obtain the dispersion relation of collective vortex gyra-tion modes, we apply a sinc function of external magneticfield, H(t)=H 0sin[2πf(t−t0)]/[2πf(t−t0)]ˆx, (19) on one of the disks with H0=10 mT, f=20 GHz, and t0= 1n s . Then, we obtain the dispersion relation from the fast Fourier transform (FFT) of the temporal oscillations of x component of the vortex core position. Figure 3(d) shows the resonant spectrum of a vortex gyration mode in anisolated magnetic nanodisk. We find that the single vortex 134423-3GO, HONG, LEE, KIM, AND LEE PHYSICAL REVIEW B 101, 134423 (2020) FIG. 3. (a) Illustration of the 1D bipartite lattice of magnetic nanodisk with a domain wall (11th disk) and an antidomain wall (31st disk) profile. (b) A magnetic nanodisk containing a single vortex. (c) V ortex core dynamics in an isolated magnetic nanodisk. (d) Resonant spectrum of single vortex gyration in an isolated magnetic nanodisk. Dispersion relation of collective vortex gyration in the bipartite lattice when theexternal field is far away from the domain-wall position (21st disk) with interdisk distance [ d 1,d2] of (e) [16 nm, 24 nm], (f) [12 nm, 28 nm], (g) [8 nm, 32 nm], and (h) [4 nm, 36 nm], and when the external field is on the domain-wall position (11th disk) with interdisk distance of (i) [16 nm, 24 nm], (j) [12 nm, 28 nm], (k) [8 nm, 32 nm], and (l) [4 nm, 36 nm]. gyration mode has a peak at f0=ωK/2π=0.955 GHz. Figures 3(e)–3(h) show the dispersion relation of the bipartite chain when the external field is located far away from thedomain-wall position (21st disk). As the difference of inter-disk distance ( d=d 2−d1) increases, a more distinct band splitting (into upper and lower bands) is observed. Note thatthe in-gap mode between the upper and lower bands has notbeen excited in this case, because it is localized on the defectposition and thus far away from the external-field position.When the external field locates on the domain-wall position(11th disk), we find that a midgap mode is excited near f 0 without significant change of the bulk dispersion [Figs. 3(i)– 3(l)]. The simulation results coincide with the analytic results in Sec. IIwith appropriate model parameters [see Figs. 1(c) and1(d)].IV . 2D MAGNETIC WA VEGUIDE Now let us consider a 2D extension of our 1D magnetic SSH model, which will be shown to support a magneticwaveguide of excitations below. The schematic illustration ofthe 2D lattice is shown in Fig. 4(a). The 2D extended model includes additional interactions proportional to ζ y,ξ2 y/2ωK, andξ2 xy(1±¯/Delta1)/2ωK[see Fig. 4(b)]. We note that ¯/Delta1represents the staggeredness of the second-order interactions, and itssign is reversed at the defect position. In momentum spacerepresentation, we have an effective Hamiltonian (see theAppendix) H 2D(kx,ky)=/parenleftbigg HAAHAB (HAB)∗HBB/parenrightbigg , (20) 134423-4REALIZATION OF SU-SCHRIEFFER-HEEGER STATES … PHYSICAL REVIEW B 101, 134423 (2020) FIG. 4. (a) A schematic illustration a two-dimensional extension of the 1D magnetic SSH model. (b) Additional interactions of the 2D tight-binding model. Bulk and bound-state dispersions of the 2Dmodel without the domain-wall defect (c) and with the domain-wall defect (d). For calculation, we take the model parameters ω K/2π= 0.955 GHz ζ/2π=− 0.04 GHz, ξ/2π=0.13 GHz, /Delta1=0.3,ζy= ζ/4,ξxy=ξ/4,ξy=ξ/6, and ¯/Delta1=/Delta1. (e) Magnetic wave propaga- tion in the magnetic waveguide supporting signal localization. (f) Magnetic wave propagation in the magnetic waveguide supportingselective propagation of frequency. where HAA=ω2D 0−2ξ2 ωKcos2kx=HBB, (21) HAB=2ζ(coskx−i/Delta1sinkx) +2ξ2 xy ω0cosky(coskx−i¯/Delta1sinkx), (22) andω2D 0(ky)=ωK−2ξ2 y ωKcos2ky+2ζycosky. The additional interactions yield the additional dispersion along kydirection. The resultant 2D band dispersions without and with the(stringlike) domain-wall defect are shown in Figs. 4(c) and 4(d), respectively. In the 2D lattice, the pointlike defect in the 1D model is extended in the ydirection and forms a domain- wall string. In the presence of the domain-wall defect, we findthat the bound state with a frequency ω 2D 0(ky) is localized on the defect position (see the Appendix). In this 2D soliton lattice model, the topological midgap bound states are localized at the defect position and spatiallyconnected in the ydirection. Therefore, magnetic excitations on the bound state propagate well along the defect stringwith a small spread in the transverse ( x) direction. This propagation characteristic realizes a magnetic waveguide byusing magnetic solitons with signal localization and selec-tive propagation of frequency modes. Figures 4(e) and 4(f) show the schematic illustration of two functionalities of themagnetic soliton waveguide. In Fig. 4(e), the incoming wavepacket is a plane wave (i.e., uniform along the xdirec- tion) and has a frequency corresponding to the bound state.Because this frequency mode can only propagate throughthe defect string, the outgoing wave packet is localized onthe defect site. In Fig. 4(f), the incoming wave packet on the defect site is a white signal having equal intensities for allfrequencies. However, most frequency modes on the defectsite cannot propagate in the ydirection, except for the bound state. As a result, the outgoing wave packet on the defectsite has a peak at a frequency corresponding to the boundstate. Unfortunately, in our magnetic waveguide, we cannotobtain a single frequency outgoing wave packet because thegroup velocity along the bound state ( v y) is very small if the bandwidth of the bound state is too narrow. For a waveguidewith finite group velocity, we need some intermediate valuesofy-directional hopping parameters. In Figs. 4(c) and4(d),w e choose a set of parameters which results in |vy|/|vupper x|≈1 and|vy|/|vlowerx|≈0.4, where |vupper x|and|vlowerx|are the aver- aged absolute value of group velocities (along the xdirection) of the upper and lower band over the first Brillouin zone,respectively. Note that our 2D magnetic waveguide does not show the topologically protected (back-scattering free) transport. Anydisorders or defects in our 2D waveguide give rise to back-scattering for transport along the waveguide. However, theexistence of the waveguide with frequencies separated fromthe bulk bands is topological in a sense that the waveguide iscomposed of topological modes in each SSH chain. We note that the frequency of the bound state is mainly determined by the gyrotropic frequency of a single magneticsoliton, which is tunable by external perturbations. For ex-ample, in the presence of an effective magnetic field H eff perpendicular to the disk plane, the gyrotropic frequency can be described as [ 50,51] ω/similarequalK G(1+kHeff), (23) where kis a proportionality constant. This suggests that the waveguide property can be manipulated by the exter-nal magnetic field or voltage-induced magnetic anisotropychange [ 52]. V . CONCLUSION To summarize, we have studied collective dynamics in a one-dimensional bipartite chain of the magnetic vorticesor skyrmions. In our magnetic system, the domain-wall-likedefects are produced by changing the interdisk distances.We have found that the defects induce the midgap stateswhich are confined at the defect position. We also providethe micromagnetic simulation results supporting the analyticresults. Our finding on the 1D model is analogous to that of theSSH model in the electron system. In contrast to the electronicSSH model, in which it is hard to manipulate the domain-wallprofiles of atomic arrangement, the topological manipulationis feasible in our magnetic SSH model. As a technologicalapplication, we propose a two-dimensional extension of our1D model, which supports a magnetic waveguide of magneticexcitations. The magnetic waveguide provides not only a sig-nal localization but also selective propagation of the frequency 134423-5GO, HONG, LEE, KIM, AND LEE PHYSICAL REVIEW B 101, 134423 (2020) modes. Our work suggests that a spintronics device based on magnetic metamaterials can offer a way for precise control ofthe the oscillation of magnetic soliton lattice. ACKNOWLEDGMENTS G.G. was supported by the National Research Foundation of Korea (NRF) (Grant No. NRF-2019R1I1A1A01063594).S.K.K. was supported by a Young Investigator Grant (YIG)from Korean-American Scientists and Engineers Association(KSEA) and Research Council Grant No. URC-19-090 of theUniversity of Missouri. K.-J.L. acknowledges support by the NRF (Grant No. NRF-2020R1A2C3013302). G.G. and I.-S.H. contributed equally to this work. APPENDIX: COMPUTATIONAL DETAILS OF THE 2D MODEL Here, we derive the effective Hamiltonian of the 2D ex- tension of our 1D magnetic SSH model. The lattice structureof our 2D model is shown in Fig. 4(a) and the second-order interactions are shown in Fig. 4(b). By using Eq. (6) of Ref. [ 23], we write i˙ψA 2m=/parenleftbigg ωK−ξ2+ξy2 ωK/parenrightbigg ψA 2m+ζ(1+/Delta1)ψB 2m+x+ζ(1−/Delta1)ψB 2m−x+ζy/parenleftbig ψA 2m+y+ψA 2m−y/parenrightbig −ξ2 2ωK/parenleftbig ψA 2m+2x+ψA 2m−2x/parenrightbig −ξ2 y 2ωK/parenleftbig ψA 2m+2y+ψA 2m−2y/parenrightbig +ξ2 xy(1+¯/Delta1) 2ωK/parenleftbig ψB 2m+x+y+ψB 2m+x−y/parenrightbig +ξ2 xy(1−¯/Delta1) 2ωK/parenleftbig ψB 2m−x+y+ψB 2m−x−y/parenrightbig , (A1) i˙ψB 2m+x=/parenleftbigg ωK−ξ2+ξy2 ωK/parenrightbigg ψB 2m+x+ζ(1−/Delta1)ψA 2m+2x+ζ(1+/Delta1)ψA 2m+ζy/parenleftbig ψB 2m+x+y+ψB 2m+x−y/parenrightbig −ξ2 2ωK/parenleftbig ψB 2m+3x+ψB 2m−x/parenrightbig −ξ2 y 2ωK/parenleftbig ψB 2m+x+2y+ψB 2m+x−2y/parenrightbig +ξ2 xy(1−¯/Delta1) 2ωK/parenleftbig ψA 2m+2x+y+ψA 2m+2x−y/parenrightbig +ξ2 xy(1+¯/Delta1) 2ωK/parenleftbig ψA 2m+y+ψA 2m−y/parenrightbig . (A2) In Eqs. ( A1) and ( A2), we neglect ξ2/Delta1/prime2/ωKterms which are induced from the staggeredness of ξ. Taking the Fourier transformation, we obtain a momentum space Hamiltonian H2D(kx,ky)=/parenleftbigg HAAHAB HBAHBB/parenrightbigg , (A3) where HAA=ωK−2ξ2 ωKcos2kx−2ξ2 y ωKcos2ky+2ζycosky=HBB, (A4) HAB=2ζ(coskx−i/Delta1sinkx)+2ξ2 xy ω0cosky(coskx−i¯/Delta1sinkx)=(HBA)∗. (A5) In order to obtain the bound-state solution, we expand the Hamiltonian around kx=π/2 and replace kx−π/2t o−i∂x. Then we have HAA=ω2D 0(ky)=HBB, (A6) HAB=2ζ(i∂x−i/Delta1)+2ξ2 xy ω0cosky(i∂x−i¯/Delta1)=2ζ[1+α(ky)]i∂x−2iζ/Delta1[1+β(ky)]=(HBA)∗, (A7) where ω2D 0(ky)=ωK−2ξ2 y ωKcos2ky+2ζycosky,α(ky)=ξ2 xy ζωKcosky,β(ky)=¯/Delta1 /Delta1α(ky). 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PhysRevB.97.184432.pdf

PHYSICAL REVIEW B 97, 184432 (2018) Composition and temperature-dependent magnetization dynamics in ferrimagnetic TbFeCo Wei Li,1Jiaqi Yan,1Minghong Tang,2Shitao Lou,1,*Zongzhi Zhang,2,†X. L. Zhang,1and Q. Y . Jin1,2,‡ 1State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062 , China 2Department of Optical Science and Engineering, Fudan University, Shanghai 200433 , China (Received 16 November 2017; revised manuscript received 12 May 2018; published 30 May 2018) The temperature-dependent magnetization dynamics in ferrimagnetic TbFeCo alloys with various compositions of Tb is investigated by the pump-probe time-resolved magneto-optical Kerr effect (TR-MOKE) in differentgeometries. It is shown that, for the case when a magnetic field is applied noncollinearly to the easy axis atroom temperature, the decrease of the MOKE signal occurring at several tens of picoseconds (ps) after a rapiddemagnetization within a few hundred femtoseconds (fs) is in fact caused by a highly damped precessional motionwith precession lifetime shorter than the precession period. This is demonstrated by reducing the Tb content to6% in atomic ratio and measuring TR-MOKE at elevated temperatures. This has the dual effect of reducing thedamping constant allowing the observation of more precession cycles and of weakening the exchange interactionbetween Tb and FeCo, making the precession oscillations more pronounced. The results give important insightinto the ultrafast spin dynamics of rare-earth-doped transition-metal alloys, and the remarkable impact of Tb onthe damping in such alloy systems. DOI: 10.1103/PhysRevB.97.184432 I. INTRODUCTION In the past few decades, ultrafast magnetization dynamics in magnetic structures has attracted a great deal of attentiondue to its crucial importance for ultrafast magnetic recordingand ultrafast spintronic devices [ 1–7]. Using an ultrashort laser pulse incident on magnetic films, the magnetization can beperturbed, showing an ultrafast demagnetization, in certaincases followed by a precession, and even a reversal of magne-tization. The pioneering work by Beaurepaire et al. exploiting the time-resolved magneto-optical Kerr effect (TR-MOKE) onnickel thin films demonstrated an ultrafast demagnetization[1]. They introduced a phenomenological three-temperature (3T) model, describing the interaction between the electron,spin, and lattice subsystems and giving an explanation of thedemagnetization and its relaxation. The ultrafast demagneti-zation of thin films consisting of only ferromagnetic transitionmetals (TMs) is considered to follow the 3T model with ademagnetization occurring at less than 1 ps. In recent years, the magnetization dynamics of rare-earth– transition-metal (RE-TM) ferrimagnetic alloys such as GdCo,GdFeCo, TbFeCo, etc. has also been studied [ 6,7], and it has shown some different behaviors from that of ferromag-netic TM materials due to their complex magnetic struc-tures. Mekonnen et al. presented their results of the laser- induced demagnetization in GdCo and GdFeCo [ 7], where they found a second demagnetization step. They proposeda four-temperature (4T) model to describe this phenomenonwith four coupled differential equations that take the heatflow between the different heat baths (electrons, lattice, Gd *stlou@admin.ecnu.edu.cn †zzzhang@fudan.edu.cn ‡qyjin@phy.ecnu.edu.cn; qyjin@fudan.edu.cn4fspins, and FeCo 3 dspins) into consideration. The new observation of two-step demagnetization can then be wellexplained, with a result that the demagnetization of rare-earth(RE) metal is about two or three orders of magnitude slowerthan that of TMs [ 5,8–11]. In addition, some work including theoretical [ 12] and experimental [ 13–15] investigations has concerned the magnetization dynamics of precessional damp-ing for RE-metal doped permalloy. It was found that mostRE-metal atoms (such as Tb, etc.) induced a large increaseof damping constant αexcept for Eu and Gd, which have no orbital momentum. Therefore, it is expected and also has beenreported that the precession in RE-TM ferrimagnetic alloyswith Tb, Dy, etc. rare-earth dopants ( >10%) is hardly observed, due to the very large increase of damping with those REelements. In RE-TM ferrimagnetic alloys, the magnetizations of RE and TM sublattices are aligned antiparallel and cancel eachother at the magnetization compensation temperature T comp [16]. The magnetic properties of RE-TM alloys, such as net sat- uration magnetization Ms, coercive field Hc, Curie temperature Tc, andTcomp, depend on Tb composition [ 16,17]. AtTcomp,t h e net magnetization changes sign: below or above Tcomp, the net magnetization is dominated by one of the magnetic sublattices(RE-dominant or TM-dominant). To understand the magnetization dynamics of RE-TM ferrimagnets in more detail, we present here a careful andsystematic study of the ultrafast demagnetization and relax-ation of TbFeCo films, a typical RE-TM alloy with ferri-magnetic order. We investigate the magnetization dynamicbehavior for different cases relative to T comp by using Tb or FeCo-dominant alloys at room temperature (RT) and bychanging the measurement temperature through T comp at a given composition of Tb. The results show that when themagnetic field is applied noncollinearly to the easy axis, amagnetization precession takes place around the time range of 2469-9950/2018/97(18)/184432(6) 184432-1 ©2018 American Physical SocietyLI, YAN, TANG, LOU, ZHANG, ZHANG, AND JIN PHYSICAL REVIEW B 97, 184432 (2018) several tens of ps after a rapid demagnetization, with various damping constants depending on the Tb composition andtemperature. II. EXPERIMENTS A. Samples A series of amorphous thin films Tb x(Fe 0.2Co0.8)1−xwith various Tb compositions were deposited on Corning glasssubstrates in a Kurt J. Lesker magnetron sputtering systemwith a base pressure better than 1 ×10 −8Torr. The TbFeCo films were achieved by cosputtering from Tb and Fe 0.2Co0.8 targets. A 4-nm-thick Ta layer was deposited on the glass as a buffer layer and a 6.7-nm-thick Pt on the top as acapping layer protecting the magnetic layer from oxidation.The thickness of TbFeCo is about 12 nm. All of the filmswere confirmed to have a perpendicular anisotropy by vibratingsample magnetometer (VSM) measurements in both in-planeand out-of-plane directions [ 18]. B. Measurement methods The pump-probe TR-MOKE measurements were per- formed at various temperatures by a pulsed Ti:sapphire ampli-fier laser at a central wavelength of 800 nm, with a repetitionrate of 1 kHz and a pulse width of about 130 fs. The linearlypolarized laser beam was split into two parts with unequalpowers, and the ratio of pump-to-probe beams was chosen tobe approximately 40 : 1. The probe pulse beam was incidentonto the sample at a small angle, and the spot sizes of thepump and probe beam are about 0.5 and 0.1 mm, respectively,so that homogeneous heating was ensured in the probing areaof the sample. An external magnetic field H extgenerated by an electromagnet was applied at an angle of 17owith respect to the sample plane and perpendicular to the sample plane (i.e.,parallel to the direction of magnetization). In these two con-figurations, the magneto-optical Kerr rotation was measured.The ellipticity was also measured to ensure that the optical(magnetization-independent) contributions to MOKE werenegligible [ 19]. The signals were read out by a lock-in amplifier with an optical chopper that modulates the pump beam at afrequency of 108 Hz. We should mention that at a photonenergy of 1.55 eV (800 nm in wavelength), the contributionto the magneto-optical Kerr signal is predominantly given bythe FeCo subsystem [ 20,21]. III. RESULTS AND DISCUSSIONS A. Static magnetic property measurements The static magnetic properties were first measured by a vibrating sample magnetometer (VSM) at room temperature(RT). The sample with a Tb composition of 24% exhibitsno obvious magnetic signal, which means the compensationtemperature with 24% Tb is around RT, where the antiparallellyaligned Tb and FeCo magnetic sublattices compensate eachother [ 22]. The samples with Tb compositions higher than 24% are Tb-dominant, and those having Tb compositions below24% are FeCo-dominant. Figure 1(a) shows a variation of the FIG. 1. (a) The saturation magnetization Msas a function of temperature for Tb x(FeCo) 1−x(x=22%, 24%, and 30%). (b) The Kerr loops of MOKE without pump pulse for Tb x(FeCo) 1−xsamples with various Tb compositions ( x=17%, 19%, 24%, 30%, and 33%) at RT. saturation magnetization Mswith temperature below 500 K for samples with x=22%, 24%, and 30% corresponding to FeCo-dominant, compensation composition, and Tb-dominantcases, respectively. Their T comp are found to be 210, 295, and 470 K, respectively, and the measured Curie temperature Tc of the sample of x=30% is about 500 K. No measurements were made beyond 500 K, but we would expect Tcfor samples of 22% and 24% to be greater than this value. Tcomp increases with increasing Tb composition, while Tcdecreases [ 23]. Due to the strong intersublattice 3 d-5d6s-4fexchange interaction between FeCo and Tb magnetic moments, Tcis higher than RT for TbFeCo alloys, though that of pure Tb is very low(only about 220 K [ 24]). The static magnetic properties were also measured using standard Kerr loops for all samples withvarious Tb compositions at RT [typical results are shown inFig. 1(b)]. Clearly, the coercivity H cincreases close to the compensation composition of x=24%. The MOKE signal reverses when the composition increases through 24% (fromFeCo-dominant to Tb-dominant, or vice versa) due to thechange of direction of the FeCo magnetization. Moreover,magnetic hysteresis could not be observed for the concentrationx=24%, which is attributed to its high coercivity (larger than our maximum magnetic field ∼14 kOe). 184432-2COMPOSITION AND TEMPERATURE-DEPENDENT … PHYSICAL REVIEW B 97, 184432 (2018) FIG. 2. TR-MOKE curves in Tb x(FeCo) 1−xusing a pump fluence of 4.2 mJ /cm2withHextof 9.2 kOe at RT. (a) x=10%, 12%, 15%, 17%, and 19%; and (b) x=30% and 33%. The orientation of magnetization (up or down) is decided by the external magnetic field. The inset between (a) and (b) shows the schematic illustration of the TR-MOKE measurement with Hextapplied with an angle θHof 73o away from the easy-axis direction of magnetization. (c) TR-MOKE curves at RT for samples of Tb 0.12(FeCo) 0.88and Tb 0.33(FeCo) 0.67in the polar geometry ( θH=0o), i.e., Hextis perpendicular to the film plane (parallel to the perpendicular anisotropy field of the films). B. Composition-dependent magnetization dynamics Figure 2shows the TR-MOKE results of laser-induced magnetization dynamics with demagnetization and relaxationrecovery in Tb x(FeCo) 1−x(x=10%, 12%, 15%, 17%, 19%, 30%, and 33%) using a pump fluence of 4.2 mJ /cm2under an external field of 9.2 kOe at RT. In Fig. 2(a), ultrafast demagnetization occurs initially at ∼600 fs. This process refers to the demagnetization of FeCo with a rapid increaseof spin temperature of 3 delectrons. Subsequently, a fast magnetization recovery takes place within about 2 ps, cor-responding to the thermal equilibrium of the FeCo electron-spin-lattice system. With increasing Tb composition (from10% to 19%), the amplitude of the demagnetization increasesslightly [see the inset of Fig. 2(a)]. On changing from FeCo- dominant to Tb-dominant compositions, as shown in Fig. 2(b), the demagnetization amplitude for Tb-dominant samples islarger. The Kerr signal originates from the FeCo subsystemin TbFeCo alloys at a wavelength of 800 nm. In addition, theexchange interaction between FeCo spins is the most importantfactor to determine the demagnetization behavior, which isobviously Tb-composition-dependent. With the increase of Tb composition in TbFeCo, the demagnetization is thought tobe easier due to the weakening of the exchange interactionbetween FeCo spins arising from the inhibition effect inducedby Tb doping ( T cis also lower for higher Tb composition [23]). This is in accord with the result in Ref. [ 18], where it was found in TbCo alloys that the trend of magnetizationquenching increased with increasing Tb composition becauseof the decreasing Co-Co coupling constant. In addition, anotherexplanation of faster reduction of the signal with increasingTb composition is the more efficient transfer of angular mo-mentum between FeCo and Tb sublattices. The intersublatticeangular momentum transfer speeds up the demagnetization inantiferromagnetically ordered materials [ 25]. After the ultrafast demagnetization and recovery processes within 2 ps, the curves exhibit an interesting phenomenonin that the Kerr signal decreases again. However, this phe-nomenon is not observed for Tb-dominant samples shownin Fig. 2(b). The schematic geometry for this TR-MOKE measurement is illustrated in the inset of Fig. 2. The external magnetic field H extis applied with an angle θHof 73oaway from the easy-axis direction of magnetization (i.e., 17orelative to the film plane), driving the magnetization orientation away from the perpendicular easy axis. This geometry is set for thepurpose of initiating the possible precession of magnetizationwhen the external magnetic field is applied along the directionclose to the hard axis of magnetization (with a small angle).This suggests that the second decrease of the Kerr signal mightbe related to the precession of magnetization. To test this hypothesis, we carried out TR-MOKE mea- surements in a polar geometry, i.e., the magnetic field isapplied perpendicular to the film plane (along the easy axisof magnetization for TbFeCo). Figure 2(c) gives the typical results for the samples with Tb compositions of 12% and 33%,where only ultrafast demagnetization can be observed for bothFeCo-dominant and Tb-dominant samples. We did not findthe second demagnetization described in GdFeCo with thesame measurement geometry [ 7], because in TbFeCo alloys the angular momentum transfer from Tb magnetic moments tothe lattice is considerably faster than that from Gd magneticmoments to the lattice in GdFeCo alloys, due to the nonzero4forbital momentum of Tb [ 9]. The second decrease of the Kerr signal occurring after 2 ps measured in the geometry ofFig.2(a)is believed to correspond to the first oscillation of the magnetization precession of FeCo spins. Due to the very largedamping constant in TbFeCo with rare-earth Tb atoms, theprecession decays very fast and dies out before the completionof the first period of oscillation. This argument is also supportedby the evidence of the magnetic-field-dependent TR-MOKEexperiment. Figure 3(a) displays the TR-MOKE data for the sample of x=12% using the same geometry as for the data in Fig. 2(a) withH extchanging from 750 Oe to 14 kOe. The corresponding delay time of the minimum Kerr signal for thesecond decrease becomes smaller as H extincreases, which is a typical feature of precession. Therefore, we believe that thedynamics occurring at several tens of ps timescale in Fig. 2(a)is the magnetization precession, which is sensitive to the appliedmagnetic field. At the large delay (200–300 ps) shown in Fig. 2(a),t h e relative reduction of the MOKE signal is larger for smaller 184432-3LI, YAN, TANG, LOU, ZHANG, ZHANG, AND JIN PHYSICAL REVIEW B 97, 184432 (2018) FIG. 3. TR-MOKE curves at RT as a function of Hextfor Tb0.12(Fe 0.2Co0.8)0.88(a) for Tb 0.06(Fe 0.2Co0.8)0.94and pure Fe 0.2Co0.8 (b). The solid lines in (a) are fitting curves. Parts (c)–(e) are, respec- tively, the magnetic-field dependences of magnetization precession frequency f, decay time τ, and the effective Gilbert damping αeff, fitted from the TR-MOKE curves in (a). Tb compositions. We know that at that delay, the precession is completely over, and the magnetization is oriented alongthe effective field direction. The relative reduction of the z projection of the magnetization is defined by the pump-inducedtilt of the effective field, which reduces with increasing Tbcomposition. This could be caused by the relatively largerreduction of anisotropy field upon laser pumping for smallerTb compositions, also it is possible due to the demagnetizingfield effect when the external magnetic field is applied at anangle with respect to the sample plane. The demagnetizing fielddecreases with increasing Tb composition as a result of thesmaller saturation magnetization for larger Tb compositions. In addition, we know that Tb-doped FeCo films show a very strong dependence of the damping constant on the dopantcomposition. With increasing Tb composition, the dampingconstant αincreases significantly [ 13,15]. We fabricated one sample with a low Tb composition of 6% and measured itsdynamic curve at RT, as shown in Fig. 3(b), in which we see more precession oscillations than those with higher Tbcompositions. As a comparison, the TR-MOKE curve of pureFeCo film is also displayed. Apparently, a very low Tb dopingof 6% in FeCo has dramatically destroyed the precession,though still two precession oscillations could be identified. Wefit the measured dynamic Kerr signal from Fig. 3(a) by using the following formula [ 26,27]: θ k=a+be−t/t0+csin(2πf t+ϕ)e−t/τ, (1) where the first term acorresponds to the background signal and it is close to zero. The second exponential decay termrepresents the slow magnetization recovery, where bis the amplitude and t 0is the characteristic relaxation time. The third term describes the magnetization precession dynamics, wherec,f,ϕ, andτrefer to the oscillation amplitude, frequency, initial phase, and decay time, respectively. The typical valueof fit parameters a,b,c, andt 0withHextof 10 kOe is 0.1, 0.07, 0.2, and 304.4, respectively. The fitted precession frequency f and decay time τare, respectively, plotted in Figs. 3(c)and3(d) as a function of Hext. We obtain that the precession frequency fincreases monotonically with Hext. Based on the fitted f andτ, the effective Gilbert damping constant αeffis derived approximately from the simple equation of αeff=1/2πf τ [28]. As shown in Fig. 3(e),αeffdecreases dramatically with an increase of the external field Hext. We can expect that by further increasing Hext, the effective damping constant will eventually approach its intrinsic value. The high αeffvalue in the low-field region results mainly from the inhomogeneous distributionof magnetization or magnetic anisotropy, which may arisefrom the interface roughness, thin layer thickness, and otherfactors [ 29]. We next utilize a decaying exponential function [ 29], α eff=αex0exp(−Hext/H0)+α0,t ofi tt h e αeffdata, where the extracted α0corresponds to αeffat an infinite Hext.T h e fitting curve is described by the solid line in Fig. 3(e), and the extracted α0is 0.714±0.021, which is very close to the early experimental work for Tb-doped Ni 80Fe20[13]. In general, the origin of the increase of damping in TMs withRE impurities is based on the strong spin-orbit couplingor the spin-spin interaction. Theoretical work investigatingorbit-orbit coupling between the conduction electrons and theimpurity ions was presented by Rebei et al. [12]. But it was sharply contradicted by Woltersdorf et al. , who explained the temperature-dependent Gilbert damping by using the slowlyrelaxing impurity model [ 15]. Although these two models cannot be distinguished by the data in our paper, both ofthem reach the same conclusion, namely that the dampingconstant becomes large when RE is added in TM, whichis consistent with the results in Fig. 2(b) showing that the precession damping of the Tb-dominant samples of 30% and33% is so large that we cannot observe the occurrence ofoscillation. C. Temperature-dependent magnetization dynamics As we mentioned at the beginning, in addition to the Tb composition change, measuring the TR-MOKE at various tem-peratures is another way to study the magnetization dynamicsacross T comp(for some compositions of Tb). The compensation point Tcomp of our Tb 0.24(FeCo) 0.76sample is near RT. It goes to a lower temperature if Tb in TbFeCo becomes less,and it goes to a higher temperature in the opposite case.Studying temperature-dependent magnetization dynamics isalso a way to change the exchange interaction between Tband FeCo, which gives rise to a change in the precessiondamping constant. The temperature plays an important rolein the precession damping [ 13]. Figure 4(a) shows the TR-MOKE signal dependence on the temperature for Tb 0.15(FeCo) 0.85. Its static MOKE curves without a pump pulse are shown in Fig. 4(b), where a reversal of the hysteresis loop is found at 80 K after crossing thecompensation point ( T comp≈100 K). At RT, the precession starts but has not completed the first cycle, implying that thelifetime of precession is shorter than its period. This time we 184432-4COMPOSITION AND TEMPERATURE-DEPENDENT … PHYSICAL REVIEW B 97, 184432 (2018) FIG. 4. (a) TR-MOKE curves measured at various tempera- tures for Tb 0.15(FeCo) 0.85and (b) its static MOKE loops without a pump pulse. (c) TR-MOKE curves at various temperatures for Tb0.06(Fe 0.2Co0.8)0.94. Note that the sample temperature presented here does not include the temperature increase (roughly ∼40 K) induced by the pump heating. use a higher pump fluence of 6.0 mJ /cm2in order to ensure that we can measure the precession for lower temperatures,thus the recovery seems more difficult than that with a lowerpump fluence. At 200 K, we can still see the decrease ofmagnetization (part of the precession) to a minimal valueat about 20 ps and a recovery afterward. But at 80 K, noclear precession could be distinguished. So, the precessionbecomes weakened as the temperature decreases from RTto 80 K, implying that the damping constant αincreases with deceasing temperature. When we focus on the lower Tbcomposition of 6%, this phenomenon is revealed more clearly.Figure 4(c) shows the TR-MOKE results measured at various temperatures for the sample of Tb 0.06(Fe 0.2Co0.8)0.94. Several complete oscillation periods of precession are clearly observed.The higher temperature leads to more precession oscillations.Normally we know that the scattering with magnons andphonons becomes stronger when the temperature increases, leading to a higher damping at higher temperature. Indeed,a phenomenon has been observed in Ref. [ 13] in which the damping constant αshows a small increase for undoped NiFe with increasing temperature. However, the NiFe sample withTb doping exhibits a decreasing behavior in damping quitestrongly with increasing temperature. The key point to explainthis is the significant increase of damping originating fromTb dopants. When the temperature increases, the exchangeinteraction between Tb and FeCo becomes weaker [ 25]. Thus, the dissipation of precession energy becomes smaller since theefficiency of transfer of FeCo spin to the lattice through Tbmagnetic moments reduces with increasing temperature, andthe damping, which dominantly represents the dynamics ofFeCo spins for TR-MOKE curves measured at a wavelengthof 800 nm, will be lower at higher temperature. As a result, weobserved a decrease of damping with increasing temperature. IV . CONCLUSIONS In summary, laser-induced ultrafast magnetization dynam- ics in the amorphous alloy TbFeCo is investigated by pump-probe TR-MOKE experiments. In the range of Tb compositionfrom 10% to 33% in our samples, the amplitude of ultrafastdemagnetization o na1p st i m e s c a l es h o w sas l i g h t l yi n c r e a s i n g trend with an increase of Tb composition. Moreover, if themagnetic field is applied noncollinearly to the easy axis atRT, the reduction of the MOKE signal occurring in tens ofps is a magnetization precession of FeCo spins, with a largedamping constant when Tb composition is not very low. Thisis evidenced by the experiment with a magnetic field appliedalong its easy axis where the MOKE oscillation disappears. Bychanging the measurement temperature for the sample with alow Tb composition, we have found that the temperature playsan important role in helping to adjust the precession dampingconstant, so that we can see clear precession oscillationspresented in TbFeCo alloys with less Tb composition at highertemperatures. ACKNOWLEDGMENTS This work was supported by the National Key R&D Program of China (2017YFA0303403), the National BasicResearch Program of China (2014CB921104), the NationalNatural Science Foundation of China (Grants No. 11674095,No. 51671057, and No. 11474067), and the 111 project(B12024). [1] E. Beaurepaire, J. C. 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PhysRevLett.108.076604.pdf

Spin Drift Velocity, Polarization, and Current-Driven Domain-Wall Motion in (Ga,Mn)(As,P) J. Curiale,1,2,*A. Lemaı ˆtre,2C. Ulysse,2G. Faini,2and V. Jeudy1,3,† 1Laboratoire de Physique des Solides, Universite ´Paris-Sud, CNRS, 91405 Orsay, France 2Laboratoire de Photonique et de Nanostructures, CNRS, 91460 Marcoussis, France 3Universite ´Cergy-Pontoise, 95000 Cergy-Pontoise, France (Received 10 November 2011; published 17 February 2012) Current-driven domain-wall motion is studied in (Ga,Mn)(As,P) ferromagnetic semiconducting tracks with perpendicular anisotropy. A linear steady state flow regime is observed over a large temperaturerange of the ferromagnetic phase ( 0:1T c<T<T c). Close to 0 K, the domain-wall velocity is found to coincide with the spin drift velocity. This result is obtained below the intrinsic threshold for domain-wall motion which implies a nonadiabatic contribution to the spin transfer torque. The current spin polarization is deduced close to 0 K and to Tc. It suggests that the temperature dependence of the spin polarization can be inferred from the domain-wall dynamics. DOI: 10.1103/PhysRevLett.108.076604 PACS numbers: 72.25.Dc, 75.50.Pp, 75.60.Jk, 75.78.Fg A spin polarized current flowing through a domain wall (DW) exerts a torque on the DW magnetization. At suffi-ciently large current, this torque produces DW motion. In the damping limited flow regimes, the DW dynamics is commonly derived from a modified Laudau-Lifshitz-Gilbert equation [ 1,2]. Within this phenomenological de- scription, two contributions to the spin torque are usuallyintroduced: an adiabatic term, and a nonadiabatic contri- bution, proportional to the so-called /C12factor. The pre- dicted DW velocities vare proportional to the spin drift velocity of the current carriers and depend on the ratio /C12=/C11, where /C11is the Gilbert damping coefficient. Several authors have carried out microscopic derivations of /C12and /C11from spin relaxation mechanisms due to impurity scat- tering in metals [ 3], or due to spin-orbit interaction [ 4,5]. However, these predictions are rather different and canhardly be compared quantitatively to the limited numberof experimental results. Experimentally, the damping limited flow regimes are difficult to reach due to the high current density threshold Jth required to move DWs. For metallic structures, linear flow regimes vðJÞwere observed only recently, in Pt=Co=AlO xtracks [ 6] with perpendicular anisotropy (Jth¼1012A=m2). In (Ga,Mn)As ferromagnetic semicon- ductors, flow regimes were evidenced ( Jth¼109A=m2 [7]) only close to the Curie temperature in layers with perpendicular magnetic anisotropy [ 7,8]. A /C12=/C11 value close to 1 was deduced from the analysis of current- induced domain-wall dynamics [ 7], performed in the frame of the 1D model [ 1,2]. However, this result remains puz- zling since the /C11values deduced from field-driven mea- surements [ 7,9] strongly differ from theoretical predictions and ferromagnetic resonance measurements. Obviously abetter understanding of the fundamental physics of current-driven DW dynamics would benefit from a model- independent determination of the parameters governing DW motion, such as the carrier spin polarization and thespin drift velocity. Moreover, it would be particularly interesting to study DW motion at low temperature.Indeed, to our best knowledge, carrier spin polarization in (Ga,Mn)As was only estimated close to zero tempera- ture [ 10–12], from point-contact Andreev reflection mea- surements. Reducing the temperature would also decreasethe thermal fluctuations which may significantly affect DWdynamics [ 13]. In previous studies, the flow regime was only accessible in a narrow temperature range below T c. These experi- ments were performed on (Ga,Mn)As tracks grown on metamorphic (In,Ga)As substrates [ 7,8], required to pro- vide a perpendicular anisotropy. However, the metamor-phic growth mode is inherently associated with theformation of emerging dislocations [ 14] and other defects which act as pinning centers for DWs. Moreover, the lowheat conductivity of (In,Ga)As substrates [ 15,16] results in a large track temperature rise produced by Joule heatingwhich impedes the exploration at low temperature. Recently, we developed a new alloy (Ga,Mn)(As,P) grown pseudomorphically on GaAs substrate [ 17], presenting a perpendicular anisotropy. Current-driven DW motion hasbeen reported in this alloy well below T c[18]. In this Letter, we present a thorough investigation of the linear flow regime in (Ga,Mn)(As,P) tracks over the wholetemperature range, from /C240:1T c(T¼13 K )u pt o Tc.D W velocity vis shown, without any assumptions on the nature of the flow regime, to coincide with the carrier spin drift velocity u, close to 0 K. Rather interestingly, these experi- ments suggest that DW dynamics give access to the currentspin polarization P cover the whole temperature range up toTc. The micro-tracks with a perpendicular magnetic anisot- ropy were elaborated [ 17] from a 50 nm thick ðGa0:90Mn0:10ÞðAs0:89P0:11Þfilm, deposited by molecular beam epitaxy on a 375/C22mthick GaAs (001) substrate atT¼250/C14C. The film was then annealed during 1 h, atPRL 108, 076604 (2012) PHYSICAL REVIEW LETTERSweek ending 17 FEBRUARY 2012 0031-9007 =12=108(7) =076604(5) 076604-1 /C2112012 American Physical SocietyT¼250/C14C. The Curie temperature of the film is Tc¼ 119/C61K. For the details on experimental methods, see Supplemental Material [ 19]. Micro-tracks 90/C22mlong, oriented along the ½1/C2210/C138, [110], and [100] axes with differ- ent widths (0.5, 1, 2, 4/C22m), and connected to a nucleation pad, were patterned by e-beam lithography. The saturation magnetization MsðTÞwas determined from magnetometry measurements (SQUID). The estimated effective Mn spinconcentration is 5% [ 19]. Current-induced DW motion was studied in an open cycle optical cryostat with a temperatureaccuracy of 0.2 K. DW motion is produced by current pulses of different amplitudes Jand of a single duration /C1t¼1/C22s. The magnetic state of the tracks was observed by differential polar magneto-optical Kerr microscopywith a 1/C22mresolution. As the DW displacements /C1x were found to be proportional to the pulse duration andto the number of pulses, the average DW velocity is definedasv¼/C1x=/C1t[7,19]. The Joule heating of the track due to current pulses was studied extensively [ 15] and carefully taken into account (see [ 19]). In order to compare DW dynamics for a fixed track temperature T, for each Jvalue, the sample holder was set to an initial temperature T i¼ T/C0/C1TðJ;1/C22sÞ, where /C1TðJ;1/C22sÞis the temperature rise at the end of the pulse. For the lowest exploredtemperature T¼13 K , the initial temperature was set to T i¼4KforJ¼13 GA =m2. The current-driven DW velocity vis reported in Fig. 1as a function of the current density for several temperatures. DW motion is observed over a large temperature range(13 K <T< 110 K ). Three different regimes can be iden- tified [ 7,8]. At low values of the current density J(see inset of Fig. 1,T¼95 K ), DWs move in a creep regime domi- nated by pinning barriers and thermal activation. Theirvelocity is low ( v<0:1m=s) and varies exponentially withJ.F o r J>J dep/C255G A =m2(T¼95 K ),vbecomes larger than /C241m=s. DWs move in a depinning regime controlled by pinning and dissipation. For J>J fl/C25 11 GA =m2(T¼95 K ), DW motion enters a linear flow regime, only limited by dissipation. This linear regime,whose nature is discussed later, is observed for each tem-perature. The linear extrapolation to zero current yields v¼0m=s, within the experimental errors (see [ 19]). In this flow regime, the current DW mobility, defined as/C22 J¼v=J, decreases as the temperature is lowered from 105 to 42.5 K. At lower temperature, /C22Jbecomes weakly temperature dependent, as evidenced by two additionalvelocity values measured at 28 and 13 K, which fall closeto the curve obtained at 42.5 K. A characteristic slopechange indicates the transition from the depinning regime to the linear one, which gives a determination of the linear regime lower bound: J flðTÞ¼5:5, 11, 13 GA =m2for T¼105, 95, and 42.5 K, respectively. To get a better insight into the temperature variations of the DW mobility /C22J¼v=J, DW dynamics was studied as a function of the temperature for three different currentdensities. Results are reported in Fig. 2. For the lowest density J¼7:0/C60:5G A =m 2(circles), /C22Jdecreases strongly as Tis reduced. The DW dynamics crosses the boundary between the flow and the depinning regimes andeventually the DWs become pinned ( /C22 J¼0) at finite temperature ( T/C2570 K ). For the intermediate density J¼ 11:5/C61:8G A =m2(triangles), a pronounced temperature variation is also observed for T>80 K . Below 80 K, /C22Jis almost independent of temperature ( /C250:5m m3=C). As J<J flðTÞ, the flow regime threshold, this could be the FIG. 1 (color online). Current-driven DW velocity vmeasured at different temperatures T. Each point and its error bar corre- spond to the average and to the standard deviation calculatedwith more than 20 measurements, respectively. At T¼42:5K, larger error bars also reflect a slight asymmetry of DW displace- ments found as the current is reversed. Inset: Semilogarithmicplot of vmeasured at 95 K for the lowest current densities.FIG. 2 (color online). Temperature variations of the current mobility ( /C22J¼v=J). The empty symbols correspond to three different current values: 13:3/C62:0G A =m2(squares), 11:5/C6 1:8G A =m2(triangles), and 7:0/C60:5G A =m2(circles). The filled symbols correspond to the boundary between the pinning controlled and the flow linear regimes, deduced form the slope changes observed in Fig. 1. The flow linear regime is material- ized by a shaded area.PRL 108, 076604 (2012) PHYSICAL REVIEW LETTERSweek ending 17 FEBRUARY 2012 076604-2signature of a DW motion controlled by a distribution of energy barriers [ 20]. For the highest current density J¼ 13:3/C62:0G A =m2(squares) the curve goes through the /C22Jvalues already reported in Fig. 1for the linear regime (J/C21Jfl). The current-induced linear flow regime is thus evidenced over the whole temperature range ( 13 K < T<110 K ). We now focus on the origin of the temperature variations of/C22Jfor the flow regime. As the DW velocity vis proportional to the current density J, we write v¼ rflðTÞu, where rflðTÞis to be determined. The spin drift velocity is u¼JPcðTÞg/C22B 2eMsðTÞ[1,2], where g,/C22B,e, and PcðTÞ are the Lande ´factor, the Bohr magneton, the electron charge, and the current spin polarization, respectively.Close to 0 K, ucan be deduced from M sðT/C240KÞ (Fig. 3) and PcðT/C240KÞ.Pcwas estimated close to 4 K from point-contact Andreev reflection spectroscopy for(Ga,Mn)As samples with similar Mn concentrations: P c/C25 0:75,6 %M n[ 10];Pc>0:85,5 %M n[ 11], and Pc¼0:57, 7% Mn [ 12]. Taking the values from Refs. [ 10,12] we get u¼8:5–11:2m=sforJ¼13:3G A =m2. We now compare these values to the DW velocity vwith the same current density, at 13 K. v¼10:5/C60:7m=s, a value very close to the spin drift velocity uat 4 K, i. e., rfl/C241forT/C240K.I n order to determine to what extent this result is valid forother temperatures, the current spin polarization P DWcðTÞ deduced from DW dynamics is plotted in Fig. 3, assuming rflðTÞ¼1. Values of PDWcðTÞare calculated using the measured magnetization MsðTÞ(see Fig. 3) and current mobility /C22JðTÞ, forJ/C21Jfl(see Fig. 2). Results obtained close to Tcwith an annealed ðGa0:93;Mn0:07ÞAs 4/C22mwide track, with similar Mn concentration [ 21] are also reported in Fig. 3. The temperature variations for both (Ga,Mn)As and (Ga,Mn)(As,P) tracks shows similar trends close to Tc.The curve extrapolates to PDWc¼0:67/C60:03, for T!0K. As shown in Fig. 3, this value is found in between the estimations of Pcgiven in Refs. [ 10,12]. Taking those estimations as boundaries for PDWc, it follows that0:85<r¼v=u < 1:12, close to 0 K. This is a key result of this Letter. It shows that a rather accurate estima-tion of the current spin polarization can be deduced fromcurrent-induced DW dynamics. Moreover, it demonstrates,without adjustable parameter, that the domain-wall veloc-ityvis quantitatively close to the spin drift velocity u, for T/C250K. The generalization of this result far from 0 K is not straightforward due to the lack of estimations of P cvalues. However, the following shows that u/C25vis compatible with experimental results, close to the Curie temperature. As observed in Fig. 3,PDWc!0fort¼T=T c!1. Indeed the spin polarization tends to zero with the collapse of holemediated ferromagnetism. Moreover, P DWcðTÞfollows the temperature variation of MsðTÞ. This observation is con- sistent with the predictions of Dietl et al. [8,22], close to Tc, where the thermodynamic spin polarization PðTÞ¼ 6kBTC ðSþ1ÞpJpdMsðTÞ MsðT¼0Þ.kB,Jpd,S, and pare the Boltzmann constant, the exchange integral ( Jpd¼/C054 meV nm3), the Mn spin ( S¼5=2), and the carrier density, respec- tively. If we assume PDWcðTÞ¼1:0–1:8PðTÞ, (the upper boundary is proposed in Ref. [ 12], close to 0 K), the carrier density pcan be deduced from PDWcðTÞandMsðTÞ(Fig. 3) and the predictions for PðTÞ. The obtained values are p/C250:18–0:32 nm/C03andp/C250:23–0:42 nm/C03for the (Ga,Mn)(As,P) and (Ga,Mn)As tracks, respectively. The carrier densities for both materials are rather close andcompatible with the somewhat larger resistivity measuredfor (Ga,Mn)(As,P) (a factor two) [ 14,23]. The same orders of magnitude were deduced from other experimental meth-ods in samples exhibiting similar magnetic properties, asreported in Ref. [ 21]. Therefore, the measured temperature variations of /C22 Jare also compatible with v/C25u, close to Tc. An alternative analysis based on the Do ¨ring inequality leads to the same conclusion, as reported in Refs. [ 7,21]. The fact that v/C25uclose to 0 K and to Tc, strongly suggests that for the linear flow regime, the temperature variation of the current mobility /C22Jis essentially deter- mined by the ratio PcðTÞ=MsðTÞbetween the current spin polarization and the magnetization. In this respect, thecurve P DWcðTÞ(obtained for v¼u) reported in Fig. 3 does reflect the temperature variation of the current spinpolarization. We now discuss the nature of the linear flow regime and the nonadiabaticity of the spin transfer torque. In Fig. 4, the reduced DW velocity v=v wpredicted by the 1D model [1,2] is plotted as a function of the reduced spin drift velocity u=v w, where vwis the so-called Walker velocity, for different values of /C12=/C11. Two linear flow regimes are predicted to occur (see the curves obtained for /C12=/C11¼8 and1=8). For the lowest uvalues, DWs move in the steadyFIG. 3 (color online). Temperature variation of the spin polar- ization PDWcdeduced from domain-wall dynamics (left scale: same legend as for Fig. 2) and of the magnetization Ms(right scale: /C12). Values of the spin polarization Pc(left scale: down triangles) deduced from point-contact Andreev reflection mea- surements [ 10–12]. The crossed symbols ( aa) correspond to results obtained for PDWc, withðGa0:93;Mn0:07ÞAs 4 /C22mwide tracks.PRL 108, 076604 (2012) PHYSICAL REVIEW LETTERSweek ending 17 FEBRUARY 2012 076604-3state flow regime whenever the magnetization direction within the DW remains constant with time. In this regime,the velocity v¼ð/C12=/C11Þu. Above the so-called Walker limit, the motion becomes nonlinear with current: DWsare in the precessional regime (the direction of the DWmagnetization precesses during the DW motion). Forhigher uvalues, DWs follow the asymptotic precessional regime for which v¼u. Let us note that no steady state regime is predicted for /C12=/C11¼0, while no preces- sional regime should occur for /C12=/C11¼1. In that specific case, the DW motion would remain in the steady stateregime for any arbitrary large current. Figure 4also reports experimental results deduced from Fig. 1. The Walker velocities are obtained from v w¼ /C220/C13M sðTÞ/C1=2, where the wall thickness parameter /C1¼ 6:5/C61:0n m is taken from Ref. [ 24].uis estimated from the values of PDWcðTÞand of MsðTÞreported in Fig. 3.I ti s worth noting that the u=v wvalues span over more than 1 order of magnitude ( 0:3< u=v w<6) because of the large MsðTÞchange over the investigated temperature range. In the linear flow regime, the reduced domain-wall velocities(filled symbols) gather onto a single linear master curvesince we assumed that v¼u. The points departing from thev¼uline (empty symbols) correspond to DW motion occurring in the depinning regime, as discussed previously.The coincidence between the points measured T¼95and 105 K in the depinning regime (empty symbols) and theflow regime predictions for /C12=/C11¼0and1=8is therefore accidental. As seen in Fig. 4, DW motion occurs in the flow regime well below the intrinsic DW motion threshold u=v w¼1 expected for a purely adiabatic spin transfer torque. This isa clear evidence of a nonadiabatic contribution (i.e., /C12/C2220) to the spin transfer torque, in contradiction with the con-clusions of Ref. [ 8]. Two different ranges of /C12=/C11 values would reproduce our experimental data. A good agreementis obtained with the predicted steady regime, for /C12=/C11¼1. The experimental results seem to be also compatible with the asymptotic precessional regime for /C12=/C11¼8, as pre- dicted in Ref. [ 4]. A higher /C12=/C11 ratio, as proposed in Ref. [ 5], would shift the Walker peak towards lower values ofu=v wand improve the quantitative agreement. In order to discriminate between the steady state and the preces- sional regimes, experiments combining current and mag- netic field-induced motion of magnetic domains were performed, as proposed in Ref. [ 7]. Weak magnetic fields are applied ( /C05<H< 5O e ) during the current pulse. The magnetic field DW mobility /C22H¼dv=/C22 0dHis then extracted and compared to mobilities measured in experi- ments where both flow regimes have been clearly identified [9]. Close to T¼0K, the measured magnetic field mo- bility /C22H¼2:3/C60:4m=sm T . This value is close to the DW mobility in the steady state regime ( /C22H¼ 1:6/C60:5m=sm T ) deduced from field-induced DW mo- tion in (Ga,Mn)As films [ 9] with similar Mn concentration. It is far larger than the mobility ( /C22H¼0:11/C6 0:02 m=sm T ) measured in the asymptotic precessional regime. Therefore this experiment clearly supports the hypothesis of DW motion in the steady state regime (i.e., /C12/C2220) and, in the frame of the 1-D model, a ratio /C12=/C11 close to 1. Our investigations on current-induced domain-wall mo- tion have evidenced a linear domain-wall flow regime overa wide range of temperatures ( 0:1T c<T<T c). Domain walls were shown to move in the steady state regime with velocities corresponding to the carrier spin drift velocities. Hence, we inferred that the /C12term, characterizing the nonadiabatic spin transfer torque, is close to the Gilbertdamping coefficient. Moreover, our results suggest that DW dynamics give direct access to the temperature varia- tion of the current spin polarization. This parameter is crucial for understanding the spin transfer phenomena. However its estimation is not straightforward experimen-tally [ 25,26]. The authors thank A. Thiaville, J. Ferre ´, and J. Miltat for useful discussions. This work was partly supported by the French projects RTRA Triangle de la physique Grants No. 2009-024T-SeMicMac and No. 2010-033T- SeMicMagII and performed in the framework of theMANGAS project (No. 2010-BLANC-0424). *Now at Consejo Nacional de Investigaciones Cientı ´ficas y Te´cnicas, Centro Ato ´mico Bariloche Comisio ´n Nacional de Energı ´a Ato´mica, Avenida Bustillo 9500, 8400 S. C. de Bariloche, Rı ´o Negro, Argentina †vincent.jeudy@u-psud.fr [1] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004) . [2] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 (2005) .FIG. 4 (color online). Comparison between the experimental results and predictions for the domain-wall flow regimes. Filled (empty) symbols correspond to the flow (depinning) regime. Inset. Zoom for low values of the reduced spin drift velocityu=v w.PRL 108, 076604 (2012) PHYSICAL REVIEW LETTERSweek ending 17 FEBRUARY 2012 076604-4[3] G. Tatara, H. Kohno, and J. Shibata, Phys. 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PhysRevB.96.064423.pdf

PHYSICAL REVIEW B 96, 064423 (2017) Consistent microscopic analysis of spin pumping effects Gen Tatara RIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Shigemi Mizukami WPI - Advanced Insitute for Materials Research, Tohoku University Katahira 2-1-1, Sendai, Japan (Received 2 June 2017; revised manuscript received 28 July 2017; published 18 August 2017) We present a consistent microscopic study of spin pumping effects for both metallic and insulating ferromagnets. As for the metallic case, we present a simple quantum mechanical picture of the effect as due tothe electron spin flip as a result of a nonadiabatic (off-diagonal) spin gauge field. The effect of interface spin-orbitinteraction is briefly discussed. We also carry out a field-theoretic calculation to discuss on equal footing the spincurrent generation and torque effects such as an enhanced Gilbert damping constant and a shift of precessionfrequency both in metallic and insulating cases. For thick ferromagnetic metals, our study reproduces the results ofprevious theories such as the correspondence between the dc component of the spin current and the enhancementof the damping. For thin metals and insulators, the relation turns out to be modified. For the insulating case,driven locally by interface sdexchange interaction due to magnetic proximity effect, the physical mechanism is distinct from the metallic case. Further study of the proximity effect and interface spin-orbit interaction wouldbe crucial to interpret experimental results in particular for insulators. DOI: 10.1103/PhysRevB.96.064423 I. INTRODUCTION Spin current generation is of a fundamental importance in spintronics. A dynamic method using magnetization preces-sion induced by an applied magnetic field, called the spin pumping effect, turns out to be particularly useful [ 1] and is widely used in a junction of a ferromagnet (F) and a normalmetal (N) (Fig. 1). The generated spin current density (in unit of A/m 2) has two independent components, proportional to ˙n andn×˙n, where nis a unit vector describing the direction of localized spin, and thus is represented phenomenologically as js=e 4π(Arn×˙n+Ai˙n), (1) where eis the elementally electric charge and ArandAiare phenomenological constants having unit of 1 /m2.T h es p i n pumping effect was theoretically formulated by Tserkovnyaket al. [2] by use of the scattering matrix approach [ 3]. This approach, widely applied in mesoscopic physics, describestransport phenomena in terms of transmission and reflectionamplitudes (scattering matrix), and provides a quantum me-chanical picture of the phenomena without calculating explic-itly the amplitudes. Tserkovnyak et al. applied the scattering matrix formulation of general adiabatic pumping [ 4,5]t ot h e spin-polarized case. The spin pumping effect was described inRef. [ 2] in terms of spin-dependent transmission and reflection coefficients at the FN interface, and it was demonstrated thatthe two parameters, A randAi, are the real and the imaginary parts of a complex parameter called the spin mixing conduc-tance. The spin mixing conductance, which is represented bytransmission and reflection coefficients, turned out to be aconvenient parameter for discussing spin current generationand other effects like the inverse spin-Hall effect. Nevertheless,the scattering approach hides the microscopic physical pictureof what is going on, as the scattering coefficients are notfundamental material parameters but are composite quantitiesof the Fermi wave vector, the electron effective mass, and the interface properties. The effects of a slowly varying potential are described in a physically straightforward and clear manner by the use ofa unitary transformation that represents the time dependence(see Sec. II Afor details). The laboratory frame wave function under a time-dependent potential |ψ(t)/angbracketrightis written in terms of a static ground state (“rotated-frame” wave function) |φ/angbracketrightand a unitary matrix U(t)a s|ψ(t)/angbracketright=U(t)|φ/angbracketright. The time derivative ∂ tis then replaced by a covariant derivative ∂t+(U−1∂tU), and the effects of time dependence are represented by (the timecomponent of) an effective gauge field, A≡−i(U −1∂tU)[ s e e Eq. ( 12)]. In the same manner as the electromagnetic gauge field, the effective gauge field generates a current if spatialinhomogeneity is present (like in junctions), and this is thephysical origin of the pumping effect in metals. It should be noted that the effective gauge field that drives spin current is a nonadiabatic one, off-diagonal in spin, andnot the adiabatic gauge field that induces spin Berry’s phase,the spin motive force, and spin transfer effects. Nevertheless,the pumping efficiency can be calculated within an adiabaticpumping scheme, as shall be discussed in Sec. II C. In the perturbative regime or in insulators, a simple picture instead of an effective gauge field can be presented. Letus focus on the case driven by an sdexchange interaction, J sdn(t)·σ, where Jsdis a coupling constant and σis the electron spin. Considering the second-order effect of thesdexchange interaction, the electron wave function has a contribution of a time-dependent amplitude U(t 1,t2)=(Jsd)2(n(t1)·σ)(n(t2)·σ) =(Jsd)2{(n(t1)·n(t2))+i[n(t1)×n(t2)]·σ},(2) where t1andt2are the times of the interactions. The first term on the right-hand side, representing the amplitude forcharge degrees of freedom, is neglected. The spin contributionvanishes for a static spin configuration, as is natural, while for 2469-9950/2017/96(6)/064423(23) 064423-1 ©2017 American Physical SocietyGEN TATARA AND SHIGEMI MIZUKAMI PHYSICAL REVIEW B 96, 064423 (2017) FNn(t)js FIG. 1. Spin pumping effect in a junction of ferromagnet (F) and normal metal (N). Dynamic magnetization n(t) generates a spin current jsthrough the interface. the slowly varying case, it reads U(t1,t2)/similarequal−i(t1−t2)(Jsd)2(n×˙n)(t1)·σ. (3) As a result of this amplitude, spin accumulation and spin current is induced proportional to n×˙n. This fact indicates thatn×˙nplays a role of an effective scalar potential or voltage in electromagnetism, as we shall demonstrate in Sec. VII B for insulators. [The factor of time difference is written in terms ofa derivative with respect to energy or angular frequency in arigorous derivation. See, for example, Eqs. ( E6) and ( E9).] The essence of the spin pumping effect is therefore thenoncommutativity of spin operators. The above picture in theperturbative regime naturally leads to the effective gauge fieldpicture in the strong-coupling limit [ 6]. The same scenario applies for cases of spatial variation of spin, and an equilibrium spin current proportional to n×∇ in emerges, where idenotes the direction of spatial variation [ 7]. The spin pumping effect is therefore the time analog of theequilibrium spin current induced by vector spin chirality.Moreover, a charge current emerges from the third-orderprocess from the identity [ 6] tr[(n 1·σ)(n2·σ)(n3·σ)]=2in1·(n2×n3), (4) and this factor, a scalar spin chirality, is the analog of the spin Berry phase in the perturbative regime. The spin pumpingeffect, spin Berry’s phase, and the spin motive force have thesame physical root, namely, the noncommutative spin algebra. From the scattering matrix theory view point, the cases of metallic and insulating ferromagnet make no differencesince what the conduction electrons in the normal metal seeis the interface. From the physical viewpoint, such treatmentappears too crude. Unlike the metallic case discussed above, inthe case of an insulator ferromagnet, the coupling between themagnetization and the conduction electron in a normal metaloccurs due to a magnetic proximity effect at the interface,as is experimentally indicated [ 8]. Thus the spin pumping by an insulator ferromagnet is a locally induced perturbativeeffect rather than a transport induced by a driving force dueto a generalized gauge field. We therefore need to applydifferent approaches for the two cases. In the insulatingcase, one may think that a magnon spin current is generatedinside the ferromagnet because the magnons couple to aneffective gauge field [ 9] similarly to the electrons in metallic case. This is not, however, true, because the gauge field formagnons is Abelian [U(1)], and has no off-diagonal “spin-flip” component. Although the scattering matrix approachapparently seems to apply to both metallic and insulating cases,it would be instructive to present in this paper a consistent microscopic description of the effects to see the differentphysics governing the two cases. A. Brief overview of theories and scope of the paper Before carrying out calculations, let us overview the history of theoretical studies of the spin pumping effect. Spin currentgeneration in a metallic junction was originally discussed bySilsbee et al. [10] before Tserkovnyak et al. It was shown there that dynamic magnetization induces spin accumulation at theinterface, resulting in a diffusive flow of spins in the normalmetal. Although at that time the experimental interest wasfocused on the interface spin accumulation, which enhancesthe signal of conduction electron spin resonance, it would befair to say that Silsbee et al. pointed out the “spin pumping effect”. In Ref. [ 2], the spin pumping effect was originally argued in the context of enhancement of Gilbert damping in anFN junction, which had been a hot issue after the study byBerger [ 11], who studied the case of FNF junctions based on a quantum mechanical argument. Berger discussed that whena normal metal is attached to a ferromagnet, the damping ofthe ferromagnet is enhanced as a result of spin polarizationformed in the normal metal, and the effect was experimentallyconfirmed by Mizukami et al. [12]. Tserkovnyak et al. pointed out that the effect can be interpreted as the counteractionof spin current generation, because the spin current injectedinto the normal metal indicates emergence of a torque forthe ferromagnet. In fact, the equation of motion for themagnetization of ferromagnet reads ˙n=−γB×n−αn×˙n−a 3 eSdjs, (5) where γis the gyromagnetic ratio, αis the Gilbert damping coefficient, dis the thickness of the ferromagnet, Sis the magnitude of localized spin, and ais the lattice constant. The spin current of Eq. ( 1) thus indicates that the gyromagnetic ratio and the Gilbert damping coefficient are modified by thespin pumping effect to be [ 2] ˜α=α+a 3 4πSdAr, ˜γ=γ/parenleftbigg 1+a3 4πSdAi/parenrightbigg−1 . (6) The spin pumping effect is therefore detected by measuring the effective damping constant and gyromagnetic ratio. For-mula ( 6) is, however, based on a naive picture neglecting the position dependence of the damping torque and the relationbetween the pumped spin current amplitude and damping, orγwould not be so simple in reality (see Sec. V). The issue of damping in an FN junction was formu- lated based on linear-response theory by Simanek andHeinirch [ 13,14]. They showed that the damping coefficient is given by the first-order derivative with respect to the angularfrequency ωof the imaginary part of the spin correlation function and argued that the damping effect is consistent withTserkovnyak’s spin pumping effect. Recently, a microscopicformulation of spin pumping effect in metallic junctions was 064423-2CONSISTENT MICROSCOPIC ANALYSIS OF SPIN . . . PHYSICAL REVIEW B 96, 064423 (2017) provided by Chen and Zhang [ 15] and one of the authors of Ref. [ 16] by use of the Green’s functions, and a transparent microscopic picture of pumping effect was provided. Thescattering representation and the Green’s function one arerelated [ 15] because the asymptotic behaviors of the Green’s functions at long distance are governed by the transmissioncoefficient [ 17]. In the study of Ref. [ 16], the uniform ferromagnet was treated as a dot having only two degrees offreedom of spin. Such simplification neglects the dependenceon electron wave vectors in ferromagnets and thus cannotdiscuss the case of inhomogeneous magnetization or positiondependence of spin damping. The aim of this paper is to provide a microscopic and consistent theoretical formulation of spin pumping effect formetallic and insulating ferromagnets. We do not rely onthe scattering approach. Instead, we provide an elementaryquantum mechanical argument to demonstrate that spin currentgeneration is a natural consequence of magnetization dynam-ics (Sec. II). Based on the formulation, the effect of interface spin-orbit interaction is discussed in Sec. III. We also provide a rigorous formulation based on the field-theoretic approachemployed in Ref. [ 16] in Sec. IV. We also reproduce within the same framework Berger’s result [ 11] that the spin pumping effect is equivalent to the enhancement of the spin damping(Sec. V). The effect of inhomogeneous magnetization is briefly discussed in Sec. VI. The case of insulating ferromagnet is studied in Sec. VII assuming that the pumping is induced by an interface ex-change interaction between the magnetization and conductionelectrons in a normal metal, namely, by the magnetic proximityeffect [ 8]. The interaction is treated perturbatively similarly to Refs. [ 18,19]. The dominant contribution to the spin current, the one linear in the interface exchange interaction, turns outto be proportional to ˙n, while the one proportional to n×˙nis weaker if the proximity effect is weak. The contribution from the magnons, magnetization fluc- tuations, is also studied. As has been argued [ 9], a gauge field for magnons emerges from magnetization dynamics. Itis, however, an adiabatic one, diagonal in spin, which acts as achemical potential for magnons, giving rise only to adiabaticspin polarization proportional to n. This is in sharp contrast to the metallic case, where electrons are directly driven bythe spin-flip component of the spin gauge field, resulting inperpendicular spin accumulation, i.e., along ˙nand n×˙n. The excitation in a ferromagnet when the magnetization istime-dependent is therefore different for the metallic andthe insulating cases. We show that a magnon excitationnevertheless generates perpendicular spin current, n×˙n,i n the normal metal as a result of annihilation and creation at theinterface, which in turn flips the electron spin. The result of themagnon-driven contribution agrees with the one in the previousstudy [ 20] carried out in the context of thermally driven spin pumping (“spin Seebeck” effect). It is demonstratedthat the magnon-induced spin current depends linearly on thetemperature at high temperature compared to magnon energy.The amplitude of magnon-driven spin current provides themagnitude of the magnetic proximity effect. In our analysis, we calculate consistently the pumped spin current and change of the Gilbert damping and resonantfrequency and obtain the relations among them. It is shownthat the spin mixing conductance scenario saying that the magnitude of spin current proportional to n×˙nis given by the enhancement factor of the Gilbert damping constant [ 2], applies only in the case of thick ferromagnetic metals. For thethin metallic and insulator cases, different relations hold (seeSec. VIII). II. QUANTUM MECHANICAL DESCRIPTION OF METALLIC CASE In this section, we derive the spin current generated by the magnetization dynamics of a metallic ferromagnet by aquantum mechanical argument. It is sometimes useful forintuitive understanding, although the description may lackclearness as it cannot handle many-particle aspects like particledistributions. In Sec. IV, we formulate the problem in the field-theoretic language. A. Electrons in ferromagnet with dynamic magnetization The model we consider is a junction of a metallic fer- romagnet (F) and a normal metal (N). The magnetization(or localized spins) in the ferromagnet is treated as spatiallyuniform but changing with time slowly. As a result of strongsdexchange interaction, the conduction electron’s spin follows instantaneous directions of localized spins, i.e., the system isin the adiabatic limit. The quantum mechanical Hamiltonianfor the ferromagnet is H F=−∇2 2m−/epsilon1F−Mn(t)·σ, (7) where mis the electron mass, σis a vector of Pauli matrices, Mrepresents the energy splitting due to the sdexchange interaction, and n(t) is a time-dependent unit vector denoting the localized spin direction. The energy is measured from theFermi energy /epsilon1 F. As a result of the sdexchange interaction, the electron’s spin wave function is given by [ 21] |n/angbracketright≡cosθ 2|↑/angbracketright + sinθ 2eiφ|↓/angbracketright, (8) where |↑/angbracketrightand|↓/angbracketrightrepresent the spin-up and -down states, respectively, and ( θ,φ) are polar coordinates for n. To treat slowly varying localized spins, we switch to a rotated framewhere the spin direction is defined with respect to an instan-taneous direction n[7]. This corresponds to diagonalizing the Hamiltonian at each time by introducing a unitary matrix U(t) as |n(t)/angbracketright≡U(t)|↑/angbracketright, (9) where U(t)=/parenleftBigg cos θ 2sinθ 2e−iφ sinθ 2eiφ−cosθ 2/parenrightBigg , (10) where states are in vector representation, i.e., |↑/angbracketright = (1 0) and |↓/angbracketright = (0 1). In the rotated frame, the Hamiltonian is diagonalized as (in the momentum representation) /tildewideHF≡U−1HFU=/epsilon1k−Mσz, (11) 064423-3GEN TATARA AND SHIGEMI MIZUKAMI PHYSICAL REVIEW B 96, 064423 (2017) HF HFU A±s,t FIG. 2. Unitary transformation Ufor conduction electrons in a ferromagnet converts the original Hamiltonian HFinto a diagonalized uniformly spin-polarized Hamiltonian /tildewideHFand an interaction with a spin gauge field, As,t·σ. where /epsilon1k≡k2 2m−/epsilon1Fis the kinetic energy in the momentum representation (Fig. 2). In general, when a state |ψ/angbracketrightfor a time-dependent Hamiltonian H(t), satisfying the Schrödinger equation i∂ ∂t|ψ/angbracketright=H(t)|ψ/angbracketright, is written in terms of a state |ψ/angbracketright connected by a unitary transformation |φ/angbracketright≡U−1|ψ/angbracketright, the new state satisfies a modified Schrödinger equation: /parenleftbigg i∂ ∂t+iU−1∂ ∂tU/parenrightbigg |φ/angbracketright=˜H|φ/angbracketright, (12) where ˜H≡U−1HU. Namely, there arises a gauge field −iU−1∂ ∂tUin the new frame |φ/angbracketright. In the present case of dynamic localized spin, the gauge field has three components(suffix tdenotes the time component): A s,t≡−iU−1∂ ∂tU≡As,t·σ, (13) explicitly given as [ 7] As,t=1 2⎛ ⎜⎝−∂tθsinφ−sinθcosφ∂tφ ∂tθcosφ−sinθsinφ∂tφ (1−cosθ)∂tφ⎞ ⎟⎠. (14) Including the gauge field in the Hamiltonian, the effective Hamiltonian in the rotated frame reads /tildewideHeff F≡/tildewideHF+As,t·σ=/parenleftBigg /epsilon1k−M−Az s,t A− s,t A+ s,t /epsilon1k+M+Az s,t/parenrightBigg , (15) where A± s,t≡Ax s,t±iAy s,t. We see that the adiabatic ( z) component of the gauge field, Az s,t, acts as a spin-dependent chemical potential (spin chemical potential) generated bydynamic magnetization, while the nonadiabatic ( xandy) components cause spin mixing. In the case of the uniformmagnetization we consider, the mixing is between the electronswith different spin ↑and↓but with the same wave vector k, because the gauge field A ± s,tcarries no momentum. This leads to a mixing of states having an excitation energy of Mas shown in Fig. 3. In low-energy transport effects, what matters are the electrons at the Fermi energy; the wave vector kshould be chosen as kF+andkF−, the Fermi wave vectors for ↑and ↓electrons, respectively. (Effects of finite momentum transfer are discussed in Sec. VI.) Hamiltonian ( 15) is diagonalized to obtain energy eigenval- ues of ˜ /epsilon1kσ=/epsilon1k−σ√ (M+Az s,t)2+|A⊥ s,t|2, where |A⊥ s,t|2≡ A+ s,tA− s,tandσ=± represents spin ( ↑and↓correspond to + and−, respectively). We are interested in the adiabatic limit, FIG. 3. For uniform magnetization, the nonadiabatic components of the gauge field A± s,tinduce a spin flip conserving the momentum. and so the lowest order contribution, namely, the first order, in the perpendicular component A⊥ s,t, is sufficient. In the present rotated-frame approach, the gauge field is treated as a staticpotential, since it already includes the time derivative to thelinear order [see Eq. ( 14)]. Moreover, the adiabatic component of the gauge field, A z s,t, is neglected, as it modifies the spin pumping only at the second order of the time derivative. Theenergy eigenvalues /epsilon1 kσ/similarequal/epsilon1k−σMare thus unaffected by the gauge field, while the eigenstates to the linear order read |k↑/angbracketrightF≡|k↑/angbracketright−A+ s,t M|k↓/angbracketright, (16) |k↓/angbracketrightF≡|k↓/angbracketright+A− s,t M|k↑/angbracketright, corresponding to the energy of /epsilon1k+and/epsilon1k−, respectively. For low-energy transport, the states that we need to consider arethe following two, having spin-dependent Fermi wave vectorsk Fσforσ=↑,↓, namely, /vextendsingle/vextendsinglekF↑↑/angbracketrightbig F=/vextendsingle/vextendsinglekF↑↑/angbracketrightbig −A+ s,t M/vextendsingle/vextendsinglekF↑↓/angbracketrightbig , /vextendsingle/vextendsinglekF↓↓/angbracketrightbig F=/vextendsingle/vextendsinglekF↓↓/angbracketrightbig +A− s,t M/vextendsingle/vextendsinglekF↓↑/angbracketrightbig . (17) B. Spin current induced in the normal metal The spin pumping effect is now studied by taking account of the interface hopping effects on the states of Eq. ( 17)t ot h e linear order. The interface hopping amplitude of electrons inF to N with spin σis denoted by ˜t σand the amplitude from N to F is ˜t∗ σ. We assume that the spin dependence of the electron state in F is governed by the relative angle to the magnetizationvector, and hence the spin σis the one in the rotated frame. Assuming, moreover, that there is no spin-flip scattering atthe interface, the amplitude ˜t σis diagonal in spin. (Interface spin-orbit interaction is considered in Sec. III.) The spin-wave function formed in the N region at the interface as a result ofthe state in F [Eq. ( 17)] is then |k F↑/angbracketrightN≡˜t|kF↑/angbracketright=˜t↑|kF↑/angbracketright−˜t↓A+ s,t M|kF↓/angbracketright |kF↓/angbracketrightN≡˜t|kF↓/angbracketright=˜t↓|kF↓/angbracketright+˜t↑A− s,t M|kF↑/angbracketright, (18) 064423-4CONSISTENT MICROSCOPIC ANALYSIS OF SPIN . . . PHYSICAL REVIEW B 96, 064423 (2017) where kFis the Fermi wave vector of an N electron. The spin density induced in N region at the interface is therefore /tildewides(N)=1 2(N/angbracketleftkF↑|σ|kF↑/angbracketrightNν↑+N/angbracketleftkF↓|σ|kF↓/angbracketrightNν↓),(19) where νσis the spin-dependent density of states of F electrons at the Fermi energy. It reads /tildewides(N)=1 2/summationdisplay σνσTσσˆz−ν↑−ν↓ M(Re[T↑↓]A⊥ s,t +Im[T↑↓](ˆz×A⊥ s,t)), (20) whereA⊥ s,t=(Ax s,t,Ay s,t,0)=As,t−ˆzAz s,tis the transverse (nonadiabatic) components of spin gauge field and Tσσ/prime≡˜t∗ σ˜tσ/prime. (21) The spin density of Eq. ( 20) is in the rotated frame. The spin polarization in the laboratory frame is obtained by a rotationmatrix R ij, defined by U−1σiU≡Rijσj, (22) as s(N) i=Rij/tildewides(N) j. (23) Explicitly, Rij=2mimj−δij, where m≡ (sinθ 2cosφ,sinθ 2sinφ,cosθ 2)[7]. Using Rij(A⊥ s,t)j=−1 2(n×˙n)i, (24) Rij(ˆz×A⊥ s,t)j=−1 2˙ni, andRiz=ni, the induced interface spin density is finally obtained as s(N)=ζs 0n+Re[ζs](n×˙n)+Im[ζs]˙n, (25) where ζs 0≡1 2/summationdisplay σνσTσσ,ζs≡ν↑−ν↓ 2MT↑↓. (26) Since the N electrons contributing to induced spin density are those at the Fermi energy, the spin current is simplyproportional to the induced spin density as j sN=kF ms(N), resulting in j(N) s=kF mζs 0n+kF mRe[ζs](n×˙n)+kF mIm[ζs]˙n.(27) This is the result of spin current at the interface. The pumping efficiency is determined by the product of hoppingamplitudes t ↑andt∗ ↓. The spin mixing conductance defined in Ref. [ 2] corresponds to T↑↓. In the scattering approach [ 2] based on adiabatic pumping theory [ 3–5], the expression for the spin mixing conductance in terms of scatteringmatrix element is exact as for the adiabatic contribution. Ourresult ( 27), in contrast, is a perturbative one valid to the second order in the hopping amplitude. To take full account of thehopping in the self-energy is possible numerically in a fieldtheoretical approach. In bulk systems without spin-orbit interaction and magnetic field, the hopping amplitudes t σare chosen as real, while at interfaces, this is not the case because inversion symmetryis broken. Nevertheless, in metallic junctions such as Cu/Co,Cr/Fe, and Au/Fe, first-principles calculations indicate thattn(t)(a)2M tn(t)(b) FIG. 4. Schematic figures of electron energy /epsilon1under precessing localized spin n(t) in the adiabatic limit (a) and with nonadiabaticity (b). Top figures represent energy levels with separation of 2 Min the rotated frame. In the perfectly adiabatic case (a), the electronstate keeps the minimum energy state as n(t) changes. Spin pumping does not occur in this limit. Case (b) is with nonadiabaticity taken into account, where temporal change of localized spin ˙ninduces a perpendicular spin polarization along n×˙n. This nonadiabatic effect is represented by the nonadiabatic gauge field A ± s,tand causes spin flip in the rotated frame, leading to a high-energy state (shown in red)and spin current generation. the imaginary part of spin mixing conductance (our ζs)i s smaller than the real part by 1–2 orders of magnitude [ 22,23]. A large spin current proportional to ˙nwould therefore suggest existence of strong interface spin-orbit interaction, as shall bediscussed in Sec. III. C. Adiabatic or nonadiabatic? In our approach, the spin pumping effect at the linear order in time derivative is mapped to a static problem of spinpolarization formed by a static spin-mixing potential in therotated frame as was mentioned in Ref. [ 16]. The rotated-frame approach employed here provides a clear physical picture, asit grasps the low-energy dynamics in a mathematically propermanner. In this approach, it is clearly seen that pumping ofspin current arises as a result of off-diagonal components ofthe spin gauge field that causes electron spin flip. If so, is spin pumping an adiabatic effect or nonadiabatic one? Conventional adiabatic processes are those where thesystem under a time-dependent external field remains to be thelowest energy state at each time [Fig. 4(a)]. In the spintronics context, an electron passing through a thick domain wall seemsto be in the adiabatic limit in this sense; the electron spinkeeps the lowest energy state by rotating it according to themagnetization profile at each spatial point [ 7] (see Table I). In contrast, as is seen from the above analysis, the spin pumpingeffect does not arise in the same adiabatic limit; it is inducedby the nonadiabatic (off-diagonal) spin gauge field A ± s,t, which changes electron spin state in the local rotated frame with acost of sdexchange energy [Fig. 4(b)]. For the spin pumping effect, therefore, nonadiabaticity is essential, as indicated alsoin a recent full counting statistics analysis [ 24]. In spite of this fact, the spin pumping effect appears to be treated within an adiabatic pumping theory [ 3–5]. In fact, a nonadiabatic gauge field serves just as a driving field for spincurrent, while the pumping efficiency is determined solely by 064423-5GEN TATARA AND SHIGEMI MIZUKAMI PHYSICAL REVIEW B 96, 064423 (2017) TABLE I. Comparison of electron transmission through a domain wall and spin pumping effect. In the figures, large arrows represent the localized spins, n, as a function of position xor time t, and the electron spin is denoted by a small arrow with a circle. A nonadiabatic spin polarization δsinduced by the nonadiabatic gauge field A± s,μis essential in both cases (represented by yellow arrows). the static (adiabatic) response of the system. This feature is the same as the linear response theory; the response function to anexternal field can be calculated within an equilibrium scheme,although the system is out of equilibrium as a result of theexternal field. Such separation of a driving field and a responsefunction is possible only by a microscopic formulation, andhas not been clearly identified in theories so far. A careful microscopic description indicates that a nonadi- abaticity is essential even in the spin transfer effect. In fact,an electron spin injected into a domain wall along xdirection gets polarized along n×∇ xnas a result of a nonadiabatic gauge field [ 7,25]. This nonequilibrium spin polarization is perpendicular to the wall plane, and thus induces a translationalmotion of the wall. This is the physical mechanism of spintransfer effect. At the same time, the spin transfer effect can bediscussed phenomenologically using the conservation law ofangular momentum [ 26]. One should not forget, however, that nonadiabaticity is implicitly assumed because spin rotation iscaused only by a perpendicular component. Physically, the spinpumping effect is essentially the same as electron transmissionthrough the domain wall if we replace a spatial coordinatexand the time, as summarized in Table I. In the case of a domain wall, including the nonadiabatic gauge field to thenext order leads to consideration of domain wall resistanceand nonadiabatic βtorque [ 27–29], while such a nonadiabatic regime has not been explored in the context of pumping. III. EFFECTS OF INTERFACE SPIN-ORBIT INTERACTION In this section, we discuss the effect of spin-orbit interaction at the interface, which modifies hopping amplitude ˜tσ.W e particularly focus on that linear in the wave vector, namely theinteraction represented in the continuum representation by aHamiltonian H so=a2δ(x)/summationdisplay ijγijkiσj, (28) where γijis a coefficient having the unit of energy representing the spin-orbit interaction, ais the lattice constant, and theinterface is chosen as at x=0. Assuming that spin-orbit interaction is weaker than the sdexchange interaction in F, we carry out a unitary transformation to diagonalize the sd interaction to obtain Hso=a2δ(x)/summationdisplay ij/tildewideγijkiσj, (29) where/tildewideγij≡/summationtext lγilRlj, with Rijbeing a rotation matrix defined by Eq. ( 22). This spin-orbit interaction modifies the diagonal hopping amplitude ˜tiin the direction iat the interface to become a complex as /tildewideti=˜t0 i−i/summationdisplay j/tildewideγijσj. (30) (In this section, we denote the total hopping amplitude including the interface spin-orbit interaction by /tildewidetand the one without by /tildewidet0.) We consider the hopping amplitude perpendicular to the interface, i.e., along the xdirection, and suppress the suffix irepresenting the direction. In the matrix representation for spin, the hopping amplitude is /tildewidet(≡/tildewidetx)=/parenleftBigg /tildewidet↑/tildewidet↑↓ /tildewidet↓↑/tildewidet↓/parenrightBigg , (31) where /tildewidet↑=˜t0 ↑−i/tildewideγxz,/tildewidet↓=˜t0 ↓+i/tildewideγxz, /tildewidet↑↓=i(/tildewideγxx+i/tildewideγxy),/tildewidet↓↑=i(/tildewideγxx−i/tildewideγxy). (32) Let us discuss how the spin pumping effect discussed in Sec. II Bis modified when the hopping amplitude is a matrix of Eq. ( 31). The spin pumping efficiency is written as in Eqs. ( 21) and ( 26). In the absence of spin-orbit interaction, the hopping amplitude ˜tis chosen as real, and thus the contribution proportional to n×˙nin Eq. ( 27) is dominant. The spin-orbit interaction enhances the other contribution proportional to ˙n because it gives rise to an imaginary part. Moreover, it leads tospin mixing at the interface, modifying the spin accumulationformed in the N region at the interface. 064423-6CONSISTENT MICROSCOPIC ANALYSIS OF SPIN . . . PHYSICAL REVIEW B 96, 064423 (2017) The electron states in the N region at the interface are now given instead of Eq. ( 18) by the following two states (choosing the basis as (|kF↑/angbracketright |kF↓/angbracketright)): |kF↑/angbracketrightN≡/tildewidet|kF↑↑/angbracketrightF=⎛ ⎝/tildewidet↑−/tildewidet↑↓A+ s,t M /tildewidet↓↑−/tildewidet↓A+ s,t M⎞ ⎠, |kF↓/angbracketrightN≡/tildewidet|kF↓↓/angbracketrightF=⎛ ⎝/tildewidet↑↓+/tildewidet↑A− s,t M /tildewidet↓+/tildewidet↓↑A− s,t M⎞ ⎠. (33) The pumped (i.e., linear in the gauge field) spin density for these two states are N/angbracketleftkF↑|σ|kF↑/angbracketrightN=−2 M⎛ ⎜⎝A⊥ s,tRe[Ttot ↑↓]+(ˆz×A⊥ s,t)Im[Ttot ↑↓] +Re[(/tildewidet↑↓)∗/tildewidet↓↑]⎛ ⎜⎝Ax s,t −Ay s,t 0⎞ ⎟⎠ +Im[(/tildewidet↑↓)∗/tildewidet↓↑]⎛ ⎝Ay s,t Ax s,t 0⎞ ⎠⎞ ⎠ −ˆz/parenleftbig Ax s,tRe[(/tildewidet↑)∗/tildewidet↑↓−/tildewidet↓(/tildewidet↓↑)∗] −Ay s,tIm[(/tildewidet↑)∗/tildewidet↑↓−/tildewidet↓(/tildewidet↓↑)∗]/parenrightbig , (34) N/angbracketleftkF↓|σ|kF↓/angbracketrightN=2 M⎛ ⎜⎝A⊥ s,tRe[Ttot ↑↓]+(ˆz×A⊥ s,t)Im[Ttot ↑↓] +Re[(/tildewidet↑↓)∗/tildewidet↓↑]⎛ ⎜⎝Ax s,t −Ay s,t 0⎞ ⎟⎠ +Im[(/tildewidet↑↓)∗/tildewidet↓↑]⎛ ⎝Ay s,t Ax s,t 0⎞ ⎠⎞ ⎠ +ˆz/parenleftbig Ax s,tRe[(/tildewidet↑)∗/tildewidet↑↓−/tildewidet↓(/tildewidet↓↑)∗] −Ay s,tIm[(/tildewidet↑)∗/tildewidet↑↓−/tildewidet↓(/tildewidet↓↑)∗]/parenrightbig . (35) We here focus on the linear effect of interface spin- orbit interaction and neglect the spin polarization along themagnetization direction, n. The expression for the pumped spin current then agrees with Eq. ( 27) with the amplitude ζ s written in terms of hopping including the interface spin orbit, T↑↓=((˜t0 ↑)∗+i(/tildewideγxz)∗)(˜t0 ↓+i/tildewideγxz). (36) In metallic junctions of Cu/Co, Cr/Fe, and Au/Fe, Im[ T↑↓]i s orders of magnitude smaller than Re[ T↑↓][22,23], suggesting that the imaginary part of bare hopping amplitude ˜t0 σis small. According to Eq. ( 36), large Im[ T↑↓] is expected if strong interface spin-orbit interaction exist. If the imaginary part of˜t0 σis neglected, we obtain (using /tildewideγxz=niγxi) Im[ζs]=ν↑−ν↓ 2M(˜t0 ↑+˜t0 ↓)γxini. (37) The measurement of the amplitude of the spin current is proportional to ˙n, thus, it works as a probe for the interface spin-orbit interaction strength γxi. Let us discuss some examples. Of recent particular interest is the interface Rashba interaction, represented by the anti-symmetric coefficient γ(R) ij=/epsilon1ijkαR k, (38) where αRis a vector representing the Rashba field. In the case of an interface, αRis perpendicular to the interface, i.e., αR/bardblˆx. Therefore the interface Rashba interaction leads to γ(R) xj=0 and does not modify the spin pumping effect at the linear order. (It contributes at the second order as discussedin Ref. [ 15].) In other words, the vector coupling between the wave vector and spin in the form of k×σexists only along thexdirection, and does not affect the interface hopping (i.e., does not include k x). In contrast, a scalar coupling η(D)(k·σ)(η(D)is a coef- ficient), called the Dirac type spin-orbit interaction, leads to γ(D) ij=η(D)δij. The spin current along ˙nthen reads j˙n s=η(D)kF(ν↑−ν↓) 2mM(˜t0 ↑+˜t0 ↓)nx˙n. (39) For the case of in-plane easy axis along the zdi- rection and magnetization precession given by n(t)= (sinθcosωt,sinθsinωt,cosθ), where θis the precession angle and ωis the angular frequency, we expect to have a dc spin current along the ydirection, as nx˙n=−ω 2sin2θˆy (nx˙ndenotes time average). Recently, spin pumping effects are discussed including a phenomenological “spin-memory loss” parameter δsml,t o represent the interface spin-flip rate [ 30,31]. The parameter corresponds roughly to δsml=|/tildewidet↑↓|2/(|/tildewidet↑|2+|/tildewidet↓|2) in our scheme [see Eq. ( 31)]. IV . FIELD THEORETIC DESCRIPTION OF METALLIC CASE Here we present a field-theoretic description of the spin pumping effect of a metallic ferromagnet. The many-bodyapproach has an advantage of taking into account the particledistributions automatically. Moreover, it describes the propa-gation of particle density in terms of the Green’s functions,and thus is suitable for studying spatial propagation as wellas for intuitive understanding of transport phenomena. All thetransport coefficients are determined by material constants. The formalism presented here is essentially the same as in Ref. [ 16], but treats ferromagnets of finite size and takes account of electron states with different wave vectors. Interfacespin-orbit interaction is not considered here. Conduction electrons in ferromagnetic and normal metals are denoted by field operators d,d †andc,c†, respectively. These operators are vectors with two spin components, i.e.,d≡(d ↑,d↓). The Hamiltonian describing the F and N electrons 064423-7GEN TATARA AND SHIGEMI MIZUKAMI PHYSICAL REVIEW B 96, 064423 (2017) t rr1 r2 Ga NGr N r4r3 G< (a) rr1 r2Ga NGr N Σ< (b)t Ga NGr N G< (c)r FIG. 5. (a) Schematic diagrammatic representations of the lesser Green’s function for an N electron connecting the same position r, G< N(r,r)/similarequalGr N/Sigma1< NGa Nrepresenting the propagation of electron density. It is decomposed into a propagation of N electrons from rto the interface at r2, then hopping to r4in the F side, a propagation inside F, followed by a hopping to N side (to r1) and propagation back to r. [Position labels are as in Eqs. ( 43)a n d( 44).] (b): The self-energy /Sigma1< Nrepresents all the effects of the ferromagnet. (c) Standard Feynman diagram representation of lesser Green’s function for N at r,E q s .( 46)a n d( 44). isHF+HN, where HF≡/integraldisplay Fd3rd†/parenleftbigg −∇2 2m−/epsilon1F−Mn(t)·σ/parenrightbigg d, HN≡/integraldisplay Nd3rc†/parenleftbigg −∇2 2m−/epsilon1F/parenrightbigg c. (40) We set the Fermi energies for the ferromagnet and the normal metal equal. The hopping through the interface is described bythe Hamiltonian H I≡/integraldisplay IFd3r/integraldisplay INd3r/prime(c†(r/prime)t(r/prime,r,t)d(r) +d†(r)t∗(r/prime,r,t)c(r/prime)), (41) where t(r/prime,r,t) represents the hopping amplitude of electrons from rin the ferromagnetic regime to a site r/primein the normal region and the integrals are over the interface (denoted by I F and I Nfor F and N regions, respectively). The hopping ampli- tude is generally a matrix that depends on the magnetizationdirection n(t), and thus depends on time t. Hopping is treated as energy conserving. Assuming a sharp interface at x=0, the momentum perpendicular to the interface is not conservedon hopping. We are interested in the spin current in the normal region, given by j α s,i(r,t)=−1 4m(∇(r)−∇(r/prime))itr[σαG< N(r,t,r/prime,t)|r/prime=r,(42) where G< N(r,t,r/prime,t/prime)≡i/angbracketleftc(r,t)c†(r/prime,t/prime)/angbracketrightdenotes the lesser Green’s function for the normal region. It is calculated from the Dyson’s equation for the path-ordered Green’s function defined for a complex time along a complex contour C: GN(r,t,r/prime,t/prime)=gN(r−r/prime,t−t/prime)+/integraldisplay cdt1/integraldisplay cdt2/integraldisplay d3r1 ×/integraldisplay d3r2gN(r−r1,t−t1) ×/Sigma1N(r1,t1,r2,t2)GN(r2,t2,r/prime,t/prime),(43) where g< Ndenotes the Green’s function without interface hopping and /Sigma1N(r1,t1,r2,t2) is the self-energy for N elec- trons, given by the contour-ordered Green’s function in theferromagnet as /Sigma1N(r1,t1,r2,t2)≡/integraldisplay IFd3r3/integraldisplay IFd3r4t(r1,r3,t1) ×G(r3,t1,r4,t2)t∗(r2,r4,t2).(44) Here, r1andr2are coordinates at the interface I Nin N region andr3andr4a r et h o s ei nI Ffor F. Gis the contour-ordered Green’s function for F electrons in the laboratory frame including the effect of spin gauge field. We denote the Green’sfunctions of F electrons by Gandgwithout suffix and those of N electrons with suffix N. The lesser component of thenormal metal Green’s function is obtained from Eq. ( 43)a s (suppressing the time and space coordinates) G < N=/parenleftbig 1+Gr N/Sigma1r N/parenrightbig g< N/parenleftbig 1+/Sigma1a NGaN/parenrightbig +Gr N/Sigma1< NGaN.(45) For pumping effects, the last term on the right-hand side is essential, as it contains the information of excitations in Fregion [ 16]. We thus consider the second term only, G < N/similarequalGr N/Sigma1< NGaN, (46) and neglect the spin dependence of the normal region Green’s functions, Gr NandGa N. The contribution is diagrammatically shown in Fig. 5. A. Rotated frame To solve for the Green’s function in the ferromagnet, it is convenient to use the rotated frame we used in Sec. II A.I nt h e field representation, the unitary transformation is representedas [Fig. 6(c)] d=U˜d, c=U˜c, (47) FNdct t∗ (a) FNd cUt t∗U−1 (b) FNd ct t∗ (c) FIG. 6. Unitary transformation Uof F electrons converts the original system with field operator d[shown as (a)] to the rotated one with field operator ˜d≡U−1d(b). The hopping amplitude for representation in (b) is modified by U. If N electrons are also rotated as˜c≡U−1c, hopping becomes ˜t≡U−1tU, while the N electron spin rotates with time, as shown as (c). 064423-8CONSISTENT MICROSCOPIC ANALYSIS OF SPIN . . . PHYSICAL REVIEW B 96, 064423 (2017) where Uis the same 2 ×2 matrix defined in Eq. ( 10). We rotate N electrons as well as F electrons, to simplify thefollowing expressions. The hopping interaction Hamiltonianreads H I=/integraldisplay IFd3r/integraldisplay INd3r/prime(˜c†(r/prime)˜t(r/prime,r)˜d(r) +˜d†(r)˜t∗(r/prime,r)˜c(r/prime)), (48) where ˜t(r/prime,r)≡U†(t)t(r/prime,r,t)U(t) (49) is the hopping amplitude in the rotated frame. The rotated am- plitude (neglecting interface spin-orbit interaction) is diagonalin spin: ˜t=/parenleftbigg˜t ↑0 0 ˜t↓/parenrightbigg . (50) Including the interaction with a spin gauge field, the Hamil- tonian for F and N electrons in the momentum representationis H F+HN=/summationdisplay k˜d† k/parenleftBigg /epsilon1k−M−Az s,t A− s,t A+ s,t /epsilon1k+M+Az s,t/parenrightBigg ˜dk +/summationdisplay k/epsilon1(N) k˜c† k˜ck. (51) As for the hopping, we consider the case the interface is atomically sharp. The hopping Hamiltonian is then writtenin the momentum space as H I=/summationdisplay kk/prime(˜c†(k)˜t(k,k/prime)˜d(k/prime)+˜d†(k/prime)˜t∗(k,k/prime)˜c(k)),(52) where k=(kx,ky,kz),k/prime=(k/prime x,ky,kz), choosing the interface as the plane of x=0. Namely, the wave vectors parallel to the interface are conserved while kxandk/prime xare uncorrelated. B. Spin polarization and current in N Pumped spin current in N is calculated by using Eqs. ( 42) and ( 46). The lesser component of the self-energy connecting Green’s functions with wave vectors kandk/primeis written using Eq. ( 44) as (in the matrix notation) /Sigma1< N(k,k/prime)=/summationdisplay k/prime/prime˜t(k,k/prime/prime)G<(k/prime/prime)˜t∗(k/prime/prime,k/prime). (53) The lesser Green’s function in F in the rotated frame is calculated including the spin gauge field (a driving field of spinpumping) to the linear order by use of the Dyson’s equation G <=g<+gr(As,t·σ)g<+g<(As,t·σ)ga, (54) where gα(α=<,r,a) represents Green’s functions without a spin gauge field. The lesser Green’s function satisfies for static case g<=F(ga−gr), where F≡(f↑ 0 0f↓)i st h e spin-dependent Fermi distribution function. We thus obtain the Green’s function at the linear order, written as δG<,as [16] δG<=gr[As,t·σ,F]ga+gaF(As,t·σ)ga −gr(As,t·σ)Fgr. (55) The last two terms of the right-hand side are rapidly oscillating as a function of position and are neglected. The commutatoris calculated as (sign ±denotes spin ↑and↓) [A s,t·σ,F]=(f+−f−)/summationdisplay ±(±)A± s,tσ∓. (56) The self-energy linear in the spin gauge field is thus /Sigma1< N(k,k/prime)=/summationdisplay ±σ∓/summationdisplay k/prime/prime(fk/prime/prime±−fk/prime/prime∓)A± s,t˜t∓(k,k/prime/prime) טt∗ ±(k/prime/prime,k/prime)gr ∓(k/prime/prime,ω)ga ±(k/prime/prime,ω). (57) The spin polarization of an N electron therefore reads (diagram shown in Fig. 7) −itr[σ±G< N(r,t,r/prime,t)] =−i/summationdisplay kk/primek/prime/primeeik·re−ik/prime·r/primegr N(k,ω)ga N(k/prime,ω)(fk/prime/prime±−fk/prime/prime∓) ×A± s,t˜t∓(k,k/prime/prime)˜t∗ ±(k/prime/prime,k/prime)gr ∓(k/prime/prime,ω)ga ±(k/prime/prime,ω). (58) We assume that the dependence of N Green’s functions on ωis weak and use/summationtext keik·rgr N(k,ω)=−iπν NeikFxe−|x|//lscript≡ gr N(r), where /lscriptis the elastic mean free path, νNandkFare the density of states at the Fermi energy and Fermi wave vector,respectively, whose ωdependencies are neglected. (For an infinitely wide interface, the Green’s function becomes onedimensional.) As a result of summation over wave vectors, theproduct of hopping amplitudes ˜t ∓(k,k/prime/prime)˜t∗ ±(k/prime/prime,k/prime) is replaced by the average over the Fermi surface, ˜t∓˜t∗ ±≡T±∓, i.e., ˜t∓(k,k/prime/prime)˜t∗ ±(k/prime/prime,k/prime)→T±∓. (59) The spin polarization of N electrons induced by magnetization dynamics (the spin gauge field) is therefore obtained in therotated frame as [with correlation function χ ±defined in Eq. ( A5)] ˜s(N) ±(r,t)=−/vextendsingle/vextendsinglegr N(r)/vextendsingle/vextendsingle2/summationdisplay ±A± s,tχ±T±∓, (60) A±s,tgr∓ ga±˜s(N) ±grN gaN˜t∓ ˜t∗± FIG. 7. Feynman diagram for electron spin density of normal metal driven by the spin gauge field of ferromagnetic metal As.T h e spin current is represented by the same diagram but with spin currentvertex. 064423-9GEN TATARA AND SHIGEMI MIZUKAMI PHYSICAL REVIEW B 96, 064423 (2017) or using χ∗ +=χ−, ˜s(N)(r,t)=−2/vextendsingle/vextendsinglegr N(r)/vextendsingle/vextendsingle2[A⊥ s,tRe[χ+T+−] +(ˆz×A⊥ s,t)Im[χ+T+−]]. (61) In the laboratory frame, we have (using s(N) i=Rij˜s(N) j) s(N)(r,t)=/vextendsingle/vextendsinglegr N(r)/vextendsingle/vextendsingle2[Re[χ+T+−](n×˙n)+Im[χ+T+−]˙n]. (62) The spin current induced in N region is similarly given by (neglecting the contribution proportional to n) js(r,t)=kF m/vextendsingle/vextendsinglegr N(r)/vextendsingle/vextendsingle2[Re[χ+T+−](n×˙n)+Im[χ+T+−]˙n] =e−|x|//lscript(Re[ζs](n×˙n)+Im[ζs]˙n), (63) where ζs≡π2kFν2 N 2mM(n+−n−)T+−. (64) The coefficient ζsis essentially the same as the one in Eq. ( 27) derived by a quantum mechanical argument, as the quantummechanical dimensionless hopping amplitude corresponds toν N˜tof the field representation. For a 3 dferromagnet, we may estimate the spin current by approximating roughly M∼1/νN∼/epsilon1F∼1 eV and nσ∼ kF3. The hopping amplitude |T+−|in the metallic case would be of order of /epsilon1F. The spin current density then is of the order of (including electric charge eand recovering ¯ h)js∼e¯hkF mh¯hω /epsilon1F∼ 5×1011A/m2if the precession frequency is 10 GHz. V . SPIN ACCUMULATION IN FERROMAGNET The spin current pumping is equivalent to the increase of spin damping due to magnetization precession, as wasdiscussed in Refs. [ 2,11]. In this section, we confirm this fact by calculating the torque by evaluating the spin polarization of theconduction electron spin in F region. (The spin accumulationwithout taking into account an interface is calculated inAppendix A.) There are several ways to evaluate the damping of magneti- zation. One way is to calculate the spin-flip probability of theelectron as in Ref. [ 11], which leads to damping of localized spin in the presence of strong sdexchange interaction. The second is to estimate the torque on the electron by use of the equation of motion [ 32]. The relation between the damping and spin current generation is clearly seen in this approach.In fact, the total torque acting on conduction electrons is (¯ h times) the time derivative of the electron spin density, ds dt=i(/angbracketleft[H,d†]σd/angbracketright+/angbracketleftd†σ[H,d]/angbracketright). (65) At the interface, the right-hand side arises from the interface hopping. Using the hopping Hamiltonian of Eq. ( 41), we have ds dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle interface=i(/angbracketleftc†tσd/angbracketright−/angbracketleftd†σt†c/angbracketright), (66) as the interface contribution. As is natural, the right-hand side agrees with the definition of the spin current passingthrough the interface. Evaluating the right-hand side, we obtainin general a term proportional to n×˙n, which gives the Gilbert damping, and a term proportional to ˙n, which gives a renormalization of the magnetization. In contrast, awayfrom the interface, the commutator [ H,d] arises from the kinetic term H 0≡/integraltext d3r|∇d|2 2mdescribing electron propagation, resulting in dsα dt=i(/angbracketleft[H0,d†]σd/angbracketright+/angbracketleftd†σ[H,d]/angbracketright) =∇· jα s, (67) where jα s(r)≡−i 2m(∇r−∇ r/prime)/angbracketleftd†(r/prime)σαd(r)/angbracketright|r/prime=ris the spin current. Away from the interface, the damping therefore occurs if the spin current has a source or a sink at the site of interest. Here we use the third approach and estimate the torque on the localized spin by calculating the spin polarizationof electrons as was done in Refs. [ 7,33]. The electron spin polarization at position rin the ferromagnet at time tis s (F)(r,t)≡/angbracketleftd†σd/angbracketright, which reads in the rotated frame s(F) α= Rαβ˜s(F) β, where ˜s(F) β(r,t)=−itr[σβG<(r,r,t,t)], (68) where G< σσ/prime(r,r/prime,t,t/prime)≡i/angbracketleft˜d† σ/prime˜dσ/angbracketrightis the lesser Green’s function in F region, which is a matrix in spin space ( σ,σ/prime=±). We are interested in the effect of the N region arising fromthe hopping. We must note that the hopping interaction ofEq. ( 48) is not convenient for integrating out N electrons, since the ˜celectrons’ spins are time-dependent as a result of a unitary transformation U(t). We thus use the following form [Fig. 6(b)], H I=/integraldisplay IFd3r/integraldisplay INd3r/prime(c†(r/prime)U˜t(r/prime,r)˜d(r) +˜d†(r)˜t∗(r/prime,r)U†c(r/prime)), (69) namely, the hopping amplitude between ˜dandcelectrons includes the unitary matrix U. Let us briefly argue in the rotated frame why the effect of damping arising from the interface. In the totally rotatedframe of Fig. 6(c), the spin of an F electron is static, while that of N electron varies with time. When an F electron hopsto N region and comes back, therefore, the electron spin getsrotated with the amount depending on the time it stayed in Nregion. This effect is in fact represented by a retardation effectof the matrices UandU −1in Eq. ( 69). If the off-diagonal nature of UandU−1is neglected, the interface effects are all spin-conserving and do not induce damping for F electrons(see Appendix B). The spin density is calculated by evaluating the lesser Green’s function in F. Including the effect of interface in termsof self-energy, it reads G <(r,t,r/prime,t)=gr/Sigma1rga+gr/Sigma1<ga+g</Sigma1aga, (70) where the self-energy of an F electron arising from the hopping to N region reads ( r1andr2are in F and a=r,a,<) /Sigma1a(r1,r2,t1,t2)=/integraldisplay INd3r/prime 1/integraldisplay INd3r/prime 2˜t(r1,r/prime 1)U−1(t1) ×ga N(r/prime 1,r/prime 2,t1−t2)U(t2)˜t†(r2,r/prime 2).(71) 064423-10CONSISTENT MICROSCOPIC ANALYSIS OF SPIN . . . PHYSICAL REVIEW B 96, 064423 (2017) ˜tU−1 gN U˜t∗σg˜t gN ˜t∗σgAs,t ˜s(F)== FIG. 8. Diagrammatic representation of the spin accumulation in a ferromagnetic metal induced as a result of coupling to the normal metal [Eqs. ( 68)a n d( C1)]. Conduction electron Green’s functions in a ferromagnet and normal metal are denoted by gandgN, respectively. Time-dependent matrix U(t), defined by Eq. ( 10), represents the effect of dynamic magnetization. Expanding UandU−1with respect to the slow time dependence of magnetization, we obtain a gauge field representation, see Eq. ( C3). Expanding to the linear order in the spin gauge field arising from the time dependence of unitary matrix U, we obtain G<(r,t,r/prime,t)=2πiν Na2/integraldisplaydω 2πf/prime N(ω)gr(r,ω) טtAs,t˜t†ga(−r,ω). (72) (Diagrammatic representation of the contribution is in Fig. 8. For calculation detail, see Appendix C). For damping, off- diagonal contributions, A± s,t, are obviously essential. The result of the spin density in F in the rotated frame, Eq. ( 68), is therefore ˜s(F) α(r,t)=2πiν Na2/integraldisplaydω 2πf/prime N(ω)Aβ s,ttr[σαgr(r,ω) טtσβ˜t†ga(−r,ω)] =2πiν Na2/integraldisplaydω 2πf/prime N(ω)Aβ s,t/summationdisplay kk/primeei(k−k/prime)·r ×tr[σαgr(k,ω)˜tσβ˜t†ga(k/prime,ω)]. (73) Evaluating the trace in spin space, we obtain ˜s(F)(r,t)=−νN[A⊥ s,tγ1(r)+(ˆz×A⊥ s,t)γ2(r)], (74) where γ1(r)≡/summationdisplay σ˜t−σ˜t† σgr −σ(r)ga σ(−r), γ2(r)≡/summationdisplay σ(−iσ)˜t−σ˜t† σgr −σ(r)ga σ(−r). (75) We consider an interface with infinite area and consider spin accumulation averaged over the plane parallel to the interface.The wave vectors contributing are then those with finite k x but with ky=kz=0 and the Green’s function becomes one- dimensional-like: /summationdisplay keik·rgr σ(k)=im kFσeikFσ|x|e−|x|/(2/lscriptσ), (76) where /lscriptσ≡vFστσ(vFσ≡kFσ/m) is the electron mean free path for spin σ. The induced spin density in the ferromagnetis finally obtained from Eq. ( 74)a s s(F)(r,t)=m2νNa2 2kF+kF−/summationdisplay σ[(n×˙n)Tσ,−σe−iσ(kF+−kF−)x +˙n(−iσ)Tσ,−σe−iσ(kF+−kF−)x] =m2νNa2 2kF+kF−/summationdisplay σ{(n×˙n)[Re[T↑,↓] cos(( kF+−kF−) ×x)+Im[T↑,↓]s i n ( (kF+−kF−)x)] +˙n[Im[T↑,↓] cos(( kF+−kF−)x) −Re[T↑,↓]s i n ( (kF+−kF−)x)]} (77) and the torque on the localized spin −Mn×s(F)is τ(r,t)=−m2νNa2M 2kF+kF−/summationdisplay σ{−˙n[Re[T↑,↓] cos(( kF+−kF−)x) +Im[T↑,↓]s i n ( (kF+−kF−)x)] +(n×˙n)[Im[T↑,↓] cos(( kF+−kF−)x) −Re[T↑,↓]s i n ( (kF+−kF−)x)]}. (78) A. Enhanced damping and spin renormalization of ferromagnetic metal The total induced spin accumulation density in a ferromag- net is s(F)≡1 d/integraldisplay0 −ddxs(F)(x) =1 M{(n×˙n)[−Im[δ](1−cos˜d)+Re[δ]s i n ˜d] +˙n[Re[δ](1−cos˜d)+Im[δ]s i n ˜d]}, (79) where ˜d≡(kF+−kF−)d,dis the thickness of the ferromagnet and δ≡m2νNa2M kF+kF−(kF+−kF−)dT↑,↓. (80) As a result of this induced electron spin density, s(F),t h e equation of motion for the averaged magnetization is modifiedto be [ 11] ˙n=−αn×˙n−γB×n−Mn× s(F), (81) where Bis the external magnetic field. Let us first discuss the thick ferromagnet case, d/greatermuch |kF+−kF−|−1, where the oscillating part with respect to˜dis neglected. The spin density then reads s(F)/similarequal 1 M(−Im[δ](n×˙n)+Re[δ]˙n) and the equation of motion becomes (1+Imδ)˙n=− ˜αn×˙n−γB×n, (82) where ˜α≡α+Reδ, (83) is the Gilbert damping including the enhancement due to the spin pumping effect. The precession angular frequency ωBis modified by the imaginary part of T↑,↓, i.e., by the spin current 064423-11GEN TATARA AND SHIGEMI MIZUKAMI PHYSICAL REVIEW B 96, 064423 (2017) proportional to ˙n,a s ωB=γB 1+Imδ. (84) This is equivalent to the modification of the gyromagnetic ratio (γ)o rt h e gfactor. For most 3D ferromagnets, we may approximate m2νNaM/epsilon1 F2 2kF+kF−(kF+−kF−)/similarequalO(1) (as kF+−kF−∝M), resulting in δ∼a dT↑,↓. As discussed in Sec. III, when the interface spin-orbit interaction is taken into account, we have T↑,↓= ˜t0 ↑˜t0 ↓+i/tildewideγxz(˜t0 ↑+˜t0 ↓)+O((/tildewideγ)2), where ˜t0 σand/tildewideγxzare assumed to be real. Moreover, ˜t0 σcan be chosen as positive and thus T↑,↓>0. (˜t0 σhere is field representation, and has unit of energy.) Equations ( 83) and ( 84) indicate that the strength of the hopping amplitude ˜t0 σand interface spin-orbit interaction /tildewideγxzare experimentally accessible by measuring the Gilbert damping and shift of resonance frequency as has beenknown [ 2]. A significant consequence of Eq. ( 83) is that the enhancement of the Gilbert damping, δα∼a d1 /epsilon1F2˜t0 ↑˜t0 ↓, (85) can exceed in thin ferromagnets the intrinsic damping pa- rameter α, as the two contributions are governed by different material parameters. In contrast to the positive enhancementof damping, the shift of the resonant frequency or gfactor can be positive or negative, as it is linear in the interface spin-orbitparameter/tildewideγ xz. Experimentally, the enhancement of the Gilbert damping and frequency shift has been measured in many systems [ 12]. In the case of Ni 80Fe20(Py)/Pt junction, the enhancement of damping is observed to be proportional to 1 /din the range of 2n m<d< 10 nm, and the enhancement was large, δα/α/similarequal4 atd=2n m[ 12]. These results appears to be consistent with our analysis. Same 1 /ddependence was observed in the shift of the gfactor. The shift was positive and the magnitude is about 2% for Py/Pt and Py/Pd with d=2nm, while it was negative for Ta/Pt [ 12]. The existence of both signs suggests that the shift is due to the linear effect of spin-orbit interaction,and the interface spin-orbit interaction we discuss is one of thepossible mechanisms. For thin ferromagnet, ˜d/lessorsimilar1, the spin accumulation of Eq. ( 79) reads s(F)=1 M((n×˙n)Re[δthin]+˙nIm[δthin]), (86) where δthin≡δ˜d=m2νNa2M 2kF+kF−T↑,↓. (87) Equation ( 86) indicates that the roles of imaginary and real part of T↑,↓are interchanged for thick and thin ferromagnets, resulting in ˜α=α+Imδthin,ω B=γB 1−Reδthin, (88) for thin ferromagnets. Thus, for weak interface spin-orbit inter- action, a positive shift of the resonance frequency is expected(as Re δ thin>0). A significant feature is that the damping can be decreased or even become negative if strong interfacespin-orbit interaction exists with a negative sign of Im δthin.O u r result indicates that the “spin mixing conductance” descriptionof Ref. [ 2] breaks down in thin metallic ferromagnets (and the insulator case as we shall see in Sec. VII D ). In this section, we have discussed spin accumulation and enhanced Gilbert damping in a ferromagnet attached to anormal metal. In the field-theoretic description, the dampingenhancement arises from the imaginary part of the self-energydue to the interface. Thus a randomness like the interfacescattering changing the electron momentum is essential forthe damping effect, which sounds physically reasonable.The same is true for the reaction, namely, the spin currentpumping effect into the N region, and thus the spin currentpumping requires randomness too. (In the quantum mechanicaltreatment of Sec. II, change of electron wave vector at the interface is essential.) The spin current pumping effecttherefore appears different from general pumping effects,where randomness does not play essential roles apparently[3]. The spin accumulation and enhanced Gilbert damping was discussed by Berger [ 11] based on a quantum mechanical argument. There, 1 /ddependence was pointed out and the damping effect was calculated by evaluating the decay rateof magnons. A comparison of enhanced Gilbert dampingwith experiments was carried out in Ref. [ 2]b u ti na phenomenological manner. VI. CASE WITH MAGNETIZATION STRUCTURE The field theoretic approach has an advantage that the generalization of the results is straightforward. Here wediscuss briefly the case of a ferromagnet with spatially varyingmagnetization. The excitations in a metallic ferromagnetconsist of spin waves (magnons) and Stoner excitation. Whilespin waves usually have a gap as a result of magneticanisotropy, Stoner excitation is gapless for a finite wave vector,(k F+−kF−)<|q|<(kF++kF−), and it may be expected to have significant contribution for magnetization structureshaving wavelength larger than k F+−kF−. Let us look into this possibility. Our result of spin accumulation in a ferromagnet, repre- sented in the rotated frame, Eq. ( A3), indicates that when the magnetization has a spatial profile, the accumulationis determined by the spin gauge field and spin correlationfunction depending on the wave vector qas /summationdisplay qA± s,t(q)χ±(q,0), (89) where χ±(q,/Omega1)≡−/summationdisplay kfk+q,±−fk,∓ /epsilon1k+q,±−/epsilon1k,∓+/Omega1+i0(90) is the correlation function with finite momentum transfer qand finite angular frequency /Omega1. For the case of free electron with quadratic dispersion, the correlation functionis [34] χ ±(q,/Omega1)=Aq+i/Omega1B qθst(q)+O(/Omega12), (91) 064423-12CONSISTENT MICROSCOPIC ANALYSIS OF SPIN . . . PHYSICAL REVIEW B 96, 064423 (2017) where Aq=ma3 8π2/bracketleftbigg (kF++kF−)/parenleftbigg 1+(kF+−kF−)2 q2/parenrightbigg +1 2q3((kF++kF−)2−q2)(q2−(kF+−kF−)2) ×ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleq+(kF++kF−) q−(kF++kF−)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg , Bq=m2a3 4π|q|, (92) and θst(q)≡/braceleftbigg1(kF+−kF−)<|q|<(kF++kF−) 0 otherwise,(93) describes the wave vectors where Stoner excitation exists. As we see from Eq. ( 91), the Stoner excitation contribution vanishes to the lowest order in /Omega1, and thus the spin pumping effect in the adiabatic limit ( /Omega1→0) is not affected. Moreover, the real part of the correlation function, Aq, is a decreasing function of qand thus the spin pumping efficiency would decrease when the ferromagnet has a structure. However, forrigorous argument, we need to include the spatial componentof the spin gauge field arising form the spatial derivative of themagnetization profile. As for the effect of the Stoner excitation on spin damping (Gilbert damping), it was demonstrated for the case of adomain wall that the effect is negligibly small for a wide wallwith thickness λ/greatermuch(k F+−kF−)−1(Refs. [ 34,35]). Simanek and Heinrich presented a result of the Gilbert damping as thelinear term in the frequency of the imaginary part of the spincorrelation function integrated over the wave vector [ 13]. The result is, however, obtained for a model where the ferromagnetis an atomically thin layer (a sheet), and would not beapplicable for most experimental situations. A discussion ofthe Gilbert damping including a finite wave vector and theimpurity scattering was given in Ref. [ 36]. Inhomogeneity effects of damping of a domain wall were studied recentlyin detail [ 37]. The effective Gilbert damping constant in the presence of a domain wall was numerically studied inRefs. [ 38,39]. A quadratic dependence on the inverse of the wall thickness appears to be consistent with the quadraticbehavior of A qat small q, while the linear behavior found for an out-of-plane extremely narrow wall [ 39] seems not to be covered by the simple argument here. VII. INSULATOR FERROMAGNET In this section, we discuss the case of a ferromagnetic insulator. It turns out that the generation mechanisms for spincurrent in the insulating and metallic cases are distinct. A. Magnon and adiabatic gauge field The Lagrangian for the insulating ferromagnet is LIF=/integraldisplay d3r/bracketleftbigg S˙φ(cosθ−1)−J 2(∇S)2/bracketrightbigg −HK, (94) where Jis the exchange interaction between the localized spin SandHKdenotes the magnetic anisotropy energy.We first study low-energy magnon dynamics induced by slow magnetization dynamics. For separating the classicalvariable and fluctuation (magnon), the rotated coordinatedescription used in the metallic case is convenient. Formagnons described by the Holstein-Primakov boson, the uni-tary transformation is a 3 ×3 matrix defined as follows [ 40]: S=U/tildewideS, (95) where U=⎛ ⎝cosθcosφ−sinφsinθcosφ cosθsinφ cosφ sinθsinφ −sinθ 0 cos θ⎞ ⎠ =/parenleftbige θeφn/parenrightbig . (96) The diagonalized spin /tildewideSis represented in terms of annihilation and creation operators for the Holstein-Primakov boson, band b†,a s[ 41] /tildewideS=⎛ ⎜⎜⎝/radicalBig S 2(b†+b) i/radicalBig S 2(b†−b) S−b†b⎞ ⎟⎟⎠. (97) We neglect the terms that are third- and higher-order in boson operators. Derivatives of the localized spin then read ∂μS=U(∂μ+iAU,μ)/tildewideS, (98) where AU,μ≡−iU−1∇μU (99) is the spin gauge field represented as a 3 ×3 matrix. The spin Berry’s phase of the Lagrangian ( 94) is written in terms of magnons as (derivation is in Appendix D) Lm=2Sγ2/integraldisplay d3ri/bracketleftbig b†/parenleftbig ∂t+iAz s,t/parenrightbig b−b†/parenleftbig← ∂t−iAz s,t/parenrightbig/parenrightbig b/bracketrightbig , (100) namely, the magnons interact with the adiabatic component of the same spin gauge field for electrons, Az s,t, defined in Eq. ( 14). As the magnon is a single-component field, the gauge field is also single-component, i.e., a U(1) gauge field.This is a significant difference between insulating and metallicferromagnets; in the metallic case, a conduction electroncouples to an SU(2) gauge field with spin-flip components,which turned out to be essential for spin current generation. Incontrast, in the insulating case, the magnon has a diagonalgauge field, i.e., a spin chemical potential, which simplyinduces diagonal spin polarization. Pumping of magnon wasdiscussed in a different approach by evaluating the magnonsource term in Ref. [ 42]. The exchange interaction at the interface is represented by a Hamiltonian H I=JI/integraldisplay d3rIS(r)·c†σc, (101) where JIis the strength of the interface sdexchange interaction and the integral is over the interface. We consider a sharp in-terface at x=0. Using Eq. ( 95), the interaction is represented 064423-13GEN TATARA AND SHIGEMI MIZUKAMI PHYSICAL REVIEW B 96, 064423 (2017) in terms of magnon operators up to the second order as HI=JI/integraldisplay d3rI/braceleftbig (S−b†b)c†(n·σ)c +/radicalbigg S 2[b†c†/Phi1·σc+bc†/Phi1∗·σc]/bracerightBigg , (102) where /Phi1≡eθ+ieφ=⎛ ⎝cosθcosφ−isinφ cosθsinφ+icosφ −sinθ⎞ ⎠. (103) Equation ( 102) indicates that there are two mechanisms for spin current generation; namely, the one due to themagnetization at the interface (the term proportional to n) and the one due to the magnon spin scattering at the interface(described by the term linear in magnon operators). Let us briefly demonstrate based on the expression of Eq. ( 102) that spin-flip processes due to magnon creation or annihilation lead to generation of spin current in the normalmetal. At the second order, the interaction induces a factor onthe electron wave function ( /Phi1 ∗(t)·σ)(/Phi1(t/prime)·σ) for magnon creation and ( /Phi1(t)·σ)(/Phi1∗(t/prime)·σ) for annihilation (we allow an infinitesimal difference in time tandt/prime). The factor for the creation has charge and spin contributions, ( /Phi1∗(t)·σ)(/Phi1(t/prime)· σ)=/Phi1∗(t)·/Phi1(t/prime)+iσ·(/Phi1∗(t)×/Phi1(t/prime)). For magnon anni- hilation, we have ( /Phi1∗(t)×/Phi1(t/prime))∗, and thus the sum of the magnon creation and annihilation processes leads to a factor /summationdisplay q[(nq+1)(/Phi1∗(t)×/Phi1(t/prime))+nq(/Phi1∗(t)×/Phi1(t/prime))∗] =/summationdisplay q{(2nq+1)Re[/Phi1∗(t)×/Phi1(t/prime)] +iIm[/Phi1∗(t)×/Phi1(t/prime)]}. (104) For adiabatic change, the amplitude is expanded as (/Phi1∗(t)×/Phi1(t/prime))=2i(1+i(t−t/prime) cosθ˙φ)n−(t−t/prime) ×(n×˙n−i˙n)+O((∂t)2), (105) where we see that a retardation effect from the adiabatic change of magnetization (represented by the second term on the right-hand side) gives rise to a magnon state change proportionalton×˙nand˙n. The retardation contribution for the spin part [Eq. ( 104)] is (t−t /prime)/summationdisplay q[−(2nq+1)(n×˙n)+i˙n]. (106) We therefore expect that a spin current proportional to n×˙n emerges proportional to the magnon creation and annihilationnumber,/summationtext q(2nq+1). (As we shall see below, the factor t−t/prime reduces to a derivative with respect to the angular frequency of the Green’s function.) A rigorous estimation using the Green’sfunction method is presented in Sec. VII C . In Eq. ( 106), the last term proportional to ˙nis an imaginary part arising from the difference of magnon creation andannihilation probabilities of vacuum, n q+1 andnq.T h et e r m is, however, an unphysical one corresponding to a real energyshift due to magnon interaction, and is removed by redefinitionof the Fermi energy.n js FIG. 9. Feynman diagrams for spin current pumped by interface sdexchange interaction. B. Spin current pumped by the interface exchange interaction Here, we study the spin current pumped by the classical magnetization at the interface, namely, the one driven bythe term proportional to Snin Eq. ( 102). We treat the exchange interaction perturbatively to the second order asthe exchange interaction between a conduction electron andthe insulator ferromagnet is localized at the interface and isexpected to be weak. The weak-coupling scheme employedhere is in the opposite limit as the strong-coupling (adiabatic)approach used in the metallic ferromagnet (Sec. IV). A recent experiment indicates that the insulator spin pumping effect isdriven by local magnetization induced in the normal metalby the magnetic proximity effect [ 8], supporting perturbative treatment. In the perturbative regime, the issue of adiabaticity needs to be argued carefully. In the strong sdcoupling limit, the adiabaticity is trivially satisfied, as the time needed for theelectron spin to follow the localized spin is the fastest timescale. In the weak-coupling limit, this time scale is long.Nevertheless, the adiabatic condition is satisfied if the electronspin relaxation is strong so that the electron spin relaxesquickly to the local equilibrium state determined by thelocalized spin. Thus the adiabatic condition is expected tobeM Iτsf/¯h/lessmuch1, where MIandτsfare the interface spin splitting energy, and the conduction electron spin relaxationtime, respectively. In the following calculation, we considerthe case of /epsilon1 Fτsf/¯h/greatermuch1, i.e., ¯ h(τsf)−1/lessmuch/epsilon1F, as the spin-flip lifetime is by definition longer than the elastic electron lifetimeτ, which satisfies /epsilon1 Fτ/¯h/greatermuch1 in a metal. The perturbative results therefore can apply to both adiabatic and nonadiabaticcases. The calculation is carried out by evaluating the Feynman diagrams of Fig. 9, similar to the study of Refs. [ 18,19]. A difference is that while Refs. [ 18,19] assumed a smooth magnetization structure and used a gradient expansion, theexchange interaction we consider is localized. The lesser Green’s function for a normal metal including the interface exchange interaction to the linear order is G (1)< N(r,t,r,t)=MI/integraldisplaydω 2π/integraldisplayd/Omega1 2π/summationdisplay kk/primee−i/Omega1tei(k/prime−k)·r ×/bracketleftbig (f(ω+/Omega1)−f(ω))gr k/prime,ω+/Omega1ga kω −f(ω)gr k/prime,ω+/Omega1gr kω+f(ω+/Omega1)ga k/prime,ω+/Omega1ga kω/bracketrightbig ×(n/Omega1·σ), (107) where MI≡JISis the local spin polarization at the interface. Expanding the expression with respect to /Omega1and keeping the dominant contribution at long distance, i.e., the terms containing both gaandgr.U s i n g/summationtext kga kωeik·r/similarequalim kFeikre−|x| /lscript(≡ 064423-14CONSISTENT MICROSCOPIC ANALYSIS OF SPIN . . . PHYSICAL REVIEW B 96, 064423 (2017) ga(r)), the result of spin current is j(1) s(r,t)=−MIm kF˙ne−|x|//lscript. (108) The second-order contribution is similarly calculated to obtain G(2)< N(r,t,r,t)/similarequal(MI)2/integraldisplaydω 2π/integraldisplayd/Omega1 1 2π/integraldisplayd/Omega1 2 2π ×/summationdisplay kk/primek/prime/primee−i(/Omega11+/Omega12)tei(k/prime−k)·rf/prime(ω)gr k/prime,ωga kω ×/parenleftbig /Omega11ga k/prime/primeω+/Omega12gr k/prime/primeω/parenrightbig/parenleftbig n/Omega11·σ/parenrightbig/parenleftbig n/Omega12·σ/parenrightbig =−2πiν(MI)2|gr(r)|2(n×˙n)·σ. (109) The corresponding spin current at the interface ( x=0) is thus j(2) s(x=0,t)=ν(MI)2m kF(n×˙n), (110) and the total spin current reads js(x=0,t)=−MIm kF˙n−2ν(MI)2m kF(n×˙n). (111) In the perturbation regime, the spin current proportional to ˙n is dominant (larger by a factor of ( νM I)−1) compared to the one proportional to n×˙n. An expression of the spin current induced by the interface exchange interaction was presented in Ref. [ 43] in the limit of strong spin relaxation, MIτsf/lessmuch1, where τsfis the spin relaxation time of electrons. By solving the Landau-Lifshitz-Gilbert equation for the electron spin, they obtained Eq. ( 111) withνM Ireplaced by MIτsf. C. Calculation of magnon-induced spin current Here, the magnon-induced spin current due to the magnon interaction in Eq. ( 102) is calculated. As a magnon is a small fluctuation of magnetization, the contribution here is a smallcorrection to the contribution of Eq. ( 111). Nevertheless, the magnon contribution has a typical linear dependence on thetemperature, and is expected to be experimentally identifiedeasily. The spin current induced in a normal metal is evaluated by calculating the self-energy arising from the interface magnonscattering of Eq. ( 102). The contribution to the path-ordered Green’s function of N electron from the magnon scattering tothe second order is G N(r,t,r/prime.t/prime)=/integraldisplay Cdt1/integraldisplay Cdt2/summationdisplay r1r2gN(r,t,r1,t1) ×/Sigma1I(r1,t1,r2,t2)gN(r2,t2,r/prime,t/prime),(112) where /Sigma1I(r1,t1,r2,t2)≡iSJ2 I 2Dαβ(r1,t1,r2,t2)σαgN(r1,t1,r2,t2)σβ (113) represents the self-energy. Here, Dαβ(r1,t1,r2,t2)≡−i/angbracketleftTCBα(r1,t1)Bβ(r2,t2)/angbracketright (114)Φ† ΦgN jsgN D FIG. 10. Feynman diagrams for spin current pumped by magnons at the interface. Green’s functions for magnons and electrons in thenormal metal are denoted by Dandg N, respectively. /Phi1represents the effects of magnetization dynamics [Eq. ( 103)]. is the Green’s function for a magnon dressed by the magneti- zation structure [ /Phi1is defined in Eq. ( 103)], Bα(r,t)≡/Phi1α(t)b†(r,t)+/Phi1† α(t)b(r,t). (115) The diagrammatic representation is in Fig. 10. In the present approximation including the interface scattering to the secondorder, the electron Green’s function in Eq. ( 113) is treated as spin-independent, resulting in a self-energy (defined oncomplex time contour) /Sigma1 I(r1,t1,r2,t2)=iSJ2 I 2(δαβ+i/epsilon1αβγσγ) ×Dαβ(r1,t1,r2,t2)gN(r1,t1,r2,t2).(116) We focus on the spin-polarized contribution containing the Pauli matrix. The self-energy is then /Sigma1I,γ(r1,t1,r2,t2)≡−SJ2 I 2/tildewideDγ(r1,t1,r2,t2)gN(r1,t1,r2,t2), (117) where/tildewideDγ≡/epsilon1αβγDαβ, and the lesser Green’s function, Eq. ( 112), reads G< N=σγG< N,γ, (118) where (time and spatial coordinates partially suppressed) G< N,γ(r,t,r/prime.t/prime)≡/integraldisplay∞ −∞dt1/integraldisplay∞ −∞dt2/bracketleftbig gr N(t−t1)/Sigma1r I,γ(t1,t2) ×g< N(t2−t/prime)+gr N/Sigma1< I,γga N+g< N/Sigma1a I,γga N/bracketrightbig . (119) The dominant contribution long distance is (see Appendix E for detail) G< N,γ(r,t,r/prime,t)/similarequal/integraldisplaydω 2π/summationdisplay kk/primegr N,kωga N,k/primeωeik·re−ik/prime·r/prime/tildewide/Sigma1I,γ (120) with /tildewide/Sigma1I,γ/similarequali/Psi1γπν /epsilon1FSJ2 I 2/summationdisplay qk/prime/prime(1+2nq)(2fk/prime/prime−fk−fk/prime).(121) The spin current pumped by the magnon scattering is therefore jm s(r,t)=πν /epsilon1FSJ2 I 2|gr(r)|2/summationdisplay q(1+2nq)(n×˙n).(122) 064423-15GEN TATARA AND SHIGEMI MIZUKAMI PHYSICAL REVIEW B 96, 064423 (2017) At high temperature compared to magnon energy, βωq/lessmuch1, 1+2nq/similarequal2kBT ωq, and the magnon-induced spin current de- pends linearly on temperature. The result ( 122) agrees with a previous study carried out in the context of thermally inducedspin current [ 20]. D. Correction to Gilbert damping in the insulating case In this section, we calculate the correction to the Gilbert damping and gfactor of an insulating ferromagnet as a result of the spin pumping effect. We study the torque onthe ferromagnetic magnetization arising from the effect ofconduction electrons of a normal metal, given by τ I=BI×n=MI(n×sI), (123) where BI≡−δHI δn=−MIsI (124) is the effective magnetic field arising from the interface electron spin polarization, sI(t)≡−itr[σG< N(0,t)]. The con- tribution to the electron spin density linear in the interfaceexchange interaction, Eq. ( 101), is s(1),α I(t)=−i/integraldisplay dt1MInβ(t1)tr[σαgN(t,t1)σβgN(t1,t)]<, (125) where the Green’s functions connect positions at the interface, i.e., from x=0t ox=0, and are spin unpolarized. (The Feynman diagrams for the spin density are the same as theone for the spin current, Fig. 9, with the vertex j sreplaced by the Pauli matrix.) The pumped contribution proportional to thetime variation of magnetization is obtained as s(1) I(t)=−MI˙n/integraldisplaydω 2π/summationdisplay kk/primef/prime(ω)/parenleftbig ga N,k/prime−gr N,k/prime/parenrightbig/parenleftbig ga N,k−gr N,k/parenrightbig =−MI(πν)2˙n. (126) The second-order contribution similarly reads s(2),α I(t)=−i 2/integraldisplay dt1/integraldisplay dt2(MI)2nβ(t1)nγ(t2)tr[σαgN(t,t1) ×σβgN(t1,t2)σγgN(t2,t)]< /similarequal−2(MI)2(πν)3(n×˙n). (127) The interface torque is therefore τI=−(MIπν)2(n×˙n)+2(MIπν)3˙n. (128) Including this torque in the LLG equation, ˙n=−αn×˙n− γB×n+τ,w eh a v e (1−δI)˙n=−αI(n×˙n)−γB×n, (129) where δI=2μd(πM Iν)3,α I=α+μd(πM Iν)2,(130) where μd∼dmp/dis the ratio of the length of magnetic proximity ( dmp) and thickness of the ferromagnet, d.T h e Gilbert damping constant therefore increases as far as theinterface spin-orbit interaction is neglected. The resonance frequency is ω B=γB 1−δI, and the shift can have both signs depending on the sign of interface exchange interaction, MI. There may be a possibility that magnon excitations induce a torque that corresponds to effective damping. In fact, sucha torque arises if /angbracketleftb/angbracketrightor/angbracketleftb†/angbracketrightare finite, i.e., if the magnon Bose condensation glows. Such condensation can in principledevelop from the interface interaction of magnon creation orannihilation induced by electron spin flip, Eq. ( 102). However, conventional spin relaxation processes arising from the secondorder of random spin scattering do not contribute to suchmagnon condensation and additional damping. Comparing the result of pumped spin current, Eq. ( 111), and that of damping coefficient, Eq. ( 130), we notice that the “spin mixing conductance” argument [ 2], where the coefficients for the spin current component proportional to n×˙nand the enhancement of the Gilbert damping constant are governed bythe same quantity (the real part of a spin mixing conductance)does not hold for the insulator case. In fact, our result indicatesthat the spin current component proportional to n×˙narises from the second-order correction to the interaction (the seconddiagram of Fig. 9), while the damping correction arises from the first-order process (the first diagram of Fig. 9). Although the magnitudes of the two effects happen to be both secondorder of the interface spin splitting, M I, the physical origins appear to be distinct. From our analysis, we see that the spinmixing conductance description is not general and applies onlyto the case of a thick metallic ferromagnet (see Sec. VA for the metallic case). VIII. DISCUSSION Our results are summarized in Table II. Let us discuss experimental results in the light of our results. In the early fer-romagnetic resonance (FMR) experiments, consistent studiesofgfactor and the Gilbert damping were carried out on metallic ferromagnets [ 12]. The results appear to be consistent with theories (Refs. [ 2,11] and the present paper). Both the damping constant and the gfactor have 1 /ddependence on the thickness of the ferromagnet in the range of 2 nm <d< 10 nm [ 12]. The maximum additional damping reaches δα∼0.1a td=2 nm, which exceeds the original value of α∼0.01. The g-factor modulation is about 1% at d=2 nm, and its sign depends on the material; the gfactor increases for Pd/Py/Pd and Pt/Py/Pt, while decreases for Ta/Py/Ta. These results appear consistentwith ours, because δω Bis governed by Im T+−, whose sign depends on the sign of interface spin-orbit interaction. Incontrast, damping enhancement proportional to Re T+−is positive for thick metals. However, other possibilities like theeffect of a large interface orbital moment playing a role in thegfactor, cannot be ruled out at present. Recently, inverse spin Hall measurement has become com- mon for detecting the spin current. In this method, however,only the dc component proportional to n×˙nis accessible so far and there remains an ambiguity for qualitative estimatesbecause another phenomenological parameter, the conversionefficiency from spin to charge, enters. Qualitatively, the valuesofA robtained by the inverse spin Hall measurements [ 44] and FMR measurements are consistent with each other. The cases of insulating ferromagnets have been studied recently. In the early experiments, orders of magnitudesmaller values of A rcompared to metallic cases were re- ported [ 43], while those small values are now understood as due to poor interface quality. In fact, FMR measure-ments on epitaxially grown samples like yttrium iron garnet 064423-16CONSISTENT MICROSCOPIC ANALYSIS OF SPIN . . . PHYSICAL REVIEW B 96, 064423 (2017) TABLE II. Summary of essential parameters determining the spin current js, corrections to the Gilbert damping δα, and the resonance frequency shift δωBfor metallic and insulating ferromagnets. Coefficients AiandArare for the spin current, defined by Eq. ( 1). Label “ −” indicates that it is not discussed in the present paper. “∗” is for the strong spin relaxation case, the density of states νis replaced by the inverse of electron spin-flip time τsf[43]. Ferromagnet (F) Ai Ar δα δω B Assumption Equations ReT+− ImT+− Thick F ( 27)(63)(83)(84) Metal Im T+− ReT+− ImT+− ReT+− Thin F ( 88) Insulator MIν (MIν)2(MIν)2(MIν)3Weak spin relaxation∗(111)(130) –( MIν)2/summationtext q(1+2nq) – – Magnon ( 122) (Y3Fe5O12,YIG)/Au/Fe turned out to show Arof 1–5 × 1018m−2(Refs. [ 45,46]), which is the same order as in the metallic cases. Inverse spin Hall measurements on YIG/Ptreport similar values [ 47], and the value is consistent with the first-principles calculation [ 48]. Systematic studies of YIG/NM with NM =Pt, Ta, W, Au, Ag, Cu, Ti, V , Cr, Mn, etc., were carried out with the result of A r∼1017–1018m−2 (Refs. [ 49–52]). If we use a naive phenomenological relation, Eq. ( 6),Ar=1018m−2corresponds to δα=3×10−4if a=2˚A,S=1, and d=20˚A. Assuming interface sdex- change interaction, the value indicates MIν∼0.01, which appears reasonable at least by the order of magnitude fromthe result of x-ray magnetic circular dichroism (XMCD)suggesting spin polarization of interface Pt of 0 .05μ Bwithin a proximity length of less than 1 nm [ 53]. A recent experiment indicates that the spin pumping effect of an insulator is inducedlocally in the normal metal as a result of the magnetic proximityeffect [ 8], supporting our perturbative treatment. On the other hand, FMR frequency shift of insulators cannot be explained by our theory. In fact, the shift for YIG/Pt isδω B/ωB∼1.6×10−2, which is larger than δα∼2×10−3, while our perturbation theory assuming weak interface sd interaction predicts δωB/ωB<δ α . We expect that the discrep- ancy arises from the interface spin-orbit interaction that wouldbe present at the insulator-metal interface, which modifies themagnetic proximity effect and damping torque significantly. Itwould be necessary to introduce an anomalous sdcoupling at the interface like the one discussed in Ref. [ 54]. Experimen- tally, the influence of interface spin-orbit interaction [ 55] and proximity effect needs to be carefully characterized by using amicroscopic technique such as MCD to compare with theories. IX. SUMMARY We have presented a microscopic study of spin pumping effects, the generation of spin current in a ferromagnet-normal metal junction by magnetization dynamics, for bothmetallic and insulating ferromagnets. As for the case of ametallic ferromagnet, a simple quantum mechanical picturewas developed using a unitary transformation to diagonalizethe time-dependent sdexchange interaction. The problem of dynamic magnetization is thereby mapped to the one withstatic magnetization and off-diagonal spin gauge field, whichmixes the electron spin. In the slowly varying limit, the spingauge field becomes static, and the conventional spin pumpingformula is derived simply by evaluating the spin accumulationformed in the normal metal as a result of interface hopping. The effect of interface spin-orbit interaction was discussed.A rigorous field theoretical derivation was also presented,and the enhancement of spin damping (Gilbert damping)in the ferromagnet as a result of spin pumping effect wasdiscussed. The case of an insulating ferromagnet was studiedbased on a model where the spin current is driven locallyby the interface exchange interaction as a result of magneticproximity effect. The dominant contribution turns out to be theone proportional to ˙n, while the magnon contribution leads to n×˙n, whose amplitude depends linearly on the temperature. Our analysis clearly demonstrates the difference in the spincurrent generation mechanism for metallic and insulatingferromagnets. The influence of atomic-scale interface structureon the spin pumping effect is an open and urgent issue, inparticular for the case of ferrimagnetic insulators which havetwo sublattice magnetic moments. ACKNOWLEDGMENTS G.T. thanks H. Kohno, C. Uchiyama, K. Hashimoto, and A. Shitade for valuable discussions. S.M. thanks the Centerfor Spintronics Network (CSRN) for supporting collaborationworks. This work was supported by a Grant-in-Aid forExploratory Research (Grant No.16K13853) and a Grant-in-Aid for Scientific Research (B) (Grant No. 17H02929) fromthe Japan Society for the Promotion of Science, and a Grant-in-Aid for Scientific Research on Innovative Areas (Grant No.26103006) from The Ministry of Education, Culture, Sports,Science and Technology (MEXT), Japan. A±s,t ˜s(F) ±gr∓ ga± FIG. 11. Feynman diagram for electron spin density of ferromag- net induced by magnetization dynamics (represented by spin gauge fieldAs) neglecting the effect of normal metal. The amplitude is essentially given by the spin-flip correlation function χ±[Eq. ( A3)]. 064423-17GEN TATARA AND SHIGEMI MIZUKAMI PHYSICAL REVIEW B 96, 064423 (2017) APPENDIX A: SPIN DENSITY INDUCED BY MAGNETIZATION DYNAMICS IN F Let us here calculate the spin density in a ferromagnet induced by magnetization dynamics neglecting the effect of interface, HI. (Effects of HIare discussed in Sec. V.) In the rotated frame, the spin density in F pumped by the spin gauge field is therefore (diagrams shown in Fig. 11) ˜s(F) α(k,k/prime)≡−i/integraldisplaydω 2πtr[σαδG<(k,k/prime,ω)] =−i/integraldisplaydω 2π/summationdisplay k/prime/prime(fk/prime/prime+−fk/prime/prime−)/summationdisplay ±(±)A± s,ttr[σαgr(k,k/prime/prime,ω)σ∓ga(k/prime/prime,k/prime,ω)] =/braceleftBigg ∓i/integraltextdω 2π/summationtext k/prime/prime(fk/prime/prime+−fk/prime/prime−)A± s,tgr ∓(k,k/prime/prime,ω)ga ±(k/prime/prime,k/prime,ω)(α=±) 0( α=z). (A1) Let us here neglect the effects of interface in discussing the spin polarization of F electrons, then the Green’s functions aretranslationally invariant, i.e., g a(k,k/prime)=δk,k/primega(k)(a=r,a). Using the explicit form of the free Green’s function, ga σ(k,ω)= 1 ω−/epsilon1k,σ−i0, and /integraldisplaydω 2πgr ∓(k,k/prime/prime,ω)ga ±(k/prime/prime,k/prime,ω)=i /epsilon1k,±−/epsilon1k,∓+i0,(A2) the spin density in the rotated frame then reduces to ˜s(F) ±(k)=−A± s,tχ±, (A3) where χ±≡−/summationdisplay kfk,±−fk,∓ /epsilon1k,±−/epsilon1k,∓+i0(A4) is the spin correlation function with spin flip, +i0 meaning an infinitesimal positive imaginary part. Since we focus onthe adiabatic limit and spatially uniform magnetization, thecorrelation function is at zero momentum and frequencytransfer. We thus easily see that χ ±=n+−n− 2M, (A5) where n±=/summationtext kfk±is the spin-resolved electron density. The spin polarization of Eq. ( A3) in the rotated frame is proportional to A⊥ s,t, and represents a renormalization of total spin in F. In fact, it corresponds in the laboratory frame to s(F)∝n×˙n, and exerts a torque proportional to ˙nonn. It may appear from Eq. ( A5) that a damping of spin, i.e., a torque proportional to n×˙n, arises when the imaginary part for the Green’s function becomes finite, because1 Mis replaced by1 M∓iηi, where ηiis the imaginary part. This is not always the case. For example, nonmagnetic impurities introduce a finite imaginary part inversely proportional to the elastic lifetime(τ), i 2τ. They should not, however, cause damping of spin. The solution to this apparent controversy is that Eq. ( A1)i s not enough to discuss damping even including lifetime. Infact, there is an additional process called vertex correctioncontributing to the lesser Green’s function, and it gives riseto the same order of small correction as the lifetime does,and the sum of the two contributions vanishes. Similarly,we expect damping does not arise from the spin-conservingcomponent of spin gauge field, A z s,t. This is indeed true as we explicitly demonstrate in Appendix B. We shall show inSec. Vthat damping arises from the spin-flip components of the self-energy. APPENDIX B: EFFECT OF SPIN-CONSERVING SPIN GAUGE FIELD ON SPIN DENSITY Here we calculate the contribution of spin-conserving spin gauge field, Az s,t, on the interface effects of spin density in F. It turns out that a spin-conserving spin gauge field combinedwith interface effects does not induce damping. This result isconsistent with a naive expectation that only the nonadiabaticcomponents of spin current should contribute to damping. The contribution to the lesser Green’s function in F from the interface hopping (lowest, the second order in the hopping) atthe linear order in the spin gauge field reads (diagramaticallyshown in Fig. 12) δG <=δG< (a)+δG< (b)+δG< (c), δG< (a)=gr(As,t·σ)gr/Sigma1r 0g<+gr(As,t·σ)gr/Sigma1< 0ga +gr(As,t·σ)g</Sigma1a 0ga+g<(As,t·σ)ga/Sigma1a 0ga, δG< (b)=gr/Sigma1r 0gr(As,t·σ)g<+gr/Sigma1< 0g<(As,t·σ)ga +gr/Sigma1a 0ga(As,t·σ)ga+g</Sigma1a 0ga(As,t·σ)ga, δG< (c)=gr/Sigma1rg<+gr/Sigma1<ga+g</Sigma1aga. (B1) Here, /Sigma1a≡˜tU−1ga NU˜t†(a=a,r,<), /Sigma1a 0≡˜tga N (B2) are the self-energy due to the interface hopping, where /Sigma1a is the full self-energy including the time-dependent unitary matrix U, which includes the spin gauge field. /Sigma1a 0is the As,t t t∗N (a)As,tt∗t N (b)tU−1 N Ut∗ (c) FIG. 12. Diagrammatic representation of the contribution to the lesser Green’s function for F electron arising from the interface hopping (represented by tandt∗) and spin gauge field ( As,t). The diagram (c) includes the spin gauge field implicitly in unitary matricesUandU −1. 064423-18CONSISTENT MICROSCOPIC ANALYSIS OF SPIN . . . PHYSICAL REVIEW B 96, 064423 (2017) contribution of /Sigma1awith the spin gauge field neglected. We here focus on the contribution of the adiabatic ( z) component, Az s,t. Using g<=F(ga−gr)f o rF( Fis a 2×2 matrix of the spin-polarized Fermi distribution function) and g< N=fN(ga N−gr N) and noting that all the angular frequencies of the Green’s function are equal, we obtain δG< (a)+δG< (b)/similarequalAz s,tσz/braceleftbig −2F/bracketleftbig (gr)3/Sigma1r 0−(ga)3/Sigma1a 0/bracketrightbig −(F−fN)[(gr)2ga+gr(ga)2]/parenleftbig /Sigma1a 0−/Sigma1r 0/parenrightbig/bracerightbig . (B3) The contribution δG< (c)is calculated noting that ˜tU−1ga NU˜t†=ga N˜t˜t†−dga N dω˜t(As,t·σ)˜t†+O((As,t)2). (B4) The linear contribution with respect to the zcomponent of the gauge field turns out to be δG< (c)/similarequalAz s,tσz/braceleftbigg F/bracketleftbigg (gr)2∂ ∂ω/Sigma1r 0−(ga)2∂ ∂ω/Sigma1a 0/bracketrightbigg +(F−fN)grga∂ ∂ω/parenleftbig /Sigma1a 0−/Sigma1r 0/parenrightbig/bracerightbigg . (B5) We therefore obtain the effect of spin-conserving gauge field as δG<=Az s,tσz∂ ∂ω/braceleftbigg F/bracketleftbig (gr)2/Sigma1r 0−(ga)2/Sigma1a 0/bracketrightbig +(F−fN)grga/parenleftbig /Sigma1a 0−/Sigma1r 0/parenrightbig/bracerightbigg , (B6) which vanishes after integration over ω. Therefore the contribution from the spin-conserving gauge field and interface hopping vanishes in the spin density, leaving the damping unaffected. APPENDIX C: DERIV ATION OF EQ. ( 72) We show here the details of the calculation of the induced spin density in the ferromagnetic metal, diagrammatically represented in Fig. 8. Writing the spatial and temporal positions explicitly, the self-energy of F electrons arising from the hopping to N region reads ( r1andr2a r ei nF ) /Sigma1a(r1,r2,t1,t2)=/integraldisplay INd3r/prime 1/integraldisplay INd3r/prime 2˜t(r1,r/prime 1)U−1(t1)ga N(r/prime 1,r/prime 2,t1−t2)U(t2)˜t†(r2,r/prime 2), (C1) where a=r,a,<. We assume the Green’s function in N region is spin-independent, i.e., we neglect higher-order contributions of hopping. Moreover, we treat the hopping to occur only at the interface, i.e., at x=0. The self-energy is then represented as /Sigma1a(r1,r2,t1,t2)=a2δ(x1)δ(x2)˜tU−1(t1)U(t2)˜t†/summationdisplay kga N(k,t1−t2), (C2) where ais the interface thickness, which we assume to be the order of the lattice constant. The diagrammatic representations of Eqs. ( 68) and ( C1) are in Fig. 8. Expanding the matrix using a spin gauge field as U−1(t1)U(t2)=1−i(t1−t2)As,t+O((As,t)2), we obtain the gauge field contribution of the self-energy as /Sigma1a(r1,r2,t1,t2)=a2δ(x1)δ(x2)/integraldisplaydω 2πde−iω(t1−t2) dω˜tAs,t˜t†/summationdisplay kga N(k,ω) =−a2δ(x1)δ(x2)/integraldisplaydω 2πe−iω(t1−t2)˜tAs,t˜t†/summationdisplay kd dωga N(k,ω). (C3) The linear contribution of the lesser component of the off-diagonal self-energy is G<(r,t,r/prime,t)=gr/Sigma1rga+gr/Sigma1<ga+g</Sigma1aga =a2/integraldisplaydω 2π/summationdisplay k/bracketleftbigg gr(r,ω)dgr N(k,ω) dω˜tAs,t˜t†g<(−r,ω) +gr(r,ω)dg< N(k,ω) dω˜tAs,t˜t†ga(−r,ω)+g<(r,ω)dga N(k,ω) dω˜tAs,t˜t†ga(−r,ω)/bracketrightbigg . (C4) For a finite distance from the interface r, the dominant contribution arises from the terms containing both gr(r,ω) andga(−r,ω), as they do not contain a rapid oscillation like ei(kF++kF−)rande2ikFσr. Using an approximation/summationtext kgr N(k,ω)∼−iπν Nand partial integration with respect to ω,E q .( C4) finally reduces to Eq. ( 72). 064423-19GEN TATARA AND SHIGEMI MIZUKAMI PHYSICAL REVIEW B 96, 064423 (2017) APPENDIX D: MAGNON REPRESENTATION OF SPIN BERRY’S PHASE TERM Here we derive the expression for the spin Berry’s phase term of the Lagrangian ( 94) in terms of a magnon operator. The time integral of the term is written by introducing an artificial variable uas [56] /integraldisplay dtL B=S/integraldisplay dt˙φ(cosθ−1)=S−2/integraldisplay dt/integraldisplay1 0duS·(∂tS×∂uS), (D1) where S(t,u) is extended to a function of tandu, but only S(t,u=1) is physical. Noting that the unitary transformation matrix element of Eq. ( 96) is written as Uij=(ej)i, (D2) where r1≡eθ,e2≡eφande3≡n, we obtain S·(∂tS×∂uS)=/tildewideS·[(∂t+iAU,t)/tildewideS×(∂u+iAU,u)/tildewideS)]. (D3) Evaluating to the second order in the magnon operators, we have ∂t/tildewideS×∂u/tildewideS=2iγˆz[(∂ub†)(∂tb)−(∂tb†)(∂ub)]. (D4) Using the explicit form of AU,μ, the gauge field contribution is ∂u/tildewideS·[/tildewideS×iAU,t/tildewideS)]=S2γ[(∂ub†)(−sinθ˙φ+i˙θ)+(∂ub)(−sinθ˙φ−i˙θ)]−2Sγ2cosθ(∂tφ)∂u(b†b). (D5) The terms linear in the boson operators vanish by the equation of motion, and the second-order contribution is S·(∂tS×∂uS)=2Sγ2{i∂u[b†(∂tb)−(∂tb†)b]−∂u[cosθ(∂tφ)b†b]+∂t[cosθ(∂uφ)b†b] +sinθ((∂tθ)(∂uφ)−(∂uθ)(∂tφ))b†b}. (D6) Integrating over tandu, the total derivative with respect to tof Eq. ( D6) vanishes, resulting in /integraldisplay dt/integraldisplay1 0duS·(∂tS×∂uS)=2Sγ2/integraldisplay dt{i[b†(∂tb)−(∂tb†)b]−cosθ(∂tφ)b†b+sinθ((∂tθ)(∂uφ)−(∂uθ)(∂tφ))b†b}.(D7) The last term of Eq. ( D7) represents the renormalization of spin Berry’s phase term, i.e., the effect S→S−b†b, which we neglect below. The Lagrangian for magnons thus reads Lm=2Sγ2/integraldisplay d3ri/bracketleftbig b†/parenleftbig ∂t+iAz s,t/parenrightbig b−b†/parenleftbig← ∂t−iAz s,t/parenrightbig/parenrightbig b/bracketrightbig , (D8) namely, magnons interacts with the adiabatic component of the spin gauge field, Az s,t. APPENDIX E: DERIV ATION OF EQS. ( 120) AND ( 121) For the self-energy type of the Green’s functions, depending on two times as g(t1−t2)D(t1−t2)[ E q .( 117)], the real-time components are written as (suppressing time and suffix of N) (see Appendix F) [g(t1−t2)D(t1−t2)]r=grD<+g>Dr=g<Dr+grD>, [g(t1−t2)D(t1−t2)]a=gaD>+g<Da=gaD<+g>Da, [g(t1−t2)D(t1−t2)]<=g<D<. (E1) The Green’s function /tildewideDis that of a composite field Bαdefined in Eq. ( 115), and is decomposed to the elementary magnon Green’s function Das /tildewideDγ(r1,t1,r2,t2)=[/Phi1†(t1)×/Phi1(t2)]γD(r1,t1,r2,t2)−[/Phi1†(t2)×/Phi1(t1)]γD(r2,t2,r1,t1), (E2) where D(r1,t1,r2,t2)≡−i/angbracketleftTCb(r1,t1)b†(r2,t2)/angbracketright. (E3) The spin-dependent factor in Eq. ( E2) is calculated for adiabatic dynamics as /Phi1†(t1)×/Phi1(t2)=2in(t1)+(t2−t1)[/Psi1+i˙n]+O((∂t)2), (E4) where /Psi1≡2 cosθ˙φn+n×˙n. (E5) 064423-20CONSISTENT MICROSCOPIC ANALYSIS OF SPIN . . . PHYSICAL REVIEW B 96, 064423 (2017) The real-time Green’s functions are therefore [ D(1,2)≡D(r1,t1,r2,t2)] /tildewideD< γ(r1,t1,r2,t2)=2in(t1)[D<(r1,t1,r2,t2)−D>(r2,t2,r1,t1)]+(t2−t1){/Psi1[D<(r1,t1,r2,t2)+D>(r2,t2,r1,t1)] +i˙n[D<(r1,t1,r2,t2)−D>(r2,t2,r1,t1)]}, (E6) /tildewideDr γ(1,2)=θ(t1−t2)/parenleftbig/tildewideD< γ(1,2)−/tildewideD> γ(1,2)/parenrightbig ,/tildewideDa γ(1,2)=−θ(t2−t1)/epsilon1αβγ(D< αβ(1,2)−D> αβ(1,2)), and/tildewideD< γis obtained by exchanging <and>in/tildewideD< γ. Elementary Green’s functions are calculated as D<(r1,t1,r2,t2)=−i/summationdisplay qeiq·(r1−r2)nqe−iωq(t1−t2),D>(r1,t1,r2,t2)=−i/summationdisplay qeiq·(r1−r2)(nq+1)e−iωq(t1−t2), (E7) where ωqis the magnon energy and nq≡1 eβωq−1. In our model, the interface is atomically flat and has an infinite area, and thus ri(i=1,2) are at x=0. The Fourier components, defined as ( a=r,a,<,> ) /tildewideDa γ(x1=0,t1,x2=0,t2)≡/summationdisplay q/integraldisplayd/Omega1 2πe−i/Omega1(t1−t2)/tildewideDa γ(q,/Omega1), (E8) are calculated from Eq. ( E6)a s /tildewideD< γ(q,/Omega1)=−i/braceleftbigg 2n(D< −−D> +)+d d/Omega1/bracketleftbig /Psi1(D< −+D> +)+i˙n(D< −−D> +)/bracketrightbig/bracerightbigg , /tildewideDr γ(q,/Omega1)=−i/braceleftbigg 2n(Dr −+Dr +)+d d/Omega1/bracketleftbig /Psi1(Dr −−Dr +)+i˙n(Dr −+Dr +)/bracketrightbig/bracerightbigg , (E9) /tildewideDa γ(q,/Omega1)=−i/braceleftbigg 2n(Da −+Da +)+d d/Omega1/bracketleftbig /Psi1(Da −−Da +)+i˙n(Da −+Da +)/bracketrightbig/bracerightbigg , where Da ±≡1 /Omega1±ωq−i0,Dr ±≡1 /Omega1±ωq+i0 D< −≡nq(Da −−Dr −),D> +≡(1+nq)(Da +−Dr +). (E10) The spin part of the Green’s function, Eq. ( 118), is G< N,γ(r,t,r/prime,t)=−SJ2 I 2/integraldisplaydω 2π/integraldisplayd/Omega1 2π/summationdisplay kk/prime/summationdisplay k/prime/primeq/bracketleftbig gr N,kω/parenleftbig/tildewideDr γ(q,/Omega1)g> N,k/prime/prime,ω−/Omega1+/tildewideD< γ(q,/Omega1)gr N,k/prime/prime,ω−/Omega1/parenrightbig g< N,k/primeω +gr N,kω/tildewideDr γ(q,/Omega1)g> N,k/prime/prime,ω−/Omega1ga N,k/primeω+g< N,kω/parenleftbig/tildewideDa γ(q,/Omega1)g> N,k/prime/prime,ω−/Omega1+/tildewideD< γ(q,/Omega1)ga N,k/prime/prime,ω−/Omega1/parenrightbig ga N,k/primeω/bracketrightbig .(E11) The contribution surviving at long distance is the one containing gr N,ω(r) andga N,ω(−r), obtaining Eq. ( 120), i.e., G< N,γ(r,t,r/prime,t)/similarequal/integraldisplaydω 2π/summationdisplay kk/primegr N,kωga N,k/primeωeik·re−ik/prime·r/prime/tildewide/Sigma1I,γ, where /tildewide/Sigma1I,γ≡−SJ2 I 2/integraldisplayd/Omega1 2π/summationdisplay k/prime/primeq/bracketleftbig/parenleftbig fk/prime/tildewideDr γ(q,/Omega1)−fk/tildewideDa γ(q,/Omega1)/parenrightbig (fk/prime/prime−1)/parenleftbig ga N,k/prime/prime,ω−/Omega1−gr N,k/prime/prime,ω−/Omega1/parenrightbig +/tildewideD< γ(q,/Omega1)/parenleftbig fk/primegr N,k/prime/prime,ω−/Omega1−fkga N,k/prime/prime,ω−/Omega1+fk/prime/prime/parenleftbig ga N,k/prime/prime,ω−/Omega1−gr N,k/prime/prime,ω−/Omega1/parenrightbig/parenrightbig/bracketrightbig . (E12) We focus on the pumped contribution, containing a derivative with respect to /Omega1in Eq. ( E9). The result is, using partial integration with respect to /Omega1(/tildewide/Sigma1Iis a vector representation of /tildewide/Sigma1I,γ), /tildewide/Sigma1I/similarequal−iSJ2 I 2/integraldisplayd/Omega1 2π/summationdisplay k/prime/primeq/braceleftbigg (fk/prime/prime−1)d d/Omega1/parenleftbig ga N,k/prime/prime,ω−/Omega1−gr N,k/prime/prime,ω−/Omega1/parenrightbig (fk/prime[/Psi1(Dr −−Dr +)+i˙n(Dr −+Dr +)] −fk[/Psi1(Da −−Da +)+i˙n(Da −+Da +)])+[/Psi1(D< −+D> +) +i˙n(D< −−D> +)]d d/Omega1/parenleftbig (fk/prime/prime−fk)ga N,k/prime/prime,ω−/Omega1−(fk/prime/prime−fk/prime)gr N,k/prime/prime,ω−/Omega1/parenrightbig/bracerightbigg . (E13) 064423-21GEN TATARA AND SHIGEMI MIZUKAMI PHYSICAL REVIEW B 96, 064423 (2017) Usingd d/Omega1ga k/prime/prime,ω−/Omega1=(ga k/prime/prime,ω)2+O(/Omega1) and an approximation, we obtain/summationtext k/prime/prime(ga k/prime/prime,ω)2/similarequal−πiν 2/epsilon1F, /tildewide/Sigma1I/similarequalπν /epsilon1FSJ2 I 2/integraldisplayd/Omega1 2π/summationdisplay qk/prime/prime/parenleftbigg /Psi1/braceleftbigg (fk/prime/prime−1)[fk/prime(Dr −−Dr +)−fk(Da −−Da +)]+1 2(2fk/prime/prime−fk−fk/prime)(D< −+D> +)/bracerightbigg +i˙n/braceleftbigg (fk/prime/prime−1)[fk/prime(Dr −+Dr +)−fk(Da −+Da +)]+1 2(2fk/prime/prime−fk−fk/prime)(D< −−D> +)/bracerightbigg/parenrightbigg . (E14) As argued for Eq. ( 106), only the imaginary part of self-energy contributes to the induced spin current, as the real part, the shift of the chemical potential, is compensated by redistribution of electrons. We therefore obtain Eq. ( 121). We further note that the component of /Psi1proportional to n[Eq. 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PhysRevApplied.11.044028.pdf

PHYSICAL REVIEW APPLIED 11,044028 (2019) Magnetization Dynamics Induced by Nanoconfined Magnetic-Field Pulse Generated by Resonant Plasmonic Nanoantennas B.C. Choi* Department of Physics and Astronomy, University of Victoria, Victoria V8P 5C2, Canada (Received 16 January 2019; revised manuscript received 14 March 2019; published 10 April 2019) Finite-difference time domain (FDTD) calculations reveal that resonant plasmonic nanoantennas are capable of generating intense magnetic field pulses in the midinfrared frequency region. The magnetic field pulse generated by nanoantennas is spatially concentrated within a nanoscale region, with its intensityincreased by more than two orders of magnitude compared to that of the incident light. Given the highly localized confinement of the intense magnetic field, the nanoantenna can be used as a nanoscale source of magnetic field pulse at optical frequencies, which can locally manipulate the magnetism within a very shorttime scale. Micromagnetic numerical results demonstrate the excitation of the magnetization oscillation in a ferromagnetic nanoelement, which is coherently coupled to the magnetic field. After the termination of magnetization oscillation, propagating spin waves emerge from the excitation magnetic field region and travel perpendicular to the static magnetization direction. The result demonstrates the potential of resonant plasmonic nanoantennas in optically triggering magnetization dynamics and subsequently generating spinwaves in the GHz frequency domain. The result opens up an interesting perspective for applying plasmonic nanoantennas in the ultrafast optical manipulation of magnetism on the nanometer scale. DOI: 10.1103/PhysRevApplied.11.044028 I. INTRODUCTION The study of nonequilibrium magnetization phenom- ena in magnetic elements has attracted much attention due to the fundamental interest in ultrafast magnetiza- tion dynamics and a variety of potential applications, including high-speed magnetoelectronic devices. Conven- tionally, the magnetization dynamics in magnetic elements have been triggered by applying magnetic field pulses or spin torque [ 1,2]. With the latest advances in laser technology, the possibility of the optical excitation of mag- netism by employing high-power lasers has been exten- sively explored. In their pioneering work of laser-induced ultrafast demagnetization, Beurepaire et al. demonstrated that ferromagnetic (FM) thin films excited by femtosecond laser pulses underwent ultrafast demagnetization within several hundred femtoseconds due to rapid energy trans- fer from thermalized electrons to the spin system [ 3]. The demagnetization was followed by a slow recovery over the picosecond timescale as electrons equilibrate with the phonons, and eventually to complete cooling via nanosec- ond lattice diffusion. The timescale of the laser-induced demagnetization process is orders of magnitude below the limit imposed by conventional switching of magnetic ordervia magnetic field pulse, and has been the focus of con- siderable research since its discovery. The laser-induced ultrafast magnetization process is, however, incoherent in *bchoi@uvic.canature, and it has been challenging to coherently controlmagnetic states. Over the past years, significant research efforts have been devoted to exploring new phenomena in the interac- tion of light with magnetism. Recently, an increasing num- ber of THz magnetization dynamics studies have emerged in which the magnetic field component of optical pulses directly couples to the magnetization via Zeeman cou- pling [ 4]. Kampfrath et al., for example, reported on the coherent control of spin waves in antiferromagnetic NiO with highly intense terahertz pulses [ 5]. Recent work by Shalaby et al. also demonstrated the magnetization dynam- ics in ferromagnets triggered by intense THz magnetic pulses, and found that the excited magnetization oscilla- tion was phase locked to the magnetic field [ 6]. Another rapidly emerging field of research is magneto-plasmonics, which combines both magnetic and plasmonic functional- ities [ 7]. Of particular relevance to information technol- ogy is that the incorporation of plasmonic nanostructures to magnetic systems can lead to a significant enhance- ment of the magneto-optical (MO) response. Recently, V. Bonanni et al . reported the magneto-plasmonic effect in FM nanostructures, in which a strong and tunable correlation between the localized surface plasmons and magneto-optical effect was observed [ 8]. In this study, we explore an alternative venue for optically manipulating magnetism by directly applying a nanoconfined magnetic field pulse generated by plasmonicnanoelements. Plasmonic nanostructures, such as split-ring 2331-7019/19/11(4)/044028(6) 044028-1 © 2019 American Physical SocietyB.C. CHOI PHYS. REV. APPLIED 11,044028 (2019) resonators [ 9–11], dielectric dimers [ 12], and nanoanten- nas [ 13–16], have been widely studied due to their capabil- ities to convert optical radiation to intense magnetic fields. In particular, T. Grosjean et al. introduced an alternative nanoantenna concept, which was based on the diabolo nanostructure and could generate significantly enhanced magnetic field in the optical frequency range [ 14]. In our numerical study, the intense magnetic field pulses in the midinfrared frequency region are generated by employing plasmonic diabolo nanoantennas. The strongly confined magnetic field is used to focus intense magnetic field pulses on a FM nanowire to excite magnetization dynam- ics. It is found that the local perturbation of the magnetic order induced by the direct coupling between the magneti- zation and nanoconfined magnetic field pulses leads to the emission of propagating spin waves. II. PLASMONIC NANOANTENNA FOR OPTICAL MAGNETIC FIELD ENHANCEMENT The properties of a plasmonic nanoantenna combined with FM thin films are studied using finite-difference time domain (FDTD) simulations, in which Maxwell’s equa- tions are numerically solved by iteration over time [ 17]. The hybrid plasmonic-FM structure consists of a pair of Ag nanoantennas, SiO 2spacer layers, and FM thin film. In the modeling, a 10-nm thick Ni film is used as the magnetic medium since the optical effect in nickel-rich Permalloy is very similar to that in Ni in the visible and near-infrared regions [ 18]. The dielectric permittivity values of Ni are taken from Johnson and Christy data [ 19]. As shown in the schematic in Fig. 1(a), the nanoantenna is based on the diabolo structure, which is a pair of metallic triangular nanostructures connected by a junction [ 16]. The bottom nanoantenna is embedded in a SiO 2substrate, whereas the top counterpart is surrounded by air. The geometry ofthe nanoantenna is 240 ×240×100 nm3, with the junc- tion of 50 ×50×100 nm3. The 10-nm thick Ni layer is placed between nanoantennas in which the 5-nm SiO 2 spacer layers electrically insulate the Ni layer from theAg antennas. The maximum magnetic field achievable for the given geometry of nanoantenna is optimized by vary- ing the SiO 2spacer layer thickness at a fixed Ni thickness of 10 nm. The nanoantennas are illuminated with a plane wave propagating downward along the −zdirection with its electric field linearly polarized along the ydirection. At resonance, free electrons in the metallic nanoantennas are collectively excited by the electric field of the incident light, and, consequently, a highly localized magnetic field is generated by the electrical current flowing through the junction. Figure 1(b) shows the spectral response of the normalized magnetic field intensity |BR|2/|Bo|2.BRcorre- sponds to the magnetic field at resonance generated inside the magnetic medium between the nanoantenna junction, whereas Bois the magnetic field of the incident light. It is found that the nanoantennas support a strong magnetic resonance, which occurs in the infrared regime with a cen- tral wavelength of 2.74 µm. At the magnetic resonance, the magnetic field intensity is enhanced by more than two orders of magnitude compared to that of the incident light. Figure 2(a)illustrates the spatial distribution of the nor- malized magnetic field intensity, |BR|2/|Bo|2, calculated inside the magnetic medium under the magnetic resonance condition. A magnetic hotspot, where the intense mag- netic field is spatially concentrated, is generated between nanoantenna junctions. This strong confinement provides an opportunity to focus intense magnetic fields into a small region while overcoming the restriction of the diffraction limit defined by the Rayleigh criterion in optical excita- tions. Figure 2(b) represents the time trace of the normal- ized magnetic field BRas compared to the optical magnetic field component Boof the incident light. It reveals the (a) (b) FIG. 1. (a) Schematic diagram of a sandwich diabolo nanoantenna with w=240 nm, d=240 nm, h=100 nm, and j=50 nm. The layer between diabolo nanostructures is composed of SiO 2(5 nm)/Ni (10 nm)/SiO 2(5 nm). A plane wave is incident from the top with its electric field Ealong the ydirection. The magnetic field is generated in the junction along the xdirection. (b) Spectral response of the normalized magnetic field intensity |BR|2/|Bo|2as a function of the wavelength of incident light. The intensity is calculated inside the magnetic medium between the junctions. The magnetic resonance occurs in the infrared regime, centered at 2.74 µm. 044028-2MAGNETIZATION DYNAMICS INDUCED. . . PHYS. REV. APPLIED 11,044028 (2019) (a) (b) FIG. 2. (a) Spatial distribution of the normalized magnetic field intensity, |BR|2/|Bo|2,i nt h e xyplane inside the magnetic medium. The position of the diabolo nanoantenna is indicated by the dotted line. Intense magnetic field is spatially concentrated between nanoantenna junctions. (b) Temporal profile of the normalized magnetic field, BR/Bo, at resonance. Red curve represents the optical magnetic field component Boof the incident light. characteristic of damped harmonic oscillator with the fre- quency of 108 THz, and its peak magnetic field is increased more than 12 times compared to the optical magnetic field of the incident light (shown as red). It is noteworthy that the duration of the induced magnetic field BRis signifi- cantly longer compared to that of the incident field Bo. This is an intrinsic property of the plasmonic resonator, in which the electric current continues to oscillate in the junction even after the termination of the incident light. This res- onating behavior of the plasmonic nanoantenna provides both the magnetic field enhancement and the elongation of the magnetic pulse duration. Given the strong confine- ment of the intense magnetic field, the nanoantenna can be used as a nanoscale-sized source of magnetic fields, which can locally manipulate the magnetism on ultrashort time scales. In order to explore this potential, the dynamic properties of FM nanoelements in response to the appli- cation of a locally confined magnetic field is numerically investigated. III. MAGNETIZATION EXCITATION BY NANOCONFINED MAGNETIC FIELD PULSE The ultrafast manipulation of the magnetization with magnetic fields and its effect on the magnetization dynamics on a longer time scale are studied using micromagnetic finite-element modeling based on the Lan- dau–Lifshitz–Gilbert equation [ 20,21] dM/dt=− |γ/(1+α2)|(M×Beff)−|(αγ)/ [M S(1+α2)]| ×[M×(M×Beff)]. Here, γis the gyroscopic ratio and αis a phenomeno- logical damping constant. Beffis the total effective field acting on the magnetization M, which mainly includes the applied external field, the exchange interaction, and the demagnetizing field. In modeling, a 10-nm thick Permalloy (Ni 80Fe20) waveguide with lateral dimensionsof 120 nm ×4µm is subdivided into homogeneously mag- netized unit cells of the dimension 5 ×5×5n m3.T h e unit cell size is comparable to the exchange length of Permalloy [ 22]. The material parameters used in the mod- eling are: saturation magnetization ( M S=800 emu/cm3), exchange stiffness ( A=1.05×10−6ergs/cm), and damp- ing constant ( α=0.008). The modeling is carried out at 0 K. A damping boundary condition, in which the damping constant is gradually increased at both ends of the waveg- uide, is applied in order to suppress the reflection of spin waves [ 23]. A uniform bias magnetic field of 400 mT is applied along the ydirection so that the magnetization in the waveguide is aligned perpendicular to the long axis of the magnetic element. In order to excite the magnetiza- tion, the temporally varying magnetic field BR, which has the same temporal profile as that in Fig. 2(b), is applied along the xdirection in the middle of the waveguide. In order to model the nanoconfinement of BR, the magnetic field is focused within the area of 120 ×60 nm2in the middle section of the magnetic waveguide. The position of the localized BRregion is marked with a dashed box, as shown in the inset of Fig. 3(b).BRis assumed to be uni- form throughout the thickness of the magnetic film within the area. The peak magnetic field of 50 mT is used in the modeling, which is experimentally achievable by employ- ing diabolo nanoantennas with laser fluence as low as 0.1 mJ/cm2. From a practical application point of view, it is important to control the plasmonic enhanced electric fieldand heat generation in nanodevices under critical limits in order to avoid an extreme influence of electromigration and heat dissipation from the metallic element to the adjacent SiO 2and FM layers [ 24]. FDTD calculation confirms a considerable electric field enhancement, mainly at the end sides of the nanoantenna elements, up to 8.8 MV/m with a fluence of 0.1 mJ/cm2. This enhanced electric field, how- ever, is much smaller compared to the previously reported field of 100 GV/m, which was estimated in a tunneling gap with dimensions of a few nanometers and did not cause 044028-3B.C. CHOI PHYS. REV. APPLIED 11,044028 (2019) (a) (b) FIG. 3. Temporal responses of out-of-plane magnetization components M z(t) and in-plane components M x(t) for the time intervals: (a) from 0 to 450 fs and (b) from 1 to 800 ps. A mag-netic field pulse is applied along the xdirection at t=0f s .C u r v e s are vertically shifted for comparison. ( inset ) Dashed box marks the position of the localized magnetic near field region, whilethe probing areas P 1and P2are represented with dotted circles with diameters of 20 nm. P2is located 200 nm away from the center of P1. Red and blue curves in (a) and (b) correspond to the local magnetic responses averaged over the areas P1and P2, respectively. damage of nanodevices [ 25]. The plasmonic enhancement of temperature in nanostructures is also calculated using the general expression of heat generation [ 26]. The cal- culation with the fluence value of 0.1 mJ/cm2predicts a maximum temperature of 1030 K at electric hotspots, which is well below the melting temperature of 1235 K for Ag. The conductive heat transfer from Ag nanoantennas to SiO 2and FM layers is investigated using the finite-element analysis modeling software COMSOL MULTIPHYSICS [27]. The resulting temperature distribution indicates that the temperature increase in the magnetic layer is negligible due to the low thermal conductivity of the SiO 2layer [ 28]. Figure 3(a) shows the dynamic responses of the local magnetization M x(t)a n d M z(t) for the first 450 fs after applying BRat time t=0 fs. The response of the magneti- zation is spatially averaged over the areas with a diameter of 20 nm, which are located at P1, that is, within the exci- tation field region, and at P2, that is, 200 nm away from the center of P1along the long axis of the waveguide, respec- tively. An important observation is the drastic difference in the dynamic response between the components M xand M zmeasured over P1, in which the out-of-plane magne- tization component M zexhibits a large change whereas the response of the in-plane component M xis insignifi- cant. The change of the M ycomponent is also very small and is not shown. The dominance of the M zcomponent in the magnetization dynamics is attributed to the fieldconfiguration of M⊥B R, in which the magnetization vec- torMexperiences the Zeeman torque ( M×BR) that leads to a significant out-of-plane magnetization contribution to the dynamics. A distinct feature found in M z(t) is that themagnetization oscillation with the frequency of approxi- mately 100 THz is directly coupled to the driving magnetic field. The magnetization oscillates coherently with BR, and the oscillation amplitude decays with decreasing field amplitude of BR. This result is in a qualitative agreement with the previous report by Shalaby et al. in which a THz- induced coherent oscillation of the magnetization in 15-nm thick Co film was observed [ 29]. The direct coupling of the magnetization dynamics with BRis further corrobo- rated by comparing the magnetic response averaged over the probing area P2, which is located outside the spatially confined magnetic field. As expected, no measurable mag- netic response is found outside the BRregion due to the absence of the Zeeman torque. Interesting features of the magnetization dynamics appear after the near complete relaxation of the BRfield- driven coherent magnetization oscillation. Figure 3(b) shows the temporal changes of M xand M zcomponents, averaged over P1and P2, for longer time scales t>1p s .I n contrast to the BRfield-induced magnetization dynamics shown in Fig. 3(a), dynamic components of the magneti- zation appear not only within the BRexcitation region, but also in the adjacent region. Moreover, oscillatory behav- iors become noticeable in both the M xand M zcomponents. This implies that the BR-induced perturbation in the mag- netic order within the P1region acts as the source of the delayed magnetic response in the form of magnetization precession in the vicinity of the excitation field region. From a microscopic point of view, the observation of the magnetization precession outside the BRfield region can be interpreted to be the result of the excitation of propagat- ing spin waves, in which the waves travel perpendicular to the direction of the static magnetization. The spin waves excited in this static magnetization-magnetic field con- figuration correspond to the Damon-Eshbach (DE) mode [30]. The spin wave excitation is attributed to the rapid relaxational process of the magnetization dynamics from BR-induced nonequilibrium state, in which the excess of magnetic energy is transformed into spin waves. In gen- eral, the magnetization relaxation from magnetic nonequi- librium is accompanied by the change of the energy in the magnetic system. It was discussed by Suhl and Safonov that the energy released during relaxation can be accom- modated by spin waves [ 31,32]. Since the modulation of spin waves is mediated by the short-range exchange inter- action of precessing magnetic moments, the characteristic time of such a modulation is given by the precessional fre- quency, which is typically in the range of a few gigahertz in FM materials [ 33]. Further insight into the details of propagating spin waves can be obtained by analyzing the spatiallydependent dynamic components of the magnetization. Figure 4(a)shows the temporal scans of the M xcomponent averaged over the areas of 20 nm in diameter at vari- ous distances from the BRexcitation region. The probing 044028-4MAGNETIZATION DYNAMICS INDUCED. . . PHYS. REV. APPLIED 11,044028 (2019) (a) (b) FIG. 4. (a) Temporal properties of spin waves measured at var- ious probing positions along the long axis of the waveguide. Numbers indicate the distance between BRexcitation and prob- ing regions. A burst of spin waves is launched from the excitationregion and propagates along the waveguide. Curves are vertically shifted for comparison. (b) Frequency spectrum reveals a main peak corresponding the spin wave mode at 20 GHz. The peak at13 GHz is associated with the spin wave mode propagating along the waveguide edge. ( Inset ) The FFT amplitude distributions are shown for the frequencies of 13 and 20 GHz, respectively. positions are equidistantly separated by 200 nm along the long axis of the waveguide. At the probing position at a distance of 200 nm from the center of the BRexcitation region, one observes that a burst of spin waves is emit- ted from the excitation field region and travels away with time. The propagating spin waves lead to the formation of a spin wave packet. The amplitude of spin waves gradu- ally decreases due to the intrinsic damping of the magnetic medium. The spin wave velocity of approximately 1 km/s is estimated from the time delay of the shift of the wave packets probed at different positions. One also observes the formation of the higher-frequency wave packets with very small amplitudes, which can be clearly seen at the positions of 400 and 600 nm. The propagation veloc- ity of these high-frequency spin waves is slightly higher and estimated at approximately 1.2 km/s. These values are in the comparable range as those previously reported spin wave velocities measured in Permalloy microstrips [34]. Figure 4(b) is the result of the fast Fourier trans- form (FFT) analysis of the data shown in Fig. 4(a).T h e frequency spectrum reveals a predominant peak at approx- imately 20 GHz, which corresponds to the major spin wave mode. The images shown in the inset of Fig. 4(b) are the FFT amplitude distributions captured at frequencies of 13 and 20 GHz, respectively. The spin waves at 20 GHz are strongly focused in the middle region of the waveguide. In contrast, the spin wave mode at 13 GHz is localized along the edges. The 13-GHz mode is associated with thespin waves propagating along the waveguide edge, which is attributed to the presence of the narrow waveguiding channels induced by the edge potential wells due to the nonuniform distribution of the internal static magnetic fieldnear the edges [ 35]. The excitation of such an edge mode in submicrometer magnonic waveguides has been previously observed by Xing et al.[36]. IV . CONCLUSIONS We demonstrate that plasmonic nanoantennas are capa- ble of generating intense magnetic fields in the mid-IR region. Since the magnetic fields are highly confined, they are ideally suited as nanoscale-sized sources of magnetic fields, which can effectively manipulate the magnetism within a very short time scale. Micromagnetic numerical results demonstrate the coherent oscillation of the magne- tization upon excitation by nanoconfined magnetic fields,which is followed by the emission of propagating spin waves with a few GHz frequencies. Considering the sig- nificant current challenges of effectively coupling spins in nanoscale magnetic elements to the magnetic component of the electromagnetic fields, the result offers the strong potential of plasmonic nanoantennas in the application of spintronics. ACKNOWLEDGMENTS The author acknowledges funding from the NSERC (Canada) Discovery Grants program. It is a pleasure to acknowledge fruitful discussions with Professor Reuven Gordon. [1] M. Freeman and B. C. Choi, Advances in magnetic microscopy, Science 294, 1484 (2001). [2] J. C. Slonczewski, Current-driven excitation of magnetic multilayers, J. Magn. Magn. Mater. 159, L1 (1996). [3] E. Beurepaire, J. Merle, A. Daunois, and J. Bigot, Ultrafast Spin Dynamics in Ferromagnetic Nickel, P h y s .R e v .L e t t . 76, 4250 (1996). [4] S. Wienholdt, D. Hinzke, and U. Nowak, THz Switching of Antiferromagnets and Ferrimagnets, P h y s .R e v .L e t t . 108, 247207 (2012). [5] T. Kampfrath, A. Sell, G. Klatt, O. Pashkin, S. Mährlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, Coherent terahertz control of antiferromagnetic spinwaves, Nat. Photonics 5, 31 (2010). [6] M. Shalaby, A. Donges, K. Carva, R. Allenspach, P. Oppe- neer, U. Nowak, and C. Hauri, Coherent and incoherentultrafast magnetization dynamics in 3dferromagnets driven by extreme terahertz fields, Phys. Rev. B 98, 014405 (2018). [7] G. Armelles and A. Dmitrev, Focus on magnetoplasmonics, New J. Phys. 16, 045012 (2014). [8] V. Bonanni, S. Bonetti, T. Pakizeh, Z. Pirzadeh, J. Chen, J. Nogués, P. Vavassori, R. Hillenbrand, J. Åkerman, and A. Dmitriev, Designer Magnetoplasmonics with Nickel Nanoferromagnets, Nano Lett. 11, 5333 (2011). 044028-5B.C. CHOI PHYS. REV. APPLIED 11,044028 (2019) [9] H. Y. Feng, F. Luo, L. Henrad, F. Garcia, G. Armelles, and A. Cebollada, Active magnetoplasmonic split-ring/ringnanoantennas, Nanoscale 9, 37 (2017). [10] Y. Yang, H. T. Dai, and X. W. Sun, Split ring aperture for optical magnetic field enhancement by radially polarizedbeam, Opt. Express 21, 6845 (2013). [11] J. Zhou, T. Koschny, M. Kafesaki, E. Economou, J. Pendry, and C. Soukoulis, Saturation of the Magnetic Response ofSplit-Ring Resonators at Optical Frequencies, Phys. Rev. Lett. 95, 223902 (2005). [12] P. Albella, M. Poyli, M. Schmidt, S. Maier, F. Moreno, J. Sanez, and J. Aizpurua, Low-loss electric and magnetic field-enhanced spectroscopy with subwavelength silicon dimers, J. Phys. Chem. C 117, 13573 (2013). [13] V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier, Plasmonic nanoantennas: Fundamentals and their use in controlling the radiative properties of nanoemitters,Chem. Rev. 111, 3888 (2011). [14] T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, Diabolo nanoantenna for enhancing and confiningthe magnetic optical field, Nano Lett. 11, 1009 (2011). [15] P. Muhlschlegel, H. Eisler, O. Martin, B. Hecht, and D. Pohl, Resonant optical antennas, Science 308, 1607 (2005). [16] M. Mivelle, T. Grosjean, G. Burr, U. Fischer, and M. Garcia-Parajo, Strong modification of magnetic dipole emission through diabolo nanoantennas, ACS Photonics 2, 1071 (2015). [17] Lumerical Solutions, www.lumerical.com [18] T. Yoshino and S. Tanaka, Longitudinal magneto-optical effect in Ni and nickel-rich Ni-Fe films in visible and near infrared regions, Opt. Commun. 1, 149 (1969). [19] P. Johnson and R. Christy, Optical constants of the noble metals, Phys. Rev. B 6, 4370 (1972). [20] Q. F. Xiao, J. Rudge, B. C. Choi, Y. K. Hong, and G. Donohoe, Dynamics of ultrafast magnetization reversal in submicron elliptical Permalloy thin film elements, Phys. Rev. B 73, 104425 (2006). [21] LLG Micromagnetic Simulator, www.llgmicro.mindspring. com [22] G. Abo, Y. Hong, J. Park, J. Lee, W. Lee, and B. Choi, Definition of magnetic exchange length, IEEE Trans. Magn. 49, 4937 (2013).[23] S. Bance, T. Schrefl, G. Hrkac, A. Goncharov, D. 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PhysRevB.103.094430.pdf

PHYSICAL REVIEW B 103, 094430 (2021) Spin to charge conversion in Si/Cu/ferromagnet systems investigated by ac inductive measurements Ei Shigematsu,1Lukas Liensberger,2,3Mathias Weiler ,2,3,*Ryo Ohshima ,1Yuichiro Ando,1 Teruya Shinjo,1Hans Huebl ,2,3,4and Masashi Shiraishi1,† 1Department of Electronics Science and Engineering, Kyoto University, 615–8510 Kyoto, Japan 2Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 3Physik-Department, Technische Universität München, 85748 Garching, Germany 4Munich Center for Quantum Science and Technology (MCQST), 80799 München, Germany (Received 17 June 2020; revised 7 December 2020; accepted 2 March 2021; published 19 March 2021) Semiconductor/ferromagnet hybrid systems are attractive platforms for investigation of spin conversion physics, such as the (inverse) spin Hall effect. However, the superimposed rectification currents originating fromanisotropic magnetoresistance have been a serious problem preventing unambiguous detection of dc spin Hallelectric signals in semiconductors. In this study, we applied a microwave frequency inductive technique immuneto such rectification effects to investigate the spin to charge conversion in heterostructures based on Si, oneof the primitive semiconductors. The Si doping dependence of the spin-orbit torque conductivity was obtainedfor the Si/Cu/NiFe trilayer system. A monotonous modulation of the spin-orbit torque conductivity by dopingand relative sign change of spin to charge conversion between the degenerate n-a n d p-type Si samples were observed. These results unveil spin to charge conversion mechanisms in semiconductor/metal heterostructuresand show a pathway for further exploration of spin-conversion physics in metal/semiconductor heterostructures. DOI: 10.1103/PhysRevB.103.094430 I. INTRODUCTION Spin to charge conversion [ 1–3] has been one of the central research topics in spintronics, evoking both scientific interestand expectation for industrial applications. This phenomenonenables an observation of spin current as an electromotiveforce by using the spin-orbit interaction (SOI) and spin-dependent momentum scattering, even though spin currentis not a conservative quantity and one cannot measure it di-rectly. Therefore, spin to charge conversion has been regardedas an important research target in the field of spintronics,and its efficiency factors, i.e., spin Hall conductivity, spinHall angle, and Rashba-Edelstein length, have been identi-fied as crucial indices in spintronic materials. Most reportson spin to charge conversion are limited to metallic materi-als, some of which exhibit high conversion efficiency due totheir large SOI [ 4]. Besides investigations of primitive spin conversion characteristics, control over the spin to chargeconversion properties is also an intriguing research issue. Inthis viewpoint, semiconductors are a promising research field,which unites flourishing spintronic physics with conventionalsemiconductor physics since carrier concentration in semicon-ductors can be modulated by doping and gating. For example,strong SOI in heavily doped semiconductor silicon [ 5] and modulation of the inverse spin Hall effect in GaAs [ 6]w e r e demonstrated. *Present address: Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany. †mshiraishi@kuee.kyoto-u.ac.jpA typical experimental implementation of spin to charge conversion consists of (i) injection of spin current from aferromagnetic material and (ii) spin to charge conversion inan adjacent material. To realize this scheme, spin pumping indetector material/ferromagnet bilayer systems is widely em-ployed, where the detector material is spin Hall active. Variousnonmagnetic [ 2,7,8] or ferromagnetic materials [ 9–11] can be used as a spin detector. Spin pumping is the phenomenonwhich induces spin current flow driven by exciting ferro-magnet resonance (FMR) in the magnetically ordered layer[2,12–14]. Most reports on spin-pumping experiments em- ployed dc detection of spin to charge conversion electromotiveforces with in-plane magnetization of the ferromagnet. Thisexperimental scheme has been the basis for many reports onspintronic properties of various materials such as nonmagneticmetals [ 2,7,8], semimetals [ 15,16], semiconductors [ 5,6], and topological insulators [ 17]. However, an influence of the rec- tification effects in the ferromagnetic metal [ 18–21] hinders precise evaluation of spin to charge-conversion-related dcsignals. Additionally, in some combinations of nonmagneticand ferromagnetic materials, a contribution of the thermo-electric signal caused by spin-wave dynamics gives rise tothermally induced spurious signals [ 22–24]. Complementary to the dc voltage detection technique, a new method whichis immune to the aforementioned spurious signals, the acinductive method, was proposed by Berger et al. [25,26]. In this experimental approach, a static magnetic field is appliedalong the normal of a thin-film nonmagnetic/ferromagneticbilayer and an ac magnetic field is applied using an adjacentmicrostrip line. An ac spin current is injected into the non-magnetic material, which gets converted into an ac electriccurrent via the spin to charge conversion. The generated ac 2469-9950/2021/103(9)/094430(8) 094430-1 ©2021 American Physical SocietyEI SHIGEMATSU et al. PHYSICAL REVIEW B 103, 094430 (2021) TABLE I. Specifications of the silicon wafers regarding the ion implantation: dopant, acceleration voltage, and area dose. According to the targeted doping concentration, we adjusted the acceleration voltage and area dose based on the SRIM simulations. No. Dopant/targeted concentration [cm−3] Dose (10 keV) [cm−2] Dose (15 keV) [cm−2] Dose (30 keV) [cm−2] 1 Phosphorus /1×10201×10145×1014 2 Phosphorus /1×10191×10135×1013 3 Phosphorus /1×10181×10125×1012 4 Nondoped 5 Boron /1×10183×10125×1012 6 Boron /1×10193×10135×1013 7 Boron /1×10203×10145×1014 current causes inductive voltages in the microstrip line, which result in perturbation of the transmission signal. By analyz-ing the transmission signal, the spin-orbit torque conductivity (σ SOT) in, e.g., Pt /Ni80Fe20(Permalloy, Py) and Cu/Py bi- layers can be calculated in a self-consistent way [ 26]. The spin-orbit torque conductivity quantifies the charge conver-sion efficiency starting from the precession dynamics of spinsin the ferromagnet, including all the intermediating processes:spin current generation, spin current transmission through theinterfaces, and spin to charge conversion. We employed this method to investigate the spin to charge conversion physics in semiconductor-metal-ferromagnet hy-brid devices. We thereby chose silicon, the vital material ofmodern electronics, and study the Si doping dependence ofthe spin-orbit torque conductivity of Si/Cu/Py trilayers. II. EXPERIMENT Dopant ions are implanted in commercially available Silicon-on-Insulator (SOI) wafers, which consist of a Si baselayer (nominal resistivity is 1 ∼2/Omega1m) and a 200-nm-thick SiO 2layer, and the top 100-nm-thick Si layer (nominal resis- tivity is 30 ∼40/Omega1m). The acceleration voltage was set to be 10 and 30 keV for phosphorus, and 10 and 15 keV for boron,respectively. Each dose was determined by SRIM (Stoppingand Range of Ions in Matter) simulations beforehand to forma uniform doping profile along the depth direction. The de-tailed recipe of the ion implantation is presented in Table I. The doped wafers were treated in a rapid thermal annealingsystem for activation of the dopants. The measured resistivityof each implanted wafer is shown in Fig. 1. The heavier doping yielded the smaller resistivity in both phosphorus andboron doping. We cut the wafers into chips of 9 ×8m m in size. After removing the natural oxidation layer by 10%hydrogen fluoride (HF) solution, a 3-nm-thick Cu interlayerand a 7-nm-thick Py layer were deposited by an electron-beamdeposition system. The inserted Cu layer prevents the inter-mixing between Si and Py, enabling more qualified interfacepreparation. Cu is also known for being a good conductorof spin current, the spin-diffusion length of which is ca. 500nm [ 27], which allows for a transparent spin current channel between Py and Si. The observed effective magnetization ofthe Py layer is comparable to that of intrinsic Py. After fabrication of the samples, we followed the measure- ment procedure described in the literature [ 25]. As shown in Fig. 2, the ground signal ground (GSG)-type coplanar waveguide (CPW) was connected to the vector network an-alyzer (VNA, N5225B, Keysight Technologies). The sample was placed on the CPW. A dc static magnetic field was appliedperpendicular to the sample plane by an electromagnet. Whilethe rf signal was transmitted from one port of the VNA to ex-cite the FMR of the Py layer, the dc magnetic field was sweptaround the FMR resonance field of Py. The transmission sig-nalS 21was measured at fixed frequency while stepping the dc magnetic field. These experiments were carried out for fixedfrequencies from 10 to 30 GHz. III. MODELING The resonance field of the Py film follows the out-of-plane- type Kittel equation, ω γ=μ0(Hres−Meff). (1) Here, ω,γ,μ0,Hres, and Meffare angular frequency, gyromagnetic ratio, vacuum permeability, resonance mag-netic field, and effective magnetization, respectively. Theaforementioned measurement scheme yields the complextransmission signal ( S 21) as a function of the external dc magnetic field. Under the FMR condition of the Py layer,an ac spin current is injected into the adjacent nonmagneticlayers consisting of the Cu (3 nm) layer and the Si (100 nm)layer. When the ac spin current is converted to an ac chargecurrent in the direction parallel to the CPW, the correspondingcharge carriers give rise to an ac voltage response in the CPW.This can be understood as a change of the inductance of the FIG. 1. Resistivity of the 100-nm-thick Si layers of the implanted SOI wafers probed by four-terminal resistance measurements. 094430-2SPIN TO CHARGE CONVERSION … PHYSICAL REVIEW B 103, 094430 (2021) FIG. 2. Schematic illustration of the setup for the complex trans- mission measurement S21. The CPW is connected with the two ports of the VNA using rf cables. The sample was placed on the CPW facing the Py. In addition, the figure shows the lumped elementcircuit model of the system, where the rf cables, the CPW, and the inductance lump of the sample are connected in series. composite system of the CPW and the sample causing the modulation of S21. By considering the continuity of voltage and current in a lumped element model of the whole systemconsisting of the serially connected rf cables, one can formu-late an equation describing the inductive signal generation.Under the off-resonant condition, the continuity of the voltagebetween point A and point B in Fig. 2gives v i+vr−vt=jωLi. (2) The continuity at the points A and B yields, vi Z0−vr Z0=i, (3a) vt Z0=i, (3b) respectively. Here, vi,vr, and vtare entering, reflecting, and transmitting voltage amplitude (complex value). Landiare the off-resonant inductance and the current in the region be-tween point A and point B. The characteristic impedance ofthe cables and the CPW is nominally 50 /Omega1. After solving these equations, the transmission S 21is expressed as below, S21=vt vi=2Z0 2Z0+jωL. (4) When Lchanges to L+/Delta1Lunder the resonance condition, the perturbation of the transmission signal ( /Delta1S21) should be obtained by the partial derivative and its ratio to the baselinebecomes a simple expression, /Delta1S 21 S21=∂S21 ∂L/Delta1L S21=−jω/Delta1L 2Z0+jωL≈−jω/Delta1L 2Z0. (5) Note that we neglected the relatively small contribution of jωLin the denominator. One may be careful about the dissipation and phase-delay factors through the CPW and thetwo rf cables. These factors, however, are constant in off-and on-resonant states of the ferromagnet and eliminated bydividing by S 21. The change in the inductance /Delta1Lis induced by (i) the ac dipolar magnetic field under the FMR, (ii) the spinto charge conversion current in odd phase, and (iii) the Fara-day effect and the spin to charge conversion current in even phase with time reversal. In Fig. 1(d) in Ref. [ 25], the phase relation among the magnetic amplitude in the ydirection ( m y), the odd and even current ( jSOT o,jSOT e) via the spin to charge conversion, and the Faraday current ( jF e) are shown. Note that jSOT ois at the phase quadrature to that of jSOT eandjF e, hence we can extract the inductance purely from jSOT oby decomposing an entire observed inductance into real and imaginary parts.The three components which contribute to the on-resonantinductance change have the same origin: precession of themagnetization. Therefore, /Delta1Lis proportional to the polder’s susceptibility tensor χ(ω,H), /Delta1L=˜Lχ(ω,H). (6) The complex value, ˜L, is the normalized inductance, rep- resenting the dipolar contribution and the spin to chargeconversion. The value of ˜Lcan be determined by a curve fitting of the S 21spectra as a function of the magnetic field, /Delta1S21 S21≈−jω˜Lχ(ω,H) 2Z0. (7) From the spectrum fitting using the measured values of /Delta1S21 S21(ω,H), one can determine ˜L(ω) and χ(ω,H). The polder’s susceptibility tensor χ(ω,H) contains the resonance field and the linewidth of the spectra, from which the magneticparameters of the Py layer were calculated. We emphasizethat the measurement observable is a frequency- and magneticfield-dependent complex microwave transmission. As suchonly signals in the microwave domain are analyzed and hencemake this technique immune to dc voltage signals, as observedin dc spin pumping and rectification experiments. IV . RESULTS AND DISCUSSION The gfactor and the effective saturation magnetization, μ0Meff, of each sample determined by analyzing the reso- nance field and frequency of the FMR [Fig. 3(a)] are shown in Figs. 3(b) and3(c), respectively. Whereas the deviations ofgfactor in all samples are within 0.5%, a notable de- crease of μ0Meffwas observed for the highly doped samples, which suggests effects of the adjacent conductive layer onthe saturation magnetization. The Gilbert damping constant,α, and the inhomogeneity broadening, μ 0/Delta1H0, were deter- mined by the frequency dependence of the linewidth of thespectrum [Fig. 3(d)], where the measured linewidth equals μ 0/Delta1H0+2αω/γ . As shown in Fig 3(e), the Gilbert damping constant does not show a discernible trend with doping, but itis scattered within 20% range. Only the highly doped p-type sample showed a relatively high μ 0/Delta1H0, as shown in Fig. 3(f). Though some of the magnetic parameters thus exhibit dopingconcentration dependence, the normalization by χ(ω,H)i n Eq. ( 7) accounts for the possible influence of the small modu- lation in the magnetic dynamics on ˜L(ω). Considering the geometry of the CPW, the sample and the spacing between these two components, ˜Lis expressed as [ 25] Re(˜L)=μ 0l 4/bracketleftbiggdFM Wwgη2+η2L21 μ0lMs¯hω eσSOT Re/bracketrightbigg , (8a) Im(˜L)=μ0l 4·η2L21 μ0lMs¯hω eσSOT Im. (8b) 094430-3EI SHIGEMATSU et al. PHYSICAL REVIEW B 103, 094430 (2021) FIG. 3. Magnetic parameters were obtained by the VNA-FMR for each sample with different doping condition of the Si layer. The frequency dependence of the FRM resonance field is shown in (a) with the linear fittings, from which (b) gfactor and (c) the effective magnetization μ0Meffwere determined. The frequency dependence of the linewidth is shown in (d) with the linear fittings, from which (e) Gilbert damping constant and (f) the inhomogeneity broadening of linewidth, μ0/Delta1H0were determined. Here, μ0,¯h,e,Msare vacuum magnetic permeability, the Dirac constant, elementary charge, and the saturation mag-netization of the ferromagnetic film. The geometrical factors:l,W wg,dFM,η,L21, are the length of the sample, the width of the CPW signal line, the thickness of the ferromagneticfilm, the spacing loss factor, and the mutual inductance be-tween the CPW and the sample. The real and imaginaryspin-orbit torque conductivity, σ SOT ReandσSOT Im, originate from the frequency-dependent current generation in the sample.Following Ref. [ 25],σ SOT Re comes from the spin to charge conversion in the even phase and the Faraday current, andσ SOT Im from the spin to charge conversion only in the odd phase. Thus, σSOT Imcorresponds to the dampinglike conversionfrom magnetization dynamics in the ferromagnetic metal to charge currents oscillating at the precession frequency. Bothreal and imaginary parts of ˜Lare linear functions of frequency. Therefore, we can determine σ SOT ReandσSOT Imby linear fitting of˜Lvs frequency. In Figs. 4(a) and4(b), the frequency dependence of the real and imaginary parts of the inductances ˜Lof each sample are shown. The phase-error correction [ 25] by imposing the prerequisite that ˜Lshould be a real-valued number at the zero- frequency limit was already applied here. According to Eqs.(8), the coefficients of linear proportion consist of the geomet- rical parameters, the magnetic properties of the Py film, andmore importantly, σ SOT ReandσSOT Im. Because the geometrical 094430-4SPIN TO CHARGE CONVERSION … PHYSICAL REVIEW B 103, 094430 (2021) FIG. 4. (a) Real and (b) imaginary parts of the normalized inductances as a function of the rf frequency measured with the sample group of different doping conditions. The solid lines are linear fits. From the slopes of these fits, we can calculate the real and imaginary spin-orbittorque conductivities, σ SOT ReandσSOT Im. parameters are in the same range in the measured samples, a rough estimation of σSOT ReandσSOT Imis given by the steepness of the linear slopes of Re( ˜L) and Im( ˜L). To determine the exact value of σSOT ReandσSOT Im, a compre- hensive linear fitting was conducted for Re( ˜L) and Im( ˜L)b y using the geometrical parameters and the effective saturationmagnetization, M eff, obtained from the FMR resonance field, as a saturation magnetization, Ms, appearing in Eqs. ( 8). We note that the spacing dbetween the CPW and the sample changes in each measurement, altering η(l,d) and L21(l,d) defined in Ref. [ 25], but dis analytically determined by the zero-frequency limit of Re( ˜L), which represents the dipolar contribution from the magnetic precession of the Py film. Werepeated the determination process of σ SOT ReandσSOT Imfor the seven samples, with results shown in Figs. 5(a)and5(b). We first focus on the results for the reference sample with nondoped Si. Here, we find σSOT Imof comparable magnitude to that reported in Ref. [ 26] for a Py/Cu(4.5-nm) bilayer. TheσSOT Im for the reference sample may originate from (i) the inverse spin Hall effect (ISHE) in the Cu interlayer and(ii) the self-induced ISHE [ 28] in the Py layer due to a pos- sible imbalance of spin absorption at the top and the bottomsurface, and (iii) sizable spin-orbit torques in a ferromagnetitself [ 29–37]. We assume that this spin charge conversion effect is present in all our samples. To discuss the influenceof doping on spin charge conversion in our Si/Cu/Py trilayers,we then calculate /Delta1σ SOT Re/Im=σSOT Re/Im−σSOT Re/Im(nondoped Si) shown in Figs. 5(c) and5(d). The observed /Delta1σSOT Re/Imare also on the order of 104/Omega1−1m−1(e. g.,/Delta1σSOT Imin the most heavily doped p-type sample), which is in the same magnitude range as Py/Cu systems [ 25]. Using the Pt-based systems, where enhanced spin to charge conversion efficiency is expected,previous studies observed a significantly larger spin-orbit torque conductivity [ 25,26]. We focus on /Delta1σSOT Imoriginating from the spin to charge conversion in the odd phase, i.e., with symmetry of theISHE. The calculated /Delta1σ SOT Im for each measured sample is shown in Fig. 5(d). A decreasing trend of /Delta1σSOT Im with the transition from n-type to p-type doping was observed. We note that /Delta1σSOT Im with the opposite sign relative to that in the nondoped samples was observed in the n-type and p-type samples. The minimum change of /Delta1σSOT Im between them is 1 .4×104/Omega1−1m−1, considering the fit errors. The doping concentration for these two samples, 1 ×1020cm−3, exceeds the effective densities of states of Si in the con-duction band (2 .8×10 19cm−3) and valence band (2 .65× 1019cm−3)[38]. In these doping levels, no depletion layer be- tween Si and Cu is formed at the Si/Cu interfaces in the n-type andp-type samples allowing carriers to transverse through the interface. In this situation [Fig. 6(a) forntype and Fig. 6(b) forptype), the spin current in the Cu layer can travel through the interface between the Cu layer and the degenerate Si. Inthis case, a possible ISHE in Si can contribute to spin chargeconversion. In the spin-scattering process associated with theISHE, the directions of the scattered charge are governed byits spin polarization irrespective of its charge. Therefore, whenthe carrier of the Si layer is switched by change of dopant,the sign of θ SHis also switched. This mechanism can explain the measured σSOT Imin the n-type and p-type samples with the doping concentration of 1 ×1020cm−3. Next, we focus on the nondegenerate Si samples. Considering the work function ofCu (4.5 eV) [ 39] and the electron affinity (4.05 eV) [ 38]o fS i , the ideal band alignment model teaches us the barrier height is0.45 eV . Furthermore, experimental studies reported that the 094430-5EI SHIGEMATSU et al. PHYSICAL REVIEW B 103, 094430 (2021) FIG. 5. (a) Real spin-orbit torque conductivity ( σSOT Re) of the measured samples. (b) Imaginary spin-orbit torque conductivity ( σSOT Im)o f the measured samples. (c) Change of real spin-orbit torque conductivity ( σSOT Re) from that of the nondoped sample. (d) Change of imaginary spin-orbit torque conductivity ( σSOT Im) from that of the nondoped sample. Beneath the xaxis, the dopant type and doping concentration of each sample are described. Fermi level located around the middle of the band gap of Si in a Cu/Si system [ 40]. Hence, a schematic viewgraph at the interface can be described as shown in Fig. 6(c) forn-type Si and Fig. 6(d) forp-type Si. Existence of the depletion layer FIG. 6. Spatial band diagrams of the interfaces between (a) n-type degenerate Si/Cu, (b) p-type degenerate Si/Cu, (c) n-type nondegenerate Si/Cu, (d) p-type nondegenerate Si/Cu. The Fermi level of Si is indicated with dashed lines and the conduction/valenceband of Si is indicated with solid lines. In the degenerate cases, carriers can flow into the Si side, where the ISHE takes place. In the nondegenerate case, carriers are partially blocked at the interface.hinders the Ohmic conduction of spin current through the Cu/Si interface, resulting in the decrease of the spin-mixingconductance, G ↑↓accompanied by a decrease of /Delta1σSOT Im.I n Fig. 5, magnitudes of the /Delta1σSOT Im of the nondegenerate Si samples ( n- and ptype, 1018and 1019cm−3) are smaller than those of the degenerate samples, indicating insufficient ISHEcurrent generation in the Si layer due to the decreased G ↑↓.W e note that a slight but clear shift of /Delta1σSOT Imfrom the baseline of nondoped Si can be seen with the p-type nondegenerate samples, which is attributed to the fact that the SOI in p-Si is stronger than that in n-Si at the same doping concentration as suggested by its band structures. Finally, we comment on doping dependence of σSOT Reshown in Fig. 5(a). According to Ref. [ 25],σSOT Reequals σF e−σSOT e, where σF eis the Faraday conductivity and σSOT eis the spin- orbit torque conductivity, both of which appear in the evenphase expected for spin to charge conversion by the inverseRashba-Edelstein effect (IREE). Because the Faraday currentdensity depends on the total conductivity of the sample, irre-spective of the carrier type, σ F eshould be constant considering the conductance difference of the stack of the 7-nm-thick Pylayer and the 3-nm-thick Cu layer to the 100-nm Si layer.Therefore, the deviations from the baseline of the /Delta1σ SOT Refor the nondoped sample are tentatively attributed to σSOT eby the IREE. In Fig. 5(c), nonzero /Delta1σSOT e within fit error is only observed for the degenerate n-doped sample and could be caused by a Rashba electric field at the Cu/Si interface. For thenondegenerate samples and the degenerate p-doped sample, no significant change of σ SOT e is observed with doping. It is most likely that the Rashba electric field intensity is not 094430-6SPIN TO CHARGE CONVERSION … PHYSICAL REVIEW B 103, 094430 (2021) sufficiently strong in the Cu/Si interface for these samples to induce observable σSOT eby the IREE. V . CONCLUSION In this study, we conducted inductive ac measurements of Si/Cu/Py trilayer samples with different doping con-centrations in Si. The obtained results indicated successfulmodulation of the spin-orbit torque conductivity of theSi/Cu/Py systems by controlling the Si carrier type and dopingconcentration. A doping dependence of σ SOT Im, compatible with the ISHE in the Si, was observed in the transition from n-type top-type doping. In the degenerate Si samples, the relative sign of σSOT Imchanged between n-type and p-type doping. Our results are in qualitative agreement with the doping depen-dence of the formation of the depletion layer and its thickness,and by the impurity scattering rate of carriers. This system-atic study of σ SOT oof Si/Cu/Py systems with various doping concentrations provides insight towards exploration for spincurrent physics of semiconductors and demonstrates the ap- plication of a technique to experimentally determine spin tocharge conversion in ferromagnet/semiconductor hybrids. ACKNOWLEDGMENTS E.S. acknowledges the JSPS Research Fellowship for Young Researchers. E.S. also acknowledges travel supportof Mazume Award (Dept. of Eng., Kyoto University). L.L.and M.W. acknowledge financial support by the Germanresearch foundation (DFG) via Project No. WE 5386/4-1, H.H. acknowledges financial support from the DeutscheForschungsgemeinschaft via Germany’s Excellence StrategyNo. EXC-2111-390814868. This work is partially supportedby a Grant-in-Aid for Scientific Research (S) No. 16H06330,a Grant-in-Aid for Young Scientists (A) No. 16H06089, JSPSKAKENHI Grant No. 17J09520, and Grant-in-Aid for Re-search Activity Start-up No. 20K22413. 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PhysRevB.77.104433.pdf

Origin of the magnetic-field dependence of the nuclear spin-lattice relaxation in iron G. Seewald, E. Zech, and H.-J. Körner Physik-Department, Technische Universität München, D-85748 Garching, Germany D. Borgmann Institut für Physikalische und Theoretische Chemie, Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany M. Dietrich Technische Physik, Universität des Saarlandes, D-66041 Saarbrücken, Germany ISOLDE Collaboration CERN, CH-1211 Geneva 23, Switzerland /H20849Received 6 December 2006; revised manuscript received 23 January 2008; published 24 March 2008 /H20850 The magnetic-field dependence of the nuclear spin-lattice relaxation at Ir impurities in Fe was measured for fields between 0 and 2 T parallel to the /H20851100 /H20852direction. The reliability of the applied technique of nuclear magnetic resonance on oriented nuclei was demonstrated by measurements at different radio-frequency /H20849rf/H20850 field strengths. The interpretation of the relaxation curves, which used transition rates to describe the excitationof the nuclear spins by a frequency-modulated rf field, was confirmed by model calculations. The magnetic-field dependence of the so-called enhancement factor for rf fields, which is closely related to the magnetic-fielddependence of the spin-lattice relaxation, was also measured. For several magnetic-field-dependent relaxationmechanisms, the form and the magnitude of the field dependence were derived. Only the relaxation viaeddy-current damping and Gilbert damping could explain the observed field dependence. Using reasonablevalues of the damping parameters, the field dependence could perfectly be described. This relaxation mecha-nism is, therefore, identified as the origin of the magnetic-field dependence of the spin-lattice relaxation in Fe.The detailed theory, as well as an approximate expression, is derived, and the dependences on the wave vector,the resonance frequency, the conductivity, the temperature, and the surface conditions are discussed. The theoryis related to previous attempts to understand the field dependence of the relaxation, and it is used to reinterpretprevious relaxation experiments in Fe. Moreover, it is predicted that the field dependences of the relaxation inFe and Co, on one side, and in Ni, on the other side, differ substantially, and it is suggested that the literaturevalues of the high-field limits of the relaxation constants in Fe are slightly too large. DOI: 10.1103/PhysRevB.77.104433 PACS number /H20849s/H20850: 76.60.Es, 75.50.Bb, 75.30.Ds, 76.80. /H11001y I. INTRODUCTION The magnetic-field dependence of the nuclear spin-lattice relaxation in Fe, Co, and Ni had been an unsolved problemfor more than 30 years. 1–3The effect typically manifests it- self at low applied magnetic fields by relaxation rates that are2–10 times larger than in the high-field limit, which is essen-tially reached within applied fields of the order of 1 T. Sincethere is a close relation between the spin-lattice relaxationand low-frequency magnetic-moment fluctuations, 4,5the lack of an explanation would point to a fundamental deficiency inour understanding of the moment fluctuations in Fe, Co, andNi. This was the motivation to obtain more information onthe effect. A phenomenological description of the effect had been proposed by Kopp and Klein: According to their enhance-ment factor model /H20849EFM /H20850, the field-dependent part of the spin-lattice relaxation is proportional to the square of theNMR enhancement factor. 6In this way, the magnetic-field dependence of the relaxation is attributed to the magnetic-field dependence of the enhancement factor. The EFM pro-vided a description of the field dependence of the relaxationin polycrystalline samples, 6,7and it was consistent with the main features of the field dependence in single-crystalsamples, in particular, with the occurrence of peaks for cer- tain directions of the magnetic field.2 However, a critical experimental test of the EFM was still missing, because in polycrystalline samples, the field depen-dence of the enhancement factor is not well known, and thefew relaxation experiments on single-crystal samples 8,9had not been interpreted quantitatively by the EFM. In this work,a single-crystal sample was used and the magnetic field wasapplied along the /H20851100 /H20852direction. The field dependence of the enhancement factor is well known for that geometry.Moreover, it was also determined experimentally. This en-abled us to establish the actual relationship between the re-laxation and the enhancement factor. It turned out to differfrom the postulated quadratic dependence on the enhance-ment factor. In context with the field dependence of the spin-lattice relaxation in Fe, Co, and Ni, several relaxation mechanismshad been discussed, but none of those could explain theeffect. 1,8,10–12It had been speculated that this failure might not be due to the inadequacy of the proposed relaxation mechanisms, but due to an incomplete knowledge of themagnetization behavior, the band structure, or the spin-wavedispersion. 9,13,14The precise data and the close examination of those mechanisms in this work show, however, that thosespeculations are not true. In contrast, it turns out that anPHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 1098-0121/2008/77 /H2084910/H20850/104433 /H2084920/H20850 ©2008 The American Physical Society 104433-1important mechanism has been ignored so far, the relaxation via eddy currents and Gilbert damping. The theory of thisrelaxation mechanism is derived and it is shown that it canexplain the observed field dependence. Since the reliability of relaxation measurements by nuclear magnetic resonance on oriented nuclei /H20849NMR-ON /H20850, the technique that was used in this work, had been ques-tioned in the past, 3,7,14the theory of NMR-ON was also re- examined. In particular, the practice to use transition rates todescribe the effect of a coherent rf field on the sublevelpopulations had been doubted. Therefore, model calculationsthat showed under which conditions that practice is justifiedwere performed. In addition, the reliability of the techniquewas tested by measurements at different rf-field strengths. The relaxation measurements were performed on radioac- tive 186Ir and189Ir nuclei, which were coimplanted into the Fe sample. One part of the experiment, the determination ofthe electric-quadrupolar contribution to the relaxation by thecomparison of the relaxations of both isotopes, was alreadytreated in Ref. 15. The present work is mainly concerned with the form and the magnitude of the field dependence of the relaxation, which were deduced from the data on 186Ir. II. ENHANCEMENT FACTOR The NMR enhancement factor /H9257in ferromagnets is de- fined as the ratio of the effective rf magnetic field at thenuclear site to the applied rf field. 16It takes into account that the magnetization and the hyperfine field are slightly dis-placed toward the instantaneous direction of the rf field. Theresulting transverse component of the hyperfine field acts atthe nuclear site as an additional rf field, which is much largerthan the applied rf field. It can be shown that /H9257=1+BHF B/H9257/H11015BHF B/H9257, /H208491/H20850 where BHFis the hyperfine field and B/H9257is the effective field that holds the magnetization in its equilibrium position. Anappropriate expression to calculate B /H9257as a function of the applied magnetic field Bextis given in Refs. 8and17. Within the EFM of Ref. 6, the relaxation rate Ris the sum of a high-field limit and a field-dependent contribution that isproportional to /H92572. To increase the flexibility of the model, we assume that the latter contribution is proportional to /H9257/H9264, where the exponent /H9264is not necessarily 2: R/H20849Bext/H20850=R/H20849/H11009/H20850+/H20851R/H208490/H20850−R/H20849/H11009/H20850/H20852/H20875/H9257/H20849Bext/H20850 /H9257/H208490/H20850/H20876/H9264 . /H208492/H20850 The original idea behind the EFM was that the internal fields that are responsible for the field-dependent part of therelaxation are similarly enhanced as the rf field. The weakpoint of that idea was that those internal fields had neverbeen specified. Nevertheless, it makes sense to try to de-scribe the field dependence in terms of /H9257, since /H9257can be viewed just as a synonym of B/H9257−1. In this sense, /H9257is relevant for the long-wavelength magnetic excitations of the systemin several ways: For example, /H9257is essentially equivalent to the transversal susceptibility, which describes the displace-ment of the magnetization in response to forces that act on the magnetization as a whole. However, /H9257is also inversely proportional to the lowest frequency of the spin-wave spec-trum. In this work, the magnetic field was applied along the /H20851001 /H20852direction in the /H20849110 /H20850plane of a Fe single-crystal disk. The rf field was also applied within the sample plane. Forthat geometry, B /H9257is well known: B/H9257=BaforBext/H33355Bdem/H208490/H20850, B/H9257=Ba+Bext−Bdem/H208490/H20850forBext/H11022Bdem/H208490/H20850. /H208493/H20850 Here, Bais the anisotropy field /H208490.059 T in Fe /H20850andBdem/H208490/H20850is the magnitude of the demagnetization field for the fully mag-netized sample. The independence from B extforBext/H33355Bdem/H208490/H20850is due to the shielding of Bextby the demagnetization field: The shielding is complete during the magnetization of the sample when thedomains with the magnetization parallel to B extgrow at the expense of the other domains. The magnetization of the sample is completed at Bext=Bdem/H208490/H20850, which thus marks the transition from the multidomain to the one-domain regime. Two features of Eqs. /H208491/H20850and /H208493/H20850deserve special attention. First, the frequency dependence of /H9257is neglected, because the relevant electronic resonance frequency, which is of theorder of /H20849 /H9253e/2/H9266/H20850/H20849B/H92574/H9266M/H208501/2/H3335610.6 GHz, is much larger than the frequencies applied in this work. Second, to obtain the correct dependence on Bdem/H208490/H20850, it must be taken into account that, due to the skin effect, the magneti-zation Mis displaced by the rf field only in a very thin surface layer. The demagnetization field in that layer is notdisplaced, since it originates largely from the rest of thesample. Therefore, the demagnetization field of the undis-turbed sample acts on Mof the surface layer like an external field. This gives in the end the dependence on B dem/H208490/H20850of Eq. /H208493/H20850. III. FIELD-DEPENDENT RELAXATION MECHANISMS Spin-lattice relaxation rates in metals are specified by the reciprocal value of the Korringa constant, R=/H20849T1T/H20850−1. Usu- ally, the dominant relaxation mechanism is the scattering ofconduction electrons via the hyperfine interaction at thenuclear site, 18,19and Ris magnetic-field independent, be- cause the involved matrix elements and densities of states arepractically field independent. In this section, we discuss sev-eral mechanisms by which the ferromagnetism can introducea field dependence. They have in common that they arisefrom the coupling of the nuclear spin to the magnetizationvector. Since, in this case, the susceptibility formalismproves to be convenient, it is discussed first. A. Susceptibility formalism 1. Formalism Within the susceptibility formalism,4,11,20SEEWALD et al. PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-2R=kB /H6036K2 /H6036/H9275n/H20849/H6036/H9253e/H2085022V /H208492/H9266/H208503/H20885Im/H20851/H9273/H20849q,/H9275n/H20850/H20852d3q. /H208494/H20850 Here, Kis the coupling constant between the nuclear and the electronic spins, /H9273/H20849q,/H9275/H20850is the transversal dynamical suscep- tibility in units of the induced magnetic moment per atomand magnetic-field unit, Vis the volume per atom, /H9253eis the gyromagnetic ratio of the electron spins, and /H9275nis the nuclear precession frequency. The magnitude of /H9275nis 2/H9266/H9263nand the sign is given by sgn /H20849/H9275n/H20850= − sgn /H20849BHF/H20850sgn /H20849/H9253n/H20850, /H208495/H20850 whereas the nuclear resonance frequency /H9263nis defined in this work as a positive quantity. /H9253nis the nuclear gyromagnetic ratio. In connection with the sign of /H9275n, it should be noted that the decisive sign is sgn /H20849/H9275n/H9253e/H20850= − sgn /H20849BHF/H20850sgn /H20849/H9253n/H20850sgn /H20849/H9253e/H20850, /H208496/H20850 which is negative if the nuclear and the electronic spins pre- cess in the same sense and positive if they precess in theopposite sense. If several different coupling constants and electronic mag- netic moments contribute to the spin-lattice relaxation, eachcontribution is given by an expression of the form of Eq. /H208494/H20850. In this section, almost exclusively the contribution from thecoupling to the magnetization vector via the static hyperfineinteraction is discussed. R swdenotes the respective contribu- tion to the relaxation constant. The coupling constant in thecase of R swis the static hyperfine coupling constant11,21,22 K=/H20849/H6036/H9275HF/H20850/S/H11015/H20849/H6036/H9275n/H20850/S, /H208497/H20850 where S=/H20849MV /H20850//H20849/H6036/H20841/H9253e/H20841/H20850is the electronic spin and /H9275HFis the precession frequency due to the static hyperfine field. Forsimplicity, in this context, /H9275HFis approximated by /H9275n, as- suming BHF/H11271Bext. This is a good approximation for ferro- magnetic transition metals, where the hyperfine fields are ofthe order of 10–100 T. The susceptibility in the case of R sw, /H9273/H20849q,/H9275/H20850=/H6036/H20841/H9253e/H20841S1 2/H20873/H11509 /H11509bx/H11032+i/H11509 /H11509by/H11032/H20874mx−imy M, /H208498/H20850 describes the displacement of the magnetization in response to a small, complex, space- and time-periodic, transversalfield b /H11032that is proportional to exp /H20849iqr−i/H9275t/H20850. Here, mxand myare the transversal components of the displaced magneti- zation, which are also proportional to exp /H20849iqr−i/H9275t/H20850. /H9273is obtained from the linearized equation of motion of m, which turns out to be of the form d dtmx M=+/H9275xmy M−/H9253eby/H11032, d dtmy M=−/H9275ymx M+/H9253ebx/H11032. /H208499/H20850 This equation has the solution mx M=/H9275x/H9253ebx/H11032+i/H9275/H9253eby/H11032 /H9275x/H9275y−/H92752,my M=−i/H9275/H9253ebx/H11032+/H9275y/H9253eby/H11032 /H9275x/H9275y−/H92752. /H2084910/H20850 Combining Eqs. /H208494/H20850,/H208497/H20850,/H208498/H20850, and /H2084910/H20850, one obtains Rsw=kB/H9275nV /H6036S/H208492/H9266/H208503sgn /H20849/H9253e/H20850/H20885Im/H20875/H9275x+/H9275y−2/H9275n /H9275x/H9275y−/H9275n2/H20876d3q, /H2084911/H20850 where /H9275xand/H9275yare functions of qand/H9275=/H9275n. In this way, the susceptibility formalism relates all relax- ation mechanisms that arise from the coupling to the magne-tization vector to the equation of motion of the magnetiza-tion. Note that this equation is naturally closely related to thespin-wave spectrum, since displacements of the magnetiza-tion that are proportional to exp /H20849iqr−i /H9275t/H20850are just spin waves. The problem is now to find the equation of motion. 2. Equation of motion of the magnetization The magnetization precesses around an effective field that is the sum of the magnetic field B, the anisotropy field Ba, the exchange field, internal fields b/H20849j/H20850due to the coupling to other excitation modes, and b/H11032: d dtM=/H9253eM/H11003/H20873B+Ba+D/H9004M /H6036/H20841/H9253e/H20841M+/H20858 jb/H20849j/H20850+b/H11032/H20874,/H2084912/H20850 where Dis the spin-wave stiffness constant. This equation must be solved together with Maxwell’s equations and theequations of motion of the other excitation modes. If the two explicitly time-dependent Maxwell equations are combined and the displacement current is neglected, oneobtains −/H9004B+4 /H9266/H9268 c2d dtB=4/H9266/H20851−/H9004M+/H11612/H20849/H11612M/H20850/H20852, /H2084913/H20850 where /H9268is the conductivity. Since b/H11032describes the hyperfine interaction acting on the electron spin, it is not a “true” mag-netic field and does not appear in Maxwell’s equations. Tolinearize the equation of motion, MandBare decomposed into large, static, and uniform zcomponents and small trans- versal components mand b, which are proportional to exp /H20849iqr−i /H9275t/H20850. The longitudinal components are approxi- mately given by Mz=M,/H20849B+Ba/H20850z=B/H9257+4/H9266M. /H2084914/H20850 bandmare related by Eq. /H2084913/H20850. Making use of the period- icities of those quantities, one obtains bx=4/H9266mxq2/H92542cos2/H9258 q2/H92542−2isgn /H20849/H9275/H20850, by=4/H9266myq2/H92542 q2/H92542−2isgn /H20849/H9275/H20850, /H2084915/H20850 where /H9254is the skin depth:ORIGIN OF THE MAGNETIC-FIELD DEPENDENCE OF … PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-3/H9254=c /H208492/H9266/H9268/H20841/H9275/H20841/H208501/2. /H2084916/H20850 /H9258is the angle between qand the direction of the magnetiza- tion, and xdenotes the transversal component parallel to q, which gives qx=qsin/H9258,qy=0 , qz=qcos/H9258. /H2084917/H20850 If Eqs. /H2084912/H20850,/H2084914/H20850, and /H2084915/H20850are combined, and if only terms of first order in mare retained, one obtains the linear- ized equation of motion /H208499/H20850with the parameters /H9275x=/H9275x/H208490/H20850+/H20858 j/H9275x/H20849j/H20850, /H9275y=/H9275y/H208490/H20850+/H20858 j/H9275y/H20849j/H20850. /H2084918/H20850 Here, /H9275x/H208490/H20850=/H9253e/H20873B/H9257+Dq2 /H20841/H9253e/H20841/H6036/H20874, /H9275y/H208490/H20850=/H9253e/H20873B/H9257+4/H9266Msin2/H9258+Dq2 /H20841/H9253e/H20841/H6036/H20874 /H2084919/H20850 are the parameters without eddy-current damping and cou- pling to other excitation modes. One set of /H9275/H20849j/H20850’s, /H9275x/H20849ed/H20850=/H9253e4/H9266M2 2+isgn /H20849/H9275/H20850/H92542q2, /H9275y/H20849ed/H20850=/H9253e4/H9266M2/H20849cos/H9258/H208502 2+isgn /H20849/H9275/H20850/H92542q2, /H2084920/H20850 are the contributions from the eddy-current damping. The other/H9275/H20849j/H20850’s are related to the internal fields from other exci- tation modes by /H9275x/H20849j/H20850=−/H9253eM myby/H20849j/H20850, /H9275y/H20849j/H20850=−/H9253eM mxbx/H20849j/H20850. /H2084921/H20850 In Fe, D=280 meV Å2,/H9253e=184 GHz T−1,4/H9266M =2.219 T, V=11.7 Å3, and S=1.06.23The expressions for /H9275x/H208490/H20850and/H9275y/H208490/H20850are well known from treatments of the spectrum of the spin-wave resonance frequencies,24–26which are given by /H20849/H9275x/H208490/H20850/H9275y/H208490/H20850/H208501/2 2/H9266. Note the dependence of /H9275x/H208490/H20850and/H9275y/H208490/H20850onB/H9257. It is the source of the field dependence of the spin-lattice relaxation for allthe relaxation mechanisms that are discussed below. Also note that, in general, /H9275x/H208490/H20850/HS11005/H9275y/H208490/H20850due to the demagnetization fields of the spin waves in the xdirection. As a result, the precession of the magnetization is, in general, elliptic. The expressions for /H9275x/H20849ed/H20850and/H9275y/H20849ed/H20850should be comple- mented by the qdependence of /H9254, since /H9268and/H9254become qdependent, when the wavelength becomes shorter than the mean free path /H9011of the conduction electrons. /H9268and/H9254are given in terms of the normal conductivity /H92680and the normal skin depth /H92540, which represent the limit /H9011q/H112701, by the fol- lowing relation, which is well known from treatments of theanomalous skin effect: 27,28 /H9268 /H92680=/H925402 /H92542=3 2/H20849/H9011q/H208502/H20877/H208511+ /H20849/H9011q/H208502/H20852arctan /H20849/H9011q/H20850 /H9011q−1/H20878./H2084922/H20850 At this point, it is also useful to introduce the length scales /H9254mand ld./H9254m−1is defined as that qthat fulfills the relation q/H9254=/H208738/H9266M B/H9257/H208741/2 . /H2084923/H20850 /H9254mcan be interpreted as an effective rf penetration depth that takes the magnetic permeability into account. For q/H9254m/H112711, the eddy-current term is a small modification of the equationof motion of the magnetization; for q /H9254m/H333551, it dominates that equation. ld=/H20873D /H6036/H20841/H9253e/H20841B/H9257/H208741/2 /H2084924/H20850 is a typical length scale of spatial variations of the direction of the magnetization, as they occur, for example, at domain walls. /H9275x/H208490/H20850and/H9275y/H208490/H20850are independent of qforqld/H112701 and are proportional to q2forqld/H112711. Usually, ld/H11270/H9254m. For example, forB/H9257=0.059 T and the parameters that are used in this work to describe the relaxation of186IrFe, typical numbers are /H9254m=0.14 /H9262m and ld=0.020 /H9262m. 3. Virtual excitation of spin waves If/H9263nlies within the spin wave resonance spectrum, the nuclear spins can emit and absorb spin waves. This relax-ation mechanism is discussed in Sec. III B. In contrast, if /H9263n is smaller than the lowest spin-wave resonance frequency,only a virtual excitation of spin waves takes place, which canbe viewed as a dynamic displacement of the magnetization inthe vicinity of the nuclear spin or as an admixture of spinwaves to the magnetic sublevels of the nuclear spin. It con-tributes to the relaxation, if the virtually excited spin wavesdecay to some other excitation mode that can be excited at /H9263n. This relaxation mechanism can be viewed in different ways: /H20849i/H20850It can be viewed as an excitation of the final exci- tation mode, where the virtual excitation of spin waves actsas an additional, indirect coupling between the nuclear spinsand that mode. /H20849ii/H20850It can be viewed as an excitation of spin waves, where the spin-wave resonance spectra are broadenedby the decay of the spin waves to the final excitation modeso that the tails of the spectra extend down to /H9263n./H20849iii/H20850Within the susceptibility formalism, the equation of motion of themagnetization is modified by the coupling of the spin wavesto the final excitation mode in such a way that the imaginarypart of the susceptibility no longer vanishes at /H9263n. Since there are several decay modes of the spin waves, several relaxation mechanisms via the virtual excitation ofspin waves can be distinguished. To find potentially relevantSEEWALD et al. PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-4decay modes, one can proceed in different ways: In previous work, it was looked for excitations that couple to the spinwaves and that have a resonance spectrum that extends downto zero. The obvious elementary excitations in this contextare sound waves and various single electron-hole excitations.The relaxation via the indirect coupling to those excitationsis discussed in Secs. III C and III D, respectively. As an al-ternative approach, in this work, we also consider the equa-tion of motion that is commonly used to describe the ferro-magnetic resonance. That equation contains two dampingterms, the eddy-current and the Gilbert damping. The relax-ation via those terms is discussed in Secs. III E and III F,respectively. 4. Approximations The integrand of Eq. /H2084911/H20850, F=I m/H20875/H9275x+/H9275y−2/H9275n /H9275x/H9275y−/H9275n2/H20876, can often be simplified. One starting point is the observation that, often, the /H9275/H20849j/H20850’s are small with respect to the /H9275/H208490/H20850’s, or, as far as the real parts are concerned, are already taken intoaccount by the /H9275/H208490/H20850’s, since they are contained in the experi- mental values of /H9253e,Ba, and D. In the case of the relaxation via the real excitation of spin waves, the consequence is thatthe density of spin-wave states at /H9275nis not decisively changed by the /H9275/H20849j/H20850’s. The /H9275/H20849j/H20850’s essentially only ensure that the imaginary parts of /H9275xand/H9275ydo not vanish, but the exact form of those imaginary parts is not decisive. In this case, itis a good approximation to assume /H9275x/H11015/H9275x/H208490/H20850−isgn /H20849/H9275n/H9253e/H20850/H9280x, /H9275y/H11015/H9275y/H208490/H20850−isgn /H20849/H9275n/H9253e/H20850/H9280y, /H2084925/H20850 where /H9280xand/H9280yare arbitrarily small positive numbers. Tak- ing the limit /H9280x,/H9280y→0, one obtains F/H110152/H9266sgn /H20849/H9275n/H9253e/H20850/H9254/H20851/H20849/H9275x/H208490/H20850/H9275y/H208490/H20850/H208501/2−/H20841/H9275n/H20841/H20852ca2, /H2084926/H20850 where /H9254/H20851¯/H20852denotes the /H9254function and not the skin depth, and ca=1 2/H20875/H20873/H9275x/H208490/H20850 /H9275y/H208490/H20850/H208741/4 − sgn /H20849/H9275n/H9253e/H20850/H20873/H9275y/H208490/H20850 /H9275x/H208490/H20850/H208741/4/H20876. /H2084927/H20850 In the case of the relaxation via the virtual excitation of spin waves, the nonvanishing spin-wave density of states at /H9275nis due to the /H9275/H20849j/H20850’s. In this case, the form of the Im /H20851/H9275/H20849j/H20850/H20852’s is decisive and must be taken into account. However, one canat least expand the real and the imaginary parts in the nu-merator and the denominator of Finto powers of /H9275/H20849j/H20850//H9275/H208490/H20850 and retain only the lowest nonvanishing order. The result is F/H11015/H20858 j−cx2Im/H20851/H9275x/H20849j/H20850/H20852−cy2Im/H20851/H9275y/H20849j/H20850/H20852 /H20849/H9275x/H208490/H20850/H208502, /H2084928/H20850 where cx=/H9275x/H208490/H20850/H20849/H9275y/H208490/H20850−/H9275n/H20850 /H9275x/H208490/H20850/H9275y/H208490/H20850−/H9275n2,cy=/H9275x/H208490/H20850/H20849/H9275x/H208490/H20850−/H9275n/H20850 /H9275x/H208490/H20850/H9275y/H208490/H20850−/H9275n2. /H2084929/H20850 Whether the condition /H9275/H20849j/H20850/H11270/H9275/H208490/H20850is fulfilled depends on the/H9275/H20849j/H20850’s and on q. For q/H9254m/H333551, it is not fulfilled, because /H9275/H20849ed/H20850is of the order of or larger than /H9275/H208490/H20850. However, for q/H9254m/H112711, it is fulfilled, at least for the spin-wave damping mechanisms and parameters that are considered in this work,and Eqs. /H2084926/H20850and /H2084928/H20850are expected to be good approxima- tions. Further possibilities to simplify Fconcern the coefficients c xandcyin Eq. /H2084928/H20850. For many isotopes in Fe, it is a good approximation to take the limit /H9275n/H11270/H9275x/H208490/H20850, which leads to cx/H110151,cy/H11015/H9275x/H208490/H20850 /H9275y/H208490/H20850. /H2084930/H20850 However, for the relatively high resonance frequency of 186IrFe in this work, deviations of the order of several per- cent are expected. A further simplification can be achieved, if one takes the limit /H9275n/H11270/H9275x/H208490/H20850/H11270/H9275y/H208490/H20850, which leads to cx/H110151,cy/H110150. /H2084931/H20850 That limit applies only in the range qld/H112701 and sin2/H9258 /H11271B/H9257//H208494/H9266M/H20850, which is, however, responsible for a major part of the field dependence of the relaxation. Since it sim-plifies the discussion considerably, this approximation mayalso be applied beyond that range, but deviations from theexact result of the order of several percent are then to be expected. A further limit of interest is /H9275n/H11270/H9275x/H208490/H20850/H11015/H9275y/H208490/H20850, where cx/H11015cy/H110151, /H2084932/H20850 because it is a good approximation for qld/H112711, that is, for the vast majority of the wave vectors. When the formalism is applied to the spin-lattice relax- ation of impurity isotopes, the question arises to which ex-tent the modifications of the solid-state properties in the vi-cinity of the impurity must be taken into account. Theanswer follows from the involved length scales: The range ofthe modifications is typically of the order of a lattice constantor less. In contrast, the wave vectors that are responsible for the field dependence of R sware of the order of ld−1or less, which corresponds to an effective range of the relevant in-teraction between the nuclear spin and the lattice of the orderofl dor larger. The interaction thus takes place essentially in the host and is expected to be only little affected by theimpurity. Accordingly, the /H9275/H208490/H20850’s and /H9275/H20849j/H20850’s are approximated in this work by their values in the undisturbed host. Rsw depends on the impurity only via the hyperfine coupling con- stant /H20849/H6036/H9275n/H20850/S. B. Excitation of spin waves If the nuclear resonance frequency /H9263nlies within the range of the spin-wave resonance frequencies, Rswis essentially due to the following mechanism: The nuclear spins emit andabsorb spin waves. In this work, a contribution from thisrelaxation mechanism can be excluded, since the lowestspin-wave frequency, /H20849 /H9253eBa/H20850//H208492/H9266/H20850=1.72 GHz, was muchORIGIN OF THE MAGNETIC-FIELD DEPENDENCE OF … PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-5larger than /H9263n. Nevertheless, the mechanism is of interest, since under special conditions, much smaller spin-wave fre-quencies can occur. 8 If it is assumed that the majority of the wave vectors of the emitted spin waves are much larger than /H9254m−1, which should be a good approximation in many cases, Eqs. /H2084911/H20850and /H2084926/H20850can be combined to Rsw=kB4/H92662/H9263n S/H20885Vca2 /H208492/H9266/H208503/H6036/H9254/H20851/H20849/H9275x/H208490/H20850/H9275y/H208490/H20850/H208501/2−/H20841/H9275n/H20841/H20852d3q, /H2084933/H20850 where cais given by Eq. /H2084927/H20850. Without the factor ca2, which arises from the ellipticity of the precession of the magneti-zation, the integral is just the density of spin-wave states ath /H9263n. Since, often, relaxation constants are derived via Fer- mi’s golden rule instead of via the susceptibility formalism,we mention that, in this case, the ellipticity of the precessionof the magnetization must already be taken into account when the spin waves are quantized. Otherwise, the factor c a2 is not reproduced. To estimate the expected order of magnitude, Rswwas calculated according to Eq. /H2084933/H20850forB/H9257=0, the most favor- able case, and /H9263n=0.79 GHz of186IrFe. The result Rsw =3.55 /H11003103/H20849sK /H20850−1is about 400 times larger than R/H20849/H11009/H20850for that system. This shows that, if /H9263nlies within the spin-wave resonance frequencies, Rswincreases the relaxation by orders of magnitude. That increase is strongly field dependent, sincethis is the case at all only for special choices of the directionand the magnitude of B ext.8 C. Excitation of sound waves In this section, the following relaxation mechanism is dis- cussed: The nuclear spins virtually excite spin waves, which,in turn, decay via the excitation of sound waves. R phdenotes the respective contribution to the relaxation constant. Thismechanism can also be described in terms of the mixing ofsound and spin waves in ferromagnets by themagnetostriction: 22,29,30The coupling of the nuclear spin to the spin-wave component of the sound-wave-like mode leadsto spin-lattice relaxation via the excitation of the sound-wave-like modes. To derive R phwithin the susceptibility formalism, one has to solve the coupled equations of motion of the amplitude m of the displacement of the magnetization, which is propor-tional to exp /H20849iqr−i /H9275t/H20850, and of the amplitude uof the dis- placement of the atoms, which is also proportional toexp /H20849iqr−i /H9275t/H20850. The equation of motion of uis assumed to be of the form Ad2 dt2u=− /H20849Av2q2/H20850u−/H9280d dtu+f, /H2084934/H20850 where Ais the mass of the atom, vis the speed of sound, /H9280is an arbitrarily small positive number, and fis the force, which is also proportional to exp /H20849iqr−i/H9275t/H20850. For simplicity, it is as- sumed that the velocity of sound is the same for all wavevectors and polarizations.Sound and spin waves are coupled by the magnetoelastic energy, which is, for small displacements of the magnetiza-tion from the /H20851001 /H20852direction, of the form B 2/H20849eyz/H9251y+exz/H9251x/H20850, /H2084935/H20850 where B2is the magnetoelastic coupling constant, the eij’s are the components of the strain tensor, and the /H9251i’s are the direction cosines of the magnetization. As a result, uandm are coupled by the following energy per atom: Eme=−iqB 2/H20875/H20849uy*cos/H9258/H20850my M+/H20849ux*cos/H9258+uz*sin/H9258/H20850mx M/H20876. /H2084936/H20850 It leads to the coupling of the equations of motion by the following forces and fields: fj=−/H11509Eme /H11509uj*, /H2084937/H20850 bj/H20849ph/H20850=−1 V/H20875/H11509Eme /H11509mj/H20876* . /H2084938/H20850 With the force from Eq. /H2084937/H20850, the solution of Eq. /H2084934/H20850is uj=/H11509Eme//H11509uj* A/H20849/H92752−v2q2/H20850+i/H9280/H9275. /H2084939/H20850 Combining Eqs. /H2084936/H20850,/H2084938/H20850, and /H2084939/H20850, one obtains b/H20849ph/H20850as a function of m. Applying, in addition, Eq. /H2084921/H20850and MV =/H6036/H20841/H9253e/H20841S, one obtains the following contribution of the sound waves to the equation of motion of the magnetization: /H9275x/H20849ph/H20850=sgn /H20849/H9253e/H20850q2B22cos2/H9258 /H6036S/H20851A/H20849/H92752−v2q2/H20850+i/H9280/H9275/H20852, /H9275y/H20849ph/H20850=sgn /H20849/H9253e/H20850q2B22 /H6036S/H20851A/H20849/H92752−v2q2/H20850+i/H9280/H9275/H20852. /H2084940/H20850 To obtain a compact expression of Rph, some additional assumptions are necessary: Neglecting all other dampingmechanisms of the spin waves, all /H9275/H20849j/H20850’s can be set equal to zero with the exception of the /H9275/H20849ph/H20850’s. Moreover, assuming that the dispersion relation of the sound waves is not deci-sively changed by the damping and the mixing with the spin waves, one can take the limits /H9280→0 and B22//H20849Av2/H20850/H11270/H6036/H9275x/H208490/H20850. Finally, assuming that qld/H112701, the q→0 limit of the /H9275/H208490/H20850’s can be used. If those approximations are applied togetherwith Eq. /H2084911/H20850, the result is R ph=kB2/H92662VB22h/H9263n4 /H6036S2/H20849/H6036/H9253eB/H9257/H208502Av5/H20885 q→0/H20849cx2cos2/H9258+cy2/H20850d/H9024 4/H9266,/H2084941/H20850 where the integration is over all directions of q. In contrast to similar expressions in the literature,22,29,30Eq. /H2084941/H20850takes the elliptic precession of the magnetization into account. The decisive point is the magnitude of the effect. In Fe, B2=0.57 meV and vranges from 0.26 to 0.65 /H11003106cm s−1.23For B/H9257=0.059 T and /H9263n=0.79 GHz of 186IrFe, the prefactor in front of the integral in Eq. /H2084941/H20850SEEWALD et al. PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-6ranges from 4.1 /H1100310−5to 4.3/H1100310−3/H20849sK /H20850−1. This is more than 3 orders of magnitude smaller than R/H20849/H11009/H20850of186IrFe. Since the integral is of the order of unity, Rphis a negligible contribution to the spin-lattice relaxation. This conclusionhas already been drawn in Ref. 1, although without explicit derivation. An examination of the numerical factors showsthat this result is less due to the weakness of the magneto-elastic coupling than due to the small sound-wave density ofstates at small q’s as a result of the linear dispersion relation. It should be added that two of the assumptions that were used to derive Eq. /H2084941/H20850are not good approximations. First, a uniform velocity of sound was assumed, although there is adistinct dependence of von the polarization and the propa- gation direction of the sound wave. The main effect of thatdependence is that, actually, the factor v−5must be evaluated for each polarization and wave vector separately and that thedifferent weighting of the polarizations must be taken intoaccount. Without detailed account of the sound-wave disper-sion, only an upper and a lower limit of R phcan be obtained by inserting the minimum and the maximum vinto Eq. /H2084941/H20850. Second, the neglect of the eddy-current damping is not jus-tified. The wave vectors of the excited sound waves are inthe range q /H9254m/H333551, where the eddy-current damping domi- nates the dispersion relation of the spin waves. It can beshown that R phis actually considerably smaller than implied by Eq. /H2084941/H20850, because the spin-wave amplitude is suppressed by the eddy-current damping. Thus, Eq. /H2084941/H20850is useful to present, in a compact expression, the decisive factors that areresponsible for the negligibility of R ph, but a more elaborate expression would be needed to calculate Rph. D. Indirect spin-wave mechanism In this section, the following relaxation mechanism is dis- cussed: The nuclear spins virtually excite spin waves, which,in turn, decay via the scattering of conduction electrons. R in denotes the respective contribution to the relaxation constant. This relaxation mechanism is known as the indirect spin-wave mechanism, the Weger mechanism, or the second-orderspin-wave mechanism. It is the dominant contribution to thespin-lattice relaxation in the rare earths. 30Its contribution to the spin-lattice relaxation in Fe has been discussed in Refs.1,11, and 21. To make the relationship to the other contri- butions to R swapparent, we rederive Rinwithin the formalism that was developed in Sec. III A. Since in transition metals the spin waves are excitations of the conduction electrons, it is useful to remember that atsmall /H9275’s and q’s, the spin waves, as collective rotations of all spins, can be well distinguished from the scattering of theconduction electrons, which describes single electron-holeexcitations. To derive R inwithin the susceptibility formalism, one has to solve the coupled equations of motion of thetransverse magnetization mdue to the spin waves and of the transverse magnetization m /H20849s/H20850due to the single electron-hole excitations. The equation of motion of m/H20849s/H20850is solved by m/H20849s/H20850=/H9273/H20849s/H20850 Vb/H20849s/H20850, /H2084942/H20850 where /H9273/H20849s/H20850is the transversal susceptibility of the conduction electrons and b/H20849s/H20850is the transversal field acting on m/H20849s/H20850. The coupling energy per atom is of the form−JV2 /H9253e/H9253s/H60362mm/H20849s/H20850, /H2084943/H20850 where Jis the coupling constant per unit of the involved spins and /H9253sis the gyromagnetic ratio of the conduction elec- trons. With respect to this coupling term, the interaction viathe demagnetization field is negligible. The coupling givesrise to the fields b /H20849s/H20850=JS /H20841/H9253s/H20841/H6036m M, b/H20849in/H20850=JV /H9253e/H9253s/H60362m/H20849s/H20850, /H2084944/H20850 where b/H20849in/H20850is the field acting on m. If Eqs. /H2084921/H20850,/H2084942/H20850, and /H2084944/H20850are combined, one obtains the following contribution of the scattering of the conductionelectrons to the equation of motion of m: /H9275x/H20849in/H20850=/H9275y/H20849in/H20850= − sgn /H20849/H9253e/H20850J2S /H9253s2/H60363/H9273/H20849s/H20850. /H2084945/H20850 The respective contribution to the spin-lattice relaxation fol- lows from Eqs. /H2084911/H20850and /H2084928/H20850, which is expected to be a good approximation. The final result is Rin=kB/H9275nJ2V /H60364/H9253s2/H208492/H9266/H208503/H20885cx2+cy2 /H20849/H9275x/H208490/H20850/H208502Im/H20851/H9273/H20849s/H20850/H20852d3q. /H2084946/H20850 A comparison with Eq. /H208494/H20850shows that Kis replaced in Eq. /H2084946/H20850by the factor J/H20849/H9275n//H9275x/H208490/H20850/H20850, which can thus be interpreted as the q-dependent coupling constant of the indirect coupling to the conduction electrons via the magnetization. The effec-tive range of that indirect coupling follows from the qde- pendence of /H9275x/H208490/H20850: It is of the order of ld. To calculate Rin, one has to know Im /H20851/H9273/H20849s/H20850/H20852, which, in turn, requires a detailed knowledge of the band structure. Sincethis is outside of the scope of this work, the magnitude of R in is left as an open problem. The estimates of Rinin Ref. 31are of little use, because they are unrealistic at least in the fol-lowing two respects: First, those estimates are based on over-estimates of the spin-lattice relaxation via the direct scatter-ing of selectrons. 31,32Second, in the case of impurity isotopes, those estimates are based on the assumption thatIm/H20851 /H9273/H20849s/H20850/H20852in Eq. /H2084946/H20850refers to the local susceptibility of the conduction electrons at the impurity. However, since the in-direct coupling to the conduction electrons takes place essen-tially in the host, the appropriate Im /H20851 /H9273/H20849s/H20850/H20852is that of the un- disturbed host. Nevertheless, some conclusions are possible without cal- culation. The decisive point in this work is the form of thefield dependence, which can already be deduced from R in/H11008/H20885Im/H20851/H9273/H20849s/H20850/H20852 /H20849/H9275x/H208490/H20850/H208502d3q, /H2084947/H20850 where cx=1 and cy=0 was assumed for simplicity. Since /H9275x/H208490/H20850 is appreciably field dependent only for small q’s, the knowl- edge of the qdependence of Im /H20851/H9273/H20849s/H20850/H20852in the limit q→0 and /H9275→0 is already sufficient in this context. Im /H20851/H9273/H20849s/H20850/H20852is a mea-ORIGIN OF THE MAGNETIC-FIELD DEPENDENCE OF … PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-7sure of the resonant absorption by the scattering of conduc- tion electrons, if a field proportional to exp /H20849iqr−i/H9275t/H20850is ap- plied. Therefore, it is proportional to the available phasespace for the scattering and to the square of the matrix ele-ment. For a given scattering mechanism and constellation ofthe involved bands, the resulting qdependence can reason- ably well be predicted. In the following, several scenarios arediscussed. Since the occurrence of small momentum trans-fers is the prerequisite for an appreciable field dependence,only the most favorable situations in this respect are consid-ered. /H20849i/H20850One scenario, the spin-flip scattering between bands that are shifted in energy relative to each other by the ex-change splitting, has already been discussed in Refs. 1,11, and21: In this case, there is a minimum momentum transfer q m.I m /H20851/H9273/H20849s/H20850/H20852is proportional to q−1forq/H11022qmand zero for q/H11021qm. The field dependence is of the form Rin/H110081 B/H9257+Bc. /H2084948/H20850 The constant Bcis given in terms of the exchange splitting JS and the gradient /H11509/H9280//H11509qof the electron energy dispersion at the Fermi energy by Bc=D /H6036/H9253e/H20849JS/H208502 /H20849/H11509/H9280//H11509q/H208502. /H2084949/H20850 Without detailed knowledge of the band structure, the fol- lowing can be said about Bc:I fJSand /H20849/H11509/H9280//H11509q/H20850are of the order of magnitude that can be expected for dbands in Fe, Bc is of the order of 103T. If nearly-free-electron bands are involved, Bccan be much smaller: Assuming a free electron energy dispersion, a Fermi energy of 8 eV,33and JS =0.15 eV,33one obtains Bc=0.5 T. However, this estimate of Bcis already rather a lower limit. /H20849ii/H20850In the case of spin-flip scattering between bands with intersecting Fermi surfaces, the available phase space ap-proaches a nonzero value in the limit q→0. The matrix ele- ment of the spin-flip operator may or may not vanish in thelimit q→0. In the latter case, Im /H20851 /H9273/H20849s/H20850/H20852is approximately con- stant for small q’s. Applying Eq. /H2084947/H20850, it can be shown that in this case, the field dependence is of the form Rin/H11008B/H9257−1 /2. /H20849iii/H20850Scattering that involves a change of the orbital mag- netic quantum number instead of a spin-flip can also contrib-ute to R in. However, the required coupling between the mag- netization and the orbital moment cannot be the spin-orbitcoupling, since it must be an electron-electron interaction.Instead, it may arise from the intra-atomic interaction be-tween the orbital moment that is admixed to the magnetiza-tion by the spin-orbit coupling and the orbital moment of thescattered electron. The most favorable constellation is thescattering into the same band. In this case, the availablephase space is proportional to q −1, whereas the square of the matrix element of the orbital moment raising operator is pro-portional to q 2. As a result, Im /H20851/H9273/H20849s/H20850/H20852/H11008q. Applying Eq. /H2084947/H20850,i t can be shown that in this case,Rin/H11008log/H20875Bc B/H9257/H20876, /H2084950/H20850 where Bcis at least of the order of 50 T. For comparison, according to our measurements, the field-dependent part of the spin-lattice relaxation can be well described by a term that is proportional to B/H9257−/H9264, where /H9264is close to 1.4, if B/H9257is of the order of 0.1 T. This observed field dependence is much stronger than any of the predicted fielddependences of R in. Therefore, the conclusion is that the in- direct spin-wave mechanism cannot explain the magnetic-field dependence of the spin-lattice relaxation, at least notwith the scattering mechanisms and band structure constella-tions that are known to us. Finally, it should be mentioned that, for simplicity, our derivation of the field dependence of the indirect spin-wavemechanism neglects the following two effects, which maysomewhat modify the form of the field dependence: The el-lipticity of the precession of the magnetization is neglectedby the assumptions c x=1 and cy=0. Moreover, in addition to the contributions from the direct and from the indirect cou-pling to the conduction electrons, the superposition of bothcouplings also contributes to the spin-lattice relaxation. Thiscontribution, which is discussed in Refs. 1and 11,i sn e - glected in this work. However, it can be shown that botheffects, at most, lead to an even weaker field dependence.Our conclusion that the field dependence of the indirect spin-wave mechanism is too weak is thus not affected. E. Eddy-current damping In this section, the following relaxation mechanism is dis- cussed: The nuclear spins virtually excite spin waves, which,in turn, induce eddy currents, which, in turn, decay via theprocesses that are summarized by the term electrical resistiv-ity.R eddenotes the respective contribution to the relaxation constant. The contribution of the eddy currents to the equa-tion of motion of the magnetization has already been derivedin Sec. III A, where it is specified in terms of the /H9275/H20849ed/H20850’s by Eq. /H2084920/H20850. To obtain Red, one has to add the /H9275/H20849ed/H20850’s to the /H9275/H208490/H20850’s, whereas the influence of other /H9275/H20849j/H20850’s can be neglected, because for q/H9254m/H112711, all contributions to the relaxation add independently, and for smaller q’s, the other /H9275/H20849j/H20850’s are negli- gible with respect to the /H9275/H20849ed/H20850’s. It follows that Redis given by Eq. /H2084911/H20850with /H9275x=/H9253e/H20873B/H9257+Dq2 /H20841/H9253e/H20841/H6036+4/H9266M2 2+isgn /H20849/H9275/H20850/H92542q2/H20874, /H9275y=/H9253e/H20873B/H9257+Dq2 /H20841/H9253e/H20841/H6036+4/H9266M2+isgn /H20849/H9275/H20850/H20849sin/H9258/H208502/H92542q2 2+isgn /H20849/H9275/H20850/H92542q2/H20874, /H2084951/H20850 where /H9254is given as a function of qand/H9275=/H9275nby the Eqs. /H2084916/H20850and /H2084922/H20850. 1. Approximate expression To reproduce Redwithin 1%, the numerical evaluation of the set of Eqs. /H2084911/H20850,/H2084916/H20850,/H2084922/H20850, and /H2084951/H20850is unavoidable. How-SEEWALD et al. PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-8ever, a less involved, though only approximate expression was also derived. To understand the used approximations, itis useful to discuss first the qdependence of the integrand in Eq. /H2084911/H20850. In the decisive range of q’s, that integrand is largely given by Im /H20849 /H9275x−1/H20850, which is proportional to /H20849q/H9254/H208502for q/H9254m/H112701, passes through a maximum at q/H9254m=1, and is pro- portional to /H20849q/H9254/H20850−2/H208511+/H20849qld/H208502/H20852−2forq/H9254m/H112711. It follows that Redis mainly due to momentum transfers in the range /H9254m−1/H11021q/H11021ld−1. The following three approximations were applied: First, the integrand was approximated by Eq. /H2084928/H20850, the appropriate expression for q/H9254m/H112711, and was integrated from q/H9254m=1 to q=/H11009. Second, cx=1 and cy=0 was assumed, which corre- sponds to taking the limits /H9275n/H11270/H9275x/H208490/H20850and/H9275x/H208490/H20850/H11270/H9275y/H208490/H20850. Third, /H9268 was approximated by its expression in the limit q/H9011/H112711: /H9268/H11015/H926803/H9266 4/H9011q. /H2084952/H20850 This is justified for large resistivity ratios, the criterion being /H9011/H11022/H9254m. If all three approximations are combined, Redcan be ex- pressed in closed form. For convenience, we give the finalresult in terms of the numerical values of the involved quan-tities: R ed/H110156.18/H1100310−3V4/H9266M /H20849/H20841/H9253e/H20841/2/H9266/H20850S/H9263n2 B/H92572/H92680 /H9011 /H11003/H20877log/H208752.73/H11003105/H20849/H20841/H9253e/H20841/2/H9266/H20850 D/H208494/H9266M/H208502/3B/H92575/3 /H9263n2/3/H20873/H92680 /H9011/H20874−2 /3/H20876−1/H20878. /H2084953/H20850 This expression applies if Ris expressed in /H20849sK /H20850−1,Vin Å3, 4/H9266Mand B/H9257in T,/H9263nin GHz, /H20849/H9253e/2/H9266/H20850in GHz T−1,Din meV Å2, and /H20849/H92680//H9011/H20850in/H20849/H9262/H9024cm/H9262m/H20850−1. To examine the typi- cal agreement of Eq. /H2084953/H20850with the exact expression, the re- sults were compared for the parameter sets that were used in this work to describe the relaxation of186Ir. The deviation was in all cases less than 10%. 2. Properties In the following, the properties of the spin-lattice relax- ation via the eddy-current damping are discussed. The firstproperty is the magnitude of the effect. It can be inferredusing Eq. /H2084953/H20850. The main uncertainty arises from /H92680//H9011. That parameter, which is independent of the mean free path, isessentially the product of the mobility and the density of theconduction electrons. It is an intrinsic property of the host.Still, the best way to obtain a realistic estimate for Fe is toconsider the values for other metals: The order of magnitudeof /H92680//H9011should be the same for transition metals as for other metals, because the larger densities of electrons at the Fermienergy are expected to be compensated by correspondingly smaller mobilities. For Al, Sn, Cd, Pb, Cu, Ag, and Au, val-ues of /H92680//H9011between 5.7 and 20.4 /H20849/H9262/H9024cm/H9262m/H20850−1have been reported.34Assuming that /H92680//H9011of Fe lies within that range, Red=16–41 /H20849sK /H20850−1is obtained from Eq. /H2084953/H20850for/H9263n =0.79 GHz and B/H9257=0.059 T, which applies to186Ir in Fe at zero applied field. This has to be compared with the observedmagnitude of the field-dependent part of the relaxation inthat case, R/H208490/H20850−R/H20849/H11009/H20850=24 /H20849sK /H20850 −1.Redis thus of the right order of magnitude to explain the field dependence of therelaxation. The second property that we discuss is the independence from the impurity. R eddepends only on properties of the host and/H9263n, but not on the element to which the particular isotope belongs or on the lattice site that it occupies. That indepen-dence from the local electronic structure reflects the longeffective range of the interaction with the lattice, which alsomanifests itself in the dominance of small momentum trans-fers. However, it is not a distinctive feature of R ed, since every close relation between the relaxation and B/H9257, a quan- tity that describes the response of the system to macroscopicperturbations, suggests a long-range interaction with theelectrons. The third property that we discuss is the form of the magnetic-field dependence. That dependence is actually a B /H9257 dependence, since Bextenters only via that quantity. There is a proportionality to B/H9257−2, which is, however, weakened by the field dependences of /H9254mandld. That weakening, represented by the B/H9257dependence of the log term in Eq. /H2084953/H20850, increases with decreasing magnetic field but also depends on the otherparameters. For example, for the parameters that were used in this work, R edis proportional to B/H9257−1.47atB/H9257=0.059 T, to B/H9257−1.59atB/H9257=0.12 T, and to B/H9257−1.75atB/H9257=1.0 T, if the field dependence is described over small field ranges as a powerlaw in B /H9257. The fourth property that we discuss is the dependence on the nuclear resonance frequency. Redis roughly proportional to/H9263n2. This corresponds to the usual scaling of the nuclear spin-lattice relaxation with the square of the relevant hyper-fine coupling constant, which is, in our case, the static hy-perfine interaction. However, there are also slight, but distinct deviations from R ed/H11008/H9263n2. Three effects can be distinguished in this re- spect: First, the skin effect, which suppresses displacementsof the magnetization with wavelengths larger than /H9254, is less effective at smaller frequencies. Due to that effect, whichgives rise to the /H9263ndependence of the logarithmic term in Eq. /H2084953/H20850,Red//H9263n2increases with decreasing /H9263n. Second, the inte- grand of Eq. /H2084911/H20850becomes almost singular at /H9275n2=/H9275x/H208490/H20850/H9275y/H208490/H20850. Therefore, Red//H9263n2increases when /H20841/H9275n/H20841approaches the range of spin-wave precession frequencies at /H20849/H9275x/H208490/H20850/H9275y/H208490/H20850/H208501/2. Third, there is an asymmetry with respect to the sign of the fre-quency: The relaxation is faster if the free precessions of thenuclear and the electron spins have the same sense. The rela-tive sense of those precessions was specified in terms of thesigns of /H9253e,/H9253n, and BHFin connection with Eq. /H208496/H20850. The combined effect of all three effects is illustrated in Fig. 1. The fifth property that we discuss is the dependence on the conductivity, which is the product of the host-specificparameter /H92680//H9011and the mean free path /H9011, which varies withORIGIN OF THE MAGNETIC-FIELD DEPENDENCE OF … PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-9the temperature and the sample preparation. As can be seen from Eq. /H2084953/H20850, the dependence of Redon/H92680//H9011is essentially given by a proportionality to /H92680//H9011, which is only slightly weakened by the /H92680//H9011dependence of the logarithmic term. This reflects that the relaxation is proportional to the eddy-current damping, apart from small momentum transfers /H20849q/H11021 /H9254m−1/H20850, where the damping becomes so large that it sup- presses the susceptibility. The dependence on /H9011is more complex, since the conduc- tivity becomes independent of /H9011,i f/H9011becomes larger than the wavelength. Therefore, /H9011must be compared with the relevant length scales of the problem, ldand/H9254m. As long as /H9011/H11270ld,Redis largely proportional to /H9011. In the range ld/H11021/H9011 /H11021/H9254m, the increase with /H9011becomes ever weaker, and in the opposite limit, /H9011/H11271/H9254m,Redis independent of /H9011. The resulting dependence of Redon/H9011is shown in Fig. 2. There, the resistivity ratio /H92680//H92680/H20849300 K /H20850serves as the mea- sure of /H9011. It can be seen that Redis rather insensitive to /H9011,a s long as /H92680//H92680/H20849300 K /H20850/H1102230, which is fulfilled for well pre- pared samples at low temperatures.The sixth property that we discuss is the temperature de- pendence. The usual proportionality of the spin-lattice relax-ation in metals to Tis already taken into account by the definition of R ed. Additional temperature dependences arise from the temperature dependences of /H9011,Ba, and D. Since the last two parameters vary only weakly up to room tempera-ture, the temperature dependence of R edis essentially deter- mined by the temperature dependence of /H9011. Accordingly, the curve in Fig. 2can also be viewed as a plot of the tempera- ture dependence if /H92680/H20849T/H20850//H92680/H20849300 K /H20850is interpreted as a mea- sure of T. The basic effect is that /H9011andReddecrease with increasing temperature. However, due to the insensitivity of Redto/H9011at high resistivity ratios, the decrease of Redsets in later than the decrease of /H9011. Taking the example of Fig. 2, the onset of an appreciable temperature dependence of Redis expected at /H92680/H20849T/H20850//H92680/H20849300 K /H20850/H1101130, which corresponds to T/H1101165 K.35At room temperature, Redis already reduced by more than a factor of 5. The last property that we discuss is the influence of the surface. Surface effects come into play when the distance tothe surface becomes smaller than the skin depth, which isnecessarily the case with NMR measurements. They arisebecause additional magnetic surface anisotropy terms and themissing magnetic volume anisotropy at the other side of thesurface modify the susceptibility, and because the truncationof the free path of the conduction electrons at the surfacemodifies the conductivity. We do not give a detailed treat-ment because the required mathematical techniques, such asthe Wiener-Hopf technique, 27are beyond our scope. More- over, decisive parameters, such as the magnetic surface an-isotropy, are, in general, not known. However, several general conclusions can already be drawn assuming strongly simplified boundary conditions. Ifsurface effects on the conductivity are completely ignored,whereas the magnetic surface anisotropy is assumed to beeither absent /H20849free-spin boundary condition /H20850or so strong that the magnetization at the surface cannot be displaced at all/H20849pinned-spin boundary condition /H20850, the problem can be solved by the introduction of a mirror nuclear spin. The result is thatthe integrand in Eq. /H2084911/H20850must be multiplied by an extra factor 1/H11006cos/H208492qd/H20850, /H2084954/H20850 where dis the distance to the surface, and the plus and the minus signs apply to the free-spin and the pinned-spin limits,respectively. This shows that the surface contribution to R ed /H20849i/H20850can become of the same order of magnitude as the volume contribution, /H20849ii/H20850can enhance or reduce the relaxation, and /H20849iii/H20850depends on the surface conditions and thus on the sample preparation. With regard to the range of the surface effects into the interior of the sample, the following can be said withoutdetailed theory: The characteristic length scales of the inter-action between the nuclear spin and the lattice are /H9254mandld. Accordingly, the surface effects are largest for d/H11021ld, dimin- ish with increasing distance to the surface in the rangel d/H11021d/H11021/H9254m, and can be neglected for d/H11022/H9254m.FIG. 1. Dependence of Red//H9263n2on/H9263nforB/H9257=0.059 T, /H92680//H9011 =6.7 /H20849/H9262/H9024cm/H9262m/H20850−1,/H92680=20 /H20849/H9262/H9024cm/H20850−1, and Fe as the host. /H9275/H110220 denotes that the electronic and the nuclear spins precess in the op-posite sense, and /H9275/H110210 that they precess in the same sense. FIG. 2. Dependence of Redon the resistivity ratio for B/H9257 =0.059 T, /H9263n=0.79 GHz, /H92680//H9011=6.7 /H20849/H9262/H9024cm/H9262m/H20850−1,/H92680/H20849300 K /H20850 =0.1 /H20849/H9262/H9024cm/H20850−1, and Fe as the host. The region of small resistivity ratios is shown enlarged in the inset.SEEWALD et al. PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-10F. Gilbert damping In this section, the following relaxation mechanism is dis- cussed: The nuclear spins virtually excite spin waves, which,in turn, decay via the Gilbert damping. R gidenotes the re- spective contribution to the relaxation constant. Since themechanisms of the Gilbert damping are not known, in thisway, the relaxation mechanism is specified only in part, inpart it is described only phenomenologically. For example, itmay well be that the scattering of conduction electrons bythe spin waves is part of the Gilbert damping and that R inis part of Rgi. The advantage of the phenomenological concept of the Gilbert damping is that it is the generally accepteddescription of the damping of the precession of the magne-tization, which has been applied, for example, to manyferromagnetic-resonance experiments. The contribution of the Gilbert damping to dM /dtis −G /H9253eM2/H20873M/H11003dM dt/H20874, /H2084955/H20850 where Gis the Gilbert damping parameter. Here, Gis as- sumed to be independent of q. This is in accord with the use ofGin the literature, where it is treated as a constant, irre- spective of the length scales of the problem, such as the skindepth or the thickness of thin films. However, one should beaware that the qindependence of Gis not well established and that a qdependence would distinctly alter the properties ofR gi. In passing, we note that the form of the Gilbert damping might appear somewhat peculiar, if compared, for example,to a Bloch-type damping: /H20849i/H20850The relaxation of a displace- ment of the magnetization is proportional to the velocity andnot to the magnitude of the displacement. /H20849ii/H20850In the case of an elliptic precession of the magnetization, the relaxation is,in general, not directed toward the equilibrium position. /H20849iii/H20850 If the magnetization precesses in a sense that is opposite tothe sense of the free precession, which may occur in re-sponse to an external rf field, the damping term increases thedisplacement of the magnetization. In this context, it is ofinterest that it can be shown that the eddy-current dampingalso shows all those peculiarities. It can be shown that the contributions to /H9275xand/H9275yfrom the Gilbert damping are /H9275x/H20849gi/H20850=/H9275y/H20849gi/H20850=−iG/H9275 /H9253eM. /H2084956/H20850 Since it turns out that Rgiis mainly due to momentum trans- fers of the order of q/H11011ld−1, approximation /H2084928/H20850can be ap- plied. The result is Rgi=kB/H9275n2V /H6036S/H208492/H9266/H208503G /H20841/H9253e/H20841M/H20885cx2+cy2 /H20849/H9275x/H208490/H20850/H208502d3q. /H2084957/H20850 If/H9275n/H11270/H9275x/H208490/H20850, Eq. /H2084957/H20850can be further simplified. In that limit, the integrand reduces to /H20849/H9275x/H208490/H20850/H20850−2+/H20849/H9275y/H208490/H20850/H20850−2, /H2084958/H20850which can be integrated by standard integrals. The result is Rgi/H11015kB/H60361/2V/H9275n2G 8/H9266S/H20849D/H9253e/H208503/2MB/H92571/2Fc/H20873B/H9257 4/H9266M/H20874, /H2084959/H20850 where Fc/H20849x/H20850=1+ x1/2arcsin/H208731 /H208491+x/H208501/2/H20874. /H2084960/H20850 The characteristic length and wavelength scales of the in- teraction between the nuclear spin and the medium are re-flected by the qdependence of the integrand in Eq. /H2084957/H20850, which is largely proportional to /H208511+/H20849ql d/H208502/H20852−2. It follows that the length scale is essentially given by ldand that mainly momentum transfers of the order of q/H11011ld−1are involved. In comparison to Red, where the relevant length scales are /H9254m andld, very small momentum transfers and very large dis- tances are less involved. The magnitude of Rgican be calculated taking G =0.053–0.076 GHz from the literature.36For/H9263n=0.79 GHz andB/H9257=0.059 T, which applies to186Ir in Fe at zero applied field, one obtains Rgi=3.2–4.6 /H20849sK /H20850−1. The comparison with the experimental relaxation constants, R/H208490/H20850−R/H20849/H11009/H20850 =24 /H20849sK /H20850−1and R/H20849/H11009/H20850=8 /H20849sK /H20850−1, shows that Rgiis a non- negligible contribution to the field-dependent part of the re-laxation, although it is not the main contribution. Theelement- and lattice-site-specific local electronic structure atthe impurity does not enter except via the parameter /H9275n. That impurity independence results, as in the case of Red, from the long range of the interaction between the spin and the me-dium. The magnetic-field dependence of R giis determined by the factors B/H9257−1 /2and Fc/H20851B/H9257//H208494/H9266M/H20850/H20852in Eq. /H2084959/H20850. The last factor distinctly weakens the proportionality to B/H9257−1 /2at mod- erate field strengths. Between B/H9257=0.059 T and B/H9257=2 T, for example, the field dependence of Rgican be well described byB/H9257−/H9264with/H9264close to 0.40. The dependence of Rgion the nuclear resonance fre- quency largely follows Rgi/H11008/H9263n2. The deviations from that pro- portionality are distinctly smaller than in the case of Red:A t B/H9257=0.059 T, for example, Rgi//H9263n2increases by 5.9% between /H9263n=0 GHz and /H9263n=1 GHz if the electronic and the nuclear spins precess in the same sense and decreases by 3.6% if theelectronic and the nuclear spins precess in the opposite sense. The temperature dependence of R giis only weak up to room temperature, since the parameters G,Ba,D,M,/H9253e, and Vare only weakly temperature dependent. Surface effects are introduced by the magnetic surface anisotropy and the miss-ing magnetic volume anisotropy beyond the surface. Theymay become as important as in the case of R ed. However, the distance to the surface where they become important is of theorder of l dand thus much smaller than in the case of Red, where it is of the order of /H9254m. G. Domain walls The nuclear spin-lattice relaxation in the domain walls is known to be by up to 2 orders of magnitude faster than therelaxation in the domains. 21,37,38This had no consequencesORIGIN OF THE MAGNETIC-FIELD DEPENDENCE OF … PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-11for the experiments in this work, since the used measurement technique is insensitive to the nuclei in the domain walls.Nevertheless, a short comment is appropriate, since this is acase where the ferromagnetism causes a strong enhancementof the relaxation and where the origin of this enhancement isthought to be known. It has been proposed that the nuclear spins in the domain walls couple to vibrations of the domain walls, which, inturn, are damped by eddy currents. 21,39Thus, it might be possible to develop a unified treatment of the relaxation indomain walls and the relaxation via eddy currents. One dif-ficulty will be that in the case of the domain walls, theirspatial distribution and their restoring forces play an impor-tant role. The poor knowledge of those parameters will makea detailed comparison with the experiment difficult. IV . RELAXATION MEASUREMENT BY NUCLEAR MAGNETIC RESONANCE ON ORIENTED NUCLEI In nuclear magnetic resonance on oriented nuclei /H20849NMR- ON /H20850, the resonant depolarization of the radioactive probe nuclei is detected via the resulting change in the anisotropicemission of the /H9253radiation.40To measure the nuclear spin- lattice relaxation by NMR-ON, the frequency modulation/H20849FM /H20850of the rf field is periodically switched on and off. 8,41 Due to the inhomogeneous broadening of the resonance, the nuclear spins are excited only if the FM is enabled and relaxback to thermal equilibrium if it is switched off. Essentially, three parameters can be obtained from a least squares fit to the relaxation curve of the /H9253anisotropy during the FM on-off cycle: the relaxation constant R, the rf transi- tion rate Rrf, which is defined below, and the fraction frfof the probe nuclei that are excited by the FM. frf/H110211 occurs, for example, if some probe nuclei are located on slightlydisturbed lattice sites, with resonance frequencies that lieoutside of the bandwidth of the FM. At the low temperatures of NMR-ON experiments, a mul- tiexponential relaxation behavior is observed, which can bedescribed by a set of rate equations /H20849the master equation /H20850for the sublevel populations: d dtpm=/H20858 n/H20849Wm,npn−Wn,mpm/H20850, /H2084961/H20850 where pmis the population of the level with the magnetic quantum number mandWm,nis the transition rate from the level nto the level m. The transition rates are given by Wm+1,m=cm,m+1/H20875h/H9263n 2kB/H208491−b/H20850R+Rrf/H20876, Wm,m+1=cm,m+1/H20875h/H9263nb 2kB/H208491−b/H20850R+Rrf/H20876, /H2084962/H20850 where cm,m+1=I/H20849I+1/H20850−m/H20849m+1/H20850,b= exp/H20873−h/H9263n kBT/H20874, and Iis the nuclear spin. When the FM is not applied, Rrf=0. We only mention that actually more sophisticated ex- pressions for the transition rates were used, which are givenin Ref. 42. However, the differences are not decisive in the context of this work. The solution of the master equation andthe relationship between the sublevel populations and the /H9253 anisotropy are described in detail in Refs. 3,41, and 43. The description of the relaxation behavior of dilute nuclear spins by transition rates is well established in theabsence of a rf field. The use of the rf transition rate R rf, however, has been discussed controversially: On one hand, ithas been argued that the effect of a coherent rf field cannotcorrectly be treated in that way. 2,3,12,14On the other hand, it has been argued that in NMR-ON experiments, the coher-ence is sufficiently disturbed to justify such a treatment. 9,44,45 To clarify that point, in the remainder of this section, the excitation process is analyzed in more detail. Due to the FM, the rf field induces transitions between the sublevels only during small time intervals when the rf fre-quency passes the resonance frequency of the particular spin.Fast passages, therefore, alternate with intervals of nearlyfree precession. Within the rotating frame, each passagecauses a rotation of the spins by an angle /H9258around the yaxis, whereas in the time until the next passage, the spins precessaround the zaxis by the angle /H9278./H9258depends on the rf-field strength. Usually, /H9258/H11270/H9266. In that limit, /H92582is proportional to the applied rf power per FM bandwidth. /H9278is given by /H9278=2/H9266/H20885 tptp+/H9004t /H20851/H9263n−/H9263rf/H20849t/H20850/H20852dt, /H2084963/H20850 where /H9263rf/H20849t/H20850is the frequency of the rf field as a function of the time, tpis the time of the passage, and /H9004tis the time between successive passages. Contributions of the fast pas-sages to /H9278are neglected here for simplicity, because they change the dependences of /H9278on/H9263nand/H9263rfnot decisively. To describe the sequence of the rotations of the spins, it is convenient to expand the spin density matrix /H9267into irreduc- ible tensor operators Tlmof rank land order maccording to /H9267=/H20858 l=02I /H20858 m=−ll blmTlm, /H2084964/H20850 where the blm’s are complex coefficients.46,47The coefficients blm/H20849j+1/H20850before the /H20849j+1/H20850th passage are then given in terms of the coefficients before the jth passage by48 blm/H20849j+1/H20850= exp /H20849−im/H9278/H20850/H20858 m/H11032=−ll dmm/H11032/H20849l/H20850/H20849/H9258/H20850blm/H11032/H20849j/H20850. /H2084965/H20850 Here, the dmm/H11032/H20849l/H20850/H20849/H9258/H20850’s are the elements of the reduced rotation matrix, which are given, for example, in Ref. 48. /H9278is the sum of its nominal value /H92780and of a fluctuating part/H9278f, which varies from passage to passage, because the instability of the rf generator leads to small fluctuations of /H9263rf around its nominal value. For example, for a sawtooth modu-lation, Eq. /H2084963/H20850givesSEEWALD et al. PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-12/H92780=2/H9266/H9263n−/H9263c /H9263FM, /H2084966/H20850 where /H9263FMis the modulation frequency and /H9263cthe center frequency. Since the inhomogeneous broadening of /H9263nis usu- ally much larger than /H9263FM, all values of /H92780between 0 and 2 /H9266 occur almost equally frequently. The mean square deviation/H9004 /H9278fof/H9278fcan be related to the full width of half maximum /H9004/H9263rfof the frequency spectrum of the fluctuations of /H9263rf.I f the correlation time of the fluctuations is much smaller than both/H9263FM−1and /H20849/H9004/H9263rf/H20850−1, the spectrum is motion narrowed,49 and the following relation can be derived from Eq. /H2084963/H20850: /H9004/H9278f=/H208732/H9266/H9004/H9263rf /H9263FM/H208741/2 . /H2084967/H20850 This detailed description of the excitation process can be used to calculate the temporal evolution of the density matrixafter the FM is switched on. Initially, the density matrix isdiagonal, which implies that only the b l0’s are different from zero. The subsequent changes of the blm’s from passage to passage follow from Eq. /H2084965/H20850. To take the variation of /H92780 between 0 and 2 /H9266into account, one must either repeat the calculation for different /H92780’s and take the average as the density matrix of the entire spin system. Or one expands theb lm’s into powers of exp /H20849i/H92780/H20850, calculates the temporal evolu- tion of the expansion coefficients by a correspondingly ex-tended version of Eq. /H2084965/H20850, and takes the coefficients of the zeroth power as the density matrix of the entire spin system. To simulate the fluctuations of /H9278f, before each passage, a new value of /H9278fwas determined by a random number gen- erator in such a way that the probability distribution of /H9278f was Gaussian with mean square deviation /H9004/H9278f. For simplic- ity, it was assumed that /H9278fis the same for all spins. Due to the fluctuations of /H9278f, the excitation curves fluctuate too. However, in the measurements, these fluctuations are re-duced, because the average over several FM on-off cycles istaken. Accordingly, the evolution of /H9267was calculated a num- ber of nrtimes with different random numbers, and the av- erage was adopted as the final result. In this way, the exact temporal evolution of the spin den- sity matrix was calculated for different values of l,/H9258,/H9004/H9278f, andnr. The comparison with the predictions of the rate equa- tions /H2084961/H20850and /H2084962/H20850showed under which conditions those equations are a good approximation. If the spin-lattice relax-ation is neglected, the rate equations, which describe only thediagonal part of /H9267,47predict41 bl0/H20849t/H20850=bl0/H208490/H20850exp /H20849−klt/H20850, kl=l/H20849l+1/H20850Rrf. /H2084968/H20850 This turned out to be a good approximation under the fol- lowing conditions. /H20849i/H20850The fluctuations of the bl0/H20849j/H20850’s due to the fluctuations of /H9278fare roughly proportional to /H20849nlnr/H20850−1 /2, where nlis the num- ber of the fast passages after which bl0is reduced by a factor ofe.nlnrcan be interpreted as the number of /H9278f’s that con- tribute to the essential part of the excitation curve. In orderthat the temporal evolution of b l0is reasonably smooth and well defined, that number must be large enough./H20849ii/H20850The coherence between the rf field and the spin sys- tem is disturbed by the random variations of /H92780from spin to spin and of /H9278ffrom passage to passage. In order that the coherence gets essentially lost, /H9004/H9278fmust be at least of the order of /H9266. In that case, the individual contributions to the nondiagonal elements of /H9267cancel each other, and the diago- nal elements decay, in the limit nr→/H11009, exponentially accord- ing to bl0/H20849j+1/H20850=bl0/H208491/H20850/H20851d00/H20849l/H20850/H20849/H9258/H20850/H20852j. /H2084969/H20850 /H20849iii/H20850In order that the respective decay constants are pro- portional to l/H20849l+1/H20850and to the applied rf power, the condition /H9258/H11270/H9266, which is equivalent to nl/H112711, must be fulfilled. In that limit the decay constant of bl0can be approximated by − log /H20851d00/H20849l/H20850/H20849/H9258/H20850/H20852 /H9004t/H11015l/H20849l+1/H20850/H92582 4/H9004t, /H2084970/H20850 which is identical to klof Eq. /H2084968/H20850,i fRrfis identified with /H92582//H208494/H9004t/H20850. A quantitative analysis revealed that the deviations from prediction /H2084968/H20850of the rate equations are less than 3% of bl0/H208490/H20850if/H9004/H9278f/H333560.55/H9266,nl/H3335610, and nlnr/H3335680. Typical num- bers that apply to the experiments in this work are /H9004/H9263rf =750 Hz, /H9263FM=100 Hz, /H9004/H9278f=2.2/H9266,nr=1000, and nl=20. The conclusion is, therefore, that the interpretation of theNMR-ON relaxation curves in this work by the rate equa-tions is justified. Finally, it should be mentioned that, in order to confine the number of the parameters to a minimum, this analysis ofthe excitation process neglects several involvements: /H20849i/H20850The spin-lattice relaxation is completely neglected. /H20849ii/H20850A single resonance frequency for each spin is assumed. However, dueto the small electric hyperfine interaction in cubic ferromag-nets, the resonance is actually split into 2 Isubresonances. /H20849iii/H20850 /H9278fis assumed to be the same for all spins. However, since the moment of a particular fast passage is not exactlythe same for all spins, /H9278factually also varies from spin to spin, though much less than from passage to passage. /H20849iv/H20850 The actual pattern of the modulation of the rf frequency maybe more involved than a sawtooth modulation. However, all those effects only further disturb the coher- ence between the rf field and the spins. The agreement withthe rate equations should, therefore, be still better than dem-onstrated above. V . EXPERIMENTAL DETAILS The Fe sample was a circular single-crystal disk with /H20849110 /H20850plane, 2.2 mm thick, and 12 mm in diameter. The pu- rity of the sample and the flatness of the surface benefitedfrom the fact that the sample was originally prepared forexperiments on surface chemistry: For example, the bulkconcentration of sulfur was reduced by baking at tempera-tures of 970–1120 K in flowing hydrogen for three weeks.The segregation of contaminants at the surface was reducedin an UHV chamber by hundred cycles of heating /H208491000 K, 10–30 min /H20850and Ar +sputtering /H20849500 K, 750 eV, 1 /H9262Ac m−2, 30–10 min /H20850. The final examination of the purity at the sur-ORIGIN OF THE MAGNETIC-FIELD DEPENDENCE OF … PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-13face by Auger-electron and photoelectron spectroscopies re- vealed only about 0.4 at. % phosphorus, 0.4 at. % sulfur,3 at. % carbon, and 1 at. % oxygen. Hg precursors of the probe nuclei 186Ir and189Ir were coimplanted at the on-line mass separator ISOLDE at CERN /H20849about 3 /H110031012nuclei of186Hg and 8 /H110031012of189Hg, im- plantation voltage of 60 kV /H20850. To delimit the variation of position-dependent parameters such as the demagnetizationfield or the rf-field strength, the Hg beam was confined by adiaphragm to a spot of 4 mm diameter in the center of thedisk. After the implantation, the sample was annealed forabout 1 /2 h at 970 K and slowly cooled down to room tem- perature. The sample was then loaded into a 3He-4He-dilution re- frigerator and cooled down to temperatures in the 20 mKrange. The magnetic field was applied along the /H20851100 /H20852direc- tion in the sample plane. The orientation of the crystallo-graphic axes relative to the magnetic field was accurate to 1°.The /H9253anisotropy was measured by four Ge detectors, placed at 0°, 90°, 180°, and 270° with respect to the direction of themagnetic field. The count rate ratio /H9280=N/H208490°/H20850+N/H20849180 ° /H20850 N/H2084990 ° /H20850+N/H20849270 ° /H20850−1 /H2084971/H20850 was used to analyze the data. The temperature was primarily determined by a60Co Co /H20849hcp /H20850nuclear orientation thermom- eter. However, because of the low sensitivity of that ther-mometer at temperatures above 20 mK, in most relaxationmeasurements, the temperature was determined via the equi- librium /H9253anisotropy of186Ir, which was calibrated with re- spect to the primary thermometer at lower temperatures. The rf frequency was modulated in the following way: An external triangular FM was applied with bandwidth of /H110065 MHz and frequency /H9263FM/H208491/H20850=100 Hz. In addition, to rein- force the disturbance of the coherence between the rf fieldand the spins, a second internal triangular FM was applied with bandwidth of /H11006200 Hz and frequency /H9263FM/H208492/H20850=1 Hz. The half-width /H9004/H9263rfof the frequency fluctuations of the rf signal generator in the external modulation mode was about750 Hz. This was measured at nominally zero applied modu-lation voltage by a rf frequency analyzer. The magnetic dipolar and the electric-quadrupolar parts of the relaxation were determined from the combined relaxation data on 186Ir and189Ir, as discussed in Ref. 15. In this work, only the magnetic relaxation constants of186Ir are quoted. Anyway, for that isotope, the quadrupolar contribution to therelaxation was only of the order of 1%. VI. MEASUREMENTS The static hyperfine interactions were determined by NMR-ON and modulated adiabatic fast passage on orientednuclei /H20849MAPON /H20850. 50,51Figure 3shows the NMR-ON spec- trum at Bext=0.1 T. The magnetic resonance frequency and the subresonance separation were /H9263n=794.68 /H2084920/H20850MHz and /H9004/H9263Q= +0.838 /H208492/H20850MHz, respectively. Additional NMR-ON spectra were measured at Bext=0.5 and 1.0 T. From the field dependence of the resonance, Bdem/H208490/H20850=0.274 /H2084917/H20850T was de-duced. To excite all subresonances in the relaxation measure- ments, the frequency was modulated, for example, atB ext=0.1 T between 789.4 and 799.4 MHz. The magnetization behavior was monitored via the /H9253an- isotropy of186Ir, which remained constant at 95 /H208491/H20850%o fi t s saturation value for Bext/H333550.25 T, increased slightly between 0.25 and 0.40 T, and remained at its saturation value forhigher fields. This confirmed that the magnetization was es-sentially aligned along the /H20851100 /H20852direction within the sample plane. Other alignments of the magnetization apparently onlyoccurred at low fields in a small fraction of the sample. Thatfraction remained constant in the multidomain regime up to B ext=Bdem/H208490/H20850but disappeared at higher fields. The angular-distribution coefficients of the most intense /H9253 transitions of186Ir, which were needed for the description of the relaxation curves and for the thermometry via186Ir, were determined by measurements of the /H9253anisotropy as a func- tion of the temperature between 10 and 23 mK. For example,A 2=−0.311 /H208492/H20850and A4=−0.136 /H208494/H20850were obtained for the 297 keV transition at Bext=0.5 T. /H20849Here, Aicorresponds to AiUiin the notation of Ref. 43./H20850 The reliability of the relaxation measurement technique was tested by measurements at different rf-power levels. For example, at Bext=0.5 T, the applied rf power Prfwas varied in five measurements by a factor of 16. Thereby the relativeresonant reduction of the nuclear magnetization varied be-tween 9% and 71%, whereas the temperature varied betweenk BT/h/H9263n=0.60 and kBT/h/H9263n=1.78. Figure 4shows three of the relaxation curves. All relaxation curves could perfectlybe described by the theory. Moreover, the least squares fitresults for R,R rf, and frfwere all consistent, demonstrating the reliability of the measurement technique and of the inter-pretation of the relaxation curves. The fit results are shown inFig. 5as a function of P rf. Similarly consistent results were obtained at Bext=0.1 T, where Prfwas varied in six measurements by a factor of 32. The only deviation from the theory was that at the two high-est rf-power levels, the increase of R rfwith Prfwas smaller than expected. However, at those high power levels, the timescale of the excitation by the rf field was extraordinarilyshort, of the order of 2–3 periods of the FM. It is not sur-prising that in this case, the picture of a continuous excitationprocess begins to fail. Further tests examined the disturbance of the coherence of the rf field. According to Sec. IV , that disturbance manifestsFIG. 3. NMR-ON spectrum of186Ir at Bext=0.1 T. T =46 /H208492/H20850mK, FM bandwidth /H110060.5 MHz. The interpretation of the only partly resolved subresonance structure made use of the knowl-edge of the distribution of /H9004 /H9263Qfrom the MAPON measurements.SEEWALD et al. PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-14itself in fluctuations of /H9278=/H92780+/H9278f, the relative phase of sub- sequent fast passages. As shown in Sec. IV , the fluctuationsare expected to be large enough that the excitation of thespins can be described by rf transition rates as in the case ofan incoherent irradiation. In this case, additional variationsof /H9278should have no effect. This was confirmed in the fol- lowing ways: /H20849i/H20850Relaxation measurements with and without the second FM, which varies /H92780periodically by more than2/H9266, gave identical results. /H20849ii/H20850Applying only a single modu- lation, the resonance effect was measured as a function of /H9263FM. In this way, /H9004/H9278fwas varied according to Eq. /H2084967/H20850, but the resonance effect remained essentially constant between /H9263FM=50 Hz and /H9263FM=1 kHz. Only for /H9263FM=5 kHz and larger /H9263FM’s the resonance effect was significantly reduced as a result of the reduction of /H9004/H9278f. The resonance effect was also reduced for /H9263FM=20 Hz and smaller /H9263FM’s, because the time between the fast passages became of the order of the relaxation time. A similar depen-dence of the NMR-ON resonance effect on /H9263FMand a similar interpretation had already been reported in Ref. 41for60Co in Fe. The magnetic-field dependences of the spin-lattice relax- ation and the enhancement factor were determined by relax-ation measurements at 17 different fields between 0.05 and2.0 T. RandR rf/Prf, which is, apart from a prefactor, equiva- lent to /H92572, are shown as a function of Bextin Figs. 6and7, respectively. frfwas in all cases consistent with the average value frf=0.88 /H208492/H20850. The field dependence of /H9257was described by Eqs. /H208491/H20850and /H208493/H20850, assuming Ba=0.059 T. This resulted in a perfect descrip- tion of the field dependence of Rrf/Prfover 3 orders of mag- nitude. Only Bdem/H208490/H20850and the proportionality constant between /H92572andRrf/Prfwere adjusted via least squares fit. The solidFIG. 4.186IrFe, Bext=0.5 T: NMR-ON relaxation curves at dif- ferent rf-power levels /H20849and temperatures /H20850.Prfin arbitrary units. The increase and the decrease of /H9280reflect the temporal evolution after switching the FM on and off, respectively. FIG. 5.186IrFe, Bext=0.5 T: R,Rrf/Prf, and frffrom measure- ments at different rf-power levels. At the lowest power level, frfhad to be taken from the other measurements, because frfandRrfproved to be too correlated to determine both parameters independently.FIG. 6.186IrFe: Magnetic-field dependence of the nuclear spin- lattice relaxation. FIG. 7.186IrFe: Magnetic-field dependence of the square of the enhancement factor.ORIGIN OF THE MAGNETIC-FIELD DEPENDENCE OF … PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-15line in Fig. 7shows the respective theoretical curve. If Ba was also fitted to the data, Ba=0.0607 /H2084911/H20850T was obtained, in agreement with the literature value. Bdem/H208490/H20850=0.287 /H208492/H20850T was obtained from the fit to the field dependence of /H92572, in agreement with the value that was de- duced from the field dependence of /H9263n, but in disagreement with the calculated value, Bdem/H208490/H20850/H110110.20 T. The deviation from the calculated value could not be resolved. In any case, the sharp bend of the field dependences at Bext=Bdem/H208490/H20850showed that the assumption of a uniform value of Bdem/H208490/H20850for all probe nuclei was essentially justified. The field dependence of Rcould be described in different ways. The EFM in the form of Eq. /H208492/H20850has the virtue that the magnitude and the form of the field dependence can be char-acterized by a simple expression without binding oneself to aparticular explanation. The dashed line in Fig. 8shows the fit of that model to the data. Using B a=0.059 T, an almost per- fect description was obtained with R/H208490/H20850= 32.20 /H2084920/H20850/H20849sK /H20850−1, R/H20849/H11009/H20850= 8.69 /H2084922/H20850/H20849sK /H20850−1, /H9264= 1.39 /H208494/H20850. If, as it is done in this work, the field dependence of the relaxation is attributed to eddy-current and Gilbert dampings,Ris the sum of a field-independent part R/H20849/H11009/H20850and of the field-dependent contributions R edand Rgi. The data could perfectly be described in this way. The solid lines in the Figs.6and8show the respective theoretical curve. The following parameters were determined via least squares fit: /H92680//H9011= 7.1 /H208496/H20850/H20849/H9262/H9024cm/H9262m/H20850−1, G= 0.075 /H2084917/H20850GHz, R/H20849/H11009/H20850= 7.4 /H208495/H20850/H20849sK /H20850−1. The quoted errors already take the uncertainty in /H92680into account. /H92680was assumed to be in the range 3.0–1000 /H20849/H9262/H9024cm/H20850−1, which corresponds to a residual resis- tivity ratio between 30 and 104. The composition of the field- dependent part of the relaxation changes strongly with thefield: At B ext=0 T, for example, the quoted damping param- eters imply Red=20.2 /H20849sK /H20850−1andRgi=4.6 /H20849sK /H20850−1, whereas atBext=1 T, for example, Red=0.29 /H20849sK /H20850−1and Rgi =1.60 /H20849sK /H20850−1. In the past, the field dependence of the relaxation had often been described by the EFM assuming /H9264=2, and Bahad been fitted to the data. To assess the results that had beenobtained in this way, we also applied that traditional variantof the EFM. The following parameters were obtained vialeast squares fit: B a= 0.091 /H208494/H20850T, R/H20849/H11009/H20850= 9.41 /H2084918/H20850/H20849sK /H20850−1. The dotted line in Fig. 8shows the respective theoretical curve. The field dependence of Ris remarkably well repro-duced, at least between 0 and 1 T, although Bais clearly wrong. VII. DISCUSSION A. Origin of the magnetic-field dependence For all the relaxation mechanisms that were discussed in Sec. III, the magnetic-field dependence is actually a depen-dence on B /H9257. The knowledge of the magnetic-field depen- dence of B/H9257is thus indispensable for the comparison be- tween experiment and theory. Therefore, the experimentaldetermination of the field dependence of B /H9257in this work via the quantity Rrf/Prfwas particularly important. It confirmed thatB/H9257was indeed given by Eq. /H208493/H20850with Ba=0.0607 /H2084911/H20850T. This showed, in particular, that the value of Baand the mag- netization behavior were not modified at the site of the probenuclei by the presence of the impurity, by the closeness to thesurface, or by other effects. The discussion of the various potentially field-dependent relaxation mechanisms in Sec. III showed that most of themcan be excluded as the source of the observed field depen-dence: The direct excitation of spin waves is not possible,since the spin-wave frequencies are larger than /H9263n. The exci- tation of sound waves is negligible. The field dependences ofthe various variants of the indirect spin-wave mechanism aregiven in Sec. III D. They are all too weak to explain theobserved field dependence. A connection to domain wallscan also be excluded, since the domain walls just disappear when the field dependence sets in at B ext=Bdem/H208490/H20850. In contrast, the relaxation via eddy-current and Gilbert dampings explains both the magnitude and the form of thefield dependence. It provides a perfect description of thedata. The used values of the damping parameters /H92680//H9011and Gare well within the expected ranges, which were specifiedFIG. 8.186IrFe: Comparison of the field dependence of the spin- lattice relaxation with the descriptions by R=R/H20849/H11009/H20850+Red+Rgi/H20849solid line /H20850, by the EFM with Ba=0.059 T and /H9264=1.39 /H20849dashed line /H20850, and by the EFM with Ba=0.091 T and /H9264=2 /H20849dotted line /H20850. In order to show the low-field part more clearly, a double logarithmic scale wasused, and B extwas converted into B/H9257, assuming Bdem/H208490/H20850=0.287 T and Ba=0.059 T.SEEWALD et al. PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-16in Secs. III E and III F. This strongly suggests that the relax- ation via eddy-current and Gilbert dampings is indeed thesource of the magnetic-field dependence of the nuclear spin-lattice relaxation in Fe. Of course, further, hitherto unknown,sources cannot completely be excluded, since /H92680//H9011andG had to be adjusted via least squares fit. One should also beaware that the used values of /H92680//H9011andGmay not represent the actual values of those damping parameters, because thesurface effects were not taken into account. Having identified the relaxation mechanism that is re- sponsible for the magnetic-field dependence, we can now usethe theory that has been worked out in Secs. III E and III F toestablish the connection to previous theoretical work, to re-interpret the results of other experiments, to compare thefield dependences in Fe, Co, and Ni, and to judge the validityof the literature values of R/H20849/H11009/H20850. B. Previous concepts The theory of the spin-lattice relaxation via eddy-current and Gilbert dampings contains several concepts that havealready been suggested previously in the context with thefield dependence of the relaxation in Fe: For example, thevirtual excitation of spin waves, which was proposed to-gether with the indirect spin-wave mechanism, is also onestep in the relaxation via eddy-current and Gilbert damping.Moreover, the indirect spin-wave mechanism itself can beunderstood as one contribution to R gi. It also turns out that the original concept of the EFM, the proportionality between the field-dependent part of the spin-lattice relaxation and the square of the enhancement factor,indeed applies in some sense to the relaxation via the virtualexcitation of spin waves. However, it is the enhancementfactor /H9257/H20849q/H20850=BHF B/H9257+/H20849Dq2/H20850//H20849/H6036/H20841/H9253e/H20841/H20850 of transversal fields with wave vector qthat enters quadrati- cally via the factor /H20851/H9275x/H208490/H20850/H20852−2in Eq. /H2084928/H20850. The field dependence thus essentially combines /H20851/H9257/H20849q/H20850/H208522terms from all momentum transfers, whereas within the EFM, it is approximated by a /H9257/H9264term, where /H9257is the q=0 limit of /H9257/H20849q/H20850. Since the field dependence of /H9257/H20849q/H20850decreases with increasing q,/H9264is smaller than 2. To which extent depends on the weighting of theindividual momentum transfers.This relationship to the EFM also reveals that the form of the field dependence is indeed a signature of the proposedrelaxation mechanism: It was first a puzzle that the field-dependent part of the relaxation seemed to be proportional to /H92572, as if only q=0 would contribute. That puzzle is now solved: On one hand, our experiment shows that /H9264is indeed smaller than 2. On the other hand, the eddy-current dampingimplies a particularly strong weighting of small momentum transfers: Im /H20851 /H9275x/H20849ed/H20850/H20852and Im /H20851/H9275y/H20849ed/H20850/H20852are proportional to q−3for q/H9254/H112711 and q/H9011/H112711. Such a strong preference of small q’s is required to explain the strong field dependence with /H9264close to 1.4. Moreover, it is not readily reproduced by other relax-ation mechanisms, as the discussion in Sec. III D showed:All discussed variants of the indirect spin-wave mechanismhave distinctly weaker field dependences, because the small momentum transfers are less strongly weighted. C. Other experiments A major problem of the interpretation of previous experi- ments is the fact that the field dependence of B/H9257is not well known in most cases. B/H9257is reasonably well known only for the experiments on Fe single crystals where the magneticfield was applied along the /H20851100 /H20852direction in the sample plane. The data of those experiments were redescribed byboth R/H20849/H11009/H20850+R ed+Rgiand the EFM. B/H9257was assumed to be given by Eq. /H208493/H20850with Ba=0.059 T. Bdem/H208490/H20850was fitted to the data. Table Isummarizes the results. The form of the field dependences supports the interpre- tation by Red+Rgi. The parameter /H9264is a measure of the re- spective agreement with the theory. It is, within the error, inall cases close to 1.4, as expected for R ed+Rgi. In contrast, the magnitudes of the field dependences are inconsistent inso far as they cannot be described by the same set of damp-ing parameters /H92680//H9011andG. This also manifests itself by the differences in /H20851R/H208490/H20850−R/H20849/H11009/H20850/H20852//H9263n2, which cannot be explained by the weak /H9263ndependence of /H20849Red+Rgi/H20850//H9263n2. This inconsis- tency may be attributed to surface effects. This would implythat differences in the surface preparation or in the locationof the probe nuclei had changed the field-dependent part ofthe relaxation by up to a factor of 2. Relaxation measurements on Fe single crystals were also performed with the magnetic field applied along otherTABLE I. Parameters of the field dependence of the spin-lattice relaxation in Fe from different experi- ments in the /H20851100 /H20852geometry. /H92680//H9011was determined by a fit of R/H20849/H11009/H20850+Red+Rgi, assuming a residual resistivity ratio between 30 and 104.G=0.075 GHz was taken from the186Ir experiment, because the precision of the data was not sufficient for its determination in the other cases. /H9264,R/H208490/H20850, and R/H20849/H11009/H20850were determined by a fit of the EFM. Isotope/H9263n /H20849GHz /H20850 Ref./H92680//H9011 /H20851/H20849/H9262/H9024cm/H9262m/H20850−1/H20852 /H9264/H20851R/H208490/H20850−R/H20849/H11009/H20850/H20852//H9263n2 /H20851/H20849sK/H20850−1GHz−2/H20852R/H20849/H11009/H20850//H9263n2 /H20851/H20849sK/H20850−1GHz−2/H20852 110mAg 0.205 9 2.6/H208495/H20850 1.60 /H2084934/H20850 21.0 /H2084924/H20850 10.3 /H2084912/H20850 131I 0.683 8and9 5.2/H208496/H20850 1.36 /H2084917/H20850 30.9 /H2084919/H20850 8.3/H2084912/H20850 186Ir 0.795 This work 7.1 /H208496/H20850 1.39 /H208494/H20850 37.2 /H208495/H20850 13.8 /H208493/H20850 191Pt 0.320 52 2.7/H208493/H20850 1.13 /H2084930/H20850 21.9 /H2084920/H20850 9.9/H2084914/H20850ORIGIN OF THE MAGNETIC-FIELD DEPENDENCE OF … PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-17crystallographic directions.8,9,14In most cases, geometries were investigated where B/H9257vanishes, at least nominally, for certain values of the magnetic field. Those experimentsnicely demonstrated that it is indeed the quantity B /H9257that has to become small to obtain a large relaxation rate. However,quantitative conclusions are not possible, because the fielddependence of B /H9257was not determined experimentally, nor can it reliably be calculated. The problem with the calcula-tion is that, when B /H9257becomes very small, the magnetization behavior becomes extremely sensitive to misalignments andinhomogeneities, which are always unavoidable to some ex-tent. Most of the previous relaxation experiments were per- formed on cold-rolled polycrystalline foils of dilute Fe al-loys, where a well-founded description of the field depen-dence of B /H9257is not feasible. Nevertheless, the previous interpretation of those experiments is of interest, because itseemed to support the assumption /H9264=2: The field depen- dence of the relaxation could be described by the EFM and /H9264=2, if B/H9257was parametrized by Ba+Bext, and Bawas ad- justed via least squares fit.6,7,53,54However, Baand/H9264are strongly correlated, and over a fairly large magnetic-fieldregion, a wrong choice of one of those parameters can becompensated by a wrong choice of the other parameter. This was demonstrated in this work for 186IrFe: The field depen- dence of Rcould almost equally well be described by /H9264 =1.39 and the “correct” Baor by /H9264=2 and a “wrong,” dis- tinctly larger Ba. Therefore, the values of /H9264andBafrom the previous interpretation of those experiments are meaningless. The theory of Red+Rgiis also important for the under- standing of experiments where the relaxation rates of differ-ent isotopes of the same element in the same sample arecompared, because it predicts deviations from the usually expected proportionality to /H9263n2, which is well established, for example, for R/H20849/H11009/H20850.53An experiment of this kind on60Co /H20849/H9263n=0.166 GHz /H20850and58Co /H20849/H9263n=0.442 GHz /H20850in Fe was re- ported in Ref. 14. From the low-field measurements of that work, /H20849R/H11032//H9263n2/H20850/H2084958Co/H20850 /H20849R/H11032//H9263n2/H20850/H2084960Co/H20850= 0.70 /H2084912/H20850 can be deduced, where R/H11032denotes the field-dependent part of R.B/H9257was presumably in the range 0.02–0.06 T. The devia- tion from R/H11032/H11008/H9263n2is, at least in part, explained by the theory ofRed+Rgi, according to which /H20851/H20849Red+Rgi/H20850//H9263n2/H20852/H2084958Co/H20850 /H20851/H20849Red+Rgi/H20850//H9263n2/H20852/H2084960Co/H20850= 0.81 – 0.89, if the damping parameters are similar to those of the186IrFe experiment. Unfortunately, the statistical significance of thequoted data is poor and technical details of the respectivemeasurements have been questioned. 7 Relaxation measurements on different isotopes of the same element in the same sample were also reported in Refs.15and 42. They were used to deduce the electric- quadrupolar part of the spin-lattice relaxation, which ispossible, if the quadrupole moment of one of the isotopes is sufficiently large. However, to separate the magnetic-dipolarand the electric-quadrupolar parts of the relaxation, a scaling of the magnetic part with /H9263n2was assumed. Therefore, the deduced field dependences of the quadrupolar relaxation areinvalid. In contrast, the deduced high-field limits should es-sentially be correct. In the case of the measurements of this work on 186Ir and 189Ir, the data were reanalyzed, taking the /H9263ndependences of RedandRgiinto account. The revised result for the ratio of the low-field and the high-field quadrupolar relaxation con-stants is R q/H208490T /H20850/Rq/H208492T /H20850=0.97 /H2084915/H20850. Thus, there is no sig- nificant field dependence of the quadrupolar relaxation, incontrast to our previous conclusion in Ref. 15. This field independence of the quadrupolar relaxation is in accord with the theory. Indeed, R edandRgialso contribute to the quadrupolar relaxation, because the magnetization is alsocoupled to the nuclear quadrupole moment via the spin-orbit-induced electric-field gradient, but the contributions are toosmall to be observable. The form of these contributions iswell known from the similar but much stronger contributionof the indirect spin-wave mechanism to the quadrupolar re-laxation in the rare earth metals: 30The net effect is that Red and Rgimust be calculated for each transition probability Wm+1,mseparately with /H9263nreplaced by the respective transi- tion frequency /H9263m+1,mof the quadrupolar-split resonance spectrum. In Fe, the effect is negligible, because the /H9263m+1,m’s differ only slightly from /H9263n. D. Magnetic-field dependence in Co and Ni Distinct magnetic-field dependences of the nuclear spin- lattice relaxation have also been observed in Co /H20849hcp /H20850,55 Co/H20849fcc/H20850,56and Ni.9A detailed comparison with the theory is not possible, because the field dependence of B/H9257is not suf- ficiently well known for those experiments. However, esti-mates of the typical magnitudes of R ed,Rgi, and R/H20849/H11009/H20850can show at least whether major differences to the situation in Feare to be expected. To estimate R edandRgiin Co and Ni, D andGwere taken from Refs. 36and57–59. As discussed below, in the case of G, the room-temperature value should be used, which is, for Co and Ni, distinctly smaller than thelow-temperature value. /H92680//H9011should be of the same order of magnitude in Fe, Co, and Ni. Data on R/H20849/H11009/H20850in Fe, Co, and Ni are available, for example, from Refs. 3,9,53,55, and 60–63. In the case of Co as the host, Red,Rgi, and R/H20849/H11009/H20850turn out to be of the same order of magnitude as in Fe. Thus, decisivedifferences to the situation in Fe are not expected, apart fromdifferences in the field dependence of B /H9257. In contrast, for Ni as the host, Redis of the same order of magnitude as in Fe, but Rgiis larger by a factor of 20 and R/H20849/H11009/H20850by typically 1 order of magnitude, if the comparison is made for the same values of /H9263nandB/H9257. Thus, if B/H9257is of the order of 0.1 T, the relaxation is faster than in the high-field limit by factors that are similar to those in Fe. However, inNi, this is largely due to R giand not to Red. This has the following consequences: The field dependence of the relax-ation is much weaker than in Fe, with /H9264close to 0.4. More-SEEWALD et al. PHYSICAL REVIEW B 77, 104433 /H208492008 /H20850 104433-18over, this implies that even at relatively large fields of the order of 1 T, a large fraction of the relaxation may be due tothe field-dependent part. It was also calculated which value of R giin Ni is expected if the low-temperature value of Gis used in combination with Eq. /H2084957/H20850. It turned out that the use of the low- temperature value is in contradiction to the experiment: For 110mAg in Ni at Bext=0, for example, in this way, Rgi =1.1 /H20849sK /H20850−1is predicted, which is much larger than the ob- served relaxation constant of about R/H110110.2 /H20849sK /H20850−1in this case.9In contrast, if the room-temperature value of Gis used, Rgi=0.19 /H20849sK /H20850−1is predicted, which is of the right order of magnitude. This finding confirms the following in-terpretation of the temperature dependence of G: According to Refs. 58and64,Gis the sum of a largely temperature- independent contribution and of a low-temperature contribu-tion, which is negligible for T/H11022150 K and which shows similar temperature and wave-vector dependences as theconductivity. The wave-vector dependence implies that therespective contribution to R giis several orders of magnitude smaller than suggested by the literature values of G, because the wave vectors that are relevant for Rgiare much larger than those that are relevant for the ferromagnetic-resonanceexperiments that are used to determine G. The consequence is that the low-temperature contribution to Gmakes only a negligible contribution to the spin-lattice relaxation.E. High-field limits The high-field limits of the spin-lattice relaxation are im- portant for the comparison with the ab initio calculations, because the available calculations only take account of es-sentially field-independent contributions. Most literature val-ues of R/H20849/H11009/H20850in Fe were deduced by the EFM assuming /H9264=2.53If the data of this work are interpreted in this way, one obtains R/H20849/H11009/H20850=9.41 /H2084918/H20850/H20849sK /H20850−1. This is close to R/H20849/H11009/H20850 =8.97 /H2084925/H20850/H20849sK /H20850−1, which follows from the literature value for IrFe from Ref. 53, if that value is corrected for the qua- drupolar contribution to the relaxation42and if a consistent set of nuclear moments is used to convert that value to186Ir. In contrast, if the data are interpreted by R/H20849/H11009/H20850+Red+Rgi, the parameter R/H20849/H11009/H20850is about 20% smaller. This suggests that the actual high-field limits are smaller than the literature val-ues by amounts of the order of 20%. ACKNOWLEDGMENTS We appreciate very much the effort which was put by the Orsay group into the development of the liquid Pb target atISOLDE. We also wish to thank E. Smolic for experimentalhelp and H. D. Rüter for communication of unpublishedwork. 1M. Kontani, T. Hioki, and Y . Masuda, J. Phys. Soc. Jpn. 32, 416 /H208491972 /H20850. 2E. Klein, Hyperfine Interact. 15/16 , 557 /H208491983 /H20850. 3E. 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PhysRevB.101.134430.pdf

PHYSICAL REVIEW B 101, 134430 (2020) Interplay of large two-magnon ferromagnetic resonance linewidths and low Gilbert damping in Heusler thin films W. K. Peria,1T. A. Peterson,1A. P. McFadden,2T. Qu,3C. Liu,1C. J. Palmstrøm,2,4and P. A. Crowell1 1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 2Department of Electrical & Computer Engineering, University of California, Santa Barbara, California 93106, USA 3Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA 4Department of Materials, University of California, Santa Barbara, California 93106, USA (Received 6 September 2019; revised manuscript received 14 December 2019; accepted 7 April 2020; published 28 April 2020) We report on broadband ferromagnetic resonance linewidth measurements performed on epitaxial Heusler thin films. A large and anisotropic two-magnon scattering linewidth broadening is observed for measurementswith the magnetization lying in the film plane, while linewidth measurements with the magnetization saturatedperpendicular to the sample plane reveal low Gilbert damping constants of (1 .5±0.1)×10 −3,( 1.8±0.2)× 10−3,a n d <8×10−4for Co 2MnSi/MgO, Co 2MnAl/MgO, and Co 2FeAl/MgO, respectively. The in-plane measurements are fit to a model combining Gilbert and two-magnon scattering contributions to the linewidth,revealing a characteristic disorder length scale of 10–100 nm. DOI: 10.1103/PhysRevB.101.134430 I. INTRODUCTION The theoretical understanding of the damping mechanism believed to govern longitudinal magnetization relaxation inmetallic ferromagnets, originally due to Kamberský [ 1,2], has in recent years resulted in quantitative damping estimates forrealistic transition metal band structures [ 3–5]. Although of great interest where engineering of damping is desired [ 6], these calculations remain largely uncompared to experimentaldata. Kamberský damping may be characterized by the so-called Gilbert damping constant αin the Landau-Lifshitz- Gilbert macrospin torque equation of motion, and formallydescribes how the spin-orbit interaction in itinerant electronsystems results in damping of magnetization dynamics [ 2]. Schoen et al. [7] have reported that αis minimized for Co-Fe alloy compositions at which the density of states at the Fermilevel is minimized, in reasonable agreement with Kamber-ský model predictions [ 8]. Furthermore, half-metallic or near half-metallic ferromagnets such as full-Heusler compoundshave been predicted to demonstrate an ultralow Kamberskýα(/lessorequalslant10 −3) due to their spin-resolved band structure near the Fermi level [ 9]. Finally, anisotropy of the Kamberský damping in single crystals has been predicted, which is morerobust for Fermi surfaces with single-band character [ 5,10]. The Gilbert damping constant is often reported through measurements of the ferromagnetic resonance (FMR)linewidth /Delta1H, which may be expressed as a sum of individual contributions /Delta1H=2αf γ+/Delta1H0+/Delta1HTMS, (1) where the first term is the Gilbert damping linewidth ( f is the FMR frequency; γis the gyromagnetic ratio), /Delta1H0 is a frequency-independent inhomogeneous broadening, and/Delta1HTMSrepresents an extrinsic two-magnon scattering (TMS) linewidth contribution [ 11,12] that is, in general, a nonlinear function of frequency. In recent years it has been realized thatTMS linewidths are pervasive for the conventional in-planegeometry of thin film FMR measurements, requiring eitherthe perpendicular-to-plane FMR geometry [ 13] (for which TMS processes are suppressed) or sufficiently broadbandmeasurements [ 14] to extract the bare Gilbert α. For instance, recent FMR linewidth studies on Heusler compounds havereported distinct TMS linewidths [ 15,16], which challenged simple inference of the Gilbert α. In this article, we present FMR linewidth measurements for epitaxial Heusler thin films for all principal orientations ofthe magnetization with respect to the symmetry axes. For thein-plane configuration, large and anisotropic TMS-dominatedlinewidths are observed. In the perpendicular-to-plane con-figuration, for which the TMS process is inactive [ 11], the Gilbert αand inhomogeneous broadening are measured. We find evidence of a low ( ∼10 −3) Gilbert αin these Heusler thin films, accompanied by a large and anisotropic TMScontribution to the linewdith for in-plane magnetization. Weconclude by discussing the interplay of low Gilbert αand large TMS, and we emphasize the nature by which the TMSmay conceal the presence of anisotropic Kamberský α. II. SAMPLES The Heusler alloy films used for these measurements were grown by molecular beam epitaxy (MBE) by coevap-oration of elemental sources in ultrahigh vacuum (UHV).The MgO(001) substrates were annealed at 700 ◦C in UHV followed by growth of a 20 nm thick MgO buffer layerby e-beam evaporation at a substrate temperature of 630 ◦C. The 10 nm thick Co 2MnAl and Co 2MnSi films were grown 2469-9950/2020/101(13)/134430(7) 134430-1 ©2020 American Physical SocietyW. K. PERIA et al. PHYSICAL REVIEW B 101, 134430 (2020) FIG. 1. (a) Wide-angle x-ray diffraction φscans of /angbracketleft202/angbracketright(blue) and/angbracketleft111/angbracketright(red) peaks for the CMS film. (b) Typical derivative susceptibility line shapes for these samples at different microwave excitation frequencies. The fits are shown as solid lines. (c) In- plane hysteresis loops for CFA obtained with a vibrating-samplemagnetometer (VSM). (d) Atomic force microscopy (AFM) image of surface topography for CFA. RMS roughness is 0.2 nm. on the MgO buffer layers at room temperature and then annealed at 600◦C for 15 min in situ in order to improve crystalline order and surface morphology. The 24 nm thickCo 2FeAl sample was grown using the same MgO substrate and buffer layer preparation, but at a substrate temperatureof 250 ◦C with no postgrowth anneal. Reflection high energy electron diffraction (RHEED) was monitored during and aftergrowth of all samples and confirmed the expected epitaxialrelationship of MgO(001) /angbracketleft110/angbracketright| | Heusler (001) /angbracketleft100/angbracketright.X - ray diffraction (XRD) demonstrated the existence of a sin-gle phase of (001)-oriented Heusler, along with the pres-ence of the (002) reflection, confirming at least B2ordering in all cases. In addition, for the Co 2MnSi film only, the (111) reflection was observed, indicating L21ordering [see Fig. 1(a)]. All of the films were capped with several nm of e-beam evaporated AlO xfor passivation prior to atmospheric exposure. The effective magnetization for the 24 nm thickCo 2FeAl film was determined from the anomalous Hall effect saturation field to be 1200 emu /cm3, which is consistent with measurements of Ref. [ 17]f o r L21-o r B2-ordered films, along with 990 emu /cm3and 930 emu /cm3for the Co2MnSi and Co 2MnAl films, respectively. Hereafter, we will refer to the Co 2MnSi(10 nm) /MgO as the “CMS” film, the Co2MnAl(10 nm) /MgO film as the “CMA” film, and the Co2FeAl(24 nm) /MgO film as the “CFA” film. III. EXPERIMENT Broadband FMR linewidth measurements were performed at room temperature with a coplanar waveguide (CPW)transmission setup, similar to that discussed in detail in Refs. [ 18,19], placed between the pole faces of an electro- magnet. A cleaved piece of the sample ( ∼2m m ×1m m ) was placed face down over the center line of the CPW. Arectifying diode was used to detect the transmitted microwavepower, and a ∼100 Hz magnetic field modulation was used for lock-in detection of the transmitted power, resulting ina signal ∝dχ/dH(where χis the film dynamic magnetic susceptibility). The excitation frequency could be varied from0 to 50 GHz, and a microwave power near 0 dBm wastypically used. It was verified that all measurements discussedin this article were in the small precession cone angle, linearregime. The orientation of the applied magnetic field couldbe rotated to arbitrary angle in the film plane (IP), or appliedperpendicular to the film plane (PP). We emphasize againthat TMS contributions are suppressed in the PP configuration[12]. The resonance fields were fit as a function of applied frequency in order to extract various magnetic properties ofthe films. The magnetic free energy per unit volume used to generate the resonance conditions for these samples is given by F M=−M·H+K1sin2φcos2φ+2πM2 effcos2θ, (2) where His the applied field, φandθare the azimuthal and polar angles of the magnetization, respectively, K1is a first order in-plane cubic anisotropy constant, and 4 πMeffis the PP saturation field, which includes the usual demagnetizationenergy and a first order uniaxial anisotropy due to interfacialeffects. The parameters obtained by fitting to Eq. ( 2)a r e shown in Table I. The uncertainty in these parameters was es- timated by measuring a range of different sample pieces, andusing the standard deviation of the values as the error bar. Thelong-range inhomogeneity characteristic of epitaxial samplesmakes this a more accurate estimate of the uncertainty thanthe fitting error. The magnetic-field-swept FMR line shapeswere fit to the derivative of Lorentzian functions [ 19] in order to extract the full width at half maximum linewidths /Delta1H [magnetic field units, Fig. 1(b)], which are the focus of this article. The maximum resonant frequency was determined bythe maximum magnetic field that could be applied for bothIP and PP electromagnet configurations, which was 10.6 kOeand 29 kOe, respectively. For the IP measurement, the angleof the applied field in the plane of the film was varied todetermine the in-plane magnetocrystalline anisotropy of oursamples, which was fourfold symmetric for the three filmscharacterized in this article. The anisotropy was confirmedusing vibrating-sample magnetometry (VSM) measurements,an example of which is shown in Fig. 1(c), which shows IP easy and hard axis hysteresis loops for the CFA film.For the PP measurement, alignment was verified to within∼0.1 ◦to ensure magnetization saturation just above the PP anisotropy field, thus minimizing field-dragging contributionsto the linewidth. IV . RESULTS AND ANALYSIS A. Perpendicular-to-plane linewidths First we discuss the results of the PP measurement. As stated in Sec. III, the TMS extrinsic broadening mechanism is suppressed when the magnetization is normal to the plane 134430-2INTERPLAY OF LARGE TWO-MAGNON FERROMAGNETIC … PHYSICAL REVIEW B 101, 134430 (2020) TABLE I. Summary of the magnetic properties extracted from the dependence of the resonance field on applied frequency for both field in- plane ( ||) and field perpendicular-to-plane ( ⊥) configurations, along with the Gilbert αand inhomogeneous broadening from the perpendicular- to-plane configuration. 2 K1/Msand 4πMeffare the in-plane and perpendicular-to-plane anisotropy fields, respectively [see Eq. ( 2)], and gis the Landé gfactor. Sample 2 K1/Ms(Oe) 4 πM|| eff(kOe) 4 πM⊥ eff(kOe) g||g⊥α001(×10−3) /Delta1H0(Oe) CMS 280 12.3 13.3 2.04 2.04 1 .5±0.19 ±1 CMA 35 11.3 11.7 2.06 2.08 1 .8±0.21 2 ±3 CFA 230 15.1 15.5 2.06 2.07 <0.8 100 ±6 CFA 500◦C anneal N /AN /A 15.1 N /A 2.07 1 .1±0.14 5 ±1 of the film. We can thus fit our data to Eq. ( 1) with /Delta1HTMS= 0, greatly simplifying the extraction of the Gilbert dampingconstant αand the inhomogeneous broadening /Delta1H 0.P r i o r knowledge of /Delta1H0is particularly important for constraining the analysis of the IP measurements, as we shall discuss. The dependence of /Delta1Hon frequency for the CMS, CMA, and CFA films in the PP configuration is summarized inFig. 2, in which fits to Eq. ( 1) are shown with /Delta1H TMS set to zero. For the CMS film, α001=(1.5±0.1)×10−3 and/Delta1H0=9 Oe, while for the CMA film α001=(1.8± 0.2)×10−3and/Delta1H0=12 Oe. Co 2MnSi 2/3Al1/3/MgO and Co2MnSi 1/3Al2/3/MgO films (both 10 nm thick) were also measured, with Gilbert damping values of α001=(1.8± 0.2)×10−3andα001=(1.5±0.1)×10−3, respectively (not shown). For CFA, we obtained a damping value of α001=3× 10−4with an upper bound of α001<8×10−4and/Delta1H0= 100 Oe. These fit parameters are also contained in Table I. The source of the large inhomogeneous broadening for theCFA film is unclear: AFM measurements [Fig. 1(d)] along FIG. 2. Linewidths as a function of frequency with the field applied perpendicular to plane, for which two-magnon scattering is inactive. The black squares are data for the CMS film, the red circlesare for the CMA film, and the blue triangles are for the CFA film. In addition, linewidths are shown for a CFA film that was annealed at 500 ◦Cex situ (magenta diamonds). Corresponding linear fits are shown along with the extracted Gilbert damping factor α. The blue dashed lines indicate an upper bound of α001=8×10−4and a lower bound of α001=0f o rC F A .with XRD indicate that the film is both crystalline and smooth. Note that the range of frequencies shown in Fig. 2are largely governed by considerations involving the Kittel equation [ 20]: measurements below 10 GHz were not used due to the increas-ing influence of slight misalignment on /Delta1H(through field dragging) for resonant fields just above the saturation value. Apiece of the CFA sample was annealed at 500 ◦Cex situ , which reduced the inhomogeneous broadening to ∼45 Oe (still a relatively large value) and increased the Gilbert dampingtoα 001=1.1×10−3(similar behavior in CFA was seen in Ref. [ 21]). The constraint of α001<8×10−4is among the lowest of reported Gilbert damping constants for metallicferromagnets, but the α∼10 −4range is not unexpected based on Kamberský model calculations performed for similar full-Heusler compounds [ 9] or other recent experimental reports [22,23]. It should be noted that Schoen et al. [7] have recently reported α=5×10 −4for Co 25Fe75thin films, where spin pumping and radiative damping contributions were subtractedfrom the raw measurement. Spin pumping contributions tothe intrinsic damping are not significant in our films, as noheavy-metal seed layers have been used and the films havethicknesses of 10 nm or greater. For the radiative dampingcontribution [ 13] in the geometry of our CPW and sample, we calculate a contribution of α rad/lessorsimilar1×10−4, which is below the uncertainty in our damping fit parameter. B. In-plane linewidths With the intrinsic damping and inhomogeneous broadening characterized by the PP measurement, we turn our attention tothe IP linewidth measurements, for which TMS contributionsare present. For hard-axis measurements, frequencies /lessorsimilar5 GHz were not used due to the influence of slight magnetic fieldmisalignment on the linewidths. For easy-axis measurements,the lower limit is determined by the zero-field FMR frequency.Figure 3shows the dependences of the resonance fields and linewidths on the angle of the in-plane field. An importantobservation seen in Fig. 3is that the linewidth extrema are commensurate with those of the resonance fields and thereforethe magnetocrystalline anisotropy energy. This rules out field-dragging and mosaicity contributions to the linewidth, whichcan occur when the resonance field depends strongly on angle[24]. We note that similar IP angular dependence of the FMR linewidth, which was attributed to an anisotropic TMS mech-anism caused by a rectangular array of misfit dislocations, hasbeen reported by Kurebayashi et al. [25] and Woltersdorf and Heinrich [ 14] for epitaxial Fe /GaAs(001) ultrathin films. 134430-3W. K. PERIA et al. PHYSICAL REVIEW B 101, 134430 (2020) FIG. 3. Azimuthal angular dependence of the linewidths (left ordinate, blue circles) and resonance fields (right ordinate, black squares) for (a) CMS, (b) CMA, and (c) CFA. The excitation fre-quency was 20 GHz for CMS, 15 GHz for CMA, and 20 GHz for CFA. The solid lines are sinusoidal fits. To further study the anisotropy of the IP /Delta1Hin our films, we have measured /Delta1Hat the angles corresponding to the extrema of HFMR (and/Delta1H)i nF i g . 3over a range of frequencies. These data are shown in Fig. 4, along with the PP ([001]) measurements for each sample. A distinguish-ing feature of the data shown in Fig. 4is the significant deviation between IP and PP linewidths in all but one case(CMS/angbracketleft100/angbracketright). Large and nonlinear frequency dependence of TABLE II. Summary of the fitting parameters used to fit the in- plane data of Fig. 4(black squares and red circles) to Eqs. ( 1)a n d (3). CFA refers to the unannealed Co 2FeAl sample. Sample (field direction) α(×10−3) ξ(nm) H/prime(Oe) CMS/angbracketleft110/angbracketright 1.6±0.24 0 ±25 55 ±30 CMS/angbracketleft100/angbracketright 1.5±0.14 0 ±25 30 ±15 CMA/angbracketleft110/angbracketright 3.1±0.27 0 ±20 30 ±5 CMA/angbracketleft100/angbracketright 4.7±0.45 5 ±10 90 ±5 CFA/angbracketleft110/angbracketright 2.0±0.32 0 ±10 175 ±60 CFA/angbracketleft100/angbracketright N/AN /AN /AFIG. 4. Linewidths along all three principal directions for CMS (a), CMA (b), and CFA (c). Heusler crystalline axes are labeled by/angbracketleft100/angbracketright(black), /angbracketleft110/angbracketright(red), and [001] (blue). In all three cases, /angbracketleft110/angbracketright is the in-plane easy axis and /angbracketleft100/angbracketrightis the in-plane hard axis. The corresponding fits are shown as the solid curves, where the in-planelinewidths are fit using Eq. ( 3) and the out-of-plane linewidths are fit to the Gilbert damping model. The fit parameters are given in Table II. the IP linewidths is strongly suggestive of an active TMS linewidth broadening mechanism. In the presence of TMS,careful analysis is required to separate the Gilbert dampingfrom the TMS linewidth contributions. We therefore describethe TMS mechanism in more detail in the following sectionin order to analyze the IP linewidths in Fig. 4and extract the Gilbert damping. C. Two-magnon scattering model The TMS mechanism leads to a characteristic nonlinear frequency dependence of /Delta1H[11,12]. In Fig. 4,t h eI P /Delta1His not a linear function of frequency, but possesses the “knee” behavior characteristic of the frequency dependenceof linewidths dominated by the TMS mechanism. We havefit our data to the TMS model described by McMichaeland Krivosik [ 12], in which the TMS linewidth /Delta1H TMS is 134430-4INTERPLAY OF LARGE TWO-MAGNON FERROMAGNETIC … PHYSICAL REVIEW B 101, 134430 (2020) given by [ 26,27] /Delta1HTMS=γ2ξ2H/prime2 df/dH|fFMR/integraldisplay /Gamma10qCq(ξ)δα(ω−ωq)d2q,(3) where /Gamma10qis the defect-mediated interaction term between magnons at wave vector 0 and q,Cq(ξ)=[1+(qξ)2]−3/2is the correlation function of the magnetic system with correla-tion length ξ, and H /primeis the magnitude of the characteristic inhomogeneity (units of magnetic field). The δαfunction in Eq. ( 3) selects only the magnon scattering channels that con- serve energy. In the limit of zero intrinsic damping, it is iden-tical to the Dirac delta function, but for finite αit is replaced by a Lorentzian function of width δω=(2αω/γ )dω/dH. The magnon dispersion relation determining ω qis the usual Damon-Eshbach thin film result [ 26,28] with the addition of magnetocrystalline anisotropy stiffness field terms extractedfrom the dependence of the resonance field on the appliedfrequency for the IP configuration. The film thickness d affects the states available for two-magnon scattering throughthe dispersion relation, namely, the linear term which givesrise to negative group velocity for small q(∝−qd). The IP FMR linewidth data shown in Fig. 4were fit to Eq. ( 1) [with Eq. ( 3)u s e dt oe v a l u a t e /Delta1H TMS] with ξ,α, and H/primeas fitting parameters (shown in Table II). The correlation length ξremains approximately constant for different in-plane direc- tions, while the strength H/primeis larger for the /angbracketleft100/angbracketrightdirections in the CMA and CFA samples and the /angbracketleft110/angbracketrightdirections in the CMS sample. Some degree of uncertainty results fromthis fitting procedure, because for linewidth data collectedover a limited frequency range, ξandαare not completely decoupled as fitting parameters. In absolute terms, however,the largest systematic errors come from the exchange stiffness,which is not well known. The error bars given in Table II were calculated by varying the exchange stiffness over therange 400 meV Å 2to 800 meV Å2, and recording the change in the fit parameters. This range of values was chosen basedon previous Brillouin light scattering measurements of theexchange stiffness in similar Heusler compounds [ 29,30]. In addition, we note that in Eq. ( 1)/Delta1H 0is taken to be isotropic, with the value given by the PP linewidth measurements shownin Fig. 2. Although certain realizations of inhomogeneity may result in an anisotropic /Delta1H 0(see Ref. [ 14] for a good discussion), doing so here would only serve to create anadditional fitting parameter. D. Effect of low intrinsic damping The effect of low intrinsic damping on the two-magnon linewidth can be seen in Fig. 5(a).A sαdecreases, with all other parameters fixed, /Delta1HTMS steadily increases and becomes increasingly nonlinear (and eventually nonmono-tonic) with frequency. In particular, a “knee” in the frequencydependence becomes more pronounced for low damping [see,e.g., Fig. 5(a) curve for α=10 −4]. The physics giving rise to the knee behavior is illustrated in Fig. 5(b).T h eT M S process scatters magnons from zero to nonzero wave vectorat small q. There is assumed to be sufficient disorder to allow for the momentum qto be transferred to the magnon system. There will always be, however, a length scale ξbelow which the disorder decreases, so that the film becomes effectivelyFIG. 5. (a) Two-magnon scattering linewidth contribution for values of Gilbert damping α=10−2,5×10−3,10−3,and 10−4.T h e inset shows magnon dispersions for an applied field of H=1k O e . (b) Contours of the degenerate mode wave number q2Min the film plane as a function of wave vector angle relative to the magnetizationforf FMR=16, 24, and 32 GHz. The dashed circle indicates the wave number of a defect with size ξ=100 nm. more uniform at large wave vectors. The corresponding FMR frequencies are those for which the contours of constantfrequency (the figure eights in Fig. 5)i nqspace have extrema atq∼ξ −1. The TMS rate is also determined by the interplay of the magnon density of states, the effective area in qspace occupied by the modes that conserve energy, and the Gilbertdamping. The knee behavior is more pronounced for lowαdue to the increased weight of the van Hove singularity coming from the tips of the figure eights, in the integrandof Eq. ( 3). Although a larger window of energies, set by the width of δ α, is available for larger α, this smears out the sin- gularity in the magnon density of states, removing the sharpknee in the TMS linewidth as a function of frequency. ThePP measurement confirms that all of these epitaxial Heuslerfilms lie within the range α< 2×10 −3. Ferromagnetic films with ultralow αare therefore increasingly prone to large TMS linewidths (particularly for metals with large Ms). The TMS linewidths will also constitute a larger fraction ofthe total linewidth due to a smaller contribution from the 134430-5W. K. PERIA et al. PHYSICAL REVIEW B 101, 134430 (2020) Gilbert damping. In practice, this is why experimental re- ports [ 7,22,23] of ultralow αhave almost all utilized the PP geometry. E. Discussion The results of the IP linewidth fits to Eqs. ( 1) and ( 3) are summarized in Table II. In the case of CMS, the high- frequency slopes in Fig. 4(a) approach the same value along each direction, as would be expected when the frequency islarge enough for the TMS wave vector to exceed the inverseof any defect correlation length. In this limit, αis isotropic (within error limits). Next, we discuss the CMA IP data shown in Fig. 4(b) and Table II. It is clear from this figure that a good fit can be obtained along both /angbracketleft100/angbracketrightand/angbracketleft110/angbracketrightdirections. In Table II it can be seen that the value of the defect correlation length ξ is approximately the same along both directions. However, thevalues of αwe obtain from fitting to Eqs. ( 1) and ( 3) do not agree well with the PP value of α 001=1.8×10−3(Fig. 2). Anisotropic values of αhave been both predicted [ 5,10] and observed [ 31], and an anisotropic αis possibly the explanation of our best-fit results. The in-plane /angbracketleft100/angbracketrightand [001] directions are equivalent in the bulk, so the anisotropy would necessarilybe due to an interface anisotropy energy [ 31] or perhaps a tetragonal distortion due to strain [ 32]. Finally, we discuss the CFA linewidths shown in Fig. 4(c) and Table II. This sample has by far the largest two-magnon scattering contribution, which is likely related to the anoma-lously large inhomogeneous broadening and low intrinsicdamping [see Fig. 5(a)] observed in the PP measurement. A good fit of the data was obtained when the field was applied along the /angbracketleft110/angbracketrightdirection. Notably, the IP /angbracketleft110/angbracketrightbest fit value of 2.1×10 −3is nearly a factor of 3 larger than the α001upper bound on the same sample (Table I), strongly suggesting an anisotropic Gilbert α. A striking anisotropy in the IP linewidth was revealed upon rotating the magnetization to the /angbracketleft100/angbracketright orientation. For the /angbracketleft100/angbracketrightcase, which yielded the largest TMS linewidths measured in this family of films, we were not able to fit the data to Eq. ( 3) using a set of physically reason- able input parameters. We believe that this is related to theconsideration that higher order terms in the inhomogeneousmagnetic energy (see Ref. [ 26]) need to be taken into account. Another reason why this may be the case is that the model ofMcMichael and Krivosik [ 12] assumes the inhomogeneities to be grainlike, whereas the samples are epitaxial [see Fig. 1(a)]. Atomic force microscopy images of these samples [Fig. 1(d)] imply that grains, if they exist, are much larger than the defectcorrelation lengths listed in Table II, which are of order 10’s of nm. We also note that there does not appear to be a correlationbetween the strength of two-magnon scattering H /primeand the cubic anisotropy field 2 K1/Ms, which would be expected for grain-induced two-magnon scattering. V . SUMMARY AND CONCLUSION We conclude by discussing the successes and limitations of the McMichael and Krivosik [ 12] model in analyzing our epi- taxial Heusler film FMR linewidth data. We have shown thattwo-magnon scattering is the extrinsic linewidth-broadeningmechanism in our samples. Any model which takes this asits starting point will predict much of the qualitative behavior we observe, such as the knee in the frequency dependenceand the large linewidths IP for low αfilms. The TMS model used in this article (for the purpose of separating TMS andGilbert linewidth contributions) is, however, only as accu-rate as its representation of the inhomogeneous magneticfield and the underlying assumption for the functional formofC q(ξ). Grainlike defects are assumed, which essentially give a random magnetocrystalline anisotropy field. We didnot, however, explicitly observe grains in our samples withAFM, at least below length scales of ∼10μm [Fig. 1(d)]. Misfit dislocations, a much more likely candidate in ouropinion, would cause an effective inhomogeneous magneticfield which could have a more complicated spatial profileand therefore lead to anisotropic two-magnon scattering (seeRef. [ 14]). The perturbative nature of the model also brings its own limitations, and we believe that the CFA /angbracketleft100/angbracketrightdata, for which we cannot obtain a satisfactory fit, are exemplaryof a breakdown in the model for strong TMS. Future workshould go into methods of treating the two-magnon scatteringdifferently based on the type of crystalline defects present,which will in turn allow for a more reliable extraction of theGilbert damping αand facilitate the observation of anisotropic Gilbert damping, enabling quantitative comparison to first-principles calculations. Regardless of the limitations of the model, we emphasize three critical observations drawn from the linewidth measure-ments presented in this article. First, in all cases we observelarge and anisotropic TMS linewidth contributions, whichimply inhomogeneity correlation length scales of order tensto hundreds of nanometers. The microscopic origin of theseinhomogeneities is the subject of ongoing work, but is likelycaused by arrays of misfit dislocations [ 14]. The relatively large length scale of these defects may cause them to be easilyoverlooked in epitaxial film characterization techniques suchas XRD and cross-sectional HAADF-STEM, but they stillstrongly influence magnetization dynamics. These defects andtheir influence on the FMR linewidth through TMS compli-cate direct observation of Kamberský’s model for anisotropicand (in the case of Heusler compounds) ultralow intrinsicdamping in metallic ferromagnets. Second, we observed lowintrinsic damping through our PP measurement, which was<2×10 −3for all of our samples. Finally, we have presented the mechanism by which FMR linewidths in ultralow dampingfilms are particularly likely to be enhanced by TMS, theanisotropy of which may dominate any underlying anisotropicKamberský damping. ACKNOWLEDGMENTS This work was supported by NSF under Grant No. DMR- 1708287 and by SMART, a center funded by nCORE, aSemiconductor Research Corporation program sponsored byNIST. The sample growth was supported by the DOE un- der Grant No. DE-SC0014388 and the development of the growth process by the Vannevar Bush Faculty Fellowship(ONR Grant No. N00014-15-1-2845). Parts of this work werecarried out in the Characterization Facility, University ofMinnesota, which receives partial support from NSF throughthe MRSEC program. 134430-6INTERPLAY OF LARGE TWO-MAGNON FERROMAGNETIC … PHYSICAL REVIEW B 101, 134430 (2020) [1] V . Kamberský, On the Landau-Lifshitz relaxation in ferromag- netic metals, Can. J. Phys. 48,2906 (1970 ). [2] V . 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PhysRevApplied.13.061002.pdf

PHYSICAL REVIEW APPLIED 13,061002 (2020) Letter Editors’ Suggestion Manipulation of Coupling and Magnon Transport in Magnetic Metal-Insulator Hybrid Structures Yabin Fan,1,*,‡P. Quarterman ,2,†,‡Joseph Finley,1Jiahao Han,1Pengxiang Zhang,1Justin T. Hou,1 Mark D. Stiles ,3Alexander J. Grutter,2and Luqiao Liu1 1Microsystems Technology Laboratories, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2NIST Center for Neutron Research, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899, USA 3Physical Measurement Laboratory, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899, USA (Received 9 March 2020; accepted 1 May 2020; published 15 June 2020) Ferromagnetic metals and insulators are widely used for generation, control, and detection of magnon spin signals. Most magnonic structures are based primarily on either magnetic insulators or ferromagneticmetals, while heterostructures integrating both of them are less explored. Here, by introducing a Pt/yttrium iron garnet (YIG)/permalloy (Py) hybrid structure grown on a Si substrate, we study the magnetic cou- pling and magnon transmission across the interface of the two magnetic layers. We find that within thisstructure, Py and YIG exhibit an antiferromagnetic coupling field as strong as 150 mT, as evidenced by both magnetometry and polarized neutron reflectometry measurements. By controlling individual layer thicknesses and external fields, we realize parallel and antiparallel magnetization configurations, whichare further utilized to control the magnon current transmission. We show that a magnon spin valve with an on:off ratio of approximately 130% can be realized out of this multilayer structure at room temper- ature through both spin pumping and spin-Seebeck-effect experiments. Owing to the efficient control ofmagnon current and the compatibility with Si technology, the Pt/YIG/Py hybrid structure could potentially find applications in magnon-based logic and memory devices. DOI: 10.1103/PhysRevApplied.13.061002 Heterostructures that integrate magnetic insulators and ferromagnetic metals are drawing widespread attention due to their rich magnonic physics. Specifically, stand- ing spin waves (SSWs) and interlayer magnon-magnon coupling have been detected in such hybrid structures [ 1– 3], with coupled layers of the magnetic insulator yttrium iron garnet (YIG) and a soft ferromagnetic metal [such as Co, (Co,Fe)B, and Ni]. In these structures, the dynamic torques generated from interlayer exchange coupling can lead to anticrossings between magnon modes unlocking functionalities with critical implications both in the clas- sical [ 4,5] and quantum domains [ 1–3,6]. In addition to magnon-magnon coupling, the interlayer exchange inter- action in YIG/ferromagnetic metal bilayers can enable additional magnonic functions [ 7] such as the magnon spin-valve effect [ 8,9]. In magnon spin valves, the trans- mission coefficient of magnons propagating through the heterostructure is tuned by the parallel and antiparallel *yabin_fan@hotmail.com †patrick.quarterman@nist.gov ‡These authors contributed equally to this work.orientations between the magnetization of two magneticlayers. One advantage of such a magnonic spin-valve device is that information is encoded in the form of magnons and a net charge current is not required in prin- ciple, avoiding Joule-heating-related dissipation in con- ventional spin-valve structures. In existing studies, YIG layers epitaxially grown on Gd 3Ga5O12(GGG) substrate are generally utilized. However, for practical device appli- cations, spin-valve structures grown on silicon substrates are preferred [ 10,11]. Stronger coupling between YIG and ferromagnetic metals may provide easier customization of the magnetization orientation in the magnon spin-valve structures. In this work, we demonstrate strong magnetic coupling in the Si/Pt/YIG/permalloy (Py) multilayer structures. We show that a pronounced antiferromagnetic coupling exists between polycrystalline YIG and Py layers in the low- field regime (defined as 0 to 50 mT), with the two layers aligning along the same direction only when the external field exceeds 150 mT. Moreover, through spin-pumping and spin-Seebeck experiments, we demonstrate that this YIG/Py hybrid structure could serve as an efficient magnon spin valve. The YIG/Py hybrid structures grown on Si 2331-7019/20/13(6)/061002(6) 061002-1 © 2020 American Physical SocietyYABIN FAN et al. PHYS. REV. APPLIED 13,061002 (2020) (a) (c) (b) FIG. 1. (a) Schematic of the Pt (10)/YIG(40)/Py(20)/Ru(3) hybrid structure grown on the Si/SiO 2substrate. (b) Atomic force microscopy image of the YIG surface for the Si/Pt (10)/YIG(40)film, indicating a roughness of around 1 nm. (c) Vibrating sample magnetometry measurements of the Si/Pt (10)/YIG(40)/Py(20)/Ru(3) sample and Si/Pt (10)/YIG(40)/MgO(3)/Py(20)/Ru(3) sample. The inset shows results from the control samples of Si/Py (20)/Ru(3)and Si/Pt (10)/YIG(40). Schematics show the magnetization ori- entation of the YIG and Py layers in the Si/Pt (10)/YIG(40)/Py(20)/Ru(3)hybrid structure in different magnetic field regions. M sis integrated over the multilayer thickness. represent a semiconductor industry-compatible technique for implementing magnonic spin valves and thus have broad application in ultralow-power magnonic devices and circuits. We first deposit Pt (10)/YIG(40)thin films (units in nanometers) on Si/SiO 2substrates by magnetron sputter- ing [12–15], followed by rapid thermal annealing (RTA) in an oxygen environment. To characterize the film quality, atomic force microscopy (AFM) measurements are per- formed [Fig. 1(b)], which indicate a surface roughness of approximately 1 nm. The polycrystalline nature of the YIG layer is verified by x-ray diffraction (Fig. S1 within the Supplemental Material) [ 16]. After annealing, a 20-nm Py thin film is grown on top of the YIG layer, followed by a 3-nm Ru passivation layer [see Fig. 1(a)]. To characterize the magnetic properties of the hybrid structure, we collect vibrating sample magnetometry (VSM) data at room temperature with magnetic fields applied within the sample plane. As shown in Fig. 1(c), the M-H curve of the Pt/YIG/Py sample shows seg- mented switching features. After the magnetization sharply switches polarity near Bx=0 T, it does not immediately reach the saturated magnetization state. Instead, it gradu- ally increases, reaching saturation at around Bx=150 mT. In order to understand this peculiar behavior, we measure a set of control samples. For the control samples of Py(20)and Pt (10)/YIG(40), the M-Hcurves exhibit typ- ical easy-axis hysteresis loops with low coercive field and square switching shape (the ratio of remanent over satu- ration magnetization ,M r/M s≈1), as plotted in the inset of Fig. 1(c). Comparing the magnetization of these threesamples, we find that in the low-field region, the net magnetic moment Mtotalfrom the Pt/YIG/Py sample is equal to the value of M(Py)- M(YIG), suggesting antifer- romagnetic coupling between these two layers. When the applied field is increased, the net moment Mtotalfrom the hybrid structure gradually increases until it reaches the sum of M(Py) and M(YIG) at approximately 150 mT, where both the Py and YIG magnetizations align with the field. To examine the detailed mechanisms of the observed antiferromagnetic coupling, we grew a control sample of Si/Pt (10)/YIG(40)/MgO(3)/Py(20), where the MgO layer serves as a spacer to prevent direct exchange coupling between the YIG and Py layers. In contrast to the Pt/YIG/Py sample, the M-Hcurve of this control sample shows full switching near Bx=0 T [Fig. 1(c)], which sug- gests that exchange interaction rather than the dipolar field is responsible for the observed coupling. In order to directly measure the magnetization of individual layers, we used polarized neutron reflectom- etry (PNR) [ 17] to probe the depth dependence of the composition and in-plane magnetization. Figure 2(a) shows a typical set of PNR data obtained from the Si/Pt(10)/YIG(40)/Py(20)sample under 4 mT of exter- nal magnetic field (reached by first saturating to 700 mT and then lowering the field). R++and R−−represent neu- tron reflectivity for the non-spin-flip channels and Qis the neutron-beam wave-vector transfer during the reflec- tion. The solid lines represent theoretical reflectivity curves generated from the scattering length density depth profiles shown in Fig. 2(c). A series of data sets obtained under fields from 700 to 1.5 mT are illustrated in Figs. S2 061002-2MANIPULATION OF COUPLING AND. . . PHYS. REV. APPLIED 13,061002 (2020) (a) (c)(b) FIG. 2. (a) Polarized neutron reflectivity for the spin-polarized R++and R−−channels. Points represent experimental results and solid lines are theoretical fits. Error bars indicate single standard deviation uncertainties. The results are obtained at room temperature with a 4 mT in-plane field. (b) Spin asymmetry between the two channels for data shown in (a). (c) Structural (nuclear) and magnetic scattering length density profiles for the multilayer structure under different in-plane field conditions. and S3 within the Supplemental Material [ 16]. Figure 2(b) shows the calculated spin-asymmetry result, which is defined as SA=(R++−R−−)/(R+++R−−) and highlights the magnetic components of the reflectometry. As described in the experimental methods section [ 16], we obtain the scattering length density (SLD) profiles [Fig. 2(c)], which provide information on the orientation and magnitude of in-plane magnetization as a function of depth from the sample surface. Under high fields, YIG and Py layers both align parallel to the applied field. Upon the reduction of applied magnetic field, the Py magnetization remains roughly unchanged while that from YIG decreases significantly. When the field is lowered to 15 mT, the mag- netization of the YIG layer aligns such that approximately 70% of its saturated magnetization ( M s) is antiparallel to the magnetic field (and Py magnetization). The PNR results, including the onset field for YIG magnetization reversal as well as the relative magnitude of the magnetic moment of the different layers during the switching, are in good agreement with the M-Hcurve shown in Fig. 1(c). We also use PNR to characterize the magnetization switching process on the control sample of Si/Pt (10)/YIG(40)/MgO(3)/Py(20). Consistent with the VSM results, with MgO insertion, the YIG and Py layers remain aligned parallel to the applied magnetic field under both high- and low-field regimes in this sample (Fig. S5 within the Supplemental Material) [ 16], indicating the exchange interaction as the coupling mechanism in Si/Pt/YIG/Py. In addition to the Si/Pt (10)/YIG(40)/Py(20)sam- ple, whose net magnetization is dominated by the Py layer at low field, we also measure a sample of Si/Pt(10)/YIG(40)/Py(2)/Ru(4), in which the magnetic moment from YIG dominates. From both VSM and PNR measurements, we observe that in contrast to the Si/Pt(10)/YIG(40)/Py(20)sample, in this control sample the YIG magnetization remains parallel to the external in- plane field, while the Py magnetization aligns opposite to the field direction in the low-field domain, as is shown in Fig. S4 within the Supplemental Material. The full PNR data with theoretical fits can be found in Sec. 4 within the Supplemental Material [ 16]. Previously the magnon spin-valve effect has been real- ized in magnetic multilayers. In these experiments, in order to isolate the coupling between two ferromagnetic layers 061002-3YABIN FAN et al. PHYS. REV. APPLIED 13,061002 (2020) and allow both parallel and antiparallel configurations, an insertion layer made from an antiferromagnetic insulator or a paramagnetic metal [ 8,9] has been employed. Because of the intrinsic, antiferromagnetic coupling between the YIG and the Py layers, their relative magnetic orientation can be toggled between the two opposite states without the need for a spacer layer. We perform both spin-pumping and spin-Seebeck effect (SSE) measurements to study the mod- ulation on magnon current transport in this hybrid structure [Figs. 3(b) and3(d)]. As shown in Fig. 3(a), a spin-pumping device is fab- ricated out of the Si/Pt (10)/YIG(40)/Py(20)/Ru(3)stack with electrical contacts made only onto the Pt layer (see Methods) [ 16]. The device is mounted onto a rf waveguide,and two dc electrodes are connected to the two sides of the Pt layer to measure the magnon spin current injected into Pt through inverse spin Hall effect (ISHE) [ 18–21]. As shown in Fig. 3(b), spin-pumping signals are observed under the driving rf frequencies between 3 and 9 GHz. By plot- ting the relationship between rf frequency and resonance field, we identify that the detected resonance signal corre- sponds to the contribution from the Py layer. This is further verified with separate ferromagnetic resonance measure- ments, where no obvious resonance peaks are observed from the YIG layer due to its polycrystalline nature (see Sec. 7 within the Supplemental Material) [ 16]. Moreover, a large dc resistance (up to 100 M /Omega1, see Fig. S6 within the Supplemental Material) [ 16] is measured between the (a) (b) (c) (d) FIG. 3. (a) Schematics of the Py spin-pumping process when the Py and the YIG magnetizations are in the antiparallel (upper panel) and parallel (lower panel) configurations under the low-field and high-field regimes, respectively. (b) Spin-pumping voltages measured from the ISHE in the Pt layer when the Py magnetization is excited to ferromagnetic resonance by external rf field in the Si/Pt(10)/YIG(40)/Py(20)/Ru(3)hybrid structure. The spin-pumping voltages are normalized by the microwave power under different frequencies. Inset: resonance field versus frequency. (c) Comparison of the field-dependent spin-pumping voltages measured in the Si/Pt(10)/YIG(40)/Py(20)/Ru(3)structure and the control structure of Si/Pt (10)/Py(20)/Ru(3). (d) Spin-Seebeck voltages measured in the Si/Pt (10)/YIG(40)/Py(20)/Ru(3)hybrid structure, when the top Py (20)/Ru(3)is in contact with a ceramic electrical heater (maintained at 50 °C) and the bottom substrate is attached to a Peltier cooler (maintained at 25 °C). Spin-Seebeck data measured in a Si/Pt(10)/YIG(40)control sample is also plotted. 061002-4MANIPULATION OF COUPLING AND. . . PHYS. REV. APPLIED 13,061002 (2020) Py/Ru top layer and the Pt underlayer in our experiment, suggesting that the thick YIG layer can completely iso- late the direct electrical current flow from Py to Pt. This allows us to exclude additional contributions from the rf rectification effect within the Py layer [ 22–24]. Therefore, the obtained signals can be directly attributed to the spin- pumping mechanism without relying on detailed analysis of the resonance lineshape [ 25]. We characterize the spin-pumping signal as a function of the rf signal frequency (or equivalently, the resonance field Bres). We note that under the lowest applied field (rf fre- quency f=3 GHz), the spin-pumping voltage VSPremains small. With the increase of f(from 3 to 9 GHz) and Bres,VSPincreases from 15 to 34.2 nV/mW. To understand the evolution of VSP, we carry out a control experiment on a simple Pt/Py bilayer film. As is illustrated in Fig. 3(c), a different trend has been observed in the Pt/Py sam- ple, where VSPdecreases with the increase of resonance frequency. This latter trend is also consistent with previ- ous reports [ 26–28] in similar spin-pumping experiments, which can be explained by the reduction of precession cone angle under a higher driven frequency (or equiv- alently, a larger external magnetic field). The observed monotonic increase of VSPas a function of frequency in Si/Pt/YIG/Py hybrid structure is consistent with the magnon spin-valve mechanism as schematically illustrated in Fig. 3(a), where the antiparallel configuration between the two magnetic layers blocks part of the magnon spin transport by lowering the spin transmission coefficient at the interface. In addition to the spin-pumping experiment, we carry out spin-Seebeck-effect measurements in which a tem- perature gradient of 25 K is created along the vertical direction in the Si/Pt/YIG/Py structure. As plotted in Fig. 3(d), the spin-Seebeck voltage VSSEdetected in the Pt layer increases monotonically with the in-plane magnetic field from 0 to 0.1 T, consistent with the scenario that the par- allel configuration between Py and YIG magnetizations allows more magnon transmission from Py through the YIG layer than the antiparallel case. Importantly, we notice that even in the low-field regime (from 0 to 50 mT), where Py and YIG are mostly antiparallel, the VSSEin Pt/YIG/Py is greater than the VSSEmeasured in a Pt/YIG control sam- ple, suggesting that magnons generated from the Py layer dominate. The measured antiferromagnetic coupling between Py and YIG corresponds to an interfacial exchange energy of approximately 8.6 ×10−4J/m2, which is orders of mag- nitude stronger than the value reported in single-crystal YIG/Py hybrid structure [ 29]. The strong intrinsic antifer- romagnetic coupling between Py and YIG layers in ourstructure directly facilitates the realization of magnonic spin-valve effect. The elimination of extra spacer lay- ers avoids additional spin scattering during magnon con- versions, which not only enhances the efficiency butalso removes the constraints set by the spacer layer, such as antiferromagnetic Néel transition temperature [30–32]. In our spin-pumping experiment, the magnonic spin-valve effect can be evaluated as (V↑↑ SP−V↑↓ SP)/V↑↓ SP= 130%, which is comparable to the value measured in the YIG/CoO/Co structure reported previously [ 9], except that our measurement is carried out at room temperature while previous results are obtained under 160 K. In our experi- ment, the magnon spin valve switches under high and low magnetic field. Further nanoscale fabrication can intro- duce shape anisotropy into the magnetic layers, which will allow the realization of bistability between the parallel and antiparallel states and work as a nonvolatile switch. The fact that the magnonic spin valve operates efficiently at room temperature and it can be integrated with other Si-based electronics suggests that this material system can provide a nice platform for realizing magnon-based spin logic and memory devices. Additional references cited within the Supplemental Material are included here [33–35]. Acknowledgments. We thank Julie Borchers for invalu- able discussions. This work is supported in part by the National Science Foundation under Grant No. ECCS- 1808826 and by SMART, one of seven centers of nCORE, a Semiconductor Research Corporation program,sponsored by National Institute of Standards and Tech- nology (NIST). The authors also acknowledge sup- port from AFOSR under award FA9550-19-1-0048. 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PhysRevB.101.214432.pdf

PHYSICAL REVIEW B 101, 214432 (2020) Enhanced skyrmion motion via strip domain wall Xiangjun Xing ,1,*Johan Åkerman,2,3and Yan Zhou4,† 1School of Physics & Optoelectronic Engineering, Guangdong University of Technology, Guangzhou 510006, China 2Department of Physics, University of Gothenburg, Fysikgränd 3, 412 96 Gothenburg, Sweden 3Material & Nano Physics, School of ICT, KTH Royal Institute of Technology, 164 40 Kista, Sweden 4School of Science & Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China (Received 13 March 2020; revised manuscript received 23 May 2020; accepted 28 May 2020; published 19 June 2020) When magnetic skyrmions move under spin-orbit torque in magnetic nanowires, they experience a skyrmion Hall effect, which pushes them towards the nanowire edge where they risk being annihilated; this puts an upperlimit on how fast they can be driven. However, the same magnetic multilayer harboring skyrmions can sustaina Néel-type strip domain wall along the nanowire length, potentially keeping the skyrmions separated fromthe edge. Here we study the interplay between current driven skyrmions and domain walls and find that theyincrease the annihilation current and allow the skyrmions to move faster. Based on the Thiele formalism, weconfirm that the emergent longitudinal repulsive force and the modified energy landscape linked to the domainwall are responsible for the enhanced skyrmion motion. Furthermore, we identify that the longitudinal repulsiveforce emerges because of the broken axisymmetry in the local magnetization in front of the skyrmion. Our studyuncovers key aspects in the interplay between two topological magnetic textures from different homotopy groupsand may inspire new device concepts. DOI: 10.1103/PhysRevB.101.214432 I. INTRODUCTION Magnetic skyrmions are localized topological solitons that have a spin structure with integer topological charge [ 1–3]. Apart from the topological stability and accessible very smallsize, skyrmions exhibit emergent electrodynamics [ 4–11]. As such, magnetic skyrmions have attracted intense researchactivities in the past few years in the hope to bring novel spin-based data storage and information processing applicationsto market. Flowing magnetic skyrmions tend to deflect theirtrajectories from the current direction, experiencing what iscalled a skyrmion Hall effect, owing to a transverse Mag-nus force associated with the nonzero topological charge[12–14]. Accordingly, in confined magnetic nanostructures, e.g., nanowires, magnetic skyrmions usually move steadilyalong one of the two long edges, where the confining force andthe Magnus force are balanced at moderate current densities[10,12,15]. However, once the current density surpasses a critical value, the skyrmions will touch the sample boundaryand be annihilated [ 16–18]. Apart from Néel skyrmions [ 12,13,19–23], chiral Néel domain walls [ 24,25] can stably exist in magnetic multi- layer nanowires with interfacial Dzyaloshinskii-Moriya inter-action (iDMI) [ 26–28] and perpendicular magnetic anisotropy (PMA). Previous studies revealed that Slonczewski-type spin-transfer torque could drive the motion of Néel-type skyrmionsand domain walls very efficiently [ 10,21,24,25,29,30]. A re- cent study [ 31] demonstrated that a Néel-type strip domain *xjxing@gdut.edu.cn †zhouyan@cuhk.edu.cnwall aligned along the nanowire length can be stabilized by the Slonczewski-type spin torque if the current used is notexcessively large. Elongated strip domain walls in magneticnanowires have proven robust magnonic waveguides that en-hance spin-wave transmission [ 32]. A method of controllably writing strip domain walls into magnetic nanowires has alsobeen established [ 31]. Using the strip domain wall as a buffer layer, it appears possible to improve the dynamic behavior ofmagnetic skyrmions subject to Slonczewski-type spin torque. In this work, we use micromagnetic simulations alongside theoretical modeling to the current-driven motion of magneticskyrmions in a magnetic multilayer nanowire containing astrip domain wall (mediated skyrmions). For comparison, themotion of magnetic skyrmions in the same nanowire withoutincluding strip domain wall (bare skyrmions) is also consid-ered. To ensure the general validity of our results, we examinea wide range of values of those material parameters that aresensitive to the multilayer interfacial condition [ 19,20,33]. Throughout the considered range of material parameters,we find that the skyrmion motion under Slonczewski spintorque is enhanced by the domain wall and the accompanyingskyrmion Hall effect is suppressed. By virtue of the Thieleapproach [ 12,20,34], we clarify the mechanism behind the observed behaviors. Our study opens a paradigm for theinterplay and manipulation of different topological magnetictextures. II. RESULTS A. Device structure, model, and simulations The platform of this study is magnetic nanowires pat- terned from an ultrathin multilayer film, which has a 2469-9950/2020/101(21)/214432(11) 214432-1 ©2020 American Physical SocietyXIANGJUN XING, JOHAN ÅKERMAN, AND YAN ZHOU PHYSICAL REVIEW B 101, 214432 (2020) HM1/FM/AO(HM2)-like structure to generate iDMI and PMA, where FM is a ferromagnetic layer, HM1 and HM2represent heavy-metal layers with strong spin-orbit coupling,and AO stands for a metal oxide layer. Experimentally, thepossible combinations of materials could be Pt /Co/AlO x [22,23,33], Pt/CoFeB/MgO [ 13,21,24,35], Ta/CoFeB /TaO x [7,12,36], Pt/Co/Ta [ 21], etc. Depending on the interfacial environment, layer thickness, and specific combination ofmaterials, the interface-sensitive material parameters, i.e., theiDMI and PMA can vary over a large range. Practically, mag-netic skyrmions can be written into the multilayer nanowirethrough a local nanocontact spin valve or magnetic tunneljunction [ 37], and strip domain walls can be injected into the multilayer nanowire from the wire terminals using anestablished approach [ 31]. Overall, the architecture of an op- erational device is analogous to that in Ref. [ 31], but here we only consider straight magnetic nanowires and concentrate onthe magnetic dynamics induced by Slonczewski spin torque.As usual, the Slonczewski spin torque is provided by a verticalspin current resulting from the spin Hall effect or a magnetictunnel junction [ 7,10,21,23–25,29,30,31,35,38]. We perform micromagnetic simulations to find the solution to the formulated question by numerically integrating the ex-tended Landau-Lifshitz-Gilbert equation with a spin-transfertorque [ 10,29,30], ∂m/∂t=−γ(m×H eff)+α(m×∂m/∂t)+T,(1) where mis the unit vector of the magnetization Mnormal- ized by its saturation value Msandtis the time; Heff= −(1/μ0)δE/δMis the effective field in the FM layer with μ0 denoting the vacuum permeability and Eis the total energy incorporating the contributions of magnetostatic, PMA, ex-change, iDMI, and Zeeman interaction terms; Trepresents the Slonczewski spin torque [ 39];γis the gyromagnetic ratio and αis the Gilbert damping constant. For simplicity, we do not take into account the out-of-plane fieldlike torque. Also, theZhang-Li torques were not included in our model since theyare negligible even for the current-in-plane geometry [ 31]. The finite-difference code MuMax3 [ 40] was used to im- plement all the numerical calculations, in which only the FMlayer is explicitly addressed. We do not directly incorporatethe HM and AO layers in our simulations, but instead takeaccount of the physical effects arising from them. In a realdevice, the HM1 layer is responsible for the generation ofspin currents in the FM layer via spin Hall effect and forthe creation of iDMI together with the AO or HM2 layer viaforming asymmetric interfaces. The thickness of the FM layer,d FM, is set to be 1 nm. The width of the nanowire is 100 nm in most cases and other values are also considered for specialpurposes. The nanowire length varies with the wire widthbut has a minimum of 1 μm. We examine the equilibrium magnetic configurations and their current-induced dynamicsin a device with either or both of the skyrmion and stripdomain wall over a broad range of K uand D. The pre- sented results are based on the following material parametersunless otherwise specified: saturation magnetization M s= 580 kA m−1, exchange stiffness A=15 pJ m−1, perpendic- ular magnetic anisotropy Ku=0.7M Jm−3, iDMI strength D=3.0m Jm−2, and Gilbert damping constant α=0.3. These parameters are typical experimental values reportedfor the HM1/FM/AO(HM2) multilayer systems [ 7,12,13,21– 24,33,35,36]. For computation, the FM layer is divided into an array of 1 ×1×1n m3cubic cells, which are much smaller than the exchange length lex=/radicalBig 2A/μ0M2 S≈8.4n m ( t h e maximum length beyond which the short-range exchange interaction cannot keep all the magnetic moments parallel),and open boundary conditions are assumed. In this study, we suppose that the Slonczewski spin torque stems from spin Hall effect [ 38], so that T= −γτ H(m×σ×m), where τH=¯hJ/Phi1H/2eμ0MsdFMand σ=ˆJ׈z is the spin current polarization direction with ¯ h denoting the reduced Planck constant, Jis the electrical cur- rent density, /Phi1His the spin Hall angle, eis the elementary charge, ˆJ is the unit vector in the electrical current direction, and ˆ z is the unit vector along the +zaxis. For dynamic simulations, the spin Hall angle /Phi1His set to be 0.13 [ 35,38] and the electric current in the nanowire is along −x.I nt h e multilayer structure, we assume that the FM layer is on top ofthe HM1 layer. Then, the electrons’ spin orientation σ=ˆJ׈z will orient along +y. For each pair of K uandD, the dynamic simulations are done for a series of current densities with aninterval of 0 .1×10 11Am−2. B. Domain-wall dynamics To control the skyrmion motion under spin-orbit torque, the strip domain wall must be stable against the same torque.Therefore, first of all, we need identify the stability window ofstrip domain wall with respect to the driving electric currentby multiple sets of simulations. Our calculations indicate that,for all the ( K u,D) combinations, the strip domain wall is not affected if the current density does not exceed 4 .0× 1012Am−2, and otherwise it will collapse once a much larger current is applied. The two different situations are illustratedin Fig. 1. Figures 1(a)and1(g) display the initial static strip domain wall, which serves as the starting point of the dynamic sim-ulations. In two separate simulations, the initial strip domainwall is supplied with lower and higher currents, respectively,and the subsequent temporal evolutions of the domain wallare recorded. The corresponding results are presented inFigs. 1(b)–1(f) and1(h)–1(l). Clearly, at the lower current, the strip domain wall maintains its original profile, whereas atthe higher current, the original narrow domain wall expandsimmediately after the current action, and meanwhile, its leftend starts to divide into two branches. Rapidly, the stripdomain wall develops a stripy substructure and the divisionextends deep into the interior of the nanowire. As depicted in Figs. 1(a) and1(g), for the applied current I, the electrons’ spin orientation σin the FM layer and the magnetization orientation min the domain wall are parallel at the center of the domain wall, and thus the torque T∼ m×σ×mvanishes therein regardless of the strength of the current density [ 31]. However, in the upper and lower mag- netic domains, the magnetic moments are aligned along the z axis and thereby the torque T∼m×σ×m=ˆz׈y׈z= ˆy. When the current becomes considerably large, the torque will overcome the PMA and make the out-of-plane magneticmoments near the domain wall rotate to the yaxis, leading 214432-2ENHANCED SKYRMION MOTION VIA STRIP DOMAIN … PHYSICAL REVIEW B 101, 214432 (2020) FIG. 1. Domain-wall dynamics under various current densities. (a)–(f) J=4.0×1012Am−2and (g)–(l) J=7.0×1012Am−2. The time elapsed from current action is indicated in each plot. The arrows in green, magenta, and yellow denote the electric current direction, the spin polarization direction of electrons, and the magnetization distribution in the strip domain wall, respectively. Ku=0.7M Jm−3andD= 3.0m Jm−2.B e l o w J=4.0×1012Am−2, the application of a current has no effect on the strip domain wall, but when J>4.0×1012Am−2, the strip domain wall becomes unstable because of the large-angle precession of the magnetization in the magnetic domains triggered by the spin torque. The color scale is used throughout this paper. to the substructure inside the strip domain wall. Of course, once the current direction reverses, the strip domain wall willimmediately collapse even for a small current density, becauseat this time the electron spins in the FM layer and the magneticmoments in the domain walls are in opposite directions, asdemonstrated in Ref. [ 31]. The threshold value J cd=4.0×1012Am−2is consider- ably large with regard to the skyrmion motion, since at Jcd no skyrmions can stay in the nanowire for any ( Ku,D). In fact, the maximal current density, which allows a skyrmionto stably exist in the nanowire, is slightly smaller than 1 .0× 10 12Am−2. This fact ensures that the strip domain wall can act as a tool for mediating the skyrmion dynamics. It is worth noting that, according to an early paper [ 41], the Walker solution of a domain wall under an external fieldis unstable: a domain wall could move very slowly under afield, which is hard to detect in a micromagnetic simulation.If the same thing happens for a strip domain wall under acurrent, there will be a time scale beyond which the stripdomain wall itself may disappear. However, as shown aboveand in our previous paper [ 31], the conventional Walker- type domain-wall motion cannot happen in the present study.Therefore, the stability problem of a domain wall identifiedin Ref. [ 41] is safely avoided. In fact, the dynamics of a magnetic domain wall depends critically on the matching ofdomain-wall configuration and spin-orbit torques, as revealedin Ref. [ 29]. C. Bare skyrmion dynamics We check the current-induced skyrmion dynamics in the nanowire. Here, the starting point is a single static skyrmion.We carry out a series of simulations for each ( K u,D)t os e e the skyrmion dynamics under various current densities. Ineach simulation, the magnetization distributions are recorded as time sequences with a fixed temporal interval. These dataallow us to identify how the skyrmion velocity depends onthe current density, and at which current density the skyrmionis annihilated due to the skyrmion Hall effect. Two sets ofrepresentative results are shown in Fig. 2. Figures 2(a) and 2(g) display the initial steady-state skyrmion that situates near the left end of the nanowire. Atthe time t=0 ns, an electric current is sent to the nanowire and then the skyrmion motion is initiated. Figures 2(b)–2(f) depict the skyrmion dynamics for a lower subthreshold currentdensity, at which the skyrmion moves through the nanowireand stops in front of the right edge, whereas at a highersuprathreshold current density, the skyrmion’s topologicalstructure is destroyed when it contacts the sample bound-ary, as shown in Figs. 2(h)–2(l). Eventually, this skyrmion is expelled from the nanowire. The dynamics of the bareskyrmion, demonstrated here, well agrees with what is knownin previous research [ 10,15,17,18,30]. Apparently, in both cases, the skyrmion transverse displacement accompanies itslongitudinal drift motion along the nanowire. For the subthreshold current densities, the confining force from the sample boundary equilibrates with the Magnusforce imposed by the current, and thereafter the skyrmionmoves steadily along the nanowire. Nevertheless, for thesuprathreshold current densities, the confining force is notstrong enough to counteract the Magnus force, resulting in anet force that drives the skyrmion to move outward. Accordingto the Thiele equation, the longitudinal skyrmion velocityv xand the Magnus force Fgsatisfy the relations vx∝Jand Fg∝J, respectively [ 12,30]. In this context, the permitted maximal Magnus force determines the critical current density,which in turn defines the maximum skyrmion longitudinalvelocity. 214432-3XIANGJUN XING, JOHAN ÅKERMAN, AND YAN ZHOU PHYSICAL REVIEW B 101, 214432 (2020) FIG. 2. Skyrmion motion along a nanowire without including strip domain wall. Two situations are considered: one corresponds to small current density and the other to big current density. (a)–(f) J=1.0×1011Am−2and (g)–(l) J=2.4×1011Am−2. The time elapsed from current action is indicated in each plot. The arrows in green denote the electric current. Ku=0.7M Jm−3andD=3.0m Jm−2. At small current densities, the skyrmion moves through the entire length of the nanowire and stops in the right terminal, whereas at big current densities, theskyrmion moves through only a short distance and then is expelled from the side edge of the nanowire. D. Mediated skyrmion dynamics To extend the upper theoretical limit of the skyrmion velocity, one has to suppress or avoid the skyrmion trans-verse motion. To this end, several classes of strategies havebeen proposed that use specially designed potential barriers[18,42], modified effective spin torque [ 17,43], or topologi- cally compensated hybrid skyrmions, e.g., magnetic bilayerskyrmions [ 16,22], antiferromagnetic skyrmions [ 44–47], and skyrmionium [ 48], to suppress the skyrmion Hall effect. These approaches can indeed give rise to increased skyrmionvelocities, however, their realization requires rare materials,complex structures, and/or delicate operation. Especially theadoption of antiferromagnetic skyrmions imposes a difficultyin the detection of information bits [ 22]. Therefore, other alternative ideas should be exploited for the development offast spintronic devices. The skyrmion motion driven by theSlonczewski spin torque through the mediation of a stripdomain wall manifests intriguing features, which are com-peting for use in spintronic technology and offer a basis forcomprehending the dynamics of interacting magnetic textures. Figures 3(a) and3(g) show the coexisting skyrmion and strip domain wall in the steady state prepared for the dynamicstudy. In the following, two situations are considered: onecorresponds to the subthreshold current densities [Figs. 3(b)– 3(f)], and the other to the suprathreshold current densities [Figs. 3(h)–3(l)]. In either case, the skyrmion moves forward and simultaneously the strip domain wall maintains its majorstructure. Specifically, the skyrmion moves along the stripdomain wall and just locally distorts the domain-wall string.The whole process seems like a ball sliding along an elasticbelt. The strength of the applied current distinguishes twokinds of dynamic behaviors. For a subthreshold current den-sity, the skyrmion can safely pass through the nanowire withits size fixed, but for a suprathreshold current density, theskyrmion approaches the domain wall and contracts gradually, vanishing when its radius shrinks to zero. Whether the skyrmion can move steadily with a stable size relies on if the confining force can cancel out the Magnusforce experienced by the skyrmion. For the Magnus force onthe mediated skyrmion, the relation F g∝J[12,30] still holds, and accordingly, the higher the applied current, the larger theMagnus force. In this way, a larger current density will lead toa shorter distance and a stronger repulsive force between theskyrmion and domain wall. In principle, the threshold currentdensity can be defined as the value at which the strip domainwall is maximally distorted by the skyrmion and meanwhilethe skyrmion reaches its minimal stable size, and additionallythe repulsive force is just able to offset the Magnus force.Once a bigger current is used, the Magnus force will con-tinue to increase but the repulsive force will not, producinga nonzero net force that destroys the skyrmion. At smallercurrent densities, the repulsive force can always offset theMagnus force with the skyrmion stabilized in the transversedirection, enabling the steady drift motion of the skyrmionalong the strip domain wall. In this respect, the strip domainwall serves to generate a confining force, playing the samerole as the sample boundary. However, there are some funda-mental differences between the strip domain wall and sampleboundary, which will be discussed in the following sections. E. Skyrmion velocity versus current density Now, we would like to describe the skyrmion motion quan- titatively in terms of the skyrmion velocity versus current den-sity (Fig. 4). Without loss of generality, multiple sets of differ- ent (K u,D) were considered. Figure 4(a) shows the skyrmion velocity as a function of the current density for all the con-sidered parameter combinations. For a direct comparison, the 214432-4ENHANCED SKYRMION MOTION VIA STRIP DOMAIN … PHYSICAL REVIEW B 101, 214432 (2020) FIG. 3. Skyrmion motion along a nanowire including strip domain wall. Two situations are considered: one corresponds to small current density and the other to big current density. Panels (a)–(f) J=1.0×1011Am−2and panels (g)–(l) J=4.8×1011Am−2. The time elapsed from current action is indicated in each plot. The arrows in green, magenta, and yellow denote the electric current direction, the spin polarization direction of electrons, and the magnetization distribution in the strip domain wall, respectively. Ku=0.7M Jm−3andD=3.0m Jm−2.A ts m a l l current densities, the skyrmion steadily slides along the domain-wall string and eventually stops near the right terminal, whereas at big current densities, the skyrmion gradually shrinks during sliding along the domain-wall string and finally vanishes when its radius reduces to zero. data are divided into two groups: one group is for the mediated skyrmion and the other for the bare skyrmion. Two strikingcharacteristics are visible from this figure: the curves for themediated skyrmion lie above those for the bare skyrmion andthe upper curves extend to the higher-current density region.To make it clear, we plot the curves for each ( K u,D) in sepa- rate panels [Figs. 4(b)–4(h)]. In each curve, the rightmost data point corresponds to the skyrmion motion at the current den-sity just below the threshold value, above which the skyrmioncannot move steadily in the nanowire and will be annihilated.Then, the mentioned features of the curves reveal the follow-ing aspects: first, at an identical current density, the mediatedskyrmion has a higher velocity than the bare skyrmion, andsecond, the mediated skyrmion can withstand stronger cur-rents than the bare one. Consequently, the maximum velocityof the mediated skyrmion corresponding to the threshold cur-rent density is approximately twice that of the bare skyrmionat its own threshold current density irrespective of the ( K u,D), as shown in Figs. 4(b)–4(h) and separately in Fig. 4(i). The simulation results in Figs. 1–4substantiate that the strip domain wall can indeed act as a buffer layer to me-diate the skyrmion dynamics, and, furthermore, the medi-ated skyrmion moves faster and permits using much strongercurrents compared to the bare skyrmion. Nevertheless, thesenumerical results do not reflect what governs the observedbehaviors. Next, we resort to the Thiele force equation to gainsome insights. F. A Thiele model of the skyrmion motion Assuming that the skyrmion has a rigid structure and pro- jecting the extended Landau-Lifshitz-Gilbert equation ontothe skyrmion translational mode, one obtains the generalizedThiele equation as follows [ 12,30]: G×v−α↔ D·v+4πB↔ R·J+Fp=0, (2) which describes the balance of the Magnus force Fg, dissipa- tive force FD, driving force FST, and confining force Fpacting on the skyrmion. In this work, we concentrate on the steady-state drift motion of a skyrmion along a nanowire, i.e., v= (v x,vy)=(vx,0).G=(0,0,−4πQ) is the gyromagnetic coupling vector with the topological charge Q=(1/4π)∫m· (∂xm×∂ym)dxdy , ↔ D=/parenleftbigg D 0 0D/parenrightbigg is a dissipation tensor, Bquantifies the efficiency of the spin texture of a skyrmion absorbing the Slonczewski spin torque, and↔ R=(cos 0 sin 0 −sin 0 cos 0) is an in-plane rotation matrix. J= (J,0) is along the nanowire. Generally, Fprepresents the force due to the confining potential associated with certaintype of magnetic features such as boundaries, impurities, andmagnetic objects [ 10,12,15,21,23,31]; here we intentionally assume that it incorporates two in-plane components, i.e.,F p=(Fx,Fy). Then, substituting these quantities into the vector equation ( 2), one finds −αDvx+4πBJ+Fx=0, −4πQvx+Fy=0. (3) After some simple algebra, one gets Fy=4πQ αD(4πBJ+Fx), vx=1 αD(4πBJ+Fx). (4) For the steadily moving bare skyrmion, the confining force due to the sample boundary is simply along the yaxis, 214432-5XIANGJUN XING, JOHAN ÅKERMAN, AND YAN ZHOU PHYSICAL REVIEW B 101, 214432 (2020) FIG. 4. Skyrmion velocity vs current density. The solid and empty symbols correspond to skyrmion motion in magnetic nanowires with and without including a strip domain wall, respectively. The lines across symbols are only guides to eyes. A series of ( Ku,D) are considered and the plots are shown in (a)–(h), respectively. Panel (i) plots the critical skyrmion velocity against critical current density, where the skyrmio n is annihilated. pointing from the sample boundary to skyrmion, i.e., Fp= (0,Fy)=[0,F⊥ p(bSK)]. Thus, for the bare skyrmion, one has F⊥ p(bSK)=4πQ αD4πBJ, vx(bSK)=1 αD4πBJ. (5) For the steadily moving mediated skyrmion, the confining force no longer simply points to the yaxis as for the bare skyrmion, and instead it has both xandycomponents, i.e., Fp=(Fx,Fy)=[F/bardbl p(mSK),F⊥ p(mSK)]. Then, for the medi- ated skyrmion, one sees F⊥ p(mSK)=4πQ αD[4πBJ+F/bardbl p(mSK)], vx(mSK)=1 αD[4πBJ+F/bardbl p(mSK)]. (6)It is easily noticed that, for the same current density J,vx(mSK)=vx(bSK)+1 αDF/bardbl p(mSK). This result explains one of the main numerical findings, namely, the mediatedskyrmion has bigger velocities than the bare one (Fig. 4). For the steadily moving mediated skyrmion, the longitudi- nal repulsive force originates from the asymmetric distortionin the strip domain wall [refer to Fig. 5(a)]. Such asymmet- ric distortion destroys the axisymmetric local magnetizationdistribution with respect to y, enabling the emergence of an x component in the repulsive force. However, for the steadilymoving bare skyrmion, the local magnetization distributionfrom the skyrmion to the sample boundary always keepsaxisymmetric relative to y, when the skyrmion approaches the boundary, not allowing the existence of a net x-directed com- ponent in the repulsive force. As a result, the repulsive forcearising from the boundary always orients along y[Fig. 5(b)]. The detailed mechanism for the formation of the asymmetric 214432-6ENHANCED SKYRMION MOTION VIA STRIP DOMAIN … PHYSICAL REVIEW B 101, 214432 (2020) FIG. 5. Force balance on a steadily flowing skyrmion. (a) Skyrmion motion along the strip domain wall. (b) Skyrmion motion along the sample boundary. Fg,FD,FST,a n d F⊥ prepresent the Magnus, dissipative, driving, and confining forces, respectively. Jis the current density andVdis the skyrmion drift velocity, i.e., vx=Vd. In (a), an extra longitudinal repulsive force F/bardbl pis exerted upon the skyrmion by the domain wall. (c) Magnetization distribution between the mediated skyrmion and sample boundary. “SK border,” “DW center,” and “Boundary” denote the skyrmion border, domain-wall center, and sample boundary, respectively. (d) Magnetization distribution between the bare skyrmion and sample boundary. “SK border” and “Boundary” denote the skyrmion border and sample boundary, respectively. domain-wall distortion and the creation of the longitudinal repulsive force is clarified, as shown in the SupplementalMaterial, Fig. S1 [ 49]. Generally, the forces exerted on a skyrmion by the phys- ical boundary can be introduced in a first approximation asF ⊥ p(bSK)=Fy=−k(y−y0)[10,50], where k>0 and y0 is the skyrmion equilibrium position along the yaxis, by assuming a harmonic potential. For the forces imposed ona skyrmion by the strip domain wall, we assume that theabove approximation still holds. Then, it is easy to obtainF ⊥ p(mSK)=Fy=−β(η−η0), where β> 0 and η=ySK− yDWrepresents the interval along the yaxis between the skyrmion ( ySKdenotes the ycoordinate of the skyrmion cen- ter) and the strip domain wall ( yDWsignifies the ycoordinate of the bottom of the bent domain wall) after a certain currentis applied, and η 0=η|J=0represents the initial equilibrium interval without the current application. Equation ( 6) suggests that F⊥ p(mSK)−4πQ αD[F/bardbl p(mSK)]= (4π)2QB αDJ, from which it is reasonable to suppose that F/bardbl p(mSK) follows a similar relation, i.e., F/bardbl p(mSK)=Fx=−ζ(η−η0), (7) where ζ> 0. Now, letting /Delta1vx=vx(mSK)−vx(bSK), one has/Delta1vx=1 αDF/bardbl p(mSK). Considering that D=π2dSK 8λDW(where dSKandλDWare the skyrmion diameter and domain-wall thickness, respectively) [ 12], one finally obtains the following formula: /Delta1vx=−8ζλDW απ2dSK(η−η0), (8) where λDW=√A/Keffwith Keff=Ku−1 2μ0M2 scan be cal- culated directly from the material parameters, and dSK,η, andη0can be derived from the simulation results. Leaving ζas the free parameter, we fit the simulated velocity differencebetween the mediated and bare skyrmions (shown in Fig. 4) using Eq. ( 8). The fitting results are presented in Fig. 6, which contains five sets of data corresponding to variouscombinations of K uandD. Overall, the agreement between theory and simulations is good, considering that there existsonly one free parameter; this fact indicates that the assumptionof the form of the longitudinal force [i.e., Eq. ( 7)] is a very good approximation. From Eqs. ( 5) and ( 6), for the same current density J, F ⊥ p(mSK)=F⊥ p(bSK)+4πQ αDF/bardbl p(mSK), suggesting that the repulsive force imposed by the strip domain wall is strongerthan that exerted by the sample boundary. In fact, the confin-ing potential on the mediated skyrmion can be much largerthan the one on the bare skyrmion, which makes the mediatedskyrmion be able to withstand a much stronger Magnus force.The mechanisms are clarified from comparing the ways ofannihilation of the mediated and bare skyrmions. For the an-nihilation of the mediated skyrmion, there exist three optionalroutes: ( 1) The skyrmion could at first push a portion of the strip domain wall out of the boundary and then leave the sam-ple from the boundary. In this case, because the strip domainwall is an extended entity, when even a piece of it approachesto the boundary, a large number of magnetic moments willjoin the strong local interaction magnetostatically causing avery strong repulsive force between the domain-wall centerand sample boundary [Fig. 5(c)]. Consequently, driving the domain wall to touch the boundary must overcome a hugeenergy barrier linked to the strong repulsive force. ( 2)T h e skyrmion may penetrate the strip domain wall and merge intothe magnetic domain. However, the extending character of thestrip domain wall together with the self-locking feature of the 214432-7XIANGJUN XING, JOHAN ÅKERMAN, AND YAN ZHOU PHYSICAL REVIEW B 101, 214432 (2020) FIG. 6. Comparison of the simulation and theoretic results of /Delta1vx=vx(mSK)−vx(bSK) as function of current density. Different combinations of KuandDare considered as indicated in each panel from (a) to (e). The simulation results of /Delta1vxare derived from the data in Fig. 4. The parameters used for the fittings are summarized in the Supplemental Material, Tables S1–S5 [ 49]. spin configuration between the skyrmion border and domain- wall center lead to a still high-energy barrier [Fig. 5(c)]. (3) The skyrmion may shrink gradually by contracting its bor-der and vanish finally by absorbing a magnetic singularity[10,51]. Owing to the relatively small size of a skyrmion, the energy barrier associated with its annihilation is the lowestamong the three situations. Therefore, a mediated skyrmion isalways seen to annihilate through route 3. Nevertheless, only two possible pathways exist for the annihilation of the bare skyrmion. ( 4)A si nr o u t e3f o rt h e mediated skyrmion, the bare skyrmion could also be anni-hilated by shrinking its size and then absorbing a magneticsingularity. Here, however, the energy barrier is relativelyhigh owing to the topological protection of the skyrmion andthe requirement to inject a topological singularity [ 10,51]. (5) Alternatively, the bare skyrmion could be annihilated by touching the boundary. In this case, unlocking of the spin con-figuration between the skyrmion border and sample boundarycan be simply launched by the reversal of those magneticmoments situating on the boundary [Fig. 5(d)], such that the entire skyrmion is easily erasable; the associated energybarrier is small. Thus, a bare skyrmion tends to be annihilatedthrough route 5. Obviously, the three annihilation routes for the mediated skyrmion require overcoming larger energy barriers comparedwith the two annihilation routes for the bare skyrmion, deter-mining that the mediated skyrmions have higher annihilationcurrent densities than the bare ones and naturally can experi-ence stronger Magnus forces. III. DISCUSSION To check the stability of the mediated skyrmion motion, we numerically study the process in longer magnetic multi-layer nanowires. The computational results indicate that themediated skyrmion can propagate steadily over a considerablylarge distance with the shape and size fixed, as shown inFig. S2 of the Supplemental Material [ 49]. We also con- sider the current-induced dynamics of an array of mediatedskyrmions (Fig. S3 [ 49]) and found that the entire array moves concertedly when the interval is adequately large orthe applied current is exceedingly low. Compared with thebare-skyrmion array, the smallest interval between two adja-cent mediated skyrmions, which permits orderly motion, islarger, because, for the mediated-skyrmion array, a skyrmionis readily affected by its neighbors through bending of thedomain wall. The results presented in this study do not rely on special material parameters and are universally valid for theHM1/FM/AO(HM2)-like multilayer system. As an example,the mediated skyrmion motion for a different ( K u,D)i sd i s - played in Fig. S4 [ 49], where the entire process is essentially t h es a m ea si nF i g . 3. A skyrmion, once driven to move, will adjust itself to a moderate stable size before reaching thesteady-state motion; this is especially clear for a big skyrmiona ss h o w ni nF i g .S 4[ 49]. These manifested characteristics of mediated skyrmions form the basic prerequisite for anyrealistic implementation of a device using them. The extended Thiele equation including F /bardbl pprovides in- sight into the numerical results, both qualitatively and quanti-tatively; it clearly demonstrates that the longitudinal repulsiveforce functions as an active driving force for the mediatedskyrmion. According to Refs. [ 50,52], an inertia term is expected to enter the Thiele equation because of the edgeconfining potential. Since the strip domain wall also imposesa confining potential on the skyrmion as the physical borders,the effect of the mass term must have been involved in oursimulation results. Despite the absence of the inertia term inour theoretical model, the agreement between the theory andsimulations seems good overall, as demonstrated in Fig. 6. Therefore, in our opinion, the influence of the mass term isnegligible in this scenario, and the model without consideringthe inertia has captured the key physics in the dynamics of themediated skyrmion. The proposed use of mediated skyrmions can suppress skyrmion Hall effect, namely, increasing the skyrmion mobil-ity and expanding the effective working range of the current,and eliminate the random scattering of edge roughness onskyrmion motion. Nevertheless, it cannot avoid the skyrmionHall effect, and thus there still exists a threshold current den-sity∼1.0×10 12Am−2, above which the mediated skyrmion will be annihilated. Analogously, a threshold current densityalso exists in most of the previously suggested schemes[16,18,42,48]; when the employed current density becomes exceedingly large, the skyrmions will be destructed by theuncompensated Magnus force. Actually, recent literature [ 44] argued that the skyrmion Hall effect will still occur in thecase of spin-polarized currents even for the skyrmions inantiferromagnets. Comparatively, our proposed scheme hasremarkable advantages: First, it simply requires writing astrip domain wall into the original skyrmion device withoutincorporating the fabrication of complex hard structures and isthus naturally reconfigurable. Second, apart from skyrmionic 214432-8ENHANCED SKYRMION MOTION VIA STRIP DOMAIN … PHYSICAL REVIEW B 101, 214432 (2020) devices, the hardware is also applicable to domain-wall race- track devices [ 53] and magnonic waveguides [ 32], without significant variation in the key parts, implying good repro-grammability. Although the scattering on skyrmion motion by edge pin- ning sites can be prevented by using strip domain wall, ourpreliminary numerical calculations suggest that the impactof pinning centers [ 7,12,21,23] in the interior region of the nanowire is unavoidable, since the randomly distributed point-like impurities pin the strip domain wall locally and modifyits profile (see the Supplemental Material, Fig. S5 and MoviesS1–S3 [ 49]). While the nanowire width is not crucial for the steady-state motion of a mediated skyrmion, the distance between the stripdomain wall and sample boundary has a decisive role. Anincreased spacing will result in enhanced distortion of thestrip domain wall and change the relative strength of F /bardbl pand F⊥ p. For instance, in wider samples with a strip domain wall situating along the central axis, the steadily moving mediatedskyrmion acquires higher velocities [as shown in Fig. 4(c)], because the heavier local bending of the strip domain wall,permitted by the bigger spacing between the domain walland sample edge, results in a larger longitudinal componentof the repulsive force. Fortunately, one can displace the stripdomain wall using, for example, a magnetic field to reach anappropriate position. Different from that of bare skyrmions, the motion of me- diated skyrmions under spin-orbit torque is unidirectional. Areversed direction of the applied current will at first causethe strip domain wall to deform randomly, and then thechaotic domain-wall dynamics destructs the skyrmion leadingto erroneous operation. IV . CONCLUSION In conclusion, we point out theoretically the possibility to control current-induced skyrmion dynamics utilizing a stripdomain wall. Through micromagnetic simulations, we studythe dynamics of strip domain wall, bare skyrmion, and coex-isting skyrmion and strip domain wall under spin-orbit torqueover a wide range of interface-sensitive material parameters.The computational results attest our theoretical conjectureand suggest that the skyrmion mediated by strip domain wallbecomes faster and more stable, which is explained by thegeneralized Thiele equation with a two-component confin-ing force. A symmetry analysis reveals that the longitudi-nal component of the confining force originates from localasymmetric distortion of the strip domain wall. The designof skyrmionic devices might benefit from these discoveries.More importantly, the study implies that, overall, the Thieleequation is robust in describing the dynamics of magnetic soli-tons [ 10], and specifically, the skyrmion velocity and Magnus force can be harnessed by a longitudinal force regardless ofits origin. ACKNOWLEDGMENTS X.J.X. acknowledges support from the National Natural Science Foundation of China (Grant No. 11774069). Y .Z.acknowledges support by the President’s Fund of CUHKSZ, the Longgang Key Laboratory of Applied Spintronics, theNational Natural Science Foundation of China (Grants No.11974298 and No. 61961136006), the Shenzhen Fundamen-tal Research Fund (Grant No. JCYJ20170410171958839),and the Shenzhen Peacock Group Program (Grant No.KQTD20180413181702403). X.J.X. initiated and designed the study. Y .Z. coordinated the project. All authors contributed to the analysis of theresults and wrote the manuscript. APPENDIX A: MICROMAGNETIC SIMULATIONS The public-domain micromagnetic codes MuMax3 [ 40] are used to implement the micromagnetic simulations, inwhich the Landau-Lifshitz-Gilbert equation is numericallyintegrated, by means of the explicit Runge-Kutta methodwith an adaptive time step, to find the equilibrium magneticconfigurations and trace the dynamics of the aimed magneticconfigurations under the applied current. For the simulations of equilibrium magnetic configura- tions, the original Landau-Lifshitz-Gilbert equation is modi-fied by including the iDMI in the free energy E. The RK23 (the Bogacki-Shampine version of the Runge-Kutta method)solver is chosen. In each simulation, the solver keeps advanc-ing until the MaxErr, /epsilon1=max|τ high−τlow|/Delta1t(where τhigh andτloware the estimated high-order and low-order torques and/Delta1tis the time step), decreases to 10−9. The initial spin configuration is a numerically conjectured structure, in whicha 20-nm-wide bubblelike spin texture centered at a site 40 nmfar from the nanowire’s left edge and 1 /4 the wire width far from the top edge is accompanied by a domain wall alignedalong the nanowire’s central axis. For the simulations of current-induced dynamics, the con- ventional LLG equation is extended by the Slonczewski spin-transfer torque. The RK45 (the Dormand-Prince version ofthe Runge-Kutta method) solver is adopted, and in eachsimulation, the solver stops advancing when the MaxErrreaches 10 −5. The equilibrium spin configurations obtained from static simulations are used as the input for dynamicsimulations. APPENDIX B: THEORETICAL MODEL The Landau-Lifshitz-Gilbert equation is well established as a general-purpose tool for describing the spin dynam-ics of continuous ferromagnetic systems. From this generalequation, the special-purpose Thiele equation can be derivedto describe the characteristic of the mediated skyrmion dy-namics. Here, to capture the main feature and for simplicity,we regard a skyrmion as a rigid soliton with zero mass,i.e., neglecting the skyrmion’s structural deformation duringthe motion and the skyrmion mass due to the confiningpotential. In our model, the strip domain wall existing ina nanowire does not manifest itself explicitly in the Thieleequation but enters implicitly into the confining force F p, and thereby the derivation of the Thiele equation follows that inRefs. [ 12,30]. 214432-9XIANGJUN XING, JOHAN ÅKERMAN, AND YAN ZHOU PHYSICAL REVIEW B 101, 214432 (2020) [1] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Skyrmion lattice in a chiralmagnet, Science 323, 915 (2009) . [2] X. Z. Yu, Y . Onose, N. Kanazawa, J. H. Park, J. H. Han, Y . Matsui, N. Nagaosa, and Y . 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PhysRevB.77.094434.pdf

Creep of current-driven domain-wall lines: Effects of intrinsic versus extrinsic pinning R. A. Duine *and C. Morais Smith† Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands /H20849Received 4 February 2008; revised manuscript received 3 March 2008; published 27 March 2008 /H20850 We present a model for the current-driven motion of a magnetic domain-wall line, in which the dynamics of the domain wall is equivalent to that of an overdamped vortex line in an anisotropic pinning potential. Thispotential has both extrinsic contributions due to, e.g., sample inhomogeneities, and an intrinsic contributiondue to magnetic anisotropy. We obtain results for the domain-wall velocity as a function of current for variousregimes of pinning. In particular, we find that the exponent characterizing the creep regime strongly depends onthe presence of a dissipative spin transfer torque. We discuss our results in the light of recent experiments oncurrent-driven domain-wall creep in ferromagnetic semiconductors and suggest further experiments to corrobo-rate our model. DOI: 10.1103/PhysRevB.77.094434 PACS number /H20849s/H20850: 72.25.Pn, 72.15.Gd, 75.60.Ch, 85.75. /H11002d I. INTRODUCTION The driven motion of line defects through a disordered potential landscape has attracted considerable attention, forexample, in the context of vortices in superconductors, 1wet- ting phenomena,2crack fronts,3and domain walls in ferromagnets.4,5The competition and interplay among the elasticity of the line, the pinning forces due to the disorderpotential, and thermal fluctuations lead to a wealth of physi-cal phenomena. Topics discussed are, for example, the uni-versality class of the roughening of the line, the nature of thepinning-depinning transition at zero temperature, 6and the slide, depinning, and creep regimes of motion of the line thatoccur for decreasing driving field. 1,7 The creep regime was experimentally observed with the field-driven motion of domain walls in ferromagnets.4,5This low-field regime is characterized by a nonlinear dependence of the domain-wall drift velocity /H20855X˙/H20856on the external mag- netic field Hext, which is given by /H20855X˙/H20856/H11008exp/H20877−Ec kBT/H20873Hc Hext/H20874/H9262f/H20878, /H208491/H20850 where Ecis a characteristic energy scale and Hcis a critical field. The thermal energy is denoted by kBTand the exponent /H9262f=/H208492/H9256−1/H20850//H208492−/H9256/H20850is given in terms of the equilibrium wan- dering exponent /H9256of the static line.1,4,7The phenomenologi- cal creep formula /H20851Eq. /H208491/H20850/H20852, which is an Arrhenius law in which the energy barrier diverges for a vanishing drivingfield, is “glassy.” The underlying assumption is that there is acharacteristic length scale that determines the displacementof the domain-wall line. The validity of Eq. /H208491/H20850was con- firmed both numerically 8and with functional renormaliza- tion group methods.9It turns out that Eq. /H208491/H20850is also valid in situations where roughening plays no role. For example, forad-dimensional manifold driven through a periodic potential ind+1 dimensions, we have /H9262f=d−1 /H20849ford/H113502/H20850.1More- over, in the regime where the line defect moves via variable-range hopping, we have that /H9262f=1 /3 if the motion is in two dimensions.1,10,11 In addition to magnetic-field-driven motion, a lot of re- cent theoretical and experimental research was devoted tomanipulating domain walls with electric current 12–21via theso-called spin transfer torques.22–25Domain-wall motion driven by a current is quite distinct from the field-drivencase. For example, it has been theoretically predicted that, incertain regimes of parameters, the domain wall is intrinsi-cally pinned at zero temperature, which means that thereexists a nonzero critical current even in the absence ofdisorder. 12In clean samples, the phenomenology of current- driven domain-wall motion turns out to crucially depend onthe ratio of the dissipative spin transfer torque parameter 26/H9252 and the Gilbert damping constant /H9251G.13Although theoretical predictions27–30indicate that, at least for model systems, this ratio differs from 1, it turns out to be difficult to extract itsprecise value from experiments on current-driven domain-wall motion to a large extent because disorder and nonzero-temperature effects 21,31complicate theoretical calculations of the domain-wall drift velocity for a given current. This is thefirst motivation for the work presented in this paper. Previous work on current-driven domain-wall motion at nonzero temperature focused on rigid -domain walls. Tatara et al. 32found that ln /H20855X˙/H20856was proportional to the current den- sity j. The discrepancy between this result and experiments21 that did not observe this exponential dependence of wall ve- locity on current motivated the more systematic inclusion ofnonzero-temperature effects on rigid-domain-wall motion by Duine et al. , 31who found that ln /H20855X˙/H20856/H11008/H20881jin certain regimes. Although the latter was an important step in qualitativelyunderstanding the experimental results of Yamanouchi et al., 21,33a detailed understanding of these experiments is still lacking and this is the second motivation for this paper. Forcompleteness, we also mention the theoretical work by Mar-tinez et al., 34,35who considered thermally assisted current- driven rigid-domain-wall motion in the regime of large an-isotropy, where the chirality of the domain wall plays no roleand the pinning is essentially dominated by extrinsic effects.Furthermore, Ravelosona et al. 36observed the thermally as- sisted domain-wall depinning, and Laufenberg et al.37deter- mined the temperature dependence of the critical current fordepinning the domain wall. In this paper, we present a model for a current-driven elastic -domain-wall line transversely moving in one dimen- sion in the presence of disorder and thermal fluctuations .A crucial ingredient in the description of current-driven motionPHYSICAL REVIEW B 77, 094434 /H208492008 /H20850 1098-0121/2008/77 /H208499/H20850/094434 /H208496/H20850 ©2008 The American Physical Society 094434-1is the chirality of the domain wall, which acts like an extra degree of freedom. This enables a reformulation of current-driven domain-wall motion as a superfluid vortex line trans-versely moving in an anisotropic potential in two dimensions /H20849see Fig. 1/H20850, which we present in detail in Sec. II. By using this physical picture, in Sec. III, we analyze the differentregimes of pinning within the framework of collective pin-ning theory. 1We present results on the velocity of the domain-wall line as a function of current, both in the regimewhere intrinsic pinning due to magnetic anisotropy domi-nates and in the extrinsic-pinning-dominated regime. Finally,in Sec. IV , we discuss our theoretical results in relation torecent experiments on current-driven domain walls inGaMnAs. 33In our opinion, although these experiments re- main not fully understood, we suggest that they may be ex-plained by assuming a specific form of the pinning potentialfor the domain-wall line. We propose further experimentsthat could corroborate this suggestion. II. DOMAIN WALL AS A VORTEX LINE The equation of motion for the magnetization direction /H9024 in the presence of a transport current jis, to the lowest order in temporal and spatial derivatives, given by /H20873/H11509 /H11509t+vs·/H11612/H20874/H9024−/H9024/H11003/H20849H+Hext+/H9257/H20850 =−/H9251G/H9024/H11003/H20873/H11509 /H11509t+/H9252 /H9251Gvs·/H11612/H20874/H9024. /H208492/H20850 The left-hand side of this equation contains the reactive26 spin transfer torque,38which is proportional to the velocity vs=Pj//H20849e/H9267s/H20850. The latter velocity characterizes the efficiency of spin transfer. Here, Pis the polarization of the current in the ferromagnet, eis the carrier charge, and the spin densityis denoted by /H9267s/H110132/a3, with aas the lattice constant. The other terms on the left-hand side of Eq. /H208492/H20850describe preces- sion around the external field Hextand the effective field H =−/H9254E/H20851/H9024/H20852//H20849/H6036/H9254/H9024/H20850, which is given by a functional derivative of the energy functional E/H20851/H9024/H20852with respect to the magneti- zation direction. The stochastic magnetic field /H9257incorporates thermal fluctuations, and it has a zero mean and correlationsdetermined by the fluctuation-dissipation theorem, 39 /H20855/H9257/H9268/H20849x,t/H20850/H9257/H9268/H11032/H20849x/H11032,t/H11032/H20850/H20856=2/H9251GkBT /H6036/H9254/H20849t−t/H11032/H20850a3/H9254/H20849x−x/H11032/H20850/H9254/H9268/H9268/H11032. /H208493/H20850 It can be shown that this equation still holds in the presence of current, at least to first order in the applied electric field30 that drives the transport current. The fluctuation-dissipationtheorem also ensures that in equilibrium the probability dis-tribution for the magnetization direction is given by theBoltzmann distribution P/H20851/H9024/H20852/H11008exp /H20853−E/H20851/H9024/H20852/k BT/H20854. The right- hand side of Eq. /H208492/H20850contains only dissipative terms. The Gilbert damping term is proportional to the damping param-eter /H9251G, and the dissipative26spin transfer torque is charac- terized by the dimensionless parameter /H9252.13 We consider a ferromagnet with magnetization direction /H9024=/H20849sin/H9258cos/H9278,sin/H9258sin/H9278,cos/H9258/H20850that depends only on the xand zdirections. In addition, we take the current in the x direction and the external magnetic field in the zdirection. The size of the ferromagnetic film in the /H9251direction is de- noted by L/H9251/H20849/H9251/H33528/H20853x,y,z/H20854/H20850and we assume that Ly/H11270Lz. The latter assumption allows us to model the domain wall as aline. Furthermore, we take the ferromagnet to have an easy z axis and a hard yaxis, with anisotropy constants Kand K /H11036, respectively. The spin stiffness is denoted by J. With these assumptions, static domain walls have a width /H9261=/H20881J/Kand are, for the simplest model, to be discussed in more detailbelow /H20851see Eq. /H2084910/H20850/H20852, described by the solutions /H92580/H20849x/H20850 =cos−1/H20851tanh /H20849x//H9261/H20850/H20852and/H9278/H20849x/H20850=0. To arrive at a description of the dynamics of the domain wall, we use two collective co-ordinates which may depend on the zcoordinate so that the domain wall is modeled as a line. The collective coordinatesare the position of the wall X/H20849z,t/H20850and the chirality /H92780/H20849z,t/H20850. The latter determines the sense in which the magnetizationrotates upon going through the domain wall. The result ofRef. 31is straightforwardly generalized to the case of a domain-wall line. This amounts to solving Eq. /H208492/H20850variation- ally with the ansatz /H9258dw/H20849x,t/H20850=/H92580/H20853/H20851x−X/H20849z,t/H20850/H20852//H9261/H20854and /H9278dw/H20849x,t/H20850=/H92780/H20849z,t/H20850, which yields the equations of motion, /H11509/H92780 /H11509t+/H9251G /H9261/H11509X /H11509t=−a3 2/H6036Ly/H9261/H9254V /H9254X+/H9252vs /H9261−Hext+/H9257X/H20849z,t/H20850, 1 /H9261/H11509X /H11509t−/H9251G/H11509/H92780 /H11509t=a3 2/H6036Ly/H92612/H9254V /H9254/H92780+vs /H9261+/H9257/H9278/H20849z,t/H20850, /H208494/H20850 where the domain-wall energy, V/H20851X,/H92780/H20852/H11013E/H20851/H9258dw,/H9278dw/H20852, /H208495/H20850 FIG. 1. /H20849Color online /H20850Mapping of current-driven domain-wall dynamics to that of a vortex line. The position of the domain wallX/H20849z,t/H20850and its chirality /H92780/H20849z,t/H20850become the position /H20849ux,uy/H20850of the vortex via /H20849ux,uy/H20850/H11013/H20849 X//H9261,/H92780/H20850. The potential landscape for this vor- tex is, in general, anisotropic. In particular, the tilting in the ux direction is set by the external magnetic field and the dissipative spin transfer torque. The tilting in the uydirection is determined by the reactive spin transfer torque.R. A. DUINE AND C. MORAIS SMITH PHYSICAL REVIEW B 77, 094434 /H208492008 /H20850 094434-2and the stochastic forces are determined from /H20855/H9257/H9278/H20849z,t/H20850/H9257/H9278/H20849z/H11032,t/H11032/H20850/H20856=/H20855/H9257X/H20849z,t/H20850/H9257X/H20849z/H11032,t/H11032/H20850/H20856 =/H20873/H9251GkBT /H6036/H20874/H20873a3 /H92612Ly/H20874/H9254/H20873z−z/H11032 /H9261/H20874/H9254/H20849t−t/H11032/H20850. /H208496/H20850 The above equations are derived using a variational methodfor stochastic differential equations based on their path- integral formulation.31,40Their validity is confirmed a poste- riori by noting that in equilibrium the probability distribution function for the position and chirality of the domain wall isthe Boltzmann distribution. That is, the Fokker-Planck equa-tion for the probability distribution P/H20851X, /H92780/H20852of the domain- wall position and the chirality that follows from Eqs. /H208494/H20850and /H208496/H20850is given by41 /H208491+/H9251G2/H208502/H6036Ly/H92612 a3/H11509P/H20851X,/H92780/H20852 /H11509t=/H20885dz/H9254 /H9254X/H20849z/H20850/H20873/H9251G/H9261/H9254V /H9254X/H20849z/H20850−/H9254V /H9254/H92780/H20849z/H20850/H20874P/H20851X,/H92780/H20852+/H20885dz /H9261/H9254 /H9254/H92780/H20849z/H20850/H20873/H9261/H9254V /H9254X/H20849z/H20850+/H9251G/H9254V /H9254/H92780/H20849z/H20850/H20874P/H20851X,/H92780/H20852 +/H9251GkBT/H20885dz /H9261/H20873/H92542 /H9254/H927802/H20849z/H20850+/H92612/H92542 /H9254X2/H20849z/H20850/H20874P/H20851X,/H92780/H20852. /H208497/H20850 Upon insertion of the Boltzmann distribution Peq/H20851X,/H92780/H20852 /H11008exp /H20853−V/H20851X,/H92780/H20852//H20849kBT/H20850/H20854into this equation, one straightfor- wardly verifies that it is indeed a stationary solution. By rewriting the equations of motion for the domain-wall position and chirality in terms of the dimensionless coordi-nateu/H20849z,t/H20850/H11013/H20851 X/H20849z,t/H20850//H9261, /H92780/H20849z,t/H20850/H20852, we find from Eq. /H208494/H20850that the domain wall is described by /H9280/H9251/H9251/H11032u˙/H9251/H11032/H20849z,t/H20850=−/H9251Gu˙/H9251/H20849z,t/H20850−/H9254V˜/H20851u/H20852 /H9254u/H9251/H20849z,t/H20850+/H9257/H9251/H20849z,t/H20850, /H208498/H20850 with/H9280/H9251/H9251/H11032as the two-dimensional Levi-Civita symbol. /H20849Sum- mation over repeated indices /H9251,/H9251/H11032/H33528x,yis implied. Note that/H9257/H9251=/H9257X,/H9278for/H9251=x,y./H20850The above equation of motion /H20851Eq. /H208498/H20850/H20852corresponds to the overdamped limit of vortex-line dy- namics in an anisotropic potential V˜/H20851u/H20852. The left-hand side of Eq. /H208498/H20850corresponds to the Magnus force on the vortex. We emphasize that a mass term is missing, which indicates thatwe are indeed dealing with the overdamped limit of vortexmotion. /H20849Note that the mass of the fictitious vortex is not related to the Döring domain-wall mass 42that arises from eliminating the chirality from the domain-wall description,which is valid provided the latter is small. 43As the dynamics of the domain-wall chirality is essential for a current-drivendomain-wall motion, this latter approximation is not suffi-cient for our purposes. /H20850The right-hand side of the equation of motion contains a damping term proportional to /H9251Gand a term representing thermal fluctuations. The force is deter-mined by the potential V˜=a3V/H20851/H9261ux,uy/H20852 2/H6036Ly/H92612+/H20885dz /H9261/H20875/H20873/H9252vs /H9261−Hext/H20874ux+vs /H9261uy/H20876./H208499/H20850 The tilting of this potential in the uxdirection is determined by the parameter /H9252, the current vs, and the external field Hext. The tilting in the uydirection is determined only by the cur- rent. The model in Eqs. /H208498/H20850and /H208499/H20850, which is illustrated in Fig. 1, is the central result of this paper. In the followingsection, we obtain the results from this model for the domain-wall velocity in different regimes of pinning, spe-cializing to the case of a current-driven domain-wall motion/H20849H ext=0/H20850. III. DOMAIN-WALL CREEP In this section, we obtain the results for the average drift velocity of the domain wall as a function of applied current.First, we discuss the situation without disorder; hereafter, weincorporate the effects of disorder. A. Intrinsic pinning In this section, we make two assumptions that do not necessarily imply each other from a microscopic point ofview. First, we consider a homogeneous system, i.e., a sys-tem without disorder potential V pin=0. Second, we take /H9252 =0. As a result of these assumptions, the domain wall is intrinsically pinned.12This comes about as follows. For the magnetic nanowire model discussed in the previous section,the energy functional in the clean limit is given by E/H20851/H9024/H20852=/H20885dx a3/H20877J 2/H20851/H20849/H11612/H9258/H208502+ sin2/H9258/H20849/H11612/H9278/H208502/H20852 +K 2sin2/H9258+K/H11036 2sin2/H9258sin2/H9278/H20878. /H2084910/H20850 Upon insertion of the domain-wall ansatz into the above en- ergy functional, we find that V˜/H20851u/H20852=/H20885dz /H9261/H20875J 2/H6036/H20873/H11509u /H11509z/H208742 −K/H11036 4/H6036cos/H208492uy/H20850+vs /H9261uy/H20876./H2084911/H20850 Because the above potential does not explicitly depend on z, the domain wall remains straight at zero temperature, i.e., /H11509u//H11509z=0. By solving the equations of motion in Eq. /H208498/H20850for the potential in Eq. /H2084911/H20850at zero temperature and for a straightCREEP OF CURRENT-DRIVEN DOMAIN-WALL LINES: … PHYSICAL REVIEW B 77, 094434 /H208492008 /H20850 094434-3domain wall, one finds that /H20855/H20841u˙/H20841/H20856/H11008/H20881vs2−/H20849/H9261K/H11036/2/H6036/H208502so that the domain wall is pinned up to a critical current given by jc=/H9261K/H11036e/H9267s/2/H6036P./H20849The brackets /H20855¯/H20856denote time and ther- mal average. /H20850This intrinsic pinning is entirely due to the anisotropy energy,12which is determined by K/H11036, and does not occur for field-driven domain-wall motion or current-driven domain-wall motion with /H9252/HS110050. Physically, it comes about because, for the model of a domain wall that we con-sider here, the reactive spin transfer torque causes the mag-netization to rotate in the easy plane. This corresponds to aneffective field that points along the hard axis. Because theGilbert damping causes the magnetization to precess towardthe effective field, the current tilts the magnetization out ofthe easy plane. This leads to a cost in anisotropy energy,which stops the drift motion of the domain wall if the currentis too small. By solving the equations of motion for the po-tential in Eq. /H2084911/H20850at nonzero temperature in the limit of a straight wall, one recovers the result of Ref. 31. At nonzero temperature, the domain wall is, however, no longer straight. Since only the chirality is important, ourmodel for current-driven domain-wall motion in Eq. /H2084911/H20850 then corresponds to the problem of a string in a tilted-washboard potential, which was studied before 44in different contexts. At nonzero temperature, the string propagatesthrough the tilted-washboard potential by nucleating a kink-antikink pair in the zdirection of the domain-wall chirality /H92780/H20849z,t/H20850. The kink and antikink are subsequently driven apart, which results in the propagation of the string. In the limit when the current is close to the critical 1, a typical energy barrier is determined by the competition be-tween the elasticity of the string and the tilted potential. 1For /H20849jc−j/H20850/jc/H112701, the cosine in the energy functional in Eq. /H2084911/H20850 may be expanded around one of its minima, which yields V˜/H20851u/H20852=/H20885dz /H9261/H20875J 2/H6036/H20873/H11509/H9254uy /H11509z/H208742 +K/H11036 /H6036/H208811−/H20873j jc/H208742 /H9254uy2+2vs 3/H9261/H9254uy3/H20876, /H2084912/H20850 where we have omitted an irrelevant constant. In the above expression, /H9254uydenotes the displacement from the minimum. Note that we have dropped the dependence of the potentialonu x, which is allowed because the potential is not tilted in theuxdirection /H20849provided that /H9252=0/H20850. The potential in Eq. /H2084912/H20850has a minimum for /H9254uymin =0 /H20849by construction /H20850and a maximum for /H9254uymax =−vsK/H11036/H208811−/H20849j/jc/H208502//H9261/H6036. The pinning potential energy barrier, i.e., the pinning potential evaluated at the maximum, scalesas/H9004V/H11008/H208511−/H20849j/j c/H208502/H208523/2. Consider now the situation that a seg- ment of length Lof the string is displaced from the minimum and pinned by the potential. The length Lis then determined by the competition between the elastic energy /H11011J/H20849/H9254uymax/L/H208502, which tends to keep the domain wall straight, and the pin-ning potential /H9004V. Equating these contributions yields the following for the length L: L/H11008/H208751−/H20873j jc/H208742/H20876−1 /4 . /H2084913/H20850 The typical energy barrier that thermal fluctuations have to overcome to propagate the domain wall is then given byevaluating Eq. /H2084912/H20850for a segment of this length. This yields a typical energy barrier /H11008/H208511−/H20849j/jc/H208502/H208525/4. By putting these re- sults together and assuming an Arrhenius law, we find thatthe domain-wall velocity is ln/H20855/H20841u˙/H20841/H20856/H11008−1 kBTJLy a3/H20881K/H11036 K/H208751−/H20873j jc/H208742/H208765/4 /H2084914/H20850 for /H20849jc−j/H20850/jc/H112701. In the regime of small currents j/H11270jc, the problem must be treated in the so-called “thin-wall” approximation.45For the case of a one-dimensional line, however, it turns out that thedependence of domain-wall velocity on current is qualita-tively similar to the rigid domain-wall situation. B. Extrinsic pinning We now add extrinsic pinning, i.e., a disorder potential Vpinto the potential in Eq. /H2084911/H20850. Following Ref. 12,w ea s - sume, in the first instance, that it only couples to the positionof the domain wall u xand not to its chirality uy. This assump- tion is made mainly to simplify the problem. By now con-sidering the general case that /H9252/HS110050, we have V˜/H20851u/H20852=/H20885dz /H9261/H20875J 2/H6036/H20873/H11509u /H11509z/H208742 −K/H11036 4/H6036cos 2 uy+Vpin/H20849ux,z/H20850 +/H9252vs /H9261ux+vs /H9261uy/H20876. /H2084915/H20850 We estimate a typical energy barrier using the collective pin- ning theory.1,7Therefore, we assume that we are in the re- gime where the pinning energy grows sublinearly with thelength of the wall, and that there exists a typical length scaleLat which domain-wall motion occurs. 1/H20849Note that we con- sider Las dimensionless since the coordinate uis dimension- less. /H20850The energy of a segment of this length that is displaced is given by E/H20849L/H20850=/H9280elux2 L+/H9252vs /H9261Lux+vs /H9261Luy. /H2084916/H20850 The first term is the elastic energy with /H9280el=J/2/H6036/H92612. The second and third terms correspond to the dissipative and re-active spin transfer torques, respectively. Note that since thedissipative spin transfer torque acts like an external magneticfield, we are able to incorporate it in the above energy. Thepotential V pin/H20849ux,z/H20850leads to a roughening in the uxdirection. Following standard practice,1,4,7we assume a scaling law ux/H20849L/H20850=ux0L/H9256, with /H9256as the equilibrium wandering exponent, which is already mentioned in the Introduction, and ux0as a constant. The displacement in the uydirection is not rough- ened because we have assumed that Vpin/H20849ux,z/H20850does not de- pend on uy, i.e., the domain-wall chirality. Rather, the dis- placement in this direction is determined by the minima ofthe potential in Eq. /H2084911/H20850and we have u y=uy0independent of Lforj/H11270jc. Note that in this limit the elastic energy due to displacement in the uydirection can also be neglected.1 Hence, we find thatR. A. DUINE AND C. MORAIS SMITH PHYSICAL REVIEW B 77, 094434 /H208492008 /H20850 094434-4E/H20849L/H20850=/H9280elux02L2/H9256−1+/H9252vs /H9261ux0L/H9256+1+vs /H9261Luy0. /H2084917/H20850 Minimizing this expression with respect to Lthen leads to a typical energy barrier. By assuming an Arrhenius law,1,4,7we find the following for the domain-wall velocity: ln/H20855/H20841u˙/H20841/H20856/H11008−/H9280el kBT/H20873jc j/H20874/H9262c . /H2084918/H20850 For/H9252=0, we have /H9262c=/H208492/H9256−1/H20850//H208492−2/H9256/H20850. For /H9252/HS110050, we find /H9262c=/H208492/H9256−1/H20850//H208492−/H9256/H20850. In particular, for /H9256=2 /3, which is appli- cable to domain walls in ferromagnetic metals,4we have /H9262c=1 /2 for /H9252=0 and /H9262c=1 /4 for /H9252/HS110050. Since the dissipa- tive spin transfer torque, which is proportional to /H9252, acts like an external magnetic field on the domain wall /H20851see Eq. /H208498/H20850/H20852, we recover the usual results for a field-driven domain-wallmotion 4from our model. This result is also understood from the fact that an external magnetic field does not tilt thedomain-wall potential in the chirality direction, as opposedto a current, so that the domain-wall chirality plays no role ina field-driven domain-wall creep. We observe that if wewould take the potential for the chirality of the domain wallto be a disorder potential instead of the washboard potential,we would find that /H9256=3 /5 and /H9262c=1 /7 for both /H9252=0 and /H9252/HS110050. Finally, we note that Eq. /H208494/H20850, or equivalently Eq. /H208498/H20850, contains a description of Walker breakdown46in the clean zero-temperature limit and is also able to describe the tran-sition from the creep regime to the regime of precessionalfield-driven domain-wall motion, which was recentlyobserved. 47 IV. DISCUSSION AND CONCLUSIONS In very recent experiments on domain walls in the ferro- magnetic semiconductor GaMnAs, Yamanouchi et al.33ob- served field-driven domain-wall creep with exponent /H9262f/H112291 and current-driven creep with /H9262c/H112291/3 over 5 orders of mag- nitude of domain-wall velocities. The fact that these two ex-ponents are different could imply that /H9252is extremely small for this material. For /H9252=0 and the specific pinning potentialdiscussed in the previous section, it is, however, impossible to find a single roughness exponent that yields both /H9262f=1 and/H9262c=1 /3./H20849Note that the theoretical arguments in Ref. 33 give/H9262f=1 and /H9262c=1 /2./H20850 Although it is extremely hard to determine the micro- scopic features of the pinning potential, we emphasize that ifpinning is not provided mainly by pointlike defects /H20849as con- sidered in this paper and argued by Yamanouchi et al. 33to be the case in their experiments /H20850but consists of random ex- tended defects, the creep exponents would dramatically change. Indeed, the latter type of disorder, which could occurin samples if there are, e.g., steps in the height of the film,allows for a variable-range hopping regime for creep, inwhich the exponent /H9262=1 /3 in the two-dimensional case. Moreover, upon increasing the driving force, a crossover oc-curs in the so-called half-loop regime, where the exponent /H9262=1.1,10An alternative explanation for the experimental re- sults of Yamanouchi et al.33would be that /H9252/HS110050 so that the behavior for field- and current-driven motion is similar. If thepinning potential is random and extended, it would be pos-sible that the current-driven experiment is probing thevariable-range hopping regime with /H9262=1 /3, whereas the field-driven case probes the half-loop regime with /H9262=1. This scenario would also reconcile the results of Ref. 33with previous ones,21which yielded a critical exponent of /H9262 /H112290.5, as the latter could be in a different regime of pinning. In conclusion, further experiments are required to clarify thisissue. 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PhysRevB.95.144412.pdf

PHYSICAL REVIEW B 95, 144412 (2017) Synchronization of spin torque nano-oscillators James Turtle,1,*Pietro-Luciano Buono,2,†Antonio Palacios,1,‡Christine Dabrowski,2,§Visarath In,3,/bardbland Patrick Longhini3,¶ 1Nonlinear Dynamical Systems Group, Department of Mathematics, San Diego State University, San Diego, California 92182, USA 2Faculty of Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, Ontario, Canada L1H 7K4 3Space and Naval Warfare Systems Center Pacific, Code 71730, 53560 Hull Street, San Diego, California 92152-5001, USA (Received 12 October 2016; revised manuscript received 17 March 2017; published 12 April 2017) Synchronization of spin torque nano-oscillators (STNOs) has been a subject of extensive research as various groups try to harness the collective power of STNOs to produce a strong enough microwave signal at thenanoscale. Achieving synchronization has proven to be, however, rather difficult for even small arrays while inlarger ones the task of synchronization has eluded theorists and experimentalists altogether. In this work we solvethe synchronization problem, analytically and computationally, for networks of STNOs connected in series. Theprocedure is valid for networks of arbitrary size and it is readily extendable to other network topologies. Theseresults should help guide future experiments and, eventually, lead to the design and fabrication of a nanoscalemicrowave signal generator. DOI: 10.1103/PhysRevB.95.144412 I. INTRODUCTION The synchronization phenomenon of spin torque nano- oscillators (STNOs) has been the subject of extensive researchfor many years due to the potential of networks of STNOsto generate microwave signals at the nanoscale [ 1–3]. In the last few years, Adler-type [ 4] injection locking has emerged as the most promising method to achieve synchronization, eitherthrough an external microwave current [ 5–7] or through a microwave magnetic field [ 8,9]. In particular, it was shown recently that a record number of five nanocontact STNOs[10] can synchronize via spin-wave beams [ 11]. Non-Adlerian approaches to synchronization of nanopillar STNOs havealso been considered. Georges et al. [12] found the critical coupling strength and minimum number of STNOs for theonset of synchronization analytically by describing the STNOsas phase oscillators in the framework of Kuramoto [ 13]. Later, Iacocca and Akerman [ 14] provided conditions for the onset of phase instability that may be caused, surprisingly, by strongcoupling in identical STNOs. It is well known, however,that amplitude can affect synchronization, especially near theonset of a Hopf bifurcation [ 15]. In fact, in STNOs amplitude and phase are intrinsically coupled by the dependence of theeffective field on the magnetization [ 16]. Thus, if the Hopf bifurcation parameter is of the same scale as the couplingparameter then the amplitude is no longer negligible and theKuramoto model reduction is no longer valid. Furthermore,when the amplitude dynamics are not negligible and the naturaloscillation frequencies are not homogeneous, synchronizationmay be enhanced regardless of the topology of the network[17]. Consequently, a complete understanding of synchro- nization of nanopillar-based STNOs, via non-Adlerian type,requires an analysis that incorporates the amplitude dynamics. *jturtle@predsci.com †Pietro-Luciano.Buono@uoit.ca ‡apalacios@mail.sdsu.edu §christinedabrowski15@gmail.com /bardblvisarath@spawar.navy.mil ¶patrick.longhini@navy.milIn 2005, back-to-back publications in Nature Letters (Kaka [2], a collaboration between NIST and Hitachi GST and Mancoff [ 18] from Freescale Semiconductor) showed that two STNOs tend to phase lock when they are in close proximityof one another. The coupling in these cases resulted from spinwaves propagating through the continuous free layers, leadingto phase locking. Soon after, Grollier et al. [1] investigated computationally the behavior of a one-dimensional (1D) seriesarray of N=10 electrically coupled STNOs. Their study showed that the ac produced by each individual oscillator leadsto feedback between the STNOs, causing them to synchronize,and that, collectively, the microwave power output of thearray increases as N 2. In a follow-up study, Persson et al. [3] mapped out numerically the region of synchronization of the 1D serially connected array considered by Grollier et al. for the special case of N=2 STNOs. Their work shows that the region of parameter space where synchronizationexists is rather small, thus explaining the difficulty (alreadyobserved by experimentalists) to achieve synchronization.Liet al. [19] showed that this difficulty was due, mainly, to the coexistence of multiple stable attractors, suggestingthat the synchronization regime is highly sensitive to initialconditions. Persson et al. [3] also investigated numerically the effect of including a time delay between the magnetization-induced change in voltage and the current variation. Theyhighlight that this increases significantly the parameter regionof synchronization, especially with respect to differencesin anisotropy fields between the STNOs. We determinenumerically that the synchronization for 1000 STNOs is robustto nonhomogeneities in the anisotropy field on the order of4–5%, as Persson et al. also observes in the absence of delay. It will be worthwhile to investigate in future work the effectsof time delay and to find out whether the synchronization isrobust to larger anisotropy in the network. O nas i n g l eS T N O[ s e eF i g . 1(a)], an originally unpolarized electric current I, in amperes, is applied to the fixed magnetic layer whose magnetization is represented by ˆM.A st h e electrons pass through the layer, their spins become aligned tothat of the fixed layer, thus creating a spin-polarized current. Then the polarized current exerts a torque on the magnetization of the free layer, which can lead to steady precession. We 2469-9950/2017/95(14)/144412(8) 144412-1 ©2017 American Physical SocietyJAMES TURTLE et al. PHYSICAL REVIEW B 95, 144412 (2017) I0 RCIjR1RNR2 Mˆ Mˆ Mˆ FIG. 1. Left: Schematic representation of a nanopillar STNO. A spin-polarized current can exert a torque on the magnetization of the free layer and lead to steady precession. Right: A circuit array ofSTNOs connected in series. consider a circuit array of Nidentical STNOs coupled in series [see Fig. 1(b)] and study the conditions to synchronize the individual precessions. Our approach employs the dc current,I dc, flowing in each STNO and the angle θhof the applied magnetic field as the bifurcation parameters. No injection ofac current is required. The all-to-all coupling of the networkof identical STNOs implies a complete permutation symmetrywhich we exploit using equivariant bifurcation theory [ 20]. We search for fully synchronized periodic oscillations in the network of NSTNOs, first by finding implicit analytical expressions for Hopf bifurcation curves, in ( I dc,θh) space, at a synchronized equilibrium that yield symmetry-preservingin-phase oscillations (see Fig. 2). We calculate the stability of the synchronization manifold near a synchronous equilibriumand combine Hopf criticality results to determine regions ofparameter space where the fully synchronized periodic stateis asymptotically stable near bifurcation. More importantly,the results are valid for networks of arbitrary size N.N o r m a l hyperbolicity [ 22,23] guarantees the synchronization manifold is robust to small nonhomogeneities in the STNOs. Numerical simulations show that synchronization is preserved to approx- imately ±5% variations in anisotropy strength. Results are illustrated with arrays of up to N=1000 nano-oscillators (see Fig. 3). The analysis also captures symmetry-breaking patterns of oscillations, but we do not pursue the study ofthose cases here. These patterns are described as “multiplesynchronization attractors” in Ref. [ 24]. II. LOCI OF STABLE SYNCHRONIZED OSCILLATIONS The free-layer magnetization vector, ˆm=[m1,m2,m3]T, for an individual nanopillar oscillator is governed bythe Landau-Lifshitz-Gilbert-Slonczewski (LLGS) [ 25–28] equation dˆm dt=−γˆm×−→Heff+αˆm×dˆm dt−γμI ˆm×(ˆm׈M), (1) where γis the gyromagnetic ratio, αis the Gilbert damping term,μcontains material parameters, and /vectorHeffis the effective magnetic field. The term /vectorHeffconsists of an anisotropy field, /vectorHan=κ(ˆm·ˆe||)ˆe||, where κis the strength of the anisotropy (we set κ=45 Oe in our simulations [ 21]) ande||= [sinθ||cosφ||,sinθ||sinφ||,cosθ||]Tis a preferred direction of FIG. 2. Top: Loci of Hopf bifurcations of synchronized oscilla- tions. Bottom: Stability of synchronization manifold (red, supercriti- cal Hopf and stable synchronization manifold; black, subcritical Hopf and unstable synchronization manifold; and blue, supercritical Hopf and unstable synchronization manifold). The combined results ofthese two plots reveal the optimal region to synchronize a series array of nanopillar STNOs: the first quadrant of parameter space ( I dc,θh). Parameters [ 21]a r eN1=1,N2=0,γ=2.2×105mA−1s−1,α= 0.008,κ=45 Oe, μ=0.992,ha=300 Oe, β/Delta1R=5.95×10−4. FIG. 3. Locking into synchronization with N=1000 STNOs. Start at high Idcand let the system lock into the common equilibrium. Then sweep down Idcuntil the common equilibrium vanishes and synchronized oscillations appear. Top inset: Zoom-in on the top partof the oscillation showing a high level of synchronization between all the STNOs. Bottom inset: Zoom-in on the set of random initial conditions for the N=1000 STNOs and evolution for small time values showing rapid convergence to a synchronized equilibrium. 144412-2SYNCHRONIZATION OF SPIN TORQUE NANO-OSCILLATORS PHYSICAL REVIEW B 95, 144412 (2017) magnetization. /vectorHdis a demagnetization field and we set /vectorHd= −4πS0(N1m1ˆx+N2m2ˆy+N3m3ˆz), where S0=8400/4πis the constant magnitude of the average magnetization vectorS(t) (in units of oersted) so that ˆm=S/S 0,N1,N2, and N3are dimensionless constants satisfying N1+N2+N3=1, and{ˆx,ˆy,ˆz}are the orthonormal unit vectors. /vectorHapplis an applied magnetic field given by /vectorHappl=ha[0,sinθh,cosθh]T, which we assume to lie on the yzplane at some angle θh instead of the zaxis, and note that hais in units of oersted. ˆMis the fixed-layer magnetization vector that defines the spin-polarization direction of the current. In what follows weassume θ ||=0 so that e||=[0,0,1], which produces an easy axis in the zdirection. Finally, we assume the direction of polarization of the spin-polarized current to remain constantalong the zdirection, i.e., ˆM=ˆz.For an array of STNOs, coupling occurs if the input current Iis replaced by I j. First, we assume the STNOs to be identical. Later, we consider the effects of nonhomogeneitiesas perturbations of the synchronization manifold. ApplyingKirchhoff’s laws we obtain the current through the jth STNO: I j=Idc/parenleftBigg 1+N/summationdisplay i=1β/Delta1Ricosθi(t)/parenrightBigg , (2) where Idcis a constant dc, β/Delta1Riis a parameter that depends on the resistances in the parallel and antiparallel magnetizationstates, and θ i(t) is the angle between the magnetization of the fixed and free ferromagnetic layers. We substitute Eq. ( 2) into Eq. ( 1) and, for convenience, we convert to complex stereographic coordinates through the change of variablesz j=(mj1+imj2)/(1+mj3). Direct calculations yield ˙zj=γ(1+iα) 1+α2/bracketleftBigg iha3zj+ha2 2/parenleftbig 1+z2 j/parenrightbig +iκ1−|zj|2 1+|zj|2zj−μIDCzj−μIDCβ/Delta1RN/summationdisplay k=11−|zk|2 1+|zk|2zj −4πS0 1+|zj|2/parenleftbiggN1−N2 2/parenleftbig z3 j−¯zj/parenrightbig +/parenleftbigg 1−3N1+3N2 2/parenrightbigg (zj−zj|zj|2)/parenrightbigg/bracketrightBigg , (3) where ha2=hasin(θh) andha3=hacos(θh). For the special case N1=N2=0.5, Eq. ( 3)i sm o r e amenable to analysis, and thus we can find, via MAPLE , implicit analytic expressions for the Hopf loci that yield synchronizedperiodic states for arbitrary arrays of size N. Although the synchronized periodic oscillation is unstable, we can stilluse these analytical expressions to follow, via the automaticnumerical continuation software AUTO [29], the movement of the Hopf loci as a function of the continuation parameters, where N 1=0.5+sandN2=0.5−s.F o rs=0.5, we arrive at the physically relevant configuration of easy-planeanisotropy or x-axis demagnetization. The Hopf loci curves for s=0.5 are shown in Fig. 2(top) for various sizes of networks. In addition, we determine the criticality of each Hopf loci pointthrough the Lyapunov constant formula [ 30] as well as the local asymptotic stability of the synchronization manifold near theHopf point, via AUTO . This process yields, for s=0.5, the red Hopf loci curves located in the first quadrant of ( Idc,θh) space from which stable synchronized periodic solutions bifurcate(see Fig. 2, bottom). Observe that the location of these curves implies that less current is required to synchronize larger arrays. Thisobservation suggests that synchronization in series arrays ofnanopillar STNOs depends more on the dynamical parametersthan on the coupling strength. Similar results have beenobserved in studies of power grids, which can also be treatedas Kuramoto oscillator networks [ 31]. We wish to emphasize that the aim of this paper is strictly the theoretical analysis to determine regions of existence of stablesynchronization. Effects of noise, such as linewidth reduction,are briefly addressed in Sec. VII, but a detailed analysis is ongoing and deferred to a future publication. Next we presentan outline of the analysis that was carried out to obtain theimplicit solutions of the Hopf loci.III. HOPF BIFURCATION CURVES This section summarizes the mathematical analysis of how one can exploit the symmetry of the network to obtain themain results shown in Fig. 2. Details of these calculations can be found in Appendix. Due to the all-to-all coupling that appears in Eq. ( 3)a s a consequence of Kirchhoff’s law, and the assumption ofidentical STNOs, any permutation of the STNOs in the arrayleaves the coupling term invariant; thus, the series array hassymmetry group S N, the group of all permutations of N objects. To find analytical expressions for the Hopf loci ofsynchronized solutions we study the linearized system nearthe origin. Let z=(z 1,..., z N)∈CNand denote Eq. ( 3)b y ˙zj=fj(z). Since we assume all the STNOs to be identical, we havef1=f2=···= fN. We rewrite the system of Eq. ( 3)i n abbreviated form ˙z=f(z), (4) where f=[f1,..., f N]T.L e t z0=(z0,..., z 0) be an equilib- rium solution of Eq. ( 4) with isotropy subgroup SN[20]. Then the linearization at z0is given by L:=⎡ ⎢⎢⎢⎢⎣AB ··· B B......... .........B B··· BA⎤ ⎥⎥⎥⎥⎦, (5) where A=(df jj)z=z0andB=(dfjk)z=z0are 2×2 Jacobian matrices of fj, with j/negationslash=k. Using symmetry methods, we block-diagonalize Lto a form which respects symmetry- invariant subspaces. Let Pbe the change-of-coordinates matrix. Applying PtoL, we obtain a block diagonalization of 144412-3JAMES TURTLE et al. PHYSICAL REVIEW B 95, 144412 (2017) the linear part of the coupled STNO array, /tildewideL:=P−1LP=diag{A+(N−1)B,A−B,..., A−B}. (6) From the diagonal structure, the eigenvalues of the blocksare also eigenvalues of /tildewideL. It follows that Hopf bifurcations in Eq. (4) occur if and only if A+(N−1)BorA−Bhave purely imaginary eigenvalues. In the former case, the eigenspaceassociated with A+(N−1)Bisv 0=[v,..., v ]Tand the symmetry group SNacts trivially on v0. This corresponds to a symmetry-preserving Hopf bifurcation in which allSTNOs oscillate in synchrony, i.e., the same wave form, sameamplitude, and same phase. In the latter case, the eigenvalueshave, generically, multiplicity N−1 (from the N−1 blocks A−B) and the emerging patterns of oscillations arise via symmetry-breaking Hopf bifurcations [ 20]. For instance, the case reported in Ref. [ 24], in which two pairs of STNOs are in phase with one another and half a period out-of-phase withrespect to each pair, corresponds to a Hopf symmetry-breakingpattern that emerges from the A−Bblock with N=4. A complete description of the possible patterns of oscillationsthat can appear for each value of Ncan be found via equivariant Hopf bifurcation [ 20]. The emphasis of this paper is, however, on the symmetry-preserving synchronization state. Combining the equilibrium conditions with the trace condition of purely imaginary eigenvalues for the blockA+(N−1)Band using polar coordinates, z 0=r(cosθ+ isinθ), we get the following set of equations as a function of (r,cosθ,Idc,θh): Re(fj)=0 Im(fj)=0 Tr(A+(N−1)B)=0. (7) To find the desired analytical expressions for the Hopf boundary curves, we solve Eqs. ( 7) implicitly for the state variables ( r,θ) as functions of the parameters Idcandθh.W e setN1=N2=0.5 as a starting point to facilitate the analysis. Through a series of substitutions we are able to reduce thissystem of three equations with four unknowns, ( r,θ,I dc,θh), to a single expression with two variables ( r,θh). To plot the boundary curves, we first extract the coordinate points fromthe solution sets, and back-substituting gives the actual pointvalues ( I dc,θh) along the curves. Varying Nwe can then trace the movement of the synchronous Hopf bifurcation curves.We verify along the curves obtained that det( A−B)>0 and det( A+(N−1)B)>0. The results just described are then extended using AUTO to the case N1=1,N2=N3=0 by continuing the Hopf loci curves in ( Idc,θh) space using N1=0.5+sandN2=0.5−sand letting the continuation parameter sevolve from 0 to 0.5. IV . STABILITY The Hopf bifurcation can be supercritical or subcritical, leading to stable or unstable synchronized oscillations, re-spectively. Which one appears is determined by the Lyapunovconstant [ 30]. If the Lyapunov constant is negative, the Hopf bifurcation is supercritical, whereas if it is positive, it leads to asubcritical Hopf bifurcation. Now, the stability property of thesynchronization manifold is determined by the eigenvalues transverse to the manifold. Those eigenvalues are given byN−1 copies of the eigenvalues of the block A−Band since the synchronization manifold is computed near an equilibrium,then normal hyperbolicity follows from the eigenvalues of theA−Bblock. The actual calculations of the Lyapunov constant and that of the transverse eigenvalues are technical and lengthyand appear in Appendix under nonlinear analysis. V . LOCKING INTO SYNCHRONIZATION Numerical simulations indicate the common equilibrium state of large arrays has a large basin of attraction for largevalues of dc, about 15 mA. This suggests a possible strategyto achieve synchronization in actual experiments: start theexperiments at high I dccurrent and let the system lock into the common equilibrium. Then sweep down Idcuntil the common equilibrium vanishes at a saddle-node bifurcationand stable synchronized oscillations appear, created via Hopfbifurcation from a coexisting common equilibrium found atlower I dcvalues. This strategy was tested with nonhomo- geneities introduced through variations in the anisotropy fieldconstant κ. As a consequence of the normal hyperbolicity of the synchronization manifold, we expect the synchronizationstate to be robust under small perturbations, such as thenonhomogeneities in κ. Indeed, numerical simulations confirm that the STNOs are able to synchronize with up to ±5% variations in anisotropy strength if the values are chosenrandomly from a uniform distribution (see Fig. 3), and up to±4% with a Gaussian distribution. VI. FREQUENCY RESPONSE We now employ the fast Fourier transform (FFT) to characterize the frequency response in networks of Nnon- identical oscillators coupled in series. The plots in Fig. 4show the frequency of oscillation for N=1, 10, 100, and 1000. The observed “dips” for small values of Idccorrespond to the switch from out-of-plane oscillations to in-plane oscillations. −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5246810121416182022 IDC (mA)Frequency (GHz)N=1 N=10 N=100 N=1000 FIG. 4. Frequency response of an array of NSTNOs connected in series. The observed dips in frequency correspond to switching between out-of-plane and in-plane oscillations. Parameters are thesame as in Fig. 2, with θ h=3π/4. 144412-4SYNCHRONIZATION OF SPIN TORQUE NANO-OSCILLATORS PHYSICAL REVIEW B 95, 144412 (2017) Forθh=0, the switch is characterized by a gluing bifurcation, that is, a global bifurcation where a pair of homoclinic loops(symmetrically related in this case) are connected to a saddleequilibrium; see Ref. [ 32] for an example in the context of STNOs. For θ h=3π/4, which is the value used in Fig. 4,t h e switch involves two homoclinic bifurcations. In both cases, theswitch from out-of-plane to in-plane oscillations explains whythe frequency approaches 0 Hz. In general, lines terminatingat nonzero frequency correspond to known Hopf bifurcations,and lines terminating at or near 0 Hz correspond to suspected(not verified for every value of N) homoclinic bifurcations. These results suggest that the range of I dcvalues for which oscillations are present increases with the number of STNOs;however, the interval of possible frequencies decreases withincreased N. VII. LINEWIDTH We now consider (briefly) the effects of thermal noise on the oscillations of the synchronized solutions by adding a stochastic thermal field term /vectorHthto/vectorHeff[33,34] in the original LLGS Eq. ( 1), becoming dˆm dt=−γˆm×(−→Heff+−→Hth)+αˆm×dˆm dt −γμI ˆm×(ˆm׈M), (8) where−→Hth=[hx(t),hy(t),hz(t)]T, in which hx(t),hy(t), and hz(t) are Gaussian distributed random functions, uncorrelated, of zero mean. The added term also carries to the complexform of Eq. ( 3). Linewidth was computed as full width of the power spectral decomposition (PSD) of the synchronizedoscillations, via FFT, at half maximum of main frequency inthe PSD. The computation was carried out as a function ofI dc, on the same interval of the frequency response of Fig. 4, and for a few different values of array size N. The results are shown in Fig. 5. The spikes in linewidth that are observed near the end points of the interval of synchronization are due to the 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 IDC (mA)050100150200250300Linewidth (MHz)N=1 N=10 N=100 024 IDC (mA)5101520Frequency (GHz) FIG. 5. Linewidth. The observed dips in frequency correspond to switching between out-of-plane and in-plane oscillations. Parametersare the same as in Fig. 2, with θ h=3π/4.oscillations having different characteristics. More specifically, for small Idcthe spikes are due to a change to out-of-plane oscillations and for large Idc(and large arrays) the spikes are due to loss of synchronization; i.e., for large arrays thesynchronized oscillations give way to out-of-phase oscillationsbefore eventually converging to an equilibrium point. Butfor the most part of the interval of synchronization, thelinewidth remains relatively small. These results suggest,again, that the synchronized solution is significantly robustagainst the effects of noise. However, one would have to carryout a complete analysis of the stochastic properties of thecoupled network equations as a function of coupling strengthand noise intensity, for instance. We also wish to point outthat temperature is assumed to be implicitly included in thestochastic thermal field. Future experimental works shouldprovide a more explicit contribution of temperature variationsand material properties towards the stochastic field. Thoseissues are important but they are beyond the scope of thepresent work. Instead, our emphasis is, mainly, on findingthe conditions for the existence and stability of synchronizedoscillations in the deterministic system. We expect to carry outthe stochastic analysis in future work. In particular, it would beinteresting to obtain theoretical formulas (possibly asymptoticfor large N) for the half linewidth for serially coupled STNOs using the theory developed by Slavin and Tiberkevich [ 34]. VIII. DISCUSSION AND CONCLUSIONS To date, the strongest microwave power that has been produced by a single STNO is on the order of 0.28 μW[35]. As mentioned in the introduction, Grollier et al. [1]s h o w e d that for an array of N=10 electrically coupled STNOs, the synchronized array microwave power output increases as N2. Thus, if the N2law holds in general, 1000 synchronized nano-oscillators, as simulated in this paper, should produceabout 0.28 W. Communication systems, which require poweron the order of milliwatts, e.g., wireless devices, radar, airtraffic control, weather forecasting, and navigation systems,would only require about 188 nano-oscillators. In Ref. [ 32], we showed computationally the nature of the bifurcations leading to these attractors and discovered thatchanging the angle of the applied magnetization field couldenlarge the basin of attraction of the synchronized oscillations.In this work we extended the bifurcation analysis of nanopillar-based STNOs connected in series arrays of arbitrary size.We use equivariant bifurcation theory to find the region ofexistence and stability of the synchronization manifold forwhich all STNOs oscillate with the same frequency, phase,and amplitude. Our approach to achieve synchronization, via non-Adlerian dynamics, employs only the dc flowing in each STNO and the angle of the applied magnetic field.The main results include implicit solutions of the Hopf locias a function of the dc and the applied magnetic field.Normal hyperbolicity of the synchronization manifold impliesrobustness of the synchronization state to small perturbations,such as those caused by nonhomogeneities or imperfectionsduring the manufacturing process. Computer simulations withnonidentical STNOs indicate robustness up to ±5% variations, which is well within typical fabrication processes. It is ourhope that the theoretical results and simulations provided in 144412-5JAMES TURTLE et al. PHYSICAL REVIEW B 95, 144412 (2017) this paper will help guide ongoing experiments. The STNOs are currently fabricated using the 50-nm technology wherelarge arrays can be configured on a substrate. Each oscillatoris independently isolated and unconnected at the fabricationstage. Once the devices are finished, the STNOs are bondedand connected in a series array. The postfabrication bondingand connection will afford us the opportunity to verify theresults established in this paper. ACKNOWLEDGMENTS We acknowledge support from ONR Code 30. J.T. and A.P. were supported in part by NSF Grant No. CMMI-1068831.P.L.B. (Discovery Grant) and C.D. (USRA) acknowledgefunding from NSERC Canada. APPENDIX: HOPF CURVES This appendix describes the mathematical analysis that was carried out to obtain the boundary curves that lead an array ofSTNO into and out of synchronization, as is shown in Fig. 2in the main text. We start by considering again the array dynamicsin stereographic coordinates captured by Eq. ( 3) with the full network in abbreviated form given by Eq. ( 4). 1. Linear analysis Letz0=(z0,..., z 0) be an equilibrium solution of Eq. ( 4) with isotropy subgroup SN[20]. Then, as described in the text, the linearization at z0is given by L:=⎡ ⎢⎢⎢⎢⎣AB ··· B B......... .........B B··· BA⎤ ⎥⎥⎥⎥⎦, where A=(df jj)z=z0andB=(dfjk)z=z0are 2×2 Jacobian matrices of fj, with j/negationslash=k. To diagonalize L,w ee m p l o yt h e SNisotypic decomposition of the phase space CN, which is given by CN=V1⊕CN,0, where V1={(z,..., z )|z∈C}, CN,0={(z1,..., z N)∈CN|z1+···+ zN=0} are absolutely irreducible representations of SN[20]. Let vj=[v,ζjv,ζ2jv,..., ζ(N−1)jv]T, where ζ=exp (2πi/N ) andv∈R. The vector v0is a basis forV1while the remaining vectors vj,j=1,..., N −1, form a basis for CN,0.N o wl e t P=[Re{v0},Im{v0},Re{¯v0},Im{¯v0},..., Re{vN−1},Im{vN−1},Re{¯vN−1},Im{¯vN−1}]T. Applying PtoL, we obtain the following block diagonal- ization of the linear part of the coupled STNO array: /tildewideL:=P−1LP=diag{A+(N−1)B,A−B,..., A−B}. (A1)From the diagonal structure, the eigenvalues of the blocks are also eigenvalues of /tildewideL. It follows that Hopf bifurcations in Eq. (4) occur if and only if A+(N−1)BorA−Bhave purely imaginary eigenvalues. In the former case, the eigenspaceassociated with A+(N−1)Bis v 0=[v,..., v ]T, where the symmetry group SNacts trivially. This corresponds to a symmetry-preserving Hopf bifurcation in which allSTNOs oscillate in synchrony, i.e., with the same wave form,the same amplitude, and the same phase. In the latter case,the eigenvalues have, generically, multiplicity N−1 (from the N−1 blocks A−B) and the emerging patterns of oscillations arise via symmetry-breaking Hopf bifurcations [ 20]. Com- bining the equilibrium conditions with the trace condition ofpurely imaginary eigenvalues for the block A+(N−1)B(or equivalently A−Bfor symmetry-breaking Hopf bifurcation) and using polar coordinates, z 0=r(cosθ+isinθ), we get the following set of equations as a function of ( r,cosθ,Idc,θh): Re(fj)=0, Im(fj)=0, Tr(A+(N−1)B)=0(A2) and require Tr(A−B)<0, det(A−B)>0, det(A+(N−1)B)>0, on the solution set of Eqs. ( A2) to guarantee no eigenval- ues with positive real parts. To find the desired analyticalexpressions for the Hopf boundary curves, we solve Eqs. ( A2) implicitly for the state variables ( r,θ) as functions of the parameters I dcandθh.W es e t N1=N2=0.5 as a starting point to facilitate analysis. Through a series of substitutionswe are able to reduce this system of three equations withfour unknowns, ( r,θ,I dc,θh), to a single expression with two variables ( r,θh). Using MAPLE ’simplicitplot function 16 times, curves are found in the ( r,θh) domain to account for all possible solutions. Combining results produces the desiredzero solution set of Eqs. ( A2). To plot the Hopf curves, we first extract the coordinate points from the solution sets, andback-substituting gives the actual point values ( I dc,θ) along the curves. Then we substitute these points to verify thatdet(A−B)>0 and det( A+(N−1)B)>0. By varying N in the implicit solver, we are then able to trace the movement of the synchronous Hopf bifurcation curves. As mentionedabove, the Hopf curves are extended using AUTO to the caseN1=1,N2=N3=0, and those are the curves plotted in Fig. 2. 2. Nonlinear analysis We set again N1=N2=0.5 as a starting point and assume A+(N−1)Bhas a pair of purely imaginary eigenvalues and translate the equilibrium z0of Eq. ( 4) to the origin using v= z−z0, leading to ˙v=f(v+z0), 144412-6SYNCHRONIZATION OF SPIN TORQUE NANO-OSCILLATORS PHYSICAL REVIEW B 95, 144412 (2017) where fjis given by fj=γ(1+iα) 1+α2/bracketleftBigg iha3(vj+z0)+ha2 2(1+(vj+z0)2) +iκ1−|vj+z0|2 1+|vj+z0|2(vj+z0)−μIdc(vj+z0) −μIdcβ/Delta1RN/summationdisplay k=11−|vk+z0|2 1+|vk+z0|2(vj+z0) +2πiS 0 1+|vj+z0|2(vj+z0−(vj+z0)|vj+z0|2)/bracketrightBigg . (A3) To determine criticality of the Hopf bifurcation we set g(v,v)=(1+|v+z0|2)−1and Taylor expand Eq. ( A3)a t (0,0) up to cubic order [ 30], which yields ˙vj=H1(vj,vj,v,v)+N(vj,vj,v,v), (A4) where N(vj,vj,v,v)=H2(vj,vj,v,v)+H3(vj,vj,v,v) with H/lscripta homogeneous polynomial of degree /lscript. That is, H1(v,v)=a10vj+a01vj+n/summationdisplay k=1b10vk+b01vk, H2(v,v)=a20v2 j+a11|vj|2+a02v2 j+n/summationdisplay k=1b20v2 k +b11|vk|2+b02vk+c110vjvk+c101vjvk, H3(v,v)=a30v3 j+a21|vj|2vj+a12|vj|2vj+a03v3 j +n/summationdisplay k=1b30v3 k+b21|vk|2vk+b12|vk|2vk+b03v3 k +/parenleftbig c120v2 k+c111|vk|2+c102v2 k/parenrightbig vj. For brevity, we list only a few of the coefficients: b10τ=μIdcβ/Delta1R(2g(0,0)2|z0|2), a10τ=iha3+z0ha2+iκg(0,0)2(1−2|z0|2−|z0|4)−μIdc −μIdcβ/Delta1Rg(0,0)2(N(1−|z0|4)−2|z0|2) +2πiS 0g(0,0)2(1−2|z0|2−|z0|4)−b10τ, b11τ=−2μIdcβ/Delta1Rz0(|z0|2−1)g(0,0)3, c101τ=2μIdcβ/Delta1Rz0g(0,0)2, a11τ=−4z0g(0,0)3/parenleftbigg iκ+i 2−μIdcβ/Delta1R/parenrightbigg −b11τ−c101τ, where τ=(1+α2)/(γ(1+iα)). We now rewrite Eq. ( A4) using the same matrix Pgiven by the decomposition of CN=CN,0/circleplustextV1intoSNirreducible representations and letting v=Pu, yielding ˙u=/tildewideLu+PTN(Pu,Pu), where /tildewideL=PTLPare the linear terms given by Eq. ( A1) and the nonlinear terms are N(v,v)= (N(v1,v1,v,v),..., N(vN,vN,v,v))T.An important observation is that the center manifold is V1= Fix(SN) and so the flow-invariant center manifold is in fact a subspace for Eq. ( 4). Thus we can compute the criticality of the Hopf bifurcation directly from the equation for ˙u1evaluated atu/lscript=u/lscript=0f o r/lscript=2,..., N , which yields ˙u1=G10u1+G01u1+G20u2 1+G11|u1|2+G02u2 1 +G30u3 1+G21|u1|2u1+G12|u1|2u1+G03u3 1,(A5) where G10=a10+Nb 10, G01=a01+Nb 01, G20=[a20+N(b20+c110)]/√ N, G11=[a11+N(b11+c101)]/√ N, G02=(a02+Nb 02)/√ N, G30=[a30+√ N(b30+c120)]/√ N, G21=[a21+√ N(b21+c111)]/√ N, G12=[a12+√ N(b12+c102)]/√ N, G03=(a03+√ Nb 03)/√ N. Now, at a Hopf bifurcation, Re( G10)=0 and the eigenval- ues are ±iρwith ρ:=/radicalbig |G10|2−|G01|2. We use the linear transformation Q=/parenleftbiggG01 iIm(G10)−iρ −iIm(G10)+iρ G01/parenrightbigg and the change of coordinates [ w1,¯w1]=Q[u1,¯u1]Tto diagonalize the linear part of Eq. ( A5)t od i a g ( iρ,−iρ). Let /tildewideH/lscript(w1,w1)=Q−1H/lscript(Q(w1,w1)T)f o r/lscript=2,3, then ˙w1=iρw 1+ρ+Im(G10) 2G01ρ(/tildewideH2(w1,w1)+/tildewideH3(w1,w1)) −i 2ρ(/tildewideH2(w1,w1)+/tildewideH3(w1,w1)). (A6) We denote by gijthe coefficients of the quadratic and cubic terms; i+j=/lscriptand/lscript=2,3. For the quadratic terms, the coefficients are g20=[ρ+Im(G10)] 2G01ρ/bracketleftbig 4G20G2 01+G11(−2G10G01i+2iG01ρ) +G02/parenleftbig −G2 10+2G10ρ−ρ2/parenrightbig/bracketrightbig −i 2ρ/parenleftbig 4G20G2 01+G11(−2G10G01i+2iG01ρ) +G02/parenleftbig −G2 10+2G10ρ−ρ2/parenrightbig/parenrightbig , g11=[ρ+Im(G10)] 2G01ρ/bracketleftbig 8G20G2 01+G11(−4G10G01i) +G02/parenleftbig −2G2 10+2ρ2/parenrightbig/bracketrightbig −i 2ρ/parenleftbig 8G20G2 01+G11(−4G10G01i) +G02/parenleftbig −2G2 10+2ρ2/parenrightbig/parenrightbig , 144412-7JAMES TURTLE et al. PHYSICAL REVIEW B 95, 144412 (2017) g02=[ρ+Im(G10)] 2G01ρ/bracketleftbig 4G20G2 01+G11(−2G10G01i−2iG01) +G02/parenleftbig −G2 10−2G10ρ−ρ2/parenrightbig/bracketrightbig −i 2ρ/parenleftbig 4G20G2 01+G11(−2G10G01i−2iG01) +G02/parenleftbig −G2 10−2G10ρ−ρ2/parenrightbig/parenrightbig , and the cubic coefficient is g21=[ρ+Im(G10)] 2G01ρW−i 2ρW, where W:=/braceleftbig 12G30G3 01+G21/parenleftbig −6G10G2 01i+2iG2 01ρ/parenrightbig +G12[4G10G01(−G10+ρ)−2G10(G10−ρ)] +2G01ρ(G10+ρ) +G03/bracketleftbig/parenleftbig G2 10−2G10ρ+ρ2/parenrightbig (G10+ρ)i +2i(ρ2−G2 10)(−G10+ρ)/bracketrightbig/bracerightbig .3. Lyapunov constant and stability Using the coefficients just listed above, we then obtain the Lyapunov constant from the formula [ 30] Re(c1)=Re/parenleftbiggi 2ρ/parenleftbigg g20g11−2|g11|2−1 3|g02|2/parenrightbigg +g21 2/parenrightbigg . (A7) The Hopf bifurcation is supercritical if Re( c1)<0 and subcritical if Re( c1)>0. However, this condition only de- termines the stability of the synchronized periodic solutionon the center manifold. Thus, we also need to considerthe eigenvalues transverse to the center manifold. Thoseeigenvalues are given by N−1 copies of the eigenvalues of the block A−Bwith real parts 1 2Tr(A−B)=Re(a10−b10). It follows that the synchronized oscillations are asymptoticallystable if Re( a 10−b10)<0. ForN1=N2=0.5, subcritical Hopf bifurcations are ob- tained. We change the direction of demagnetization to N1=1, N2=N3=0 by numerical continuation using AUTO and we obtain that Hopf bifurcation curves in the first quadrantof (I dc,θh) space are supercritical and the synchronization manifold is asymptotically stable near z0. This leads to an asymptotically stable periodic solution near bifurcation.See Fig. 2. [1] J. Grollier, V . Cros, and A. Fert, P h y s .R e v .B 73,060409(R) (2006 ). [2] S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, Nat. 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Amiri, I. N. Krivorotov, H. Zhao, G. Finocchio, J. P. Wang, J. A. Katine, Y . Huai, J. Langer, K. Galatsis, K. L.Wang, and H. W. Jiang, ACS Nano 6,6115 (2012 ). 144412-8

PhysRevB.102.144419.pdf

PHYSICAL REVIEW B 102, 144419 (2020) Editors’ Suggestion Evaluation of the switching rate for magnetic nanoparticles: Analysis, optimization, and comparison of various numerical simulation algorithms Elena K. Semenova , Dmitry V . Berkov , and Natalia L. Gorn General Numerics Research Lab, Moritz-von-Rohr-Straße 1A, D-07745 Jena, Germany (Received 4 August 2020; revised 22 September 2020; accepted 23 September 2020; published 14 October 2020) In this paper, we present a detailed comparative study of various analytical and numerical methods intended for the evaluation of the escape rate over high-energy barriers (transition rate or, equivalently, switching times)in magnetic systems, using the archetypal application-relevant model of a biaxial macrospin. First, we derivea closed-form analytical expression of the transition rate for such a particle, using the general formalismof Dejardin et al. [Phys. Rev. E 63, 021102 (2001) ], and define a parameter which determines whether the system is in the low, intermediate, or high damping regimes. Then we carry out a comprehensive analysis ofthree numerical algorithms: time-temperature extrapolation method, “energy bounce” methods [S. Wang and P.Visscher, J. Appl. Phys. 99, 08G106 (2006) ], and the forward-flux sampling [R. J. Allen et al. ,P h y s .R e v .L e t t . 94, 018104 (2005) ], which appear to be the most promising candidates for evaluating the transition rate using computer simulations. Based on underlying physical principles and peculiarities of magnetic moment systems,we suggest several optimization possibilities, which strongly improve the performance of these methods forour applications. For energy barriers /Delta1Ein the range 10 k BT/lessorequalslant/Delta1E/lessorequalslant60kBTwe compare the switching times, which correspondingly span more than 20 orders of magnitude, obtained with all the above-mentioned analyticaland numerical techniques. We show that although for relatively small barriers all methods agree well with eachother (and with straightforward Langevin dynamics simulations), for larger barriers the differences becomesignificant, so that only the forward-flux method provides physically reasonable results, giving switching timeswhich exceed the prediction of analytical approaches (interestingly, the ratio τ FFS sw/τan swis nearly constant for a very broad interval of switching times). The reasons for the corresponding behavior of numerical methodsare explained. Finally, we discuss the perspectives of the application of the analyzed numerical techniques tofull-scale micromagnetic simulations, where the presence of several contributions to the total system energymakes the situation qualitatively different from that for the macrospin approach. DOI: 10.1103/PhysRevB.102.144419 I. INTRODUCTION During the recent two decades, a large progress by the eval- uation of escape rates over high-energy barriers in differentphysical systems in general and in magnetic systems in par- ticular has been achieved. First of all, for systems of magnetic particles with and without internal magnetization structureseveral methods for computing the height of energy barriersseparating their metastable energy minima have been imple-mented: minimization of the Onsager-Machlup functional [ 1] for an interacting system of single-domain particles [ 2], string method searching for the “minimal energy path” based on the condition that the energy gradient component perpendicular to this path should be zero along the whole path [ 3], and the closely related “nudged elastic band” (NEB) method [ 4]. The latter method, which is presently the most widely used, is the“micromagnetic” adaptation of the NEB algorithm of Jonssonet al. [5], with the main idea to connect the neighboring system states along the transition path with artificial “springs” to prevent a too large distance between these states during the path-finding procedure. However, knowledge of the energy barrier /Delta1Ealone is obviously not enough to compute the average lifetime of asystem within an energy well (or, correspondingly, the es- cape rate /Gamma1out of this well), the quantity of real interest for applications. The simplest possibility to evaluate this rate isprovided by the Arrhenius law /Gamma1=ν attexp(/Delta1E/kBT), where the “attempt frequency” νattis usually interpreted as the os- cillation frequency of the system near the energy minimum.The evaluation of this frequency by itself for systems withan internal magnetization structure is a highly nontrivial taskdue to the existence of internal eigenmodes in such systems(see, e.g., [ 6,7]). But, even with the properly evaluated ν att, the Arrhenius formula can not be considered as a satisfactoryapproach from a fundamental point of view, as stressed, e.g.,in [8,9], because it does not contain a dependence of the switching rate on the system damping, which is mandatoryaccording to the fluctuation-dissipation theorem. The problem of providing an analytical expression of the escape rate, which would explicitly contain the damping pa-rameter, was first solved in the intermediate-to-high damping(IHD) regime by Brown [ 10] and later for very low damping (VLD) by Klik and Gu ¨nther [ 11]. In the meantime, the correct analytical description of the escape rate for a system withan arbitrary damping was provided in the classical paper ofMel’nikov and Meshkov [ 12], who have evaluated both the 2469-9950/2020/102(14)/144419(17) 144419-1 ©2020 American Physical SocietySEMENOV A, BERKOV , AND GORN PHYSICAL REVIEW B 102, 144419 (2020) lifetime of a Brownian particle in a single energy well and decay rates in a double-well potential in the correspondinggeneral case. The formalism and ideas from [ 11,12]w e r e successfully applied to a single-domain magnetic particlein [13,14], resulting in an analytical formula for the escape rate out of a single well and transition rates between twoenergy minima in a double-well magnetic system valid for alldamping regimes. The comprehensive treatment of this topiccan be found in the extensive review [ 9]. Although very useful, this analytical approach has sev- eral limitations. Even for single-domain particles, the methodcannot take into account the so-called “back-hopping” trajec-tories, where the system magnetization returns back to theinitial local minimum shortly after crossing the saddle point,i.e., without reaching the (partial) thermodynamic equilibriumin the target minimum. Further, for magnetic systems with asymmetry lower than the perfect uniaxial anisotropy with twoequivalent minima, as it is the case, e.g., for particles in anexternal field (both along the easy axis [ 15] and oblique [ 16]), or for particles with the anisotropy more complicated than auniaxial one [ 17], the treatment becomes increasingly com- plicated, making corresponding final expressions difficult inpractical applications. The really serious problem of the analytical treatment, however, is that it cannot be applied to most application-relevant cases, where the particle size is larger than eitherthe exchange or demagnetizing characteristic micromagneticlength (for corresponding definitions and discussion see,e.g., [ 18]). For such systems, magnetization configuration be- comes spatially nonhomogeneous, thus making the usage ofanalytical methods nearly impossible. For this reason, thereexists a pressing demand for numerical methods comput-ing not only the energy barrier, but the actual escape rate.Straightforward Langevin dynamics (LD), being a powerfultool for short-time simulations (see, e.g., [ 19–23], is obvi- ously not applicable for studying magnetization transitionsbetween minima separated by high-energy barriers (about/Delta1E>10k BT) because waiting times become macroscopi- cally large. Numerical methods for evaluating the escape rate in sys- tems with high barriers usually employ the paradigm of a“gradual climbing” uphill the energy surface by computingthe probability p(λ i−1→λi) to reach some intermediate in- terface λifrom the previous interface λi−1. The subsequent interfaces should be positioned relatively close to each other,either in the coordinate space or in the energy space, so that p(λ i−1→λi) can be computed reasonably fast and accurately by standard LD simulations. Multiplication of these transitionprobabilities for all interface pairs between the two energyminima of interest should give (augmented by a properlydefined factor with the dimensionality 1 /t) the total transition rate. The most successful general-purpose representative of the methods outlined above is the forward-flux sampling (FFS)(see [ 24–26] for specific issues and [ 27] for a comprehensive review). In FFS, the interfaces are defined in the coordinatespace, usually by setting the desired values of the so-called“reaction coordinate” or an “order parameter,” which valuedefines whether the transition has occurred or not. In mi-cromagnetics, this method was applied for two very specificsystems in [ 28,29]. A related method, where the interfaces were defined as the system energy values used to confinethe magnetization motion, is the “energy bounce” algorithmintroduced in [ 30]; this short paper contains only the basic idea and the application example to a macrospin with onlyone value of the energy barrier. Hence, it can be seen that as far as micromagnetic appli- cations are considered, the methods for computing transitionrates over high barriers are at their infancy (what can be seenalready from a very small number of corresponding publi-cations). Physical understanding of their functioning whenapplied to micromagnetic simulations is insufficient, system-atic comparison of corresponding numerical results with theavailable analytical expressions is, up to our knowledge, notavailable, and the optimization of the algorithms with re-spect to the minimization of the computational time (whatis crucial for such time-consuming simulations) has not beenaccomplished. Further, the analysis of possible alternative al-gorithms capable of computing the switching time without agradual climbing from the minimum to the saddle point hasalso not been performed. In our study, we intend to fill in the gaps outlined above, performing detailed analytical and numerical studiesof magnetization transitions over energy barriers. We con-fine our study to purely classical processes, leaving aside thephenomenon of macroscopic quantum tunneling of magneti-zation; the latter is usually relevant at very low temperatures(according to various estimations, for T<T qt, where Tqt∼ 100 mK ÷10 K [ 31,32]), which are of no interest for appli- cations we have in mind. This paper is organized as follows:In Sec. IIwe describe our biaxial macrospin model and derive closed-form analytical expressions for its switching rate bothin the Arrhenius approximation and in the general formal-ism [ 13,14] for an arbitrary damping value. In Sec. IIIwe present results of LD simulations, to be used as a referencefor further comparisons. In this section we also discuss indetail a very important question of distinguishing between“false” and “true” transitions, when the magnetization pro-jection of interest changes its sign. In Sec. IVwe present the most straightforward method for computing switchingrates for a system with arbitrarily high barriers using onlyLD simulations, our “time-temperature” extrapolation method(related to the idea suggested in [ 33]). In this method we use the extrapolation of switching rates obtained at severalhigher temperatures toward the room temperature to obtainthe desired quantity. Next, in Sec. V, we perform the detailed analysis of the energy bounce method (EBM) and introducetwo versions of this method, which enable to strongly reducethe corresponding computation time and to prepare EBM forusage in full-scale micromagnetic simulations. In addition, wediscuss again the criterion for filtering out the false switch-ings, as the dynamics in EBM is qualitatively different fromthat by nonconstrained LD simulations. Section VIis devoted to our implementation of the FFS method, where we sug-gest the placement of interfaces in the energy space (insteadof using magnetization projections), allowing us to obtainthe best interface positions without any optimization, thusgreatly increasing the statistical accuracy of results. Finally, inSec. VIIwe compare the results obtained by all analytical and numerical methods used in our study for energy barriers in 144419-2EV ALUATION OF THE SWITCHING RATE FOR MAGNETIC … PHYSICAL REVIEW B 102, 144419 (2020) FIG. 1. Coordinate system and geometry of simulated nanoelements. the interval 10 kBT/lessorequalslant/Delta1E/lessorequalslant60kBT, so that the correspond- ing switching times span over 20 orders of magnitude. Inthe Conclusion, we summarize our findings and discuss thecomparative quality of the studied methods and perspectivesof their application of all methods to full-scale micromagneticsimulations. II. SIMULATED MODEL AND ANALYTICAL APPROXIMATIONS FOR THE ESCAPE RATE A. Macrospin approximation (MSA) In this study, we simulate magnetization switching of el- liptical nanoelements with the thickness h=3 nm, the short axis b=40 nm, and different long axes a=50–100 nm. Corresponding geometry together with Cartesian coordinatesassumed throughout the paper is shown in Fig. 1.W eu s e magnetization M s=800 G and Gilbert damping λ=0.01 and neglect the magnetocrystalline anisotropy (magnetic pa-rameters typical for Py). Shape anisotropy is introduced inthe standard way via the demagnetizing field tensor ˆNwith diagonal components N x,Ny, and Nz[23]; in our geometry, we always have Nx<Ny<Nz. For all methods presented below, we have determined the transition rate at room temperature(T=300 K). In our simulations we use the macrospin approximation, i.e., we assume that the magnetization of nanoellipses is ho-mogeneous in space and can only rotate as a whole. We pointout that from the physical point of view this approximationis not valid for nanoelements of these sizes, because at leastthe long axis of our ellipses greatly exceeds the single-domainparticle size for Py, which is estimated to be ∼10 nm. How- ever, we shall employ the macrospin model in order to focusour study on fundamental questions important for all meth-ods intended for simulation of thermally activated switching,without yet being involved into the complicated problemsrelated to internal dynamic modes of a switching system; cor- responding problems (arising by the application of methodsdiscussed below to full-scale micromagnetic simulations) willbe discussed in Sec. VII. From the four standard contributions to the micromag- netic energy (energy in an external field, magnetocrystallineanisotropy, exchange and magnetodipolar energy), only twoterms are present in frames of MSA: energy in an externalfield and the magnetodipolar energy, which in this approxi-mation is usually called the shape anisotropy energy. The firstterm is absent in our case, as we study magnetization switch-ing without an external field. The shape anisotropy energyis defined using the above-mentioned tensor ˆNand Cartesian components of the unit magnetization vector mas E an=2πM2 sV/parenleftbig Nxm2 x+Nym2 y+Nzm2 z/parenrightbig , (1) where Vdenotes the particle volume. Expression ( 1) (biaxial anisotropy) is the simplest analytical approximation for theshape anisotropy energy of a flat elliptical magnetic nanoele-ments shown in Fig. 1; this shape is widely used for many applications including, e.g., in-plane magnetic random accessmemory (MRAM) cells. In addition, this is the simplest pos-sible model where one has the easy-plane anisotropy (with0xyas the easy plane) and the energy barrier between the two equilibrium states in this plane, along +xand−xdirections. For analytical calculations of the switching rate, we shall need the expansion of the density of this energy ( /epsilon1=E/V) near the energy minima (where m x=± 1) and the saddle points (in-plane switching, hence, my=± 1) in terms of two remaining magnetization projections. Using the relation m2 x+ m2 y+m2 z=1 for the elimination of mxin the first case and my in the second case, we obtain /epsilon1min(m)=/epsilon1(0) min+2πM2 s/parenleftbig Cyxm2 y+Czxm2 z/parenrightbig , (2) /epsilon1sad(m)=/epsilon1(0) sad+2πM2 s/parenleftbig Cxym2 x+Czym2 z/parenrightbig , (3) where Cαβ=Nα−Nβ(α,β=x,y,z); note that our constants Cαβdiffer from the analogous constants ciin [9] by the factor 4πM2 s. In this paper we shall study the escape rate from the minimum corresponding to mx=+ 1 (the region around this starting point is denoted as the basin A) to the minimum with mx=− 1 (with the surrounding region denoted as the basin B). For the analysis of the behavior of different numerical methods, we shall need the density of states (number of statesper unit energy interval) D(E). Analytical evaluation of this dependence for a macrospin with a biaxial anisotropy is verytedious. For this reason, we have computed D(E) numerically by evaluating the system energy ( 1) for all moment orienta- tions on the ( θ,φ) grid in the spherical coordinate system shown in Fig. 1, with the polar axis along the xaxis and the azimuthal axis φ=0, along the yaxis of our coordinate system. Correspondingly weighted ( w∝sinθ) energy val- ues were assembled to a histogram. Resulting (normalized)D(E) for several macrospin sizes are shown in Fig. 2.N o t e that the density of states for a biaxial macrospin divergesat the saddle-point energy E sad=E(θ=π/2,φ=0o rπ) (because at this point both partial energy derivatives are zero:∂E/m x=∂E/mz=0); however, this divergence is not as strong as for a uniaxial macrospin, where the saddle on theenergy surface is represented by the whole line θ=π/2. In terms of this density of states, the probability p(E)t o observe an energy Efor a system in thermodynamic equilib- rium is p(E)=D(E)e −E/kBT. (4) B. Arrhenius approximation of the escape rate As mentioned in the Introduction, the simplest (and still most widely used) analytical approximation for the escape 144419-3SEMENOV A, BERKOV , AND GORN PHYSICAL REVIEW B 102, 144419 (2020) FIG. 2. Densities of states D(E) for macrospins with various long ellipse axis aas shown in the legend. rate resulting from the Arrhenius law is called the transition state theory rate [ 34]: /Gamma1Arr=ωatt 2πe−/Delta1E/kBT(5) (kis Boltzmann constant). In MSA, both the energy barrier /Delta1Eand the attempt frequency ωattentering this expression can be evaluated analytically: /Delta1E=KV=2πM2 sCyxπhab 4, (6) ωatt=γ4πMs/radicalbig CyxCzx (7) (for the last expression see, e.g., [ 35]), where γis the gyro- magnetic ratio. Dependencies of these quantities on the long axis of our elliptical nanoelement are shown in Fig. 3. Demagnetizing factors required in ( 6) and ( 7) have been computed in our paper [ 23] by comparing initial slopes of the hysteresis loop calculated by full-scale micromagnetic simulations (using thecell size 2 ×2n m 2in plane) with the corresponding slope ex- pected in MSA. As explained in detail in [ 23], demagnetizing factors computed this way represent a better approximation FIG. 3. Attempt frequency νatt=ωatt/2π(main plot) and energy barrier ( 6) (inset) as functions of the long ellipse axis a(short axis b=40 nm).to the demagnetizing factors of a flat elliptical nanoelement than those computed from the axis ratios of the correspondingthree-dimensional (3D) ellipsoid. Note that the dependence/Delta1E(a) is slightly nonlinear because demagnetizing coeffi- cients N x(y,z)also depend of a; however, this effect is weak compared to the linear dependence V∼a. The switching time in this approximation is τArr sw=1 2/Gamma1Arr, (8) where the additional factor1 2is due to the existence of two saddle points in our system. Note that in the interval of thelong axis lengths a=50–110 nm studied here the switching time spans about 25 orders of magnitude. The dependence oflog(τ Arr sw)v s ais also slightly nonlinear, not only due to the nonlinearity of /Delta1E(a), but also due to the nonlinear depen- dence νatt(a) (see Fig. 3). C. Magnetization escape rate for a biaxial particle by arbitrary damping The Arrhenius expression has two well-known technical drawbacks: (i) it does not take into account the curvature ofthe energy landscape around the saddle point and (ii) it doesnot consider the possibility of a reverse transition shortly afterthe particle has crossed the energy barrier (back hopping).However, a much more serious problem is that the Arrheniuslaw does not include the damping constant, meaning that inthis formalism the switching can occur without any damp-ing, which is clearly impossible (no coupling to thermal bathpresent, see [ 8] for details). Large effort has been undertaken to derive physically meaningful expressions for various damp-ing regimes [ 8,12,36]; corresponding results for the magnetic particle switching (where the precessional motion plays a veryimportant role) have been summarized in the comprehensivereview [ 9]. To proceed with our specific case, we shall need the general analytical expression for the magnetization escape rate /Gamma1 an, valid (in the limit of the high-energy barrier /Delta1E/greatermuchkBT)f o r all values of the damping parameter α[9,14]: /Gamma1an=/Omega1 ωsA(αS)/Gamma1Arr,τan sw=1 2/Gamma1an. (9) Here, the damped saddle angular frequency /Omega1 /Omega1=πM2 sV kBT1 τN/bracketleftbigg/radicalbigg (Cxy−Czy)2−4CxyCzy α2−(Cxy+Czy)/bracketrightbigg (10) [CxyandCzyare defined after the Eq. ( 3)] contains the charac- teristic diffusion time of the magnetization τN=VM s 2γkBT1+α2 α. (11) The undamped saddle angular frequency ωsis defined analo- gously to the attempt frequency ( 7): ωs=γ4πMs/radicalbig −CxyCzy (12) (note that Cxy=Nx−Ny<0). 144419-4EV ALUATION OF THE SWITCHING RATE FOR MAGNETIC … PHYSICAL REVIEW B 102, 144419 (2020) Prefactor A, called the depopulation factor because the decrease rate of the particle concentration within an energyminimum is proportional to A, has been derived in case of an arbitrary damping for the first time by Melnikov andMeshkov [ 12] and has the form A(αS)=exp/bracketleftbigg1 π/integraldisplay∞ 0ln{1−e−αS(z2+1/4)}dz z2+1/4/bracketrightbigg . (13) For small α→0, the asymptotic behavior of this prefactor isA(αS)→αS,s oE q .( 9) reduces to the result of Klik and Gunther [ 11]. For large damping α→∞ we have A(αS)→1, and Eq. ( 9) reproduces the escape rate in the intermediate-to- high (IHD) damping range [ 37]. The dimensionless action Sin the depopulation factor ( 13) is given in case of a magnetic particle by the integral ( p= cosθ)[9,11] S=Vp kBT/contintegraldisplay E=Esad/bracketleftbigg [1−p2(φ)]∂/epsilon1 ∂pdφ−1 1−p2∂/epsilon1 ∂φdp/bracketrightbigg , (14) where /epsilon1denotes the energy density, expressed as the func- tion of spherical coordinates of the magnetic moment: /epsilon1= /epsilon1(θ,φ). This integral should be taken along the trajectory where the system energy Eis equal to the saddle-point energy Esad; hence, the polar angle θcan be viewed as a function of the azimuthal angle φ, so that p≡cosθ=p(φ). To evaluate the action ( 14), we shall use the energy den- sity expression ( 2) and introduce the reduced energy density u(my,mz)a s u(my,mz)=Cyxm2 y(θ,φ)+Czxm2 z(θ,φ) (15) so that the action takes the form S=2πM2 sVp kBT/contintegraldisplay E=Esad/bracketleftbigg (1−p2)∂u ∂pdφ−1 1−p2∂u ∂φdp/bracketrightbigg =2πM2 sVp kBT[I1+I2]. (16)In the spherical coordinate system with the polar axis along the Cartesian xaxis, and φdefined as the angle between the projection of monto the yzplane and yaxis, we have mx= cosθ,my=sinθcosφ,mz=sinθsinφ, so that the energy density ( 15)i s u(θ,φ)=sin2θ(Cyxcos2φ+Czxsin2φ) =sin2θCyx(1+κsin2φ) =Cyx(1−p2)(1+κsin2φ), (17) where the ratio κ=(Czx−Cyx)/Cyx=Czy/Cyx>0i si n t r o - duced. The integration trajectory passes through the saddle point (θ=π/2,φ=0) and hence the function u(θ,φ) along this trajectory is equal to u=usad(π/2,0)=Cyx, leading to the relation (1−p2)(1+κsin2φ)=1. (18) Using this condition by calculating partial derivatives ofthe energy density ( 17) and substituting them into the inte- gral ( 16), we obtain I 1=/contintegraldisplay u=usad[1−p2(φ)]∂u ∂pdφ=− 4Cyx/integraldisplayπ 0p(φ)dφ, (19) I2=−/contintegraldisplay E=Esad1 1−p2∂u ∂φdp=−Cyxκ/contintegraldisplay u=usadsin 2φdp dφdφ. (20) Employing the same relation ( 18), we can find the derivative dp dφ=±√κcosφu (1+κsin2φ)3/2, (21) where the upper sign corresponds to the interval φ∈[0,π], the lower sign to φ∈(π,2π). Substituting this derivative into ( 20) and reducing integration limits over φto [0,π], we obtain I1+I2=− 4Cyx√κ/integraldisplayπ 0/bracketleftbiggsinφ (1+κsin2φ)1/2+κsinφcos2φ (1+κsin2φ)3/2/bracketrightbigg dφ=− 4Cyx√κ(1+κ)I(κ) (22) (the minus sign appears due to the chosen integration direction along the trajectory and hence can be ignored, as we need onlythe absolute value of the action). Forκ/greaterorequalslant0 the integral I(k)i n( 22) can be evaluated analyt- ically: I(κ)=2/(1+κ). Hence, the final result for the action Sis S=16πM 2 sVp kBT/radicalbig CyxCzy. (23) An important remark is in order. The depopulation factor A[Eq. ( 13)] depends on the product αS. Hence, it is clear already from the prefactor in the action expression ( 16) that the parameter which defines whether we are in the region ofa small, intermediate, or large damping is notthe value of the damping parameter αby itself, but the value of the product ofαand the relation of the energy barrier to the thermal energy /Delta1E/k BT(∼M2 sV/kBTin our case). We shall return to this statement below by comparing the results obtained by variousmethods. III. LANGEVIN DYNAMICS (LD) SIMULATIONS The most straightforward method to determine the switch- ing rate between the two metastable system states is thesimulation of the system dynamics in presence of thermalfluctuations using the corresponding stochastic equation ofmotion [ 38]. Taking into account that we are interested in magnetic systems by temperatures much lower than theCurie temperature (so that the magnetization magnitude 144419-5SEMENOV A, BERKOV , AND GORN PHYSICAL REVIEW B 102, 144419 (2020) M=const), we use the Landau-Lifshitz-Gilbert equation dM dt=−γ[M×(Hdet+Hfl)] −γλ Ms[M×[M×(Hdet+Hfl)]] (24) to describe the system dynamics. Here, the fluctuation field accounting for thermal fluctuations has the properties /angbracketleftHfl ζ/angbracketright= 0 and /angbracketleftHfl ζ(0)·Hfl ψ(t)/angbracketright=2Dflδ(t)δζψ, where the fluctuation power is Dfl=λ/(1+λ2)(kBT/γμ )(ζ,ψ=x,y,z),μbeing the magnetic moment magnitude. The deterministic field Hdet contains the contributions from all magnetic energy terms, what in our case reduced to the anisotropy field only (we recallthat we simulate nanoelements in the absence of an externalfield). The constant γin (24) relates to the gyromagnetic ratio γ 0and the damping αin the alternative form of this equa- tion proposed by Gilbert ˙M=−γ0[M×(H−(α/Ms)˙M)]a s γ≈γ0viaγ=γ0/(1+α2), and damping parameters λand αare equal; for further details, see, e.g., [ 39]. We have performed LD simulations using our micromag- netic package MICROMAGUS [40] where Eq. ( 24)i si n t e g r a t e d using one of the adaptive step-size algorithms (Runge-Kuttaor Bulirsch-Stoer) for the stochastic differential equations(SDE) describing the dynamics of vector fields with theconstant vector magnitudes. The possibility to apply suchmethods to SDE ( 24) is justified in [ 41], where we have shown that for M=const both Ito and Stratonovich stochastic calculi lead to identical results. For generation of the thermal noisewe have used the version of the vector statistics (VS) Gaussianrandom numbers generator from Intel MKL library, whichemploys the inverse cumulative distribution function method(ICDF-type generator) to produce a sequence of independentGaussian random numbers with the prescribed mean and dis-tribution width; this generator is known for its high quality.Cross checks with other (simpler) random number generators,l i k et h o s ef r o m[ 42], have shown that final results remain the same (in frames of statistical errors). Simulation of magnetization switching using Langevin dy- namics is possible only for systems with relatively smallenergy barriers (not higher than /Delta1E∼10k BT) because sim- ulation times grow exponentially with the energy barrier. Forthis reason we could perform LD studies only for macrospinscorresponding to nanoellipses with a=50 nm ( /Delta1E≈9k BT) anda=55 nm ( /Delta1E≈14kBT); we remind that T=300 K. Taking into account that both energy minima for our sys- tem are equivalent ( Hext=0), the average switching time τLD sw for LD simulations can be computed as τLD sw=tsim Nsw, (25) where tsimdenotes the (physical) simulation time and Nsw the number of switching events between the energy minima observed during the simulation run. However, in order to calculate τLD swproperly, we have to correctly determine whether the true switching (defined asthe transition between two metastable energy minima withm x=± 1) took place. For this purpose, it is not enough to count the number of times when the dependence mx(t) changes its sign [see Fig. 4(a)] (or, to be more careful, crosses FIG. 4. Difference between “true” and “false” switchings (see text for details). On (c) the 3D magnetization trajectory in the timeinterval 17 ns /lessorequalslantτ/lessorequalslant20 ns is depicted. some negative threshold, say, mx=− 0.2, when coming from positive values). For the correct determination of Nswwe have to distinguish between true and false switchings. A simple example where this difference is clear is illus- trated in Figs. 4(a)–4(c). Here, a true switching has occurred atτ≈13 ns; corresponding pieces on the dependencies mx(t) (which changes sign during the switching) and mz(t)a r e drawn in green. But, the (numerous!) sign changes of mx(t) in the interval τ≈18–19 ns clearly do not correspond to any real switching process. 3D representation of the corre-sponding piece of the magnetization trajectory, marked in redin Fig. 4(c), demonstrates that these sign changes of m xare due to the so-called out-of-plane (OOP) precession. By thisprecession kind the magnetization rotates in the high-energyregion of the energy landscape (because /bardblm z/bardblis relatively large), so that during this process no real switching betweenthe energy minima occurs. Hence, in Fig. 4during the time interval 17 /lessorequalslantτ/lessorequalslant20 ns only one real switching is observed. To distinguish between true and false switchings, one could in principle perform the analysis of “candidate” cases, i.e.,events when the sign change of m xhas been detected, using the time dependencies of other magnetization projections.For example, for the particular case shown in Fig. 4,t h e m z projection does not change its sign during the whole time interval marked in red, indicating that the magnetization pre-cession takes place only on one side of the easy plane ofthe nanoelement ( m z/lessorequalslant0), meaning that the OOP precession 144419-6EV ALUATION OF THE SWITCHING RATE FOR MAGNETIC … PHYSICAL REVIEW B 102, 144419 (2020) FIG. 5. Excitation of the macrospin before and its equilibration after the true switching as mx(t) dependence (a) and 3D magnetiza- tion trajectory (b) (the switching process itself is highlighted in dark red). Only the switchings after which the macrospin spends in the new basin more time than the equilibration time ( ≈3 ns) are counted as true switching events. (and not a true switching) is in process. However, consider- ation of all particular cases would make the corresponding“projection-based” differentiating algorithm too complicatedand thus unreliable. For this reason, we have adopted a more general method to identify true switching events. The method is based onthe very definition of “switching” which is understood as atransition between two metastable states, whereby the systemunder study must spend sufficient time in the vicinity of eachstate in order to achieve a partial thermal equilibrium withincorresponding energy basins. If this is not the case, the switch-ing is considered as false. The idea is illustrated in Figs. 5and 6. During a true switching (Fig. 5) the magnetic moment is first excited by thermal fluctuations so that it can overcome the energy barrier,and afterward the equilibration in the other energy well takesplace. This equilibration, as shown in Fig. 5(a), takes about t eq≈2 ns, whereby teqdepends mainly on the damping pa- rameter λand slightly on the energy landscape near the energy minima. Based on this finding, we consider a switching asbeing “true” when the time /Delta1t wellspent by the system in the target energy well after the switching is larger than teq. To further support this idea, we have collected the his- togram of time intervals between two subsequent sign changes of mx(t). This histogram shows a huge peak for small FIG. 6. (a) The region of the histogram of τswobtained from LD simulations when alltime moments when mxchanges sign are counted as switchings (note the log scale of the yaxis). (b) The region of this histogram for t<1 ns. (c) A typical mx(t) dependence for a false switching event [taken from the ρ(τsw) peak shown in (b)] and (d) the corresponding 3D magnetization trajectory. time intervals, as shown in Fig. 6(a) (note the logarithmic scale of the yaxis!); this peak is presented in Fig. 6(b) in a much higher resolution. The analysis of magnetization trajec-tories corresponding to the events attributed to this peak hasclearly demonstrated that these events typically represent an“excursion” of the magnetization toward the opposite energyminimum [see trajectories in Figs. 6(c) and 6(d)], and are clearly “false” switchings. So, in further analysis we haveused the criterion /Delta1t well>teqto identify real (true) switching events in LD simulations. In order to obtain a sufficiently accurate statistics, for nanoelements with a=50 nm, we have simulated a col- lection of 100 macrospins during tsim=150μs and for elements with a=55 nm an ensemble of 400 macrospins during tsim=5m s=5×106ns, applying the approach de- scribed in our paper [ 23]. After subtracting false switching events (using the criterion described above), we have ob-tained τ LD sw(a=50 nm) =2.36(±0.3)×103ns and τLD sw(a= 55 nm) =1.3(±0.2)×105ns. Note that for a=50 nm the switching time is ≈1.6 times and for a=55 nm about 1.4 times larger than the analytical values for these elements givenby (9). This difference is most probably due to the fact that the approximation ( 9) does not take into “return” trajectories and thus overestimates the transition rate. IV . TIME-TEMPERATURE EXTRAPOLATION METHOD (TTE) The most straightforward idea which can be used to obtain switching rates for systems with energy barriers unachievablefor standard LD simulations at room temperature is to per-form LD simulations at higher temperatures and extrapolateobtained switching rates to the temperature of interest. Fromthe quantitative point of view, this method (which we shallcall the time-temperature extrapolation method or TTE) em-ploys the assumption that the main temperature dependenceofτ swis due to the exponential factor in the expression τsw= τ0exp(/Delta1E/kBT) and all other dependencies on T, which may be hidden in the prefactor τ0, are weak. This assumption implies that we can try to overcome the inherent limitation 144419-7SEMENOV A, BERKOV , AND GORN PHYSICAL REVIEW B 102, 144419 (2020) of the Langevin dynamics’ ability to model only systems with relatively low barriers in the following way. For a system with a high-energy barrier we should perform LD simulations of the magnetization switching at several tem-peratures (all of them much larger than the temperature ofinterest), thus obtaining the dependence τ sw(T) at relatively high temperatures. Then we can use the analytical form τsw= cexp(b/T) to extrapolate the obtained dependence τsw(T) toward the desired low temperature (we note that a somewhatsimilar idea was used in [ 33] to obtain hysteresis loops in a low-frequency external field at a low temperature by sim-ulating the loops in a high-frequency field at much largertemperatures). The main physical deficiency of this idea is that by simu- lations at elevated temperatures the system will spend most ofthe simulated time in regions of the energy landscape whichare inaccessible for this system at actual transition tempera-tures. However, as long as switching events remain relativelyrare (so that the system mostly stays in the vicinity of energyminima), we can hope that the accuracy of the extrapolatedresult is reasonable. The precision of the proposed method crucially depends on (i) the lowest temperature achievable in simulations forthe given energy barrier and (ii) on the statistical accuracy ofthe switching time values (obtained in LD simulations) whichwill be used in the subsequent extrapolation. For our nanoele-ments, the lowest temperature for which LD simulations havebeen performed was chosen from the requirement that duringthe simulation time of 1 ms approximately 500 switchingevents should occur. Corresponding lowest temperature in-creases from T min=600 K for a=60 nm to Tmin=1800 K fora=100 nm. For two smallest elements a=50 (55) nm we have stopped to decrease TatTmin=400 (500) K in order to check how the extrapolation results agree with direct LDsimulations available for T=300 K for these nanoelements. For each temperature, LD simulations were performed simultaneously for 100 nanoelements using our approachdescribed in [ 23]. For each macrospin size a, averaged switching times obtained from these simulations τ sw(T) were fitted using the function τa(T)=caexp(ba/T) where data points were weighted according to their statistical er-rors. An example of the corresponding fitting is shown inFig. 7. FIG. 7. Simulated temperature dependence of the switching time (open circles) and its fitting by the function τ=caeba/T(solid line) for the macrospin corresponding to the nanoelement a=50 nm. FIG. 8. Relation /Delta1E/kBT, obtained analytically using ( 6), com- pared with the coefficients ba/Tobtained from the fitting of TTE dependencies τa(T)=caexp(ba/T). Interestingly, energy barrier /Delta1Eeff/kBT=ba/Tobtained from this fitting was always somewhat smaller than the actualbarrier /Delta1E/k BTevaluated from the analytical expression ( 6), as shown in Fig. 8. As a consequence, for large barriers the switching time evaluated by the TTE method is smaller thanthe analytical result ( 9), as it will be discussed in Sec. VII Finally, switching times for all sizes at room temperature were evaluated by extrapolating the fitting functions τ a(T)t o T=300 K as shown in Fig. 9by dashed green lines; switch- ing times obtained this way are plotted in the same figure byred circles. Analysis of these results is postponed to Sec. VII, FIG. 9. Extrapolation of switching time obtained for higher temperatures using the LD dynamics (blue open circles) to T= 300 K (red open circles) for nanoelement sizes a=50–100 nm. The extrapolated τswfor the macrospin with a=50 nm is τTTE sw= 2.18(±0.2)×103ns, for a=100 nm it is τTTE sw=1.2(±0.7)× 1023ns. 144419-8EV ALUATION OF THE SWITCHING RATE FOR MAGNETIC … PHYSICAL REVIEW B 102, 144419 (2020) FIG. 10. Energy landscape with “bounce energy” contours Ebn. where switching times obtained by all methods (analytical and numerical) will be compared. V . ENERGY BOUNCE METHOD A. Basic idea and analysis of the original methodology The main idea of the “energy bounce” algorithm [ 30]i st o enable LD simulations of the switching rate for arbitrary high-energy barriers by forcing the point representing the systemstate in the phase space to climb an “energy ladder” fromthe energy minimum to the saddle point. For this purpose,the energy interval /Delta1Ebetween the minimum and the saddle is divided into much smaller intervals (in our simulationswe have used δE=k BT). Corresponding “splitting” of the energy landscape is visualized in Fig. 10. Now, we start LD simulations from the energy minimum and continue until the energy histogram is computed with asufficient accuracy (the importance of this criterion will be ex-plained below). At the next stage, simulations start from somestate (achieved so far) with the energy E/greaterorequalslantE min+δE=E(1) bn and all LD steps which would lead to a state with an energy E<E(1) bnare rejected, i.e., the system is allowed to move only in its phase-space region defined by the condition E/greaterorequalslant E(1) bn. Again, these LD simulations run until the accumulated energy histogram in this energy interval is sufficiently accu-rate (the duration of our LD simulations above each E (i) bnis twalk=2000 ns). Then, at the next stage the minimal allowed energy is again increased by δE(i.e., LD steps are rejected if E<E(2) bn=E(1) bn+δE), etc. This procedure is repeated until the bounce energy E(i) bn is only a few kBTlower than the saddle point, so that a sufficient number of transitions over the saddle is observedby LD simulations above E (i) bn. In other words, for these values ofE(i) bnthe apparent escape time from the energy well A(see Fig. 10)τ(i) A,app=/Delta1t(i) A/N(i) swcan be computed with a reason- able precision (here /Delta1t(i) Ais the time spent in the well Aduring LD simulations with E>E(i) bn). The key question here is how to connect the time /Delta1t(i) A spent by the system within Afor trajectory obeying the condition E>E(i) bn, with the time spent in this well for un- constrained simulations. As the energy near the saddle can notbe achieved for unconstrained simulations within a reasonabletime, this connection can be established only recursively, i.e.,the time /Delta1t (i) Aspent in Aduring simulations with E>E(i) bn should be related to /Delta1t(i−1) A during the previous stage (when E>E(i−1) bn). If we denote the corresponding proportionality coefficient as Fi, i.e.,/Delta1t(i) A=Fi/Delta1t(i+1) A, then for the determi- nation of the actual switching rate we obtain the expression /Gamma1EnB n=n−1/productdisplay j=1FjN(n) sw /Delta1t(n) A, (26) where nis the total number of bounce energy levels used to climb the path from the energy minimum to the saddlepoint. Before proceeding to the detailed consideration of methods for the computation of F i, we emphasize that these coefficients have to be determined with a very high precision, wherebysystematic errors are especially dangerous. This feature fol-lows directly from the basic expression ( 26), which involves theproduct of all F i, meaning that any systematic error by their calculation will be exponentially amplified. In the original version [ 30] it was suggested to determine Fifrom the probability densities ρi(E|E>Ei bn)≡ρi(E)t o encounter the energy Eat the ith stage. Namely, in [ 30]i t was assumed that the probability density ρi+1(E)i ss i m p l y proportional to the corresponding probability density ρi(E). This would mean that ρi(E)=Fiρi+1(E), where the “transfer coefficient” Fidoes not depend on E(for E>E(i) bn, i.e., if the energy Eis accessible for both stages iandi+1). This independence of FionEis the main assumptions in this version of the energy bounce method. It can be verified onlyby the direct comparison of energy histograms obtained atstages iandi+1. According to [ 30], the proportionality ρ i(E)=Fiρi+1(E) approximately holds except for the energies close to E(i+1) bn. Basing on this finding, and in order to increase the accuracy by the calculation of Fi, Wang and Visscher [ 30] have suggested to compute Fiusing the integral ratio F(int) i=/integraltext∞ E(i+1) bn+/epsilon1offρi(E)dE /integraltext∞ E(i+1) bn+/epsilon1offρi+1(E)dE, (27) where the lower limit of both integrals is larger than E(i+1) bn by an offset energy /epsilon1off(=kBTin [30]) to exclude the above- mentioned histogram region near E(i+1) bn. Analyzing the energy histograms from our simulations, we could confirm that the ratio ρi(E)/ρi+1(E) is approximately constant, except for the regions near the bounce energies,where this ratio becomes singular [as shown in Fig. 12(b) ] because ρ i+1(E)→0f o r E→E(i+1) bn(see Fig. 11). This feature of accumulated histograms shows that the true ther-modynamic equilibrium is not achieved near the bounceenergies [we note that the density of state D(E) has no zeros or singularities near E (i) bn]. In [ 30] it was suggested that this is due to the finite size of the LD time step.Indeed, our studies have shown that the width of the dis-turbed area [where the accumulated energy histogram stronglydiffers from the true-equilibrium result, see Fig. 11(b) ] de- creases when the LD step size becomes smaller. However,this decrease is very slow so that for any reasonable time 144419-9SEMENOV A, BERKOV , AND GORN PHYSICAL REVIEW B 102, 144419 (2020) FIG. 11. (a) Energy histograms sampled at two subsequent val- ues of the bounce energy. The offset /epsilon1offmarks the regions above E(i+1) bn where the histogram does not correspond to the thermody- namic equilibrium due to the influence of the “hard” energy cutoff atEbn. (b) Energy histogram sampled by simulations (yellow line) compared to the exact analytical result ( 4) for the probability to obtain the energy Ein a true equilibrium (blue line); the ratio of the simulated histogram to the analytical result is shown by the greenline. step the width of the out-of-equilibrium energy interval re- mains substantial. To avoid this region, we have also usedthe offset energy /epsilon1 off=kBT(Fig. 11). Consequences of the existence of this disturbed region for the switchingrate computed via the expression ( 26) will be discussed in Sec. VII. In addition to the problem discussed in the previous para- graph, the ratio ρ i(E)/ρi+1(E) exhibits a small systematic decrease when the energy increases [see Fig. 12(b) ]. To study whether this systematic decrease affects the computed switch-ing time, we have tested another method for the evaluation ofF i, based as average ratio of histograms F(av) i=/angbracketleftbiggρi(E) ρi+1(E)/angbracketrightbigg Ei+1 bn+/epsilon1off<E<Ei+1max(28) computed at the interval from the offset energy to the maximal energy Ei+1 maxfor which ρi+1(E) becomes too small (typi- cally less than 5% of its maximal value) so that the ratioρ i(E)/ρi+1(E) becomes ill defined due to statistical fluctu- ations of accumulated histograms. The product of Fi(see Fig. 13) and switching times (shown in Fig. 17for two differ- ent macrospin sizes) were very close to those obtained usingthe initial definition ( 27), showing the high robustness of both methods for evaluating F ifor a macrospin. FIG. 12. (a) Energy histograms ρi(E) for subsequent stages of the energy bounce method ( δE=kBT); (b) ratios ρ(i+1)(E)/ρi(E) for some histogram pairs (for the macrospin with a=50 nm). B. Alternative method to define transition coefficients Unfortunately, the energy bounce algorithm based on the evaluation of transfer coefficients Fiemploying energy histograms (not to mention the assumption of their pro-portionality) cannot be used for full-scale micromagneticsimulations, where other energy contributions, in addition tothe shape anisotropy energy present for a macrospin, playan important role. The major problem is that the heightof an energy barrier in typical magnetic systems is deter-mined by either the magnetocrystalline anisotropy E anor the magnetodipolar energy Edip(which is responsible for the shape anisotropy introduced ad hoc in the macrospin ap- proach), whereby the energy itself is largely determined bythe exchange stiffness energy E exch. The latter contribution is especially high by simulations including thermal fluctuations FIG. 13. (a) Product of coefficients Fias the function of the bounce energy for various methods to evaluate Fi. (b) Ratios of the products F(int) iandF(av) ito the product of F/Delta1t i(for the macrospin with a=100 nm). 144419-10EV ALUATION OF THE SWITCHING RATE FOR MAGNETIC … PHYSICAL REVIEW B 102, 144419 (2020) (what is mandatory for studies of thermally activated switch- ing), and may exceed both EanandEdipby several orders of magnitude. This feature of Eexch makes the usage of histograms of the total energy completely impractical because the energy ofinterest, e.g., E dipin case of shape-anisotropic particles, would represent only a tiny contribution to this (noisy!) histogram.Further, the analysis of histograms of E diponly would also be not really helpful because in a strongly interacting systemno general statements concerning the statistical distributionof some part of the total energy (like the existence of theBoltzmann distribution) can be made, not to mention someproportionality assumptions like those used in [ 30]. For this reason, we suggest to use a qualitatively different method to compute the coefficients F i, which employs our def- inition of Fias proportionality coefficients between the time interval /Delta1t(i+1) Aspent in Aduring simulations with E>E(i+1) bn and the corresponding interval /Delta1t(i) Aduring the ith stage (when E>E(i) bn), i.e., F(/Delta1t) i=/Delta1tA/parenleftbig E>Ei+1 bn+/epsilon1off/parenrightbig /Delta1tA/parenleftbig E>Ei bn+/epsilon1off/parenrightbig. (29) Results of simulations where this definition of Fihas been used turned out to be in a very good agreement with the orig-inal method ( 27) and its modification ( 28) for all macrospin sizes studied in this paper, as shown in Fig. 13on an example fora=100 nm. The advantages of this method are twofold: (i) it is very simple and much faster than histogram-based methods be-cause one does not need to accumulate energy histogramswith the high accuracy required for the precise determinationofF i, and (ii) it can be applied to systems, where only one contribution to the total energy should be monitored, no matterwhat the distribution of this energy term looks like. Usinga slightly modified definition ( 29)o f F i, we could expand the energy bounce method toward full-scale micromagneticsimulations using the magnetodipolar energy of the spatiallyaveraged nanoelement magnetization as the energy of interestentering ( 29). Corresponding results, being out of scope of this publication, will be reported elsewhere. Next, the problem how to determine another key quantity in (26), the number of “true” switchings N (n) swover the barrier when the system stays above the bounce energy level E(n) bn, should be considered. The method described in Sec. IIIwhich is based on the criterion /Delta1twell>teqis not applicable here because no real thermal equilibration occurs after switchingdue to the artificial restriction imposed on the system energy(E/greaterorequalslantE (n) bn). For this reason, we have decided to consider a switching as being “true” if after changing sign of mx, the system completes at least one precession cycle around the new equilibrium orientation of the magnetic moment. This criterion was supported again by the analysis of his- tograms of time intervals between subsequent sign changesofm x, which always look like the example shown in Fig. 14: a large peak at very small time intervals followed by smallerbut well-distinguished peaks for larger /Delta1t’s. The visualization of magnetization trajectories corresponding to these peakshas shown that the first peak corresponds entirely to magne- FIG. 14. Histogram of switching times for the macrospin with a=70 nm for the bounce energy near the energy barrier ( Ebn= /Delta1E−5kT). The first peak corresponds to the out-of-plane preces- sion cycles [see Figs. 15(a) and15(b) ], so that these events are not counted as switchings tization “excursions” toward the opposite energy minimum, where in most cases one cycle of the OOP precession isaccomplished [see Figs. 14(a) and14(b) ]. The next peaks con- tained real switching events, where the number of precessioncycles around the energy minimum was equal to the sequencenumber of corresponding peak in the histogram (if the firstpeak is not counted) [see an example in Figs. 14(c) and14(d) ]. For these reasons, the switching was considered as being true,if the time spent in the energy basin after switching exceededthe time separating the OOP peak and the next peak on thehistogram ρ(/Delta1t well); the corresponding threshold is shown in Fig. 14by the red dashed line. The remaining question is how to choose the total num- ber of energy bounce levels nwhich is best suited for the switching rate computation. This question is briefly addressedin [30], but a more detailed discussion is clearly necessary. Namely, the stability of the evaluation of /Gamma1 EnBusing ( 26)a s - sumes the existence of a delicate balance between the productofF i’s and the number of switching events N(n) swobserved dur- ing LD simulations above the bounce energy E(n) bn. Whereas the product of Fi’s exponentially decreases with n, because Fi<1 according to its definition (see Fig. 13), the number of switchings N(n) swshould exponentially increase with nbecause we approach the saddle point. From the analytical point ofview, these two tendencies should exactly compensate eachother, providing the same answer for /Gamma1 EnBno matter how many bounce levels we use. However, in real simulations with the limited twalkfor each E(i) bna sufficiently large number of switchings ( Nsw>100) necessary to establish the exponential trend N(n) sw∝exp(n) with a sufficient accuracy (Fig. 16) is observed only for high bounce energies E(i) bn/greaterorequalslant/Delta1E−5kT. On the other hand, the ith bounce energy level should not be too close to the energybarrier because otherwise the very concept of switching asa rare transition over the barrier becomes invalid. These twoconditions leave a relatively narrow window of bounce en-ergies where we really have N (n) sw∝exp(n), as demonstrated in Fig. 16. Only in this interval of Ebnthe switching time τEnB sw=1//Gamma1EnB n[see ( 26)] is approximately independent on 144419-11SEMENOV A, BERKOV , AND GORN PHYSICAL REVIEW B 102, 144419 (2020) FIG. 15. Typical mx(t) dependencies and 3D magnetization tra- jectories for false (a), (b) and true (c), (d) switchings when Ebnis close to the energy barrier. Events shown on (a) and (b) correspond to the first peak and on (c) and (d) to the second peak on the histogramshown in Fig. 14. the number of the bounce level n(see two examples in Fig. 17, where the plateaus suitable for the determination of τsware explicitly marked). These plateaus should be determined man-ually, making the application of the whole method rathernontrivial. Further, in order to improve the statistics in determination ofN (n) sw,w eh a v es e t twalk=500 ns for Ebn</Delta1E−5kBT, and increased twalkto 105ns for Ebn/greaterorequalslant/Delta1E−5kBT.A tt h es a m e time, the bounce energy step in this interval was decreased toδE=0.5k BT. Comparison between switching times obtained using dif- ferent versions ( 27)–(29) for the evaluation of transfer coefficients is given in Fig. 18. Overall, the agreement be- tween all three versions can be considered as being fairlygood: we emphasize here that computed switching times covermore than 20 orders of magnitude, so that they had to bedivided by the exponential factor exp( /Delta1E/k BT) to enable a meaningful comparison between them on a single plot.Among all versions, our method ( 28) provides the best agree- ment with the results of LD simulations available for smallbarriers. Switching times computed according to the moreuniversal method ( 29) (yellow curve) lie systematically some- what lower than for other two versions (compare to Fig. 13). However, this difference becomes smaller than statistical er-rors when the energy barrier increases (for /Delta1E/k BT/greaterorequalslant30), FIG. 16. Number of switchings as the function of the bounce energy for the macrospin with the long axis a=100 nm. FIG. 17. Switching times as functions of the last bounce energy E(n) bnfor the macrospin with a=50 nm (a) and with a=100 nm (b) used for calculation of τswvia ( 26). Different line colors corre- spond to two approaches ( 27)a n d( 29) for calculating Fi. Plateau which can be used to establish τsware marked with curly brackets. Dashed lines show the energy barriers /Delta1E. i.e., this energy bounce version is clearly applicable for the most interesting region of energy barriers. The relation between switching times obtained with the energy bounce method and other methods (analytical and nu-merical) for all macrospin sizes studied here will be discussedin Sec. VII. VI. FORWARD-FLUX SAMPLING (FFS) The forward-flux sampling method was initially suggested as a method to evaluate switching rates between different FIG. 18. Average switching time [divided by exp( /Delta1E/kBT)f o r the presentation clarity] obtained from the three approaches ( 27)– (29) used to compute τswin the energy bounce method as the function of the energy barrier. 144419-12EV ALUATION OF THE SWITCHING RATE FOR MAGNETIC … PHYSICAL REVIEW B 102, 144419 (2020) FIG. 19. Illustration of FFS method. Transition between wells A andBover interfaces λi. metastable configurations of complex molecules in biochem- istry [ 24]. In principle, FFS can be adopted to any biological, chemical, or physical applications where transitions rates overhigh-energy barriers have to be evaluated, including micro-magnetic problems (see, e.g., [ 29]). A. General methodology of FFS The idea of FFS can be understood from Fig. 19.B a s i n s A and B (in the corresponding system coordinate space) sur-rounding the two corresponding metastable states of interestare confined by the interfaces λ AandλB. Intermediate inter- facesλi,i=1,..., n, are constructed in-between λAandλB so that transitions between two the subsequent interfaces iand i+1 can be expected during LD simulations of the system within a reasonable time. LD simulations are then started from the state mx=+ 1 and thermalization within the basin Ais carried out (i.e., simulations are performed until the average energy does notexhibit any systematic trend). Afterward, the flux per unit timeout of the basin Ais computed as the relation /Phi1 A=NA→0//Delta1tA, (30) where NA→0is the number of times when the system tra- jectory coming from the basin Ahas reached (crossed) the interface λ0during the simulation time interval /Delta1tA. System states corresponding to these N0crossings are saved as poten- tial starting states for the next stage. Next, M0trial trajectories are started from the states chosen randomly out of the set of above-mentioned N0saved states on the interface λ0. If a trial trajectory returns into the basin A, it is disregarded. If such a trajectory reaches the interface λ1, the system state corresponding to this crossing point is saved. If the total number of these crossing events is N1, then the conditional probability that a trajectory starting from theinterface λ 0will reach the interface λ1isp(λ1|λ0)=N1/M0. Repeating the same procedure starting from the subsequent interfaces, we can then compute the required transition ratestraightforwardly as /Gamma1 FFS=/Phi1Ap(λB|λ0)=/Phi1An/productdisplay i=0p(λi+1|λi), (31) where λn+1=λB; here, we have used the chain rule stating that the conditional probability p(λB|λ0) equals to product of corresponding conditional probabilities that a trajectorywill reach the interface λ i+1when having started from theinterface λi[p(λi+1|λi)=Ni+1/Mi]. In our simulations which results are presented below, we have used Natt=500 attempts for each macrospin size and Mi=500 trial trajectories for starting from each interface within the given attempt. B. Positioning the interfaces based on the energy considerations The FFS method as such does not contain any adjustable parameters like, e.g., the offset energy in the energy bouncemethod. The procedure described above leads to the unbiasedestimation ( 31) of the escape rate. Hence, the primary question is how to maximize the efficiency of FFS, meaning how tominimize the statistical error of the computed escape rate forthe fixed amount of the computer time spent by calculations. This problem has been analyzed in details in several publications [ 26,27] treating FFS in general, i.e., without a reference to any specific physical system. This analysis hasled to the intuitively expected result that the best efficiency ofFFS is achieved when the flux between the two subsequent in-terfaces M ip(λi+1|λi) is constant “along” the system, in other words, does not depend on the interface number. Taking intoaccount that the number of trial “shots” from each interfaceis usually the same, we arrive at the statement that in order tominimize the statistical error, we should construct the set ofinterfaces so that the transition probability p i≡p(λi+1|λi)= const. Several methods for the construction of the corresponding set{λi}have been suggested [ 26,27]. All these methods are iterative and provide some recipes how to shift the interfaces{λ i}based on the transition probabilities piobtained on this set. In our case, a much simpler solution is possible. We con- sider the escape of a physical system over an energy barrier,and thus have to our disposal the Boltzmann distribution p∝ exp(−E/k BT) of the probabilities to find a system with an energy Eat the temperature T. Hence, we can position the interfaces employing the idea that the transition probabilitybetween the two subsequent interfaces is roughly proportionalto the energy difference between them: p i∝exp[( Ei+1− Ei)/kBT]. This relation is not exact, as there can be small deviations due to the dependence of the density of systemstates on the system energy, but this correction is usually smallcompared to the exponential dependence of the probability onthe energy difference. Using this proportionality, we introduce for the “uphill” path the set of interfaces, which are equidistant in the energy space . Namely, we first define the boundary λ Aof the basin A (from which we start the simulation) by the energy E(λA)= Emin+kBT. Then we place the interface λ0used for the flux calculation ( 30) one kBThigher: E(λ0)=E(λA)+kBT.F i - nally, we place the uphill interfaces λi(i=1,..., n) so that Eup(λi)≡Eup i=E(λ0)+iδEif, where the number of inter- faces nis chosen so that (i) the last uphill interface is placed in the vicinity of the saddle point, but slightly beyond it, sothat Eup n≈Esad=Emin+/Delta1E, whereas mx(λn)<0 and (ii) the energy difference between the interfaces δEif≈kBT. The positioning of interfaces on the downhill path is less important because the flux toward the basin B after passingthe saddle point is large, so that corresponding conditionalprobabilities rapidly tend to 1.0. Hence, we use here only two 144419-13SEMENOV A, BERKOV , AND GORN PHYSICAL REVIEW B 102, 144419 (2020) FIG. 20. Conditional probability pas the function of the inter- face energy Efor the macrospin with a=50 nm (energy barrier /Delta1E≈9kBT), obtained for 500 attempts and Ni=500 starting points from each interface additional interfaces with the energies Edown 1=Esad−kBT andEdown 2=Esad−3kT. Placing of interfaces in the energy space requires a special discussion because usually the interfaces are posi-tioning in the coordinate space {x}of the studied system. From the mathematical point of view, assignment of inter-faces in the energy space can be considered as a specificform of placing coordinate-based interfaces with coordi-nates defined via an implicit function E if=Eif{x}. In our specific case where the energy is given by the simpleexpression ( 1), this implicit function, together with the re- lation /bardblm/bardbl= 1, defines closed ellipselike contours C yxm2 y+ Czxm2 z=Eif/2πm2 sV,mx=± (1−m2 y−m2 z)1/2on the unit sphere. The plus (minus) signs before the mxprojection cor- respond to interfaces on the uphill (downhill) path. In case ofmore complicated systems, e.g., in full-scale micromagneticsimulations, the simple recipe of placing interfaces using thetotal energy does not work for the same reason as the usage ofthis total energy as energy bounce intervals (see discussion inthe Sec. VB). Corresponding extension of the interface posi- tioning method will be discussed in the upcoming publication. Using our methodology for the interface positioning, we have introduced another optimization which strongly reducesthe total computation time. In the standard FFS version, thetrajectory is abandoned (the attempt is considered as failed),when after having started from some interface λ i, it returns to the initial basin A. We abandon a trajectory already when the corresponding energy drops below E(λi)−5kTbecause in this case it is exponentially unlikely that this trajectory everclimbs above the interface λ i. For the highest-energy barrier studied here ( /Delta1E≈60kBT), this optimization leads to ≈4× acceleration of simulations. An example for the dependence of the transition probabil- itypibetween the interfaces on the interface number i(in fact, on the interface energy Ei) is shown in Fig. 20. It can be seen that for the uphill interfaces this probability is indeednearly independent on E i, thus ensuring the smallest possible statistical error by calculating the switching rate. Switching times calculated using the FFS modified as de- scribed above are presented in Fig. 21and will be discussed in the next section. FIG. 21. Switching times in dependence on the macrospin size computed by all analytical and numerical methods (a) and ratio of switching times for all methods to the switching time obtained in the analytical approximation ( 9). VII. RESULTS AND DISCUSSION: COMPARISON OF ANALYTICAL AND NUMERICAL METHODS Results for the switching time dependencies on the macrospin size (long ellipse axis a) obtained with all an- alytical and numerical methods presented in this paper arecollected in Fig. 21. First, we point out that in the interval of energy barriers 9 </Delta1E/k BT<70 studied here, switching times span the interval of more than 20 orders of magnitude(from≈2μst o≈30 million years). For all methods τ swgrows (at least approximately) proportional to the relation /Delta1E/kBT, so that the difference between τswmeasured by various meth- ods is barely visible when τswis plotted as the function of size [Fig. 21(a) , logarithmic scale]. For this reason, in Fig. 21(b) we have plotted the ratio τsw/τan swof switching time obtained by different methods to the corresponding time calculated using the analytical ap-proximation ( 9), which is valid for /Delta1E/greatermuchk BTand should be applicable for arbitrary damping. This way we eliminatethe exponential dependence of τ swon the energy barrier (or, equivalently, on the long axis a), enabling the meaningful comparison of various approaches. First of all, we note a remarkable coincidence of the Arrhe- nius approximations ( 5)–(8) with the more general analytical result ( 9) in the whole range of switching times: τArr sw/τan sw≈1 [see the blue line in Fig. 21(b) ]. This agreement is due to the fact that for our system the product of damping α=0.01 and the ratio 10 </Delta1E/kBT<60 lies in the range 0 .1< α/Delta1E/kBT<0.6. As stated in Sec. II C, it is this product (and 144419-14EV ALUATION OF THE SWITCHING RATE FOR MAGNETIC … PHYSICAL REVIEW B 102, 144419 (2020) not the damping value αby itself) which governs the transition between various damping regimes. Hence, the values of theparameter which controls the transition from the low to thehigh damping regime lie for our macrospins in the interme-diate region, where a good agreement between the simpletransition state theory (Arrhenius law) and the sophisticatedanalytical result ( 9) is indeed expected (see, e.g., Fig. 9in [9]). Next, we discuss the relation between switching times obtained by different numerical methods and the analyticalapproximation ( 9). For relatively small energy barriers ( /Delta1E/k BT/lessorequalslant15), where a comparison with straightforward LD simulations isstill possible, all numerical methods agree with LD resultswithin the statistical errors of the latter (only the TTE resultis slightly below the LD value). Further, all numerically com-puted switching times for these small barriers are larger thanthe analytical ones ( τ num sw/τan sw>1f o r a/lessorequalslant55 nm, see Fig. 21). This relation is in accordance with the well-known featureof analytical approximations: they overestimate the transitionrate (thus underestimating the switching time) because theydo not take into account the possibility that a system trajec-tory can return to the initial basin A shortly after crossingthe saddle (i.e., without visiting the basin B) [ 9]. Note that these “back-hopping” events should not be mixed up with theout-of-plane precession analyzed in Sec. III. When the energy barrier increases, results of numerical methods exhibit considerably different trends. Time-temperature extrapolation method. Relation of switching times obtained via TTE to analytically computedtimes decreases with increasing /Delta1E, becomes smaller than 1.0 for /Delta1E/k BT/greaterorequalslant20, and drops to τTTE sw/τan sw≈0.1f o r the largest particles studied here with a=90 and 100 nm (/Delta1E/kBT>50). From the “technical” point of view, this decrease is due to the fact that “effective” energy barriers obtained by the expo-nential fitting of TTE switching times (computed by highertemperatures) are systematically lower than actual barriers,and this difference increases with the barrier height, as shownin Fig. 8. The most probable physical explanation of this behavior is that for larger energy barrier LD simulations inthe TTE method have to be conducted by higher temperatures,so that magnetic moments precess in the higher-energy rangethan by the room temperature. In this energy range the cur-vature of the energy landscape (i.e., the density of states) isconsiderably different from the curvature near the bottom ofthe energy minimum, which may result in a lower “effective”energy barrier. Still, we point out that this conceptually very simple (so that it can be easily extended to full-scale micromagnetics)and relatively fast method performs surprisingly well: In theinterval of τ swcovering more than 20 orders of magnitude, TTE switching times τTTE sw differ in the worst case only by one order of magnitude from results obtained by much moresophisticated methods. Energy bounce method. For this method, relation of its switching times to the analytical ones also decreases withthe energy barrier, although much slower than for TTE.Still, the ratio τ EnB sw/τan swdrops below unity for /Delta1E/kBT/greaterorequalslant30 and achieves the value ≈0.5f o r a=100 nm ( /Delta1E/kBT≈ 60). Taking into account that the switching time computednumerically should be larger than τan sw(see the explanation above), the reason for this systematic decrease of the ratioτ EnB sw/τan swshould be found. In the energy bounce expression for the switching rate ( 26), the number of switching events N(n) swat the nth stage of the method is determined using the same criteria as for thestraightforward LD simulations without the bounce energy.Hence, the only possible source of the systematic underes-timation of the actual switching time in the energy bouncealgorithm is the systematic error by the computation of thetransition factors F i. To explain the appearance of such deviation by computing Fi, we remind that these factors are evaluated using either the ratio of energy histograms obtained for different bounceenergy levels E (i) bn(in the initial version) or the ratio of times spent above these levels (in our version). In both versions, the interval between Ebnand Ebn+/epsilon1offwhere the system equilibrium is strongly disturbed is excluded (see Fig. 11)i n order to compute these ratios as correct as possible. However, in spite of the exclusion of this interval, the introduction of artificial energy levels Ebn(and the prohibition to visit the phase space with E<Ebn) still leads to systematic errors by the computation of Fi’s. The reason for these errors is that the true thermal equilibrium is disturbed for all ener- gies above Ebn. This perturbation can be demonstrated using thenormalized probability histograms shown in Fig. 11(b) . Here, it can be seen that for E>Ebn+/epsilon1off, the probability ρ(E|E(i) bn) is always larger compared to the true-equilibrium distribution ρ(E) because for energies close to E(i) bn, the energy bounce histogram is smaller than the actual ρ(E). This sys- tematic deviation is different for different Ebnlevels due to the energy dependence of the system density of states D(E) (see Fig. 2). Hence, the coefficients Ficomputed as ratio of any quantities derived from system trajectories above Ebn+/epsilon1off also exhibit systematic deviation from the correct transfer coefficients. The effect considered above is small because the main energy dependence of the probability p(E) is due to the Boltzmann exponent exp( −E/kBT), and not to the depen- dence D(E). Still, this small effect, accumulated in course of subsequent multiplications of Fi’s, most probably leads to the above-mentioned systematic underestimation of the switchingtime by the energy bounce method. In spite of this underestimation, the energy bounce method performs considerably better than the TTE method, leading for the highest studied barrier to the underestimation of τ sw only by two times compared to the analytical approximation and three times to the forward-flux method. Forward-flux sampling. Among all numerical methods con- sidered here, the FFS is the only one which uses neitherany far-reaching extrapolation from high- Tresults nor any artificial boundaries restricting the system motion in the phasespace. All LD simulations in frames of FFS are conductedfor an undisturbed system, so that the method should be asreliable as the LD itself. The only problem of FFS is the re-quirement to achieve a high accuracy by the evaluation of thetransition probabilities between the interfaces p(λ i+1|λi). As explained above, we have solved this problem by positioninginterfaces in the energy space and thus could obtain switching 144419-15SEMENOV A, BERKOV , AND GORN PHYSICAL REVIEW B 102, 144419 (2020) times with a very low statistical error, as demonstrated in Fig. 21(see yellow lines). Switching times computed by FFS coincide with the LD results for low-energy barriers (within the statistical errors ofthe latter method). Further, τ FFS swlie systematically above the analytical approximation ( 9), as it should be according to the consideration of back-hopping trajectories (see above). The significance of the back-hopping processes for our system can be estimated from Fig. 20. Here, it can be seen that the probability to go downhill from the interface λ7(for which we have already mx<0, so that this interface is slightly beyond the barrier) is ≈90%, meaning that about 10% of trajectories starting at this interface, go back to the initialbasin A. Further, probability to go downhill from the interface λ 8is only slightly larger than 90%, so that again ≈10% of trajectories go back from this interface to the basin A. Finally, we recall that the interface λ7is already beyond the barrier, so that some back hopping may occur between the separatrix andthis interface (note that the back-hopping probability is largerfor trajectories in the immediate vicinity of the saddle point).Hence, we can conclude that the fraction of back-hoppingtrajectories is significant (in any case much larger than 20%),which makes the systematic increase of the FFS switchingtime in Fig 21(b) over the analytical expression plausible. Interestingly, the ratio τ FFS sw/τan swis nearly constant ( ≈1.5) for a very broad interval of switching times considered here.It might be an indication that for the macrospin model, thefraction of trajectories which return to the initial basin aftercrossing once the saddle point depends only weakly on theenergy barrier height. VIII. CONCLUSION In this paper we have studied the dependence of the switch- ing time for a bistable biaxial magnetic particle in dependenceof its in-plane size, which translates into the dependence onthe energy barrier separating two energy minima. We have ap-plied two analytical methods (a simple transition state theoryleading to the Arrhenius law and the sophisticated approachbased on the Melnikov-Meshkov solution of the Kramersproblem for an arbitrary damping) and four numerical tech-niques (straightforward LD simulations, the time-temperatureextrapolation method, the energy bounce method, and theforward-flux sampling). Analyzing the results obtained by analytical methods, we have shown that the parameter which governs the transitionfrom the low damping to the high damping regime is not thedamping λin the LLG equation by itself, but rather the prod- uct of λand the reduced energy barrier /Delta1E/k BT. Taking intoaccount that the damping for a typical magnetic material used in applications is λ∼0.01 and the energy barrier required to achieve a stability during a macroscopic time interval is /Delta1E∼ 50kBT, we conclude that magnetization switching proceeds usually in the intermediate damping regime λ/Delta1E/kBT∼1. Our comparison of switching times obtained in the Arrheniusapproximation and in the Melnikov-Meshkov formalism con-firms this conclusion. Comparison of numerical methods has shown that for low- to-moderate energy barriers ( /Delta1E/lessorequalslant10k BT) where direct LD simulations are possible, results of all numerical methodsagree within statistical errors. However, when the energybarrier height increases, the relation of switching times ob-tained by the TTE and by the energy bounce methods toτ an swsystematically decreases (the decrease is slower for the energy bounce method), so that for sufficiently high barriersthe analytically computed switching time becomes larger thanthe numerical one. We could show that this artificial trendis the consequence of physical principles the correspondingmethods are based on. Hence, the quality of results obtainedby the TTE and energy bounce method is limited. For the forward-flux sampling, our recipe for choosing the interfaces which are equidistant in the energy space (forthe evaluation of transition probabilities p i) has led to nearly interface-independent probabilities piwithout any a posteriori optimization, assuring the high accuracy by the computationof switching times. Corresponding values τ FFS swcoincide with LD results for low barriers and are higher than τan swfor all en- ergy barriers studied here, demonstrating that FFS representsa reliable technique for computing switching rate in magneticsystems, and that a very high accuracy can be potentiallyachieved by this method. In this research, we have concentrated on the inherent properties of various techniques to study the escape of mag-netic systems over energy barriers, and thus have performedour studies in frames of the macrospin approximation. Forreal applications, a full-scale micromagnetic framework isclearly necessary. The corresponding generalization of ourtechniques proposed in this paper, except of the TTE method,is highly nontrivial due to the contribution of other energyterms (mainly the exchange energy). This problem and itssolution will be discussed in details in the forthcoming publi-cation. 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PhysRevB.95.184401.pdf

PHYSICAL REVIEW B 95, 184401 (2017) Autoresonant magnetization switching by spin-orbit torques Gyungchoon Go,1Seung-Jae Lee,2and Kyung-Jin Lee1,2,* 1Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea 2KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea (Received 19 October 2016; revised manuscript received 14 February 2017; published 2 May 2017) Autoresonance is a self-sustained resonance mechanism due to a driving force whose frequency monotonically varies with time. We theoretically show that the autoresonance mechanism allows an efficient switchingof perpendicular magnetization by spin-orbit spin-transfer torques. We find that a threshold current for theautoresonant switching can be much smaller than that of conventional spin-orbit torque switching driven by a DCcurrent. Moreover, the suggested scheme allows fully deterministic switching without the help of any externalfield. DOI: 10.1103/PhysRevB.95.184401 I. INTRODUCTION Since Slonczewski and Berger’s seminal works [ 1,2], numerous studies have been conducted on the spin-transfertorque [ 3–12], which is based on the transfer of spin angular momentum from conduction electron spins to local magneti-zations. The spin-transfer torque offers a way to control themagnetization of a switchable layer with an electrical currentspin-polarized by a fixed ferromagnetic layer. As this switchingmechanism requires a current passing through both fixed andswitchable layers, it is realized by flowing an electrical currentperpendicular to the plane of layered structures (i.e., current-perpendicular-to-plane geometry). Together with the tunnelmagnetoresistance [ 13–17] as a reading scheme, the spin- transfer torque provides the operation principle for magneticrandom access memories. Recently, an alternative way to manipulate the magnetiza- tion direction has been demonstrated in ferromagnet/heavy-metal bilayer structures [ 18,19]. Because of bulk [ 19–22] or interfacial spin-orbit coupling effects [ 18,23–30], or both [31–33], an in-plane current passing through bilayer structures is converted into a spin current flowing perpendicular to thefilm plane, which exerts a torque, called spin-orbit torque, onlocal magnetization of a ferromagnetic layer. The spin-orbittorque consists of two vector components: T D=γcJˆm×[ˆm×(ˆj׈z)], (1.1) TF=γdJˆm×(ˆj׈z), (1.2) where TDis a damping-like torque, TFis a field-like torque, γis the gyromagnetic ratio, ˆmis the unit vector along the magnetization, ˆjis the unit vector along the current direction, ˆzis the direction in which the structural inversion symmetry is broken (i.e., thickness direction), and cJanddJare the effective spin-orbit fields corresponding to damping-like andfield-like torques, respectively. Spin-orbit torque switching of perpendicular magnetiza- tion, which is of technological relevance, has been theoreticallyinvestigated considering the damping-like torque only [ 34,35] and both torque components [ 36]. These theoretical studies *kj_lee@korea.ac.krtogether with experimental ones [ 37,38] have found that the switching current density for the spin-orbit torque is muchlarger than that for the conventional spin-transfer torque inthe current-perpendicular-to-plane geometry [ 39]. This large switching current density of the spin-orbit torque switchingscheme is apparently detrimental for practical applications. Inthis work, as one of possible attempts to reduce the switchingcurrent, we theoretically investigate the autoresonance mech-anism [ 40] based on the spin-orbit torque. Autoresonance is a self-sustained resonance mechanism due to an external perturbation which has a monotonicallydecreasing (or increasing) frequency. Unlike usual resonant ex-citations with a fixed frequency, the phase of oscillator for theautoresonance is locked to that of an external perturbation. Thephase-locked excitation enables a highly efficient excitation ofthe oscillator even with a low-amplitude external perturbation.The autoresonance mechanism is applicable to magnetizationswitching: Klughertz et al. reported autoresonant magnetiza- tion switching using time-dependent external magnetic fields[41,42]. In this paper, we theoretically study the autoresonant magnetization dynamics driven by time-dependent spin-orbittorques. Our results show that the switching current densityfor the autoresonance mechanism can be remarkably reducedas compared to that of time-independent spin-orbit torques. The paper is organized as follows. In Sec. II, we describe the basic principle of autoresonant magnetization dynam-ics. In Sec. III, we provide a theoretical analysis of the autoresonant magnetzation dynamics driven by spin-orbittorques. In Sec. IV, we show numerical simulation results of the autoresonant magnetization excitations. In Sec. V,w e conclude with a brief summary and discussion. II. BASIC PRINCIPLE OF AUTORESONANT MAGNETIZATION DYNAMICS We consider a ferromagnet/heavy-metal bilayer system where the ferromagnetic layer has perpendicular magneticanisotropy. Macrospin dynamics for this configuration isdescribed by the Landau-Lifshitz-Gilbert (LLG) equationincluding spin-orbit torque terms: dˆm dt=−γˆm×HK,effmzˆz+αˆm×dˆm dt +γcJˆm×[ˆm×(ˆj׈z)]+γdJˆm×(ˆj׈z),(2.1) 2469-9950/2017/95(18)/184401(7) 184401-1 ©2017 American Physical SocietyGYUNGCHOON GO, SEUNG-JAE LEE, AND KYUNG-JIN LEE PHYSICAL REVIEW B 95, 184401 (2017) driving frequency resonant frequency tω(t) tω(t) tω(t) (a) (b) (c) FIG. 1. The autoresonance mechanism for different chirp rates. (a) An appropriate chirp rate makes an efficient excitation of magnetization. (b) No autoresonance occurs for a too high chirp rate. (c) A low chirp rate results in a longer switching time. where ˆm=(mx,my,mz),HK,eff [=(2K/M s−4πMs)≡ 2Keff/Ms] is the effective perpendicular anisotropy field, K stands for the magnetocrystalline anisotropy constant, Ms is the saturation magnetization, αis the Gilbert damping constant, cJ[=(¯h/2e)(θSHJ/M stF)] is the damping-like spin- orbit effective field, dJ[=−β(¯h/2e)(θSHJ/M stF)] is the field-like spin-orbit effective field, θSHis the effective spin Hall angle, tFis the thickness of the ferromagnetic layer, Jis the current density, and βis the ratio of dJtocJ. From the precession torque term of Eq. ( 2.1), one finds that the resonance frequency ω0of the system is γHK,effmz. We note that ω0depends on mzand the resonance condi- tion therefore changes as the magnetization switches froman equilibrium direction (i.e., m z=+ 1) to the other (i.e., mz=− 1). Following works of Klughertz et al. [41,42], we first describe the basic principle of autoresonant magnetic ex-citation with an oscillating driving field with the angularfrequency ωand the chirp rate ξ(>0), e.g., c J(t)=0 and dJ(t)=d0cos(ωt−πξt2). For the autoresonant excitation, we set the initial frequency of the chirped field to be abit higher than γH K,eff, that is, the resonant frequency of the initial magnetization (i.e., mz=1). Since the driving frequency decreases with increasing t(i.e.,ξ> 0), the driving frequency encounters the resonant frequency ω0at a time, and the resonance starts. Because of the resonance, themagnetization excites rapidly and m zdecreases accordingly, which in turn results in a decreased ω0because ω0depends onmz. At this moment, a small mismatch between the driving frequency and ω0emerges. However, because the driving frequency decreases monotonically with time, thismismatch soon disappears and the resonance restores again.Because of this repeated procedure (i.e., on-resonance → off-resonace →on-resonance again), the time-average phase of the magnetic excitation is locked to that of the driving field,called the autoresonance. Figure 1represents the autoresonant magnetic excitations for different chirp rates. From Fig. 1, one sees that there is an appropriate chirp rate for an efficientautoresonant excitation. If the chirp rate is too high, theautoresonance cannot occur [Fig. 1(b)]. If the chirp rate is too low, the autoresonance occurs but the switching is delayed[Fig. 1(c)]. Therefore, the autoresonance mechanism with an appro- priate chirp rate can reduce the threshold external field forthe magnetization switching. Considering spin-orbit torqueswitching where c J/negationslash=0, it may also reduce the threshold switching current compared to the case with time-independentspin-orbit torques since the autoresonant excitation maintainsthe resonance on average. III. THEORETICAL ANALYSIS In this section, we provide a theoretical analysis of the autoresonance mechanism of magnetzation excitation inthe presence of both damping-like and field-like spin-orbit torques. We note that our theoretical framework closely follows those in Refs. [ 42,43]. We consider the circularly polarized AC current as ˆj(t)=cosφ(t)ˆx+sinφ(t)ˆy,φ (t)=ωt−πξt 2.(3.1) For a small damping ( α/lessmuch1), we rewrite Eq. ( 2.1)a s dˆm dt=−γˆm×{[HK,effmz−c/prime J(mxcosφ+mysinφ)]ˆz +(c/prime Jmzcosφ+d/prime Jsinφ+αHK,effmy)ˆx +(c/prime Jmzsinφ−d/prime Jcosφ−αHK,effmx)ˆy}, (3.2) where c/prime J=cJ+αdJandd/prime J=dJ−αcJ. Neglecting the zcomponent of the oscillating field [ =c/prime J(mxcosφ+ mysinφ)ˆz], which is irrelevant to the resonant excitation, we have dˆm dt=−γˆm×{HK,effmzˆz+(c/prime Jmzcosφ +d/prime Jsinφ+αHK,effmy)ˆx +(c/prime Jmzsinφ−d/prime Jcosφ−αHK,effmx)ˆy}. (3.3) We introduce the dimensionless parameters as ˜HK,eff=γHK,eff (2πξ)1/2,˜cJ=γc/prime J (2πξ)1/2,˜dJ=γd/prime J (2πξ)1/2, τ=(2πξ)1/2t, ˜ω0=ω0/(2πξ)1/2, (3.4) and complex variables mx=A1A∗ 2+A∗ 1A2,m y=i(A1A∗ 2−A∗ 1A2), mz=|A1|2−|A2|2, (3.5) 184401-2AUTORESONANT MAGNETIZATION SWITCHING BY SPIN- . . . PHYSICAL REVIEW B 95, 184401 (2017) with|A1|2+|A2|2=1. With these parameters and variables, we rewrite Eq. ( 3.3)a s idA 1 dτ=1 2(˜ω0−2˜ω0|A2|2)A1+1 2[mz˜cJe−iφ+i˜dJe−iφ]A2 +iα˜ω0(|A1|2−|A2|2)|A2|2A1, idA 2 dτ=−1 2(˜ω0−2˜ω0|A2|2)A2+1 2[mz˜cJeiφ−i˜dJeiφ]A1 −iα˜ω0(|A1|2−|A2|2)|A1|2A2. (3.6) Introducing new complex amplitudes ¯A1=A1exp/parenleftbiggi 2/integraldisplay ˜HK,eff(τ)dτ/parenrightbigg , ¯A2=A2exp/bracketleftbigg −i/parenleftbigg φ−i 2/integraldisplay ˜HK,eff(τ)dτ/parenrightbigg/bracketrightbigg ,(3.7) and considering the weakly nonlinear excitation regime, i.e., ¯A1≈1 and|¯A2|/lessmuch 1, Eq. ( 3.6) is further simplified as idψ dτ+(τ−|ψ|2+iλ/2)ψ=/bracketleftbigg/parenleftbigg 1−|ψ|2 ˜ω0/parenrightbigg μ−iν/bracketrightbigg , (3.8) where ψ=(2 ˜ω0)1/2¯A2,μ=˜cJ(˜ω0/2)1/2,ν=˜dJ(˜ω0/2)1/2, andλ=2α˜ω0. Equation ( 3.8) is a nonlinear Schrödinger-like equation describing the autoresonance excitation with an additional source term (1 −|ψ|2 ˜ω0)μ, caused by the damping-like spin-orbit torque. Without the additional term (i.e., μ=0), the theoretical analysis for zero damping case gives the thresholdν th 0/similarequal0.413 for the autoresonance [ 42]. In the following, we investigate the effect of damping-like (μ) and field-like ( ν) torques on the autoresonant magnetiza- tion excitation process. We note that the sign of the product μν in experiments is usually negative in our convention [ 44,45]. In this paper, we thus focus on μν < 0 case and choose μ> 0 without loss of generality. For zero damping case ( λ=0), decomposing Eq. ( 3.8)i n t o real and complex parts, we have ˙a=−/parenleftbigg 1−a2 ˜ω0/parenrightbigg μsin/Phi1−νcos/Phi1, ˙/Phi1=τ−a2−/parenleftbigg 1−a2 ˜ω0/parenrightbiggμ acos/Phi1+ν asin/Phi1,(3.9) where ψ=aei/Phi1. When μ=0, Eqs. ( 3.9) are equivalent to those describing the evolution of autoresonant pendulum withamplitude aand phase mismatch /Phi1between driving field and excitation [ 43]. In terms of the oscillator action I≡a 2,w e rewrite Eqs. ( 3.9)a s ˙I=− 2I1/2(μsin/Phi1+νcos/Phi1)+2I3/2 ˜ω0μsin/Phi1, ˙/Phi1=τ−I−I−1/2(μcos/Phi1−νsin/Phi1)+I1/2 ˜ω0μcos/Phi1. (3.10)Using Eq. ( 3.10), we obtain ¨/Phi1=1+S/bracketleftbigg 2I1/2(μsin/Phi1+νcos/Phi1)−2I3/2 ˜ω0μsin/Phi1/bracketrightbigg −1 2˙I I˙/Phi1, (3.11) where S=1−1 2I−3/2(μcos/Phi1−νsin/Phi1)−I−1/2 2˜ω0μcos/Phi1. (3.12) Henceforth, we consider the weak perturbation limit (μ,ν/lessmuch˜ω0) for simplicity. We also assume that the system comes in the phase-locked state ( /Phi1=/Phi10)a tt=τ0(<0) and stays in this state for some finite time. By analyzing Eq. ( 3.10), we find that /Phi10satisfies μcos/Phi10−νsin/Phi10=−/radicalbig μ2+ν2. Therefore, in the phase-locked state, Sis approximated as S/similarequal1+1 2/radicalbig μ2+ν2I−3/2. (3.13) For the phase-locked state, we read from the second equation of Eq. ( 3.10) that I0−/radicalbig μ2+ν2I−1/2 0=τ, (3.14) where I0(>0) is the instantaneous equilibrium action which increases with time (See Ref. [ 43] and references therein). By using Eq. ( 3.13), we rewrite Eq. ( 3.11) in terms of the equilibrium action I0as ¨/Phi1=−∂Vps ∂/Phi1−γeff˙/Phi1, (3.15) where Vps=−/Phi1+/parenleftBigg 2I1/2 0+/radicalbig μ2+ν2 I0/parenrightBigg (μcos/Phi1−νsin/Phi1), (3.16) andγeff=˙I0/(2I0). Since the pseudopotential Vpshas both a /Phi1-linear term and periodic terms (combination of trigonometric functions), itbehaves as a series of tilted wells [see Fig. 2(a)]. If the /Phi1-linear term is smaller than the periodic terms, the potential well existsand the system is trapped there. As described in Ref. [ 43], the existence of the potential well is the essential conditionfor the autoresonance. When this condition is satisfied, /Phi1 remains nearly constant and I 0grows without breaking the phase-locking for a relevant time. Let us find the threshold for being captured into the autoresonance. By using the property of the trigonometricfunction, we rewrite Eq. ( 3.16)a s V ps=−/Phi1+μ/radicalbig 1+β2/parenleftBigg 2I1/2 0+μ/radicalbig 1+β2 I0/parenrightBigg cos(/Phi1+δ), (3.17) where β=−ν/μ andδ=cos−1(1//radicalbig 1+β2). From the slopes of linear and periodic terms in Eq. ( 3.17), the necessary condition of the potential well existence isgiven as ˜μ/parenleftbig 2I1/2 0+˜μI−1 0/parenrightbig >1, (3.18) 184401-3GYUNGCHOON GO, SEUNG-JAE LEE, AND KYUNG-JIN LEE PHYSICAL REVIEW B 95, 184401 (2017) 0 1 2 3 4 5 60 2 4 6Vps Φμ = 0.2 μ = 0.5μ = 0.4μ = 0.3(a) 0 2 4 6 8 10 1204812 τI0μ = 0.29μ = 0.28 μ = 0.30μ = 0.27(b) FIG. 2. (a) /Phi1dependence of pseudopotential Vpsforβ=1. Forμ=0.2 (red dotted line) and μ=0.3 (blue solid line) the potential well does not exist, whereas the potential well appears for μ=0.4 (green dashed line) and μ=0.5 (purple dot-dashed line). (b) Dynamics of I0forβ=1. For μ/greaterorequalslant0.3,I0monotonically increases as τincreases. Although there is no potential well at μ=0.3 [see Fig. 2(a)], the autoresonance excitation occurs due to the inertia effect (blue solid line). For (a), ˜I0=0.45 is used. where ˜ μ=μ/radicalbig 1+β2. Differentiating the above condition with respect to I0, we find the minimum value of I0to match the condition Ic=˜μ2/3. In order to obtain the threshold of μ, one needs to consider the effect of inertia [ 43]. When the system slightly deviates from the potential well existence condition [Eq. ( 3.18)],/Phi1 varies very slowly and I0increases accordingly (i.e., weak resonance). If the pseudopotential well is established beforeloosing the weak resonance, the system can be trapped in thepseudopotential well. Taking into account this inertia effect,Eq. ( 3.18) is modified as κ˜μ/parenleftbig 2I 1/2 0+˜μI−1 0/parenrightbig >1, (3.19) where κreflects the effect of inertia. Inserting Ic=˜μ2/3into Eq. ( 3.19), we obtain the threshold for zero damping as ˜μth 0/similarequal/parenleftbigg1 3κ/parenrightbigg3/4 . (3.20) Since it is difficult to fully solve the nonlinear equations ( 3.10), we numerically obtain κ. By comparing Eq. ( 3.20) with the numerical threshold value of Eq. ( 3.10), we obtain κ=1.08. Thus we have μth 0/similarequal0.413/radicalbig 1+β2. (3.21)The numerator of Eq. ( 3.21) is equivalent to the previously known result [ 42]. Because we are dealing with the weak excitation regime, the magnetization is forced by the vectorsum of damping-like torque ( μ) and field-like torque ( ν), which is reflected by the coefficient 1 //radicalbig 1+β2. Figure 2(b) shows the time evolution of I0forβ=1. The system enters the autoresonance when μexceeds the threshold (μth 0/similarequal0.292 for β=1). As shown in Fig. 2(a), there is no potential well for μ=0.3. Due to the inertia effect, however, the autoresonance excitation is allowed near this point (bluesolid line). We next discuss the effect of damping on the autoresonant spin-orbit torque switching. As the autoresonance is governedby the precession motion of magnetization, the damping tendsto disturb the autoresonant dynamics. As a result, the thresholdfor the autoresonant magnetic excitation increases with thedamping [ 42,43]. For a small damping (i.e., λ< 1),μ thcan be expanded as μth=μth 0(1+pλ+qλ2), (3.22) where the coefficients p=1.05 and q=0.83 are obtained by numerical comparison between Eqs. ( 3.8) and ( 3.22). In terms of the spin-orbit torque coefficient cJ, this corresponds to cth J/similarequal√ 2(2πξ)3/4 γω1/2 0μth=√ 2(2πξ)3/4 γ3/2H1/2 K,effμth. (3.23) Equation ( 3.23) is the central result of our work. Three remarks for Eq. ( 3.23) are in order. First, the threshold value scales with ξ3/4, which is a representative characteristic of the autoresonance phenomenon [ 40–43]. Second, the obtained threshold is inversely proportional to H1/2 K,eff.T h i s inverse proportionality is in contrast to conventional switchingmechanism driven by DC spin-transfer torque or spin-orbittorque, where the threshold current is proportional to H K,eff [3,4,34]. This feature would enable a significant decrease in the threshold current for the autoresonant switching mechanismwhile keeping the thermal stability of magnetization. Third,Eq. ( 3.23) is valid only for a small damping. As we show below, one has to numerically estimate the threshold for a highdamping. As further analysis is difficult because of the complexity of Eq. ( 3.16), we compute threshold AC currents for various cases using numerical simulations in the next section. IV . NUMERICAL SIMULATIONS A. Single-domain nanomagnet at zero temperature In this subsection, we consider the single-domain nano- magnet without thermal fluctuations. We use parameters ofthe W /CoFeB structure in Refs. [ 46,47]:α=0.012,K eff= 4.25×106erg/cm3,Ms=1240 emu /cm3. We assume γ= 1.76×107(Oe s)−1andξ=0.3×1018s−2. Figure 3shows the numerically computed magnetization dynamics. We notethatm zclosely follows the time-dependent change in the angular frequency ω, which is a representative feature of the autoresonant switching. Figure 4(a) shows numerical results of the threshold cth J as a function of the damping α. In the low damping regime, numerically obtained thresholds are in good agreement with 184401-4AUTORESONANT MAGNETIZATION SWITCHING BY SPIN- . . . PHYSICAL REVIEW B 95, 184401 (2017) 0 50 100 150 2001.0.50.0.51. t(ns)mx mzmy ω(t)/ω 0 FIG. 3. Single-domain magnetization dynamics. Blue, green and red (bold) lines represent mx,my,a n dmz, respectively. Purple line represents the normalized angular frequency. In this plot, dJ=−cJ (i.e.,β=1) is assumed. theoretical ones [Eq. ( 3.23)]. We note that our theoretical analysis is valid for the small damping case. There aretwo effects of damping. First, the damping enhances theminimum value of c Jto be captured into the autoresonant excitation. This is taken into account in our theoreticalanalysis [Eq. ( 3.22)]. Second, the damping interrupts the phase-locked excitation. Therefore, for high damping, oncethe magnetization dynamics is captured into the phase-lockedexcitation, soon after it looses its phase-locking. This is notconsidered in our theoretical analysis. In the high dampingregime, c th Jis linearly proportional to α. Figure 4(b) shows the dependence of cth Jonβ(=−dJ/cJ). For a nonzero β, not only damping-like torque but also field-like torque contributesto the autoresonant switching and c th Jtherefore decreases with β. Figure 4(c) shows the chirp rate ( ξ) dependence of cth J and switching time. cth Jdecreases with increasing ξand the switching time shows 1 /ξdependence. We note that the threshold cJfor the autoresonant spin-orbit torque switching is on the order of 40 Oe (Fig. 4). On the other hand, the threshold cJfor conventional DC spin-orbit torque switching is about 3400 Oe [ 34,35] for the same parameter set we use for numerical calculations of Fig. 4. Therefore, the autoresonant switching is highly efficient to reduce theswitching current in comparison to the conventional DC spin-orbit torque switching. We also note that the autoresonantswitching process allows fully deterministic spin-orbit torqueswitching without help from an external field. B. Micromagnetic simulation with thermal fluctuations We use micromagnetic simulation to consider multiple spins in a system at a finite temperature [ 48–51]. In this case, the phase of each spin becomes randomized due to thermalfluctuations. For the autoresonant switching mechanism, suchphase randomization may disturb the resonance excitation.Therefore, it is important to check whether or not theautoresonant spin-orbit torque switching allows a reducedswitching current even for multiple spins subject to thermalfluctuations. Based on micromagnetic simulations, we compute switch- ing probabilities for a nanopillar sample with area π(15 nm) 2 and thickness 1.1 nm. We use following parameters: the0. 0.005 0.01 0.015 0.020204060 Simulation Eq. (3.22) αLinear fit(a) c (Oe)Jth 0. 0.3 0.6 0.9 1.2 1.5020406080 β(b) Simulation c (Oe)Jth 0 1 2 3 4 535404550 0306090(c) c (Oe)Switching timeSwitching time (ns) (by simulation)cJthJth ξ (10 s )18-2 FIG. 4. (a) αdependence of the threshold value. In this plot, dJ=−cJis assumed. (b) βdependence of the threshold value. In this plot, α=0.012 is assumed. (c) ξdependence of the threshold value (black circle) and switching time (blue solid line). In this plot, dJ=−cJandα=0.012 are assumed. exchange stiffness constant Aex=1.6×10−6erg/cm, the temperature T=300 K, the unit cell size =1n m ×1n m ×1.5 nm. Other parameters are the same as for the macrospin simulation (Fig. 4). The switching probabilities are obtained from 20 switching trials. The simulation results are summarized in Fig. 5.W e find that the threshold cJis on the order of 100 Oe. This threshold obtained from micromagnetic simulation at T= 300 K is larger than that obtained from macrospin simulationatT=0 K (Sec. IV A ), but is still much smaller than that of DC spin-orbit torque switching ( ≈3400 Oe). We also find that the threshold value of c Jdecreases as βincreases, consistent with the macrospin results. Especially for β=1, cth Jis about 150 Oe, which corresponds to the current density 184401-5GYUNGCHOON GO, SEUNG-JAE LEE, AND KYUNG-JIN LEE PHYSICAL REVIEW B 95, 184401 (2017) 0 100 200 300 400 500 6000.0.20.40.60.81. c (Oe)Switching probability Jβ = 1.0 β = 0.2β = 0.6 β = 0.4β = 0.8 FIG. 5. Switching probabilities in the autoresonant spin-orbit torque switching process with ξ=0.3×1018s−2. Each data point is obtained from 40 trials. Parameters are T=300 K, α=0.012,Ku= 1.3×107erg/cm3,Ms=1240 emu /cm3,Aex=1.6×10−6erg/cm, andγ=1.76×107(Oe s)−1. ofJSO-STT SW /similarequal2.1×107A/cm2fortF=1.1 nm and θSH=0.3 [46]. V . CONCLUSION In this paper, we investigate spin-orbit-torque-driven au- toresonant switching of perpendicular magnetization. We findtwo interesting features of the autoresonant switching incomparison to conventional DC spin-orbit torque switching.First, the threshold current for the autoresonance mechanism isan order of magnitude smaller than that of DC spin-orbit torqueswitching. However, the switching time for the autoresonanceprocess is on the order of 10 ns, which is longer than that ofDC spin-orbit torque switching [ 34,35,38,39]. This differentswitching time is caused by the fact that the autoresonant switching inevitably involves multiple precession motion,whereas the DC spin-orbit torque switching requires only halfa precession. Second, the autoresonance mechanism allowsfield-free switching of perpendicular magnetization. We notethat conventional DC spin-orbit torque switching requiresan additional in-plane magnetic field for the deterministicswitching of perpendicular magnetization, which is detrimen-tal for device engineering. As a result, much effort has beenrecently expended in finding field-free switching schemes:Recent reports show that the field-free switching is possibleby breaking lateral symmetry [ 52] or exchange bias from an antiferromagnetic layer [ 53–56]. The autoresonant switching process provides an additional way for the field-free switching.In this respect, the present work may be expected to open upa future spintronics device which offers an efficient way forfield-free spin-orbit torque switching. We end this paper by noting that, for autoresonant spin- orbit-torque switching, low damping materials with large spinHall angle are favorable. Several low damping materials arefound in experiments for Permalloy [ 57], CoFe alloy [ 58], Fe 1−xVx[59], and Heusler alloy films [ 60]. Also, some materials with both large spin Hall angle and small dampingconstant are found in β-Ta/CoFeB [ 61],β-W/CoFeB [ 46], AuPt/NiFe [ 62], and WO x/CoFeB [ 63] bilayer structures. 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PhysRevB.45.295.pdf

PHYSICAL REVIEW B VOLUME 45,NUMBER 1 1JANUARY 1992-I Ferromagnetic resonance studyofmagnetic order-disorder phasetransition inamorphous Fe90—„CoZr,oalloys S.N.KaulandP.D.Babu Schoo!ofPhysics, Uniuersity ofHyderabad, Central Uniuersity P.O.,Hyderabad 500134,India (Received 24April1991;revisedmanuscript received 22July1991) Theutilityoftheferromagnetic-resonance (FMR)technique todetermine accurately thespontaneous magnetization andinitialsusceptibility criticalexponents Pandy,whichcharacterize theferromagnetic (FM)-paramagnetic (PM)phasetransition attheCurietemperature Tzforferromagnetic materials is demonstrated through adetailed comparative studyonamorphous Fe90Zr, oalloy,whichinvolves bulk magnetization andFMRmeasurements performed onthesamesampleinthecriticalregion.Magnetiza- tiondatadeduced fromtheFMRmeasurements takenonamorphous Fe90„Co„Zrla alloyswithx=0,1, 2,4,6,and8inthecriticalregionsatisfythemagnetic equationofstatecharacteristic ofasecond-order phasetransition. Contrary totheanomalously largevaluesoftheexponents Pandyreported earlier, thepresent values,P=0.38%0.03andy=1.38+0.06,arecomposition indepen-dent andmatchverywell thethree-dimensional Heisenberg values.Thefractionofspinsthatactually participates intheFM-PM phasetransition, c,isfoundtoincrease withtheCoconcentration asc(x)—c(0)—=axandpossessa smallvalueof11%forthealloywithx=O.The"peak-to-peak" FMRlinewidth (hH») varieswith temperature inaccordance withtheempirical relationbH»(T)=EH(0)+ [A/M,(T)],whereM,isthe saturation magnetization. BoththeLandesplitting factorgaswellastheGilbert damping parameter A, areindependent oftemperature, but,withincreasing Coconcentration (x),A,decreases slowlywhileg staysconstant atavalue2.07+0.02. I.INTRODUCTION Theferromagnetic- (FM)paramagnetic (PM)phase transition inamorphous (a-)Fesp+Zrip and Feso„(Co,Ni)„Zr, oalloyshasbecome acontroversial' is- sueeversincethebulkmagnetization (BM)measure- ments'inthecriticalregiononthesealloyshaveyielded valuesforthespontaneous magnetization and"zero- field"susceptibility criticalexponents Pandythatare roughly1.4timeslargerthantherenormalization-group estimates foranisotropic nearest-neighbor (NN)three- dimensional (3D)Heisenberg ferromagnet primarily be- causetheunphysically largevaluesforcriticalexponents havebeentakentoreflectlargefluctuation intheex- changeinteraction. Subsequently, anelaborate analysis' (reanalysis) ofhigh-precision magnetization dataona FescZrio (published'dataona-Fe»Zr9 anda-Fe92Zrs al- loys)revealed that,contrary totheearlierfinding,'the exponents P,y,and5(exponent forthecriticalisotherm) possess valueswhicharefairlyclosetothe3DNN Heisenberg values.Thisresultcastsseriousdoubtsabout thegenuineness oftheanomalously largecritical ex- ponent valuesreported fora-Fe90„(Co,Ni)„Zrio alloys andhencenecessitates adetailed studyofthecriticalbe- haviorinthesealloys.Inthispaperwedescribe anex- periinental determination oftheexponents Pandy, whichcharacterize theFM-PM phasetransition at T&,theCurietemperature, forthea-Fe9Q CoZr,Qal- loys.Although theferromagnetic-resonance (FMR)tech- niqueassuchisoldandtherelated technique called the"zero"-applied-field ferromagnetic-antiresonance (FMAR) tnicrowave transmission technique hasbeensuc-cessfully usedinthepasttodetermine theexponent P forcrystalline FeandNi,wedemonstrate thattheFMR technique canyieldaccurate estimates oftheexponents P andyforferromagnets through adetailed comparative studyonthea-Fe9QZr,Qalloywhichinvolves bulkmagne- tization andFMRmeasurements performed onthesame sampleinthecriticalregion. II.EXPERIMENTAL DETAILS Amorphous Fe9Q„CoZr,Qalloyswithx=0,1,2,4,6, and8wereprepared underargon(inert)atmosphere by thesingle-roller melt-quenching technique intheformof -2-mm-wide and(20—30)-pm-thick ribbons. Theamor- phousstateoftheribbons wasconfirmed byx-ray- diffraction andelectron microscopic methods. Usinga PAR4500vibrating-sample magnetometer, magnetiza- tion(M)versusexternal magnetic-field (H,„)isotherms weremeasured at0.1-Kintervals inthecriticalregionon several6-mm-long stripsofthea-Fe9QZr,Qalloystacked oneabovetheotherinfieldsupto15kOedirected along thelengthintheribbonplanesoastominimize the demagnetization effects. Thesample temperature was heldconstant towithin+25mKbyaLakeshore DRC 93Ctemperature controller andmonitored byaprecali- bratedcopper-constantan thermocouple incontact with thesample. Themicrowave power(P)absorption deriva- tivedP/dHfora-Fe9Q CoZr,Qalloyswithx=0,l,2, 4,6,and8wasmeasured asafunction oftheexternal staticmagnetic field(H)on4-mm-long strips,cutfrom thealloyribbons, usinghorizontal-parallel (ii")and vertical-parallel (ii")sample configurations (inwhichH 45295 1992TheAmerican Physical Society 296 S.N.KAULANDP.D.BABU liesintheribbonplaneandisdirected alongthelength andbreadth, respectively) atafixedmicrowave-field frequency of-9.225GHzonaJEOLFE-3XEPR spectrometer inthetemperature range—0.1~E=(T—Tc)/Tc~0.1at0.5-Kintervals. Acopper constantan thermocouple situated justoutside themi- crowave cavityafewcentimeters awayfromthesample wasusedasatemperature-controlling sensor, andthe temperature T'atthelocationofthissensorwasheld constant towithin+0.1Kateverytemperature setting byregulating theflowofcoldnitrogen gasaround the sample, mounted inastress-free condition insidea quartztube,bycontrolling thepowerinputtotheheater, immersed inacontainer filledwithliquidnitrogen, with theaidofaproportional, integral, andderivative (PID) temperature controller. Thesampletemperature T,in thiscasealsowasmeasured bymeansofaprecalibrated copper-constantan thermocouple andwasfoundtobe stabletowithinT+50mKwhenT*fluctuates between T—0.1andT+0. 1Katagiventemperature T.No changeinT,duetoeddycurrents wasdetected whenthe microwave powerlevelisincreased fromzeroto1mW. NotethatP=1mWforallthemeasured dP/dH-vs-H isotherms andthattheFMRandBMmeasurements were performed onthesamea-Fe90Zr, osample. Basedona detailed compositional analysis andtheobserved depen- denceofTconCoconcentration, weconclude thatthe rounding ofthetransition inzerofieldshouldoccurfor temperatures c&4X10 .Thusthedatatakeninthis temperature rangearenotincluded intheanalysis. Re- peatedFMRexperimental runsonthesamesamplehave revealed thattheresonance fieldH„,(defined asthefield wherethedP/dH=O linecutsthedP/dH vsHcurve)-- and"peak-to-peak" linewidth lelkHpparereproduced to within+1%and+10%,respectively.3 =&23338 344"( 543 3645 3747 3850 239.52 24.O.55 4260 362 III.EXPERIMENTAL RESULTS: DATAANALYSIS ANDDISCUSSION A.Bulkmagnetization Figure 1displays theM-vs-H,„isotherms takenonthe amorphous (a-)Fe9QZr, oalloyinanarrow temperature rangearoundCuriepointTcintheformofamodij7ed Arrottplot[i.e.,M'~vs(H/M)'rj.Thevaluesofthe exponents Pandyusedtoconstruct thisplotandgiven8l2 (H/M)'" FIG.1.Modified Arrottplotconstructed usingthebulk magnetization datatakenonthea-Fe9QZr, palloy. TABLEI.Valuesfortheparameters thatcharacterize thecriticalbehavior neartheFM-PM phasetransition inamorphous Fe9Q—„Co„Zr~p alloysandacomparison between theexperimentally determined andtheoretically predicted valuesforthecriticalex- ponentsPandyandfortheuniversal criticalamplitude ratiopphp/ksTc.Thenumber intheparentheses denotes theuncertainty in theleastsignificant figure,and(1),(2),and(3)denotethefirst,second, andthirdexperimental runsonthesamesample. BMstands forbulkmagnetization, whereasFMRistheabbreviation forferromagnetic resonance. Alloy concentration (x)/theory Method Tc(K) mp(G)hpImp hp(10G)pp(pg )pphpIkgTcp&p(pg )c(%) 3DHeisenbergFMR(1) FMR(2) FMR(3) BM FMR(1) FMR(2) FMR(1) FMR(2) FMR(1) FMR(2) FMR(1) FMR(2) FMR(1) FMR(2)238.55{15) 238.50(15) 238.55(15) 238.50(5) 254.50(20) 254.80(20) 284.00(25) 284.50(25) 336.00(10) 335.85(10) 377.04(10) 377.10(10) 419.50{10) 419.55(10)0.380{30) 0.370(30) 0.380(30) 0.360(20) 0.380{30) 0.380(30) 0.375(25) 0.385(25) 0.386(20) 0.380(20) 0.375(25) 0.385(25) 0.384(20) 0.382(20) 0.365(3)1.40(6) 1.39(6) 1.38(6) 1.38(3) 1.38(6) 1.38(6) 1.38(5) 1.38(5) 1.38(5) 1.39(6) 1.38(5) 1.39(6) 1.38(6) 1.38(6) 1.386(4)870(30) 870(30) 865(30) 870(25) 1030(35) 1045(25) 1070(25) 1065(25) 1075(20) 1075(20) 1160(30) 1165(30) 1215(35) 1215(35)500(50) 525(50) 500(50) 500(50) 500(75) 430(50) 400(60) 450(50) 650(50) 650(50) 950(50) 950{50) 1200(50) 1200(50)4.4(6) 4.6(6) 4.3(6) 4.4(6) 5.1(9) 4.5(7) 4.3(8) 4.8(6) 7.0(6) 7.0(6) 11.0(9) 11.0(8) 14.6(7) 14.6(7)1.386 1.386 1.386 1.386 1.50 1.50 1.61 1.61 1.68 1.68 1.70 1.70 1.70 1.700.172(17) 0.180(20) 0.168(17) 0.172(16) 0.202(35) 0.178(22) 0.164(30) 0.182(22) 0.235(16) 0.235(16) 0.333(25) 0.333(25) 0.397(18) 0.397(18) 1.5812.7(16) 12.2(17) 13.0(15) 12.7(16) 11.7(18) 13.3{15) 11.3(10) 13.9(15) 11.3(10) 11.3(10) 8.0(6) 8.0(6) 6.8(5) 6.8(5)11.0(10) 11.0(11) 11.0(10) 11.0(10) 12.8(15) 11.3(10) 10.4(16) 11.6(10) 15.0(12) 15.0(12) 21.5(15) 21.2(15) 25.0(18) 25.0(18) 'Valuesobtained fromthebulkmagnetization measurements takenat4.2Kontheglassyalloysunderconsideration. 45 FERROMAGNETIC RESONANCE STUDYOFMAGNETIC ORDER-. .. 297 inTableIhavebeendetermined bythemodified asymp- toticanalysis(AA-II;fordetailsseeRef.9),andthemag- neticfieldHhasbeencorrected forthedemagnetizing effects(i.e.,H=H,„H—~,,whereHz,isthedemagnet- izingfieldestimated fromthe"low-field" magnetization data}.Theisotherms areseentobeasetofstraight lines, andthecritical isotherm atT=Tc=238. 50Kpasses through theorigin,asexpected foracorrectchoiceofthe exponents Pandy.46K B.Ferromagnetic resonance Thevariation ofdP/dH withHinthe~~"configuration atafewselected valuesoftemperature inthecriticalre- gionisdepicted fora-Fe90Zr, oinFig.2.Thesecurvesare alsorepresentative ofthoserecorded fora-Fe90Zr, ointhe ()'configuration andforotheralloysinboth(("and()" geometries. ItisevidentfromFig.2thatasthetempera- tureisincreased throughTc,thepeakinthedP/dH-vs- Hcurvesatalowerfieldvalue-800Oedevelops intoa full-fledged resonance (secondary resonance) for T~Tc+10K,whereas themain(primary) resonance shiftstohigherfieldsandbroadens out.AdetailedFMR study'carriedoutontheglassyalloysinquestion ina widetemperature range77~T&500Krevealsthatthe secondary-resonance (whosesignature isfirstnoticed in themostsensitive settingofthespectrometer atatern- perature closetoTcoraboveTc)exhibits a"cluster spin-glass-like" behavior, whereas theprimary resonance possesses properties characteristic offerromagnets with"' (forxS2)orwithout' (forx~2}reentrant spin-glass behavior atlowtemperatures. Sincethestudy ofcriticalbehavior neartheFM-PM phasetransition in a-Fe90„Co„Zr&0 alloysisofprimeconcern inthispaper andthedatarecorded in~~"and ~~"configurations yieldex- actlythesameresultssofarasthecriticalbehavior inthe investigated glassyalloysisconcerned, henceforth we dealwiththeprimary resonance partofthedP/dH-vs-H curves,recorded inthe~~"configuration, only.Nowthat inthecriticalregionhH-=H„,/4,theobserved value ofH„,couldsignificantly differfromthe"true"reso- nancecenter,andhenceadetailed line-shape analysisfor eachresonance lineseparately iscalledfor.ThedP/dH vs-Hcurvesrecorded atdifferent temperatures inthe~~"42K 36K 34K 30K I I I I I I I I 00.51.01.52.02.53.03.540 H(kQe}I I 4.55.0 FIG.2.Powerabsorption derivative curvesfora-Fe90Zr, oat afewrepresentative temperature valuesinthecriticalregion recorded usingthe~~"sampleconfiguration. Solidcurvesdepict theobserved variation ofdP/dH withH,whereas theopencir- clesdenotethecalculated valuesbasedonEqs.(1)and(2)ofthe text.Numbers ontheleft-hand sideofthecurvesareameasure ofthesensitivity atwhichthespectraaretaken. configuration havebeenfittedtothetheoretical expres- sion'' dP [(~2+~~2)1/2+ «]1/2 dHdH withtherealandimaginary components ofthedynamic permeability givenby and[(H+H)(B+H )—I—(co/y)][(B+H )—I(co/y)]+—2I(B+H)(B+H+2H ) [(H+H)(B+H )—I2—(co/y)]+I(B+H+2H )2(2a} —2I(B+H)[(H+H}(8+H )—I—(co/y)]+I(B+H+2H )[(B+H )—I—(co/y)] [(H+H)(B+H )—I—(ei/y)]+I(B+H+2H )(2b) derivedfordP/dH intheparallel geometry andobtained bysolvingtheLandau-Lifshitz-Gilbert (LLG)equationof motion inconjunction withMaxwell's equations, by making useofanonlinear least-squares-fit computer pro-gramwhichtreatstheLandesplitting factorgandsatu- rationmagnetization M,=(BH)/4m asfreefit—ting parameters andusestheobserved valuesof bH=1.45I=1.45ico/y M,(where )1,istheGilbert 298 S.N.KAULANDP.D.BABU 45 damping parameter, y=g~e~/2mc, andv=co/2~ isthe microwave-field frequency), andvaluesofthe"in-plane" uniaxial anisotropy fieldHzdeduced fromtherelations' andres resKII= II(3a) H„,=H„,+Haec (3b) InEqs.(3a)and(3b),H,'i„andHi'„are theestimates for theresonance fieldsinthe~~"and ~~"configurations, re- spectively, obtained aftercorrectinq theobserved values forthedemagnetization fieldsHfi,andH)~,m,deter- minedfromthelow-field magnetization measurements performed onthesamples usedforthepresentFMR studywiththeexternal magnetic fieldapplied alongthe easy(ff"-configuration) andhard(ff"-configuration) direc- tionsintheribbonplane,andHII„istheresonance field intheabsenceofHz.Intheline-shape calculations lead- ingtoEqs.(1),(2a),and(2b),theexchange-conductivity contribution hasbeendropped inviewofthewell-known observation'''thatthiscontribution tothelinewidth aswellastotheresonance fieldissosmallastofallwell withintheerrorlimitsbecause thevaluesfortheex- change stiffness parameter andconductivity bothareat leastanorderofmagnitude smaller''thantheircorre- sponding valuesforcrystalline metals. Theoretical fits,depicted byopencirclesinFig.2,not onlyassertthattheline-shape analysis yields"true" valuesofthelinecentersfortheprimary resonance even inthepresence ofasecondary resonance because the baseline forthetworesonances isthesome(Fig.2),but alsoindicate thattheLLGequation adequately describes theresonant behavior inthecriticalregion.Inaddition, theline-shape calculations revealthatthesplitting factor ghasaconstant valueof2.07+0.02withintheinvestigat- edtemperature range.Thatthegfactoristemperature independent andtheLLGequation formsanadequate description ofH„,(T)andhH(T)inthecriticalregion forcrystalline ferromagnets alsohasbeenclaimed by Rodbell' andbyHaraldson andPettersson,'butthis claimhasbeenrefuted byBhagat andRothstein. The present resultsare,however, consistent withourearlier observation' thatM,(T)deduced inthisway'fromthe FMRdataareinexcellent agreement withM(T)mea- suredonthesamesampleatanexternal magnetic field whosestrength iscomparable toH„,. m=fz(h), (4)C.Critical exponents, amplitudes, andscalingequation ofstate IAHavingdetermined M,(T)andH„,(T)[=HJi„(T); the superscript~~ishenceforth dropped forconvenience] toa highprecision fromtheline-shape calculations, accurate valuesofp,y,andTcareextracted fromthe M(H,T)=M,(H„„T) databyidentifying H„,withthe ordering fieldHconjugate toM(=M,),andusingthe "range-of-fit" scaling-equation-of-state (SES}analysis, whichisbasedonthemagnetic equationofstate,Fe9QZr»c&0p p9+ + +z&0 Tc=238. 0K+g&0p +++ + + 0++ ~+ ++@&0 ++ LZ3 +0+ ++ g+c&0 Tc=239.0K +++Tc=238.5K P=038 y=138 I I I I 13151719 ln(H/icis")21 FIG.3.PlotsofIn(M/~s~s)against ln(H/~s~s+")for different valuesofTcfora-Fe9OZr, o. whereplusandminussignsrefertotemperatures above andbelowTcandm=M/~s~~ andh:H/)Is)~+~ are—the scaledmagnetization andscaledfield,respectively. Inthe conventional SESmethod, M(H,T)data,inthecritical region,aremadetofallontwouniversal curves,ffor s(0andf+fors)0,through anappropriate choiceof theparameters Tc,p,andyinanm-vs-hplot,butthis choiceisbynomeansuniqueinthesensethatnearlythe samequalityofdatacollapse ontothetwouniversal curvescanbeachieved forawiderangeofparameter values(tyPically, +2%forTcand+10%%uoforPandy}. Thisproblem is,however, effectively tackled byemploy- ingtherange-of-fit SESanalysis inwhichmoreand moreofthedatatakenattemperatures awayfromTcare excluded fromthem-vs-hplotsothattheexponents p andybecome increasingly sensitive tothechoiceofTc andthedataexhibitstrongdepartures fromthecurvesf(h}andf+(h)ifthechoiceoftheparameters differs evenslightly fromthecorrect one.Thisprocedure, therefore, goesonrefining thevaluesofthecriticalex- ponents untiltheyapproach theasymptotic values.The finalvaluesoftheparameters Tc,p,andyfora-Fe9OZr&o soobtained aregiveninTableI.Figure3servestoillus- tratetheeffectofthevariation inthevalueofT&onthe qualityofdatacollapse. Similar effectsareobserved if oneoftheexponents isvariedwhilekeepingTcandthe otherexponent fixed. Acomparison between thelnm-vs-lnh scalingplotsfor a-Fe9oZr&o constructed usingtheBMandFMRdata (recorded inthreedifferent experimental runs}takenon thesamesampleisshowninFig.4.Aperfectagreement between different setsofFMRdataandbetween there- sultsofFMRandBMmeasurements isevidentfromthis figure.However, amorerigorous meansofascertaining 45 FERROMAGNETIC RESONANCE STUDYOFMAGNETIC ORDER-. .. 299 FegPZC1O ~~~~~BMData OOOOO FMRRun 1 clouuu FMRRun2 AAAAA FMRRun3 ~O —8-~ QD@&0 e&0c&0 c&0 ~~~~ FMR BM Tc=238.55238.50K P=0.3800.360 y=1.3801.380 6I I I I I I I 121314151617181920 1n(H/(sI~'") FIG.4.1n(M/~s ~s)-vs-1n(H/~s ~s+")plotsfora-Fe9OZr, ocon- structed usingthebulkmagnetization dataandsaturation mag- netization datadeduced fromFMRspectra employing line- shapeanalysis.00 4.0 3.02468 h/m(10')101214 whether ornottheabovemethod yieldsaccurate values forthecriticalexponents andT~isprovided byaSES formthatdiffersfromEq.(4),i.e., m=+a++b+(h/m) (5)1.0 M,(s)=limM(H,s)=mo(—s)~,s(0H~O and(6a)(wheretheplusandminussignsaswellasmandhhave thesamemeaning asgivenabove}, because eventhe slightest deviations ofthedatafromtheuniversal curvesf(h)andf+(h), whichdonotshowupclearly ina lnm-vs-lnh plotbecauseoftheinsensitive natureofthe double-logarithmic scale,become easilydiscernible whenthesamedataareplotted intheformofanm-vs- (h/m)plot.Another advantage inusingEq.(5)isthat thecritical amplitudes mo=a'/ andho/mo=a+/b+, definedby0.00 6810 h/m(10')1214 FIG.5.m-vs-h/m plotfora-Fe9O„Zr,oconstructed using (a)bulkmagnetization dataand(b)saturation magnetization datadeduced fromtheFMRspectratakeninthefirst(opencir- cles),second(opensquares), andthird(opentriangles) experi- mentalrunsonthesamesample. Thedataneartheoriginare plottedonasensitive scaleintheinsetwithaviewtobringout valuesoftheintercepts moandho/mponmandh/maxes clearly. ~}M(H,s) XoBHH=O=(ho/mo)sr,s)0 (6b) aregivenbytheintercepts oftheuniversal curveswith mandh/maxes,respectively, inanm-vs-h/m plot. Suchplotsconstructed usingthechoiceofparameters Tc,P,andygiveninTableIareshowninFigs.5(a)and 5(b).Consistency amongdifferent setsofdataisnowall themoreobvious, particularly whentheBMandFMR dataareplottedonahighlysensitive scaleandonlythose BMdatathataretakenatfieldscomparable instrength tothoseusedinFMRexperiments areincluded inFig. 5(a).Fromtheobservation thatnodeviations fromthe universal curvesareevident evenatlowfieldsinFigs.5(a) and5(b),weconclude thatthevaluesforthecriticalex-ponents andTzdetermined bytherange-of-fit SES analysis arereasonably accurate. Moreover, thevalueof thespecific-heat criticalexponenta=—0.14,computed usingthepresently determined valuesoftheexponents p andy(whichconform verywellwiththosepreviously re- ported'forthisalloybasedontheBMmeasurements) in thescalingrelationa=2(1—p)—y,andthepresent value ofTc(Table I)agree closely withthose (a=—0.13+0.06T=238.6+0.1K)extracted fromre- centelectrical resistivity measurements' onasamplecut fromthesamea-Fe9oZr, oribbonasthatusedinthis work.TableIliststhevaluesofCurietemperature, criti- calexponents pandy,critical amplitudes moand (ho/mo}, andtheratiopoho/ksTc,deduced forthea- Fe9OZr&o alloyfromtheBMdataandfromthedifferent setsofFMRdatatakenonthesamesample, andcom- S.N.KAULANDP.D.BABU 45 paresthemwiththetheoretical valuespredicted foran isotropic NN3DHeisenberg ferromagnet. Anassess- mentofthedatapresented inTableIrevealsthatare- markably closeagreement existsnotonlybetween the BMandFMRresults,butalsobetween thetheoretical andexperimental valuesforthecriticalexponents. How- ever,theobserved valueoftheratiopoho/k~TC isone orderofmagnitude smallerthanthetheoretically predict- edone.Sincehoispresumably aneffective exchange in- m C) moteraction field,theproductofhoandanaverage effective elementary moment (p,s)involved intheFM-PM phase transition, i.e.,theeffective exchange energyp,go,isex- pectedtoequalthethermal energyk&TcatTc.Obvi- ously,thisisnotthecasefora-Fe90Zr, ounlessp,zistak- entobeverymuchlargerthanpo(average magnetic mo- mentperalloyatomat0K).Nowthattheexponents possess3DHeisenberg values,theratiop,golk~Tc is alsoexpected toequalthe3DHeisenberg estimateof 1.58.Thisispossible onlywhen JM,&assumes thevalues giveninTableI.Moreover, iftheconcentration ofsuch effective moments isc,thenc=po/p, &;Thevaluesofc calculated inthiswayandincluded intheTableIstrong- lyindicate thatonlyasmallfractionofmoments (i.e.,the moments onFeatomsinthecaseofa-Fe9oZr, o)partici- patesintheFM-PM phasetransition. Having demonstrated thattheFMRtechnique isa powerful toolforinvestigating thecritical behavior in ferromagnets, accurate valuesforTc,thecritical ex- ponentsPandy,critical amplitudes moand(ho/mo), andconcentration ofeffective moments participating in theFM-PM transition, c,havebeendetermined by analyzing theFMRdatatakenonthealloyswithx=1, 2,4,6,and8usingthesamemethod asmentioned above. Thevaluessoobtained arelistedinTableIandareused toconstruct them-vs-h/m scaling plotsfortheCo- containing glassyalloysshowninFig.6.Anumberofin- teresting pointsemergefromacomparison ofthepresent mo 069 h/m(lo') F~90-x«xz«101215 1.8—FeA-4~ O -2 mo EOC) mo CQ25 15-M O 2 02468 Coconcentration x10 2mo '0 69 h/m(10')12'l5 FIG.6.m-vs-h/m plotsfora-Fe9p CoZr,palloyscon- structed usingthesaturation magnetization datadeduced from theFMRspectrarecorded atdifferent temperatures inthecriti- calregion.FIG.7.Functional dependences ofCurietemperature T&, magnetic moment peralloyatomat0K,pp,Gilbert damping parameter A,,andthefractionofspinsparticipating intheFM- PMphasetransition, c,ontheCoconcentration xfora- Fe9p„Co„Zrlp alloys.Thesolidcurvesaretheleast-squares fits tothedata,whereas thedashedcurvesserveasaguidetothe eye.NotethattheerrorlimitsforA,aretypically +5%%uoofits valueatagivenx.Theerrorlimitsfortheotherquantities are giveninTableI. 45 FERROMAGNETIC RESONANCE STUDYOFMAGNETIC ORDER-.~. 301 resultswiththosepreviously obtained andwiththose predicted bythetheory; namely, (i}contrary totheear- lierfinding, theexponents Pandydonotdependonthe alloycomposition andpossess3D-Heisenberg-like values; (ii)thefractionofspinsthatactually participates inthe FM-PM phasetransition, c,issmallfortheparentalloy (x=0),butincreases withCoconcentration xas c(x)—c(0)—=ax,withc(0)=11+1% anda—=0.23,in theinvestigated composition range(Fig.7);and(iii)in conformity withthepreviously reported' result,the Curietemperature T&increases roughly linearly with thecomposition x[i.e.,Tc(x)=Tc(0)+23.2x,with Tc(0}=237.2K],whereas themoment peralloyatomat 0K,po,increases steeplywithxintherange0&x&2 andattainssaturation forxR4(Fig.7). D.FMRlinemdth Variation ofthepeak-to-peak FMRlinewidth LakHpp withtemperature isdisplayed inFig.8.Aslopechange in5Hz~(T) atTc(B=O)foralltheglassyalloysunder consideration isindicative ofawell-defined magnetic phasetransition atT&.Itshouldbeemphasized atthis stagethatthevaluesofb,Harethesame(withinerror limits)forboth ~~"and ~~"configurations atalltempera- tureswithinthetemperature rangecovered inthepresent experiments. Hence bH~~(T)observed inthe configuration anddepicted inFig.8reproduces allthe features b,H"(T)eventotheminutest detail.FMRmea- surements carriedout,inthepast,overawiderangeof microwave-field frequencies atconstant temperaturebH(T)=bH(0)+[A/Ms(T)] . (7) Ifthefirstandsecondtermsontheright-hand sideofEq. (7)areidentified withbHtandEH„„o, respectively, the coefficient AinEq.(7},determined bytheleast-squares methodfordifferent compositions, permitsastraightfor- wardcalculation ofthedamping parameter A,.Notethat Eq.(7)withthemeaning ofthetermsEH(0) and A/M,(T)sameasabovehasbeenpreviously usedto(T&Tc)onalargenumberofamorphous ferromagnetic alloysystems haverevealed thatthetwomaincon- tributions tohHareEHI,whichisnearlyindepen- dentofthemicrowave-field frequency v=co/2na. ndis mostprobably causedbythetwo-magnon scattering from spatially localized magnetization inhomogeneities,''' andAHLz6=1. 45K,co/yM„which hasalineardepen- denceonvandresultsfromaLLGrelaxation mecha- nism.Insuchmaterials, hHisfoundtoremainpracti- callyconstant''''overawiderangeoftempera- tureswellbelowTc(T50.8Tc)andtheGilbert damping parameter A,varieslinearly''''withM,.Anim- mediate consequence oftheresultX~M,isthathH„~z doesnotvarywithtemperature, sothatinviewofacon- stantvalueofLalHppevenEHIshouldnotdependontem- perature. Inthecritical region (—0.055s-0.05), hH(T)foralltheamorphous alloysinquestion, with theexception ofthosewithx=0and4,canbeverywell described (Fig.9)bytheempirical expression IFeao-xcoxzr101 -74~ x=40 ~gggs4 x=2C4 7-x=1gpQ~+p ~C++~'7tP 9P~ o~ -2DpQL)3 po&oooaoCO~(g)ooo™F000-XCOXZ110 ~Q+0 x-I2~ ~p.m~-5~I Ol x-& 4- X~2~~P EO~~Q5-3 -5 X ygjlxx4 4,44444 ~'4-4~~C] acP~~~QP -3 X~8 2 —0.06I I 0.000.03 e=(T—Tc)/TcI—0.03 0.06 FIG.8.Variation ofthepeak-to-peak FMRlinewidth (AH»)withtemperature inthecritical regionfora- Fe90„Co„Zr» alloys.TheAH»(T) datatakeninthefirst, second, andthirdexperimental runsonthesamesamplearede- pictedbythesymbols opencircles, squares, andtriangles, re- spectively.2.5I3.54.5 MB(10G)I 5.5 FIG.9.Peak-to-peak FMRlinewidth(hH„)plottedagainst inverse saturation magnetization inthetemperature interval—0.05~c+0.05forFe90„Co„Zrlo alloys. Thesolidand dashedstraight linesdrawnthrough thedatapoints(denoted by thesymbols whichhavethesamemeaning asinFig.8) represent theleast-squares fittotheAH»(T) databasedonEq. (7)ofthetextinthetemperature intervals—0.05~a~0 and—0.05~c&0.05respectively. 302 S.N.KAULANDP.D.BABU 45 describe thetemperature dependence ofhHinFe-richPPa-Fe,ooB„alloys fortemperatures wellbelowTc.The temperature-independent valuesofA,socomputed range between2X10and3X10sec'andexhibitaweakde- creasing trendwithCoconcentration asshowninFig.7. Bycontrast, theintercept bH(0)doesnotshowanysys- tematic trendwithx;i.e.,aminimum[bH;„(0)=20Oe] intheEH(0)-vs-x curveatx=2isfollowed byamax- imum[EH,„(0)-=140 Oe]atx=4.Different setsof FMRdatatakenonthesamesampledemonstrate that thevaluesforA,arereproduced towithin+10%. Another important findingwhichmeritsattention isthat thequalityoftheleast-squares fitstotheb,H(T)data basedonEq.(7)improves, asinferred byalowervalue forthesumofdeviation squares (y),iftherangeoftem- peratures overwhichsuchfitsareattempted isconfined totemperatures justbelowTc,i.e.,0.05~@.&0.Howev- er,amarked improvement inthequalityofthesefits, brought aboutbyawidelydifferent choiceoftheparame- tersEH(0)andA[i.e.,b,H(0)decreases byafactorof about1.5,whileA(andhence A,)increases bythesame factor] isobserved forthealloyswithx=0and4only; fortheremaining alloys,theslopeandintercept values change onlyslightly fromtheirprevious estimates, with theresultthatonlyamarginal decrease ingoccurs. Theappearance ofthesecondary resonance atT=—Tcfor a-Fe90Zr&o andatatemperature afewdegrees aboveTc forthealloywithx=4asagainstattemperatures well aboveTcforotheralloyscouldbeattherootofthe uniquebehavior ofthealloyswithx=0and4.Acom- pleteunderstanding ofthisaspectoftheFMRdatamust, however, awaitadetailed investigation whichshedslight ontheexactoriginofthesecondary resonance. The presently determined valuesofthedamping parameter I, areabout3timeslargerthanthosereported forawide varietyofcrystalline''''andamorphous'fer- romagnets attemperatures wellbelowtheCurietempera- tureTc.Suchalargeenhancement inthevalueofXfor temperatures closetoTcisnotuncommon.''More- over,thefindingthatkdoesnotdependontemperature in thecritical regionisaproperty whichtheinvestigated glassyalloyssharewithcrystalline ferromagnets detailed butaccurate measurements ofbH~„(T) inthe criticalregionforamorphous ferromagnets arepresently lacking. Considering thewell-known''factthatthedynamic permeability attainsitsmaximum valueatthefieldcorre- sponding toferromagnetic resonance andthemicrowave radiation penetrates onlyathinsurface layer(typical- ly''10A)inaferromagnetic metal,FMRmeasure- mentshavealsobeenperformed ontheBMsamples after etching themwithnital(10%concentrated HNO3+ 90%%uoC2H5OH) solution for30minsoastoensurethat theresultsarerepresentative ofthebulk.Fromthe weight-loss measurements, weinferthatthethickness of thesample isreduced tonearlyhalfaftertheetchingtreatment. Apartfromasystematic downward shift(up- wardshift)intheresonance field(saturation magnetiza- tion)versustemperature curvefortheetchedsamples withrespecttothesimilarcurveinthe"as-quenched" samples, nochange inthevaluesquotedfordifferent quantities inTableIhasbeendetected. Itshouldbeem- phasized atthisstagethatthefullpotential oftheFMR technique toyieldaccurate valuesforthecritical ex- ponents canbeexploited onlywhenthistechnique isused tostudythemagnetic order-disorder phasetransition in goodqualitythinfilmssincetheskindepthinthatcaseis comparable tothefilmthickness andtheconventional methods tomeasure bulkmagnetization forsamples in thin-film formlacktherequired sensitivity. IV.CONCLUSION Fromadetailed studyofcritical behavior inamor- phousFe90CoZrioalloysusingbulkmagnetization (forthealloywithx=0alone)andFMRtechniques, the following conclusions canbedrawn. (i)TheFMRtechnique canbeusedtodetermine the criticalexponents Pandyforferromagnets toanaccura- cywhichcompares wellwiththatachieved inthebulk magnetization method. (ii)Contrary totheearlierclaim, thecritical ex- ponentsPandyarecomposition independent andpossess valueswhichareclosetotherenormalization-group esti- matesforaspinsystem withspinaswellasspatial dimensionality ofthree.Alternatively, thetransition at Tciswelldefined andthequenched disorder doesnot alterthecriticalbehavior ofanordered3DHeisenberg ferromagnet; i.e.,thewell-known Harriscriterion is satisfied. (iii)Thefractionofspinsthatactually participates in theFM-PM phasetransition, c,increases from11%at x=0to25%%uoatx=8andthefunctional dependence ofc onxiswelldescribed bytheempirical relation c(x)—c(0)-=ax. (iv)bHz(T)closely followsarelationofthetype bH(T)=AH(0)+ [A/M,(T)]inthecriticalregion. (v)BoththeLandesplitting factorgandGilbertdamp- ingparameter A,aretemperature independent withinthe investigated temperature range,butwithincreasing Co concentration A,decreases whilegremains constant atthe value2.07+0.02. ACKNO%'LED GMKNTS Thefinancial support bytheDepartment ofScience andTechnology, NewDelhi, underproject No. Sp/S2/M21/86 tocarryoutthisworkisgratefully ac- knowledged. Oneofus(P.D.B.)isthankful totheUni- versityGrantsCommission, NewDelhi,forfinancial as- sistance. 45 FERROMAGNETIC RESONANCE STUDYOFMAGNETIC ORDER-. .. 303 S.N.Kaul,J.Phys.F18,2089(1988). H.Yamauchi, H.Onodera, andH.Yamamoto, J.Phys.Soc. Jpn.53,747(1984). K.Winschuh andM.Rosenberg, J.Appl.Phys.61,4401 (1987). 4L.C.LeGuillou andJ.Zinn-Justin, Phys.Rev.B21,3976 (1980). R.Reisser, M.Fahnle, andH.Kromuller, J.Magn.Magn. Mater.75,45(1988). H.Hiroyoshi, K.Fukamichi, A.Hoshi,andY.Nakagawa, in HighFieldMagnetism, editedbyM.Date(North-Holland, Amsterdam, 1983),p.113. 7J.D.CohenandT.R.Carver,Phys.Rev.B15,5350(1977);J. H.Abeles,T.R.Carver, andG.C.Alexandrakis, J.Appl. Phys.53,7935(1982). S.N.KaulandT.V.S.M.MohanBabu,J.Phys.Condens. Matter.1,8509(1989). S.N.Kaul,J.Magn.Magn.Mater.53,5(1985). S.N.KaulandV.Siruguri,J.Phys.Condens. Matter. (tobe published), andunpublished results. 'S.N.Kaul,Phys.Rev.B27,6923(1983). 'P.Deppe,K.Fukamichi, F.S.Li,M.Rosenberg, andM.Sos- tarich,IEEETrans.Magn.MAG-20, 1367(1984). S.N.KaulandV.Siruguri,J.Phys.F17,L255(1987). S.M.Bhagat,S.Haraldson, andO.Beckman,J.Phys.Chem. Solids38,593(1977). S.N.KaulandV.Srinivasa Kasyapa,J.Mater.Sci.24,3337 (1989). S.N.Kaul,J.Phys.Condens. Matter3,4027(1991). Ch.V.Mohan,P.D.Babu,M.Sambasiva Rao,T.Lucinski, andS.N.Kaul(unpublished). D.S.Rodbell, Phys.Rev.Lett.13,471(1964). S.Haraldson andL.Pettersson, J.Phys.Chem.Solids42,681 (1981). S.M.BhagatandM.S.Rothstein, SolidStateCommun. 11, 1535(1972),andreferences citedtherein. Thecustomary approach ofdetermining M,(T)eitherbyus- ingtheresonance condition forthe~~"configuration andthe resultsofFMRmeasurements performed inthesame configuration atthreewidelyspacedvaluesofmicrowave fre- quencyvorbymaking useoftheresonance conditions forthe ~~"andi"(thehorizontal-perpendicular samplegeometry, in whichtheexternal staticmagnetic fieldisappliedperpendicu- lartothesampleplane)configurations andtheFMRresults obtained forthesesampleconfigurations atasinglevalueofv hasnotfollowed inthisworkfortworeasons. First,thelack ofexperimental facilities required forsuchanexperimental investigation. Second, evenwithutmostcareexercised in sample mounting anditspositioning intheexternal static field,thelineshapeforthel"configuration andhencethe valueofH„,couldnotbereproduced withashighanaccura-cyaswasachieved inthedetermination ofH~~,andH~~„ presumbly becauseoftheextreme sensitivity ofH„,tothean- glebetween thefielddirection andsampleplane.Inviewof thisobservation, weconsider aperfect agreement observed between thevaluesofH„,extracted fromsomeexperimental runsandthosecalculated usingthenumerical estimates ofg andM„deduced fromthepresent line-shape analysisofthe FMRspectratakeninthe ~~configuration, intheresonance condition forthei"configuration, asfortuitous M.Fahnle,W.U.Kellner, andH.Kronmuller, Phys.Rev.B 35,3640(1987);W.U.Kellner, M.Fahnle,H.Kronmuller, andS.N.Kaul,Phys.StatusSolidiB144,397(1987). S.N.Kaul,Phys.Rev.B23,1205(1981). M.L.SpanoandS.M.Bhagat,J.Magn.Magn.Mater.24,143 (1981). L.Kraus,Z.Frait,andJ.Schneider, Phys.StatusSolidiA63, 669(1981). J.F.Cochran,K.Myrtle, andB.Heinrich,J.Appl.Phys.53, 2261(1982). B.Heinrich,J.M.Rudd,K.Urguhart, K.Myrtle,J.F. Cochran, andR.Hasegawa, J.Appl.Phys.55,1814(1984). D.J.WebbandS.M.Bhagat,J.Magn.Magn.Mater.42,109 (1984) ~ S.M.Bhagat,D.J.Webb,andM.A.Manheimer, J.Magn. Magn.Mater.53,209(1985). Anadditional contribution tohH»,besideshHILGandEHI, originates fromtheskin-depth effect(whichmakesthemag- netization induced bythemicrowave fieldnonuniform inthe volumeofthesurfacepenetration layer),butthiscontribution fortheinvestigated alloysturnsouttobeassmallas-=10Oe (Ref.15).Thisvaluelieswellwithintheobserved errorlimits andhenceneednotbeconsidered whilediscussing different contributions toEHpp. B.Heinrich,J.F.Cochran, andR.Hasegawa, J.Appl.Phys. 57,3690(1985). 32J.F.Cochran,R.W.Qiao,andB.Heinrich, Phys.Rev.B39, 4399(1989). 3sS.M.Bhagat, inMeasurement ofPhysical Properties Part2:. Magnetic Properties andMossbauer Egect,editedbyE.Pas- saglia(Wiley,NewYork,1973),p.79. Z.FraitandD.Fraitova, inSpinWavesandMagnetic Excita- tions,editedbyA.S.Borovik-Romanov andS.K.Sinha(El- sevier,NewYork,1988),Pt.2,p.1. Z.FraitandD.Fraitova, Phys.StatusSolidiB154,363 (1989) ~ B.Heinrich andA.S.Arrott,J.Magn.Magn.Mater.31-34, 669(1983). D.S.Rodbell, Physica1,279(1965). 38S.M.Bhagat andH.O.Stevens,J.Appl.Phys.39,1067 (1968).

PhysRevB.86.214416.pdf

PHYSICAL REVIEW B 86, 214416 (2012) Temperature dependence of the frequencies and effective damping parameters of ferrimagnetic resonance F. Schlickeiser,1,*U. Atxitia,2,3S. Wienholdt,1D. Hinzke,1O. Chubykalo-Fesenko,3and U. Nowak1 1Fachbereich Physik, Universit ¨at Konstanz, D-78457 Konstanz, Germany 2Department of Physics, University of York, Heslington, York YO10 5DD United Kingdom 3Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain (Received 3 September 2012; published 17 December 2012) Recent experiments on all-optical switching in GdFeCo and CoGd have raised the question about the importance of the angular momentum or the magnetization compensation point for ultrafast magnetization dynamics. Weinvestigate the dynamics of ferrimagnets by means of computer simulations as well as analytically. The resultsfrom atomistic modeling are explained by a theory based on the two-sublattice Landau-Lifshitz-Bloch equation.Similarly to the experimental results and unlike predictions based on the macroscopic Landau-Lifshitz equation,we find an increase in the effective damping at temperatures approaching the Curie temperature. Further results forthe temperature dependence of the frequencies and effective damping parameters of the normal modes representan improvement of former approximated solutions, building a better basis for comparison to recent experiments. DOI: 10.1103/PhysRevB.86.214416 PACS number(s): 75 .78.−n, 75.50.Gg I. INTRODUCTION The recent discovery of ultrafast, optomagnetic writing schemes using circularly polarized laser pulses,1–3pure thermal excitation,4,5or terahertz radiation6focuses much attention on the understanding of antiferromagnetic andferrimagnetic materials since all these effects have been foundonly for materials with at least two sublattices. Switching withcircularly polarized light or just with the heat pulse has beenrestricted to ferrimagnets with a rare-earth component, as, e.g.,GdFeCo 1or TbCo.7The reason for this restriction is not fully understood, though it has been speculated that the peculiaritiesof the dynamics of a ferrimagnet across the angular momentumcompensation temperature, where the effective damping andthe frequency of the normal modes are predicted to increaserapidly, 8plays a crucial role. In general, ferrimagnetic materials with two sublattices show two characteristic damped precession motions of thetotal magnetization around an external field H 0. As they can be excited experimentally by oscillating magnetic fields, theyare called resonance modes. For one mode both sublatticesstay antiparallel to each other. The dynamics related to thismode can be described as an effective ferromagnetic systemand is called the ferromagnetic mode (FMM). The othernormal mode is caused by the antiferromagnetic couplingbetween the two sublattices. In this so-called exchangemode (EXM), the sublattices are tilted at a characteristicangle. 9The characteristic motion of both modes is shown in Fig. 1. The parameters that basically define the possible switching time are the frequency and the effective damping of theresonance modes of the samples. Both need to be high inorder to enable fast magnetization reversal. The temperaturedependence of the dynamic behavior of ferrimagnets is ofspecial interest here since in earlier theories of ferrimagneticresonance 8,10–12based on the two-macrospin Landau-Lifshitz- Gilbert (LLG) equation of motion, the FMM shows a diver-gence of (or at least a rapid increase in) the frequency andthe effective damping parameter at the angular momentumcompensation temperature T A.Recently, the temperature dependence of these resonance modes was investigated experimentally for amorphous, fer-rimagnetic GdFeCo by Stanciu et al. 13and for amorphous, ferrimagnetic CoGd by Binder et al.14In both experiments it was shown that both the frequency and the effective dampingparameter of the FMM increase significantly, approachingthe angular momentum compensation point T A. Besides this partial coincidence with the analytical prediction for thefrequency of the FMM, the experimental findings in Ref. 13 also feature some disagreement with earlier theories. Thecommon approximate solution by Wangsness 8predicts that the frequency will go to 0 at the magnetization compensation pointT M, while in the experiment its value remains finite, not even with a minimum. For the experimentally observed effectivedamping parameter the disagreement with earlier theories iseven more pronounced. Unlike the theoretical predictions inthe experiment the effective damping is observed to increasesignificantly, approaching the Curie temperature T C. In this work, we present a more general analytical so- lution based on the Landau-Lifshitz-Bloch15(LLB) equation of motion for the temperature dependence of the frequencyand effective damping parameters of both modes and comparethem with our numerical findings from atomistic spin-modelsimulations. We show that the assumption of a temperature-independent sublattice damping parameter, confirmed experi-mentally for a wide range of temperatures 16far below TC, does not hold in the high-temperature regime close to TC, and we present the derivation of new temperature-dependent dampingparameters. Additionally, we recall the invalidity of somecommon approximate solutions at the compensation pointsand show the influence of the strength of a magnetocrystallineanisotropy on the properties of the resonance modes, enablingan understanding of the experimental findings. II. NUMERICAL METHODS A. Model Our numerical results are based on a spin model where we consider classical spins Sν,κ=μν,κ/μν,κon two different 214416-1 1098-0121/2012/86(21)/214416(7) ©2012 American Physical SocietyF. SCHLICKEISER et al. PHYSICAL REVIEW B 86, 214416 (2012) (a) MT MRH0(b) MT MRH0 FIG. 1. (Color online) Schematic of the two resonance modes in ferrimagnets. (a) For the ferromagnetic mode, the sublattices remain antiparallel; (b) for the exchange mode, the sublattices are tilted at acharacteristic angle. sublattices. Here, ν,κ=T,R with κ/negationslash=νrepresents either the rare-earth-metal (R) or the transition-metal (T) sublattice, andμ νis the atomic magnetic moment with μR/μT=2. The position of the spins, which are localized regularly on thetwo intertwined sublattices of a simple cubic lattice, is chosensuch that nearest neighbors (nn’s) always belong to the othersublattice respectively. The contribution to the Hamiltonianfrom one single spin is H i ν=−1 2/summationdisplay j∈nnnJνSi νSjν−1 2/summationdisplay j∈nnJνκSi νSjκ −dz ν/parenleftbig Sz,i ν/parenrightbig2−μνH0Si ν. (1) Here, the first sum represents the ferromagnetic coupling between spins in the same sublattice [next-nearest-neighbor(nnn) interaction], while the second sum represents theantiferromagnetic interaction between spins in different sub-lattices (nn). Besides the exchange interaction with reducedvalues J R/JT=0.2 and JRT/JT=− 0.1, we consider also a magnetocrystalline anisotropy in the zdirection with the anisotropy constant dz ν(which is varied) as well as the Zeeman energy from an external magnetic field H0. Our numerical results for N=323spins are generated by solving the stochastic Landau-Lifshitz17equation via Heun’s method. The equation itself and the method used are describedin detail in Ref. 18. For the gyromagnetic ratios we use γ R/γT=0.75, and as the microscopic damping constant, describing the coupling of the spin system to the heat bath,we use λ=0.01 for the nonanisotropic case and λ=0.001 for finite anisotropy. The heat bath is provided by the electronicdegrees of freedom as well as by the lattice and it defines thetemperature of our simulations in the canonical ensemble. B. Equilibrium magnetizations and transverse relaxation The sublattice equilibrium magnetizations are calculated as the spatial and time average of the (easy axis) zcomponent of the magnetic moments, Me ν=μν Nνa3·/angbracketleftBiggNν/summationdisplay i=1Sz,i ν/angbracketrightBigg , (2) withν=R, T, and Nνdefining the number of unit cells with volume a3in the system. The temperature dependencefit-functionstotalrare-earthtransition-metal temperature kBT/JTmagnetization Me(T)/MT(0) 4 3.5TC3 2.52 1.5TATM 0.501 0.5 0 -0.5 -1 -1.5 -2 FIG. 2. (Color online) Temperature dependence of the equi- librium magnetization of the sublattices. At the magnetizationcompensation point, T M, the sublattice magnetizations cancel each other, while at the angular momentum compensation point, TA,t h e angular momenta of both sublattices are equal. Above the criticalpointT Cthe system is paramagnetic. of the resulting sublattice and total magnetizations, basically determined by the respective exchange constants, are shownin Fig. 2. We note that, due to their coupling, the two sublattices have the same critical temperature. In additionto this Curie temperature, ferrimagnets may have two othercharacteristic temperatures relevant to their magnetic behavior.At the magnetization compensation point T Mthe sublattice magnetizations cancel each other, so that the total magneti-zation M e total=Me R+Me Tis 0. This point can exist in ferri- magnets where the sublattice with the larger zero-temperaturemagnetization has a weaker ferromagnetic coupling, so thatthe magnetization decays more rapidly compared to the othersublattice. Additionally, if both sublattices have different gyro-magnetic ratios there is an angular momentum compensationpointT Awhere the angular momenta of both sublattices are equal, Me T/γT=Me R/γR. The solid lines in Fig. 2correspond to fit functions, which are important when numerical simulationsare compared with analytical predictions in the followinganalysis. They are obtained via a polynomial fitting procedurewhich includes the mean-field critical behavior close to theCurie temperature. In our simulations the excitation of either the FMM or the EXM is done separately, by first loading a multispin con-figuration from equilibrium calculations for that temperature(Fig. 2) and then tilting the spin system with respect to an external magnetic field H 0. While the FMM is excited by tilting the total system by 30◦with respect to the external magnetic field, for the EXM the angles between each sublatticeand the external magnetic field are varied separately, sincethe characteristic angle between the sublattices (Fig. 1)i s temperature dependent. 11In this work the external magnetic field is always parallel to the zaxis. Therefore the time development of the xandycomponents of the magnetization for both modes follows a damped precession motion with Mx,y(t)∝exp(−bt)·cos(ωt+φ), (3) where brepresents the damping rate, ωis the frequency, and φcorresponds to a phase shift. The different frequencies and 214416-2TEMPERATURE DEPENDENCE OF THE FREQUENCIES AND ... PHYSICAL REVIEW B 86, 214416 (2012) damping rates for either the FMM or the EXM have been obtained by fitting directly to the resulting time developmentof thexandycomponents of the magnetization. Alternatively, we tried to obtain the parameters above from a Fouriertransformation of the time-dependent magnetization data.However, these results turned out to be less accurate, probablydue to the fact that our simulations are very time-consuming,and consequently, the number of oscillations is not sufficientfor an analysis via Fourier transformation. III. PROPERTIES OF THE NORMAL MODES A. Transverse relaxation within the Landau-Lifshitz (LL) equation Earlier analytical calculations of the normal modes have been based on two coupled nonthermal equations of motionfor the macroscopic magnetizations of sublattices using certainapproximations. 8,10–12Here we want to go beyond these restrictions, first avoiding approximations and in the nextsection including thermal effects via the LLB equation. Wewill see that the use of the LLB equation will only affect thetemperature dependence of the damping. Considering, in a two-sublattice micromagnetic LL equa- tion, only intersublattice exchange, the Zeeman energy, andthe magnetorystalline anisotropy, the effective fields of bothsublattices are given by H eff ν=H0+Hex ν+Han ν. With the magnetic field and the magnetocrystalline anisotropy parallelto the zaxis, the effective field contributions become H 0= H0ez,Hex ν=−AMκ, and Han ν=± 2Dz νMe,z νez. Here, Arep- resents the interlattice micromagnetic exchange stiffness andD z νis the micromagnetic anisotropy constant. By comparing these expressions with the corresponding effective fields Hi ν= −1/μν·∂Hi ν/∂Si νfor the spin model [Eq. (1)], we obtain the relations between atomistic and micromagnetic parametersA=ηJ RT/μTμRandDz ν=dz ν/μ2ν, withν,κ=T,R andκ/negationslash=ν as well as ηrepresenting the number of nn’s. In what follows we use unit vectors nν=Mν/Me ν. The equations of motion for the two sublattices read ˙nν γν=− (nν×H/prime ν)−αν[nν×(nν×H/prime ν)] +AMe κ{(nν×nκ)+ανAMe κ[nν×(nν×nκ)]},(4) where Me κrepresents the equilibrium magnetization of the re- spective other sublattice, H/prime ν=H0+Han ν,γνare the atomistic gyromagnetic ratios, and ανare the damping constants. Close to equilibrium, with nT=nT(nx T,ny T,1) and nR= nR(nx R,ny R,−1), we can consider ∂tnz T(R)=0,Mz ν≈Me,z ν, and neglect second-order terms, leading to ˙nx ν γν=/parenleftbig −ny ν∓ανnx ν/parenrightbig/parenleftbig H0±2Dz νMe ν/parenrightbig −AMe κ/parenleftbig αν/parenleftbig nx ν+nx κ/parenrightbig ∓/parenleftbig ny ν+ny κ/parenrightbig/parenrightbig (5) and ˙ny ν γν=/parenleftbig nx ν∓ανny ν/parenrightbig/parenleftbig H0±2Dz νMe ν/parenrightbig ±AMe κ/parenleftbig (nx ν+nx κ/parenrightbig −αν/parenleftbig ny ν+ny κ/parenrightbig/parenrightbig . (6) Here, the upper algebraic sign is for the transition metal, while the lower one is for the rare-earth metal. By transforminginto the variables of the rotating system n+ ν=nx ν+iny νand n− ν−=nx ν−iny νand assuming an exponential solution n± ν= n0± νexp(i˜ωt), we obtain /parenleftbig ±˜ω−γT/parenleftbig H0+2Dz TMe T+AMe R/parenrightbig/parenleftbig 1±iαT/parenrightbig/parenrightbig n± T −γTAMe R(1±iαT)n± R=0, (7) /parenleftbig ±˜ω−γR/parenleftbig H0−2Dz RMe R−AMe T/parenrightbig (1∓iαR)/parenrightbig n± R +γRAMe T(1∓iαR)n± T=0. The solution for the frequencies corresponds to the real part of the two independent solutions for the FMM and EXM,respectively, and the damping rate is given by the imaginarypart. The effective damping parameter is then given by the ratioof damping rate to frequency, 11αeff=bfm,ex/ωfm,ex.D u et o their length, we do not write down these equations, but we willuse the full solution later for comparison with numerical dataand an improved analytical approach. Note, however, that theeffective damping is the same for both modes if the sublatticedamping parameters α νare assumed to be equal. Based on this approach several approximated solutions for the frequencies and effective damping parameter have beenderived in the past. However, the most common solutions forthe frequencies, by Wangsness 10for the FMM, ωFMM=γTγR/parenleftbig Me T−Me R/parenrightbig /parenleftbig γRMe T−γTMe R/parenrightbigH0, (8) and by Kaplan and Kittel9for the EXM, ωEXM=A/parenleftbig γTMe R−γRMe T/parenrightbig , (9) make use of two main approximations: first, they neglect the influence of damping and anisotropy completely; andsecond, they include the assumptions AM e ν/greatermuchH0, which fails close to the Curie temperature TC;A(Me T−Me R)/greatermuch H0, which fails close to the magnetization compensation point TM; and A(γTMe R−γRMe T)/greatermuchH0, which fails close to the angular momentum compensation point. Thus, theseapproximations predict an erroneous behavior at and closeto these characteristic temperatures. Similar approximationsin calculations of the effective damping parameter 11and the solution for the frequency of the finite-anisotropy case byWalker 19fail here correspondingly. Note that also the solution of the effective damping parameter,11 αeff=Me RγTαR+γRαTMe T Me RγT−Me TγR, (10) predicts a divergence at TAand therefore zero switching time. As we will show in the following, in the full analytical solutionneither the frequencies nor the effective damping parametersdiverge at T A. Instead, we find only characteristic maxima at or close to the angular momentum compensation point. B. Temperature-dependent transverse relaxation within the LLB theory for ferrimagnets The recently published derivation of the LLB equation for two-component systems in Ref. 21explicitly refers to a disordered ferrimagnet. Since for this work we consideran ordered ferrimagnet, we briefly repeat the derivation and 214416-3F. SCHLICKEISER et al. PHYSICAL REVIEW B 86, 214416 (2012) present the formula for the ordered case in the explicit form. In the following we derive the macroscopic equationfor the thermally averaged spin polarization m ν=/angbracketleftSν i/angbracketrightin each sublattice ν=T,R, following the theory of the LLB equation for ferromagnets.15The derivation uses a mean-field approximation (MFA). Since in the present article we are notinterested in longitudinal motion, observed on the time scale of100 fs to 1 ps, we focus our attention on the LLB equation withtransverse motion only. Additionally, the longitudinal normalmodes are decoupled from the transverse ones, which allowsfor their separate consideration. Such an approximation leadsto the following sets of coupled LLB equations: 15 ˙mν=γν/bracketleftbig mν×HMFA ν/bracketrightbig −/Gamma1ν ⊥[mν×[mν×mν,0]] m2ν(11) with mν,0=B(ξν,0)ξν,0 ξν,0,ξν,0≡μνHMFA ν kBT, (12) where HMFA ν is the average mean field acting on the spin, and the relaxation rates are given by /Gamma1ν ⊥=γνλνkBT μν/parenleftbiggξν,0 B(ξν,0)−1/parenrightbigg , (13) where B(ξ)=coth (ξ)−1/ξis the Langevin function. In Eq.(11) the first term describes the magnetization precession and the second term the transverse relaxation. The next step isto use the MFA in Eqs. (12). The MFA expression for fields in a ferrimagnet are well known; see also recent results forFeCoGd. 20Defining H/prime eff,T(R)=H+HA,T(R)as the sum of the external and anisotropy fields in each sublattice, we can writethe average molecular field acting at each sublattice spin as μ RHMFA R=μRH/prime eff,R+J0,RmR+J0,TRmT, (14) μTHMFA T=μTH/prime eff,T+J0,TmT+J0,TRmR, (15) where J0,T=ηJT,ηis the number of nn’s of transition-metal type for the transition-metal spin, and J0,TRandJ0,Rhave similar definitions. The minimum condition for the free energy,∂F/∂m R=0 and ∂F/∂mT=0, leads to the coupled Curie- Weiss equations, mR=B(ξR,0)ξR,0 ξR,0,mT,0=B(ξT,0)ξT,0 ξT,0, (16) the self-consistent solutions of which are the equilibrium magnetization of each sublattice. We treat the most general case where the continuous approximation in each sublattice can be used. In orderto simplify the problem we decompose the magnetizationvector m νinto two components, mν=/Pi1ν+τν, where /Pi1ν is perpendicular to mκ, so that it can be expressed as /Pi1ν= −[mκ×[mκ×mν]]/m2 κ, andτνis parallel to mκ, so that it can be expressed as τν=mκ(mν·mκ)/m2 κ, where κ/negationslash=ν. Similarly, the MFA exchange field HMFA EX,νin Eqs. (15) and (14) can be written as the sum of the exchange field parallel and perpendicular to magnetization of the sublattice ν, HMFA EX,ν=H/bardbl EX,ν+H⊥ EX,ν=/tildewideJ0,ν μνmν+J0,νκ μν/Pi1κ,where we have defined a new function, /tildewideJ0,ν(mκ,mν),a s/tildewideJ0,ν= J0,ν+J0,νκ(mν·mκ)/m2 κ. Note that /tildewideJ0,νis not a constant but a function of both sublattices’ magnetizations. In the following, we consider the case where the transverse contribution in the exchange field is small in comparison to the longitudinal one, |H/bardbl EX,ν|/greatermuch| H⊥ EX,ν|, i.e., where the non- collinearities between sublattices are small. Finally, HMFA ν/similarequal H/bardbl EX,ν+H/prime/prime eff,ν, where H/prime/prime eff,ν=H+HA,eff,ν+H⊥ EX,ν.W en o w expand mν,0up to the first order in H/prime/prime eff,ν, under the assumption |H/bardbl EX,ν|/greatermuch| H/prime/prime eff,ν|.F r o mE q s . (16) the value of mν,0in the above conditions21can be substituted into Eq. (11), leading to the following equation of motion: ˙mν=γν[mν×H/prime/prime eff,ν]−/Gamma1ν ⊥Bνμν mν˜J0,ν[mν×[mν×H/prime/prime eff,ν]] m2ν. (17) In the same approximation we have Bν/similarequalmν,B(ξ0,ν)/ξ0,ν/similarequal (kBT)/˜J0,ν, and, finally, /Gamma1ν ⊥=γναν ⊥kBT μν/parenleftbiggξ0,ν B(ξ0,ν)−1/parenrightbigg /similarequalγναν ⊥/parenleftbigg˜J0,νmν μνBν/parenrightbigg ,(18) where αν ⊥=λν(1−kBT /tildewideJ0,ν). Hence the final form of the LLB equation is ˙mν=γν[mν×H/prime/prime eff,ν]−γναν ⊥[mν×[mν×H/prime/prime eff,ν]] m2ν.(19) The temperature dependence of the damping parameters is obtained in the first order in deviations of magnetizationfrom their equilibrium value. Note that in Eq. (17) all the terms are of the first order in the parameter H /prime/prime eff,ν/H|| EX,νso that the damping parameters should be evaluated in the zeroorder in this parameter. As a result, the effective dampingparameter depends on the temperature Tvia the equilibrium magnetization values as /tildewideJ 0,ν/similarequalJ0,νme ν−J0,νκme κ meν. (20) Note also that the field H/prime/prime effcould be substituted in the pre- cession and the transverse damping terms with Heff(including the exchange field coming from the opposite sublattice), sincethe action of the component of this field parallel to themagnetization m νis 0. Note also that Eq. (19) does not have exactly the LL form due to the presence of the m2 νterm in the denominator. The difference between the LL and the LLBdamping is discussed for the ferromagnetic case in Refs. 22 and23. For a comparision with the results in Sec. III A ,E q . (19) can be written in terms of the variable n=m/m e. After renor- malizing the equation and linearizing it close to equilibriumat a given temperature, one gets a similar result as for the LLequation [Eqs. (5)and (6)] but with temperature-dependent parameters, ˜α T ⊥(T)=αT me T(T)/parenleftbigg 1−me T(T)kBT J0,Tme T(T)−J0,TRme R(T)/parenrightbigg ,(21) 214416-4TEMPERATURE DEPENDENCE OF THE FREQUENCIES AND ... PHYSICAL REVIEW B 86, 214416 (2012) and ˜αR ⊥(T)=αR me R(T)/parenleftbigg 1−me R(T)kBT J0,Rme R(T)−J0,RTme T(T)/parenrightbigg , (22) where the parameters λR,λTwere substituted by αR,αTto comply with the standard notations of the micromagneticequation. Here, similarly to the procedure described in Ref. 20, we have renormalized the exchange parameters within theMFA. The replacement of Eqs. (21) and(22) in Eq. (7)leads to an increase in the effective damping parameters for bothmodes at high temperatures, which agrees with the numericalfindings. It is that combination of equations that we call theanalytical solution in the following. IV . RESULTS AND DISCUSSION Let us start with a discussion of the zero-anisotropy case. Since in our simulations an external magnetic field H0= 0.02JT/μTis constantly switched on, for zero anisotropy this magnetic field will have to change its sign at the magnetizationcompensation point T Min order to avoid a switching of the whole system. This change leads to the discontinuity of theanalytical solutions [Eq. (7)] shown in Fig. 3atT M.F o r the frequencies [Fig. 3(a)] as well as the effective damping parameters [Fig. 3(b)] we obtain a very good agreement between analytical and numerical solutions in both modes andfor the whole temperature range. We note that the value of thefrequency of the FMM first tends to 0 below the magnetizationcompensation point T M, where it starts to increase to its exchange (EXM)ferromagnetic (FMM)frequency ωμT/JTγT0.3(a) (b)0.25 0.2 0.15 0.1 0.05 0 αν ⊥=constEXMFMM temperature kBT/JTeffective damping αeff TC3 2.52 1.5TATM 0.500.15 0.1 0.05 0 FIG. 3. (Color online) Frequencies and effective damping param- eters in the zero-anisotropy case. Temperature dependence of (a) frequencies and (b) effective damping parameters αeff. Numerically obtained data points are compared with analytical solutions. The switching of the external magnetic field H0l e a d st oag a pi nt h e solutions at the magnetization compensation point TM.maximum above the angular momentum compensation point TA. After decreasing with higher temperatures the value of the frequency of the FMM converges to a constant level.For the EXM the effect of changing the relative directionof the external field is stronger, since, in comparison tothe approximated solution 9[Eq. (9)], the value of the EXM frequency is constantly shifted proportionally to the strengthofH 0. Above the angular momentum compensation point TA the frequencies of both modes reach the same value, where the FMM has its maximum and the EXM reaches a localminimum. Note that there is an increase in the effective damping parameter at high temperatures that is much stronger forthe EXM. Interestingly, without considering the temperaturedependence of the sublattice damping parameters [Eqs. (21) and (22)] in the analytical solution [Eq. (7)] and assuming simply the microscopic damping constant λ=0.01 to describe the relaxation dynamics of the sublattice magnetizations, theeffective damping parameters α efffor both modes are equal. This solution, plotted as the dashed line in Fig. 3(b), does not coincide with our numerical data. Only considering thetemperature dependence of the sublattice damping parameters,the effective damping parameters α effof both modes become different and describe the increase for both modes at hightemperatures correctly [Fig. 3(b)]. Note also that the influence of the temperature dependence for ˜ α T ⊥(T) and ˜ αR ⊥(T)i s negligible at low temperatures but becomes very importantwith increasing temperatures. For the finite-anisotropy case (Fig. 4)w eh a v eu s e dt h e following values as atomistic damping parameters, external (a) EXMdz=0.01FMM dz=0.01frequency ωμT/JTγT0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 (b) EXMdz=0.01FMM dz=0.01αν ⊥=const temperature kBT/JTeffective damping αeff TC3 2.52 1.5TATM 0.500.014 0.012 0.01 0.008 0.006 0.004 0.002 FIG. 4. (Color online) Frequencies and effective damping param- eters in the finite-anisotropy case. Temperature dependence of (a) frequencies and (b) effective damping parameters αeff. Numerically obtained data points are compared with analytical solutions. 214416-5F. SCHLICKEISER et al. PHYSICAL REVIEW B 86, 214416 (2012) field, and anisotropy constant: λ=0.001,H0=0.01,JT/μT, anddz=0.01JT. Here, we have used smaller atomistic damping compared to the previous simulations, since theconsideration of an anisotropy leads to an increase in thediscrepancy between simulated and ideal damped modes, sothat more cycles had to be fitted in order to obtain goodresults. With anisotropy, consequently, the resulting anisotropyfield compensates the external field and avoids switching atthe magnetization compensation point T Min our simulations. Therefore we have not switched the direction of H0in this case. For higher temperatures, however, due to thermalexcitation as well as the decaying anisotropy field, the systemstarts switching anyway. Therefore the effective dampingparameters could not be obtained for the high-temperaturerange. For the frequencies [Fig. 4(a)] the consideration of a uniaxial anisotropy leads to the fact that the minimum at themagnetization compensation point vanishes. Again, above theangular momentum compensation temperature T Awe obtain a maximum for the FMM frequency and a minimum for theEXM frequency. We note that the shift of this characteristicpoint from T Ato higher values is proportional to the strength of the external magnetic field as well as the anisotropy. Regarding the effective damping parameters [Fig. 4(b)]a finite anisotropy leads to a less pronounced maximum at TA. Once again, the dashed line in Fig. 4(b) corresponds to the analytical solution without consideration of the temperaturedependence of the sublattice damping, leading to the equalityof the effective damping parameters of both modes, notshowing the increase in α efffor higher temperatures. Besides the good agreement between numerical and analytical resultswhen the temperature dependence of the sublattice dampingis taken into account, we now also obtain a good agreementwith the experimental findings of Stanciu et al. 13Thus we are able to reproduce these experimental findings qualitatively byconsidering a uniaxial, magnetocrystalline anisotropy as wellas temperature-dependent sublattice damping parameters asderived within the framework of the LLB equation. Thesecoinciding findings clearly demonstrate the failure of theanalytical solutions based on the LL and LLG equations ofmotion 8,10–12for high temperatures. In Fig. 5the analytical solutions [Eqs. (7),(21), and (22)]o f the frequencies of both modes as well as the effective dampingparameter for the FMM are shown for different strengthsof the uniaxial anisotropy. Here, with H 0=0.01JT/μTand λ=0.01, we have also not switched the external field at TM. First, we note that due to the temperature dependence of the anisotropy field in the high-temperature regime, theinfluence of the strength of the anisotropy constant d zbecomes smaller with increasing temperatures. This effect leads to theconvergence of all sets of curves for different anisotropieswith increasing temperatures up to T C, where the anisotropy fields vanish and the different curves join. Second, we seethat the frequencies of both modes increase with increasinganisotropy. This effect is much stronger for the FMM.Additionally, the maximum of the frequency of the FMMas well as the minimum of the frequency of the EXM areshifted from the the angular momentum compensation point T A towards higher temperatures with increasing anisotropy. For the effective damping parameter, with increasing anisotropy(a) dz=0.04dz=0.02dz=0.01dz=0.005frequency(FMM) ωμT/JTγT0.12 0.1 0.08 0.06 0.040.02 0 (b) dz=0.04dz=0.02dz=0.01dz=0.005frequency(EXM) ωμT/JTγT 0.3 0.25 0.2 0.15 0.1 0.05 0 (c) dz=0.04dz=0.02dz=0.01dz=0.005 temperature kBT/JTeffective damping(FMM) αeff TC3 2.52 1.5TATM 0.500.1 0.08 0.06 0.04 0.02 FIG. 5. (Color online) Frequencies and effective damping param- eters in the finite-anisotropy case. Temperature dependence of (a) theferromagnetic mode frequency, (b) the exchange mode frequency, and (c) the effective damping parameter α effof the FMM for different strengths of the magnetocrystalline anisotropy. Analytical results asexplained in the text. we obtain a decrease and a washing-out of the maximum close toTA. V . CONCLUSIONS A detailed investigation of the dynamics of ferrimagnets was performed by means of computer simulations as wellas analytically. Formulas were derived for the frequenciesand effective damping parameters of bot, the FMM and theEXM. We show that a correct calculation does not predict anydivergence either of the effective damping parameters or ofthe frequencies close to the angular momentum compensationpoint, but only a finite maximum. Nevertheless, both thefrequencies and the effective damping parameters stronglydepend on the temperature, with that explaining the largevariations of relaxation times in ferrimagnets, especially inoptomagnetic experiments with pronounced heating effects. Similarly to the experimental results (see Fig. 3 in Ref. 13) and unlike predictions based on the macroscopic, 214416-6TEMPERATURE DEPENDENCE OF THE FREQUENCIES AND ... PHYSICAL REVIEW B 86, 214416 (2012) two-sublattice LLG8,10–12equation, we find an increase in the effective damping at a temperature approaching theCurie temperature. This stresses the importance and validityof the recently derived two-sublattice LLB equation forfinite-temperature micromagnetics. The latter builds a newbasis for finite-temperature micromagnetic calculations offerrimagnets.ACKNOWLEDGMENTS This research received funding from the European Commis- sion via the 7th Framework Programme grant FEMTOSPIN.The authors in Madrid also acknowledge funding by theSpanish Ministry of Science and Innovation under Grant No.FIS2010-20979-C02-02. *Correspondence author: frank.schlickeiser@uni-konstanz.de 1C. D. Stanciu, F. Hansteen, A. V . Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and Th. Rasing, P h y s .R e v .L e t t . 99, 047601 (2007). 2A. V . Kimel, A. Kirilyuk, P. A. Usachev, R. V . Pisarev, A. M.Balbashov, and Th. Rasing, Nature 435, 655 (2005). 3K. Vahaplar, A. M. Kalashnikova, A. V . Kimel, D. Hinzke, U. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, andTh. Rasing, Phys. Rev. Lett. 103, 117201 (2009). 4I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. D¨u r r ,T .A .O s t l e r ,J .B a r k e r ,R .F .L .E v a n s ,R .W .C h a n t r e l l , A. Tsukamoto, A. Itoh, A. Kirilyuk, Th. Rasing, and A. V . Kimel,Nature 472, 205 (2011). 5T .A .O s t l e r ,J .B a r k e r ,R .F .L .E v a n s ,R .W .C h a n t r e l l ,U .A t x i t i a , O. Chubykalo-Fesenko, S. El Moussaoui, L. Le Guyader,E. Mengotti, L. J. Heyderman, F. Nolting, A. Tsukamoto, A. Itoh,D. Afanasiev, B. A. Ivanov, A. M. Kalashnikova, K. Vahaplar,J. Mentink, A. Kirilyuk, Th. Rasing, and A. V . Kimel, Nature Commun. 3, 666 (2012). 6S. Wienholdt, D. Hinzke, and U. Nowak, Phys. Rev. Lett. 108, 247207 (2012). 7S. Alebrand, A. Hassdenteufel, D. Steil, M. Bader, A. Fischer,M. Cinchetti, and M. Aeschlimann, Phys. Status Solidi A 209, 2589 (2012). 8R. Wangsness, Phys. Rev. 91, 1085 (1953). 9J. Kaplan, and C. Kittel, J. Chem. Phys. 21, 760 (1953). 10R. Wangsness, Phys. Rev. 93, 68 (1954).11A. G. Gurievich and G. A. Melkov, Magnetisation Oscillations and Waves (CRC Press, Boca Raton, FL, 1965). 12B. Lax and K. J. Button, Microwave Ferrites and Ferrimagnets (McGraw–Hill, New York, 1962). 13C. D. Stanciu, A. V . Kimel, F. Hansteen, A. Tsukamoto, A. Itoh,A. Kiriliyuk, and Th. Rasing, P h y s .R e v .B 73, 220402 (2006). 14M. Binder, A. Weber, O. Mosendz, G. Woltersdorf, M. Izquierdo, I. Neudecker, J. R. Dahn, T. D. Hatchard, J.-U. Thiele, C. H. Back,and M. R. Scheinfein, P h y s .R e v .B 74, 134404 (2006). 15D. A. Garanin, Phys. Rev. B 55, 3050 (1997). 16S. M. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974). 17L. Landau and E. Lifshitz, Phys. Z. Sowjet. 8, 153 (1935). 18U. Nowak, Handbook of Magnetism and Advanced Magnetic Materials , edited by H. Kronm ¨uller and S. Parkin (John Wiley & Sons, Chichester, UK, 2007), V ol. 2. 19N. Geschwind and L. R. Walker, J. Appl. Phys. 30, 163S (1959). 20T. A. Ostler, R. F. L. Evans, R. W. Chantrell, U. Atxitia,O. Chubykalo-Fesenko, I. Radu, R. Abrudan, F. 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PhysRevLett.126.026601.pdf

Spin Fluctuations in Quantized Transport of Magnetic Topological Insulators Yu-Hang Li1,*and Ran Cheng1,2,† 1Department of Electrical and Computer Engineering, University of California, Riverside, California 92521, USA 2Department of Physics, University of California, Riverside, California 92521, USA (Received 1 October 2020; accepted 23 December 2020; published 13 January 2021) In magnetic topological insulators, quantized electronic transport is intertwined with spontaneous magnetic ordering, as magnetization controls band gaps, hence band topology, through the exchangeinteraction. We show that considering the exchange gaps at the mean-field level is inadequate to predict phase transitions between electronic states of distinct topology. Thermal spin fluctuations disturbing the magnetization can act as frozen disorders that strongly scatter electrons, reducing the onset temperature ofquantized transport appreciably even in the absence of structural impurities. This effect, which has hitherto been overlooked, provides an alternative explanation of recent experiments on magnetic topological insulators. DOI: 10.1103/PhysRevLett.126.026601 The inquiry into topological materials has recently mingled with the quest for low-dimensional magnets,giving birth to an emerging frontier known as magnetic topological insulators (TIs) where a topologically nontrivial band gap is controllable by spontaneous magnetic ordering[1–4]. Therefore, manipulating magnetization becomes a new tuning knob of the quantized electronic transport. Forexample, in a TI with coexisting ferromagnetic order, thesystem should exhibit the quantum anomalous Hall (QAH)effect when a finite magnetization is established below theCurie temperature ( T c)[5]. However, the QAH effect was first realized in a magnetically doped TI in which themagnetic moments are embedded randomly [6], leading to strong disorder effects that significantly reduce the electron mobility and hence inhibit the appearance of quantizedtransport [7–10]. As a result, the actual onset temperature of the QAH effect in such a material is much lower than themagnetic ordering temperature. Removing this roadblock calls for magnetic TIs in which the magnetic moments are arranged periodically on alattice. This can be achieved in either an intrinsic magnetic TI[11–14]or a heterostructure with a TI sandwiched between two magnetic thin films [15,16] . However, the quantized transports in these systems turned out to be asvulnerable to an increasing temperature as those studied inmagnetic doped TIs [17]. While this discouraging obser- vation might still be attributed to structural impurities, itremains an open question what is responsible for thedisappearance of QAH effect at a temperature far below T c. In this Letter, we introduce an alternative mechanism in magnetic TIs that can substantially reduce the onset tempera-ture of quantized transport even in the absence of structuralimpurities. Contrary to the electrons governed by anformidably high Fermi temperature, spin fluctuations (SF)disturbing the magnetic order are very susceptible to thermalagitations [18]. Because spin fluctuations take place on a timescale that is orders of magnitude larger than the electronrelaxation time [19], the electron dynamics can adjust adiabatically to the instantaneous configuration of magnetic moments, seeing the instantaneous spin fluctuations as arandom potential almost frozen in time. For this reason, thermal spin fluctuations in the magnetic degree of freedom can manifest as effective disorders affecting the electrontransport, even though magnetic atoms are arranged per- fectly on a lattice free of structural impurities. As schematically illustrated in Fig. 1(a), we model the system as a magnetic trilayer where topological electrons FMFM TI FMFM(a) (b)(c) FIG. 1. (a) Schematic of a magnetic TI in the presence of spin fluctuations. (b) The mean field scaled by Ms≡hMðT→0Þifor B→0and the susceptibility χas functions of temperature. (c) Probabilities of different Szon an individual spin versus temperature for S¼5=2.PHYSICAL REVIEW LETTERS 126, 026601 (2021) 0031-9007 =21=126(2) =026601(5) 026601-1 © 2021 American Physical Societyare confined between two magnets, which applies to not only a heterostructure but also an intrinsic magnetic TI with uniform magnetic ordering [20]. To ensure the relative orientation of the two magnetic layers, we include an auxiliary magnetic field Balong zaxis to stabilize the system, but the B→0limit will be taken at the end. Now let us quantify the magnetization dressed with spin fluc-tuations in an individual magnetic layer, which is supposed to be independent of all other layers as schematically illustrated in Fig. 1. Based on recent experiments [12–14,16] , the magnet under consideration can be cap- tured by the minimal Hamiltonian H M¼−JX hijiSi·Sj−κX iS2 i;z−gμBBX iSi;z;ð1Þ where J>0is the (intralayer) Heisenberg exchange coupling, κis the uniaxial anisotropy, gis the Land´ e factor, μBis the Bohr magneton, and hijienumerates all nearest neighbors. The spin vector Siis dimensionless. In the mean-field approximation [18], spins become effectively decoupled while the exchange interaction that entangles different spins recasts as an effective mean field hMi¼JhP iSi;ziT=ðgμBNÞwhere Nis the total number of spins and h/C1 /C1 /C1iTdenotes the thermal average. Consequently, the system becomes a paramagnet interact- ing with a total magnetic field Btot¼BþhMias if there is no exchange interaction. In the limit J≫κ, the effective Zeeman energy is E¼−gμBðBþ<M> ÞP iSiz, from which the mean field hMican be solved self-consistently [18]. Figure 1(b) shows the mean field hMiand the susceptibility χ≡limB→0½hMðBÞi−hMð0Þi/C138=Bfor S¼5=2as a function of temperature scaled by the Curie temperature Tc¼aJS ðSþ1Þ=3kBon a simple square lattice with the coordination number a¼4.A s every spin is now isolated from all other spins, theprobability of an individual spin S itaking Szperpendicular to the plane is determined straightforwardly by the Boltzmann distribution PðSzÞ¼expð−ε=kBTÞ=Zwhere ε¼ −gμBSzðBþhMiÞand the partition function Z¼ sinh ½ð2Sþ1Þy/C138=sinhywithy¼a J<M>= 2T. As plot- ted in Fig. 1(c), the spin is fully polarized to Sz¼Sat T¼0, whereas when T→Tcall possible quantized values ofSztend to be equally probable, destroying the magneti- zation completely at Tc. The mean-field approach enables us to determine the projection of a given spin Sionzdirection probabilistically. With the spherical parametrization Si¼Sðsinθicosϕi; sinθisinϕi;cosθiÞ, it amounts to determining θiprobabil- istically. The azimuthal angle ϕi, on the other hand, cannot be captured by the mean-field picture. Because we only consider the incoherent thermal spin fluctuations, ϕi should be uniformly distributed within the range ½0;2πÞ. Moreover, because different modes of spin excitation superimpose with completely random phases, ϕishouldbe independent of its neighbors. In other words, the variable ϕis spatially uncorrelated, or hϕiðtÞϕjðtÞi∼δijat any instant of time. In contrast, the temporal correlation of ϕis much larger than the electron relaxation time. Specifically, hϕiðtÞϕiðt0Þi∼e−jt−t0j=τs, where the characteristic decay time τsmay depend on the mode of excitation, but a qualitative estimation is that τs∼1=αωwhere αis the Gilbert damping and ωis the frequency of ferromagnetic resonance. So a typical value of τsis on the order of 10–100 ns. Comparatively, the electron relaxation time τe determined by the Fermi energy is on the order of 1 –10 fs, which is 7 orders of magnitude smaller than τs. A similar argument applies to the correlation of θas well. Therefore, while spin fluctuations are spatially uncorrelated, they exhibit extremely long temporal correlation, which amounts to a random potential frozen in time acting onthe electrons [21]. This justifies the adiabatic approxima- tion essential to our following discussions. Even though Dirac electrons and magnetic layers repeat periodically in an intrinsic magnetic TI, the system can be simplified as a trilayer heterostructure consisting of onlyone TI layer sandwiched between two magnetic layersas illustrated in Fig. 1(a) [20] . Under the basis ψ k¼ ðct k↑;ct k↓;cb k↑;cb k↓ÞTwith ctðbÞ kσannihilating an electron of momentum kand spin σon the top (bottom) surface, the magnetic TI can be described by the Hamiltonian HMTI ¼HTIþHex, where [5,22,23] HTI¼vFðkyτz⊗σx−kxτz⊗σyÞþmðkÞτx; ð2Þ Hex¼JexX iSi·σ: ð3Þ Here, vFis the Fermi velocity, Jexis the exchange coupling between the Dirac electrons and the magnetic moments, mðkÞ¼m0þm1k2describes the overlap of Dirac electrons in the top and bottom surfaces, and σandτare the vectors of Pauli matrices acting on the spin and layer degree of freedom, respectively. The lattice wave vectors kx;yare defined in the first Brillouin zone of a L×Wsquare lattice with the lattice constant a0≡1. Since the Fermi temperature TFis orders of magnitude larger than Tc, the electron dynamics is effect- ively in the zero temperature regime as we focus on T<T c [24,25] . Unless otherwise stated, we will take vF¼1 as the energy unit and assume m1¼1,kBTc¼0.002, Jex¼0.035,a n d S¼5=2. However, our theory is universal and not limited to these special parameters. To demonstrate the influence of spin fluctuations on the electron transport more clearly, it is instructive to first lookinto the homogeneous case without any spin fluctuations, in which S zis described by the mean field while SxandSyare completely ignored. In this situation, the lattice periodicityis restored in the exchange field, so we can transform the exchange Hamiltonian in Eq. (3)into the momentum space, andH MTIðkÞ¼HTIþλτ0⊗σz, where λ¼gμBJexhMiisPHYSICAL REVIEW LETTERS 126, 026601 (2021) 026601-2the homogeneous exchange field that depends on tempera- ture through the mean field hMi. Diagonalizing HMTIðkÞ gives the band dispersion and the corresponding eigen- states, based on which we can calculate the Chern numbers characterizing different topological phases. At low temper-atures, λ>m 0, the system is a QAH insulator with a Chern number C¼1. By contrast, the system becomes a normal insulator (NI) with C¼0when λ<m 0at high temper- atures. Setting λ¼m0solves the critical temperature Thm for the homogeneous case. Therefore, the system under- goes a topological phase transition at finite temperature below Tconly if m0is less than the maximum exchange field δ≡gμBJexMswithMs¼hMðT→0Þithe saturated mean field. In Fig. 2, the critical temperature Thmfor the homogeneous case is marked by the black arrows fordifferent ratios of m 0=δ. Next, we turn to the transport property in the presence of spin fluctuations, which, as discussed above, act on electrons as a frozen random potential. In the considered magnetic TI, the appearance of topological edge states canbe minimally revealed in a two-terminal junction, where the longitudinal conductance is σ¼e 2=h(σ¼0) in the QAH (NI) phase. We calculate σthrough the Landauer-Büttiker formula [26] σ¼Tr½ΓLGrΓRGa/C138, where Γβ¼i½Σr β−ðΣr βÞ†/C138 with β¼LorR, and Gr¼ðGaÞ†¼ðEF−HMTI− Σr L−Σr RÞ−1with EFthe Fermi energy and Σr βthe self- energy due to the coupling with metallic leads.To simulate the random potential, we generate a set of L×W¼200×200 random numbers representing Sz¼ Scosθon each lattice according to the probability distri- bution PðSzÞ¼expð−ε=kBTÞ=Zdetermined by the mean- field approach. We also assign each spin a random phase ϕ specifying its transverse component as discussed previ-ously. Then we calculate the conductance σunder this particular configuration of random potential. Repeating this procedure for 160 times, we obtain the ensemble average of σ, which is shown in Figs. 2(a)–2(c) as a function of temperature for different m 0. We see that σchanges from e2=hto 0 (i.e., transition from the QAH to NI phase) at a critical temperature TSFmanifestly below what it would be without spin fluctuations (i.e., Thmdetermined by solving λ¼m0), as indicated by the red arrows. The reduction of critical temperature appears to be more striking for largerm 0in Fig. 2.F o r m0¼0.8δ[Fig. 2(c)],σeven becomes ill quantized in the QAH phase due to the finite-size effect [27]. If the system is infinite, σwould be a step function across the critical point. Finite-size effects will be discussedin more detail later. The topological phase transition between the QAH insulator and the NI can be alternately characterized by the current noise SðωÞ¼ 1 2R dτeiωτhδˆIðtÞδˆIðtþτÞþ δˆIðtþτÞδˆIðtÞi, where δˆIðtÞ¼ ˆIðtÞ−hˆIðtÞiwith ˆIðtÞthe current operator [28,29] . Using the nonequilibrium Green ’s function [30], we calculate the zero-frequency current noise S0. Figures 2(d)–2(f)show the ensemble average of S0 corresponding to Figs. 2(a)–2(c). The noise S0peaks at the critical point and extends over a finite range of temperature due to finite-size effects; it will become infinitely sharp atthe critical point if the system is infinite. We see that σand S 0plotted in Fig. 2perfectly agree with the relation S0¼ 2e3Vσð1−σÞ=hwhere Vis the bias voltage across the junction, affirming that the QAH edge states can bedescribed by a one-channel ballistic tunneling model [29]. Without spin fluctuations, the mean field hMi, hence the exchange field λ, decreases as temperature is raised. When λbecomes comparable to m 0, the chiral edge states on opposite transverse edges start to overlap, merging into thebulk states [27]. This destroys the electron transport and diminishes the conductivity. Spin fluctuations as random potential, on the other hand, brings about scatteringof the chiral edge states, which facilitates their overlapping and merging into the bulk states, so the phase transition takes place at a reduced temperature. This subtle mecha-nism can be unraveled by studying the nonequilibrium current distribution inside the magnetic TI. Under a bias voltage Vacross the system, the local current flowing from site ito its neighbor jis given by J ne i→j¼ ImfTr½ˆtijðGrΓLGaÞji/C138g2e2V=h where ˆtijis the hoping matrix [31]. Figure 3shows the distributions of nonequilibrium currents in the TI at three representative temperatures for m0¼0.5δ[(a)–(c)] and m0¼0.8δ[(d)–(f)], respectively.(a) (d) (b) (e) (c) (f) FIG. 2. (a) –(c) Ensemble average of the two-terminal conduct- ance σas a function of temperature for different m0and fixed δ¼gμBJexMs(the maximum exchange field). (d) –(f) The corresponding zero-frequency current noise S0. The red arrows mark the critical temperature TSFobtained by the finite-size scaling shown in Fig. 4. The black arrows mark where m0¼λ, representing the critical temperature Thmin the absence of spin fluctuations. The system size is L¼W¼200and the error bars are magnified ten times for visual clarity.PHYSICAL REVIEW LETTERS 126, 026601 (2021) 026601-3AtT≪TSFandm¼0.5δ[Fig. 3(a)], the electron flow is fully confined to one edge, so the conductance is quantized —a hallmark of the QAH effect. For m¼0.8δ[Fig. 3(d)], however, the edge current becomes much wider so that it partially leaks into the opposite edge and flows backwards, leading to an ill-quantized conductance as shown in Fig. 2(c). At the true critical point T¼TSF[(b) and (e)] where λ>m 0, spin fluctuations strongly scatter the electrons from one edgeto the other, because of which electrons cannot propagate in one direction dictated by the applied bias voltage; they are instead back-scattered to the left lead. Accordingly, the chiraledge states become indistinguishable from the bulk states. AtT¼T hm[(c) and (f)] where λ¼m0, the edge states completely disappear and the conductance is identically zero.Integrating the current density over the full width Wyields a conductance that quantitatively agrees with the results shown in Fig. 2, confirming the validity of the nonequilibrium distribution. In Fig. 4, we draw a full phase diagram on the m 0−T plane. Because the specific profiles of σandS0depend on the system size, the actual critical temperature TSFcan be extracted by finite-size scaling. To this end, for a given setof variables, we calculate σas a function of Tfor three different system sizes and identify the intersection of the three curves as T SF(see the inset of Fig. 4). The criticaltemperature TSF(Thm) calculated in the presence (absence) of spin fluctuations is depicted by red dots (dashed lime curve). We see that both TSFandThmdecreases monoton- ically with an increasing ratio of m0=δ. However, the discrepancy ΔT¼Thm−TSF, which measures the reduc- tion of critical temperature due to spin fluctuations, reachesmaximum around m 0=δ¼0.75;ΔTvanishes for both m0=δ→0andm0=δ→1limits. Finally, we check the consistency of our conclusion by calculating the Hall conductance σxyusing the noncom- mutative Kubo formula with periodic boundary conditions,in which the Chern number is obtained directly from thereal space rather than a momentum-space integral [32,33] . For a system of L¼W¼50, we numerically calculate σ xy and superimpose the result in Fig. 4, where it exhibits a phase boundary that matches TSFremarkably well. We stress that the mechanism of spin fluctuations studied in this Letter is entirely different from the ordinarymagnon-electron scattering. First of all, we have consideredthe adiabatic regime such that spin fluctuations are frozenin time, whereas magnons are propagating spin waves. Second, spin fluctuations form a background random potential that scatters the electrons passively, whilereversely, the excitation of spin fluctuations by electronsis ignored. Third, the physical picture of spin fluctuationspersists up to T c, whereas magnons are well defined only at low temperatures. In addition, the mechanism here is intrinsic, different from structure impurities, which canbe removed by improving the material quality. To close our discussion, we further remark that if adjacent magnetic layers are antiferromagnetically(a) (b)(d) (e) (f) (c) FIG. 3. Nonequilibrium current distributions for m0¼0.5δ (a)–(c) and m0¼0.8δ(d)–(f) at T¼0.02Tc,T¼Ttsf, and T¼Thm. Red arrows indicate local current densities and direc- tions. FIG. 4. Phase diagram of the two-terminal conductance on them 0−Tplane. The inset illustrates how TSFis obtained from finite-size scaling. The red dots plot TSFand the red curve is a guide to the eye that marks the phase boundary in the presence ofspin fluctuations. The dashed lime curve marks T hm, which is the phase boundary in the absence of spin fluctuations. The back-ground color shows the Hall conductance calculated independ-ently for a system of L¼W¼50, which conforms with T SF.PHYSICAL REVIEW LETTERS 126, 026601 (2021) 026601-4directed, the Dirac electrons will form an axion insulator rather than a QAH insulator below Tc, which has been realized in MnBi 2Te4[13]. Unlike the QAH insulators, the topological behavior in an axion insulator does not mani- fest in transport properties; instead, it leads to quantizedmagnetoelectrical responses [15,34 –36].H o w e v e r ,b y performing a similar analysis of spin fluctuations, we find that the coefficients of magnetoelectrical responses onlyexperience negligible changes. In summary, we have demonstrated that spin fluctuations can play the role of a frozen random potential that leads to asignificant reduction of the onset temperature of quantized transport in a magnetic TI. Even in the absence of structural disorders, considering the exchange gap at the mean-fieldlevel is insufficient to predict the critical temperature correctly. Our result provides an alternative explanation of the puzzling in recent experiments, and points out anunavoidable mechanism suppressing the quantized trans- port even in clean magnetic TIs. We acknowledge insightful discussions with C. Z. Chen and Y. Z. You. This work was supported in part by theUniversity of California, Riverside. *yuhang.li@ucr.edu †rancheng@ucr.edu [1] C.-X. Liu, S.-C. Zhang, and X.-L. Qi, Annu. Rev. Condens. Matter Phys. 7, 301 (2016) . [2] Y. Tokura, K. Yasuda, and A. Tsukazaki, Nat. Rev. Phys. 1, 126 (2019) . [3] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011) . [4] M. Z. Hasan and C. L. 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Kawamura, M. Mogi, A. Tsukazaki, X. Z. Yu, K. Nakajima, K. S. Takahashi, M.Kawasaki, and Y. Tokura, Appl. Phys. Lett. 115, 102403 (2019) . [17] M. Mogi, R. Yoshimi, A. Tsukazaki, K. Yasuda, Y. Kozuka, K. S. Takahashi, M. Kawasaki, and Y. Tokura, Appl. Phys. Lett. 107, 182401 (2015) . [18] W. Nolting and A. Ramakanth, Quantum Theory of Magnet- ism(Springer Science & Business Media, New York, 2009). [19] M. P. Marder, Condensed Matter Physics (John Wiley & Sons, New York, 2010). [20] L. Fu, Rational design of magnetic topological insulators, in Journal Club for Condensed Matter Physics, October 2019.Available at https://www.condmatjclub.org/?p=3833 . [21] We have also considered the case of finite spatial correlation length, in which the lattice is divided into a superlatticeconsisting of 10×10supercells. Magnetic spins are fully correlated, hence are uniform inside each supercell whereasspins from neighboring supercells are uncorrelated. We findno visible changes in our results. [22] K. Nomura and N. Nagaosa, Phys. Rev. Lett. 106, 166802 (2011) . [23] Q. Liu, C.-X. Liu, C. Xu, X.-L. Qi, and S.-C. Zhang, Phys. Rev. Lett. 102, 156603 (2009) . [24] Typically, the Curie temperature of thin-film magnetic TIs is on the order of 2 meV, while the typical exchange field canbe on the order of 100 meV. [25] M. M. Otrokov et al. ,Nature (London) 576, 416 (2019) . [26] D. A. Ryndyk, Landauer-Büttiker method, in Theory of Quantum Transport at Nanoscale: An Introduction (Springer International Publishing, Cham, 2016), pp. 17 –54. [27] Because the band gap in the QAH phase is jλj−jm 0j, the width of an edge state is w≈vf=ðjλj−jm0jÞwith vfthe Fermi velocity of the QAH edge states. When m0¼0.8δ,w is larger than the half-width of the sample, which partiallyoverlaps with the opposite chiral edge state, resulting in anill-quantized σ. [28] Y. Blanter and M. Bttiker, Phys. Rep. 336, 1 (2000) . [29] T. Martin, in Nanophysics: Coherence and Transport, Proceedings of the Les Houches Summer School , Session 81, edited by H. Bouchiat, Y. Gefen, S. Guron, G.Montambaux, and J. Dalibard (Elsevier, New York,2005), pp. 283 –359. [30] Y.-H. Li, J. Liu, H. Liu, H. Jiang, Q.-F. Sun, and X. C. Xie, Phys. Rev. B 98, 045141 (2018) . [31] H. Jiang, L. Wang, Q.-f. Sun, and X. C. Xie, Phys. Rev. B 80, 165316 (2009) . [32] E. Prodan, J. Phys. A 44, 113001 (2011) . [33] E. Prodan, Appl. Math. Res. Express 2013 , 176 (2012). [34] X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, Science 323, 1184 (2009) . [35] A. M. Essin, J. E. Moore, and D. Vanderbilt, Phys. Rev. Lett. 102, 146805 (2009) . [36] J. Li, Y. Li, S. Du, Z. Wang, B.-L. Gu, S.-C. Zhang, K. He, W. Duan, and Y. Xu, Sci. Adv. 5, eaaw5685 (2019) .PHYSICAL REVIEW LETTERS 126, 026601 (2021) 026601-5

PhysRevB.86.144417.pdf

PHYSICAL REVIEW B 86, 144417 (2012) Investigation of spin wave damping in three-dimensional magnonic crystals using the plane wave method J. Romero Vivas,1,2S. Mamica,1M. Krawczyk,1,*and V . V . Kruglyak3 1Faculty of Physics, Adam Mickiewicz University in Poznan, Umultowska 85, Poznan, 61-614 Poland 2Electrolaer, Godard # 46, Col. Guadalupe Victoria., C.P . 07890, Mexico D.F ., Mexico 3School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, United Kingdom (Received 21 August 2012; published 25 October 2012) The Landau-Lifshitz equation with a scalar damping constant predicts that the damping of spin waves propagating in an infinite homogeneous magnetic medium does not depend on the direction of propagation.This is not the case in materials with a periodic arrangement of magnetic constituents (known as magnoniccrystals). In this paper, the plane wave method is extended to include damping in the calculation of the dispersionand relaxation of spin waves in three-dimensional magnonic crystals. A model material system is introducedand calculations are then presented for magnonic crystals realized in the direct and inverted structure and fortwo different filling fractions. The ability of magnonic crystals to support the propagation of spin waves ischaracterized in terms of a figure of merit, defined as the ratio of the spin wave frequency to the decay constant.The calculations reveal that in magnonic crystals with a modulated value of the relaxation constant, the figureof merit depends strongly on the frequency and wave vector of the spin waves, with the dependence determinedby the spatial distribution of the spin wave amplitude within the unit cell of the magnonic crystal. Bands anddirections of exceptionally long spin wave propagation have been identified. The results are also discussed interms of the use of magnonic crystals as metamaterials with designed magnetic permeability. DOI: 10.1103/PhysRevB.86.144417 PACS number(s): 75 .30.Ds, 75 .75.−c, 75.78.−n, 76.90.+d I. INTRODUCTION In photonics and phononics, periodic patterning has proven itself as an effective way to obtain materials with custom-made properties. Analogously, materials with a periodicarrangement of magnetic constituents [i.e., magnonic crystals(MCs)], can show properties not found in bulk samples. Thesecrystals can be used for the fabrication of new devices inwhich spin waves (SWs) act as information carriers. Thus,the investigation of properties of three-dimensional (3D) MCswith nanoscale lattice constants is of both scientific andpractical interest. Reviews of possible applications of MCs with modulation at different length scales can be found, for example, in Refs. 1–5. Loss is an unavoidable property of materials. Hence, it has to be taken into account in the design of magnonic devicesand MCs. Studies of the damping of spin waves travelingin thin ferrite films have already been presented. 6,7Some of the damping effects in one-dimensional MCs have alsobeen discussed in the literature. 8–10In two-dimensional (2D) MCs, the considerations to include damping effects in the plane wave method (PWM) have been published in Ref. 11. As a continuation of that work, we have implemented losscalculations within the PWM in the case of 3D MCs with theaim of exploring the options for tailoring the intrinsic spinrelaxation. It was shown that 3D MCs with lattice constantsin nanoscale should have magnonic gaps when constituent ferromagnetic materials are chosen properly. 12,13This gap can be obtained for many crystal structures, including cubic,simple hexagonal, and close packed lattices. 12,14,15 The possibility for tailoring damping in magnetic ma- terials is intensively studied in the literature. This direc-tion of research is developed not only in the context ofpotential applications [e.g., within spintronics (in particular, spin transfer torque devices) and magnonic devices] butalso to understand fundamental experimental results in thephysics of magnetism. 16–21There are two contributions to spin relaxation usually identified, intrinsic and extrinsic. Theformer is usually related to the spin-orbit coupling and isdescribed by the phenomenological Gilbert damping term.In principle, the damping constant can be anisotropic but in3D metallic ferromagnets under the conditions used in SWcalculations this anisotropy is usually averaged out. 22,23For the extrinsic damping the main contribution is usually attributedto two-magnon scattering processes, 24,25which can be related to the scattering on defects26including also their periodic distribution.19,27The influence of the periodic modulation on extrinsic damping processes (i.e., two-magnon scatterings)were studied in Ref. 27. It was shown that a periodic scattering potential for magnons can result in a significant increase inthe spin relaxation rates and can enable the tailoring of theanisotropy of damping. 18,27In another study, the influence of retardation effects on the effective damping was investigated.28 It was shown that the lifetime of SWs depends on theretardation time and long-living SWs for selected wave vectorswere found. In this paper we show that 3D MCs offer the possibility of tailoring the strength and anisotropy of the intrinsic dampingof SWs. We neglect the extrinsic damping as well as anycontribution from the surroundings 29as we assume that the MCs fill the whole space. We also neglect magnetic relaxationdue solely to interfaces. 30The results of our calculations using the above mentioned model are presented for 3D MCscomposed of ferromagnetic spheres in a ferromagnetic matrixin a simple cubic (sc) lattice. We assume that in each ofthe two constituent materials the coefficient of the Gilbert 144417-1 1098-0121/2012/86(14)/144417(9) ©2012 American Physical SocietyROMERO VIV AS, MAMICA, KRAWCZYK, AND KRUGLY AK PHYSICAL REVIEW B 86, 144417 (2012) damping is isotropic, which is a common assumption in the linear approximation used in our calculations.31 The layout of this work is as follows. Section IIpresents the general theory of our calculation method. In Sec. III,w e report the calculations using our method for a MC consistingof spherical scattering centers forming a sc lattice. We consideralso the effect of material and structural parameter values onthe wave damping in this section. The interpretation of thenumerical results with the effective and estimated dampingcoefficients is presented in Sec. IV. Finally in Sec. Vwe detail how the anisotropy in the effective damping can be used as atool for designing practical devices. II. CALCULATION METHOD The equation of motion of the space- and time-dependent magnetization vector M(r,t), [i.e., the Landau-Lifshitz-Gilbert equation (LLG)], is the starting point for the study of spinwaves in the classical approach. When written in internationalsystem of units, it reads ∂M(r,t) ∂t=γμ 0M(r,t)×Heff(r,t)+α MS/parenleftbigg M×∂M(r,t) ∂t/parenrightbigg . (1) In this equation γis the gyromagnetic ratio, Heffdenotes the effective magnetic field acting on the magnetization. Weassume that the effective magnetic field is composed of threecontributions: bias external magnetic field, exchange field, andmagnetostatic field. 32MSis the saturation magnetization, μ0 denotes the permeability of vacuum and the last term on the right describes damping. The dimensionless damping factor α in the rightmost term is Gilbert’s phenomenological dampingparameter. The LLG equation is nonlinear and to obtain thespin wave spectrum we need to obtain a linear approximationof this equation: we decompose the magnetization vector intoa static part (parallel to the external bias magnetic field H 0 with a value equal to the MS) and a small dynamic part m(r,t) (themvector is perpendicular to H0). In addition, because we are considering a periodic system, the Bloch theorem isapplied. The Fourier transform is used to obtain a frequencydomain solution [i.e., we assume m(r,t)∝exp(i/Omega1t)]. This method, called the plane wave method (PWM), has alreadybeen described in the literature, for details see, e.g., Refs. 12 and13. The methodology for extending the method to consider damping has been described for the 2D case in Ref. 11. Here, we implement the calculation of damping for 3Dstructures. In the traditional PWM, an eigenvalue problemis obtained and the eigenvalues represent the frequencies,/Omega1. In the implementation extended to consider damping effects, a generalized eigenvalue problem is obtained and theeigenvalues can adopt complex values (i.e., /Omega1=/Omega1 /prime+i/Omega1/prime/prime) where i is the imaginary unit. The real part of these eigenvalues,/Omega1 /primeis the frequency, and the imaginary part /Omega1/prime/primegives the inverse of the SW lifetime (i.e., the decay rate).33In the calculations we used 1331 plane waves to obtain reasonable convergencefor three low-frequency magnonic bands analyzed in thispaper. The value of the decay rate alone could be used to evaluate the damping at a specific frequency and direction in a wavevector space. On the other hand, considering Eq. (2), whichis a known result in the theory of ferromagnetic resonance measurements, we can see that the Gilbert damping parameterin the LLG equation models damping as proportional to thefrequency 34 /Delta1BG=1.16α/Omega1/prime γ, (2) where the BGis a half width of a ferromagnetic resonance line. It makes sense, therefore, to define a similar quantitythat is not proportional to the frequency and therefore allowsus to compare the damping modification in the spin wavepropagation for different magnonic bands. This quantity willdepend on the profile of the magnetization distribution butwill not be proportional to the frequency. This quantity, calledfigure of merit (FOM), was used already in Ref. 11as the ratio of the real to the imaginary part of the eigenvalues FOM=/Omega1 /prime /Omega1/prime/prime. The FOM allows us to compare the degree of damping variation between bands and between different directions inspace. In the following section, Sec. III, we define the physical system used to exemplify the use of this 3D PWM includingdamping and provide representative results. III. RESULTS OF THE PLANE WA VE METHOD CALCULATIONS The first system under study is a 3D MC obtained by arrang- ing spherical ferromagnetic scattering centers (material A) in asimple cubic (sc) lattice, as shown in Fig. 1(a). The [001] axis of the crystal is parallel to the zaxis. The assumed value of the sc lattice constant is a=10 nm; the magnetic parameters of the matrix material (i.e., material B) in Fig. 1(a) are sat- uration magnetization M S=0.194×106A/m and exchange constant A=3.996×10−12J/m; the magnetic parameters of the spherical scattering centers, material A, are MS=1.752× 106A/m and A=2.1×10−11J/m. The values assumed for the Gilbert damping parameter were chosen arbitrarily asα A=0.0019 and αB=0.064 for spherical scattering centers and the matrix, respectively, unless stated otherwise. Thisstructure will be called direct crystal . We will study also the aaA B2R xyz H0(a) (b) XMM’RX’ FIG. 1. (Color online) (a) The sc structure of the considered MC (direct crystal). The MC consists of spherical scattering centers (material A of radius R) immersed in the host matrix (material B). The lattice constant is aand the external static magnetic field, H0is directed along the zaxis. (b) The first BZ of the sc lattice. The dark (orange) color shows the part of the BZ over which the calculations of the magnonic structure are performed. 144417-2INVESTIGATION OF SPIN W A VE DAMPING IN THREE- ... PHYSICAL REVIEW B 86, 144417 (2012) 0100200300400500600 X’M’X ΓXM ΓRXband 1 band 2 band 3 0102030 X’M’X ΓXMΓRX)zHG(''( MOF'' '(a) (b) (c) )zHG(' 15.81616.216.4 X’M’X ΓXM ΓRX FIG. 2. (Color online) The frequency and decay rate of the SWs are shown for the direct MC (spheres of material A in matrix B) in (a) and (b), respectively. A filling fraction of 0.2 was assumed in the calculations. (c) The figure of merit (FOM) is shown for the same structure. corresponding inverted crystal (i.e., a crystal with scattering centers made of materials B and A serving as the matrixmaterial). A constant external magnetic field μ 0H0=0.3 T is applied in the zdirection to saturate the crystal. The dynamic part of the precessing magnetization has only xandynonzero components. The magnonic band structure resulting from the numerical solution of the eigenproblem for the direct MC is shown inFig. 2. The filling fraction, defined as a ratio of the volume of scattering centers in the unit cell (it is a sphere in our case) tothe volume of the unit cell: f=4πR 3/3a3, was assumed to bef=0.2. It corresponds to a sphere radius R=3.628 nm. The magnonic band structure was calculated along a path inthe irreducible part of the first Brillouin zone (BZ). The pointsalong this path are defined in Fig. 1(b). We limit the spectra presented in this paper to low frequencies only (i.e., to thefirst three bands). We show the frequency ( /Omega1 /prime) and decay rate (/Omega1/prime/prime) in dependence on the wave vector in Figs. 2(a) and2(b), respectively. We can see that there is not any magnonic bandgap in the frequency spectra. We also observe similar wavevector dependencies for the frequencies and decay rates. Thisimplies that the FOM should be quite uniform in the whole BZ.This is confirmed by Fig. 2(c) in which the FOM is shown along the path in the first BZ. The FOM has values in the range from15.8 to 16.4. In this case the FOM can therefore be regarded asnearly isotropic for all of the considered low-frequency bands.Let us now increase the filling fraction in the direct structure. In Figs. 3(a) and 3(b) we show the corresponding magnonic band structure (i.e., the frequency and decay rate)respectively, for the direct MC with f=0.5 (R=4.92 nm). We found the magnonic band spectrum to be quite different tothat for f=0.2. In particular, the first band is separated from the upper bands in most of the first BZ except for the R-X direction, thereby forming a partial band gap. The wave vectordependence of the decay rate for the first band follows thatof the frequency. Consequently, we obtain an almost constantFOM [note the scale of the vertical axis in Fig. 3(c)]. For the second band, we found that around the /Gamma1point, where /Omega1 /primehas a maximum, /Omega1/prime/primehas the minimum. As a result of this, the FOM is very large near the center of the BZ, reaching nearly 160. Inthe rest of the BZ, the FOM is below 30. The FOM is stronglydependent on the value of the wave vector but remains almostindependent on the direction of propagation. Thus, we have found that a partial band gap in the magnonic spectrum and significant values of the FOM coexist at somepoints in the BZ. We have performed calculations for otherfilling fractions from 0 up to the value corresponding to theclose-packed structure ( f=0.523) for the direct crystal and no full gap was found. Nevertheless, a full band gap is observedfor the inverted crystal structure. In Figs. 4(a) and 4(b) we show/Omega1 /primeand/Omega1/prime/primeas a function of the wave vector along a path in the first BZ for the inverted MC with filling fraction of 0.5. 0100200300400 X’M’X ΓXM ΓRX01020 X’M’X ΓXM ΓRX2060100140 X’M’X ΓXMΓRXband 1 band 2 band 3 )zHG(''FOM ('' ' )zHG('(a) (b) (c) FIG. 3. (Color online) The frequency and decay rate of SWs in the first BZ for a direct MC (spheres of material A in a matrix of material B) are shown in (a) and (b), respectively. A filling fraction 0.5 was assumed in calculations. (c) Figure of merit (FOM) for the same structure. 144417-3ROMERO VIV AS, MAMICA, KRAWCZYK, AND KRUGLY AK PHYSICAL REVIEW B 86, 144417 (2012) 0100200300400 XM’X’ΓXMΓRX0510152025 XM’X’ΓXMΓRX2060200 XM’X’ΓXMΓRXband 1 band 2 band 3 100 Magnonic gap)zHG(''( MOF'' ' )zHG('(a) (b) (c) FIG. 4. (Color online) The frequency and decay rate of SWs in the first BZ for an inverted MC (spheres of material B in a matrix of material A) are shown in (a) and (b), respectively. A filling fraction of 0.5 was assumed in the calculations. (c) The FOM is shown for the same structure. We found a complete magnonic band gap between the first and second bands. We also found that, in this case, the FOM hasvery small values for the first band. These values are nearlyindependent on the propagation direction or the magnitude ofthe wave vector. On the other hand, the second band reachesa significantly higher value of the FOM at the /Gamma1point (up to 250). This value is much larger than that observed for thedirect structure [Fig. 3(c)]. However, the FOM is again almost isotropic and has a sharp peak exactly at the center of the BZ(i.e., because the dispersion curve is flat and the group velocityof SWs goes through zero). From the results presented so far, we can see that significant values of the FOM are associated with the second band (i.e.,two absolute magnonic band gaps are found). In the previouslypresented spectra, the lowest band is separated from the secondone by a magnonic gap. Figure 5shows the magnonic spectra for the inverted crystal with a filling fraction of 0.2, with /Omega1 /prime and/Omega1/prime/primeplotted as a function of the wave vector in panels (a) and (b), respectively. Both band gaps are marked in yellowcolor in the figure. These two band gaps separate the secondband from the other magnonic bands. The imaginary partof the frequency shows features that are not present in theother crystals investigated here. For the second band, /Omega1 /primehas a maximum at the /Gamma1point, and consequently, the FOM has a minimum in the same point. The maximal values of the FOM,larger than 400, are found at corners and edges of the firstBZ [i.e., at the points M=π/a(1,1,0),M /prime=π/a(1,0,1), andR=π/a(1,1,1)]. At the borders of the first BZ along the principal axis [ X=(1,0,0) and X/prime=(0,0,1)] we found the FOM to reach only values that are smaller than 200. Wecan conclude that in this crystal, the FOM is anisotropic andstrongly dependent on the magnitude of the wave vector. In summary, we have found:(i) A low value of the FOM is observed for the first band both for direct and inverted crystals irrespectively of thefilling fraction. The FOM is almost isotropic and only weaklydependent on the absolute value of the wave vector. This meansthat this band can be described using effective parameters. (ii) Only for the inverted crystal with f=0.2 we found strong anisotropy in the FOM for the second band. (iii) There is apparently a cause-and-effect relationship between the isolation of the second band (due to the presenceof band gaps directly above and below) and the observation ofhigh values of the FOM. 0100200300 X'M'X ΓXMΓRX01234567 X'M'X ΓXMΓRX0100200300400500 X'M'X ΓXMΓRXband 1 band 2 band 3 400)zHG(''FOM ('' ' )zHG(' Magnonic gap(a) (b) (c) FIG. 5. (Color online) The real and imaginary parts of the frequency in the first BZ for inverted MC (spheres of material B in matrix of material A) in (a) and (b), respectively. A filling fraction of 0.2 was assumed in the calculations. (c) The FOM is shown for the same structure. 144417-4INVESTIGATION OF SPIN W A VE DAMPING IN THREE- ... PHYSICAL REVIEW B 86, 144417 (2012) 0 0 010 10 1020 20 2030 30 300200400 05f= 0.2 f= 0.5 0 0200 200400 400latsyrc tceriD latsyrc detrevnI 0 200 400(a) (c)(b) (d) pag cinonga Mpag cinonga M)zHG( )zHG('''' ' (GHz) ' (GHz) FIG. 6. (Color online) Decay rate of SWs versus its frequency for wave vectors randomly chosen from the first BZ. Direct crystals (Aspheres in the B matrix) are shown in (a) and (b), inverted crystals (B spheres in the A matrix) in (c) and (d). In (a) and (c) the results forf=0.2 and in (b) and (d) for f=0.5 are shown. Magnonic gaps are colored in yellow color. In (c) the second band with the moonlike shape is marked by blue ellipse, as it is expected to have strong anisotropy in damping. IV . DISCUSSION It has already been mentioned that MCs showing a strong dependence of loss on the direction of the wave vector can beused to design effective magnonic waveguides. 11We would like to focus on the anisotropy of lifetime of SWs from anotherpoint of view and to explain the physical mechanisms thatgovern the damping of SWs in MCs. To facilitate the analysis,we propose to plot the decay rate versus frequency of theSWs as calculated for wave vectors of random direction andmagnitude in the first BZ. Figure 6shows such plots for the direct and inverted MCs with filling fractions f=0.2 and 0.5. From these figures we find two linear dependencies: a linearfunction /Omega1 /prime/primevs./Omega1/primefor the first mode and a linear dependence of the upper limit of /Omega1/prime/prime≡/Omega1/prime/prime maxon/Omega1/primefor all the considered structures.35 The linear relation between the decay rate of SWs and the frequency for the first band can be described by the followingrelation: /Omega1 /prime/prime=FOM−1×/Omega1/prime. We found the inverse of FOM to be the slope of the straight line obtained by regression of thedata presented in Fig. 6. To explain this feature let us consider SWs propagating in uniform materials. To have a goodmodel for comparison we have to choose a proper structure.Because we are studying 3D MCs filling the whole space,the proper choice seems to be the ferromagnetic uniformlymagnetized sphere with free boundary conditions imposedon the dynamic component of the magnetization vector. Thesphere is considered in order to avoid shape anisotropy effects.If the sphere is small enough to separate higher harmonicsfrom the uniform excitation, such results should be useful forinterpretation of the dependencies found for low-frequencymodes in 3D MCs, at least. In uniformly magnetized spheres,FOM =1/α, where αis a Gilbert damping constant of the uniform sphere. 33This means that the lifetime of SWs fromthe first band of 3D MCs behaves like the one from uniform materials. This allows us to introduce the effective dampingof the low-frequency mode in 3D MCs as α eff=1/FOM, where the inverse of FOM is the slope of the line fitted tothe dependencies shown in Fig. 6for the first band. From the PWM solutions we have found α effequal: 0.062, 0.058, 0.052, and 0.059 for the direct crystal f=0.2 and 0.5, and inverted crystals with f=0.2 and 0.5, respectively. These values are between the values of the Gilbert damping coefficient of theconstituent materials ( α AandαB) but in fact all of them are very close to the highest value (i.e., 0.064). This behavior would bereasonable if the SW modes from the first band in both kinds ofthe investigated crystals concentrated their amplitude mainlyin the material with higher value of damping. Two-dimensionalcolor maps of the modulus of the dynamical components of the magnetization vector (i.e., |m|=√ m2 x+m2 y) are shown in Fig. 7, confirming our hypothesis. The amplitude is shown in two cross sections perpendicular to the zaxis: one plane crossing the centers of the spheres [plane (001)] and thesecond crossing the space in the middle between the spheres[plane (002)]. Red color marks maximum values while bluecorresponds to zeros of the amplitude. To have a quantitative measure of the damping of SW modes we can integrate the mode profiles in Fig. 7weighted with the respective damping. The derivation of the formula for anestimated damping α estin one-dimensional periodic structures can be found in Ref. 36. A similar procedure can be applied to 3D structures and the final expression will have a similar formwith integrals over volume of the material A or B in the unitcell, now in 3D, α est(k,n)=αsph MS,sph/integraltext sph|mk,n|2dv+αmat MS,mat/integraltext mat|mk,n|2dv 1 MS,sph/integraltext sph|mk,n|2dv+1 MS,mat/integraltext mat|mk,n|2dv,(3) where the indices “sph” and “mat” make reference to the sphere and matrix, respectively; mk,nis the dynamical component of the magnetization vector for the band nand wave vector k. This formula allows us to calculate an estimated value of damping for each band ( n) and each wave vector ( k). The calculated values of the estimated damping parameters for thefirst band in /Gamma1andRpoints in the BZ are collected together with the effective damping constants obtained from the slopesin Fig. 6in Table I. For the direct crystal the damping constant from both methods match very well. For the inverted crystal,there is significant variation of αin dependence on the wave vector value [see also Fig. 5(c)] but the arithmetic average of estimated values also match well with the α eff. Now we will discuss the results obtained for the second band, where in the case of the inverted crystals, a large FOMwas found in the RandMpoints in the BZ. The amplitude of the dynamical components of the magnetization vector and therespective estimated damping parameters are shown in Fig. 7 for the wave vector from the BZ center and BZ edge (i.e., forthe/Gamma1andRpoints) respectively, for the direct crystal ( f= 0.5) and inverted crystal ( f=0.2). The estimated damping parameters α estfrom the profiles are given also in this figure. For the direct crystal we have found that the FOM reaches ahigh value at the /Gamma1point ( ∼=160) and a very small value for the Rpoint (less than ∼=18), as shown in Fig. 3(c). The respective damping values obtained from the profiles are 0.006 and 0.063, 144417-5ROMERO VIV AS, MAMICA, KRAWCZYK, AND KRUGLY AK PHYSICAL REVIEW B 86, 144417 (2012) Direct crystal = 0.5fInverted crystal = 0.2fenalP(001) enalP)200(R R 1st banddnab dn2enalP)100( enalP)200(est,= 0.006 est,= 0.058est,= 0.016 est,= 0.045est,R= 0.063 est,R= 0.06est,R= 0.002 est,R= 0.0590max FIG. 7. (Color online) The amplitude of the dynamical components of the magnetization vector across the planes perpendicular to the z axis and crossing it at 0 [plane (001)] and at a/2 [plane (002)]. The profiles from the first and the second band in /Gamma1andRpoint are shown for the direct crystal with f=0.5 (left columns) and for the inverted crystal and f=0.2 (right columns). The estimated value of the damping constant of the related mode [calculated according to Eq. (3)] is also given for each profile. for the /Gamma1andRpoints, respectively. For the inverted structure a strong change in the FOM, which is eight times lower atthe/Gamma1point as compared to its value at the Rpoint can be seen in Fig. 5(c). This fact is also supported by the damping coefficient values obtained from the SW profiles: α est(/Gamma1,2)= 0.016 and αest(R,2)=0.002 at /Gamma1andRpoint in the BZ, respectively. TABLE I. The effective damping parameters ( αeff)f o rt h efi r s t band obtained by assuming a linear dependence /Omega1/prime/prime(/Omega1/prime) and fitting the slope from Fig. 6are shown. The estimated damping coefficients (αest) extracted according to Eq. (3)from the profiles of SWs at the /Gamma1andRpoints in the first BZ for the first band shown in Fig. 7are also presented. Structure αeff αest(/Gamma1,1) αest(R,1) Direct f=0.5 0.058 0.058 0.06 Inverted f=0.2 0.052 0.045 0.059We have already established the relation between the value of the FOM, estimated values of the damping coefficients,and the distribution of the mode profiles over the constituentmaterials. It remains still unattended, however, how changesin the damping parameters of the constituent materials ( α Aand αB) influence the lifetime of the different modes. To get some insight we propose to take a look at the frequency and decayrate as a function of the relative loss parameter (RLP), whichtakes values from 0 to 1 and we define as α sph=0.0659×RLP, (4) αmat=0.0659×(1.0−RLP), where the coefficient 0.0659 is chosen equal to αA+αB. According to this definition, for RLP =0 the SWs in spheres will be undamped while the damping will reach its maximalvalue (i.e., 0.0659) in the matrix. For RLP =1, the reverse situation occurs (i.e., no damping is present in the matrix)while it reaches its maximum value in the spheres. In Figs. 8(a) and8(b) the frequency and decay rate of SW modes from the 144417-6INVESTIGATION OF SPIN W A VE DAMPING IN THREE- ... PHYSICAL REVIEW B 86, 144417 (2012) 04812 0 0 1 1140160180200220 0 1RL P RLP RLPΓ Γ ΓX XXR RR Γ XR(a) (b) 00,020,040,06 est'xest(c) (d) 04812 0 1 RLP)zHG('')zHG(' FIG. 8. (Color online) (a) Frequency and (b) decay rate of the SWs are shown as a function of the RLP for the inverted crystal with filling fraction 0.2 calculated with PWM. The frequencies from the second band for the /Gamma1,R,a n d Xpoints in the BZ are shown. (c) Estimated values of the damping coefficient are shown in dependence on RLP for the /Gamma1,X,a n d Rpoints for the second band. αestare calculated according to Eq.(3). (d) The product of αestand/Omega1/primeis shown for the /Gamma1,X,a n d Rpoints from the second band. There is a close relation between αest×/Omega1/prime [shown in (d)] and /Omega1/prime/prime[shown in (b)]. second band for three points from the first BZ (i.e., for /Gamma1,X, andR) are shown in dependence on the RLP. The calculations were performed for the inverted crystal with filling fraction0.2. We have found that the frequency is virtually independenton the RLP [Fig. 8(a)]. This behavior can be expected due to the fact that in thin films the dependence of damping onfrequency is a second-order effect. 33The decay rate is a linear function of the RLP with negative slope depending on the wavevector: at the Rpoint the slope reaches its highest value while at the /Gamma1point, it reaches its lowest value. We can understand this behavior, because we already showed that the amplitudeof the SW modes for the second band is concentrated mainly inthe spheres. Consequently, it is expected to observe the lowestvalues of /Omega1 /prime/primefor RLP =1. The RLP of the inverted crystal [the corresponding band structure is shown in Fig. 5(a)] is 0.97. We see that for this RLP, the imaginary part of a frequency at RandXhas almost the same value but less than half of that at /Gamma1point. This shows from another point of view the main features alreadyobserved in Fig. 5(c) (i.e., highest FOM at Rand smallest at /Gamma1). Here we see that the anisotropy of the FOM is dependent onthe distribution of damping among the constituent materialsof the MC. In particular, no anisotropy is observed for thecase when RLP =0.5 (i.e., when the damping coefficients in spheres and matrix are equal). This result can also be obtaineddirectly from the PWM [i.e., by plotting FOM (RLP) ≡/Omega1 /prime (RLP) /Omega1/prime/prime(RLP)]. The linear dependencies presented in Fig. 8(b) provide evidence that the estimated damping coefficient calculatedfrom the Eq. (3)should also preserve a linear dependence on the RLP. The α est(RLP) can be calculated from the SW profiles obtained from PWM (the profiles have to be calculated onlyonce for selected RLP) according to Eq. (3)with damping coefficients defined by Eqs. (4). The results for the second band for a few selected points in the BZ are shown inFig. 8(c). We see that for RLP =0.5t h e α estis the same independently of the wave vector. To have a quantitativecomparison between α estand the numerically calculated /Omega1/prime/prime we need to multiply αestby/Omega1/prime. The product αest×/Omega1/primeis shown in Fig. 8(c). A good qualitative agreement with /Omega1/prime/primeas shown in part (b) of this figure is clear. Quantitatively, the differ-ences are largest near RLP =0 and decrease when the RLP increases. V . APPLICATION OF THE PROPOSED THREE-DIMENSIONAL MAGNONIC CRYSTAL In Fig. 6(c) a shape similar to a crescent moon can be observed for the second band lying between two magnonicband gaps. This is interesting because it means that, dependingon the direction, we can identify regions of low and highdamping for the same frequency and nothing in between. Forapplication as beam shaper, this is just what we need. Theregion in which this happens is easy to identify using theintroduced style of plotting. Also, it allows us to identifypropagation directions that correspond to low and highdamping at the same frequency. We can use Fig. 5for this purpose, where the respective band structure is shown. We cannote that on the paths going through /Gamma1point (i.e., from X /primeto /Gamma1toXand from Mto/Gamma1toR, the decay rate is higher (and consequently the FOM is smaller) than in the rest of the path.We can also notice that for frequencies above approximately200 GHz there is an allowed SW band only around the Rpoint. In this point also the decay rate is very low, giving high FOM.This functionality can be combined with changes controlledby the external magnetic field, which would have the effect ofshifting up or down the range of frequencies where the abovementioned conditions are fulfilled. If we want to use a mode for transmitting information, in other words, if we want to use our infinite MC as a waveguide,a necessary condition for usefulness is to show that the groupvelocity (i.e., its magnitude) is greater than zero. The extremalvalues of the FOM are found for values of the wave vectors inthe BZ border ( R,X,o rMpoints) or at the BZ center ( /Gamma1point). At these points the dispersion curves reach extreme values andthe group velocity is 0. To have a qualitative measure of theusefulness of a given mode we propose to look at the product ofthe group velocity and the FOM. A large value of this productwill occur at points with low loss and high group velocity. InFig.9we show the product, v g×FOM (calculated directly from the dispersion relation) for the second band along the path in 144417-7ROMERO VIV AS, MAMICA, KRAWCZYK, AND KRUGLY AK PHYSICAL REVIEW B 86, 144417 (2012) XM’ X’ ΓXM Γ RX0246810vgFOM (10 )4m/s FIG. 9. The absolute value of the product of a group velocity ( vg) and FOM for the second band for the inverted crystal (with f=0.2) along the path in the first BZ. the first BZ for the inverted crystal with f=0.2. We see, that on the path along, /Gamma1-RandX-Rthis product has maxima, which define the optimal wave vectors for possible applications. Atthese points the group velocity of SWs is around 200 m /s. This speed is rather low, but still a lifetime around 0.5 GHzwill allow a transport of the signal for a distance around2μm. This application would also face the challenge that the anisotropy of the damping in 3D MCs will depend on theRLP as shown in Figs. 8(b) and 8(c). This means that the properties depend on the distribution of damping among theconstituent materials. This however could also be consideredas an opportunity from the designer point of view. In Ref. 37, Mruczkiewicz et al. showed that stacks of 2D all-ferromagnetic magnonic crystals could be used to designmetamaterials with negative permeability at frequencies ofseveral tenths of GHz. In particular, negative permeability wasobserved in the vicinity of high-order resonances for whichthe magnonic mode amplitude was preferentially distributedwithin one of the two constituent materials. The resultspresented in this paper allow us to speculate that if this materialin which the magnonic amplitude is concentrated is in additioncharacterized by a low damping coefficient, then the resonancewill be even stronger and the quality factor will be evenhigher than that obtained in Ref. 37. A rigorous proof of this hypothesis is however beyond the present study.VI. CONCLUSION Using numerical calculations based on the PWM we have shown that magnonic crystals enable us to tailor the effectiveintrinsic damping of spin waves. A proper choice of theMC structure and its filling fraction allows us to designa magnonic band structure with anisotropic and stronglywave-vector-dependent effective damping. We introduced theplots of the decay rate versus frequency for randomly chosenwave vectors from the first BZ. With the help of these plotswe have shown that it is possible to obtain for the samefrequency two different directions of SW propagation withlow and high damping, where propagation takes place at afinite group velocity. We have proposed a qualitative explanation of the depen- dencies observed in our numerical results based on the analysisof the SW amplitude distribution among the constituentmaterials. The formula for the estimated effective dampingcoefficient, introduced here for 3D MCs, is wave vector andband number dependent and describes adequately numericalresults. We have shown that the decay rate of SW in 3D MCsis a linear function of the relative loss parameter. This is animportant result, which allows for a reduction of the time ofcomputations. We have shown also that large values of theFOM in MCs coexist with magnonic gaps in the spin wavespectra as both effects are influenced by the distribution of theSW amplitude in the unit cell in a similar way. This modelallows us to understand the effective behavior of damping ofthe first mode in the magnonic spectrum, irrespective of thedirection and magnitude of the wave vector. ACKNOWLEDGMENTS The research leading to these results has received funding from the European Community’s Seventh Framework Pro-gramme (Grant No. FP7/2007-2013) under Grant AgreementsNo. 247556 (People), NoWaPhen and No. 233552, DYNA-MAG project. V .V .K. also acknowledges funding receivedfrom EPSRC of the UK under Project No. EP/E055087/1.The calculations presented in this paper were performed inPoznan Supercomputing and Networking Center. *krawczyk@amu.edu.pl 1A. A. Serga, A. V . Chumak, and B. Hillebrands, J. Phys. D: Appl. Phys. 43, 264002 (2010). 2S. Neusser and D. Grundler, Adv. Mater. 21, 2927 (2009). 3V . V . Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D 43, 260301 (2010). 4A. Khitun, M. Bao, and K. L. Wang, J. Phys. D 43, 264005 (2010). 5S.-K. Kim, J. Phys. D 43, 264004 (2010). 6D. D. Stancil, J. 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M ¨oller, C. Hassel, O. Posth, M. Farle, and H. Wende, IEEE Trans. Magn. 46, 2252 (2010). 18I. Barsukov, P. Landeros, R. Meckenstock, J. Lindner, D. Spoddig,Z.-A. Li, B. Krumme, H. Wende, D. L. Mills, and M. Farle, Phys. Rev. B 85, 014420 (2012). 19P. Landeros and D. L. Mills, P h y s .R e v .B 85, 054424 (2012). 20K. Sekiguchi, T. N. Vader, K. Yamada, S. Fukami, N. Ishiwata, S. M. Seo, S. W. Lee, K. J. Lee, and T. Ono, Appl. Phys. Lett. 100, 132411 (2012). 21L. Lu, J. Young, M. Wu, C. Mathieu, M. Hadley, P. Krivosik, andN. Mo, Appl. Phys. Lett. 100, 022403 (2012). 22J. Kune ˇs and V . Kambersk ´y,Phys. Rev. B 65, 212411 (2002). 23J. Seib, D. Steiauf, and M. F ¨ahnle, P h y s .R e v .B 79, 092418 (2009). 24R. Arias and D. L. Mills, P h y s .R e v .B 60, 7395 (1999). 25J. Dubowik, K. Zaleski, H. Glowinski, and I. Goscianska, Phys. Rev. B 84, 184438 (2011). 26G. Woltersdorf and B. Heinrich, P h y s .R e v .B 69, 184417 (2004). 27I .B a r s u k o v ,F .M .R ¨omer, R. Meckenstock, K. Lenz, J. Lindner, S. Hemken to Krax, A. Banholzer, M. K ¨orner, J. Grebing, J. Fassbender, and M. Farle, P h y s .R e v .B 84, 140410 (2011). 28T. Bose and S. Trimper, Phys. Rev. B 83, 134434 (2011).29Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, P h y s .R e v .L e t t . 88, 117601 (2002). 30B. Heinrich, R. Urban, and G. Woltersdorf, J. Appl. Phys. 91, 7523 (2002). 31K. Gilmore, M. D. Stiles, J. Seib, D. Steiauf, and M. F ¨ahnle, Phys. Rev. B 81, 174414 (2010). 32We use here the same formulations for the exchange and magneto- static fields as in Ref. 12. 33A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves (CRC Press, Boca Raton, 1996). 34K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M. Farle, U. von H ¨orsten, H. Wende, W. Keune, J. Rocker, S. S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, and Z. Frait, P h y s .R e v .B 76, 104416 (2007). 35For the inverted crystal with f=0.2 shown in the Fig. 6(c) due to limiting range of presented points the linear dependence of theupper limit is not clear. But for higher frequencies (decaying rates)this linear dependence is still present. 36A. M. Zyuzin, A. G. Bazhanov, S. N. Sabaev, and S. S. Kidyaev,Phys. Solid State 42, 1279 (2000). 37M. Mruczkiewicz, M. Krawczyk, R. V . Mikhaylovskiy, and V . V . Kruglyak, Phys. Rev. B 86, 024425 (2012). 144417-9

PhysRevLett.92.027201.pdf

Direct-Current Induced Dynamics in Co90Fe10=Ni80Fe20Point Contacts W . H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva National Institute of Standards and T echnology, Boulder, Colorado 80305, USA (Received 23 June 2003; published 15 January 2004) W e have directly measured coherent high-frequency magnetization dynamics in ferromagnetic films induced by a spin-polarized dc current. The precession frequency can be tuned over a range of severalgigahertz by varying the applied current. The frequencies of excitation also vary with applied field, resulting in a microwave oscillator that can be tuned from below 5 to above 40 GHz. This novel method of inducing high-frequency dynamics yields oscillations having quality factors from 200 to 800. W ecompare our results with those from single-domain simulations of current-induced dynamics. DOI: 10.1103/PhysRevLett.92.027201 P ACS numbers: 75.47.–m, 75.75.+a, 85.75.–d Since the initial predictions of Slonczewski [1] and Berger [2] that a spin-polarized current can induce mag-netic switching and dynamic excitations in ferromagneticthin films, a great deal of work has focused on under-standing the interactions between polarized currents andferromagnetic nanostructures [3]. It was predicted, and later confirmed, that this effect can lead to current- controlled hysteretic switching in magnetic nanostruc-tures in moderate applied magnetic fields [4,5]. Thisbehavior is not only of scientific interest but also findspotential applications in devices such as current-controlled switching of magnetic random access memoryelements and has implications for the stability of mag-netic hard-disk read heads. Another prediction is that the spin torque can drive steady-state magnetization preces- sion in the case of applied fields large enough to opposehysteretic switching [1,2]. Numerous applications existfor such current-controlled microwave oscillators that areintegrable with semiconductor electronics [6]. However,with one recent exception in nanopillar devices [7], todate no direct measurements of these high-frequencydynamics have been reported [4,5,8]. Here we report direct measurements of spin-torque induced magnetiza- tion dynamics for in-plane and out-of-plane applied fieldsas a function of field strength Hand current I,a n d compare the results with simulations based on the theo-retical model of Ref. [1]. Studies discussed here were performed on lithograph- ically defined point contacts to spin valve mesas ( 8/.0022m/.0002 12/.0022m). The point contacts are nominally 40 nm diame- ter circles, have resistances between 4 and 10/.0010 , and show no indications of tunneling in their transport character-istics. Top and bottom electrical contacts to the devicesare patterned into 50/.0010 planar waveguides. Fabrication details will be presented elsewhere. Specifically, thespin valve structures are Ta/.01332:5nm/.0134=Cu/.013350nm /.0134= Co 90Fe10/.013320 nm /.0134=Cu/.01335n m /.0134=Ni80Fe20/.01335n m /.0134=Cu /.01331:5nm/.0134=Au/.01332:5nm/.0134and show typical magnetoresis- tance (MR) values of 80m/.0010 .T h eCo90Fe10is the ‘‘fixed’’ layer in terms of the spin torque due to its larger volume,exchange stiffness, and saturation magnetization com-pared withNi 80Fe20[9]. The device is contacted with microwave probes and a dc current is injected through abias-tee, along with a 20/.0022Aac current (500Hz ), allow- ing simultaneous measurement of the dc resistance, dif-ferential resistance, and microwave output. The devicesare current biased so that changes in the alignment between theNi 80Fe20andCo90Fe10layers appear as volt- age changes across the point contact. The high-frequencyvoltage signal is measured with either a 50 GHz spectrumanalyzer or a 1.5 GHz real-time oscilloscope. The band-width of the circuit is 0.1 to 40 GHz. Measurements wereperformed at room temperature. All results discussedhere occur for only one direction of current, correspond-ing to electrons flowing from the top contact into the spin valve. Figure 1(a) shows a differential resistance dV=dI curve of a device taken with an in-plane field /.0022 0H/.01360:1T.T h e nonhysteretic peak in the dV=dI curve, at I/.01364mA in Fig. 1(a), has been taken as indirect evidence of current-induced magnetization dynamics [4,5,8]. Tsoi et al. dem- onstrated changes in the dc transport properties of pointcontacts under the influence of external radiation, imply- ing a relationship between spin dynamics and dc resis- tance [8]. Here we observe these dynamics directly, asshown in Fig. 1(b). For low currents, no peaks are ob-served in the spectra. As Iis increased to 4 mA a peak appears at f/.01367:9GHz .A sIis further increased, the peak frequency decreases, a trend observed for all in-plane fields measured. This frequency redshift is linear inI(inset) and typically varies from /.00250:2GHz =mAat low fields ( /.002550mT )t o/.00251GHz =mAat fields of /.00250:8T.A t higher values of I, the excitations decrease in magnitude until no peaks are observed, as shown in the I/.01369mA spectrum. Assuming the high-frequency signals resultfrom a MR response, we estimate a maximum excursionangle between the layers of approximately 20 /.0014.A st h e measured peak amplitude does not increase linearly withI(as it would for fixed excursion angle), we infer that the orbit traversed by the magnetization changes with I.T h e dynamics here are strongly correlated with the peak inthedV=dI curve. This is not the case for all devices:PHYSICAL REVIEW LETTERSweek ending 16 JANUARY 2004 VOLUME 92, N UMBER 2 027201-1 0031-9007 =04=92(2)=027201(4)$22.50 027201-1Typically the onset of the dynamics occurs only in the vicinity of a feature (step, peak, or kink) in dV=dI ,a n d the relative position of this onset varies with H. To better understand the possible trajectories of these excitations, we compare our results with simulations thatassume an isolated single-domain particle ( 40nm /.0002 40nm ) whose behavior is described by a modified Landau-Lifshitz-Gilbert (LLG) equation proposed by Slonczewski [1]. This only approximates the point contact geometry, where the region undergoing dynamic excita-tions is coupled to a continuous film by intralayerexchange. For example, effects associated with the for-mation of domain walls between the region under thecontact area and the rest of the free layer are not included,nor are effects of spin-wave radiation damping [1]. Finite-temperature effects are included through a randomly fluctuating field [10]. The simulations show two basic regimes of motion for in-plane fields. At low current, when oscillations begin,the magnetization Mprecesses in a nearly elliptical mode aboutHand the time-averaged magnetization hMilies parallel to H.A s Iincreases, the trajectories become nonelliptical and have greater excursion angles withrespect to H. However, Mcontinues to precess about the applied field, while hMichanges from parallel to antiparallel alignment with H. Within this regime, the simulated excitation frequency decreases approximately linearly with I, in agreement with the data shown in Fig. 1(b). Furthermore, jdf=dI jincreases with increasing H, also in agreement with our measurements. As Iisfurther increased, the second regime is reached and the simulations show Mprecessing out-of-plane with the precession frequency increasing with current. Con- sequently, we infer that the observed excitations corre- spond only to in-plane precession, perhaps due to a lack ofstability of the trajectories in our devices, or because thedevices are unable to support sufficient current densities.It may also indicate a need to incorporate micromagneticeffects in the modeling. The measured linewidths are quite narrow, indicating that the excitations can be considered coherent single- mode oscillations. The peaks in Fig. 1(b) have full-width- at-half-maximum (FWHM) of /.002520MHz and voltage (power) quality factors Q/.0136f=/.0133FWHM /.0134of/.0025350/.0133600/.0134, with particular values depending on I. The FWHMs of the excitations only weakly depend on H, leading to values of Q>500 forf>30GHz . Analogous line- widths in ferromagnetic resonance (FMR) measurementswould give damping parameters of /.0011/.01361–5/.000210 /.02554, with the particular value depending on H[11]. Our modeling requires /.0011/.01360:5–1/.000210/.02553to produce similar linewidths at300K . Either analysis gives values of /.0011much smaller than values obtained through field-induced excitations ofNi 80Fe20thin films ( /.0011/.01360:01to 0.005) [12,13]. Line- widths we have measured in nanopillar devices (notshown here) are about a factor of 5larger than those measured in point contacts, showing that the narrowness of these peaks is not a general result for current-induced excitations. The lack of physical magnetic edges in pointcontact devices may account for their narrow linewidthsin comparison to nanopillars. Increased linewidths andeffective damping are often found in magnetic nano-structures, resulting from Mat the edges of patterned devices lagging Mat the center of the device during large-angle oscillations [13]. Figure 2(a) shows the measured frequencies as a func- tion of in-plane field. The data correspond to the highest-frequency (lowest-current) excitation observed at a givenH. Below /.0022 0H/.013650mT no excitations are seen. Around /.00220H/.01360:6Tthe excitation amplitude begins to drop and by/.00220H>1Tis below our noise floor. The data are fit using the Kittel equation for in-plane magnon generation,excluding dipole effects, appropriate for the thin-film limit [14]: f/.0133H/.0134/.0136/.0133g/.0022 B/.00220=h/.0134/.0137/.0133H/.0135Hsw/.0135Hk/.0135Meff/.0134 /.0002/.0133H/.0135Hsw/.0135Hk/.0134/.01381=2; (1) where Hsw/.0136Dk2=/.0133g/.0022B/.00220/.0134,Dis the exchange stiffness, gis the Lande ´factor, kis the magnon wave number, Meff is the effective magnetization, Hkis the anisotropy field, /.00220is the permeability of free space, his Planck’s con- stant, and /.0022Bis the Bohr magneton. In fitting the data, k andgare treated as free parameters while fixed values of/.00220Meff/.01360:8Tand/.00220Hk/.01360:4mT are used, as de- termined from magnetometry measurements. The fit FIG. 1. (a) dV=dI vsIwith/.00220H/.01360:1T. (b) High frequency spectra taken at several different values of current through thedevice, corresponding to the symbols in (a). V ariation of fwith I(inset).PHYSICAL REVIEW LETTERSweek ending 16 JANUARY 2004 VOLUME 92, N UMBER 2 027201-2 027201-2yields g/.01361:78/.00060:01and a magnon wavelength of /.0021/.0136390/.000680nm . W e note Eq. (1) is strictly valid only in the limit of small amplitude spin-waves, a limit notstrictly met in the present measurements, as discussedabove. From both the above fit and from the linear portionof the data for /.0022 0H>0:4Twe determine g/.01361:78/.0006 0:01, smaller than the value of g/.01362:0determined on analogous films by other methods [12,13]. However, nu-merical simulations of the LLG equation show that fittingEq. (1) to oscillations of large amplitude results in anapparently suppressed value of gas found here. It was initially predicted that the lowest-order excited modewould have a wavelength of roughly twice the contactdiameter [1]. However, the excitation wavelengths deter- mined from fits to these and other data are much larger than the nominal or calculated contact sizes, which rangefrom 25 to 40 nm from a Sharvin-Maxwell calculation[15].W e infer that the excitations are ones with negligiblewave vector, i.e., the uniform FMR mode, although thisdoes not exclude the presence of excitations outside ourmeasurement bandwidth. Device-to-device variation ofthe measured fat a given His<10%, while the calcu- lated contact size varies by 60%, consistent with the excitation of a long-wavelength mode. Spectra taken over a wider range of frequencies show a peak at twice the frequency of the one discussed above asshown in Fig. 2(b). The frequencies, along with theirvariations with both IandH, differ by a factor of 2:00/.00060:01, and are observed in fields much larger than any anisotropies in the film. W e have not observed higher-harmonic signals. The ratios of the fto2famplitudes depend on both IandH, and show a nonmonotonic dependence on Iand a slight increase with H(inset). For precession symmetric about the fixed layer direction,the signal from a MR-derived voltage should be twice thephysical oscillation frequency of M. However, any mis- alignment between the layers would result in the detec-tion of a signal at fas well as2f. W e estimate a misalignment of a few degrees would give the fto2f amplitude ratios observed. The limiting slope of the data in Fig. 2(a) is 26GHz =T, in good agreement with the value expected from Eq. (1) for a first harmonic signal,indicating that the lower-frequency peaks correspond tothe physical precessional frequency of M. The devices also emit power at lower frequency. Figure 3 shows I/.01365mA and11mA spectra of the device along with the corresponding dV=dI curve. At lowI, no signal is found, but as Iis increased to 8mA , a shoulder in the dV=dI curve appears and a signal is observed, the strength of which increases with current. Inthis device, by I/.01368mA the coherent dynamics dis- cussed above have already turned off. However, we havemeasured other devices where both the high-frequencysingle-mode oscillations and the low-frequency signalhave been simultaneously observed over a range of cur- rents. From real-time measurements of the voltage fluc- tuations in these devices and nanopillars, we find that thislow-frequency signal results from two-state switching inthe device, as has also been reported in Ref. [5]. Thespectral shape follows a Lorentzian function, as expectedfor stochastic switching between two well-defined energystates [16]. At higher currents this switching typicallyceases, although this is not always the case before the highest Isupported by a contact ( /.002514mA ) is reached. The dynamics change dramatically with applied field direction. In Fig. 4(a) is a two-dimensional plot showingfas a function of Ifor the device discussed above, but FIG. 3. Low-frequency power spectra for two different cur- rents along with a fit to the data at I/.013611mA to a Lorentzian function. The fitted center frequency is f0/.01360/.000650MHz . (inset) The corresponding dV=dI curve. FIG. 2. (a) In-plane fvsHdispersion curve along with a fit to Eq. (1). Error bars (FWHM) are smaller than the data points.(b) Frequency spectra for I/.01365mA to 9 mA in 0.5 mA steps with /.0022H/.01360:06T showing responses at both fand2f. (Inset) fto2famplitude ratios as a function of Ifor two different fields.PHYSICAL REVIEW LETTERSweek ending 16 JANUARY 2004 VOLUME 92, N UMBER 2 027201-3 027201-3with an out-of-plane field of 0.9 T. Along the xaxisI varies from 4 to 12 mA and back to 4 mA. Avertical slicethrough the plot yields a frequency spectrum at a fixed I. This field aligns the Ni 80Fe20layer with Hwhile canting theCo90Fe10layer about30/.0014out of the film plane. For /.00220H>0:6T, a blueshift in fwith increasing Iis seen. More complicated behavior is also found, e.g., jumps in f occur at I/.01366mA and 7.5 mA. These jumps are not hysteretic and occur in all devices for out-of-plane fields.According to our modeling of this geometry, the Ni 80Fe20 magnetization precesses in a nearly circular orbit about H, with frequency increasing with I, the trend seen in our measurements. However, abrupt changes of fwith in- creasing Iare not found in our modeling. As shown in Fig. 4(b), dynamics persist to /.00220H/.0136 1:3T andf/.013638GHz , and spectrally can be well fit with a Lorentzian function (inset). Even at these frequen-cies, the voltage (power) linewidths are /.002560MHz (40MHz ), and have Q>650/.0133950/.0134. Because of band- width limitations we were not able to follow the oscil-lations to higher frequencies. At least for point contacts,the two-state switching behavior found with in-plane fields is largely suppressed in this geometry. As seen inFig. 4(b), the highest frequencies at a given field vary linearly in Hwith a slope of 32GHz =Tand give g/.0136 2:1/.00060:01, differing from the value determined from the in-plane measurements. It may be that His not yet large enough for fto be a truly linear function of H, leading to an inflated value of g. Finally, in contrast with FMR measurements, we note the excited frequencies here in-crease continuously in fields ranging from H<M NiFeto H>MNiFe, and persist even for H/.0136MNiFe when the FMR resonance frequency is nominally zero. W e thank J. A. Katine for supplying nanopillar devices, and M. D. Stiles and F . B. Mancoff for helpful discussions.This work was supported by the DARP A SPinS and NISTNano-magnetodynamics programs. [1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); 195, L261 (1999). [2] L. Berger, Phys. Rev. B 54, 9353 (1996). [3] M. D. Stiles and A. Zangwill, Phys. Rev. B 66,0 1 4 4 0 7 (2002); Y a. B. Bazaliy, B. A. Jones, and S. C. Zhang,J. Appl. Phys. 89, 6793 (2001); S. Zhang, P . M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); C. Heide, P . E. Zilberman, and R. J. Elliott, Phys. Rev. B 63, 064424 (2001); X. W aintal et al. , Phys. Rev. B 62, 12 317 (2000). [4] J. A. Katine et al. , Phys. Rev. Lett. 84, 3149 (2000); E. B. Myers et al. , Science 285, 867 (1999); F . B. Mancoff et al. , Appl. Phys. Lett. 83, 1596 (2003); J.-E. W egrowe et al. , Phys. Lett. 80, 3775 (2002); J. Grollier et al. , Appl. Phys. Lett. 78, 3663 (2001); J. Z. Sun et al. , Appl. Phys. Lett. 81, 2202 (2002); B. O ¨zyilmaz et al. , Phys. Rev. Lett. 91, 067203 (2003); W . H. Rippard, M. R. Pufall, and T . J. Silva, Appl. Phys. Lett. 82, 1260 (2003); Y . Ji, C. L. Chien, and M. D. Stiles, Phys. Rev. Lett. 90, 106601 (2003); S. M. Rezende et al. , Phys. Rev. Lett. 84, 4212 (2000). [5] S. Urazhdin et al. , Phys. Rev. Lett. 91, 146803 (2003). [6] J. C. Slonczewski, U.S. Patent No. 5695864, 1997. [7] S. I. Kiselev et al. , Nature (London) 425, 308 (2003). [8] M. Tsoi et al. , Phys. Rev. Lett. 80, 4281 (1998); M. Tsoi et al. , Nature (London) 406,4 6(2 0 0 0 ) . [9] M. R. Pufall, W . H. Rippard, and T. J. Silva, Appl. Phys. Lett. 83, 323 (2003). [10] W . F. Brown, Phys. Rev. 130, 1677 (1963); Jian-Gang Zhu, J. Appl. Phys. 91, 7273 (2002). [11] V . Kambersky and C. E. Patton, Phys. Rev. B 11, 2668 (1975). [12] J. P . Nibarger, R. Lopusnik, and T. J. Silva, Appl. Phys. Lett. 82, 2112 (2003). [13] S. Kaka et al. , J. Appl. Phys. 93, 7539 (2003); R. H. Koch et al. , Phys. Rev. Lett. 81, 4512 (1998). [14] Charles Kittel, Introduction to Solid State Physics (John Wiley, New Y ork, 1986), 6th ed., p. 454. [15] G. W exler, Proc. Phys. Soc. London 89, 927 (1966). [16] S. Machlup, J. Appl. Phys. 25, 341 (1954).18202224 Current (mA)8 8 12 4 4Frequency (GHz) 0.50 0.75 1.00 1.25203040 (b) 32 ± 0.8 GHz/T Frequency (GHz) Field (T)37.6 37.8 38.00.00.20.4∆f = 58 MHz Freq. (GHz)(a) 0.0nV/Hz0.50.9 Ampl. (nV/Hz1/2) FIG. 4 (color). (a) Plot of fvsIwith amplitude shown in a linear color scale from 0 (blue) to 0:9nV=Hz1=2(red), discre- tization results from measuring spectra in 500/.0022Aintervals. (b) Out-of-plane fvsHdispersion curve. Data correspond to the highest fat a given H. Error bars (FWHM) are smaller than the data points. (inset) Spectral peak at 1.3 T and I/.0136 11mA along with a fit.PHYSICAL REVIEW LETTERSweek ending 16 JANUARY 2004 VOLUME 92, N UMBER 2 027201-4 027201-4

PhysRevB.91.214435.pdf

PHYSICAL REVIEW B 91, 214435 (2015) Quantum mechanism of nonlocal Gilbert damping in magnetic trilayers Ehsan Barati and Marek Cinal Institute of Physical Chemistry of the Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland (Received 16 April 2015; published 30 June 2015) A fully quantum-mechanical calculation of the Gilbert damping constant αin magnetic trilayers is done by employing the torque-correlation formula within a realistic tight-binding model. A remarkable enhancement of α in Co/NM 1/NM 2trilayers is obtained due to adding the caps NM 2=Pd, Pt, and it decays with the thickness of the spacers NM 1=Cu, Ag, Au in agreement with experiment. Nonlocal origin of the Gilbert damping is visualized with its atomic layer contributions. It is shown that magnetization in Co is damped remotely by strong spin-orbitcoupling in NM 2via quantum states with large amplitude in both Co and NM 2. DOI: 10.1103/PhysRevB.91.214435 PACS number(s): 75 .78.−n,75.40.Gb,75.70.Tj I. INTRODUCTION Employing magnetic layered structures in spintronic de- vices such as hard-disk read heads and magnetic random access memories is the key ingredient in data storage technologyand its ongoing developments. This followed the discoveriesof interlayer exchange coupling [ 1,2], giant [ 3] and tunnel- ing [ 4,5] magnetoresistance, and spin-transfer torque [ 6–8] in trilayers built of ferromagnetic layers separated by non-magnetic spacers. Metallic trilayers are also commonly usedto investigate magnetization dynamics in view of potentialspintronic applications like racetrack memories [ 9] and spin torque nano-oscillators [ 10]. The dynamical processes in mag- netic nanodevices and, in particular, magnetization switchingare profoundly affected by magnetic damping due to spin-flipscattering and transfer of spin angular momentum. Magnetic relaxation in ferromagnetic metals is governed by the Gilbert damping, which enters the phenomenologicalLandau-Lifshitz-Gilbert (LLG) equation [ 11,12]. The Gilbert damping plays a crucial role in magnetization dynamics ofmagnetic layered systems. In particular, it affects the thresholdspin current required for magnetization switching [ 13] and the domain wall velocity in current-carrying domain wallstructures [ 9]. In the last two decades, extensive research activities have been devoted to magnetization dynamics in magneticfilms [ 14–25]. The Gilbert damping constant αin ferromag- net/nonmagnet (FM/NM) metallic bilayers is found to be ap-preciably enhanced in comparison with its bulk value [ 21,22]. The damping is also enhanced in FM /NM 1/NM 2trilayer structures with spacer layers of NM 1=Cu due to adding the NM 2=Pd, Pt, Ru, and Ta caps [ 16–18,21,23–25]. This experimental evidence clearly shows that the enhancement ofthe Gilbert damping in magnetic layered systems is of nonlocalorigin. An early theoretical paper on nonlocal magnetic damping is due to Berger [ 26]. He argued that the exchange coupling between itinerant spelectrons passing through the FM/NM interface and delectrons in the FM yields an enhanced Gilbert damping due to spin-flip electron transitions in which spinwaves are emitted or absorbed near the interface. The enhanced Gilbert damping in FM/NM layered systems is explained in a semiphenomenological way in Refs. [ 27,28] by pumping spin angular momentum from the FM into theadjacent nonmagnetic (normal metal) layers. According tothe spin pumping theory, the predicted damping enhancement in FM /NM 1/NM 2trilayers decays with the thickness of the NM 1spacer with low spin-flip rate. Although this theory gives a plausible general explanation of spin relaxationin magnetic layered systems, it does not provide a fullyquantum-mechanical description of this phenomenon. Sucha description can be achieved using Kambersk ´y’s torque correlation model [ 29] on which the present calculations are based. Despite numerous experiments no quantum calculations of magnetic damping in magnetic trilayer systems have been re-ported, except a recent paper [ 30] which addresses the Gilbert damping only in NM /Py/NM symmetric trilayers within an ab initio scattering formalism. In our recent work [ 31]t h e damping constant in bulk ferromagnets, ferromagnetic films,and FM/NM bilayers was calculated with the torque corre-lation formula within a realistic tight-binding (TB) model.Therein, it has been shown that magnetic damping in Co /Pd and Co /Pt bilayers has large nonlocal contributions from their nonmagnetic parts adjacent to the ferromagnetic Co layer.This paper is devoted to FM /NM 1/NM 2trilayers in which a significant damping contribution comes from the secondnonmagnetic part NM 2separated from the ferromagnetic layer by a magnetically inactive spacer. The aim of the present workis to establish the quantum mechanism of the nonlocal Gilbertdamping in such trilayers. Calculations have been performed for Co /NM 1/NM 2tri- layers with NM 1=Cu, Ag, and Au as the spacer and NM 2=Pd and Pt as the cap. The dependence of αon the spacer thickness and the electron scattering rate is investigated. The nonlocalorigin of the Gilbert damping in such systems is visualizedvia atomic layer contributions to α. To better understand the mechanism of the nonlocal damping we investigate the spatialdistribution of contributing electron states. II. THEORY A phenomenological description of magnetization dynam- ics in magnetic systems is given by the LLG equation ∂m ∂t=−γm×Heff+αm×∂m ∂t(1) that represents the time evolution of the unit vector mpointing along magnetization M. The first term in Eq. ( 1), with the gyromagnetic ratio γ, describes the Larmor precession of 1098-0121/2015/91(21)/214435(5) 214435-1 ©2015 American Physical SocietyEHSAN BARATI AND MAREK CINAL PHYSICAL REVIEW B 91, 214435 (2015) magnetization around the effective magnetic field Heff, applied externally and /or due to magnetic anisotropy. The second term, proportional to the Gilbert damping constant α, describes the relaxation of magnetization towards the direction of thefield. A pioneering quantum-mechanical description of the Gilbert damping dates back to 1976 when Kambersk ´yp r o - posed his torque correlation model [ 29]. The expression for αwithin this model takes the following form for a magnetic layered system [ 31] α=π NFMμs1 /Omega1BZ/integraldisplay dk/summationdisplay n,n/prime|Ann/prime(k)|2Fnn/prime(k). (2) It includes the integration over the wave vector kin the two-dimensional (2D) Brillouin zone (BZ) of the volume /Omega1BZ and the sum over band indices n,n/prime. The parameters μsand NFMstand for the atomic magnetic moment (in units of the Bohr magneton μB) and the number of atomic layers in the ferromagnetic part of the system, respectively. The matrixelements A nn/prime(k)=/angbracketleftnk|A−|n/primek/angbracketrightare found for the torque A−=[S−,HSO] due to the spin-orbit (SO) interaction HSO where the spin operator S−=1 2(σx−iσy) is given by the Pauli matrices σx,σy. The factor Fnn/prime(k) is defined as the integral over energy Fnn/prime(k)=/integraldisplay∞ −∞d/epsilon1η(/epsilon1)L(/epsilon1−/epsilon1n(k))L(/epsilon1−/epsilon1n/prime(k)).(3) Here, η(/epsilon1)=−dfFD/d/epsilon1 is the negative derivative of the Fermi-Dirac function fFD(/epsilon1), and the two Lorentzians L depend on the energies /epsilon1n(k),/epsilon1n/prime(k) of the electron states |nk/angbracketright,|n/primek/angbracketright, respectively. The width of the Lorentz function L(x)=(/Gamma1/2π)/(x2+/Gamma12/4) is the average electron scattering rate/Gamma1, treated here as an independent parameter. The present calculations are based on the TB model of the electronic structure in magnetic layered systems [ 31,32]. The TB Hamiltonian, including the SO interaction, is constructedwithin the Slater-Koster formalism [ 33,34]. The expression ( 2) is employed to calculate αin FM /NM 1/NM 2trilayers with out-of-plane magnetization; cf. Ref. [ 35] for a discussion on an arbitrary direction of M. The calculations are done for a wide range of scattering rates 0 .001 eV /lessorequalslant/Gamma1/lessorequalslant2.0 eV (expressed as/planckover2pi1/τwith the lifetime τin Refs. [ 36,37]). The integral in Eq. ( 3) is evaluated efficiently by summing over the Matsubara frequencies and the poles of the two Lorentz functions [ 31]. Since the calculated αis weakly dependent on temperature T entering fFD(/epsilon1)[31,37], finite T=300 K is used to obtain a fast convergence of αwith (60)2kpoints in the 2D BZ. The calculations are further speeded up by limiting the integrationto 1/8 of the 2D BZ. III. RESULTS In this paper, we particularly concentrate on calculation of αin Co/NM 1/NM 2trilayers. The considered NM 1=Cu, Ag, and Au spacers are poor spin sinks as possessing long spin-diffusion lengths λ sd(Refs. [ 18,25,38,39]) and the NM 2=Pd and Pt caps with short λsd(Ref. [ 25]) are known as perfect spin sinks.FIG. 1. (Color online) The Gilbert damping constant αin an Co(6 ML) film, Co(6 ML) /NM 1bilayers (NM 1=Cu, Ag and Au), and Co(6 ML) /NM 1/NM 2(4 ML) trilayers (NM 2=Pd, Pt) vs NM 1 thickness; the scattering rate /Gamma1=0.01 eV . In Fig. 1we depict the damping constant αversus the NM 1 spacer thickness in Co(6 ML) /NM 1/NM 2(4 ML) trilayers for the scattering rate /Gamma1=0.01 eV . For comparison, αfor the corresponding Co(6 ML) /NM 1bilayers and the Co(6 ML) film ( α/similarequal0.0026 for /Gamma1=0.01 eV) are also shown. The calculated αin the Co /NM 1/NM 2trilayers declines almost monotonically, while slightly oscillating, with increasing thethickness Nof the NM 1spacer layer; the oscillation periods are 5 ML for Cu and 5–7 ML for Ag. These oscillations areattributed to quantum well states with energies close to theFermi level /epsilon1 F. The damping constant αfound for the Co/Cu /Pt trilayer is larger than that of the Co /Cu bilayer in accord with experiment [ 18,23]. The enhancement is over threefold at the Cu thickness of 3 and 5 ML and more than twofold for5M L<N/lessorequalslant70 ML. Using Pd as the cap instead of Pt also results in significant damping enhancement though with muchsmaller values of αdue to the weaker SO coupling in Pd. Almost the same results are obtained if Ag is used insteadof Cu as the spacer, whereas the Au spacer leads to a higherdamping due to its strong SO coupling. The presently obtained 1 /N Codependence of αon the Co thickness NCoin Co/NM 1/NM 2trilayers (not shown) is also in agreement with experiment on FM /Cu/NM 2het- erostructures [ 16,17]. Other experimental reports on Py/Cu/Ta trilayers [ 21] and an Cu /Py/Cu/Pt system [ 18]h a v es h o w n that the contribution from the second nonmagnetic layer (i.e.,NM 2=Ta and Pt, respectively) vanishes for the spacer layer thicker than its λsd. However, such spacers are too thick for calculating αin the present model. For a spacer with thickness Nmuch smaller than its spin-diffusion length, the analytical formula for αderived in the spin pumping theory [ 25,28] yields the following simple dependence of α=A+B N+ConN.H e r e A,B, andCare expressed with NCo, the spin mixing conductance of the Co/NM 1interface and the parameters of both nonmagnetic metals: λsdand the electrical conductance. Our results for 214435-2QUANTUM MECHANISM OF NONLOCAL GILBERT DAMPING . . . PHYSICAL REVIEW B 91, 214435 (2015) FIG. 2. Gilbert damping constant αin Co(6 ML) /Cu(NML)/ Pt(4 ML) trilayers against the scattering rate /Gamma1for different Cu spacer thicknesses N. Co/NM 1/NM 2trilayers (Fig. 1) are perfectly fitted with this general formula within the considered range of the NM 1spacer thicknesses (up to 70 ML ≈1 2 . 5n m )w h i c ha r es m a l l e rb ya t least one order of magnitude than λsdof NM 1(200±50 nm for Cu [ 18,25]). Figure 2illustrates that the Gilbert damping in Co(6 ML)/Cu(NML)/Pt(4 ML) trilayers alters with the scattering rate/Gamma1in a similar way for different thicknesses of the Cu spacer. The minimum of αoccurs at /Gamma1∈[0.01 eV,0.1 eV] de- pending on N. Such a minimum occurs for bulk ferromagnets in the same range of /Gamma1[37,40]. As seen, the damping constant is almost independent of the spacer thickness for /Gamma1/greaterorequalslant0.05 eV . The experimentally observed decrease in αwith increasing N is obtained for the range /Gamma1< 0.05 eV , including /Gamma1=0.01 eV used in the present work. We attribute the obtained enhancement of the Gilbert damping in Co /NM 1/NM 2trilayers to the strong SO coupling in the NM 2cap as well as the high density of states at /epsilon1Fin NM 2. The effect of the former has already been confirmed for Co /Pt bilayers by switching off the SO coupling in the Pt cap [ 31]. The composition of the quantum states contributing most to αis discussed in more detail below. A deeper understanding of the nonlocal enhancement of the Gilbert damping can be achieved by analyzing its spatialdistribution. In our recent paper [ 31], an analytical expression for the damping constant α=N −1 FM/summationtext lαlrepresented by a sum of contributions αlfrom individual atomic layers lhas been derived and applied to ferromagnetic films and Co/NMbilayers. Therein, it has been shown how the Gilbert dampingwhich stems from the ferromagnetic (Co) part is also dampednonlocally in the nonmagnetic part of the bilayers. Here, theanalysis of layer contributions is utilized to investigate thenonlocal Gilbert damping in the Co /NM 1/NM 2trilayers. Figure 3presents the layer contributions to the damping constant for Co(6 ML) /NM 1/Pt(4 ML) trilayers with different thicknesses of the NM 1spacer. It is seen that the distribution of the Gilbert damping within such trilayer structures is similarfor different Cu spacer thicknesses. There are significant layercontributions in the Co part and almost no contributions fromatomic layers inside the Cu spacer. Dominating contributionsFIG. 3. Layer contributions to the damping constant αin Co/NM 1/Pt trilayers (NM 1=Cu, Au) with different NM 1thick- nesses; /Gamma1=0.01 eV . come from the Pt layers in a similar way as previously reported for Co /NM bilayers with NM =Pd and Pt [ 31]. As the Cu spacer gets thicker the contributions from the Pt layer getsmaller, in accordance with experiment [ 18] and prediction of the spin pumping theory [ 28]. However, even for the thickest considered spacers (70 ML thick) the total contribution from the Pt cap is larger than the contribution from the Co film.Such spatial distribution of the Gilbert damping is due to thelack and presence of dbands with energies very close to /epsilon1 F in Cu and Pt, respectively, as well as the strong SO coupling in Pt. Similar damping distributions (not shown) are obtainedfor Ag spacers and for the NM 2=Pd cap whose top of the d band lies above /epsilon1Fas in Pt. Since the SO coupling in Pd is weaker than in Pt the layer contributions inside the Pd cap aresmaller than in the Pt cap. This pattern is noticeably changed ifAu is used as the spacer instead of Cu since there are nonzerocontributions from the Au atomic layers at both the Co/Au andAu/Pt interfaces as well as the modified contributions from the Co and Pt interface atomic layers. The obtained results prove the nonlocal nature of the relax- ation process in the investigated trilayers where magnetizationprecesses in the ferromagnetic Co film, but it is damped inthe distant nonmagnetic cap separated from the Co film by amagnetically inactive spacer. To understand this mechanismon an even more fundamental level we examine the quantumstates that contribute to the Gilbert damping. The Gilbert damping in the torque-correlation model stems from two kinds of electron transitions: intraband ( n=n /prime) transitions within a single energy band and interband ( n/negationslash=n/prime) transitions between different energy bands [ 29,37]. The main source of the damping enhancement in Co(6 ML) /Cu(N ML)/Pt(4 ML) trilayers with /Gamma1=0.01 eV is the intraband transitions though the interband transitions also give a signifi-cant contribution to the damping constant as shown in Fig. 4(a). The spatial composition of quantum states contributing tothe intraband term of αis visualized in Figs. 4(b)–4(d).I ti s found that, while large contributions come from states almostentirely localized inside the Co film, the majority of states thatsignificantly contribute to the Gilbert damping span throughout 214435-3EHSAN BARATI AND MAREK CINAL PHYSICAL REVIEW B 91, 214435 (2015) FIG. 4. (Color online) (a) Intraband and interband terms of the damping constant αin the Co(4 ML)/Cu( NML)/Pt(4 ML) trilayers with/Gamma1=0.01 eV vs the Cu spacer thickness N; (b)–(d) contributions to the intraband term of αin the trilayers with N=3,9,30 ML from quantum states |nk/angbracketrightwith various fractions in the Co film and the Pt cap. The inclined yellow lines correspond to states with a fixedfraction in the Cu spacer. the whole trilayer. Such states have a substantial fraction in each of its three constituent parts: Co, Cu, and Pt. In thetrilayers with the Cu spacer a few ML thick [Fig. 4(b)] these fractions range from 0.2 to 1 in Co, from 0.0 to 0.8 in Pt, and upto 0.2 in Cu while summing up to 1 for each state. For thickerspacers the states giving predominant contributions to αhave smaller fractions in Pt, and they tend to be split into two groups.AtN=30 ML the group of states with fractions between 0.1 and 0.5 in both Co and Pt gives a contribution of 0.010to the total α=0.022. These states are responsible for thedamping enhancement due to the combination of their sizable amplitude in the Pt cap and the large SO coupling strength ofPt. The fraction of these states in Cu grows with increasingthe Cu spacer thickness however the average probability perCu atomic layer is similar for all investigated Cu thickness(3 ML, 9 ML, 30 ML) and it is around 0 .02 ML −1. Thus, the states leading to enhanced αin the Co /Cu/Pt trilayers with thick Cu spacer are composed of bulklike spstates in Cu and, attached to them, dstates in Co and Pt with amplitudes up to a few times larger than in Cu. IV . CONCLUSIONS We present a quantum-mechanical calculation of the Gilbert damping constant αin Co/NM 1/NM 2trilayers within the torque-correlation model. The damping is found to be remark-ably enhanced due to adding Pt as the second nonmagneticlayer NM 2, and it decreases with increasing the thickness of the NM 1=Cu, Ag, and Au spacers in agreement with experiment. The analysis of atomic layer contributions to αelucidates the nonlocal nature of the Gilbert damping in magnetic trilayers.The spatial decomposition of quantum states contributing tothe damping shows that its enhancement is due to delocalizedelectrons whose wave functions are sizable in all parts ofthe trilayers. The spins of such electrons contribute to themagnetization in the ferromagnetic Co layer, but they alsostrongly interact, via the SO coupling, with heavy atoms in thePt layer. Therefore, the precession of these spins is dampedefficiently, and this leads to enhanced damping of the totalmagnetic moment, although it is almost entirely confined tothe Co layer. This paper thus provides insight into quantummechanisms of magnetic damping in metallic layered systems. ACKNOWLEDGMENTS We acknowledge the financial support of the Foundation for Polish Science within the International PhD Projects Pro-gramme, cofinanced by the European Regional DevelopmentFund within Innovative Economy Operational Programme“Grants for innovation”. [1] P. Gr ¨unberg, R. Schreiber, Y . Pang, M. B. Brodsky, and H. Sowers, P h y s .R e v .L e t t . 57,2442 (1986 ). [2] S. S. P. Parkin, N. More, and K. P. Roche, Phys. Rev. Lett. 64, 2304 (1990 ). [3] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. 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PhysRevB.100.235453.pdf

PHYSICAL REVIEW B 100, 235453 (2019) Controlling spins with surface magnon polaritons Jamison Sloan,1,*,†Nicholas Rivera,2,*John D. Joannopoulos,2Ido Kaminer,3and Marin Solja ˇci´c2 1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3Department of Electrical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel (Received 11 February 2019; revised manuscript received 11 November 2019; published 30 December 2019) Polaritons in metals, semimetals, semiconductors, and polar insulators can allow for extreme confinement of electromagnetic energy, providing many promising opportunities for enhancing typically weak light-matterinteractions such as multipolar radiation, multiphoton spontaneous emission, Raman scattering, and materialnonlinearities. These extremely confined polaritons are quasielectrostatic in nature, with most of their energyresiding in the electric field. As a result, these “electric” polaritons are far from optimized for enhancingemission of a magnetic nature, such as spin relaxation, which is typically many orders of magnitude slowerthan corresponding electric decays. Here, we take concepts of “electric” polaritons into magnetic materials, andpropose using surface magnon polaritons in negative magnetic permeability materials to strongly enhance spin relaxation in nearby emitters. Specifically, we provide quantitative examples with MnF 2and FeF 2, enhancing spin transitions in the THz spectral range. We find that these magnetic polaritons in 100-nm thin films can beconfined to lengths over 10 000 times smaller than the wavelength of a photon at the same frequency, allowingfor a surprising 12 orders of magnitude enhancement in magnetic dipole transitions. This takes THz spin-fliptransitions, which normally occur at timescales on the order of a year, and forces them to occur at sub-mstimescales. Our results suggest an interesting platform for polaritonics at THz frequencies, and more broadly, away to use polaritons to control light-matter interactions. DOI: 10.1103/PhysRevB.100.235453 Polaritons, collective excitations of light and matter, offer the ability to concentrate electromagnetic energy down tovolumes far below that of a photon in free space [ 1–6], holding promise to achieve the long-standing goal of low-loss con-finement of electromagnetic energy at the near-atomic scale.The most famous examples are surface plasmon polaritons onconductors, which arise from the coherent sloshing of surfacecharges accompanied by an evanescent electromagnetic field.These collective excitations are so widespread in optics thattheir manipulation is referred to as plasmonics. Plasmonsenjoy a myriad of applications, particularly in spectroscopydue to their enhanced interactions with matter. This enhance-ment applies to spontaneous emission, Raman scattering,optical nonlinearities, and even dipole-“forbidden” transitionsin emitters [ 7–16]. Beyond plasmons in metals, polaritons in polar dielectrics, such as phonon polaritons [ 17–20]a r en o w being exploited for similar applications due to their ability toconcentrate electromagnetic energy on the nanoscale in themid-IR /THz spectral range. The ability of nanoconfined polaritons to strongly enhance electromagnetic interactions with matter can ultimately beunderstood in terms of electromagnetic energy density. Anelectromagnetic quantum of energy ¯ hω, confined to a volume V, leads to a characteristic root-mean-square electric field of order√ ¯hω /epsilon10V. In the case of field interaction with an electron in an emitter, this characteristic field drives spontaneous emis-sion, and thus concentration of energy to smaller volumes *These authors contributed equally to this paper. †Corresponding author: jamison@mit.eduleads to enhanced emission. This well-studied phenomenonis best known as the Purcell effect [ 21]. Interestingly, if one looks at the electromagnetic energy distribution of a highlyconfined plasmon or phonon polariton, one finds that an over-whelming majority of this energy resides in the electric field[22–24]. For a polariton with a wavelength 100 times smaller than that of a photon at the same frequency, the magnitudeofEis then 100 times larger than that of μ 0cH. In sharp contrast to free space wave propagation, the energy residingin the magnetic field is of the order of a mere 0 .01% of the total energy ¯ hω. This largely suggests that such excitations are relatively inefficient for enhancing spontaneous emissionprocesses which couple to the magnetic field, such as spin-fliptransitions or magnetic multipole decays. As such, enablingmagnetic decays at very fast rates represents a rewarding chal-lenge, as increasing rates of spontaneous emission can provideopportunities for detectors, devices, and sources of light. The Purcell enhancement of magnetic dipole transitions has been approached by a few basic means: The use ofhighly confined resonances at optical frequencies [ 25,26], metamaterials [ 27,28], and for microwave frequencies, mate- rials with simultaneously very high quality factors and highlyconfined fields. These advances are reviewed in Ref. [ 29]. Many of these methods have the benefit of compatibility withwell-known materials and use at optical frequencies, but thePurcell enhancements in these cases are typically very farfrom maximal Purcell enhancements that can be achievedwith “electric” polaritons at similar frequencies [ 14,16,30– 33]. This prompts the question: What kind of electromagnetic response allows one to achieve a similar degree of very strongenhancement for magnetic transitions? 2469-9950/2019/100(23)/235453(11) 235453-1 ©2019 American Physical SocietyJAMISON SLOAN et al. PHYSICAL REVIEW B 100, 235453 (2019) The duality between electric and magnetic phenomena, combined with ideas from plasmonics and nano-optics, sug-gests a pathway for achieving strong magnetic transitionenhancement: Highly confined magnetic modes in materialswith negative magnetic permeability . In particular, plasmon and phonon polaritons are associated with a negative dielec-tric permittivity /epsilon1(ω). By the well-established principle of electromagnetic duality [ 34,35], if one replaces /epsilon1(ω) with the magnetic permeability μ(ω), then the electric field Ein the dielectric structure becomes the magnetic field Hin the dual magnetic structure. Thus, to very efficiently enhance magneticdecays, one desires a material with negative μ(ω) which sup- ports modes dual to “electric” surface polaritons. While likelynot the only example, AFMR is a well-studied example of aphenomenon which can provide precisely this permeability,and the corresponding modes are surface magnon polaritons(SMPs) [ 36–38]. Here, we use macroscopic quantum electrodynamics (MQED) of magnetic materials to propose extreme enhance-ment of magnetic transitions in nearby quantum emitters byusing highly confined SMPs. We find enhancement of spinrelaxation rates by over 12 orders of magnitude, showingmagnetic Purcell enhancements as large as the highest limitspredicted for electric Purcell enhancements. We discuss howthe losses present in magnetic materials impact the magneticdecay rate and argue that even with these considerations,extremely large enhancements can be achieved. Such en-hancements could provide access to extremely fast magneticdipole decays, shortening radiative lifetimes on the order of ayear to submillisecond timescales. The organization of this paper is as follows: In Sec. I,w e review the classical electrodynamics of SMPs and derive thedispersion relation and mode profile of SMPs for the exam-ple of an antiferromagnetic thin film. We briefly review thepropagation properties of these modes and, in particular, notetheir extremely large confinement. In Sec. II,w eu s eM Q E D to quantize the SMP modes and calculate the spontaneousemission rate of nearby magnetic dipole emitters into thesemodes. Finally, in Sec. III, we provide quantitative results for the spontaneous emission by spin systems near existingmagnon-polaritonic materials, such as MnF 2and FeF 2. I. SURFACE MAGNON POLARITON MODES The spin interactions in solids which give rise to different varieties of magnetic order have been studied extensively.Of particular note for our purposes is the study of the long-range order established by spin waves in (anti)ferromagnets[39–47]. These spin waves can be excited at the level of a single quantum, and the quasiparticles associated with theseexcitations are magnons [ 37]. More recently, magnons have attracted considerable attention for their ability to interactwith electric currents and electron spins, leading to the rapidlygrowing field of magnon spintronics [ 48–57]. We begin by reviewing the confined modes which exist on thin films of materials with negative magnetic permeability,denoted μ(ω). The modes we describe are well-studied SMPs [36,58–60] with Re μ(ω)/lessorequalslant0. At a microscopic level, the modes correspond to ordered precession of the spins in anantiferromagnetic lattice and are also referred to as surfaceTABLE I. Anisotropy fields, exchange fields, sublattice mag- netization, resonance frequencies, and damping constants (where known) for antiferromagnetic materials that can support SMPs. Parameters are taken from Refs. [ 66,69]. Material μ0HA(T)μ0HE(T)μ0HM(T)ω0(rad THz) τ(nsec) MnF 2 0.787 53.0 0.06 1.69 7.58 FeF 2 19.745 53.3 0.056 9.89 0.11 GdAlO 3 0.365 1.88 0.062 0.23 – spin waves [ 61]. The classical dynamics of spin-wave propa- gation are governed by the Landau-Lifshitz-Gilbert equation,which accounts for damping [ 62,63]. These microscopic inter- actions give rise to a magnetic susceptibility (or equivalentlya magnetic permeability) which dictates how macroscopicelectromagnetic fields propagate in the material. Given theclassical solutions to the Maxwell equations in a materialconfiguration, one can then quantize the magnon modes,allowing the use of quantum optics techniques to describethe interaction of magnon modes in the vicinity of emitters.We construct these classical solutions, quantize these modes,and then solve for magnetic dipole transition rates into thesemodes. For the specific case of an antiferromagnetic material near resonance, the frequency-dependent permeability which in-cludes material losses takes the form of a Lorentz oscillatorwhich depends on the microscopic magnetic properties ofthe antiferromagnetic crystal. Studies of the crystal structuresof important antiferromagnetic materials can be found inRef. [ 64]. The magnetic permeability function for antiferro- magnetic resonance (AFMR) in the absence of an externalmagnetizing field from [ 65–67]i s μ xx=μyy=1+2γ2HAHM ω2 0−(ω+i/Gamma1)2, (1) with coordinates shown in Fig. 1.I nE q .( 1),ω0is the resonance frequency, HAis the anisotropy field, HMis the sub- lattice magnetization field, γis the gyromagnetic ratio, and /Gamma1=1/τis a phenomenological damping parameter inversely proportional to the loss relaxation time τ. Furthermore, in the approximation of low damping, the resonant frequency isgiven as ω 0=γ√2HA(HA+HE), where HEis the exchange field which is representative of the magnetic field required toinvert neighboring spin pairs. For antiferromagnetic materialssuch as MnF 2and FeF 2, the resonance frequencies ωtakes values 1 .69×1012and 9.89×1012rad/s, respectively, and have negative permeability over a relatively narrow bandwidthon the scale of a few GHz. Most importantly for our purposes,Reμ(ω)<0f o rω<ω 0<ω max, which will permit surface- confined modes. Finally, we note that we have implicitly as-sumed that the magnetic permeability carries no dependenceon the wave vector through nonlocal effects. For wavelengthswhich substantially exceed the atomic lattice spacing, thisshould be an excellent approximation. A more detailed dis-cussion of nonlocality in terms of mean-field parameters fromLandau-Ginzburg phase transition theory can be found inRef. [ 68]. Table Ishows values of material parameters for a variety of antiferromagnetic materials. Figure 2(a) shows the real and imaginary parts of the magnetic permeability μ(ω) 235453-2CONTROLLING SPINS WITH SURFACE MAGNON … PHYSICAL REVIEW B 100, 235453 (2019) FIG. 1. Electromagnetically dual relationship between surface plasmon polaritons on negative permittivity materials and surface magnon polaritons on negative permeability materials. (a) Surface plasmon polariton represented as charge density oscillations in a negative /epsilon1material. These quantum fluctuations can couple strongly to an electric dipole emitter near the surface to drive enhanced spontaneous emission.(b) Surface magnon polariton represented as a spin density oscillation in a negative μmaterial. These quantum fluctuations can couple strongly to a magnetic dipole emitter near the surface to drive enhanced spontaneous emission. Both electric and magnetic surface polaritons can exhibit strong mode confinement, helping to overcome the mismatch between mode wavelength and emitter size. associated with the AFMR in MnF 2. We see that at the peak of the resonance, Re( μ)≈−40 and Im( μ)≈90. We now discuss the geometry of the thin-film configura- tions we study. Antiferromagnetic fluorides exhibit a uniaxialpermeability structure with two orthogonal components ofthe permeability tensor given by μ(ω) above, and the other orthogonal component as unity. We start by focusing oncrystal orientations in which μ=(μ(ω),μ(ω),1). It is also worthwhile to note that experiments, specifically on nonre-ciprocal optical phenomena [ 70], have been performed on these materials in a less conventional geometry where μ= (μ(ω),1,μ(ω)). The in-plane anisotropy of this configuration substantially complicates the dispersion relation and propaga-tion structure of the modes. As such, we focus primarily on theisotropic case but present results for the in-plane anisotropiccase near the end of the text. For concreteness, we focus on MnF 2, a material which has been studied in depth both in theory and experiment[71,72], and also exhibits a relatively low propagation loss. We note that FeF 2is also a promising candidate with higher resonance frequency, but also higher loss [ 73,74]. We solve for SMPs supported by optically very thin (here, submicronthickness denoted by d)M n F 2films surrounded by air. For the confined modes we consider, the effect of retardation isnegligible [ 75], and thus we can find the magnon modes using a quasimagnetostatic treatment as described in Ref. [ 66]. In the magnetostatic limit, the resulting “polaritons” are muchmore magnonlike than photonlike. Nevertheless, many ofthe applications which are considered in polaritonics arefeasible with these modes [ 2,4]. In the absence of retardation, the electric field is negligible, and the magnetic field, sincethere are no free currents, satisfies ∇×H=0. Thus the magnetic field can then be written as the gradient of a scalarpotential H=∇ψ H. This scalar potential then satisfies a scalar Laplace equation, ∂iμij(ω)∂jψH=0, (2) where we have used repeated indices to denote summation. In this paper, the absence of applied magnetic fields guaranteesthatμ ijis diagonal, and so Eq. ( 2) contains only three terms. Applying boundary conditions for the continuity of Bin the zdirection and of Hin the xyplane at the two interfaces of afilm of thickness dgives the dispersion relation qn=1 2d√−μ(ω)/bracketleftbigg tan−1/parenleftbigg1√−μ(ω)/parenrightbigg +nπ 2/bracketrightbigg , (3) where nis an integer, qnis the in-plane wave vector of mode n, andμ(ω) is the permeability given in Eq. ( 1). We see that qn is inversely proportional to the thickness of the slab d, which is anticipated, as the thickness of the material sets the scaleof the wave solution in the zdirection. Identical to confined modes on thin films of plasmonic materials (silver and goldfor instance), a thinner film results in a smaller wavelength.An extreme limiting case in plasmonics is graphene, in whichan atomically thin layer is capable of confining surface plas-mons with confinement factors of 200 [ 5]. Figure 2(c) shows plots of the scalar potential ψ Hassociated with SMP modes on MnF 2, which is proportional to the magnetic field in direction of propagation. The scalar potential solutions to the Laplaceequation take the form ψ n H(r,ω)=/braceleftBigg eiqn·ρe−qn|z||z|>d/2/parenleftbige−qnd f(qnd)/parenrightbig eiqn·ρf(qnz)|z|<d/2,(4) where ρ=(x,y) is the in-plane position, f(x)=cos(x)f o r even modes, and f(x)=sin(x) for odd modes. Taking the gra- dient of the scalar potential gives the fully vectorial magneticfield, which reveals that the SMP mode propagates in the in-plane direction ˆ qwith circular polarization ˆ ε q=(ˆq+iˆz)/√ 2. This polarization is well known to be typical of quasistaticsurface polariton modes, whether they are the transverse mag-netic modes associated with quasielectrostatic excitations ortransverse electric modes associated with quasimagnetostaticexcitations. We now discuss the key properties of these surface modes, including their dispersion, confinement, velocities,and quality factor resulting from material losses. In Fig. 2(b), we plot the material-thickness-invariant dispersion relationω(qd). The dimensionless wave vector qdindicates how the size of the in-plane wave vector compares to the thickness ofthe film. We note that we have incorporated the effect of lossinto the dispersion by finding solutions with real frequencyand complex wave vector. Our dispersion plots show the realpart of the wave vector. In the lossless limit, the dispersion 235453-3JAMISON SLOAN et al. PHYSICAL REVIEW B 100, 235453 (2019) FIG. 2. Surface magnon polariton (SMP) modes on MnF 2. (a) Frequency-dependent permeability function for MnF 2calculated using Eq. ( 1) and using the parameters given in Table I.F o rM n F 2, the resonance frequency is ω0=1.68×1012rad/s. For ω0<ω<ω max, Re(μ)<0, allowing for surface modes. (b) Dispersion relation for MnF 2of thickness d, calculated in the quasimagnetostatic limit which is valid in the range of thicknesses dwe consider. The first four modes are shown. (c) Visualization of fundamental and first harmonic mode SMP through the field component Hxshown for a d=200-nm film of MnF 2atω/ω 0=1.005. The locations of these two modes are indicated on the dispersion curve. is asymptotic to a fixed frequency in the limit that q→∞ . The introduction of loss causes the band to fold back on itself,placing a limit on the wave vectors which can be excited.Consequently, modes near the peak of this folded band exhibitthe highest attenuation. The dispersion plot shows the first four bands—the fundamental mode ( n=0) as well as three higher harmonics (n=1,2,3). Due to the the reflection symmetry of the geom- etry in the zdirection, two of these modes are even parity, and two are odd parity. We can interpret the mode index as thenumber of half oscillations which the magnetic field makes inthezdirection of the film. Higher order modes will have larger wave vectors. Once again, we can further understand thedispersion relation of these modes through analogy to existingpolaritonic systems. Specifically, MnF 2is a hyperbolic material since μ⊥>0 while μ/bardbl<0 (where the directions ⊥ and/bardblare taken with respect to the zaxis). This is much like the naturally occurring hyperbolic material hexagonal boronnitride, which has one component of its permittivity negative,while another component is positive [ 18,19]. As a result of this, these systems have a multiply branched dispersion, andthe electromagnetic fields are guided inside the crystal. Thefirst two modes ( n=0,1) are shown in Fig. 2(c), where we note the mode confinement to the slab, as well as theevanescent tails which enable interaction with surrounding emitters. The most impressive figure of merit of these modes is the size of their wavelength in comparison to the free spacewavelength at a given frequency, also known as a confinementfactor or effective index of the mode. Figure 3(b) highlights this, showing the confinement factor η=qc/ω=λ 0/λSMPfor the first four modes ( n=0,1,2,3) on d=200 nm MnF 2as a function of frequency. We see that the fundamental modereaches a peak confinement of η=2×10 4, while the first harmonic is confined to twice that with η=4×104. These values exceed by two orders of magnitude the maximum confinement values that have been observed incommon plasmonic media such as thin films of silver, gold,or titanium nitride, or doped graphene. Furthermore, sincethe confinement scales linearly with q∼1/d, decreasing the material thickness increases the achievable range of confine-ment factors. As a simple example of this, consider that amaterial thickness of d=50 nm would correspond to a wave vector four times larger than for d=200 nm, in other words a maximum fundamental mode confinement of 8 ×10 4, and a confinement above 104for much of the surface magnon band. An explanation for this high confinement in terms of most basic principles is that the frequencies at which SMPs exist FIG. 3. Propagation properties of SMP modes on MnF 2. The following dimensionless quantities are plotted for MnF 2with propagation loss τ=7.58 nsec for the first four modes indexed by n=(0,1,2,3): (a) mode quality factor Q=Re(q)/Im(q) as a function of mode frequency, (b) mode confinement factor η=qc/ωas a function of mode frequency, and (c) normalized group velocity vg/c=|dω/dk|/cas a function of mode frequency. 235453-4CONTROLLING SPINS WITH SURFACE MAGNON … PHYSICAL REVIEW B 100, 235453 (2019) (GHz-THz) are orders of magnitude lower than for plasmons which typically exist in IR to optical regimes. Simultaneously,the scale of the wave vector qin both plasmonic and magnonic media is set by the film thickness dfor electrostatic and mag- netostatic modes, respectively (this means that plasmons andmagnons will have wave vectors of similar scale, regardlessof frequency). In other words, at a fixed material thickness,lower frequency surface magnons have substantially higherpotential for geometrical squeezing than surface plasmons.We note that this is not of purely formal interest, as whenconsidering the enhancement of spontaneous emission, onefinds that the enhancement is proportional to a power ofprecisely this confinement factor. In addition to understanding the confinement of magnon polaritons, it is also important to understand their propagationcharacteristics, such as propagation quality factor, and groupvelocity. Figures 3(a) and3(c) shows the quality factor Q= Re(q)/Im(q), as well as the normalized group velocity v g/c as a function of frequency for the first four modes. We seethat propagation losses are lowest toward the middle of theallowed frequency band, showing quality factors greater than20 for the fundamental mode ( n=0). Additionally, we see that the group velocity v greaches its maximum near the lower portion of the allowed frequency range, and goes toward zeroat the other end. II. THEORY OF SPIN RELAXATION INTO MAGNON POLARITONS We now discuss how an an emitter with a magnetic dipole transition placed above the surface of a thin negative per-meability material can undergo spontaneous emission intoSMPs which is much faster than the emission into free spacephotons. First, we consider the Hamiltonian which couplesthe magnetic moment of the emitter to the quantized mag-netic field. Fluctuations in the evanescent magnetic field fromSMPs can then cause the emitter to relax via the emission of aSMP. The rate at which this process occurs is calculated usingFermi’s golden rule. Finally, we discuss the effect of materiallosses on the total decay rate, and argue that for parameters ofinterest, the effect should be small. We first discuss the mechanisms that can allow an emitter to couple to highly confined SMPs. A magnetic field can cou-ple to both the electron spin angular momentum and orbitalangular momentum, as both angular momenta contribute tothe electron’s magnetic moment. We describe this interactionquantum mechanically with an interaction Hamiltonian H int between an emitter and a magnetic field [ 76,77] Hint=−μ·B=−μB(L+gS) ¯h·B, (5) where μis the total magnetic moment of the emitter, S= ¯h 2σis the spin angular momentum operator, Lis the orbital angular momentum operator, g≈2.002 is the Landé g-factor. In this Hamiltonian, we note that Bis the quantized magnetic field operator associated with SMP modes. To provide a fully quantum mechanical description of the interactions, we use the formalism of macroscopic QED(MQED) to rigorously quantize the electromagnetic fieldmodes in a medium (in this case, a thin slab of negativepermeability material) This approach is similar to that in Ref. [ 78], which was applied to quantize electromagnetic fields in dielectric structures. We consider a geometry ofan e g a t i v e μmaterial which is translation invariant (i.e., a slab geometry). In this case, the modes are labeled by anin-plane wave vector q. We can then construct an operator which creates and annihilates excitations of the magnetic fieldwhich are normalized so each SMP carries energy ¯ hω q.T h e magnetic field operator in the evanescent region above the slab(z>d/2) takes the form B(r)=/summationdisplay q/radicalBigg μ0¯hω 2ACq(ˆεqeiq·ρe−qzaq+ˆε∗ qe−iq·ρe−qza† q),(6) where a† qandaqare creation and annihilation operators for the SMP modes satisfying the canonical commutation relation[a q,a† q/prime]=δqq/prime,ˆεqis the mode polarization, Ais the area normalization factor, and Cq=/integraltext dzH∗(z)·d(μω) dω·H(z)i sa normalization factor ensuring that the mode H=∇ψHhas an energy of ¯ hωq. The energy has been calculated according to the Brillouin formula for the electromagnetic field energyin a dispersive medium in a transparency window [ 79,80]. As a point of comparison, we note that similar quantiza-tion schemes have been implemented for surface plasmon-polariton modes on graphene [ 22] and many other systems in optics [ 78,81]. In this expression for the energy, we have also used the fact that the modes are magnetostatic in nature, so thecontribution of the electric field to the energy associated withthem is negligible. To establish the strength of the coupling between a mag- netic dipole emitter and SMPs, we calculate spontaneousemission of a spin into a thin negative μmaterial such as an antiferromagnet, using Fermi’s golden rule. The rate oftransition via the emission of a magnon of wave vector qis given as /Gamma1 (eg) q=2π ¯h2|/angbracketleftg,q|Hint|e,0/angbracketright|2δ(ωq−ωeg). (7) We specify the initial and final states of the system as |e,0/angbracketrightand|g,q/angbracketright, respectively, where e and g index the excited and ground states of the emitter, qis the wave vector of the magnon resulting from spontaneous emission, ωqis its corresponding frequency, and ωegis the frequency of the spin transition. Note that Eq. ( 7) applies generally and can capture any multipolar magnetic transition. With the magnetic field quantized appropriately and the in- teraction Hamiltonian established, obtaining the spontaneousemission rate proceeds in the usual way. Substituting Eq. ( 6) into the Hamiltonian of Eq. ( 5), and then applying Fermi’s golden rule as written in Eq. ( 7), we find that the spontaneous emission rate /Gamma1 (eg)per unit magnon in-plane propagation angleθis given by d/Gamma1(eg) dipole dθ=μ2 Bμ0ωeg 2π¯hq3(ωeg) Cq(ωeg)|vg(ωeg)|e−2q(ωeg)z0|Meg|2,(8) where |vg|=|∇qω|is the magnitude of the SMP group veloc- ity,μBis the Bohr magneton, and Meg=/angbracketleftg|ˆ/epsilon1q·(L+gS)|e/angbracketrightis the matrix element which describes the transition. Also notethat here we have made the dipole approximation for magnetic 235453-5JAMISON SLOAN et al. PHYSICAL REVIEW B 100, 235453 (2019) transitions, which comes from assuming that the evanescent field of the emitted SMP varies negligibly over the size ofthe emitter, and can thus be assumed constant. However, ifone wishes to remove this simplifying assumption to considermagnetic multipole transitions, the matrix element can benumerically evaluated. To simplify the proceeding discussion,we focus on cases where the transition corresponds only toa change of spin of the electron in the emitter from |↑/angbracketright to|↓/angbracketright, this matrix element is simply proportional to σ eg= /angbracketleft↓ |σ·ˆεq|↑ /angbracketright. Here, the angular dependence can come solely from the magnon polarization. For a spin transition orientedalong the z(i.e., out-of-plane) axis, the transition strength into modes at different θwill be the same, and thus the distribution of emitted magnons isotropic. Spin transitionsalong a different axis will break this symmetry, resultingin angle-dependent emission. In any case, the total rate of emission is obtained by integrating over all angles as /Gamma1 (eg) dipole=/integraltext2π 0(d/Gamma1(eg) dθ)dθ. We now consider the effect of material losses, and argue that the lossless approximation for decay rates presented hereshould provide a strong approximation for decay rates inthe presence of losses. The formalism of macroscopic QEDdetailed in Ref. [ 34] can be used to incorporate material losses into spontaneous emission calculations. It was foundexplicitly in Ref. [ 14] that, in general, the presence of losses does not drastically change the total decay rate of the emitter,unless the emitter is at distances from the material muchsmaller than the inverse wave vector of the modes that areemitted. For the case of relatively low losses, Fermi’s goldenr u l es h o w ni nE q .( 7) can be modified by replacing the delta function density of states with a Lorentzian of width /Delta1ω≡ 1/τ. The lossy decay rate is then obtained as a convolution of this Lorentzian frequency spread with the lossless rate as /Gamma1 (eg) dipole−→/integraldisplay /Gamma1(eg) dipole/parenleftbigg1 π1/(2τ) (ωeg−ω)2+(1/2τ)2/parenrightbigg dω. (9) In general, this correction from losses will be small provided that the range of frequencies /Delta1ω coupled by Eq. ( 9)i s small compared to the width of the magnon band, denoted/Delta1/Omega1. More succinctly, losses are negligible if /Delta1ω//Delta1/Omega1 /lessmuch1. For the MnF 2considered here, /Delta1ω≈10−8s−1, and/Delta1/Omega1≈ 1010s−1,s o/Delta1ω//Delta1/Omega1 ≈10−2, confirming that the Lorentzian distribution behaves similarly to a delta function δ(ωeg−ω) which does not mix frequencies. Having presented the generalframework for analyzing SMP emission, we now presentspecific results for SMP emission into a thin film of MnF 2. III. TRANSITION RATE RESULTS A. Dipole transition rates We first discuss the transition rates and associated Purcell factors of magnetic dipole emitters. For a z-oriented spin flip of frequency ωegplaced a distance z0from the surface of a negative μfilm, the spontaneous emission rate is given as /Gamma1(eg) dipole=μ2 Bμ0ωeg ¯hq2(ωeg) C/prime(ωeg)|vg(ωeg)|e−2q(ωeg)z0, (10) where C/prime(ω)=C(ω)/q(ω) is introduced to remove the wave- vector dependence from the normalization. We also note that FIG. 4. Dipole transition rate enhancement by SMPs. (a) Dipole transition rate for a z-oriented spin flip as a function of normalized frequency and distance z0from the emitter to the surface of a d=200 nm MnF 2film. The transition rates decay exponentially with increasing distance from the surface. (b) Line cuts of the information shown in (a) for different fixed distances z0.T h ea x i s on the left shows the total transition rate, while the axis on the right shows the Purcell factor; in other words, the transition ratenormalized by the free space transition rate. the group velocity |vg(ω)|∝1/q(ω), and thus the whole expression, carries a wave-vector dependence of /Gamma1(eg) dipole∝ q3(ωeg). We now discuss the numerical values for spin-flip tran- sition rates in nearby emitters which come directly fromEq. ( 10). We find these transition rates into SMPs to be orders of magnitude faster than the rates of transition into free-spacephotons at the same frequency. Figure 4shows the emission rate as a function of frequency ωand emitter distance z 0for ad=200 nm MnF 2film. Figure 4(b) shows line cuts of the dipole transition rate at various emitter distances z0.I n this geometry, we find that for the highest supported magnonfrequencies, the total rate of emission may exceed 10 5s−1, which corresponds to a decay time of 10 μs. This is 11 or- ders of magnitude of improvement over the free-space decaylifetime of more than a week. We see that for sufficientlyclose distances z 0, the decay rate increases with ω, spanning many orders of magnitude over a small frequency bandwidth.Furthermore, we see that with increasing distance z 0,t h e total decay rate is suppressed exponentially by the evanescenttail of the surface magnon. More specifically, we see in theexponential dependence e −2q(ωeg)z0that, for rate enhancement to be effective, z0should be comparable to or ideally smaller than 1 /q∼d. For a 200-nm film, enhancement begins to 235453-6CONTROLLING SPINS WITH SURFACE MAGNON … PHYSICAL REVIEW B 100, 235453 (2019) saturate for z0<20 nm. In terms of a potential experiment, these are promising parameters which could result in a totaltransition rate of 10 4s−1. Finally, we note that at distances z0 extremely near to the surface, effects such as material losses or nonlocality may cause the behavior of the transition rateto deviate slightly from the predicted behavior. The exactmagnitude of such effects could be taken into account directlyby solving for the dispersion with the full, nonlocal, magneticsusceptibility which is presented in Ref. [ 68]. It is also worthwhile to consider not only the total transition rates, but also the Purcell factors. The right side axis ofFig. 4(b) shows the Purcell factor for spin relaxation into SMPs, computed as the ratio between the enhanced transi-tion rate and the free-space transition rate, and denoted asF p(ω)=/Gamma1dipole//Gamma10. We note that while the transition rate in the magnonic environment is technically the sum of theSMP emission rate and the radiative rate, in our systems theradiative rate is so small that it need not be considered. Thinner films offer even more drastic capabilities for en- hancement. The dipole transition rate and Purcell factor scaleasη 3, which means that shrinking the film thickness deven by conservative factors can result in a rapid increase in themaximum transition rate achievable. This η 3scaling is exactly the same scaling found for Purcell factors of electric dipoletransition enhancement in the vicinity of highly confined elec-trostatic modes such as surface plasmon polaritons [ 14,17,33]. Having established the duality between electric and mag- netic surface polaritonics in the context of Purcell enhance-ment, other important conclusions about the scope and utilityof SMPs follow. Most notably, Purcell factors for higher ordermagnetic processes should scale with mode confinement iden-tical to those for the corresponding electric processes. Givenan emitter-material system that can support such processes,it should be possible to compute transition rates of higherorder processes such as magnetic quadrupole transitions andmultimagnon emission processes. Conveniently, electromag-netic duality implies that the confinement scaling properties ofall electric multipolar or multiphoton transitions into electricpolaritons are identical to those of their magnetic analogs. Forexample, the magnetic quadrupole transition Purcell factorshould scale as ∝η 5. For emission into modes confined to factors of 1000 or more, this enhancement factor could easilyexceed 10 15, alluding to the possibility of making highly forbidden magnetic quadrupole processes observable. B. Emission with in-plane anisotropy Thus far, we have considered geometries of MnF 2in which the anisotropy axis of the crystal is out of the plane of a thinfilm (in the zdirection). Past work has brought both theoretical interest as well as experimental studies on antiferromag-netic surface interfaces in which the magnetic permeabilityanisotropy axis lies in plane. In other words, the material hasnegative permeability in the out-of-plane direction as well asone in-plane direction, while having a permeability of 1 inthe other in-plane direction. This geometry gives rise to anrich anisotropic dispersion relation of SMP modes, which inturn result in a nontrivial angular dependence for processes ofspontaneous emission. We summarize those findings here.For the in-plane anisotropic geometry with μ= (μ(ω),1,μ(ω)), the dispersion (obtained again by solving Maxwell’s equations for a quasimagnetostatic scalarpotential) is given by solutions to e qd√β(θ,ω)=1−μ(ω)√β(θ,ω) 1+μ(ω)√β(θ,ω), (11) where β(θ,ω)=cos2θ+sin2θ/μ(ω) andθis the in-plane propagation angle measured with respect to the xaxis. When β> 0, the mode function has a zdependence of cosh( qz) or sinh( qz), dependent on the parity of the solution. When β< 0, the modes have a cos( qz)o rs i n ( qz) dependence. We note that the β< 0 solutions have a multiply branched structure which correspond to higher harmonic modes, just aswith the in-plane isotropic case discussed throughout the text.Furthermore, recalling that μ< 0 and examining β(θ,ω), we see that for angles of propagation near 0, βwill be positive, while for angles of propagation near π/2,βis negative. Based on the sign of β, we can classify the modes into two distinct types. We refer to β> 0 modes as type-I modes and β< 0 modes as type-II modes. The fundamental type-I modes prop-agate in the range θ∈(0,θ x), where θx=tan−1(√−μ(ω)), while the type-II modes with n=1 propagate in the range θ∈ (θy,π/2), with θy=cos−1(1/√−μ(ω)). The angular prop- agation ranges for the type-I modes and the lowest ordertype-II mode are nonoverlapping and the gap between θ xand θyincreases with ω. The dispersion for even type-I and type-II modes are, respectively, given as qI=−1 2d√β(θ,ω)tanh−1/parenleftbigg1 μ(ω)√β(θ,ω)/parenrightbigg , (12) qn II=1 2d√−β(θ,ω)tan−1/parenleftbigg1 μ(ω)√−β(θ,ω)+nπ 2/parenrightbigg , (13) where nis an integer. We see that for even type-I modes, only a single band of surface polariton modes exists, whilefor type-II modes, a richer structure with harmonics existsdue to the multivalued nature of the arctangent, just as inthe in-plane isotropic case. In Fig. 5, we see the isofrequency contours for the dispersion in the case of in-plane anisotropy.We clearly observe that the mode structure is anisotropic,in that type-I modes behave differently than type-II modes.We comment briefly on the polarization of the modes. Thein-slab H-field polarization of the type-I and -II modes are, respectively, given as ˆε q=/braceleftBiggˆqcosh( qz)+isinh( qz)ˆz√ 2,type I ˆqcos(qz)+isin(qz)ˆz√ 2, type II .(14) Applying the same formalism as before, the rate of emis- sion into SMPs per unit angle by a z-oriented spin flip of strength μBis given by d/Gamma1(eg) dθ=μ2 Bμ0ωeg 2π¯hq3(θ,ω eg)|σeg·ˆ/epsilon1q|2 Cq(θ,ω eg)|vg(θ,ω eg)|e−2q(θ,ω eg)z0.(15) 235453-7JAMISON SLOAN et al. PHYSICAL REVIEW B 100, 235453 (2019) FIG. 5. Dispersion for anisotropic modes. Isofrequency contours for MnF 2of thickness d=200 nm. The frequency labels are given asω/ω 0,w h e r e ω0is the resonance frequency of the material. The first type-I modes are shown in red, while the type-II modes with n=1a r es h o w ni nb l u e . The total rate is obtained by integrating over all angles: /Gamma1(eg)=μ2 Bμ0ωeg 2π¯h/integraldisplay2π 0q3(θ,ω eg)|σeg·ˆ/epsilon1q|2 Cq(θ,ω eg)|vg(θ,ω eg)|e−2q(θ,ω eg)z0dθ. (16) In Fig. 6we see the lossless differential decay rate d/Gamma1(eg)/dθplotted as a function of polar angle θfor a z- oriented spin-flip transition at different emitter frequenciesω. We see that with increasing frequency, the angular spread of type-I modes narrows, while the angular spread of type-IImodes increases. We can understand this behavior in termsof the availability and confinement of modes for differentpropagation angles θ. The most highly confined modes are the type-I modes near the angular cutoff. As ωincreases the FIG. 7. Magnetic dipole transition rate for in-plane anisotropic MnF 2. Magnetic dipole transition rate for a z-oriented dipole transi- tion a distance z0=5 nm from the surface into two different SMP modes in a d=200-nm-thick anisotropic slab of MnF 2. The type-I mode emits most strongly but over a narrower range of frequencies. The cutoff frequency is the frequency at which the first type I mode no longer satisfies the boundary conditions. The first-order type-II mode is emitted more weakly but is supported over the entire rangeof frequencies for which μ(ω)<0. confinement of type-I modes at low angles increases, while the confinement of type-II modes decreases. This systemexhibits the interesting property that tuning the frequencyof the emitter over a narrow bandwidth dramatically shapesthe angular spectrum of polariton emission. An interestingconsequence is that for an emitter with a broadened spectralline (broader than 0 .001ω 0), the angular spectrum will be a complicated mixture of the qualitatively different angularspectra in Fig. 6. In Fig. 7, we see the total transition rate /Gamma1 (eg)for a dipole emitter above MnF 2oriented with the anisotropy axis in the y direction. While the transition rates of both modes are greatlyenhanced compared to the free-space transition rate of order10 −6s−1, the type-I mode benefits approximately two orders of magnitude more than the first type-II mode. In particular,the Purcell factor for the type-I mode ranges from 10 10to 1012, and is thus quite comparable to Purcell factors obtained forthe in-plane isotropic discussed previously. In this sense, wesee that extreme enhancement of magnetic dipole transition FIG. 6. Angular distribution of SMP emission. Magnetic dipole transition rate per unit angle d/Gamma1(eg)/dθfor radiation into SMPs on a 200-nm-thick slab of MnF 2. The radial axis shows d/Gamma1(eg)/dθplotted on a log scale in units of s−1. The first type-I modes are shown in red and the first type-II modes are shown in blue. Dashed lines indicate the angular cutoffs θxandθyfor each type of mode. Note that at low frequencies, θxandθybecome very close. We additionally note that for ω/ω 0>1.0035, the type-I mode branch shown in red vanishes entirely, leaving only the type-II modes. 235453-8CONTROLLING SPINS WITH SURFACE MAGNON … PHYSICAL REVIEW B 100, 235453 (2019) rates is achievable in both crystal orientations. The dispersion relation, however, is notably different in these cases. Asan additional degree of freedom, one can consider how thedispersion, and consequently the dipole emission rate, will beinfluenced by an applied magnetic field along the anisotropyaxis of a material such as MnF 2. In this case, an effective Zeeman splitting causes the resonance frequency ω0to split into two frequencies which move away from each other inlinear proportion to the applied field, as described, for exam-ple, in Ref. [ 82]. When the anisotropy axis lies in the plane of the material, such an applied field results in nonreciprocalpropagation of waves due to the broken reflection symmetry.For these reasons, applied fields may be used to tune theAFMR frequencies or to shape the properties of the spinwaves emitted by magnetic dipole transitions. The net resultis a highly flexible platform for strong interaction betweenmagnetic transitions and matter. IV . EXPERIMENTAL CONSIDERATIONS AND OUTLOOK We have shown that highly confined SMPs, such as those on antiferromagnetic materials, could speed up magnetic tran-sitions by more than ten orders of magnitude, bridging theinherent gap in decay rates which typically separates elec-tric and magnetic processes. We predict that these confinedmagnetic surface modes in systems with realizable parametersmay exhibit confinement factors in excess of 10 4.W ed e v e l - oped the theory of magnon polaritons and their interactionswith emitters in a way that unifies this set of materials withother more well-known polaritonic materials, casting light onopportunities to use these materials to gain unprecedentedcontrol over spins in emitters. To push the field of magnon polaritonics at THz fre- quencies forward, it will be necessary to identify an idealexperimental platform for manipulating these modes and in-terfacing them with matter. For antiferromagnetic platforms,experiments will need to take place below the Néel tem-perature of the material to establish antiferromagnetic order.Importantly, we note that the only strict material requirementfor SMPs is that Re( μ)<0 over some frequency range, presenting opportunities for other types of magnetic order,2D magnetic materials, or even metamaterials which exhibitnegative permeability. The other key consideration is whatclass of emitters may be well-suited to interact with thesepolaritonic modes. In terms of existing materials, a potentialemitter system which can interact with the antiferromagneticSMPs discussed here is ErFeO 3, which has several electric and magnetic dipole transitions in the range between 0.25 and1.5 THz [ 83]. Recent work has also considered THz magnon polaritons in TmFeO 3[84]. It could also prove interesting to consider GHz-THz orbital angular momentum transitionsbetween high-energy levels in Rydberg atoms, Landau levels,or vibrational modes in molecules. In addition, one couldconsider THz transitions arising from impurity states in semi-conductors [ 85], which have the benefit of the tunability over THz scale by the application of an external magneticfield. The theoretical predictions made in this paper could be verified by fluorescence spectroscopy measurements on a thinlayered sample as shown in Fig. 8. We represent the emitter FIG. 8. Schematic for a potential fluorescence spectroscopy ex- periment to observe enhancement of magnetic dipole (MD) tran- sitions through surface magnon polaritons. We consider a lay- ered sample which contains a thin negative permeability filmwhich supports SMPs and a material containing an appropriately chosen emitter material. An external laser prepares the emitters into an excited state via an IR /optical transition. This excited state then decays via a THz transition into SMPs in the thin film and then relaxes via a photon transition into the far field. The far-field signal can be measured with a spectrometer to detect the Raman shift in thefluorescence frequency compared to the incident laser frequency. as a three-level system, where the gap between the lower level and the higher levels is in the optical /IR and is excited with an external laser via an electric dipole transition. The excitedstate can then decay into SMPs in the material below via amagnetic dipole transition. Such a magnetic dipole transitionis usually very slow in free space, but as detailed in ourpaper, will occur orders of magnitude faster due to decayinto SMPs. The emitter state populations and transition ratescan then be monitored via spectroscopy of the optical photonemitted to free space. One would expect to see a decrease influorescence at the exciting laser frequency, in conjunctionwith the appearance of a new Raman peak, shifted fromthe exciting frequency by the THz SMP frequency. Similarschemes for monitoring Purcell enhancements in plasmonicshave been implemented in Ref. [ 86]. Time-resolved measure- ments have also been made in Ref. [ 87] to directly measure the decay in excited state populations which occurs throughPurcell-enhanced emission of polaritons. Alternatively, a sub-stantial rate increase in a THz MD transition due to SMPexcitation could influence rate dynamics in a way whichproduces optical /IR far-field decays at frequencies entirely different from the exciting laser. Methods for analyzing suchmechanisms are detailed in Ref. [ 88]. Future work could also consider processes involving the emission of multiple surface magnons using the frameworkpresented in Ref. [ 33] or mixed processes with the emission of a magnon polariton in addition to one or more excitationsof another nearby material. In any case, SMPs provide an in-teresting degree of control over magnetic degrees of freedomin matter as well as a means to consider magnetic analogs atTHz frequencies of many famous effects in plasmonics andpolaritonics. 235453-9JAMISON SLOAN et al. PHYSICAL REVIEW B 100, 235453 (2019) ACKNOWLEDGMENTS The authors thank C. Roques-Carmes and N. Romeo for help reviewing the paper. Research was supported as partof the Army Research Office through the Institute for Sol-dier Nanotechnologies under Contract No. W911NF-18-2-0048 (photon management for developing nuclear-TPV andfuel-TPV mm-scale systems), also supported as part of theS3TEC, an Energy Frontier Research Center funded by the USDepartment of Energy under Grant No. DE-SC0001299 (for fundamental photon transport related to solar TPVs and solar-TEs). I.K. acknowledges support as an A. 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PhysRevB.71.094410.pdf

Longitudinal complex magnetic susceptibility and relaxation times of superparamagnetic particles with triaxial anisotropy Yuri P. Kalmykov and Bachir Ouari Laboratoire Mathématiques et Physique des Systèmes, Université de Perpignan, 52, Avenue Paul Alduy, 66860 Perpignan Cedex, France sReceived 1 October 2004; revised manuscript received 18 November 2004; published 11 March 2005 d The longitudinal relaxation time and spectrum of the complex magnetic susceptibility of single domain ferromagnetic particles with triaxial sorthorhombic danisotropy are calculated by averaging the Gilbert- Langevin equation for the magnetization of an individual particle and by reducing the problem to that ofsolving a system of linear differential-recurrence relations for the appropriate equilibrium correlation functions.The solution of this system is obtained in terms of matrix continued fractions. It is shown that in contrast to thelinear magnetic response of particles with uniaxial anisotropy, there is an inherent geometric dependence of thecomplex susceptibility and the relaxation time on the damping parameter arising from coupling of longitudinaland transverse relaxation modes. Simple analytic equations, which allow one to understand the qualitativebehavior of the system and to accurately predict the spectrum of the longitudinal complex susceptibility inwide ranges of the barrier height and dissipation parameters, are proposed. DOI: 10.1103/PhysRevB.71.094410 PACS number ssd: 75.60.Jk, 75.50.Tt, 76.20. 1q, 05.40. 2a I. INTRODUCTION The single domain ferromagnetic particles, which are characterized by an internal anisotropy potential, may haveseveral positions of local equilibrium of the magnetizationwith potential barriers separating them. If the particles aresmall s,100 Å d, thermal fluctuations may cause the magne- tization vector Mto reorient itself over the barriers from one equilibrium position to another. The instability of the mag-netization due to thermal agitation results in superpara-magnetism 1because each fine particle behaves like an enor- mous paramagnetic atom having a magnetic moment,10 4–105Bohr magnetons. The superparamagnetism of single-domain ferromagnetic nanoparticles is important inthe context of rock magnetism and technology because of theever decreasing size of the particles used in magnetic record-ing. The pioneering theory of thermal fluctuations of the mag- netization Mstdof a single domain ferromagnetic particle due to Néel 2was further developed by Brown3,4using the theory of the Brownian motion. Brown proceeded by takingas Langevin equation, Gilbert’s equation 5for the motion of the magnetization augmented by a random field hstd, viz.:3,4 u˙std=hustd3fgHefstd−au˙std+ghstdgj, s1d whereu=M/MSis the unit vector directed along M,MSis the saturation magnetization, gis the gyromagnetic ratio, a is the dimensionless damping sdissipation dparameter, Hef =−]V/]M,Vis the free energy per unit volume scharacter- izing the magnetic anisotropy and Zeeman energy density ofthe particle d, and a random field hstdwith white noise prop- erties, accounting for the thermal fluctuations of the magne- tization of an individual particle. Brown derived from theGilbert-Langevin Eq. s1d, the Fokker-Planck equation for the distribution function WsM,tdof the orientations of the mag- netization vector M: 4] ]tW=LFPW=1 2tNhbfa−1u·s„V3„Wd +„·sW„Vdg+DWj, s2d where L FPis the Fokker-Planck operator, „andDare the gradient and Laplacian operators on the surface of unitsphere,uis the unit vector directed along M, b=v/skTd, vis the volume of the particle, Tis the temperature, kis the Boltzmann constant, and tN=bMSs1+a2d/s2gadis the free diffusion time of the magnetization. A detailed discus- sion of the assumptions made in the derivation of the Fokker-Planck and Gilbert equations is given elsewhere se.g., Refs. 6 and 7 d. For the purpose of mathematical simplification, the mag- netization relaxation of superparamagnetic particles has usu-ally been considered for particles with uniaxial magnetic an-isotropy se.g., Refs. 3 and 7–13 d. For such particles, the free energy density Vis given by 3 V=−K3cos2q, s3d whereK3is the anisotropy constant and qis the polar angle. For axially symmetric potential s3d, the longitudinal relax- ation is governed by a single state variable q. The second state variable, namely, the azimuthal angle wappears merely as a steady precession of the vector M. The decoupling be- tween the transverse and longitudinal modes results in anexact single-variable Fokker-Planck equation in the colati-tude q,3,4that allows one readily to evaluate the magnetic characteristics for uniaxial particles ssuch as the complex magnetic susceptibility and relaxation times; see, e.g., Refs.4, 11, and 14 d. Although use of the axially symmetric potential consider- ably simplifies the analysis, the results obtained in this ap-proximation cannot, however, be applied to particles withnonaxially symmetric anisotropy, such as cubic or triaxialanisotropy. Nonaxially symmetrical anisotropy generates azi-muthally nonuniform energy distributions with saddle pointsthat leads to an effect, viz., strong intrinsic dependence ofPHYSICAL REVIEW B 71, 094410 s2005 d 1098-0121/2005/71 s9d/094410 s8d/$23.00 ©2005 The American Physical Society 094410-1magnetic characteristics on the value of the damping param- eter aarising from coupling of the longitudinal and trans- verse relaxation modes. For nonaxially symmetric potentials,Fokker-Planck Eq. s2dcan be solved by expanding Was a series of spherical harmonics so yielding an infinite hierar-chy of differential-recurrence equations for the statisticalmoments saveraged spherical harmonics or appropriate cor- relation functions d. 14,15The hierarchy of moment equations can also be obtained by direct averaging Gilbert’s equationwithout recourse to the Fokker-Planck equation. 14The sys- tem of moment equations can be solved by calculating theeigenvalues and eigenvectors of the system matrix se.g., Refs. 16 and 17 dor by a matrix continued fraction method. 14,18The last approach has been used recently for the study of the magnetization dynamics of particles with cubicanisotropy 19–21and uniaxial anisotropy in the presence of an external dc magnetic field, which breaks the axialsymmetry. 22,23 In the present paper, we use the continued fraction method to calculate the longitudinal complex magnetic susceptibilityand relaxation time of single domain particles with triaxialsi.e., orthorhombic danisotropy, where the free energy density is given by 24 V=−K1sin2qcos2w−K2sin2qsin2w−K3cos2q+const, s4d sK1,K2,K3are the anisotropy constants d. In spite of the practical importance of orthorhombic anisotropy, which mayyield an essential contribution to the free energy density ofmagnetic nanoparticles, 25,26the orthorhombic case is to some extent incomplete. The only available appropriate formulafor the relaxation time of the magnetization of orthorhombiccrystals has been given by Smith and de Rozario 24in the low-temperature limit and intermediate-to-high dampingfsIHDd aø1g. A very similar problem of magnetization re- versal in elongated particles swhere easy and hard-axis an- isotropy terms present din the presence of a strong dc mag- netic field has been treated by Braun27but also in the IHD limit only. Some quantum and field effects for magnetic re-laxation of biaxial particles have been treated, for instance,in Refs. 28–30. Here we present the results of a study of thelongitudinal complex magnetic susceptibility xisvdand re- laxation time tiof single domain particles with triaxial an- isotropy for wide ranges of the anisotropy energy and dissi-pation parameters. Numerical results obtained with the helpof matrix continued fractions are compared with asymptoticestimates based on Kramers’ escape rate theory. 4,14,27 II. LONGITUDINAL DYNAMIC SUSCEPTIBILITY AND RELAXATION TIMES According to linear response theory sRef. 14, Chap. 2 d, the decay of the longitudinal component of the magnetiza-tion kM ilstdof a single domain particle, when a small con- stant external field H1,bsM·H1d!1, applied along the z axisswhich is the easy axis of the particle dhas been switched off at time t=0,iskMilstd=MSE 0pE 02p cosqWsq,w,tdsinqdqdw=xiH1Cistd, s5d where Cistd=kcosqs0dcosqstdl0 kcos2qs0dl0=o kcke−lkts6d is the normalized equilibrium autocorrelation function of the longitudinal component of the magnetization, lkare the ei- genvalues of the Fokker-Planck operator L FPfrom Eq. s2d, okck=1,xi=bMS2kcos2ql0is the static longitudinal magnetic susceptibility of the particle, and the brackets kl0designate the equilibrium ensemble average defined as kAl0=1 ZE 02pE 0p Asq,wde−bVsq,wdsinqdqdw s7d sZis the partition function d. The correlation function Cistd completely determines the transient longitudinal relaxation of the magnetization. Moreover, it allows one to evaluate theac response of the system to a small ac perturbing magneticfield, namely, the longitudinal complex susceptibility xisvd =xi8svd−ixi9svd, which is given by14 xisvd/xi=1−ivE 0‘ e−ivtCistddt. s8d According to Eq. s8d, the behavior of xisvdin the frequency domain is completely determined by the time behavior of Cistd. In order to characterize quantitatively the time behav- ior ofCistd, one may formally introduce two time constants. These are the integral relaxation sor correlation dtime tide- fined as the area under Cistd, viz.14 ti=E 0‘ Cistddt=o kck/lk, s9d and the effective relaxation time tiefgiven by tief=−1/C˙is0d=So kcklkD−1, s10d fwhich yields precise information on the initial decay of Cistdin the time domain g. III. MATRIX CONTINUED FRACTION SOLUTION We can calculate numerically the relaxation time tiand the dynamic susceptibility xisvdby using the matrix contin- ued fraction approach developed in Ref. 14. According to this approach, the solution of the Gilbert-Langevin Eq. s1d foranyanisotropy potential can be reduced to the solution of an infinite hierarchy of differential-recurrence equations forthe statistical moments sequilibrium correlation functions d c l,mstd=kcosqs0dYl,mfqstd,wstdgl0fso that c1,0std/c1,0s0d =Cistdggoverning the dynamics of the magnetization.14Here Yl,msq,wdis the spherical harmonic defined as31Y. P. KALMYKOV AND B. OUARI PHYSICAL REVIEW B 71, 094410 s2005 d 094410-2Yl,msq,wd=s−1dm˛s2l+1dsl−md! 4psl+md!eimwPlmscosqd, where the Plmsxdare the associated Legendre functions. For the problem in question, one can obtain the differential- recurrence equations for cl,mstdusing a general formula de- rived in Ref. 32 ssee also Ref. 14, Chap. 7 d. Thus noting that the free energy density Vfrom Eq. s4dis expressed in terms ofYl,mas V=˛2p 15sK2−K1dfY2,2sq,wd+Y2,−2sq,wdg −4 3˛p 5SK3−K2+K1 2DY2,0sq,wd+const, we have the 15th term differential-recurrence equation tNd dtcn,mstd=vn,mcn−2,mstd+wn,mcn−1,mstd+xn,mcn,mstd +yn,mcn+1,mstd+zn,mcn+2,mstd+vn,m+cn−2,m+2std +wn,m+cn−1,m+2std+xn,m+cn,m+2std +yn,m+cn+1,m+2std+zn,m+cn+2,m+2std +vn,m−cn−2,m−2std+wn,m−cn−1,m−2std +xn,m−cn,m−2std+yn,m−cn+1,m−2+zn,m−cn+2,m−2std, s11d wherenø1,−nłmłnand the coefficients vn,m,vn,m±, etc. are defined inAppendix A. Equation s11dcan be transformedinto the three-term vector recurrence equation tNd dtCnstd=Qn−Cn−1std+QnCnstd+Qn+Cn+1std,snø1d, s12d whereCnstdare the column vectors arranged in an appropri- ate way from cn,mstd, viz. Cnstd=1c2n,−2nstd c2n,−2n+1std A c2n,2nstd c2n−1,−2n+1std c2n−1,−2n+2std A c2n−1,2n−1std2,snø1d, andQn−,Qn,Qn+are the supermatrices given in Appendix A. The exact solution of Eq. s12dfor the Laplace transform C˜1ssd=e0‘C1stde−stdtcan be given in terms of matrix contin- ued fractions14 C˜1ssd=tND1ssdHC1s0d+o n=2‘Fp k=2n Qk−1+DkssdGCns0dJ, s13d where the infinite matrix continued fraction Dnssdis defined as Dnssd=I tNsI−Qn−Qn+ I tNsI−Qn+1−Qn+1+ I tNsI−Qn+2−...Qn+2−Qn+1−, s14d sthe fraction lines denote matrix inversion d, andIare the unit matrices of appropriate dimensions. The initial conditionvectorsC ns0din Eq. s13dmay be evaluated in terms of ma- trix continued fractions Dns0d14ssee Appendix B d. Having determined C˜1ssd, one may evaluate the relaxation time ti=C˜s0d=c˜1,0s0d/c1,0s0ds 15d as well as the spectrum of the correlation function C˜svd =c˜1,0sivd/c1,0s0dand thus the complex susceptibility from Eq.s8d. Moreover, by using matrix continued fractions, one can also evaluate the smallest nonvanishing eigenvalue l1.23 The matrix continued fraction approach provides an effec- tive method of computation of the susceptibility xisvdand correlation time tisalgorithms for calculating matrix contin-ued fractions are discussed in Refs. 14 and 18 d. The advan- tage of the matrix continued fraction approach is that it ap-plies to the case, where the magnetic anisotropy energy iscomparable to the thermal energy kT. Nevertheless, its appli- cation is rather limited since the dependence of xisvdandti on the model parameters sdamping coefficient, anisotropy constants dis not obvious by this method. IV. ASYMPTOTIC FORMULAS The qualitative behavior of xisvdandtican readily be understood in the low temperature limit, where the magneti- zation relaxation is determined by the smallest nonvanishingeigenvalue l 1.14Indeed, according to Eq. s9d, the correlation time ticontains contributions from allthe eigenvalues lk.LONGITUDINAL COMPLEX MAGNETIC … PHYSICAL REVIEW B 71, 094410 s2005 d 094410-3The smallest nonvanishing eigenvalue l1is associated with the slowest overbarrier relaxation mode and so with the long-time behavior of C istd; the other eigenvalues lkcharacterize high-frequency “intrawell” modes. In general, in order to evaluate tinumerically, a knowledge of all the lkandckis required. However, in the low temperature shigh barrier d limit, l1!ulkuandc1<1@ckskÞ1dprovided the wells of the potential remain equivalent sas for the triaxial potential d so that ti<1/l1. s16d In other words, the inverse of the smallest nonvanishing ei- genvalue closely approximates the correlation time tiin the low temperature limit. The smallest nonvanishing eigenvalue l1may be esti- mated with the help of the Kramers escape rate theory33as extended to the magnetic problem by Brown,3,4Smith and de Rozario,24Klik and Gunther,6,34and Coffey et al.35We recall that in order to estimate the characteristic time of reversal ofthe magnetic moment over the internal anisotropy potentialbarrier of a uniaxial particle, Brown 3adapted an ingenious method originally proposed by Kramers33in connection with thermally activated escape of Brownian particles out of apotential well. In the fsIHDd aø1gformulas for the escape rates of magnetic systems were derived by Smith and deRozario 24and Brown.4Moreover, in 1990, Klik and Gunther6,34realized that the very low damping sVLD d,sa !1dregime also applied to magnetic relaxation of single domain ferromagnetic particles and derived the correspond- ing VLD formula for the escape rate. The conditions of ap-plicability of these IHD and VLD solutions for superpara-magnets are defined by aø1 and a,0.001, respectively. For crossover values of damping, 0.001 ,a,1, Coffey et al.35have derived a universal formula for bridging the VLD and IHD escape rates as a function of the dissipation param-eter for single-domain ferromagnetic particles having a non-axially symmetric free-energy density sfor a review of appli- cations of the Kramers method to magnetic problems seeRefs. 14, 15, 27, and 35 d. Using the approach of Coffey et al., 35the universal for- mula for the relaxation times swhich is universal in the sense that it is valid for all values of damping including IHD andVLD regions dis given by 36 l1−1,tIHDAs8as˛dd A2s4as˛dd, s17d where tIHDis the longest relaxation time of an orthorhombic crystal derived by Smith and de Rozario24in the IHD limit sin our notation d tIHD t0=pess1+a2d as˛1+1/ df1−d+˛s1+dd2+4d/a2g,s18d d=D/s,D=bsK2−K1d.0, and s=bsK3−K2d.0 are the di- mensionless anisotropy and barrier height parameters, re- spectively, t0=tNa/s1+a2d=bMS/s2gdis an a-independent characteristic time, andAsaSd=expF1 pE 0‘ lnf1−exp h−aSsx2+1/4 djg 3sx2+1/4 d−1dxG. Noting that AsaSd/a!Sasa!0,35Eq.s18dyields tVLD t0,pes 4as2˛ds1+dd, s19d which is in agreement with estimations in the context of the Klik and Gunther theory.34 In order to understand the qualitative behavior of the complex susceptibility xisvd, one can use a simple analytical equation derived in Refs. 37 and 38 ssee also Ref. 14, Chaps. 7–9d. According to Ref. 37, the correlation function Cistd fwhich in general comprises an infinite number of decaying exponentials, see Eq. s6dgmay be approximated in the IHD limit by two exponentials only, viz. Cistd<D1e−l1t+s1−D1de−t/tW, s20d where D1andtWare expressed in terms of ti,tief, and l1 as14,37 D1=ti/tief−1 l1ti−2+1/ sl1tiefd,tW=l1ti−1 l1−/tief. Thus the dynamic susceptibility xisvdgiven as an infinite series of Lorenzians may be approximated by a sum of two Lorentzians only xisvd xi<D1 1+iv/l1+1−D1 1+ivtW. s21d The parameters D1andtWin Eq. s21dare determined in such way37as to guarantee the correct asymptotic behavior of xisvdin the extreme cases of very low and very high frequencies14 xisvd xi,51−ivE 0‘ Cistddt=1−ivti,v!‘ Cis0d iv+fl=−i vtief+fl,v!0.6 s22d Equation s21dwas derived and tested for particles with uniaxial sat all damping dand cubic anisotropy sat IHD values of damping, aø1d.37Fora!1, where the interactions be- tween the longitudinal and transverse modes cannot be ig-nored, Eq. s21dmay be used at vtNł1. In practical calculations, Eq. s21drequires a knowledge of ti,tief, and l1.The smallest eigenvalue may readily be evalu- ated from Eq. s17d. The effective relaxation time tiefis given by an exact analytic equation37 tief=2tNkcos2ql0s1−kcos2ql0d−1, s23d where kcos2ql0can be calculated from Eq. s7d. Unfortu- nately, there is no simple equation for the correlation time tiY. P. KALMYKOV AND B. OUARI PHYSICAL REVIEW B 71, 094410 s2005 d 094410-4fEq.s16dis unreliable here as it yields tW=0g. However, one can overcome this problem38by noting that the intrawell relaxation time time tWcan be estimated in the low- temperature limit, s@1, from the deterministic Gilbert equa- tion as tW,vwell−1=tN/s2s˛1+dd, s24d where vwell=g/MS˛s]2V/]u12ds]2V/]u22dis the well angular frequency. Noting Eqs. s21d,s22d, and s24d, one can evaluate D1as D1<1−tW/tief. s25d Equations s21dands23d–s25dand allow one readily to calcu- latexisvd. V. RESULTS AND DISCUSSION The greatest relaxation time predicted by the universal Eq.s17dand the correlation time ticalculated numerically by the matrix continued fraction method for triaxial anisotropyare shown in Figs. 1 and 2 sas a function of the damping parameter ad, and 3 sas a function of the barrier height pa- rameter sd. Apparently, at high barriers, sø5, theasymptotic Eq. s17dprovides a good approximation of tifor all values of asFigs. 1 and 2 dandDø1sFig. 3 d. We em- phasize that Eq. s17dis not valid for d=D/s!0 correspond- ing to uniaxial anisotropy. Here tNl1is given by Brown’s formula3,4 tNl1,2s3/2e−s/˛p. s26d The uniaxial asymptote Eq. s26dis shown in Fig. 3 for com- parison. It follows that the triaxial anisotropy causes the vari-ous damping regimes sIHD andVLD dof relaxation to appear unlike in an axially symmetric potential. Results of the calculation from Eqs. s21dands23d–s25d, and those obtained using matrix continued fractions are com-pared in Figs. 4–6. Here the imaginary part of the normal- ized susceptibility xi9svdsbMS2=1dis plotted for typical val- ues of the model parameters s,D, and a.The results indicate that a marked dependence of xisvdonaexists and that three distinct dispersion bands appear in the spectrum. The char- acteristic frequency and half-width of the low-frequencyband are completely determined by the smallest nonvanish-ing eigenvalue l 1. Thus the low frequency behavior of xisvd is dominated by the barrier crossing mode. In addition, a far weaker second relaxation peak appears at high frequencies.This high frequency relaxation band is due to the intrawellmodes. The characteristic frequency of this band is vwell ,tN−1s2s˛1+dd. The third ferromagnetic resonance sFMR d FIG. 1. ti/t0vsaforD=10 and various values of s. Solid lines: exact matrix continued fraction solution; dashed lines: theVLD Eq. s19d; dotted lines: the IHD Eq. s18d; symbols: the univer- sal Eq. s17d. FIG. 2. ti/t0vsafors=10 and various values of D. Solid lines: matrix continued fraction solution; dashed lines: the VLD Eq.s19d; dotted lines: the IHD Eq. s18d; symbols: the universal Eq. s17d. FIG. 3. ti/t0vssfora=0.1 and various values of D. Solid lines: matrix continued fraction solution; symbols: the universal Eq.s17d; dotted line: Eq. s26d. FIG. 4. −Im fxisvdgvsvtNfors=10, D=10, and various values ofa. Solid lines 1–3: matrix continued fraction solution. Symbols: Eqs. s21dands23d–s25d; dotted and dashed lines: Eq. s22d.LONGITUDINAL COMPLEX MAGNETIC … PHYSICAL REVIEW B 71, 094410 s2005 d 094410-5peak due to excitation of transverse modes having the peak frequency close to the precession frequency vpr<gkHefl0of the magnetization appears only at low damping sa!1dand strongly manifests itself in the high frequency region. As a decreases, the FMR peak shifts to higher frequencies since vpr,a−1ssee Fig. 4 d.As one can see in Figs. 4–6 the agree- ment between exact matrix continued fraction calculationsand the approximate Eq. s21dis very good in the low- frequency region, vtNł1, for all values of damping because the low-frequency response is completely determined by theoverbarrier relaxation mode. The approximate Eq. s21d yields a reasonable description of the high frequency relax-ation band at IHD values of damping s aø1d, where one can ignore the interactions between the longitudinal and trans- verse modes. However, Eq. s21ddoes not allow one to de- scribe the FMR peak which appears at very low damping, a!1. One may conclude that the longitudinal magnetic suscep- tibility xisvdand relaxation time tiof systems of single do- main particles with triaxial anisotropy may by evaluated ex- actly in terms of matrix continued fractions sfor all values of model parameters das well as in terms of simple analytic equations sin the low-temperature limit d. In contrast touniaxial particles, where the damping only enters in the dif- fusion time tN, for the particles with triaxial anisotropy, there is an inherent geometric dependence of xisvdandti/tNon the value of the damping parameter aarising from coupling of the longitudinal and transverse relaxation modes. In thederivation of the above results, it was supposed that all par-ticles are identical. In order to take into account the polydis-persity of the particles of a real sample one must also aver-age the susceptibility over appropriate distribution functionsse.g., over that of particle volumes d. The approach developed can be applied with small modifications to the evaluation ofthe transverse and nonlinear responses of orthorhombic crys-tals, to the estimation of the effect of an external dc magneticfield on the relaxation behavior of the magnetization of suchcrystals, etc. ACKNOWLEDGMENTS The support of this work by INTAS sProject No. 01-2341 d is gratefully acknowledged. We thank Professor W. T. Cof-fey, Professor J. L. Déjardin, Dr. S. V. Titov, and Dr. P.-M.Déjardin for stimulating discussions and useful comments. APPENDIX A: MATRICES Q n,Qn+,Qn− The matrices Qn,Qn+,Qn−are defined as Qn=SX2nW2n Y2n−1X2n−1D,Qn+=SZ2nY2n 0Z2n−1D, Qn−=SV2n0 W2n−1V2n−1D. The dimensions of the matrices Qn,Qn+, andQn−are accord- ingly equal to 8 n38n,8n38sn+1d, and 8n38sn−1d.I n turn, the matrices Qn,Qn+,Qn−consist of submatrices. There are five distinct types of three diagonal submatrices Vl,Wl, XlYl, andZlwhich have the dimensions s2l+1d3s2l−3d, s2l+1d3s2l−1d,s2l+1d3s2l+1d,s2l+1d3s2l+3d, and s2l+1d3s2l+5d, respectively. The elements of these subma- trices are given by sVldn,m=dn−4,mvl,−l+m+3−+dn−2,mvl,−l+m+1+dn,mvl,−l+m−1+, sWldn,m=dn−3,mwl,−l+m+2−+dn−1,mwl,−l+m+dn+1,mwl,−l+m−2+, sXldn,m=dn−2,mxl,−l+m+1−+dn,mxl,−l+m−1+dn+2,mxl,−l+m−3+, sYldn,m=dn−1,myl,−l+m−+dn+1,myl,−l+m−2+dn+3,myl,−l+m−4+, sZldn,m=dn,mzl,−l+m−1−+dn+2,mzl,−l+m−3+dn+4,mzl,−l+m−5+, where FIG. 5. −Im fxisvdgvsvtNforD=10, a=0.01, and various values of s. Solid lines 1–3: matrix continued fraction solution. Symbols: Eqs. s21dand s23d–s25d; dotted and dashed lines: Eq. s22d. FIG. 6. −Im fxisvdgvsvtNfors=10, a=0.01, and various val- ues of D. Solid lines 1–4: matrix continued fraction solution. Filled circles: Eqs. s21dands23d–s25d; dotted and dashed lines: Eq. s22d.Y. P. KALMYKOV AND B. OUARI PHYSICAL REVIEW B 71, 094410 s2005 d 094410-6vn,m=Ss+D 2Dsn+1d s2n−1d˛fsn−1d2−m2gsn2−m2d s2n+1ds2n−3d, vn,m−=vn,−m+=−D 4sn+1d s2n−1d 3˛sn+m−3dsn+m−2dsn+m−1dsn+md s2n+1ds2n−3d, wn,m=−iSD 2+sDm a˛n2−m2 4n2−1, w− n,m=−w+ n,−m =−iD 4a˛sn+m−2dsn+mdfn2−sm−1d2g 4n2−1, xn,m=−nsn+1d 2+SD 2+sDnsn+1d−3m2 s2n−1ds2n+3d, xn,m−=xn,−m+=3D 4˛fn2−sm−1d2gfsn+1d2−sm−1d2g s2n−1ds2n+3d, yn,m=−iSD 2+sDm a˛sn+1d2−m2 s2n+1ds2n+3d, yn,m−=−yn,−m+ =iD 4a˛fsn+1d2−sm−1d2gs1+n−mds3+n−md s2n+1ds2n+3d, zn,m=−SD 2+sDn s2n+3d˛fsn+1d2−m2gfsn+2d2−m2g s2n+1ds2n+5d, zn,m−=zn,−m+=D 4n s2n+3d 3˛sn−m+4dsn−m+3dsn−m+2dsn−m+1d s2n+1ds2n+5d. APPENDIX B: CALCULATION OF C n0 The initial condition vectors Cns0din Eq. s13dmay be evaluated by noting that the equilibrium averages kYl,ml0sat-isfy the following recurrence equation fcf. Eq. s11dg: vn,mkYn−2,ml0+wn,mkYn−1,ml0+xn,mkYn,ml0+yn,mkYn+1,ml0 +zn,mkYn+2,ml0+vn,m+kYn−2,m+2l0+wn,m+kYn−1,m+2l0 +xn,m+kYn,m+2l0+yn,m+kYn+1,m+2l0+zn,m+kYn+2,m+2l0 +vn,m−kYn−2,m−2l0+wn,m−kYn−1,m−2l0+xn,m−kYn,m−2l0 +yn,m−kYn+1,m−2l0+zn,m−kYn+2,m−2l0=0. Thus, one can transform the above equation into the tridiago- nal vector recurrence equation Qn−Rn−1+QnRn+Qn+Rn+1=0,snø1d, sB1d whereR0=1/˛4pand Rn=1kY2n,−2nl0 A kY2n,2nl0 kY2n−1,−2n+1l0 A kY2n−1,2n−1l02,nø1. Equation sB1dhas a solution14 Rn=SnRn−1=SnSn−1flS2S1/˛4p, where Sn=Dns0dQn−. Using the identity cn,ms0d =dn+1,mkYn+1,ml0+dn,mkYn−1,ml0, where dn,m =˛sn2−m2d/s4n2−1d,14the initial condition vectors Cns0d are given by Cns0d=F00 D2n−10GRn−1+F0D2nT D2n0GRn +F00 D2n+10GT Rn+1. Here the superscript Tdenotes matrix transposition. The di- mension of the matrix Dliss2l+1d3s2l−1dand its elements are given by sDldn,m=dn−1,mdl,−l+m. 1C. P. Bean and J. D. Livingston, J. Appl. Phys. 30, 120S s1959 d. 2L. Néel, Ann. 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PhysRevLett.92.086601.pdf

Theory of Current-Driven Domain Wall Motion: Spin Transfer versus Momentum Transfer Gen Tatara Graduate School of Science, Osaka University, T oyonaka, Osaka 560-0043, Japan Hiroshi Kohno Graduate School of Engineering Science, Osaka University, T oyonaka, Osaka 560-8531, Japan (Received 22 August 2003; published 26 February 2004) A self-contained theory of the domain wall dynamics in ferromagnets under finite electric current is presented. The current has two effects: one is momentum transfer, which is proportional to the charge current and wall resistivity ( eP0026w); the other is spin transfer, proportional to spin current. For thick walls, as in metallic wires, the latter dominates and the threshold current for wall motion is determined by thehard-axis magnetic anisotropy, except for the case of very strong pinning. For thin walls, as in nanocontacts and magnetic semiconductors, the momentum-transfer effect dominates, and the thresh- old current is proportional to V 0=eP0026w,V0being the pinning potential. DOI: 10.1103/PhysRevLett.92.086601 P ACS numbers: 72.25.Pn, 72.15.Gd Manipulation of magnetization and magnetic domain wall [1] by use of electric current is of special interestrecently [2–6], from the viewpoint of application to spin- tronics, e.g., novel magnetic devices where the informa- tion is written electrically, and also as a basic physics inthat it involves fascinating angular momentum dynamics. Current-driven motion of a domain wall was studied in a series of pioneering works by Berger [7–9]. In 1984, heargued that the electric current exerts a force on thedomain wall via the exchange coupling [8]. Later, in1992, he discussed that a spin-polarized current (spin current) exerts a torque on the wall magnetization and studied the wall motion due to a pulsed spin-polarizedcurrent [9]. These theoretical works are based on his deepphysical insight but seem to lack transparency as a self-contained theory. Also, their phenomenological charactermakes the limit of applicability unclear. In viewof recent precise experiments [4–6], a general theorystarting from a microscopic description is now needed. In this Letter, we reformulate the problem of domain wall dynamics in the presence of electric currentand explore some new features such as current-induceddepinning of the wall. W e start from a microscopicHamiltonian with an exchange interaction between con-duction electrons and spins of a domain wall [10]. With akey observation that the wall position Xand polarization eP00I0 0(the angle between spins at the wall center and the easy plane) are the proper collective coordinates [11] to describe its dynamics, it follows straightforwardly thatthe electric current affects the wall motion in two differ-ent ways, in agreement with Berger’s observation. Thefirst is as a force on X, or momentum transfer, due to the reflection of conduction electrons. This effect is propor-tional to the charge current and wall resistance and,hence, is negligible except for very thin walls. The other is as a spin torque (a force on eP00I0 0), arising when an electron passes through the wall. Nowadays it is alsocalled as spin transfer [2] between electrons and wallmagnetization. This effect is the dominant one for thick walls where the spin of the electron follows the magne-tization adiabatically. The motion of a domain wall under a steady current is studied in two limiting cases. In the adiabatic case, weshow that even without a pinning force, there is a thresh-old spin current j crsbelow which the wall does not move. This threshold is proportional to K?, the hard-axis mag- netic anisotropy. Underlying this is that the angular mo-mentum transferred from the electron can be carried byboth XandeP00I0 0, and the latter can completely absorb the spin transfer if the spin current is small, js<jcrs.T h e pinning potential V0affects jcrsonly if it is very strong, V0*K?=eP00RR, where eP00RRis the damping parameter in the Landau-Lifshits-Gilbert equation. In most real systemswith small eP00RR, the threshold would thus be determined by K ?. Therefore, the critical current for the adiabatic wall will be controllable by the sample shape and, in particu-lar, by the thickness of the film and does not suffer very much from pinning arising from sample irregularities. This would be a great advantage in application. Thewall velocity after depinning is found to be h_XXi//.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 /.0133j s=jcrs/.01342/.02551p . In the case of a thin wall, the wall is driven by the momentum transfer, which is proportional to the charge current jand wall resistivity eP0026w. The critical current density in this case is given by jcr/V0=eP0026w. W e consider a ferromagnet consisting of localized spinsSand conduction electrons. The spins are assumed to have an easy zaxis and a hard yaxis. In the continuum approximation, the spin part is described by theLagrangian [12–14] L S/.0136Zd3x a3/.0020 /.0022hS_eP00I0eP00I0/.0133coseP00R8/.02551/.0134/.0255Vpin/.0137eP00R8/.0138/.0255S2 2fJ/.0133/.0133reP00R8/.01342 /.0135sin2eP00R8/.0133reP00I0/.01342/.0134/.0135sin2eP00R8/.0133K/.0135K?sin2eP00I0/.0134g/.0021 ;(1)PHYSICAL REVIEW LETTERSweek ending 27 FEBRUARY 2004 VOLUME 92, N UMBER 8 086601-1 0031-9007 =04=92(8) =086601(4)$22.50 2004 The American Physical Society 086601-1where ais the lattice constant, and we put S/.0133x/.0134/.0136 S/.0133sineP00R8coseP00I0;sineP00R8sineP00I0;coseP00R8/.0134,a n d Jrepresents the ex- change coupling between localized spins. The longitu- dinal ( K) and transverse ( K?) anisotropy constants incorporate the effect of demagnetizing field. The con-stants J,K,a n d K ?are all positive. The term Vpinrepre- sents pinning due to additional localized anisotropyenergy. The exchange interaction between localized spinsand conduction electrons is given by H int/.0136/.0255/.0001 SZ d3xS/.0133x/.0134/.0133cy/.0027c/.0134x; (2) where2/.0001andc(cy) are the energy splitting and annihi- lation (creation) operator of conduction electrons, respec-tively, and /.0027is a Pauli-matrix vector. The electron part is given by H el/.0136P keP00R5kcy kckwith eP00R5k/.0136/.0022h2k2=2m. In the absence of VpinandHint, the spin part has a static domain wall of width eP002R/.0017/.0133J=K/.01341=2as a classical solution. W e consider a wire with width smaller than eP002Rand treat the spin configuration as uniform in the yzplane, perpen- dicular to the wire direction x. The solution centered atx/.0136Xis given by eP00R8/.0136eP00R80/.0133x/.0255X/.0134,eP00I0/.01360, where coseP00R80/.0133x/.0134/.0136tanh/.0133x=eP002R/.0134,a n dsineP00R80/.0133x/.0134/.0136 /.0133cosh/.0133x=eP002R/.0134/.0134/.02551.T o describe the dynamics of the domain wall, it is crucial to observe that the weighted average of eP00I0, defined by eP00I00/.0133t/.0134/.0017R/.0133dx=2eP002R/.0134eP00I0/.0133x; t/.0134sin2eP00R80/.0133x/.0255X/.0133t/.0134/.0134plays the role of momentum conjugate to Xand, hence, must be treated as dynamical [14]. Neglecting spin-wave excitations, weobtain the Lagrangian for X/.0133t/.0134andeP00I0 0/.0133t/.0134as LS/.0136/.0255/.0022hNS eP002RX_eP00I0eP00I00/.02551 2K?NS2sin2eP00I00/.0255Vpin/.0133X/.0134;(3) where Vpin/.0133X/.0134is a pinning potential for X,a n d N/.0136 2AeP002R=a3is the number of spins in the wall. ( Ais the cross-sectional area.) The equations of motion, derivedfrom the Lagrangian, L S/.0255Hint, are given by /.0022hNS eP002R/.0018 _eP00I0eP00I00/.0135eP00RR_XX eP002R/.0019 /.0136Fpin/.0135Fel; (4) /.0022hNS eP002R/.0133_XX/.0255eP00RReP002R_eP00I0eP00I00/.0134/.0136NS2K? 2sin2eP00I00/.0135Tel;z;(5) where Fpin/.0136/.0255 /.0133 @Vpin=@X/.0134, Fel/.0017/.0255/.0001 SZ d3xrxS0/.0133x/.0255X/.0134/.0001n/.0133x/.0134; (6) Tel/.0017/.0255/.0001 SZ d3xS0/.0133x/.0255X/.0134/.0002n/.0133x/.0134: (7) HereS0denotes S/.0133x/.0134with eP00R8/.0136eP00R80/.0133x/.0255X/.0134,eP00I0/.0136eP00I00,a n d neP0022/.0017hcyeP002>eP0022ci(eP0022/.0136x; y; z ) is (twice) the spin density of conduction electrons. Felrepresents a force acting on the wall, or momentum transfer, due to the electron flow, while Telis a spin torque, or spin transfer, which comes from the directional mismatch between wall magnetiza-tionS0/.0133x/.0255X/.0134andn/.0133x/.0134. W e have added a damping term (eP00RR), which represents a standard damping torque (Gilbert damping), Tdamp/.0136/.0255eP00RR SS/.0002_SS[1]. Note that the spin- transfer effect acts as a source to the wall velocity via vel/.0017/.0133eP002R=/.0022hNS/.0134Tel;z. To estimate Felandvel, we calculate spin polarization n/.0133x/.0134in the presence of a domain wall by use of a local gauge transformation in spin space [15], c/.0133x/.0134/.0136U/.0133x/.0134a/.0133x/.0134, where a/.0133x/.0134is the two-component electron operator in the rotated frame, and U/.0133x/.0134/.0017m/.0133x/.0134/.0001/.0027is an SU(2) matrix with m/.0133x/.0134/.0136fsin/.0137eP00R80/.0133x/.0255X/.0134=2/.0138coseP00I00;sin/.0137eP00R80/.0133x/.0255 X/.0134=2/.0138sineP00I00;cos/.0137eP00R80/.0133x/.0255X/.0134=2/.0138g. The expectation value in the presence of electric current is written in termsof the Keldysh-Green function in the rotated frame.For instance, n x/.0133x/.0134/.0136/.0137 /.01331/.0255coseP00R80/.0134cos2eP00I00/.02551/.0138~nnx/.0135/.01331/.0255 coseP00R80/.0134coseP00I00sineP00I00~nny/.0135sineP00R80coseP00I00~nnz, where ~nneP0022/.0133x/.0134/.0017 /.0255iTr/.0137G<xx/.0133t; t/.0134eP002>eP0022/.0138, G< xeP002>;x0eP002>0/.0133t; t0/.0134/.0017ihay x0;eP002>0/.0133t0/.0134ax;eP002>/.0133t/.0134i, (eP002>; eP002>0/.0136/.0006 denotes spin) being the lesser component of the Keldysh-Green function. After a straightforward cal- culation, we obtain Fel/.0136/.0255eP0025/.0022h2/.0001 L2X kqeP002>u2qfkeP002>/.01332k/.0135q/.0134x 2meP002>eP00R4/.0133eP00R5k/.0135q;/.0255eP002>/.0255eP00R5keP002>/.0134; (8) and vel/.0136/.0022heP002R2/.0001 NSL2X kqeP002>u2qfkeP002>/.01332k/.0135q/.0134x 2mP eP00R5k/.0135q;/.0255eP002>/.0255eP00R5keP002>;(9) to the lowest order in the interaction (with wall) uq/.0017 /.0255Rdxe/.0255iqxrxeP00R80/.0133x/.0134/.0136eP0025=/.0137cosh/.0133eP0025eP002Rq=2/.0134/.0138. The distribu- tion function fkeP002>specifies the current-carrying nonequi- librium state, and Pmeans taking the principal value. As is physically expected, Felis proportional to the reflection probability of the electron and, hence, to the wall resistivity, as well as to the charge current. In fact, by adopting the linear-response form, fkeP002>’f0/.0133eP00R5keP002>/.0134/.0135 eE/.0001veP0028/.0133@f0=@eP00R5/.0134, as obtained from the Boltzmann equa- tion ( f0: Fermi distribution function; E: electric field; v/.0136/.0022hk=m;eP0028: transport relaxation time due to a single wall), we can write as Fel/.0136enjRwin one dimension. Here nandjare the electron density and current density, respectively, and Rw/.0136/.0133h=e2/.0134/.0133eP00252=8/.0134/.0133eP00R62=1/.0255eP00R62/.0134/.0133u2 /.0135/.0135 u2/.0255/.0134is the wall resistance [16], with eP00R6/.0017/.0133kF/.0135/.0255 kF/.0255/.0134=/.0133kF/.0135/.0135kF/.0255/.0134and u/.0006/.0017ukF/.0135/.0006kF/.0255. More generally, one can prove rigorously the relation [17,18] Fel/.0136eNeeP0026wj/.0136enRwIA; (10) using the Kubo formula, where eP0026w/.0017RwA=L is the resis- tivity due to a wall [19], I/.0017jA,a n d Ne/.0017nLA is the total electron number. Equations (4) and (5) with (9) and (10) constitute a main framework of the present Letter. W e next go on to studying them in the two limiting cases: adiabatic wall and abrupt wall.PHYSICAL REVIEW LETTERSweek ending 27 FEBRUARY 2004 VOLUME 92, N UMBER 8 086601-2 086601-2W e first study the adiabatic limit, which is of interest for metallic nanowires, where eP002R/.0029k/.02551 F.I nt h i sl i m i t ,w e take u2q!4eP0025 eP002ReP00R4/.0133q/.0134and by noting /.0133eP00R5k/.0135q;/.0255eP002>/.0255eP00R5keP002>/.0134q/.01360/.0136 2eP002>/.0001/.02220, we immediately see from Eq. (8) that Fel/.01360, whereas vel/.0136eP002R/.0022h NS1 LX keP002>eP002>kx mfkeP002>/.01361 2Sa3 ejs (11) remains finite. The spin transfer in this adiabatic limit is thus proportional to spin current flowing in the bulk(away from the wall), j s/.0017e/.0022h mVP kkx/.0133fk/.0135/.0255fk/.0255/.0134(V/.0017 LAbeing the system volume). In reality, the spin current is controlled only by controlling charge current. In the linear-response regime, it is proportional to the charge current jasjs/.0136eP00R>j,eP00R>being a material constant. This parameter can be written as eP00R>/.0136P eP00RR/.0133eP002>eP00RR /.0135/.0255 eP002>eP00RR/.0255/.0134=P eP00RR/.0133eP002>eP00RR /.0135/.0135eP002>eP00RR/.0255/.0134for a wire or bulk transport, and eP00R>/.0136P eP00RR/.0133NeP00RR /.0135/.0255NeP00RR/.0255/.0134=P eP00RR/.0133NeP00RR /.0135/.0135NeP00RR/.0255/.0134for a nanocontact and a tunnel junction, where eP002>eP00RR /.0006and NeP00RR /.0006are band ( eP00RR)a n d spin ( /.0006) resolved electrical conductivity and density of states at the Fermi energy, respectively, of a homogeneous ferromagnet. Experiments indicate that eP00R>is of the order of unity in both bulk transport [20,21] and tunnel junc-tions ( /.00240:5[22]). As seen from Eq. (15) below, the speed of the stream motion of the wall is roughly given by v el(except in the vicinity of the threshold jcr). For a lattice constant a/.0024 1:5/.0023Aand current density j/.01361:2/.00021012/.0137A=m2/.0138[6], we have a3j=e/.0024250/.0137m=s/.0138. This speed is expected for strongly spin-polarized materials ( eP00R>/.00241) including tran- sition metals, but is 2 orders of magnitude larger than theobserved value /.00243/.0137m=s/.0138[6]. This discrepancy may be due to dissipation of angular momentum by spin-waveemission, which is now under investigation [17]. Let us study the wall motion in the absence of pinning, F pin/.01360, by solving the equations of motion, (4) and (5) in the adiabatic case ( Fel/.01360). The solution with the initial condition X/.0136eP00I00/.01360att/.01360is obtained as eP0020cot/.0018eP00RR eP002RX/.0019 /.0136/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 /.0129 1/.0255eP00202p coth/.0133eP00RIt/.0134/.01351 /.0133jeP0020j<1/.0134(12) /.0136/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 /.0129 eP00202/.02551p cot/.0133eP00RIt/.0134/.01351 /.0133jeP0020j>1/.0134;(13) where eP0020/.00172/.0022hvel=/.0133SK?eP002R/.0134 and eP00RI/.0136/.0137eP00RR=/.01331/.0135 eP00RR2/.0134/.0138/.0133SK?=2/.0022h/.0134/.0002/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 /.0129 1/.0255eP00202p .F o r jvelj<vcr/.0017SK?eP002R=2/.0022h(i.e., jeP0020j<1), cot/.0133eP00RRX=eP002R /.0134remains finite as t!1 , and the wall is not driven to a stream motion but just displaced by /.0001X/.0136 eP002R 2eP00RRsin/.02551eP0020. In this case, the transferred spin is absorbed by eP00I00and ‘‘dissipated’’ through K?, as seen from Eq. (5), and is not used for the translational motion of the wall(_XX); the wall is apparently ‘‘pinned’’ by the transverse anisotropy. Thus, even without pinning force, the current cannot drive the wall if the associated spin current is smaller than the critical value [23]j cr/.01331/.0134 s/.0136eS2 a3/.0022hK?eP002R: (14) Above this threshold, js>jcr/.01331/.0134 s(jeP0020j>1), this process with K?cannot support the transferred spin and the wall begins a stream motion. The wall velocity after ‘‘depin-ning’’ is an oscillating function of time around the aver- age value (Fig. 1) h_XXi/.01361 1/.0135eP00RR21 2Sa3 e/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 /.0129 j2s/.0255/.0133jcr/.01331/.0134 s/.01342q ; (15) which is similar to the W alker’s solution for the field- driven case [1,24]. (The bracket h/.0001 /.0001 /.0001i means time aver- age.) The asymptotic behavior h_XXi/jsforjs/.0029jcr/.01331/.0134 sis governed by the angular momentum conservation (withconstant dissipation rate). W e now introduce a pinning potential V pinand study the ‘‘true’’ depinning of the wall by the spin-transfer effect in the adiabatic limit. Since spin transfer acts as aforce on eP00I0 0, the depinning can be better formulated in terms of eP00I00. W e consider a quadratic pinning potential with a range eP0024,Vpin/.0136/.0133NV0=eP00242/.0134/.0133X2/.0255eP00242/.0134eP00R8/.0133eP0024/.0255jXj/.0134, where eP00R8/.0133x/.0134is the Heaviside step function. Then the equa- tion for eP00I00reads /.01331/.0135eP00RR2/.0134/.0127eP00I0eP00I00/.0136/.0255 eP00RR_eP00I0eP00I00/.0133eP002I/.0135eP0022cos2eP00I00/.0134/.0255 eP002I/.0137/.0133eP0022=2/.0134sin2eP00I00/.0135/.0133vel=eP002R/.0134/.0138, where eP0022/.0017SK?=/.0022hand eP002I/.0017 2V0eP002R2=eP00242/.0022hS. This equation describes the motion of a classical particle in a tilted washboard potential ~VVwith (modified) friction. For vel>vcr/.0133/.0136eP0022eP002R 2/.0134,l o c a lm i n i m a disappear in ~VVandeP00I00is then ‘‘depinned.’’ Then the above equation indicates that eP00I00starts to drift with average velocity h_eP00I0eP00I00i/.0136/.0255 vel=/.0133eP00RReP002R/.0134(with oscillating components neglected). The displacement of X/.0133t/.0134inside the pinning potential is then obtained from Eqs. (4) and (5) as X’ /.0133vel=eP002IeP00RR/.0134/.0017Xmax. The depinning of the wall occurs when Xmax>eP0024, which defines another critical current, jcr/.01332/.0134 s. Thus, the critical spin current jcrswill be given by jcr/.01331/.0134 s defined above if the pinning is weak ( V0&K?=eP00RR), while it is given by jcr/.01332/.0134 s/.00174e a3/.0022heP00RRV0eP002R2=eP0024 (16) if the pinning is strong ( V0*K?=eP00RR). Since eP00RRis usually believed to be small [9], we expect that the criticalcurrent is mostly determined by K ?. This seems to be consistent with the observations that the critical current is FIG. 1. Time-averaged wall velocity as a function of spin current, js, in the weak pinning case ( V0&K?=eP00RR).PHYSICAL REVIEW LETTERSweek ending 27 FEBRUARY 2004 VOLUME 92, N UMBER 8 086601-3 086601-3larger for a thinner film [6,9] and does not depend much on pinning [25]. It would be interesting to carry outmeasurements on a wire with small K ?. Let us go on to the opposite limit of an abrupt wall, eP002R!0. As seen from Eq. (9), the spin-transfer effect vanishes. The pinning-depinning transition is thus deter-mined by the competition between F elandFpin,g i v i n g the critical current density jcr/.0136NV0 eP0024eNeeP0026w/.01362V0eP002R ena3eP0024RwA: (17) The average wall velocity after depinning is obtained as h_XXi/.0136/.0133 eP002R2Nee=/.0022heP00RRNS /.0134eP0026wj. This velocity vanishes in the limit eP002R!0due to the divergence of the wall mass Mw/.0136 /.0022h2N=K ?eP002R2. For metallic nanocontacts, where eP0024/.0024eP002R/.0024aandna3/.0024 1, experiments indicate that the wall resistance can be of the order of h=e2/.013626 k/.0010 [26]. Thus jcr/.0024/.01335/.00021010/.0002 Bc/.0137T/.0138/.0134 /.0137A=m2/.0138, where Bc/.0136V0eP002R=eP0022BeP0024Sis the depinning field ( eP0022Bis Bohr magneton). Bc/.002410/.02553/.0137T/.0138(like in Ref. [26]) corresponds to jcr/.00245/.0002107/.0137A=m2/.0138. In conclusion, we have developed a theory of domain wall dynamics including the effect of electric current.The current is shown to have two effects: spin transferand momentum transfer, as pointed out by Berger. For anadiabatic (thick) wall, where the spin-transfer effect dueto spin current is dominant, there is a threshold spincurrent j crs/.0024/.0133eeP002R=a3/.0022h/.0134maxfK?;eP00RR V0eP002R eP0024gbelow which the wall cannot be driven. This threshold is finite even in the absence of pinning potential. The wall motion is hence not affected by the uncontrollable pinning arising fromsample roughness for weak pinning ( V 0&K?=eP00RR). In turn, wall motion would be easily controlled by thesample shape through the demagnetization field andthus K ?. The wall velocity after depinning is obtained ash_XXi//.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 /.0129 /.0133js/.01342/.0255/.0133jcrs/.01342p . In contrast, an abrupt (thin) wall is driven by the momentum-transfer effect due to charge current, i.e., by reflecting electrons. In this case, the depinning current is given in terms of wall resistivityeP0026 wasjcr/V0=eP0026w. The two limiting cases considered above are both realistic. Most metallic wires fabricated by lithographyare in the adiabatic limit, as is obvious from the verysmall value of wall resistivity [27]. In contrast, a very thinwall is expected to be formed in metallic magnetic nano- contacts with a large magnetoresistance [26]. A system of recent interest is magnetic semiconductors [28], where theFermi wavelength is much longer than in metallic sys-tems. As suggested by the large magnetoresistance ob-served recently [29], magnetic semiconductors would besuitable for precise measurement in the thin wall limit. The authors are grateful to T. Ono for motivating us by showing the experimental data prior to publication. W e also thank J. Shibata and A. Y amaguchi for valuablediscussions. G.T. is grateful to Monka-shou, Japan and The Mitsubishi Foundation for financial support. [1] A. Hubert and R. Scha ¨fer,Magnetic Domains (Springer- V erlag, Berlin, 1998); F . H. de Leeuw, R. van den Doel, and U. Enz, Rep. Prog. Phys. 43, 659 (1980). [2] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [3] Y . Tserkovnyak, A. Brataas, and G. E.W . Bauer, Phys. Rev. B 66, 224403 (2002). [4] J. Grollier et al. , J. Appl. Phys. 92, 4825 (2002); Appl. Phys. Lett. 83, 509 (2003); N. V ernier et al. , Europhys. Lett. 65, 526 (2004). [5] M. Kla ¨uiet al. , Appl. Phys. Lett. 83, 105 (2003). [6] A. Y amaguchi et al. , Phys. Rev. Lett. 92, 077205 (2004). 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(14)] by use of phenomenological parameters under the assumption that the interband scattering is essential. [19] eP0026wis proportional to the wall density, i.e., 1=L, and so NeeP0026wis independent of L. [20] I. A. Campbell and A. Fert, in Ferromagnetic Materials , edited by E. P . W ohlfarth (North-Holland, Amsterdam, 1982), V ol. 3. [21] L. Piraux et al. , Eur. Phys. J. B 4, 413 (1998). [22] D. J. Monsma and S. S. P . Parkin, Appl. Phys. Lett. 77, 720 (2000). [23] The same expression was obtained in a different context as the critical current for the precession of wall spins inL. Berger, Phys. Rev. B 33, 1572 (1986) [Eq. (5)]. [24] In the case of a pulsed current, the wall displacement /.0001X was plotted as function of current in Fig. 4 of Ref. [9]. [25] S. S. P . Parkin (private communication); T. Ono (private communication). [26] N. Garcia et al. , Phys. Rev. Lett. 82, 2923 (1999); G. Tatara et al. ,ibid. 83, 2030 (1999). [27] A. D. Kent et al. , J. Phys. Condens. 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PhysRevLett.123.217201.pdf

Fast Domain Wall Motion Governed by Topology and Œrsted Fields in Cylindrical Magnetic Nanowires M. Schöbitz,1,2,3 ,*A. De Riz,1S. Martin,1,3S. Bochmann,2C. Thirion,3J. Vogel ,3M. Foerster ,4L. Aballe,4 T. O. Mente ş,5A. Locatelli ,5F. Genuzio ,5S. Le-Denmat,3L. Cagnon,3J. C. Toussaint,3D. Gusakova,1 J. Bachmann ,2,6and O. Fruchart1,† 1Univ. Grenoble Alpes, CNRS, CEA, Spintec, 38054 Grenoble, France 2Friedrich-Alexander-Universität Erlangen-Nürnberg, Inorganic Chemistry, 91058 Erlangen, Germany 3Univ. Grenoble Alpes, CNRS, Institut N´ eel, 38042 Grenoble, France 4Alba Synchrotron Light Facility, CELLS, 08290 Barcelona, Spain 5Elettra-Sincrotrone Trieste S.C.p.A., Basovizza, 34149 Trieste, Italy 6Institute of Chemistry, Saint Petersburg State University, St. Petersburg 198504, Russia (Received 9 July 2019; published 21 November 2019) While the usual approach to tailor the behavior of condensed matter and nanosized systems is the choice of material or finite-size or interfacial effects, topology alone may be the key. In the context of the motion of magnetic domain walls (DWs), known to suffer from dynamic instabilities with low mobilities, we report unprecedented velocities >600m=s for DWs driven by spin-transfer torques in cylindrical nanowires made of a standard ferromagnetic material. The reason is the robust stabilization of a DW type with a specific topology by the Œrsted field associated with the current. This opens the route to the realization of predicted new physics, such as the strong coupling of DWs with spin waves above >600m=s. DOI: 10.1103/PhysRevLett.123.217201 It is well known that specific properties in condensed- matter and nanosized systems can be obtained by eitheracting on the electronic structure by selecting an appropriatematerial composition and crystalline structure, or by makinguse of finite-size and interfacial effects, strain, gating with anelectric field, etc. [1]. These approaches have proven suitable for tailoring charge transport, optical properties, electric or magnetic polarization, etc., however, there arelimits regarding what can be achieved with materials, orrealized with device fabrication. An alternative strategyentails considering a specific topology in order to developthe desired properties of a system, yielding diverse appli-cations such as the design of wide-band-gap photoniccrystals [2]and the control of flow of macromolecules [3], or novel theoretical methods such as for the description of defects [4], or intringuing 3D vector-field textures such as hopfions and torons [5]. As regards magnetism, unusual properties resulting from topological features have beenpredicted, such as the existence of a domain wall (DW) in theground state of a Moebius ring [6], or the nonreciprocity of spin waves induced by curvature and boundary conditions in nanotubes [7]. Here, we show that topology plays a critical role in the physics of DW motion in one-dimensional conduits,a prototypical case for magnetization dynamics. For the sake of simplicity of fabrication and monitoring, DW motion under magnetic field or spin-polarized current isusually conducted in planar systems, made of stacked thinfilms patterned laterally by lithography. In them, DWs aredynamically unstable above a given threshold of field or current (Walker limit), undergoing transformations oftheir magnetization texture associated with a drastic dropin their mobility. Ways are being investigated to overcome this limitation through the engineering of microscopic properties. Two major routes are the use of theDzyaloshinskii-Moriya interaction in order to stabilizethe walls [8–10], or of natural or synthetic ferrimagnets with vanishing magnetization to decrease the angularmomentum in order to increase spin-transfer torque effi- ciency and boost the precessional frequency [11–13]. The three-dimensional nature of cylindrical nanowires (NWs) gives rise to the existence of a DW with a specific topology, which respects the rotational invariance and circular boundary conditions. It is named the Bloch-pointwall (BPW) [14] and has been experimentally confirmed only recently [15,16] . It was predicted that this wall can circumvent the Walker limit, but field-driven motion experiments disappointingly failed to confirm a topological protection [17]. Here, we report experimental results on current-induced DW motion in such NWs. We show thatalthough previously disregarded, the Œrsted field induced by the current plays instead a crucial and valuable role in stabilizing BPWs, contrary to the field-driven case. This allows them to retain their specific topology and thus reachvelocities >600m=s in the absence of Walker breakdown, which is quantitatively consistent with predictions. DWs with two distinct topologies exist in NWs: the transverse-vortex wall (TVW) and the BPWPHYSICAL REVIEW LETTERS 123, 217201 (2019) 0031-9007 =19=123(21) =217201(6) 217201-1 © 2019 American Physical Society[Figs. 1(a), 1(b) ]. The former has the same topology as all DW types known in 2D flat strips [18]. The latter is found only in NWs and exhibits azimuthal curling of magneticmoments around a Bloch point, a local vanishing ofmagnetization [19,20] . This unique topological feature of NWs is at the origin of the predicted fast speed and stabilityduring magnetic-field or current-driven motion of BPWs.This is easily explained by considering the time derivativeof the magnetization vector _mat any point, described by the Landau-Lifshitz-Gilbert equation [21]. _m¼−γ 0m×Hþαm×_m−ðu·∇Þmþβm×½ðu·∇Þm/C138; ð1Þ with γ0¼μ0jγj,γbeing the gyromagnetic ratio, α≪1the Gilbert damping parameter, and βthe nonadiabaticity parameter. H, the total effective field, is comprised of applied fields and fields originating from magnetic anisotropy,exchange, and dipolar energy. The spin-polarized part ofthe charge current induces so-called spin-transfer torques,taken into account through u,w i t h juj¼Pðjμ B=eM sÞ[21]. jandPare the charge current and its spin-polarization ratio, respectively, μBis the Bohr magneton, ethe elementary charge, and Msthe spontaneous magnetization. In purely field driven cases, the applied field favors the precession of maround the field direction. In flat strips, for applied fields above a few mT this causes repeated DWtransformations from transverse to vortex walls for in-planemagnetization, and from N´ eel to Bloch walls for out-of- plane magnetization. This so-called Walker breakdown [22] is facilitated by the fact that all these DW configurationsshare the same topology [23–25]. The mobility is high below the Walker threshold field (scaling with 1=α) and low above (scaling with α). The same physics is expected in NWs for the TVW, with the Walker field equal to zero due to therotational symmetry [7,14] . The phenomenology of current- driven cases is similar: the adiabatic term favors motion, thenonadiabatic term favors azimuthal precession [third and fourth terms in Eq. (1), respectively], and the DW velocity is expected to be ≈ðβ=αÞubelow the Walker threshold and ≈u above it [21,24] , again with a vanishing threshold for TVWs in NWs [26]. In contrast to these cases, one expects that magnetization cannot freely precess azimuthally in a BPW, since it wouldperiodically imply a head-on or tail-on configuration alongall three axes around the Bloch point, with an enormouscost in dipolar energy. Instead, the azimuthal rotationshould come to a halt and remain in a state essentiallysimilar to the static one [Fig. 1(b)]. This implies an absence of Walker breakdown, both under field [7,14] and current [27,28] , and steady-state motion of the wall. The steady circulation is expected to be clockwise (CW) with respectto the direction of motion of the DW, while the counter-clockwise (CCW) circulation may undergo a dynamics-induced irreversible switching event to recover the CWcirculation and steady state. This picture is valid both for BPWs in wires [14,27] , and vortex walls [7,28] in thick- walled tubes. Thanks to this locked topology, the mobilityof the BPW is expected to remain high under both field andcurrent. Only when a speed around ≈1000 m=s is attained, the speed is predicted to reach a plateau, with new physicsexpected to occur via interactions with spin waves, knownas the spin-Cherenkov effect [7]. However, so far there exists no experimental report of the mobility of any of thesewalls under neither magnetic field nor current. Our Letter is based on magnetically soft Co 30Ni70wires with diameter 90 nm, electroplated in anodized aluminatemplates [29]. Following the dissolution of the latter, isolated wires lying on a Si substrate are contacted withpads to allow for the injection of electric current. DWs weremonitored with both magnetic force microscopy (MFM)and x-ray magnetic circular dichroism photo-emissionelectron microscopy (XMCD-PEEM) in the shadow mode[Fig. 1(c)] to reveal the three-dimensional texture of magnetization [16,30,31] . While in MFM, sharp ns-long pulses could be sent, in XMCD-PEEM the shape of currentpulses was distorted to a minimum width of 10 –15 ns, due to long cabling, UHV feedthroughs and the sample holder contacts. Micromagnetic simulations were carried out withthe homemade finite-element code FeeLLG ood[32], based on the Landau-Lifshitz-Gilbert equation including spin-trans-fer torques. See Supplemental Material for additionaldetails on the methods [33]. Domain wall velocities were experimentally investigated primarily with MFM imaging. Figure 2(b)shows an atomic force microscopy (AFM) image of the left-hand side of thecontacted NW from Fig. 2(a). The corresponding MFM image in Fig. 2(c)shows the initial magnetic configuration, with two DWs located at 1.2 and 7.2μm from the edge of the left contact. By applying a current pulse of duration 5.8 ns and amplitude 2.2×10 12A=m2, the left hand DW moved over a distance of ≈2μm [Fig. 2(d)], corresponding to an average velocity of ≈350m=s. However, the right-hand (a) (c)(b) Photons 16°Electrons SubstrateXMCD-PEEM image Wire Shadow area FIG. 1. Schematic of (a) a TDW and (b) a BPW. (c) Schematic of shadow XMCD-PEEM and the contrast resulting from a BPW.PHYSICAL REVIEW LETTERS 123, 217201 (2019) 217201-2DW remains pinned, highlighting a common and key issue for inferring DW velocities from motion distances: pinning on geometrical or microstructural defects hampers DWmotion [44]. Depinning not only requires a current density above a critical value j dp, but repinning can also occur at another location with a deeper energy well, while the current pulse is still being applied. This results in DW propagationwith an effective time span possibly much shorter than thenominal pulse duration. Consequently, the values for DW velocity converted from motion distance and nominal pulse length are a lower bound of an unknown higher velocity (seeSupplemental Material [33] for a quantitative discussion). Furthermore, with such large current densities the effect of Joule heating may not be neglected. However, measurementsof the NW resistance during the pulse showed that the samples never exceeded the Curie temperature (see Supplemental Material [33]) and that the results described herein are not caused by thermal activation. Figure 2(e) (open circles) shows the discussed lower bound for DW velocity, as a function of applied current density, inferred from a multitude of MFM images before and after pulses with durations ranging from 5 to 15 ns.Consistent with the expected occurrence of re-pinning,lower velocities are inferred from longer pulse durations.Still, DW velocities up to >600m=s were observed for applied current densities ≈2.4×10 12A=m2. This sets a fivefold record for purely spin-transfer torque motion ofDWs in a standard ferromagnetic material, i.e., with large magnetization, with reported values hardly exceeding 100m=s[45]. Similar or higher speeds have been mea- sured recently, however, in low-magnetization ferrimag- nets, thereby enhancing the efficiency of spin-transfer torque [46]. Here, it is the topology of the wall that enhances the DW speed, not a special material.Similarly, these DW velocity measurements are not dis- torted by DW inertia, since simulations showed that this effect will only come into play in subnanosecond pulseexperiments (see Supplemental Material [33]). The black dotted lines in Fig. 2(e)act as a guide to the eye through the speed predicted by the one-dimensional model below theWalker breakdown v¼ðβ=αÞu, for three different ratios ofβ=α: 1, 2, and 3 (for Co 30Ni70Ms¼0.67MA=m2, P≈0.7, resulting in u≈60.4m=s per 1012A=m2). This is not intended as precise modeling, but rather to show that the experiments are clearly not compatible with v¼u, supporting the absence of Walker breakdown for the BPW. Instead a value of β=α⪆3is inferred. Note, however, that the adverse effects of DW pinning reappear in the form of a threshold current density jdp≈1.2×1012A=m2required to set any DW in motion. Even above this value, DW motion was not fully reproducible, with some pinning sites associated with a larger jdp. To link unambiguously the measured velocity with theory, the DW type must be identified. For this purpose, weemployed shadow XMCD-PEEM and imaged NWs beforeand after injecting a given current pulse [Figs. 3(a), 3(b) , and full symbols in Fig. 2(e)]. Note that the values for speed are lower than those measured with MFM, as expected for lesssharp pulse shapes with consequentially larger width. Returning to the DW type, the first striking fact is the following: from hundreds of DWs imaged after currentinjection, all were of the BPW type. These unambiguouslyappear as a symmetric bipolar contrast in the shadow [16], corresponding to azimuthal rotation of magnetization as on Figs. 3(a)–3(b). This sharply contrasts with all our previous observations of NWs, imaged in the as-prepared state or following a pulse of magnetic field, for which both TVWs and BPWs had been found in sizable amounts [16,17] .T h e second striking fact is that the sign of the BPW circulation is deterministically linked to the sign of the latest current pulse, provided that its magnitude is above a rather well-definedthreshold which, as shown in Fig. 3(c), lies around 1.4×10 12A=m2. In contrast with a one-time Walker event discussed previously, this holds true irrespective of whether or not the wall has moved under the stimulus of the currentpulse, and is independent of the pulse duration at the probedtimescales. We hypothesize that these two facts are related to theŒrsted field associated with the longitudinal electric current, its azimuthal direction favoring the BPW with a (a)10 µm j8 6 2 4 0µm j j (1012 A/m2)3.0 0 0.5 1.0 1.5 2.0 2.55 t < 10 ns 10 t < 15 ns 15 t < 25 ns 25 t < 35 ns 35 t nsvDW (m/s) 0100200300400500600700(b) (c) (d) (e) FIG. 2. (a) SEM, (b) AFM, and corresponding (c),(d) MFM images of a 90 nm diameter Co 30Ni70NW with Ti =Au electrical contacts. (c) Initial state, with two DWs. (d) Same wire, after a current pulse with 2.2×1012A=m2magnitude and 5.8 ns duration. (e) Domain wall velocity as a function of appliedcurrent density and duration (see inner caption), monitored withMFM (open circles) and XMCD PEEM (filled circles) from fourindividual NWs. The dashed lines are expectations from the one-dimensional model below the Walker breakdown, for v¼ðβ=αÞu with β=α¼1,2 ,3 .PHYSICAL REVIEW LETTERS 123, 217201 (2019) 217201-3given circulation. Indeed, for a uniform current density j, theŒrsted field is H¼jr=2at distance rfrom the NW axis. For the present NWs with radius R¼45nm and j¼ 1×1012A=m2this translates to 28 mT at the NW surface, which is a significant value. In order to support this claim, we conducted micro- magnetic simulations including the Œrsted field, which had not been considered in previous works. Starting from a DWat rest with R¼45nm, we used α¼1to avoid ringing effects and obtain a quasistatic picture, suitable to describe the PEEM experiments, for which the pulse rise time is several nanoseconds. We evidenced that while the addedeffect of spin-transfer torques may alter the transformationmechanisms, it is of minor importance compared to theŒrsted field and considering or disregarding these torques does not quantitatively impact switching. Accordingly,below we present only results disregarding these torques.With an applied Œrsted field, within the domains the peripheral magnetization tends to curl around the axis,while it remains longitudinal on the NW axis. We firstconsider TVWs as the initial state and find that thesetransform into BPWs with CW circulations with respect tothe current direction, if the current density exceeds 0.4×10 12A=m2. The underlying process is illustratedon Fig. 4(a), displaying maps of the radial and azimuthal magnetization components, mrandmφ, respectively, on the unrolled surface of a NW as a function of time. These highlight the locations of the inward and outward flux of magnetization through the surface, signature of a TVW [18]. While these local configurations are initially diamet- rically opposite, they approach each other until they eventually merge, expelling the transverse core of the wall from the NW. This is associated with the nucleation of aBloch point at the NW surface, which later on drifts towards the NW axis, ending up in a BPW. This process is similar to the dynamical transformation of a TVW into aBPW upon motion under a longitudinal magnetic field [17], and explains the absence of TVWs in our measurements, for which the applied current densities were always larger than 0.4×10 12A=m2. In order to understand the unique circulation observed, we now consider a BPW as the initial state. BPWs with a circulation matching that of the Œrsted field do not change qualitatively, only their width increases during the pulse. On the contrary, BPWs shrink if their initial circulation is CCW, i.e., opposite to the Œrsted field. Forj≤1.5×1012A=m2the CCW BPW reaches a narrow yet stable state, and recovers its initial state after the pulse. Beyond this value the circulation switches through a transient radial orientation of magnetization [Fig. 4(b)]. After the switching of circulation, the BPW expands and reaches a stable CW state. The value of the critical current density required for circulation switching isin quantitative agreement with the experimental one [Fig. 3(c),≈1.4×10 12A=m2], although the simulation does not incorporate thermal activation and considersα¼1. This suggests that the switching process is robust and intrinsic, in agreement with the narrow experimental distribution of critical current. In our simulations the timerequired for switching is <10ns, though switching times an order of magnitude faster are expected for realistic values of α<0.1, which explains why no dependence on the pulse width was observed in the experiments, where all pulse widths were above 5 ns. In experiments where the DW type was visible, DW motion events were observed for applied current densities larger than the critical current density required for the circulation switching event. Thus, in these the circulation isalways CCW with respect to the propagation direction, i.e., CW with respect to the current direction, because the charge of electrons is negative. Remarkably, this sense ofcirculation is opposite to the situation expected when neglecting the Œrsted field, which would select the CW circulation with respect to the propagation direction, asdictated by the chirality of the LLG equation [14,27,28] . There must therefore be a competition for the circulation sense and for the case of 90 nm diameter NWs, the Œrsted field dominates. Despite this, we find in simulations that the BPW motion still follows v≈ðβ=αÞuwhether or not the Œrsted field is considered. Notice that the βparameter is (a) (b) (c)1 µm -2 2 1 0 -1 j (1012 A/m2)01 Pswitch(j)Wire 1 Wire 2 10 t < 20 ns 20 t < 30 ns 30 t < 40 ns 40 t ns FIG. 3. (a), (b) Consecutive XMCD-PEEM images of a NW with a tilted x-ray beam (orange arrow). The azimuthal circu- lation of the four BPWs seen in the NW shadow is indicated by the white arrows, consistent with the Œrsted field of the previously applied current (blue and red arrows in the right-handschematic, respectively). From (a) to (b), a 15 ns and 1.4×10 12A=m2current pulse switches 75% of BPWs. DW displacement from (a) to (b) cannot be discussed as directlyresulting from spin-transfer torque, and the density of current liesbelow the threshold for free motion (c) BPW switching proba-bility as a function of jfor two different wire samples (squares and triangles). Pulse durations are categorized and color coded,see included labels. The gray region indicates the current densityrequired for switching in simulations.PHYSICAL REVIEW LETTERS 123, 217201 (2019) 217201-4expected to depend on the DW width, however, for widths much smaller than the ones studied here [47]. The predictions of high mobility and possibly spin-Cherenkov effect are thus probably not put into question. Surprisingly, the Œrsted field was previously only considered in a single report for NWs of square crosssection [48]. No qualitative impact was found, likely because a NW side of at most 48 nm was considered, and a simple analytical model describing magnetization in the domain and balancing Zeeman Œrsted energy with exchange energy shows that the impact of the Œrsted field scales very rapidly as R 3, a tendency confirmed by simulations. The situation closest to the present case isthe report of flat strips made of spin-valve asymmetricstacks [49]. Such strips can be viewed as the unrolled surface of a wire, the curling of the BPW translating into a transverse wall, which tends to be stabilized during motiondue to the Œrsted field. To conclude, we have shown experimentally and by simulation that the Œrsted field generated by thespin-polarized current flowing through a cylindrical NW has a crucial impact on DW dynamics, while it had been disregarded so far. This Œrsted field robustly stabilizes BPWs, in contrast with the field-driven case [17]. This stabilization allows for the key features predicted for their specific topology to apply [14,27,28] : we evidenced DW velocities in excess of 600m=s confirming the absence of Walker breakdown [7,50] and setting a fivefold record for spin-transfer-torque-driven DW motion in large magneti- zation ferromagnets [45]. This suggests that the experi- mental realization of further novel physics is at hand, such as the predicted spin-Cherenkov effect with strong coupling of DWs with spin waves. M. S. acknowledges a grant from the Laboratoire d’excellence LANEF in Grenoble (ANR-10-LABX-51-01). The project received financial support from the French National Research Agency (Grant No. JCJC MATEMAC-3D). This work was partly supported by the French RENATECH network, and by the Nanofab platform (Institut N´ eel), whose team is greatly acknowledged for technical support. We thank Jordi Prat for his technical support at the ALBA Circe beam line and Olivier Boulle for useful discussions. *Corresponding author. michael.schobitz@cea.fr †Corresponding author.olivier.fruchart@cea.fr [1] R. E. Newnham, Properties of Materials —Anisotropy, Sym- metry, Structure (Oxford University Press, Oxford, 2005). [2] Y. Lu, Y. Yang, J. K. Guest, and A. Srivastava, Sci. Rep. 7, 43407 (2017) . [3] J. Qin and S. T. Milner, Macromolecules 47, 6077 (2014) . [4] N. D. Mermin, Rev. Mod. Phys. 51, 591 (1979) . [5] P. J. Ackerman and I. I. Smalyukh, Phys. Rev. X 7, 011006 (2017) . [6] O. V. Pylypovskyi, V. P. Kravchuk, D. D. Sheka, D. Makarov, O. G. Schmidt, and Y. Gaididei, Phys. Rev. Lett. 114, 197204 (2015) . [7] M. Yan, C. Andreas, A. Kakay, F. Garcia-Sanchez, and R. Hertel, Appl. Phys. Lett. 99, 122505 (2011) . [8] I. M. Miron, T. Moore, H. Szambolics, L. D. Buda- Prejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. Vogel,M. Bonfim, A. Schuhl, and G. Gaudin, Nat. 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PhysRevD.90.042005.pdf

Constraining the gravitational wave energy density of the Universe using Earth ’s ring Michael Coughlin1and Jan Harms2 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2INFN, Sezione di Firenze, Sesto Fiorentino 50019, Italy (Received 5 June 2014; published 25 August 2014) The search for gravitational waves is one of today ’s major scientific endeavors. A gravitational wave can interact with matter by exciting vibrations of elastic bodies. Earth itself is a large elastic body whose so-called normal-mode oscillations ring up when a gravitational wave passes. Therefore, precisemeasurement of vibration amplitudes can be used to search for the elusive gravitational-wave signals. Earth ’s free oscillations that can be observed after high-magnitude earthquakes have been studied extensively with gravimeters and low-frequency seismometers over many decades leading to invaluableinsight into Earth ’s structure. Making use of our detailed understanding of Earth ’s normal modes, numerical models are employed for the first time to accurately calculate Earth ’s gravitational-wave response, and thereby turn a network of sensors that so far has served to improve our understanding ofEarth, into an astrophysical observatory exploring our Universe. In this paper, we constrain the energy density of gravitational waves to values in the range 0.035 –0.15 normalized by the critical energy density of the Universe at frequencies between 0.3 and 5 mHz, using ten years of data from the gravimeter network ofthe Global Geodynamics Project that continuously monitors Earth ’s oscillations. This work is the first step towards a systematic investigation of the sensitivity of gravimeter networks to gravitational waves. Further advances in gravimeter technology could improve sensitivity of these networks and possibly lead togravitational-wave detection. DOI: 10.1103/PhysRevD.90.042005 PACS numbers: 04.80.Nn, 91.30.Fn, 95.75.Wx I. INTRODUCTION So far, the strongest evidence for the existence of gravitational waves (GWs) comes from the observationof the binary pulsar PSR B 1913 þ16[1]. The shrinking of its orbit observed over three decades can be fully explained by the emission of GWs and associated energy loss according to the general theory of relativity. Dedicatedexperiments attempt to measure these waves as phasemodulation of laser beams (GEO600 [2], LIGO [3], Virgo [4], KAGRA [5], eLISA [6],T O B A [7]), or through their imprint on the polarization of the cosmic micro-wave background (BICEP2 [8],E B E X [9]). Furthermore, searches for GWs can be performed in data of otherhigh-precision experiments including Doppler trackingof satellites [10], monitoring arrival times of pulsar signals [11], or using the global positioning system [12]. Gravitational waves can also excite oscillations of elasticbodies. This principle is exploited for example in thedesign of spherical resonant GW detectors (MiniGRAIL [13], Mario Schenberg [14]), and more famously in past resonant-bar detectors (AURIGA [15], Allegro [16]). Also oscillations of stars can be excited, and therefore observa-tion of these modes can be used to detect GWs [17]. All these experiments combined monitor a wide range ofGW frequencies starting from waves that have oscillatedonly a few times since the beginning of the Universe, upto a few 1000 Hz.Recently, the authors of this paper have presented results from an observation of the free, flat surface response of Earth to GWs [18]. As was explained there, the method cannot be extended to frequencies below about 50 mHz since seismic motion starts to be globally coherent at lower frequencies, and the GW response is s t r o n g l ya f f e c t e db yE a r t h ’s spherical shape. The low- frequency GW response is best described in terms of Earth ’s normal-mode oscillations [19]. These oscillations are continuously monitored by a global network of low- frequency seismometers and gravimeters. Especially the superconducting gravimeters of the Global Geodynamics Project (GGP), which were used in this paper, provide excellent sensitivity below 10 mHz with data records reaching back more than 10 years [20].A sw i l lb es h o w n , the stationary noise background is almost the same for all gravimeters and uncorrelated between different instru- ments, which makes it possible to use a large fraction of the data of the entire network to search for GW signals that are significantly weaker than the stationary noise level by means of a near-optimal correlation method. Whereas previous GW searches using Earth ’s normal modes only tried to explain excess energy in normal modes [21,22] , the work in this paper is the first to combine a near- optimal analysis of gravimeter data with a detailed GWresponse model, which makes it possible to accurately calibrate normal-mode amplitudes into GW strain.PHYSICAL REVIEW D 90,042005 (2014) 1550-7998 =2014 =90(4) =042005(8) 042005-1 © 2014 American Physical SocietyThe limits obtained in this study through normal-mode observations are plotted in Fig. 1together with upper limits set in other frequency bands. The upper limits in thef r e q u e n c yr a n g e0 . 3t o5m H za r ei m p r o v e db y2t o5orders of magnitude. A brief summary of normal-modeoscillations is given in Sec. II. In Sec. III, we outline the theory of Earth ’s resonant (normal-mode) response to GWs. A characterization of gravimeter data is presented in Sec. IV. Finally, the GW search algorithm is discussed in Sec. V, and new constraints are presented on the energy density of GWs averaged over directions and wave polarizations. II. EARTH ’S NORMAL-MODE OSCILLATIONS Earth ’s free oscillations, called normal modes, can be excited by gravitational waves. Earth ’s slowest normal- mode oscillation occurs at about 0.3 mHz, and distinctmodes can still be identified up to a few millihertz. Athigher frequencies, the discrete vibrational spectrum trans-forms into a quasi-continuous spectrum of seismic vibra-tions that are increasingly dominated by local sources. Thedata used in this study were sampled once per minute, andlow-pass filtered suppressing signal response above about 5 mHz depending on the gravimeter. In addition, a few gravimeters show resonant features above 5 mHz in theirresponse. Therefore, the upper frequency bound of the GWsearch was chosen to be 5 mHz to guarantee accuratecalibration of the data. At frequencies below 5 mHz, the diameter of Earth is much smaller than the length of GWs. In this so-called long-wavelength regime, a GW can effectively be repre-sented by a quadrupole-force field that excites Earth ’s normal modes. Normal modes are divided into toroidal nTl and spheroidal nSlmodes, where n; lare non-negative integers that determine the radial and angular mode shape,respectively. The toroidal modes only produce tangential displacement. Spheroidal modes show tangential and radial displacement, and they also perturb Earth ’s gravity field. Not all normal modes are equally responsive to a quadru- pole force. In fact, only the quadrupole modes with l¼2 show significant GW response in the long-wavelengthregime [19]. The coupling mechanism of a GW to oscillations of elastic bodies is governed by variations of the shear modulus, including the shear-modulus change across the free surface. Earth shows strong internal varia- tions of the shear modulus. In the liquid outer core, theshear modulus vanishes, and therefore significant internal contributions to Earth ’s GW response can be expected at the inner-core boundary, as well as at the core-mantleboundary. Due to the complex internal structure of Earth, normal modes also show a complex radial dependence of their amplitudes. Modes with the high amplitudes at theinner-core boundary, core-mantle boundary, and free sur- face couple strongly to GWs. In order to calculate the response of Earth to GWs, normal-mode amplitudes as a function of radius need to be modeled numerically. For superconducting gravimeters,three contributions need to be modeled and added coher- ently: seismic acceleration, perturbation of the gravity potential, and lift against a static gravity gradient. For thiswork, normal-mode solutions were generated with the numerical simulation tool Minos [25]. These solutions are valid for a spherical, nonrotating, laterally homogeneousEarth, and here are based on the Earth model PREM [26]that describes variations of mass density, seismic speeds, and damping parameters from Earth ’s center to its surface. The gravimeters are designed to measure radial ground motion and gravity changes, which are caused only by spheroidal modes. Therefore, one can focus on these modes for theGW search. Of all spheroidal quadrupole modes nS2,o n l y1 4 have frequencies fnbelow 5 mHz as shown in Table I. Even though Earth also responds to GWs at off-resonance frequencies, the best sensitivity is obtained at normal-modefrequencies making use of the resonant signal amplification. The GW response at normal-mode frequencies needs to take into account the damping experienced by each mode inorder to obtain the correct signal amplification. The damping is quantified by a mode ’s quality factor, which corresponds to the ratio of a mode frequency to its natural spectrallinewidth. The quality factors of the 14 modes lie between about Q¼100and 900. The mode frequencies and quality factors used here were all taken from the numericalsimulation, but it should be emphasized that numerical estimates of the mode frequencies are very accurate, at least for the purpose of this paper, and also the quality factorsagree well with observation [27,28] . The coupling strength α nof a mode to a GW, see Eq. (3), can be expressed by a dimensionless quantity. Its values for the 14 quadrupole modes below 5 mHz are summarized in Table I. They depend on the radial as well as tangential10−810−610−410−210010210−51001051010 LIGO PulsarCassiniSeismic Normal Modes Frequency [Hz]ΩGW FIG. 1 (color online). Current upper limits on GW energy density. These limits were set by pulsar timing observations [23], Doppler-tracking measurements of the Cassini spacecraft [10], monitoring Earth ’s free-surface response with seismometers (“Seismic ”)[18], and correlating data from the first-generation, large-scale GW detectors LIGO [24]. The new limits resulting from normal-mode measurements are shown as crosses.MICHAEL COUGHLIN AND JAN HARMS PHYSICAL REVIEW D 90,042005 (2014) 042005-2displacement of each mode, and also on shear-modulus changes and mass density as functions of the distance toEarth ’s center. The coupling strength varies by more than an order of magnitude without clear pattern. This is owed to the complexity of mode solutions, which have greatly varying sensitivity to shear-modulus changes at differentdepths. In addition to the coupling strengths, the second important parameter characterizing each mode is its vertical displacement u nand gravity potential perturbation pnat the surface, which govern the gravimeter signal. These ampli- tudes are also summarized in Table I. The amplitudes of displacement and gravity potential are normalized such thattheir relative contribution to the gravimeter signal can be compared. It can be seen that the gravity perturbation is significant only for the two modes 0S2and1S2. A feature of normal modes that is not captured by the Minos simulation is mode coupling due to Earth ’se l l i p - ticity, rotation, and lateral heterogeneity. One effect is theso-called self-coupling, in which a quadrupole ( l¼2) multiplet can split into up to five resolvable modes, which are labeled by a third integer m¼−2;…;2[29]. Since each mode can therefore potentially respond to a different GW, mode splitting influences the overall GW response. Another possibility is that two modes that happen to bevery close in frequency can couple and exchange energy.The latter situation is depicted in Fig. 2taking the mode 6S2as an example. The curves are the sum of response functions of uncoupled oscillators normalized such thatthe peak value of each response corresponds to the mode ’s Qvalue. Whereas the next highest quadrupole mode 7S2is well isolated, mode6S2lies very close to other spheroidal modes, which can couple and exchange energy. Although the effect is minor on normal-mode frequencies and Q values [30,31] , a consequence is that one cannot design the GW search into too narrow frequency bands only relying on simulation predictions. An extreme narrow- band search needs to be based on a detailed characteri-zation of the quadrupole modes taking into account observed mode (self-)coupling, which has not been done in the work presented here. Concerning energy transferbetween coupled modes, the effect would generally lead toa decrease in GW response of a quadrupole mode independent of the Qvalues of the coupled modes. However, estimating the change in GW response that isconsistent with observed shifts of normal-mode frequen- cies (based on a simple coupled harmonic oscillator model), it can be concluded that the energy lost intoother modes through coupling is negligible. Therefore, the main issue with mode coupling is that the GW search needs to be designed with sufficient bandwidth aroundeach mode frequency so that it is guaranteed that the peak response of the entire quadrupole multiplet lies within this band. Further details about the impact of mode couplingon GW sensitivity are given in Sec. V. III. THEORY OF EARTH ’S RESPONSE TO GRAVITATIONAL WAVES Two response mechanisms of an elastic body to GWs have been described in detail in past publications. First, Dyson calculated the amplitude of seismic waves producedby GWs incident on a free, flat surface [32]. He found that the first time derivative of vertical surface displacement is given by _ξ zð~r; tÞ≈−β2 α~e⊤z·hð~r; tÞ·~ez: ð1ÞTABLE I. Summary of mode parameters: mode frequencies fn, quality factors Qn, coupling strengths αn, radial surface displacement un, perturbation of gravity surface potential pn(both normalized to the same, but arbitrary unit). The last row shows the upper limits on the energy density ΩGWas plotted in Fig. 6. nS2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 fn[mHz] 0.309 0.679 0.938 1.11 1.72 2.09 2.41 2.52 3.21 3.23 4.03 4.06 4.33 4.84 Qn 510 310 95.9 365 433 317 92.9 340 316 445 203 126 229 878 αn −0.645−18.3−1.78−0.696−18.913.3 4.31 −34.5−3.97−6.54 15.8 −16.9 12.7 3.12 un 0.74 −0.14−0.06−0.19−0.18−0.11−0.019−0.086−0.050 0.13 0.073 0.057 −0.021 0.086 pn −0.43 0.028 5.7e-5 −4.9e-42.1e-4 4.7e-5 −3.1e-61.7e-5 5.3e-6 3.6e-6 2.0e-7 7.4e-7 1.0e-6 1.9e-8 ΩGW 0.039 0.039 0.040 0.048 0.041 0.045 0.042 0.044 0.035 0.036 0.15 0.12 2.3 2.35 2.4 2.45 2.50100200300400500600 0S150S161S11 2S104S55S4 6S27S2 Frequency [mHz]Mode amplitudes FIG. 2 (color online). Simulated spectrum of spheroidal normalmodes around 6S2. The curves are sums of harmonic oscillator response functions (solid: all spheroidal modes, dashed: allspheroidal quadrupole modes). The values of the red markerscorrespond to the modes ’Qvalues.CONSTRAINING THE GRAVITATIONAL WAVE ENERGY … PHYSICAL REVIEW D 90,042005 (2014) 042005-3Here, ~ezdenotes the normal vector of the surface, hthe spatial part of the GW strain tensor, and α;βare the compressional and shear-wave speed. It can already be seen that the shear modulus μplays an important role in the elastic-body response since β2¼μ ρ: ð2Þ Accordingly, the GW response vanishes for vanishing shear modulus. One has to keep in mind though that the equationsof elastic deformation used to derive this result are neglecting contributions that can become important when the shear modulus is sufficiently small. For example, theGW response model of a spherical body with vanishing shear modulus has been used by Siegel and Roth [17] to propose GW measurements by monitoring oscillations ofthe Sun. The role of the shear modulus is also evident in the GW response of an elastic spherical body. This case was studiedby Ben-Menahem [19]and is used here to calculate Earth ’s resonant GW response. In the following, we will present the most important results of his work with minor reformula-tions. In terms of the amplitudes of radial displacement u nlmðrÞ, and tangential displacement vnlmðrÞ, the couplingstrength of a GW to a normal quadrupole mode ( l¼2) can be defined as αn2m≡−R β2cRRþ 0drr2μ0ðrÞðun2mðrÞþ3vn2mðrÞÞRR 0drr2ρðrÞðu2 n2mðrÞþ6v2 n2mðrÞÞ;ð3Þ where Ris Earth ’s radius, βcis the shear-wave speed at Earth ’s center, μ0ðrÞthe derivative of the shear modulus, andρðrÞthe mass density. The upper integration limit Rþ signifies that the shear-modulus change across the free surface needs to be included. In the following, only the radial order nwill be used to specify a quadrupole mode whose properties are independent of the index mneglecting mode coupling. The mode amplitudes unðrÞ;vnðrÞhave arbitrary units, since units of the mode variables cancel inthe final result. They are considered unitless in this work. It is only necessary that all mode variables including the amplitude ϕ nðrÞof the gravity potential are normalized consistently. The complete solution for the GW response also needs to take into account the angular dependence of excited oscillations. A simple first step is to consider the response to a single, plus-polarized GW. For a spherical, laterallyhomogeneous Earth, the acceleration a n2mmeasured by a gravimeter in the long-wavelength regime can be written an2mðfn;θ;ϕÞ¼ffiffiffiffiffiffiffiffi 24πp 15β2c RQnαnhðfnÞδjmj;2Ym/C3 2ðθ;ϕÞ·/C18 unðRÞþ3ϕnðRÞ Rð2πfnÞ2þ2g Rð2πfnÞ2unðRÞ/C19 ; ð4Þ where hðfnÞis the GW strain amplitude, g¼9.81m=s2, andδklthe Kronecker delta. For a quadrupole mode with l¼2, the angular parameter can take the values m¼−2;…;2. The expression in the brackets comprises the three contributions to the gravimeter signal: radial surface displacement, perturbation of the gravity potential,and lift against a static gravity gradient [33]. The second contribution corresponds to the parameter p nin Table I: pn≡3ϕnðRÞ=ðRð2πfnÞ2Þ. The angle θdenotes the relative angle between the direction of propagation of the GW and the location of the gravimeter on Earth ’s surface in a coordinate system with origin at the center of the Earth. Theangle ϕdescribes the rotation of this coordinate system with respect to the polarization frame of the GW. Accord- ingly, two modes, m¼/C6 2, of the quadrupole multiplet are excited by each GW in this choice of coordinate system. For the GW search carried out in this study, we also need to know the correlation between two gravimeters due to an isotropic GW background. Each GW that couples toquadrupole normal modes produces an angular surface vibration pattern that can, in an arbitrarily oriented Earth- centered coordinate system, be represented by a linearcombination of quadrupole spherical harmonics Y m 2ðθ;ϕÞwith m¼−2;…;2[34]. The situation is illustrated in Fig. 3. A plus-polarized GW propagates parallel to the north –south axis. The red and blue colored shapes represent Earth ’s induced quadrupole oscillation at its two maxima separated by half an oscillation period. Since the signal amplitude measured by gravimeters depends on theirlocation, coherence between two gravimeters also dependson location. For symmetry reasons, it is clear that for an isotropic GW field, coherence integrated over all polar- izations and propagation directions only depends on therelative position of the two gravimeters. This correlationfunction is known as overlap-reduction function, and normalized such that it is unity for collocated gravimeters [35]. In order to calculate it, the response as given by Eq.(4)needs to be calculated in a rotated coordinate system for one of the gravimeters. Since the GW correlation also depends on the nature of the GW field, a specific model needs to be chosen. Results in this paper are calculated foran isotropic, stationary field of GWs. Integration over allGW propagation directions and polarizations yields the overlap-reduction function, γ 12ðσÞ¼ffiffiffiffiffiffiffiffiffiffi 4π=5p Y0 2ðσ;0Þ; ð5ÞMICHAEL COUGHLIN AND JAN HARMS PHYSICAL REVIEW D 90,042005 (2014) 042005-4where σis the angle subtended by the great circle that connects the two gravimeters. All else being equal, the gravimeter pairs that contribute most significantly to the estimate of a GW energy density are either close to eachother or antipodal. Note that the overlap-reduction function can be approximated as frequency independent since the Earth is orders of magnitude smaller than the length of a GW at mHz frequencies. IV. GRAVIMETER DATA In addition to disturbances from large earthquakes including the subsequent ringdown of the normal modes [36], or local short-duration disturbances, gravimeter data also contain a stationary noise background consisting of instrumental noise, hydrological, and atmospheric disturbances [33]. The stationary noise level is very similar in almost all instruments, with a median of a fewðnm=s 2Þ=Hz1=2at 1 mHz. The medians of gravimeter spectra recorded during the year 2012 are plotted in Fig. 4. Four gravimeters show elevated medians, but in all these cases it is not the stationary background being higher, but instead the four instruments are frequently perturbed by strong local events, which therefore contribute signifi- cantly to the medians. A detailed study of gravimeter noise for most of these sites can be found in [37]. A local disturbance can produce strong broadband noise in gravimeters. Consequently, noise amplitudes at different frequencies show partial correlation. This property was exploited to subtract some of the background noise that adds to the normal-mode signals, and thereby improvesensitivity to GWs. In this way, it was possible to suppress the background noise at normal modes up to a factor 3(varying in time, and with different success for each normal mode). Using off-resonance amplitudes for noise subtrac- tion, it is possible to ensure that an insignificant amount ofGW signal is subtracted with the noise. Additional noise reduction can be achieved in some gravimeters by direct subtraction of gravity noise of atmospheric origin [38].F o r this purpose, each superconducting gravimeter is equippedwith a pressure sensor. The idea is that the pressure data contain direct information about corresponding atmos- pheric density and therefore gravity perturbations. It isfound that the correlation between pressure and gravimeter data is significant below about 1 mHz and weakly fre- quency dependent. This can be exploited to coherentlysubtract gravity noise with a conversion factor around−0.35μgal=hPa, which needs to be optimized for each gravimeter. The quality of pressure data is poor at some gravimeter sites so that good noise reduction cannot begenerally achieved. Another important property of gravimeter data is that coherence between any two gravimeters of the GGPnetwork at frequencies between 0.3 and 5 mHz producedby environmental disturbances is insignificant provided that times of high-magnitude earthquakes are excluded. Even for superconducting gravimeters that contain twolevitated spheres, strong coherence is only observedbelow about 2 mHz after removing the highest 10 percentof loudest spectra as shown in Fig. 5. The lack of environmental coherence is an important feature of thegravimeter network, which makes it a very efficient tool to search for GWs, since significant correlations of environmental origin would greatly limit the networksensitivity. FIG. 3 (color online). Earth quadrupole oscillation. The red and blue shapes correspond to the maxima of a quadrupole oscillationseparated by half an oscillation period. The green balls marklocations of some of the gravimeters of the GGP network. Herethe oscillation is induced by a GW propagating along the north-south axis.10−110010−1100101102103104 Frequency [mHz]Gravity [(nm/s2)/√ Hz]Ny−Alesund, Norway Concepcion, Chile Wuhan, China Hsinchu, Taiwan FIG. 4 (color online). Medians of gravimeter spectra measured in 2012. All gravimeters used in this study show a comparablelevel of stationary background noise represented by their spectralmedians, except for the four gravimeters highlighted in the plot.CONSTRAINING THE GRAVITATIONAL WAVE ENERGY … PHYSICAL REVIEW D 90,042005 (2014) 042005-5V. SEARCH FOR A STATIONARY GRAVITATIONAL-WAVE BACKGROUND In this section, we outline the GW search method based on correlation measurements between gravimeter pairs. Depending on the relative position of two gravimeters onEarth ’s surface, correlation of gravimeter signals arising from GWs is described by the overlap-reduction function in Eq.(5). Once the expected correlation of GW signals between different gravimeters is calculated, the measuredcorrelations are used to obtain an estimate of the energy density of GWs following the method described in [39]. The upper limit on the GW energy density presented in thispaper was obtained as a near-optimal combination of measured correlations using 10 years of data, forming pairs with gravimeters of the GGP network. The totalamount of data is divided into stretches short enough so that the spectral resolution is wider than the frequency spread of a quadrupole multiplet as discussed in Sec. II. The length of data stretches obtained in this way is different for each normal mode. Each data stretch leads to a point estimate of the GW energy density according to ˆΩ GWðfnÞ¼4π2 3H2 0ˆS12ðfnÞf3n γ12: ð6Þ Here, ˆS12ðfnÞis the measured cross-spectral density between two gravimeters in units of GW strain spectraldensity. As pointed out before, the overlap-reduction function γ 12can be approximated as frequency independent for normal-mode observations. Based on the conservative assumption that the l¼2 quadrupole mode splits into five distinct isolated modes(m¼−2;…;2) that all respond incoherently to GWs, the GW response of a quadrupole mode is obtained by adding contributions from different values of mincoherently. Furthermore, two pairs of the 14 quadrupole modes are too close in frequency to be resolvable with the chosen frequency resolution ( n¼8, 9 and n¼10, 11, see Table I). This means that in addition to the incoherent sum over amultiplet, contributions from the two quadrupole modes in each of these pairs need to be summed incoherently leading to a combined point estimate. Accurate knowledge of modesplitting would lead to sensitivity improvements. One advantage would be that the bandwidth of the search could be narrowed, which means that the measurement noisewould be decreased. The improvement factor depends on theQvalues and difference in frequencies between modes of a multiplet, and is likely modest. In addition, thesensitivity of our search was explicitly punished by a factorffiffiffi 5p , since we assumed conservatively that GW signals in five submodes of a multiplet add incoherently.Therefore, if five modes were observed separately, thenthey could all be combined to set an upper limit for the entire frequency range of the multiplet, which would then be at least a factorffiffiffi 5p lower than achieved in this paper. The exact improvement depends on the gravimeter noise at each submode (e.g. it is conceivable, but certainly highly unlikely, that the gravimeter noise at one of the submodes issubstantially lower, therefore leading to a much better upper limit). The calculation of upper limits is based on a combination of estimates from many data segments. It must take into account the point estimates of energy density from eachdata segment as well as the corresponding measurement error. A result is consistent with a nondetection, if after combining point estimates and errors of all segments, thecombined estimate of the energy density is smaller thanthe combined measurement error (or similar thresholds can be applied depending on confidence levels). Data quality changes in nonstationary gravimeter data, and thereforesegments, are vetoed based on some criteria. For this study, data segments that showed high values of the ratio “point estimate over error ”were vetoed. High correlation can be associated with increased gravimeter noise following large earthquakes, but typically gravimeter correlations show a higher number of transient features than gravimeter noise.For this reason, the veto is mostly on high-correlation segments, which favors low point estimates while having a smaller effect on average gravimeter noise. This causes theupper limits to be dominated by the errors. The final results are presented as constraints on the energy density in GWs separately for each mode. Figure 6shows the estimates of the GW energy density with error bars. All estimates areconsistent with a nondetection, and the resulting energy constraints are mostly determined by the error bars for reasons explained above. The values are listed in Table I. Energy densities can be translated into strain spectral Frequency [mHz]Percentile 0.1 0.2 0.4 0.81 2 4102030405060708090100 Coherence 00.20.40.60.81 FIG. 5 (color online). Coherence of signals from two levitated spheres in the same gravimeter at Wettzell, Germany. The result isshown as a function of percentile of gravimeter noise excludedfrom the coherence measurement. A percentile of 90 means that10% of the loudest spectra were excluded from the coherencemeasurement. Only the high- Qradial normal mode 0S0at about 0.81 mHz contributes significantly to coherence for all times.MICHAEL COUGHLIN AND JAN HARMS PHYSICAL REVIEW D 90,042005 (2014) 042005-6densities, which lie between hGW≤2.2×10−14Hz−1=2for the mode0S2andhGW≤6.2×10−16Hz−1=2for0S13.E v e n though these results demonstrate an improvement in sensitivity by a few orders of magnitude over previous searches in this frequency band (see Fig. 1), the new upper limits are still not stringent enough to constrain cosmo- logical models of GW backgrounds. A conservative esti- mate of the energy density of GWs from inflation predicts avalue of order Ω GW∼10−15, and a GW background from cosmic strings is predicted at ΩGW∼10−7, both at normal- mode frequencies [24]. Also a GW background from a cosmological distribution of unresolved compact binary stars such as white dwarfs and neutron stars is predicted at lower values, ΩGW∼10−12, at normal-mode frequencies [40]. Therefore, with achieved upper limits between ΩGW¼0.035–0.15, the stationary gravimeter noise as plotted in Fig. 4has to be lowered by 2 to 3 orders of magnitude to be able to place first constraints on cosmo- logical models. VI. CONCLUSION In this paper we showed that our understanding of Earth ’s interior can be used to accurately calculate Earth ’s resonant GW response. In this way, it was possible to calibrate gravimeter data into units of GW strain, anddirectly obtain new upper limits on the GW energy densityin the range 0.035 –0.15 at frequencies between 0.3 and 5 mHz. This was achieved by correlating data of gravimeterpairs recorded over the past ten years. Alternatively, one could make use of the same response mechanism to search for individual astrophysical signalssuch as galactic white-dwarf binaries. Millions of binaries are predicted to radiate quasi-monochromatic waves in this frequency band [41]including already discovered systems (see for example Roelofs et al. [42]). Again, about 3 orders of magnitude sensitivity improvement are required to make a detection likely. The integrated gravitational-wave signalshould be distinguishable from terrestrial sources since it is modulated due to Earth ’s rotation. The additional challenge here is that a continuous integration of the signal, as wouldbe favorable for this search, results in an extremely narrow frequency resolution, which requires a more detailed investigation of mode-coupling effects. The diversity innature of Earth ’s oscillations also makes it possible to test alternative theories of gravity. For example, a scalar component of the GW field could be searched in monopolemodes nS0as has already been attempted by Weiss and Block [21]. Further improvement in GW sensitivity may be achieved with a new generation of gravimeters. Especially atom- interferometric gravimeters are currently under active development [43]. The open question is if there will be some form of environmental noise limiting the sensitivity of gravimeters irrespective of their intrinsic acceleration sensitivity, and whether methods can be developed tomitigate this noise if necessary. Nonetheless, we havedemonstrated that gravimeter technology is a viable option to detect GWs, and that ground-based GW detection seems to be a possibility at frequencies, which are generallyconsidered accessible only for space-borne detectors. ACKNOWLEDGMENTS We want to thank David Crossley for helpful discussions on gravimeter data and Jean-Paul Montagner, Guy Masters, and Eric Clévédé for their help on normal-mode simula- tions. Thanks also for constructive feedback from RanaAdhikari, Ray Weiss, and Stan Whitcomb on an earlier version of the manuscript. M. C. was supported by the National Science Foundation Graduate Research FellowshipProgram, under NSF Grant No. DGE 1144152. Uncorrected gravimeter data (GGP-SG-MIN) used for this project were downloaded from http://isdc.gfz ‑potsdam.de/index .php?module=pagesetter&func=viewpub&tid=1&pid=54 .1 2 3 4−0.2−0.100.10.2 Frequenc y [mHz]ΩGW FIG. 6. Point estimates of GW energy density and errors. Of the 14 original modes, only 12 are plotted here since two pairs,n¼8, 9, and n¼10, 11, have been merged to one value each since the chosen frequency resolution cannot resolve them.CONSTRAINING THE GRAVITATIONAL WAVE ENERGY … PHYSICAL REVIEW D 90,042005 (2014) 042005-7[1] J. M. 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PhysRevLett.123.047204.pdf

Gigahertz Frequency Antiferromagnetic Resonance and Strong Magnon-Magnon Coupling in the Layered Crystal CrCl 3 David MacNeill,1,*Justin T. Hou,2,*Dahlia R. Klein,1Pengxiang Zhang,2Pablo Jarillo-Herrero,1and Luqiao Liu2 1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 5 February 2019; published 24 July 2019) We report broadband microwave absorption spectroscopy of the layered antiferromagnet CrCl 3.W e observe a rich structure of resonances arising from quasi-two-dimensional antiferromagnetic dynamics.Because of the weak interlayer magnetic coupling in this material, we are able to observe both optical and acoustic branches of antiferromagnetic resonance in the GHz frequency range and a symmetry-protected crossing between them. By breaking rotational symmetry, we further show that strong magnon-magnoncoupling with large tunable gaps can be induced between the two resonant modes. DOI: 10.1103/PhysRevLett.123.047204 Antiferromagnetic spintronics is an emerging field with the potential to realize high speed logic and memory devices [1–6]. Compared to ferromagnetic materials, antiferromag- netic dynamics are less well understood [7–10], partly due to their high intrinsic frequencies that require terahertz tech- niques to probe [11–13]. Therefore, antiferromagnetic materials with lower resonant frequencies are desired to enable a wide range of fundamental and applied research [5]. Here, we introduce the layered antiferromagnetic insulatorCrCl 3as a tunable platform for studying antiferromagnetic dynamics. Because of the weak interlayer coupling, the antiferromagnetic resonance (AFMR) frequenciesare within the range of typical microwave electronics (<20 GHz). This allows us to excite different modes of AFMR and to induce a symmetry-protected mode crossingwith an external magnetic field. We further show that a tunable coupling between the optical and acoustic magnon modes can be realized by breaking rotational symmetry.Recently, strong magnon-magnon coupling between two adjacent magnetic layers has been achieved [14,15] , with potential applications in hybrid quantum systems [16–18]. Our results demonstrate strong magnon-magnon coupling within a single material and therefore provide a versatile system for microwave control of antiferromagnetic dynam-ics. Furthermore, CrCl 3crystals can be exfoliated down to the monolayer limit [19] allowing device integration for antiferromagnetic spintronics. The crystal and magnetic structures of CrCl 3are shown in Fig.1 [19 –26]. Spins within each layer have a ferromagnetic nearest-neighbor coupling of about 0.5 meV , whereas spins in adjacent layers have a weak antiferromagnetic coupling of about 1.6μeV[25]. Therefore, we can consider each layer as a two-dimensional ferromagnet coupled to the adjacent layers by an interlayer exchange field of roughly 0.1 T [25]. The weak interlayer coupling implies that the field and FIG. 1. (a) Crystal structure of a single CrCl 3layer. Red and purple spheres represent chlorine and chromium atoms respec-tively. (b) Magnetic structure of bulk CrCl 3below the Ne´ el temperature, and without an applied magnetic field. Blue spheresrepresent the Cr atoms. Red arrows represent the magneticmoment of each Cr atom with parallel intralayer alignmentand antiparallel interlayer alignment. The net magnetizationdirection alternates between layers, having direction ˆm A(ˆmB) on layers in the A(B) magnetic sublattice. (c) Experimental schematic featuring a coplanar waveguide (CPW) with a CrCl 3 crystal placed over the signal line. Hjj,H⊥, and Hzare the components of the applied dc magnetic field. The microwave transmission coefficient was measured as a function of appliedmagnetic field and temperature. (d) Typical microwave trans-mission at 5 GHz and 1.56 K as a function of magnetic fieldapplied parallel (blue) or perpendicular (red) to the in-plane rffield, showing resonances due to AFMR. The two traces weretaken from different CrCl 3crystals.PHYSICAL REVIEW LETTERS 123, 047204 (2019) 0031-9007 =19=123(4) =047204(6) 047204-1 © 2019 American Physical Societyfrequency required to manipulate the antiferromagnetic order parameter (N´ eel vector) are orders of magnitude lower than in typical antiferromagnets [27–29]. Magnetic resonance measurements of CrCl 3have a long history, including one of the earliest observations of para- magnetic resonance in a crystal [30]. However, the dynam- ics below the N´ eel temperature remain largely unexplored. To study this, we first synthesized bulk CrCl 3crystals according to the method of McGuire et al. [19,31] . The crystals are transferred to a coplanar waveguide (CPW) and secured with Kapton tape [Fig. 1(c)]. The crystal caxis is normal to the CPW plane. The CPW is mounted in a cryostat and connected to a vector network analyzer by rf cables for microwave transmission measurements. A dc magnetic field is applied with the field directions illustrated in Fig. 1(c). To study the response in the linear regime and to prevent heating, we use a low power excitation signal (estimated to be −35dBm at the sample). We measure magnetic resonance by fixing the excitation frequency and sweeping the applied magnetic field. Weobserve different resonant features in different field geom-etries [Fig. 1(d)]. Only one resonance is observed when the dc magnetic field is applied perpendicular to the rf field(H ⊥), but two resonances show up when the dc magnetic field is applied parallel to the rf field ( Hjj). We plot the transmission as a function of excitation frequency and applied magnetic field (Fig. 2). Under Hjj, two modes exist with distinct field dependencies [Fig. 2(a)]: one starting from finite frequency and softening with applied field, and the other with frequency proportional to the applied field.Remarkably, the modes cross without apparent interactionleading to a degeneracy at their crossing point; as wediscuss below this crossing is protected by symmetry whenthe applied field lies in the crystal planes. With H ⊥, we see only the linearly dispersing mode [Fig. 2(b)]. To understand the origin of the two modes, we model the magnetic dynamics of CrCl 3in the macrospin approxima- tion. We assume that the magnetization direction is uniformwithin each layer, and introduce unit vectors ˆm Aand ˆmBto represent the magnetization direction on the AandB sublattices. The interlayer exchange energy is approxi- mated as μ0MsHEˆmA·ˆmB, where HEis the interlayer exchange field and Msis the saturation magnetization. Omitting damping, we get a coupled Landau-Lifshitz- Gilbert (LLG) equation [33]: dˆmA dt¼−μ0γˆmA×ðH−HEˆmB−MsðˆmA·ˆzÞˆzÞþτA; dˆmB dt¼−μ0γˆmB×ðH−HEˆmA−MsðˆmB·ˆzÞˆzÞþτB: ð1Þ Here γis the gyromagnetic ratio and ˆzis the direction perpendicular to sample plane (along the crystal caxis). τA and τBare the torques which arise from the rf field of the CPW. The term proportional to Msrepresents an easy-plane anisotropy arising from the demagnetization field of a platelet shaped crystal. Previous studies have shown thatthe magnetic anisotropy of CrCl 3is well described by this shape anisotropy [19,25] . Replacing Msby an effective value would allow for an additional uniaxial magneto-crystalline anisotropy, but we confirm below that this effect is small. We also neglect in-plane anisotropy as the energy depends weakly on the in-plane orientation of the magneticmoments [19,25] . FIG. 2. Microwave transmission as a function of frequency and in-plane magnetic field at 1.56 K with the magnetic field applied (a) parallel and (b) perpendicular to the in-plane rf field (the data for the two panels were taken from different samples). The regions oflower transmission arise from magnetic resonance. Two modes are observed in the H jjconfiguration: an optical mode that has finite frequency at zero applied field, and an acoustic mode with frequency proportional to the applied field. Only the acoustic mode isobserved in the H ⊥configuration. Blue and red dashed lines in both panels are fits of the optical and acoustic mode frequencies to Eqs. (2)and(3), respectively. Insets of (a) and (b) show the relative orientation of the dc magnetic field, the equilibrium sublattice magnetizations, and the in-plane ( hIP) and out-of-plane ( hOP) components of the rf field. (c),(d) Schematic illustrations of the precession orbits for the two sublattice magnetizations in the optical mode and the acoustic mode.PHYSICAL REVIEW LETTERS 123, 047204 (2019) 047204-2When the magnetic field His applied in the layer plane, Eq.(1)is symmetric under twofold rotation around the applied field direction combined with sublattice exchange[31]. In the linear approximation, this results in two independent modes with even and odd parity under the symmetry (optical mode and acoustic mode; see Fig. 2). The optical and acoustic modes result in Lorentzianresonances centered around the frequencies ω /C6. The frequencies have magnetic field dependence [31,34,35] : ωþ¼μ0γffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2HEMs/C18 1−H2 4H2 E/C19s ; ð2Þ and ω−¼μ0γffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2HEð2HEþMsÞp H 2HE: ð3Þ We can fit the resonance frequencies using Eqs. (2)and (3). These fits are shown by the dashed lines in Fig. 2(a) with fit parameters of μ0HE¼105mT and μ0Ms¼ 396mT at T¼1.56K, assuming γ=2π≈28GHz=T for CrCl 3[36]. The observed saturation magnetization is close to3μBper Cr atom, consistent with magnetometry [19,31] and confirming that the out-of-plane crystalline anisotropyis small. Note that the acoustic mode changes its slope at μ 0H≈200mT. This occurs because the moments of the two sublattices are aligned with the applied field when H> 2HE[31]. In this case the crystal behaves as a ferromagnet and the acoustic mode transforms into uniform ferromag- netic resonance (FMR) described by the Kittel formula ωFMR¼μ0γffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HðHþMsÞp . Figures 2(a) and 2(b) also show fits of the data for H> 2HEto the Kittel formula (dash-dotted line). (The data above and below H¼2HE are fit simultaneously to extract a consistent parameter set.)The dependence on field geometry in Fig. 2can now be understood as a consequence of selection rules. We can state the rule as follows: an rf magnetic field will excite theeven (odd) parity mode if it is even (odd) under twofoldrotation around the applied field direction [31]. The rf magnetic field generated from the CPW [Fig. 1(c)] has both in-plane and out-of-plane components. Directly over thesignal line, the rf field points in the sample plane, while inthe gap between the signal line and ground, the rf field is perpendicular to the sample plane. Our crystal is large enough to cover both regions and experience both fielddirections. In the perpendicular geometry ( H ⊥), both the in- plane and out-of-plane rf fields change sign under twofold rotation around the applied field direction [inset of Fig. 2(b)]. Therefore only the odd parity (acoustic) mode will be excited. In the parallel field geometry ( Hjj), the in- plane component is invariant under the twofold rotation and excites the even parity (optical) mode, while the out-of- plane component changes sign and excites the odd-parity(acoustic) mode [inset of Fig. 2(a)]. We will focus on measurements in the parallel field geometry because it allows simultaneous excitation of both modes. We further study the evolution of the AFMR as a function of temperature. As the temperature is increased from 1.56 K [Fig. 2(a)] to 7 K [Fig. 3(a)], the optical mode frequency decreases due to the reduction of H Eand Ms. The optical mode disappears entirely at 14 K, implying that the sample is no longer antiferromagnetic, consistent with magnetometry measurements [19,31] .A t higher temperatures, the magnetic resonance frequencydepends linearly on the applied field with a slope of 30.4GHz=T and 28.8GHz=T at 21 K and 30 K, respec- tively [Figs. 3(c) and3(d)]. This is electron paramagnetic resonance arising from Cr 3þions [36]. In this temperature range, the magnetization is proportional to the applied field through M¼χðTÞH. Then the resonant frequency is FIG. 3. Microwave transmission as a function of frequency and in-plane magnetic field at 7, 14, 21, and 30 K. At 7 K, the sample is antiferromagnetic and both acoustic and optical modes are observed; μ0HE¼89mT and μ0Ms¼323mT are determined by fits to Eqs. (2)and(3), which are smaller than those at 1.56 K [Fig. 2(a)]. Only one mode is observed at 14 K, and its frequency does not show a purely linear field dependence. At 21 and 30 K, a single mode with linear field dependence is observed, arising from electronparamagnetic resonance.PHYSICAL REVIEW LETTERS 123, 047204 (2019) 047204-3ω¼μ0γffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HðHþMðH;T ÞÞp ¼μ0γffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þχðTÞp H[37].O u r data correspond to χð21KÞ¼0.178 and χð30KÞ¼ 0.056. For the temperature range 14 –20 K, the resonant frequencies do not have a purely linear dependence onapplied field [Fig. 3(b)]. This is likely due to a nonlinear relationship MðHÞasMapproaches its saturation value, previously detected in magnetization and magneto-opticalexperiments [19,31,38] . So far, we have discussed AFMR with an in-plane applied magnetic field [34]. In this case, the system is symmetric under twofold rotation around the applied field direction combined with sublattice exchange. This pre- vents hybridization between the optical and acoustic modes, leading to a degeneracy where they cross [see Figs. 2(a) and3(a)]. In principle, breaking this symmetry can hybridize the two modes and generate an anticrossing gap. One possible approach for inducing symmetry breaking is to use different M sfor the AandBsublattices by stacking different 2D magnets. Here, we instead employ an out-of-plane field to break the 180° rotational symmetry. We measure the AFMR spectrum for a dc magnetic field applied at a range of angles, ψ, from the CPW plane. For ψ¼30°, the mode structure is largely unchanged, except that a gap opens near the crossing point [Fig. 4(b)]. Increasing the tilt angle increases the gap size as shown in Fig. 4(c). Therefore, breaking the rotational symmetry with an out-of-plane field introduces a magnon-magnon coupling between the previously uncoupled modes. To quantitatively describe the magnon-magnon cou- pling, we turn to the matrix formalism of the LLG equation [31]. The result is high and low frequency branches of antiferromagnetic resonance, continuously connected to the even and odd parity modes. The evolution of both modes and their mixing can be captured by the eigenvalue problemof a two-by-two matrix:/C12/C12/C12/C12ω 2aðH;ψÞ−ω2Δ2ðH;ψÞ Δ2ðH;ψÞ ω2oðH;ψÞ−ω2/C12/C12/C12/C12¼0 ð4Þ Here ω a¼μ0γffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þðMs=2HEÞp Hcosψis the bare acoustic mode frequency and ωo¼μ0γð2HEMs½1− ðH2=H2 FMÞ/C138 þ fð sin2ψÞ=½1þðMs=2HEÞ/C1382gH2Þ1=2is the bare optical mode frequency. HFMis the applied magnetic field required to fully align the two sublattices, satisfy- ing1=H2 FM¼cos2ψ=ð2HEÞ2þsin2ψ=ð2HEþMsÞ2.Δ¼ μ0γH½2HE=ð2HEþMsÞsin2ψcos2ψ/C1381=4represents the magnon-magnon coupling term. The solutions of Eq. (4) are the resonance frequencies of the LLG equation. Whenjω o−ωaj≫Δ, the effect of the coupling term is negligible and the solutions are approximately ω≈ωoandω≈ωa. When the optical and acoustic modes become closer infrequency, they are hybridized by the coupling termopening a gap. This coupling is zero for ψ¼0° and only becomes nonzero as we cant the applied field out-of-plane. Note that when ψ¼90° the mode coupling is zero again corresponding to rotational symmetry around the out-of- plane direction [31]; this decoupling becomes relevant for ψ>80° so we will focus on the regime ψ<80°. The dashed lines in Figs. 4(a)–4(c) indicate fits to the eigenvalues of Eq. (4), with fit parameters μ 0Ms¼409mT and μ0HE¼101mT. The coupling strength g=2πis determined as half of the minimal frequency spacing in the fits. We determine the dissipation rates of the upperand lower branches, κ UandκL, by Lorentzian fitting of the frequency dependence of the transmission, using vertical cuts on the 2D plots of 4(c). For ψ¼55°, we obtain g=2π≈0.8GHz, κU=2π≈0.5GHz, and κL=2π≈ 0.2GHz which indicates strong magnon-magnon coupling asg>κUandg>κL[14]. The cooperativity is C¼g2= ðκU×κLÞ¼6.4, which can be improved by using a more homogeneous sample. Figure 4(e) shows the angular dependence of g. By rotating the crystal alignment in an FIG. 4. Microwave transmission as a function of frequency and applied field at 1.56 K; the field is applied at an angle of (a) ψ¼0°, (b) 30°, and (c) 55° from the sample plane. When the field is applied in-plane, the mode crossing is protected by rotational symmetrycombined with sublattice exchange. Out-of-plane field breaks the symmetry and couples the two modes resulting in tunable gaps.(d) Microwave transmission vs applied field at ψ¼55° for various frequencies, showing the coupling gap. An extra small dip appears at 7.2 GHz, probably due to resonance peak splitting induced by inhomogeneities. (e) The coupling strength gincreases with ψ, and can be tuned from 0 to 1.37 GHz.PHYSICAL REVIEW LETTERS 123, 047204 (2019) 047204-4applied field, we can tune the system from a symmetry- protected mode crossing to the strong coupling regime. In summary, we have measured magnetic resonance of the layered antiferromagnet CrCl 3as a function of temper- ature and applied magnetic field, with the magnetic fieldapplied at various angles from the crystal planes. We have shown that CrCl 3possess a rich GHz-frequency AFMR spectrum due to the weak interlayer coupling. We detectboth acoustic and optical branches of AFMR and show thatan applied magnetic field can induce an accidental degen- eracy between them. Furthermore, by breaking rotational symmetry we can induce a coupling between these modesand open a tunable gap. All of these effects are captured with analytical solutions to the LLG equation. We also expect interaction between the modes in the nonlinearregime. For example, three-magnon processes could be triggered when the frequencies of the acoustic and optical modes satisfy certain relationships. There is also tremendous interest in isolating ultrathin layered magnets using mechanical exfoliation [39,40] , and incorporating them in van der Waals heterostructures [41,42] . Because CrCl 3can be cleaved to produce air- stable films down to the monolayer limit [19], we expect our results to enable device-based antiferromagnetic spin- tronics with microwave control of the N´ eel vector. Our results apply broadly within the class of transition metal trihalides, so that the frequency scale can be tuned by varying the chemical composition and thickness [19]. Using van der Waals assembly, we can combine differentmagnetic materials to induce magnon-magnon coupling without out-of-plane field by breaking sublattice exchange symmetry. Device fabrication, data analysis, and measurements by the PJH group were primarily supported by the DOE Office of Science, Basic Energy Sciences under Award No. DE- SC0018935 (D. M.), as well as the Gordon and BettyMoore Foundations EPiQS Initiative through GrantNo. GBMF4541 to P. J.-H. Crystal growth was partly supported by the Center for Integrated Quantum Materials under NSF Grant No. DMR-1231319 (D. R. K.).D. R. K. acknowledges partial support by the NSF Graduate Research Fellowship Program under Grant No. 1122374. J. T. H., P. Z., and L. 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PhysRevB.95.155139.pdf

PHYSICAL REVIEW B 95, 155139 (2017) Temperature-dependent transport properties of FeRh S. Mankovsky, S. Polesya, K. Chadova, and H. Ebert Department Chemie, Ludwig-Maximilians-Universität München, 81377 München, Germany J. B. Staunton Department of Physics, University of Warwick, Coventry, United Kingdom T. Gruenbaum, M. A. W. Schoen, and C. H. Back Department of Physics, Regensburg University, Regensburg, Germany X. Z. Chen and C. Song Key Laboratory of Advanced Materials (MOE), School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China (Received 9 June 2016; revised manuscript received 16 March 2017; published 25 April 2017) The finite-temperature transport properties of FeRh compounds are investigated by first-principles density- functional-theory-based calculations. The focus is on the behavior of the longitudinal resistivity with risingtemperature, which exhibits an abrupt decrease at the metamagnetic transition point, T=T m, between ferro- and antiferromagnetic phases. A detailed electronic structure investigation for T/greaterorequalslant0 K explains this feature and demonstrates the important role of (i) the difference of the electronic structure at the Fermi level betweenthe two magnetically ordered states and (ii) the different degree of thermally induced magnetic disorder in thevicinity of T m, giving different contributions to the resistivity. To support these conclusions, we also describe the temperature dependence of the spin-orbit-induced anomalous Hall resistivity and Gilbert damping parameter.For the various response quantities considered, the impact of thermal lattice vibrations and spin fluctuations ontheir temperature dependence is investigated in detail. Comparison with corresponding experimental data shows,in general, very good agreement. DOI: 10.1103/PhysRevB.95.155139 I. INTRODUCTION For a long time the ordered equiatomic FeRh alloy has attracted much attention owing to its intriguing temperature-dependent magnetic and magnetotransport properties. Thecrux of these features of this CsCl-structured material isthe first-order transition from an antiferromagnetic (AFM) toferromagnetic (FM) state when the temperature is increasedabove T m=320 K [ 1,2]. In this context the drop in the electrical resistivity that is observed across the metamagnetictransition is of central interest. Furthermore, if the AFMto FM transition is induced by an applied magnetic field,a pronounced magnetoresistance (MR) effect is found ex-perimentally with a measured MR ratio of ∼50% at room temperature [ 2–4]. The temperature of the metamagnetic transition as well as the MR ratio can be tuned by the additionof small amounts of impurities [ 2,5–8]. These properties make FeRh-based materials very attractive for future applicationsin data storage devices. The origin of the large MR effectin FeRh, however, is still under debate. Suzuki et al. [9] suggest that, for deposited thin FeRh films, the main mech-anism stems from the spin-dependent scattering of conductingelectrons on localized magnetic moments associated withpartially occupied electronic dstates [ 10] at grain boundaries. Kobayashi et al. [11] have also discussed the MR effect in the bulk ordered FeRh system, attributing its origin to themodification of the Fermi surface across the metamagnetictransition. So far only one theoretical investigation of theMR effect in FeRh has been carried out on an ab initio level [ 12].II. COMPUTATIONAL DETAILS The present study is based on spin-polarized electronic structure calculations using the fully relativistic multiplescattering Korringa-Kohn-Rostoker (KKR) Green’s functionmethod [ 13,14] with the framework of spin density functional theory. For the self-consistent calculations a parametrizationfor the exchange and correlation potential based on thegeneral gradient approximation (GGA) [ 15] has been used. For the charge and potential representation the atomic sphereapproximation (ASA) has been applied. For the wave functionsand corresponding matrices of the KKR formalism the cutoffvalue l max=3 has been used for the angular momentum expansion. The central advantage of the KKR formalism is that it gives direct access to the retarded single-particle Green’s functionG +(/vectorr,/vectorr/prime,E), which is given by [ 16–18] G+(/vectorr,/vectorr/prime,E)=/summationdisplay /Lambda1/Lambda1/primeZm /Lambda1(/vectorr,E)τmn /Lambda1/Lambda1/prime(E)Zn× /Lambda1/prime(/vectorr/prime,E) −δmn/summationdisplay /Lambda1Zn /Lambda1(/vectorr,E)Jn× /Lambda1(/vectorr/prime,E)/Theta1(r/prime n−rn) +Jn /Lambda1(/vectorr,E)Zn× /Lambda1(/vectorr/prime,E)/Theta1(rn−r/prime n), (1) where the spatial vectors /vectorrand/vectorr/primeare assumed to be within the atomic cell centered at sites /vectorRm,/vectorRn, respectively. Within the fully relativistic formulation used here the combined quantumnumber /Lambda1=(κ,μ) stands for the relativistic spin-orbit and magnetic quantum numbers κandμ, respectively [ 19]. 2469-9950/2017/95(15)/155139(9) 155139-1 ©2017 American Physical SocietyS. MANKOVSKY et al. PHYSICAL REVIEW B 95, 155139 (2017) Accordingly, Zn /Lambda1andJn /Lambda1are four-component wave functions obtained as regular and irregular solutions to the single-siteDirac equation for the isolated potential well V ncentered at site n, respectively. The symbol “ ×” as a superscript of Zn /Lambda1andJn /Lambda1indicates the left-hand-side solution to the Dirac equation. Dealing with a magnetically ordered system withinthe framework of spin density functional theory, the potentialV nis spin dependent. As a consequence Zn /Lambda1=/Sigma1/Lambda1/primeZn /Lambda1/prime/Lambda1(and alsoJn /Lambda1) stands for a superposition of various partial waves with spin-angular character /Lambda1/prime[20,21]. Finally, the quantity τnn/prime /Lambda1/Lambda1/primeis the so-called scattering path operator that represents the transfer of a wave coming in at site n/primewith character /Lambda1/primeto a wave outgoing from site nwith character /Lambda1and all possible scattering events taking place in between [ 17]. The scheme sketched above to calculate the retarded Green’s function gives direct access to the density of states(DOS) n(E) via the expression n(E)=−Im π/integraldisplay /angbracketleftG+(/vectorr,/vectorr,E)/angbracketrightcd3r. (2) Information on the electronic structure more detailed than that given by the DOS is given by the Bloch spectral function (BSF) AB(/vectork,E). In terms of the retarded Green’s function, this quantity is defined via AB(/vectork,E)=−Im π/summationdisplay n,mexp[ı/vectork·(/vectorRn−/vectorRm)] ×/integraldisplay /angbracketleftG+(/vectorr+/vectorRn,/vectorr+/vectorRm,E)/angbracketrightcd3r, (3) where again the angular brackets specify an appropriate configurational average. For a perfectly ordered system the BSF would be a set of Dirac delta functions, AB(/vectork,E)=/summationtext γδ(E−E/vectorkγ), and for E=EFit would trace out the Fermi surface. For a system with thermally induced spin fluctuations and lattice displacements the BSF has features with finite widthfrom which the mean-free-path length of the electrons can beinferred. The present approach used for the electronic structure calculations allows us to calculate the transport properties atfinite temperatures on the basis of the linear response for-malism using the Kubo-St ˇreda expression for the conductivity tensor [ 22,23], σ μν=¯h 4πN/Omega1Tr/angbracketleftˆjμ[G+(EF)−G−(EF)]ˆjνG−(EF) −ˆjμG+(EF)ˆjν[G+(EF)−G−(EF)]/angbracketrightc, (4) where /Omega1is the volume of the unit cell, Nis the number of sites, ˆjμis the relativistic current operator, and G±(EF) are the electronic retarded and advanced Green’s functions,respectively, calculated at the Fermi energy E F.I nE q .( 4)t h e orbital current term has been omitted as it provides only smallcorrections to the prevailing contribution arising from the firstterm in the case of a cubic metallic system [ 24–26]. The Gilbert damping parameters αare calculated using a Kubo-Greenwood-like equation [ 27]: α μμ=−¯hγ πMsTr/angbracketleftˆTμImG+(EF)ˆTμImG+(EF)/angbracketrightc,(5)with the torque operator ˆTμgiven by the expression ˆTμ=β[/vectorσ׈ez]μBxc(/vectorr), (6) with ˆezbeing the direction of magnetization and Bxc(/vectorr) being the spin-dependent part of the potential. The angular brackets /angbracketleft ···/angbracketright c(if applicable) in all expressions above specify the average over temperature-induced spinfluctuations and lattice vibrations treated within the alloyanalogy model described in the Appendix A. III. RESULTS First, we focus on the finite-temperature properties of the electrical resistivity of FeRh. In order to take into accountelectron-phonon and electron-magnon scattering effects inthe calculations, the so-called alloy analogy model [ 27,28] is used. Within this approach the temperature-induced spin(local moment) and lattice excitations are treated as local-ized, slowly varying degrees of freedom with temperature-dependent amplitudes. Using the adiabatic approximation inthe calculations of transport properties and accounting for therandom character of the motions, the evaluation of the thermalaverage over the spin and lattice excitations in Eq. ( 4)i s reduced to a calculation of the configurational average overthe local lattice distortions (averaged within the unit cell)and magnetic moment orientations, /angbracketleft ···/angbracketright c, using the recently reported approach [ 27,28], which is based on the coherent potential approximation (CPA) alloy theory [ 29–31]. To account for the effect of spin fluctuations, which we describe in a way similar to what is done within thedisordered local moment (DLM) theory [ 32], the angular distribution of thermal spin moment fluctuations is calculatedusing the results of Monte Carlo (MC) simulations. Theseare based on ab initio exchange coupling parameters and reproduce the finite-temperature magnetic properties for theAFM and FM states in both the low-temperature ( T< T m) and high-temperature ( T> T m) regions very well [ 33]. The inset in Fig. 1(a) shows the temperature-dependent magnetization M(T) for one of the two Fe sublattices aligned antiparallel (parallel) to each other in the AFM (FM) state, calculatedacross the temperature region covering both AFM and FMstates of the system. The different behavior of the magneticorderM(T) in the two phases has important consequences for the transport properties, as discussed below. Figure 1(a) shows the calculated electrical resistivity as a function of temperature ρ xx(T), accounting for the effects of electron scattering from thermal spin and lattice excitations,and compares it with experimental data. There is clearly a rather good theory-experiment agreement, especially con- cerning the difference ρ AFM xx(Tm)−ρFM xx(Tm) at the AFM/FM transition, Tm=320 K. The AFM state’s resistivity increases more steeply with temperature when compared to that of theFM state, which has also been calculated for temperaturesbelow the metamagnetic transition temperature (dotted line).Note that the experimental measurements have been performedfor a sample with 1% intermixing between the Rh and Fesublattices, leading to a finite residual resistivity at T→0 K, and as a consequence, there is a shift of the experimentalρ xx(T) curve with respect to the theoretical one [ 34]. 155139-2TEMPERATURE-DEPENDENT TRANSPORT PROPERTIES OF . . . PHYSICAL REVIEW B 95, 155139 (2017) FIG. 1. (a) Calculated longitudinal resistivity (solid circles: AFM state, open circles: FM state) in comparison with experiment [ 2]. The dashed line represents the results for Fe 0.49Rh0.51, while the dash- dotted line gives results for (Fe-Ni) 0.49Rh0.51with the Ni concentration x=0.05 to stabilize the FM state at low temperature. The inset represents the relative magnetization of one Fe sublattice as a function of temperature obtained from MC simulations (AFM: solid circles, FM: open circles) and the experimental magnetization curve M(T) (dashed line). (b) Electrical resistivity calculated for the AFM (solid symbols) and FM (open symbols) states accounting for all thermal scattering effects (circles) as well as accounting for effects of latticevibrations (diamond) and spin fluctuations (squares) separately. The inset shows the temperature-dependent longitudinal conductivity for the AFM and FM states due to only lattice vibrations. We can separate out the contributions of spin fluctuations and lattice vibrations to the electrical resistivities, ρfluc xx(T) andρvib xx(T), respectively. These two components have been calculated for finite temperatures while keeping the atomicpositions undistorted to find ρ fluc xx(T) and with fixed collinear orientations of all magnetic moments to find ρvib xx(T), respec- tively. The results for the AFM and FM states are shown inFig. 1(b), where again the FM (AFM) state has also been considered below (above) the transition temperature T m.F o r both magnetic states the local moment fluctuations have adominant impact on the resistivity. One can also see that both components, ρfluc xx(T) and ρvib xx(T), in the AFM state have a steeper increase with temperature than those of the FM state. The origin of this behavior can be clarified by referring to Mott’s model [ 35] with its distinction between delocalized spelectrons, which primarily determine the transport prop- erties owing to their high mobility, and the more localizeddelectrons. Accordingly, the conductivity should depend essentially on (see, e.g., [ 36]) (i) the carrier (essentially, sp character) concentration nand (ii) the relaxation time τ∼ [V 2 scattn(EF)]−1, where Vscattis the average scattering potential andn(EF) is the total density of states at the Fermi level. This model has been used, in particular, for qualitative discussionsof the origin of the giant magnetoresistance (GMR) effectin heterostructures consisting of magnetic layers separatedby nonmagnetic spacers. In this case the GMR effect canbe attributed to the spin-dependent scattering of conductionelectrons, which leads to a dependence of the resistivitieson the relative orientation of magnetic layers, parallel orantiparallel, assuming the electronic structure of nonmagneticspacer remains unchanged. These arguments, however, cannotbe straightforwardly applied to CsCl-structured FeRh, eventhough it can be pictured as a layered system with one-atom-thick layers, since the electronic structure of FeRhshows strong modifications across the AFM-FM transition asdiscussed, for example, by Kobayashi et al. [11] to explain the large MR effect in FeRh. We use the calculated density of states at the Fermi level as a measure of the concentration of the conducting electrons. Thechange in the carrier concentration at the AFM-FM transitioncan therefore be seen from the modification of the spDOS at the Fermi level. The element-projected spin-resolved spDOS n sp(E), calculated for both FM and AFM states at different temperatures, is shown in Fig. 2. At low temperature, for FIG. 2. Comparison of the temperature-dependent densities of states (DOSs) for the FM and AFM states of FeRh for T=40–400 K: (a) Fe sDOS, (b) Fe pDOS, (c) Rh sDOS, and (d) Rh pDOS. 155139-3S. MANKOVSKY et al. PHYSICAL REVIEW B 95, 155139 (2017) both Fe and Rh sublattices, the spDOS at EFis higher in the FM state than in the AFM state, nFM sp(EF)>nAFM sp(EF). This gives the first hint concerning the origin of the largedifference between the FM and AFM conductivities in thelow-temperature limit [see inset for σ vib xxin Fig. 1(b)]. In this case the relaxation time τis still long owing to the low level of both lattice vibrations and spin fluctuations,which determines the scattering potential V scatt. For both magnetic states the decrease in the conductivity with risingtemperature is caused by the increase of scattering processesand consequent decrease of the relaxation time. At the sametime, the conductivity difference, /Delta1σ(T)=σ vib,FM xx (T)− σvib,AFM xx (T), reduces with an increase in temperature. This effect can partially be attributed to the temperature-dependentchanges in the electronic structure (disorder smearing of theelectronic states) reflected by changes in the density of statesat the Fermi level [ 34]( s e eF i g . 2). Despite this, up to the transition temperature, T=T m, the difference /Delta1σ(T) is rather pronounced, leading to a significant change in the resistivityatT=T m. One has to stress that in calculating the contribution of spin moment fluctuations to the resistivity, the different temperature-dependent behaviors of the magnetic order in the FM and AFM states must be taken into account. This means that at the critical point, T=Tm, the smaller sublattice magnetization in the AFM state describes a more pronounced magnetic disorder when compared to the FM state, which leads to both a smaller relaxation time and shorter mean free path. The result is a higher resistivity in the AFM state. The different mean-free-path lengths in the FM and AFM states at a given temperature can be analyzed using the BSFA B(/vectork,E), calculated for E=EF, since the electronic states at the Fermi level give the contribution to the electrical conduc-tivity. For a system with thermally induced spin fluctuationsand lattice displacements the BSF has features with finite width from which the mean-free-path length of the electrons can be inferred. Figure 3shows an intensity contour plot for the BSF of FeRh averaged over local-moment configurationsappropriate for the FM and AFM states just above and justbelow the FM-AFM transition, respectively. Figure 3(a)shows the AFM Bloch spectral function, whereas Figs. 3(b) and3(c) show the sharper features of the spin-polarized BSF of the FM state, especially for the minority-spin states. This implies a longer electronic mean free path in the FM state in comparisonto that in the AFM state, which is consistent with the drop inresistivity. Finally, we discuss the behavior of the electrical resistivity of FM-ordered FeRh in the vicinity to the Curie temperature.First of all, once the temperature has been raised above the Curie temperature and the system is in a magnetically disordered state, there is no longer a contribution from thespin fluctuations to the increase in the resistivity ρ(T) when Tis increased further. The transition to the PM state results therefore in an abrupt decrease of the rate of increase of ρ(T) with temperature (see Fig. 1). This effect, observed also in Fe and Ni, has been discussed previously [ 28]. Below T C, the sharp increase of the resistivity as the Curie temperature is approached is a consequence of the fast increase in the amplitude of transverse spin fluctuations in this temperatureregion. Figure 4demonstrates the impact of thermally induced FIG. 3. Bloch spectral function (in units of states/Ry) of FeRh calculated (a) for the AFM state at T=300 K and for the FM state resolved into (b) majority-spin and (c) minority-spin electron components, calculated for T=320 K. The finite width of this feature determines the electronic mean free paths. magnetic disorder on the electronic structure, leading to an increase in smearing of the electron energy bands when thetemperature changes from 600 to 700 K. As discussed above, 155139-4TEMPERATURE-DEPENDENT TRANSPORT PROPERTIES OF . . . PHYSICAL REVIEW B 95, 155139 (2017) FIG. 4. Element-resolved BSF on (a) Fe and (b) Rh sites in FeRh calculated for the FM (left) and PM (right) states at finite temperatures T=600 K and T=700 K, respectively; (c) comparison of the element-resolved Fe (left) and Rh (right) DOS calculated for theFM (solid line) and PM (dashed line) states at finite temperatures T=600 K ( M/M 0=0.66) and T=700 K ( M/M 0=0). this observation is connected to the shortened lifetime of the electronic states that causes an increase in the electricalresistivity. Clearly, the differences in the ρ(T) behavior in the vicinity of T Cfor different systems stem from specific features of their electronic structures relevant to their PM states. For example, there is (i) magnetic “local-moment” disorder in the case of pure Fe, (ii) a Pauli paramagnetic state in the case ofpure Ni, and (iii) magnetic local-moment disorder on the Fesublattice and disappearance of spin polarization on the Rhsublattice in the case of FeRh. Figure 4(b) demonstrates the induced spin splitting of the Rh electronic states in FeRh, inparticular around the Fermi level, at T< T C(T=600 K, left panel). This splitting disappears above TC(T=700 K, right panel), so that the Rh DOS increases at the Fermi level[Fig. 4(c), right panel]. This leads in turn to the sharp increase in the resistivity as the critical temperature is approachedsince ρ(T) is inversely proportional to the relaxation time τ, i.e., ρ(T)∼[V 2 scattn(EF)] (see discussions above). It is also worth mentioning the combined effect of both scattering channels that arise from spin fluctuations and lattice vibrations. The latter contribution is rather small (see Fig. 1), and consequently, ρ(T) has a temperature dependence determined essentially by the spin fluctuations. In the case of Fe [ 28], on the other hand, both contributions are comparable, andlattice vibrations lead to a rather pronounced smearing of theelectronic states at E Fwhen the temperature approaches TC, which conceals the impact of the electron scattering from the FIG. 5. (a) The temperature dependence of the anomalous Hall resistivity for the FM state of (Fe 0.95Ni0.05)Rh in comparison with experimental data [ 11]; (b) Gilbert damping parameter as a function of temperature: theory accounting for all thermal contributions (squares) in comparison with the experimental results for a thick-film system (50 nm; open diamonds) [ 37] and for an FeRh thin film deposited on a MgO(001) surface (upward and downward triangles). Upward and downward triangles represent data for heating (h) and cooling (c) cycles, respectively (for details see Appendix B). The inset represents the results for the individual sources for the Gilbert damping, i.e., lattice vibrations (circles) and spin fluctuations (diamonds). The totalαvalues calculated for the FeRh crystal without (cub) and with tetragonal (tetra) distortions ( c/a=1.016) are shown by open and solid squares, respectively. Gilbert damping for the FM phase is shown by dashed lines in the temperature region below the metamagnetic transition temperature. spin fluctuations. As a result, the total ρ(T) has an almost linear increase up to TC. In particular concerning technical applications of FeRh, it is interesting to study further temperature-dependent responseproperties. In Fig. 5(a) we show our calculations of the total anomalous Hall resistivity for FeRh in the FM state,represented by the off-diagonal term ρ xyof the resistivity tensor, and compare it with experimental data [ 11]. As the 155139-5S. MANKOVSKY et al. PHYSICAL REVIEW B 95, 155139 (2017) FM state is unstable in pure FeRh at low temperatures, the measurements were performed for (Fe 0.965Ni0.035)Rh, for which the FM state has been stabilized by Ni doping. Thecalculations have been performed both for the pure FeRhcompound and for FeRh with 5% Ni doping, (Fe 0.95Ni0.05)Rh, which theory finds to be ferromagnetically ordered down toT=0 K. As can be seen, the magnitude of ρ xy(T) increases in a more pronounced way for the undoped system. Nevertheless,both results are in rather good agreement with experiment. In addition to temperature-dependent transport properties linear response calculations with the inclusion of relativisticeffects enable us to present results for Gilbert damping, whichplays a crucial role for spin dynamics. The experimental datashown in Fig. 5(b) by triangular symbols represent results for rather thin FeRh films ( d=10 nm) deposited on top of a MgO(001) substrate (see experimental details described inAppendix B). Upward and downward triangles in Fig. 5(b) represent the Gilbert damping obtained for heating and coolingcycles, respectively. The FeRh unit cell with a lattice constant√ 2 times smaller than that of MgO is rotated around the zaxis by 45◦with respect to the MgO cell. Because of this, a compressive strain occurs in the FeRh film. From theexperimental data [ 38], this implies a tetragonal distortion of the FM FeRh unit cell with c/a=1.016. Theαcalculations have been performed for the FM state taking into account all temperature-induced effects, i.e., spinfluctuations and lattice vibrations [ 28,39]. As one can see in Fig. 5(b), these results are in good agreement with the ex- perimental value (shown by a diamond) for a thick (bulk) filmwhere αwas measured as 0.0012 at T=420 K [ 37]. However, the calculated αvalues are smaller by a factor of 3 when compared to the experimental data measured for the thinner10-nm film. Accounting for the tetragonal distortion results ina rather weak change for the calculated α, as can be seen in the inset of Fig. 5(b) (solid squares). Therefore, the discrepancies between theory and experiment have to be attributed partially to surface and finite-size effects, as discussed, for example, byBarati et al. [40], which are not accounted for within the present calculations. Another reason for the discrepancies can beassociated with the inhomogeneities presented in the sample.Note also that the measurements represented in Fig. 5(b) have been performed in the vicinity of the metamagneticAFM-FM transition. In this temperature region the FM stateis not uniform, as discussed, for example, by Baldasseroniet al. [41], who observed the mixed-phase (FM +AFM) state close to the T mtransition temperature. Evidently, this can also lead to an increase in the Gilbert damping in this temperatureregion when compared to the pure FM state considered in thecalculations. The separate contributions to the Gilbert damping due to spin fluctuations and lattice vibrations are shown in the insetof Fig. 5(b) for a range of temperatures extended to low tem- peratures beyond those measured by experiment. As discussedin the literature, magnetization dissipation at low temperatureis well described via the breathing Fermi-surface model forpure elemental materials and ordered compounds [ 28,39,42]. In this regime the temperature dependence of the Gilbertdamping is directly connected to the relaxation time parameterof the electronic subsystem, which in turn is determined bythe dominating spin-conserving electron scattering that arisesfrom lattice vibrations, V 2 vib, and spin fluctuations, V2 flu.I n this low-temperature regime (as discussed in Appendix C), α∼[V2 vib+V2 flu]−1. The thermally induced increase in the amplitude of lattice vibrations and spin fluctuations resultsin an increase in the effective scattering cross section for theelectrons and hence a decrease in the Gilbert damping. Basedon the expressions given in Appendix C, one can consider individual contributions from different scattering channels atlow temperature. Thus, since α∼(α −1 vib+α−1 fluc)−1, one can say that the higher rate of decrease with rising temperature for α is associated with the scattering mechanism which has thelarger scattering cross section. In particular, at T≈200 K, the Gilbert damping associated with spin fluctuations isappreciably smaller than that due to lattice vibrations [seeinset in Fig. 5(b)]. This implies a large decrease at T< 200 K ofα(T) with an increase in T, as seen in the inset in Fig. 5(b). This clearly shows (see Appendix C) the dominant role of spin fluctuations for the Gilbert damping in the low-temperatureregime, leading to a similar behavior for the total Gilbertdamping [squares in Fig. 5(b)]. Moreover, it can be seen that the total αaccounting for both scattering channels is still smaller in the low-temperature regime owing to the increasedeffective scattering cross section. The “resistivitylike” behavior at higher temperatures, i.e., αgrowing with rising temperature, reflects the increasing role of the interband transitions which determines a dominatingspin-flip dissipation mechanism [ 43]. In this regime, as seen in Fig. 5(b), the increase in the total Gilbert damping with rising temperature is predominantly determined by electronscattering from lattice vibrations, demonstrating the leadingrole of this scattering channel for the Gilbert damping at hightemperatures. Note that the spin fluctuations in the temperatureregion shown in Fig. 5(b)lead to a weak decrease of α(T) with an increasing temperature, indicating a small contribution tothe spin-flip dissipation mechanism. IV. SUMMARY In summary, we have presented ab initio calculations for the finite-temperature transport properties of the FeRh compound.A steep increase in the electric resistivity has been obtainedfor the AFM state, leading to a pronounced drop in resistivityat the AFM to FM transition temperature. This effect can beattributed partially to the difference in the electronic structureof FeRh in the FM and AFM states, as well as to a fasterincrease in the amplitude of spin fluctuations caused bytemperature in the AFM state. Further calculated temperature-dependent response properties such as the anomalous Halleffect (AHE) resistivity and the Gilbert damping parameter forthe FM system also show good agreement with experimentaldata. This gives additional confidence in the model used toaccount for thermal lattice vibrations and spin fluctuations. ACKNOWLEDGMENTS Financial support from the DFG via SFB 689 (Spin- phänomene in reduzierten Dimensionen) and from the EPSRC(UK; Grant No. EP/J006750/1) is gratefully acknowledged. 155139-6TEMPERATURE-DEPENDENT TRANSPORT PROPERTIES OF . . . PHYSICAL REVIEW B 95, 155139 (2017) APPENDIX A: TREATMENT OF THERMAL LATTICE DISPLACEMENT AND SPIN FLUCTUATIONS To account for the impact of the thermal lattice vibrations and spin fluctuations, the alloy analogy model is used inthe present work. The multiple-scattering theory allows us to describe the uncorrelated local thermal atomic displacements and spin moment deviations from their equilibrium, withinthe single-site CPA alloy theory. This implies the reductionof the calculation of the thermal average to the calculationof a configurational average in full analogy with random alloysystems [ 28]. Within this approach the coherent scattering path operator is defined as τ CPA=Nvf/summationdisplay v=1xvxfτvf, (A1) with summation over all types of local lattice vibrations and spin fluctuations with the corresponding probabilities xvand xf[28]. The underline indicates matrices with respect to the combined index /Lambda1.T h eτvfoperators are defined through the corresponding single-site scattering matrices tloc vf[28]: tvf=U(/Delta1/vectorRv)R(ˆef)tR(ˆef)−1U(/Delta1/vectorRv)−1. (A2) HereR(ˆe) is a rotation matrix for the transformation from the local to the global frame of reference. The so-called U- transformation matrix U(/Delta1/vectorRq v) for each atomic qsite in the unit cell is given by [ 44,45] ULL/prime/parenleftbig /Delta1/vectorRq v/parenrightbig =4π/summationdisplay L/prime/primeil+l/prime/prime−l/primeCLL/primeL/prime/primejl/prime/prime/parenleftbig/vextendsingle/vextendsingle/Delta1/vectorRq v/vextendsingle/vextendsinglek/parenrightbig YL/prime/prime(ˆs), (A3) where L=(l,m) represents the nonrelativistic angular mo- mentum quantum numbers, jl(x) is a spherical Bessel function, YL(ˆr) is real spherical harmonics, CLL/primeL/prime/primeis the corresponding Gaunt number, and k=√ Eis the electronic wave vector. The amplitude of atomic displacements |/Delta1/vectorRq v|is represented by the temperature-dependent rms displacement ( /angbracketleftu2/angbracketrightT)1/2according to Nv/summationdisplay v=1xq v/vextendsingle/vextendsingle/Delta1/vectorRq v(T)/vextendsingle/vextendsingle2=/angbracketleftbig u2 q/angbracketrightbig T. (A4) Basically, the mean-square displacement of the atom qalong the direction μ(μ=x,y,z ) can be evaluated within phonon calculations [ 46]. However, in the present work we have used the approach based on Debye’s theory with the Debyetemperature /Theta1 Dtaken from experiment [ 47]. In this case the individual mean-square displacements for different atomic types in the unit cell are not well defined. Moreover, theirrelative magnitudes can change as a function of temperatureas a consequence of different ratios of the amplitude ofdisplacements for different types of atoms, associated withacoustic and optical phonon modes in the limit of smallwave vector /vectorq, as well as with the phonon modes with /vectorq approaching the boundary of the Brillouin zone /vectorG/2 (see, e.g., Ref. [ 46]). Because of the lack of such information, we have used an approximation based on the averagedmean-square displacement. This implies that the mean-squaredisplacements for both types, Fe and Rh, are equal and are given by the expression [ 48,49] /angbracketleftbig u 2 μ/angbracketrightbig T=1 43h2 π2Mk B/Theta1D/bracketleftbigg/Phi1(/Theta1D/T) /Theta1D/T+1 4/bracketrightbigg , (A5) with/Phi1(/Theta1D/T) being the Debye function. In spite of the simplicity, this approach gives results in rather good agreementwith experimental data for disordered alloys as well as forordered compounds, as was shown previously [ 39,50]. As a consequence of the above-mentioned temperature-dependentproperties of the mean-square displacements, the differencebetween the experimental and theoretical resistivities for theordered FeRh compound [see Fig. 1(a) in the main text] can be partially attributed to the present simplification used for theevaluation of mean-square displacements. APPENDIX B: EXPERIMENTAL DETAILS OF GILBERT DAMPING FeRh films were grown on (001)-oriented single-crystal MgO substrates using dc magnetron sputtering. The basepressure of the chamber was 2 ×10 −5Pa. The substrates were kept at 573 K for 30 min. Then 10-nm FeRh weredeposited with a growth pressure of 0.7 Pa Ar correspondingto stoichiometric Fe 51Rh49films [ 51]. The sputtering power is 30 W for 3-inch Fe 50Rh50targets. Afterwards, the films were heated to 1023 K and annealed for 100 min. When the filmswere cooled down to room temperature, they were capped with5-nm Al in situ . The experimental data were obtained by field-swept fer- romagnetic resonance measurements of a 25-nm FeRh filmgrown on MgO(001) and capped by 5-nm Al in the out-of-plane configuration for frequencies from 5 to 24 GHz. Thetemperature was controlled by heating through the substrate,and the measured absorption spectra were fitted to a Lorentzianline shape [ 52] in order to obtain the linewidth /Delta1H.T h e damping parameter αwas determined from the frequency dependence of /Delta1H, as demonstrated by Mancini et al. [37] and Heinrich et al. [53]. APPENDIX C To discuss the temperature-dependent behavior of Gilbert damping in more detail one can represent the expression in Eq. ( 5) in terms of the Bloch spectral function A(E,/vectork,n), following the corresponding discussions by Kamberský [ 54] and Gilmore et al. [55]. According to these authors, the leading contribution to the Gilbert damping in the low-temperature limit is associated with the intraband scatteringgiven by [ 54,55] α intra∼/summationdisplay n/integraldisplayd3k (2π)3|/Gamma1− nn(/vectork)|2 ×/integraldisplay dEA (E,/vectork,n)A(E,/vectork,n)/parenleftbigg −df(E) dE/parenrightbigg ,(C1) with A(E,/vectork,n)=w/vectork,n (E−E/vectork,n)2+w2 /vectork,n, 155139-7S. MANKOVSKY et al. PHYSICAL REVIEW B 95, 155139 (2017) where /Gamma1− nnis the matrix element of the transverse torque operator and w2 /vectork,nis related to the imaginary part of the the scattering self-energy [ 54]. In the present work we discuss two contributions due to various electron scattering channels,i.e., due to lattice vibrations with Im /Sigma1 vib /vectork,n∼(τvib /vectork,n)−1and due to spin fluctuations with Im /Sigma1flu /vectork,n∼(τflu /vectork,n)−1, and the relaxation times, τvib /vectork,nandτflu /vectork,n, corresponding to the different scattering channels. With this, w2 /vectork,ncan be represented by the effective relaxation time ( τeff)−1=(τvib)−1+(τflu)−1.A sw a ss h o w n in Refs. [ 54,55], after integration over the energies, Eq. ( C1) can be reduced to αintra∼τeff/summationdisplay n/integraldisplayd3k (2π)3|/Gamma1− nn(/vectork)|2. (C2) According to the discussions above, we have τflu∼ [V2 flun(EF)]−1andτvib∼[V2 vibn(EF)]−1, leading to the follow- ing dependence: α∼τeff∼[V2 vib+V2 flu]−1. The expression in Eq. ( C2) can also be reduced to the form used for discussions of the Gilbert damping within the breathing Fermi-surfacemodel [ 55–57] that describes well the temperature-dependent behavior α(T) in the low-temperature regime.The interband contribution in terms of Bloch spectral function is given by the expression [ 54,55] α inter∼/summationdisplay n/negationslash=m/integraldisplayd3k (2π)3|/Gamma1− nm(/vectork)|2 ×/integraldisplay dEA (E,/vectork,n)A(E,/vectork,m)/parenleftbigg −df(E) dE/parenrightbigg .(C3) At low temperature this contribution increases with tem- perature as αinter∼τ−1 eff∼[V2 vib+V2 flu][54,55] and above a certain temperature Tmbecomes the dominating part of the Gilbert damping. 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PhysRevB.92.060411.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 92, 060411(R) (2015) Current-induced fingering instability in magnetic domain walls J. Gorchon,1J. Curiale,1,2,3A. Cebers,4A. Lema ˆıtre,2N. Vernier,5M. Plapp,6and V . Jeudy1,7,* 1Laboratoire de Physique des Solides, Universit ´e Paris-Sud, CNRS, UMR 8502, F-91405 Orsay, France 2Laboratoire de Photonique et de Nanostructures, CNRS, UPR 20, F-91460 Marcoussis, France 3Consejo Nacional de Investigaciones Cient ´ıficas y T ´ecnicas, Centro At ´omico Bariloche-Comision Nacional de Energ ´ıa At ´omica, Avenida Bustillo 9500, 8400 San Carlos de Bariloche, R ´ıo Negro, Argentina 4University of Latvia, Zellu-8, Riga LV-1002, Latvia 5Institut d’ ´Electronique Fondamentale, Universit ´e Paris-Sud, CNRS, UMR 8622, F-91405 Orsay, France 6Physique de la Mati `ere Condens ´ee, Ecole Polytechnique, CNRS, F-91128 Palaiseau, France 7Universit ´e Cergy-Pontoise, F-95000 Cergy-Pontoise, France (Received 19 May 2015; published 25 August 2015) The shape instability of magnetic domain walls under current is investigated in a ferromagnetic (Ga,Mn)(As,P) film with perpendicular anisotropy. Domain wall motion is driven by the spin transfer torque mechanism. Acurrent density gradient is found either to stabilize domains with walls perpendicular to current lines or toproduce fingerlike patterns, depending on the domain wall motion direction. The instability mechanism is shownto result from the nonadiabatic contribution of the spin transfer torque mechanism. DOI: 10.1103/PhysRevB.92.060411 PACS number(s): 72 .25.Dc,75.50.Pp,75.60.Ch,75.78.Fg Interface instabilities are encountered in a great variety of physical systems such as liquids [ 1], liquid-gas interfaces, ferro- and ferrimagnetic films [ 2–4], electrically polarizable and magnetic liquids [ 5–7], the intermediate state in type I superconductors [ 8,9], etc. These instabilities originate from a competition between the surface tension, which tends to favorflat interfaces, and a destabilizing interaction as a gradient ofexternal driving force (temperature, gravitational field, mag-netic field, etc.), or as long-range dipolar interactions [ 10,11] for quasi-two-dimensional systems [ 12]. A crucial point for understanding the interface dynamics as well as the formationof domain pattern is to determine the parameters controllingthe instabilities and their formation mechanism. In ferromagnetic systems, it was shown recently that domain walls (DWs) can be moved by a spin polarizedcurrent [ 13–16] through the so-called spin transfer torque (STT) [ 17–20]. This has motivated an intense research effort for elucidating the physics of STT and for potentialapplications in spin electronics [ 21,22]. The STT acts as a driving force proportional to the current density. As expectedby analogy with the well-studied field-driven dynamics,essentially two dynamical regimes are observed. At lowdrive, DWs move in the pinning-dependent creep regime.Above a depinning threshold, the dynamics corresponds toflow regimes limited by dissipation [ 16]. Current-driven DW dynamics is most generally studied in narrow tracks, whereDWs remain stable over the track width. However, field-and current-driven dynamics exhibit, in extended geometry,quite different behavior. A magnetic field acts essentially asa magnetic pressure pushing DWs with an average uniformvelocity. In contrast, the current-driven creep regime was foundto result in the formation of triangular domain shapes [ 23]. In the flow regime [ 24], the DW velocity was shown to depend on the respective orientations of the DW and thecurrent flow. Those observations suggest a complex interplaybetween the DW shape and dynamics, and the STT magnitude. *vincent.jeudy@u-psud.frIn this frame, it is particularly interesting to characterizethe shape stability of DWs driven by current. To addressthis issue, we investigated DW motion under current inwide geometries where instabilities induced by current and/ordipolar interactions can develop and be visualized. We useda (Ga,Mn)(As,P) thin film with perpendicular magnetizationdue to the extraordinary weak current density that is requiredfor DW motion, which gives us access to a wide range ofdynamical regimes [ 16,25]. To get a better understanding of the role of current-induced motion on DW stability, weintroduce, on purpose, a progressive current density gradientby patterning our device in a semicircular geometry [ 26]. In this Rapid Communication, we show how the STT mechanism affects the domain pattern and the DW shapestability. We found, in particular, that a current densitygradient, depending on the DW motion direction, stabilizes ordestabilizes the DW shape. A model, also taking into accountthe surface tension and dipolar interactions, grasps the mainfeatures of DW stability. A5 0n mt h i c k( G a 0.95,Mn 0.05)(As 0.9,P0.1)fi l mw a sg r o w n by low-temperature ( T=250◦C) molecular beam epitaxy on a GaAs (001) substrate [ 25]. It was then annealed at T=250◦C for 1 h. Its magnetic anisotropy is perpendicular (saturation magnetization M=23±1k A/m) and its Curie temperature Tcis 119 ±1 K. The semicircular geometry (100μm radius) was patterned by electron beam lithography and etching and then connected to a narrow (width w=2μm) electrode at the straight edge center and to a semicircular elec-trode made of Ti (20 nm) /Au (200 nm) layers [see Fig. 1(a)]. The shape of the magnetic domains and of DWs is controlledby differential polar magneto-optical Kerr microscopy with a1μm resolution in a cryostat with a base temperature of 95 K for all the experiments presented here [see Figs. 1(b)–1(d) ]. The two gray levels correspond to opposite magnetizationdirections perpendicular to the film. Due to the semicirculargeometry, the electrical current lines are radial. The currentdensity jdecays with the distance rfrom the narrow electrode asj(r)≈I/πrh (Iis the injected current and h=50 nm the film thickness) so that the absolute value of the gradient 1098-0121/2015/92(6)/060411(5) 060411-1 ©2015 American Physical SocietyRAPID COMMUNICATIONS J. GORCHON et al. PHYSICAL REVIEW B 92, 060411(R) (2015) FIG. 1. Current-induced modification of the magnetic domain pattern. (a) Sample description. (b) Magnetic-field-driven domain pattern corresponding to the initial magnetic state. (c), (d) Modifica- tion of the domain pattern due to a dc current. The current flows fromthe narrow to the semicircular electrode ( j> 0) for 60 s. Its amplitude was (c) I=2.16 mA and (d) I=2.98 mA. The domain pattern is observed by magneto-optical Kerr microscopy. The two gray levels reflect the two opposite magnetization directions perpendicular to the (Ga,Mn)(As,P) film. T=95 K. decreases progressively with ras|dj/dr |=|I|/(πhr2). In the following, by convention, I>0 (i.e., j> 0) corresponds to a current flow from the narrow to the semicircular electrode. First evidences of domain wall shape instability are shown in Fig. 2. A set of semiconcentric magnetic domains centered on the narrow electrode [see Figs. 2(a)and2(e)] was prepared using current-induced stochastic domain nucleation and DWpropagation (see Ref. [ 26] for details) starting from a uniform magnetization state. Next, a dc current was injected betweenthe two electrodes for a fixed duration, after which an imagewas acquired. The sequence is repeated for Figs. 2(b)–2(d) with increasing current (for 60 s at I=0.7, 1.1, and 1.2 mA) and for Figs. 2(e)–2(g) with increasing duration (1 μs, 10μs, and 100 μsa tI=−1.547 mA). The DW motion observed in Figs. 2(b)–2(d) and2(e)–2(g) originates from the spin transfer torque. In ferromagnets, the electrical current is spin polarizedand carriers crossing a DW exert a torque on the local magneticmoment, which results in DW propagation. In (Ga,Mn)(As,P)films with perpendicular anisotropy, DW motion is in theopposite direction to the current [ 16], as it can be observed. In this experiment, different DW dynamical regimes are expectedto occur due to the decay of jwithr. Close to the narrow electrode, the current density ( j≈I/hw =10–20 GA /m 2, where w=2μm is the width of the narrow electrode) is sufficiently large for the flow regime to be reached [ 16] while pinning-dependent regimes are expected to occur in the otherparts of the device. The most original of the results shown in Fig. 2is the dependence of the shape of the domains on the current polarity.Forj> 0, the semicircular symmetry of the domains breaks. In Fig. 2(b), the black domain next to the narrow electrode expands toward the electrode by forming fingerlike shapes. FIG. 2. (Color online) Instability of magnetic DW produced by a gradient of current density. Left frames: Stability of a domain wall(dotted arcs) placed perpendicularly to a current density gradient. A small tilt of an elementary wall length (blue segments) produces an asymmetry of the forces due to spin transfer (thin black arrows).A current flow (thick green arrows) in the direction of the narrow electrode ( j> 0, top frame) tends to destabilize the initial orientation while it tends to be stabilized for j< 0 (bottom frame). (a)–(d) DW shape instability for j> 0. (a) Initial state. (b)–(d) A 60 s dc current flow produces a finger growth towards the narrow electrode. Increasing the current magnitude [ I=0.70, 1.10, and 1.20 mA for (b)–(d), respectively] enhances the distance at which the semicircular DWs become unstable. (e)–(g) Stable radial DW growth for j< 0. The sample that is initially in an homogeneous magnetic state issubmitted to a current pulse of amplitude −1.547 mA of increasing duration [1 μs, 10μs, and 100 μs for (e)–(g), respectively]. The propagation front remains almost semicircular. The weak anisotropyof magnetic domain growth is most probably associated with the in-plane anisotropy of the (Ga,Mn)(As,P) single crystal. T=95 K. As the current amplitude increases [Figs. 2(c) and 2(d)], the instability process also takes place in domains locatedfarther away from the electrode. In contrast, for j< 0, the semicircular geometry is conserved. The shape of the domainwalls is stable during the motion. We explain now, firstqualitatively, the contribution of the current density gradientto the domain wall stability. The left frames of Fig. 2give a schematic description of this mechanism. Let us considera slightly tilted elementary DW segment. Due to the currentdensity gradient, the two segment ends experience a differentSTT amplitude. It is larger for the one closer to the narrowelectrode. This asymmetry is the driving mechanism for theDW stabilization or destabilization. When the jis negative, the STT force points away from the narrow electrode and the DWsegment moves away from the electrode. However, the laggingsegment extremity experiences a stronger STT force than theopposite end, therefore acting as a restoring force. The DWremains stable during its motion. In turn, this mechanism isresponsible for the DW destabilization when j> 0 (opposite DW motion direction) since the STT force is stronger forthe forward end. It eventually leads to domain growth alongthe current lines. This behavior shares similarities with theRayleigh-Taylor instability [ 1], when a heavier liquid is above a lighter one. This instability mechanism has dramatic consequences on domain pattern formation up to very large radii and hence verylow DW velocities [see Figs. 1(b)–1(d) ]. Figure 1(b) shows an initial demagnetized state (obtained before applying any 060411-2RAPID COMMUNICATIONS CURRENT-INDUCED FINGERING INSTABILITY IN . . . PHYSICAL REVIEW B 92, 060411(R) (2015) magnetic field or current). The magnetic domains with oppo- site magnetization direction present a self-organized pattern,as usually observed in ferromagnetic films with perpendicularanisotropy. The typical domain width and spacing ( ≈10 and ≈20μm, respectively) results from a balance between the positive DW energy and long-range magnetic interactionsbetween domains [ 27]. The domain shape corresponds to randomly oriented corrugated lamellae. After applying a positive dc current ( j> 0) for 60 s [see Figs. 1(c) and1(d)], the domains tend to be aligned radially. For the largest current value ( I=2.98 mA), the domain pattern is modified over the full sample surface area, asobserved in Fig. 1(d). The DWs are aligned along the current lines, a consequence of the gradient-induced destabilizationmechanism described earlier. We can get insight into theDW organization dynamics when injecting lower currentvalues. In that case, the domain pattern modification remainsspatially limited by a semicircular boundary centered onthe narrow electrode, as seen in Fig. 1(c) (I=2.16 mA). Indeed, sufficiently far from the narrow electrode, DWsfollow dynamical regimes controlled by DW pinning andthermal activation. In those regimes, the DW velocity variesexponentially with the driving force. As the STT amplitudedecreases as |I|/r, the DW velocity considerably reduces as it is located at a greater distance from the high currentdensity regions close to the narrow electrode. Therefore, for alimited current pulse duration (60 s), each given current valueIdefines a clear-cut semicircular boundary separating regions with unmodified patterns (at a scale of the experimental spatialresolution ≈1μm) from regions presenting significant DW displacements, as observed in Fig. 1(c). At this point, we have shown how a current density gradient can stabilize or destabilize a DW. However, we have notconsidered yet how this mechanism competes or cooperateswith the other mechanisms involved in DW stability, suchas dipolar interactions and the DW surface tension. To thatend, we extended the experiment described in Figs. 2(e)–2(h) (I=−1.55 mA) to longer current pulses. As previously, the sample was first prepared in a fully homogeneous magnetizedstate. Negative current pulses were injected for 10 ms, 690 ms,and 29.7 s durations. In this situation, the gradient acts asa stabilization contribution. For the shortest duration, thedomains present a semicircular shape [see Fig. 3(a)] that reflects the current line symmetry, as already observed inFigs. 2(e)–2(h) . However, for the longest durations [see Figs. 3(b) and3(c)], the semicircular shape of the domains with the largest radius becomes unstable and fingerlike domaingrowth is observed. The finger width is close to the typicalsize of the domain patterns observed in the demagnetizedconfiguration [see Fig. 1(b)]. This behavior strongly points toward dipolar interactions as the destabilization mechanism.For a different injected current I, the values of the critical instability radius r cat which the finger-shaped domains start to grow were systematically deduced from images [such asthose presented in Figs. 3(a)–3(c)] obtained for a large set of increasing pulse durations. As reported in Fig. 3(d),r 2 c is found to vary linearly with I, i.e., the critical radius is associated to a well-defined critical current density gradient|dj/dr | c=|I|/(πhr2 c). Therefore, the DW shape instability observed in Figs. 3(a)–3(c) occurs when the current density FIG. 3. (Color online) Fingerlike magnetic domain growth re- sulting from domain wall instability. The images were obtained for a constant current ( I=−1.55 mA) directed towards the narrow electrode ( j< 0) and for different durations [(a) 100 μs, (b) 690 ms, and (c) 29.7 s]. T=95 K. (d) Square of the critical instability radius as a function of the bias current. The line corresponds to thebest fit of the theoretical prediction. gradient becomes too weak to stabilize the DWs perpendicular to current lines against the dipolar interactions. To get more quantitative insight on the DW shape instabil- ities, we have elaborated a model that describes the stabilitylimit of a flat DW subjected to an electrical current gradient.The model considers a ferromagnetic layer of thickness h, aligned along the zdirection. A flat DW separating two domains with opposite magnetization directions is parallelto the x-zplane. The DW is submitted to a current flow exhibiting a gradient in the ydirection. The magnetization vector is given as−→M=M(sinθcosϕ,sinθsinϕ,cosθ). In the perturbed state, the DW position is given by the equationy=q(x,t). The DW shape stability analysis is based on the Landau-Lifshitz-Gilbert equation and follows the calculationof Refs. [ 28,29]. The full calculation is detailed in the Supplemental Material [ 30]. For a weakly perturbed DW, the equations of motion are γ/parenleftbigg μ 0M+2ψ2M(q,h) h/parenrightbigg −2Aγ M/Delta1∂2q ∂x2=˙ϕ−α˙q /Delta1+βu /Delta1(1) and μ0 2γMsin 2ϕ−2Aγ M∂2ϕ ∂x2=−α˙ϕ+u /Delta1−˙q /Delta1, (2) where γ,α, andβare the gyromagnetic factor, the Gilbert damping parameter, and the so-called nonadiabatic term,respectively. /Delta1=√ A/K is the domain wall thickness param- eter, where AandKare the spin stiffness and the anisotropy constant, respectively. The parameter uis the spin drift velocity defined by u=jPcgμB 2eM, where j,Pc,g,μB, and e(<0), are the current density, the current spin polarization, theLand ´e factor, the Bohr magneton, and the electron charge, respectively. In Eq. ( 1),ψ 2M(q,h) is a potential describing 060411-3RAPID COMMUNICATIONS J. GORCHON et al. PHYSICAL REVIEW B 92, 060411(R) (2015) the dipolar interaction between the DW magnetization and the field created by the two magnetic domains with oppositemagnetization. For a small perturbation δϕ,δq of the DW, the perturbation of the spin drift velocity can be written as δu≈(du/dq )δq. Assuming a steady DW motion ( ˙ ϕ=0) and looking for solutions of the type δq∼δq 0exp(ikx), Eq. ( 1) leads to a growth rate of the instability given by d(δq0) dt=μ0Mγ/Lambda12 αh2/bracketleftbigg F+du dqβ μ0Mh2 γ/Lambda12/bracketrightbigg δq0, (3) where we have introduced the exchange length /Lambda1defined by A=μ0M2/Lambda12/2, and the magnetic Bond number [ 31]Bm= μ0(2M)2h/(4πσ) with the DW surface energy given by σ= 4√ AK.I nE q .( 3), the function Fis given by F=4Bm(γE+ log(kh/2)+K0(kh))−(kh)2, where K0(kh) is the McDonald function and γE(=0.5772) is the Euler constant. The differential equation ( 3) shows that a flat DW is unstable [i.e., d(δq0)/dt > 0] if the term in brackets on the right-hand side is positive. The instability thus results froma competition between the dipolar energy (the first terms offunction F), the DW surface tension [the term ( kh) 2inF], and the STT gradient [ ∝du/dq in Eq. ( 3)]. One should note that only the nonadiabiatic contribution ( ∝β) of the STT plays a role in DW stability [ 30]. The fastest instability growth rate corresponds to the function Fmaximum which is equal toFmax=2Bmexp (1 −2γE−2/Bm) and to a wavelength λ=πhexp (γE+1/Bm−1/2), in the limit of small kh. For the semicircular geometry considered in this work, the conservation of the current I=jπrh leads todu dq=IPcgμB πhr22|e|M. For a current flow from the narrow electrode ( j> 0, i.e., du/dq > 0), the term in brackets in Eq. ( 3) remains positive and the flat DW is always unstable. This corresponds to thecase presented in the top frames of Fig. 2for which both the current density gradient and the dipolar interactions have adestabilizing contribution. For the opposite current direction(j< 0, i.e., du/dq < 0), the DW is only stable below a critical radius given by r 2 c=IC Fmax, where C=hβPcgμB 4πAγ|e|[see Figs. 2(e)–2(g) and3(a)]. Above this critical radius, as the stabilization contribution of the gradient becomes too weakto counteract the effect of dipolar interactions, the domainwall becomes unstable. This instability leads to the growth offingerlike domains, as observed in Figs. 3(b) and3(c).Comparing those predictions to the experimental results requires the evaluation of the magnetic Bond number B m. First, Bmcan be estimated from the critical radius rc, measured in Fig. 3(d). The data best fit gives a ratio r2 c/I=C/F max= 58±3μm2/mA. Assuming β=0.3[16],Pc=0.5,g= 2,μB=9.3×10−24JT−1,γ=1.76×1011Hz T−1, and A=0.07±0.03 pJ/m[32], we have 1 /Fmax≈10 000 and Bm≈0.25 [33].Bmcan also be deduced from the number n of fingers observed in Figs. 3(b) and3(c). Indeed, assuming nto remain constant after the onset of DW instability (occurring for r=rc), the critical perturbation wavelength readsλ=πrc/n, whose value was extracted from a statistical analysis, λ=3±1μm. The prediction for λleads to Bm= 0.36±0.06, a value close to the previous estimation. Finally, Bmcan also be estimated independently from micromagnetic parameters (see Ref. [ 32]) since Bm=μ0(2M)2h/(4πσ) with σ=4√ AK. The obtained Bond number equals 0 .3±0.1 and presents a good quantitative agreement with the twoprevious estimations. This unambiguously demonstrates thatthe domain wall fingering instability, observed for j< 0, originates from a competition between the dipolar interactionsand the effect of the current gradient whose magnitude isshown to be proportional to the nonadiabatic contribution ofthe STT. In conclusion, these results show that the domain wall orientation with respect to a current flow is very sensitiveto current density gradients in current-induced DW motionexperiments. They unveil some potential weaknesses for futuredevices relying on complex circuits where these gradients areubiquitous, yet they also give us some interesting directions topropagate and manipulate DW over large surfaces, by takingadvantage of the gradient-controlled stability. The authors wish to thank J. Miltat for his careful reading of the manuscript and A. 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PhysRevB.98.134450.pdf

PHYSICAL REVIEW B 98, 134450 (2018) Nonabelian magnonics in antiferromagnets Matthew W. Daniels,1,*Ran Cheng,1,2Weichao Yu ( /ZdZ1157/ZdZ1099/ZdZ17073),3Jiang Xiao ( /ZdZ14675/ZdZ8587),3,4,5and Di Xiao1 1Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA 2Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA 3Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China 4Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China 5Institute for Nanoelectronics Devices and Quantum Computing, Fudan University, Shanghai 200433, China (Received 24 January 2018; revised manuscript received 13 August 2018; published 31 October 2018) We present a semiclassical formalism for antiferromagnetic (AFM) magnonics which promotes the central ingredient of spin wave chirality , encoded in a quantity called magnonic isospin, to a first-class citizen of the theory. We use this formalism to unify results of interest from the field under a single chirality-centricformulation. Our main result is that the isospin is governed by unitary time evolution, through a Hamiltonianprojected down from the full spin wave dynamics. Because isospin is SU(2) valued, its dynamics on the Blochsphere are precisely rotations, which, in general, do not commute. Consequently, the induced group of operationson AFM spin waves is nonabelian. This is a paradigmatic departure from ferromagnetic magnonics, whichoperates purely within the abelian group generated by spin wave phase and amplitude. Our investigation ofthis nonabelian magnonics in AFM insulators focuses on studying several simple gate operations, and offeringin broad strokes a program of study for interesting new logic families in antiferromagnetic spin wave systems. DOI: 10.1103/PhysRevB.98.134450 I. INTRODUCTION Recent years have seen a surge of interest in the generation and in-flight manipulation of magnons in antiferromagnets(AFMs). We now know that AFM magnons can couple to theangular momentum carried by electrons [ 1,2], photons [ 3–5], and other spin carriers. Detection of magnon-mediated spinsignals from AFM insulators, typically measured through theinverse spin Hall effect, has also matured to the point ofexperimental implementation [ 6–8]. It has been shown that AFM spin waves possess pointed dynamical distinctions fromtheir ferromagnetic (FM) counterpart [ 9–11], especially in the presence of spin texture [ 12–17] or broken inversion symme- try [ 11,12,18,19]. In particular, collinear AFMs possess two degenerate spin wave eigenmodes of opposite chirality [ 20]. They are often referred to as right- and left-handed modes,according to the precessional handedness of the Néel vector(Fig. 1). This notion of spin wave chirality has proved to be a useful narrative tool for understanding how AFM magnonicsdiffers from the ferromagnetic (FM) case. As a patchwork of novel results begins to populate the field of AFM magnonics, a coherent framework for under-standing their similarities, differences, and possible exten-sions becomes necessary. Our central thesis is that manyof these results can be understood in terms of spin wavechirality, through a spinor [SU(2)-valued] quantity we referto as the magnon isospin . One important corollary of this formulation is that, because isospin dynamics proceeds byintrinsically noncommutative unitary rotations on the Blochsphere, implementations of magnonic computing in AFMs *Corresponding author: danielsmw@protonmail.comwill in general be nonabelian. This fundamental departurefrom the behavior of FM magnonics calls for a seriousreinvestigation of primitive magnonic operations for AFMs;working only off analogies to extant ferromagnetic propos-als is a program restricted by commutativity, and inevitablylifts only into a small subset of available AFM computingschemes. One practical disadvantage of FM magnonics has been the need to constantly refresh the signal power in a device.This is particularly problematic in interferometric [ 21,22] spin wave logic, where the Boolean output of FM magnoniclogic gates is encoded by setting a threshold amplitudefor the spin wave power. Phase interference techniques arethen used to achieve the desired magnon amplitude. Sincehalf of the desired outputs are represented by suppressingthe power spectrum of the magnon signal, this scheme in-curs significant energy inefficiencies and requires sources ofpower to constantly refresh the signal [ 23]. Isospin com- puting resolves this problem neatly since we can encodeand manipulate data in the spin wave chirality rather thanthe spin wave amplitude. This improvement is reminis-cent of proposals for polarization-based optical computingschemes from the 1980’s [ 24]. A chief practical distinction between AFM isospin computing and optical computing isthat the former can be carried out in nanoscale solid-statesystems. Given the importance of the isospin in AFM magnonics, we consider in this paper its dynamics for a broad class ofinteractions that may manifest in AFMs, and offer an exten-sible formalism by which others can easily incorporate theeffects of new physical interactions. In the development of thisformalism, we find that there are notable differences betweenbipartite and synthetic AFMs, and we discuss the advantages 2469-9950/2018/98(13)/134450(19) 134450-1 ©2018 American Physical SocietyDANIELS, CHENG, YU, XIAO, AND XIAO PHYSICAL REVIEW B 98, 134450 (2018) SA SB− Right-HandedSA SB+ Left-Handed FIG. 1. Schematic representations of right- and left-handed modes. Red and blue arrows demonstrate the spin precession on each of the two sublattices. Because the ˆSzcomponents differ between the sublattices during a spin wave precession, each eigenmode carries an opposite sign of spin angular momentum. and disadvantages of pursuing nonabelian magnonics in these two types of systems. We then apply this formalism to anumber of examples, for the threefold purposes of illustratingits use, validating it against a set of known results, andgenerating results in a few interesting systems. With several concrete results in hand, we then propose in broad strokes a program of next-generation computing basedon nonabelian magnonics [ 25]. Although FM magnonics has been studied extensively [ 21,22,26,27], we show that the comparative richness of the AFM isospin offers dramaticallymore and different avenues for progress. The fact that isospinmanipulations do not commute offers, by purely algebraicconsiderations, a more bountiful landscape for compositionof logical operations than can be found in FMs. We emphasize that the dual-sublattice nature of AFMs does not merely amount to two copies of FM magnon sys- tems. Although one may be able to import FM magnonicschemes into the AFM architecture, one could also lookto more spinful classes of physics for inspiration in appli-cation. Spintronic [ 11] and optical [ 12] analogies to AFM magnonics have proved inspiring for novel device designs.We close by offering possibilities for future research in thisdirection. II. FORMALISM In this section, we review the AFM spin wave theory in the sublattice formalism, as we expect many of our readers aremore familiar with the staggered-order-centric approach. Webegin by exploring spin wave chirality in a minimal model: acollinear AFM with easy axis anisotropy. The description ofeasy-axis AFMs such as MnF 2, FeF 2,o rC r 2O3may follow from such a model. Using this familiar context, we reviewchirality and the way in which it encodes spin carried bythe magnon excitation. We then review a common formalismfor handling spin texture and introduce the texture-inducedgauge fields. Finally, we derive the spin wave equations ofmotion in the sublattice formalism by the variational principle.These subsections set the stage for our main results, which arepresented in the next section. A. Sublattice-centric magnonics In terms of the two sublattices, the free energy of an easy-axis collinear AFM in the continuum limitis F=Fexch+FEAA, (1a) Fexch=1 2/integraldisplay ZmA·mB−J∇mA·∇mBddx,(1b) FEAA=−K 2/integraldisplay (mA·ˆz)2+(mB·ˆz)2ddx. (1c) Here, Kis the easy-axis anisotropy (EAA), while ZandJ are the so-called homogeneous and inhomogeneous exchangeinteractions, respectively [ 28]. They have been chosen so that, under the change of variables m=m A+mB 2and n=mA−mB 2, (2) the exchange free-energy density becomes [ 29] Fexch=Z|m|2+J 2|∇n|2+O(|m|4). (3) The quantities mandnare the local magnetization and the staggered order [ 30]. We have written in Eq. ( 1) a free energy for the classic g-type antiferromagnet, but merely as a convenient concretization. Our main result generalizes to anykind of collinear AFM order, and in particular we use resultsfor synthetic AFMs later in the paper. On each sublattice of the AFM, the semiclassical spin dynamics is governed by the Landau-Lifshitz equation ˙m A=mA×1 SδF δmA, (4a) ˙mB=mB×1 SδF δmB, (4b) where Fis the free-energy functional and S=s¯hthe spin magnitude on a lattice site. Define ˆzas the easy-axis direction, and take the Néel ground state as mA=ˆzandmB=− ˆz. Spin wave fluctuations, at linear order in the cone angle bywhich precessing spins cant away from the ground state,reside entirely in the xyplane [ 31]. It is convenient to rewrite fluctuations from equilibrium as α ±=(mx A±imy A)/√ 2 and β±=(mx B±imy B)/√ 2. We will treat these four quantities as independent variables [ 32]. Collecting the equations of motion for this new basis [ 33] into matrix form, the spin wave equations for /Psi1=(α+,β+,α−,β−)a r e i(τz⊗σz)˙/Psi1=/parenleftbiggˆh 0 0 ˆh∗/parenrightbigg /Psi1=H/Psi1, (5) where τjare the Pauli matrices in isospin space and σjthe Pauli matrices in the sublattice subspace. In other words, theσ jmatrices distinguish α+fromβ+andα−fromβ−, while τjdistinguish from α+fromα−andβ+fromβ−. Our use of the term “isospin” will be introduced more fully at the end ofthis section. For the problem we outlined above, ˆhis a 2×2 Hermitian operator given by ˆh= 1 2[(Z+2K)12+σx(Z+J∇2)]. (6) For the simple free energy we have adopted in Eq. ( 1), Eq. ( 5) apparently contains two copies of the same two- level dynamics. These two copies are related by complexconjugation, which we write as the time-reversal operator T. 134450-2NONABELIAN MAGNONICS IN ANTIFERROMAGNETS PHYSICAL REVIEW B 98, 134450 (2018) The mapping of the Landau-Lifshitz-Gilbert (LLG) equation onto a Schrödinger equation is standard practice in theoreticalmagnonics [ 34,35], but note that our Eq. ( 5) differs from the usual Schrödinger equation by the appearance of τ z⊗σz on the left-hand side. The mathematical and philosophical details of Schrödinger equations with this structure have beenconsidered at length in Ref. [ 36]. Since the Hamiltonian ( 5) is block diagonal, let us first focus on the subspace governing α +andβ+. Assuming our system is stationary and translationally invariant, we can makethe ansatz ψ=ψ 0ei(k·x−ωt). The resulting eigenproblem is ¯hωσ zψ=ˆhψ. (7) For a generic 2 ×2 Hermitian operator ˆh=a12+bσx+ cσy+dσz,E q .( 7) has the solution [ 18] ψ0=/parenleftBigg coshϑ 2 −eiϕsinhϑ 2/parenrightBigg andψ1=/parenleftBigg −sinhϑ 2 eiϕcoshϑ 2/parenrightBigg , (8) where the angles ϑandϕare given through a=/lscriptcoshϑ, (9a) b=/lscriptsinhϑcosϕ, (9b) andc=/lscriptsinhϑsinϕ. (9c) The corresponding eigenvalues are ¯hω=± (d+/lscript)=±1 S/radicalbigg 1 2JZk2+ZK (10) at leading order in K. The well-known resonant energy is given then by ¯ hω0=√ ZK/S . We note that the bosonic normalization condi- tion [ 18,37,38]a2−b2−c2=± 1 implies that the space of Hamiltonians, as well as the eigenvectors themselves, liveon the hyperboloid of two sheets SU(1,1). When d=0, as in Eq. ( 6), the eigenvectors have particle-hole symmetry. ψ jexhibits eigenfrequency ( −)j|ω|. Analysis of the basis functions shows that ψ0is a right-handed precession of mA (and therefore n) while ψ1is a left-handed precession. We say that they have opposite chirality, namely, right-handedand left-handed chirality. Notice that the sister eigenproblem (for α −andβ−)i nt h e lower two rows of Eq. ( 5) has positive frequency solutions corresponding to left-handed modes and negative frequencysolutions corresponding to right-handed modes. This inver-sion from the {α +,β+}problem arises precisely due to the conjugate basis. We will take the positive-energy solutionfrom each block, /Psi1 0=⎛ ⎜⎜⎝coshϑ 2 −eiϕsinhϑ 2 0 0⎞ ⎟⎟⎠and/Psi11=⎛ ⎜⎜⎝0 0 −sinhϑ 2 eiϕcoshϑ 2⎞ ⎟⎟⎠,(11) as a chirally complete basis for the positive energy, de- generate Hilbert subspace of Eq. ( 5). Note that whereas the solutions ( 8) obey /angbracketleftψi|σz|ψj/angbracketright=(−)jδij, the solutions /angbracketleft/Psi1i|τz⊗σz|/Psi1j/angbracketright=δijare properly normalizable. We will of- ten work directly in the /Psi10and/Psi11basis, writing |0/angbracketright=(1,0) and|1/angbracketright=(0,1) as in Fig. 2. The use of bra-ket notation here FIG. 2. Linear combinations of the right- and left-handed modes |0/angbracketright∼=/Psi10and|1/angbracketright∼=/Psi11, respectively, produce an entire Bloch sphere’s worth of possible isospin states. We have labeled selected states by the polarization of the Néel order fluctuations in that state. Right- and left-handed modes correspond to right- and left-handedprecession of n, while equal linear combinations produce linearly polarized waves. The angle of linear polarization depends on the relative phase of the spin waves between the sites. Note that X-a n d Y-polarized states are orthogonal here, while in a traditional quantum spin space |X/angbracketrightis orthogonal to |−X/angbracketright, not|Y/angbracketright. Since our formalism parametrizes this space in terms of a two-level spinor, we refer to it as a Bloch sphere. Students of optics, however, will recognize that it is analogous to the Poincaré sphere that parametrizes opticalpolarization states. is a formalism of convenience arising from the close math- ematical similarities between our system and single-particlequantum mechanics. However, we emphasize early on thatthis is a purely notational convenience; it is impossible to re-alize many-body quantum phenomena, such as entanglement,in a purely semiclassical magnonic system. Since cosh x> sinhxfor all real x, the magnitude of the spin wave precession is clearly dominated by the Asublattice in/Psi1 0and the Bsublattice in /Psi11. The physical spin fluctua- tions can be recovered by taking mx A=Re[(α++α−)/√ 2], my A=Re[(α+−α−)/√ 2i], and likewise with the Bsublat- tice, so that δm(0) A=1√ 2(cos(ωt),sin(ωt))coshϑ 2, (12a) δm(0) B=−1√ 2(cos(ωt−ϕ),sin(ωt−ϕ))sinhϑ 2,(12b) δm(1) A=−1√ 2(cos(ωt),−sin(ωt))sinhϑ 2, (12c) δm(1) B=1√ 2(cos(ωt−ϕ),−sin(ωt−ϕ))coshϑ 2.(12d) We see in Eqs. ( 12) that the right-handed modes dominate on theAsublattice, as in Fig. 1. One can also see from Fig. 1 that these two modes carry opposite magnetization since the ˆSzcomponent of the sublattices must differ if one of the sublattices dominates. The reduction of magnetization on each 134450-3DANIELS, CHENG, YU, XIAO, AND XIAO PHYSICAL REVIEW B 98, 134450 (2018) sublattice is simply given by the squared magnitude of the lattice spin wave, so that the total magnetization induced by aspin wave is m z=−S/angbracketleft/Psi1|(12⊗σz)|/Psi1/angbracketright, (13) which will be negative for right-handed waves proportional to /Psi10and positive for left-handed waves proportional to /Psi11.T h i s operator 12⊗σzcorresponds to a so-called nongeometric symmetry [ 39]. It has sometimes been given as the definition of spin wave chirality. In electromagnetic analogies for AFMspin wave dynamics it corresponds to optical helicity [ 39], where the corresponding conserved quantity is the so-calledzilch [ 40]. So far, we have dealt only with a block-diagonal Hamil- tonian. Restricting to the positive energy subspace, we seethatHhas no off-diagonal terms that connect /Psi1 0and/Psi11. If such terms existed, we could manipulate the total spincarried by the spin wave in transit, rotating our spin wave statewithin the degenerate eigensubspace. We may imagine thatthe coefficients balancing these eigenvectors in a superposi-tion|η/angbracketright=η 0|0/angbracketright+η1|1/angbracketrightdefine a degree of freedom which we refer to as the magnonic isospin [ 41].The desire to exploit this internal degree of freedom motivates the remainder of thepaper . B. Spin texture, characteristic length scales, and perturbative parameters In order to control η, we must find a way to break the degeneracy between the right- and left-handed modes; thatis, we must break whatever symmetries are protecting eitherconservation of chirality (that is, the block diagonality of H) or conservation of the relative phase between right- and left-handed modes. In this paper, the main tools we consider forthis purpose are spin texture and the Dzyaloshinskii-Moriyainteraction (DMI). The latter is well known and we introducethe appropriate free energies when they are needed. Spin tex-ture, however, is somewhat more subtle, so we briefly reviewtheoretical tools for handling it. These techniques have beenused to great success in describing transport effects arisingfrom both ferromagnetic [ 34,35,42,43] and antiferromagnetic [44–46]t e x t u r e s . To describe the spin texture in our formalism, we encode the texture in a rotation matrix Rdefined by Rn=|n|ˆz.T h i s rotation matrix induces a generator of infinitesimal spin rota-tions ( ∂ μR)RT, which itself can be regarded as a collection of vector potentials Ax μJx+Ay μJy+Az μJz=(∂μR)RT,t h e decomposition being directed through the standard generators[47] of three-dimensional (3D) rotations J j. Here, μis a spacetime index, and the components Aj μdefine the (1 +d)- vectors Aj=(Aj t,Aj). Because our spin texture is described with respect to the ˆzaxis, Azwill be of paramount importance. It gives rise to an emergent magnetic field B=∇×Azthat produces a Lorentz force on magnons in Eqs. ( 27), and the temporal component Az tlikewise produces an emergent electric field. We will usually describe the influence of the other two po-tentials through the complex variable A μ=(Ax μ+iAy μ)/√ 2. For more information on these fields, the reader is referredto Appendix B. For a full discussion of this gauge fieldformalism in the treatment of spin texture, the reader may check Refs. [ 34,35]. We will soon need an approximation scheme to deal with the many perturbative effects (anisotropy, DMI, etc.) of our spin wave system. Since Aj μis a derivative of the texture- defining angles, let it define a characteristic length scale λof the system, /vextendsingle/vextendsingleAj μ/vextendsingle/vextendsingle∼1 λ. (14) In textured systems with DMI, the characteristic length scale is proportional [ 48]t oJ/D , where Dis the DMI strength [ 49] FDMI=DmA·(∇×mB). Therefore [ 50], D/J∼/vextendsingle/vextendsingleAj μ/vextendsingle/vextendsingle. (15) In systems with easy-axis anisotropy, meanwhile, the well- known characteristic length of a domain wall is√J/K , and thus K/J∼/vextendsingle/vextendsingleAj μ/vextendsingle/vextendsingle2. (16) Finally, the local magnetization [ 51]μ=(Rm)·(ˆx+ iˆy)/√ 2 scales as a derivative of the staggered order, [ 28] μ∼/vextendsingle/vextendsingleAj μ/vextendsingle/vextendsingle. (17) As it happens, the magnetization will, in our calculations, never show up as a lone linear-order term; even so, thequadratic terms O(μ 2)=O(K/J ) must be preserved. We have established a hierarchy of perturbative orders based on a single parameter |A|. In our spin wave treatment, we will keep terms up to order ∂A∼O(A2), that is, to linear order in the emergent electromagnetic field B=∇×Az. C. Matrix structure of the spin wave Hamiltonian Once we add extra terms to the free energy (spin texture, the Dzyaloshinskii-Moriya interaction, and so on) the equa-tion of motion becomes i(τ z⊗σz)˙/Psi1=/epsilon1d nSδF δ¯/Psi1−Az t(12⊗σz)/Psi1 (18) so that the spin wave Hamiltonian is given through H/Psi1= δF/δ ¯/Psi1. Here, /epsilon1is the lattice constant, dis the dimensionality of the lattice, and n=1+|μ|2is the effective index of refraction for the spin wave speed, when viewed from theperspective of the wave equation governing staggered orderdynamics. Equation ( 18) prescribes the correct harmonic spin wave theory for any free energy F, where the independent variables {α +,β+,α−,β−}are now defined as the purely in- plane fluctuations of the sublattice spin wave modes after the active rotation by Rof the ground-state texture. The detailed derivation of Eq. ( 18) is given in Appendix A. For concreteness, we now present the detailed matrix form of the exchange interaction Hamiltonian. Beginning fromEq. ( 1b), we rotate the fields by Rand change variables to the in-plane complex fluctuations α +,β+,α−, andβ−.T h e corresponding Hamiltonian for the homogeneous exchange 134450-4NONABELIAN MAGNONICS IN ANTIFERROMAGNETS PHYSICAL REVIEW B 98, 134450 (2018) interaction is Hhom=Z 2⎛ ⎜⎜⎜⎝1−3|μ|21−|μ|2μ2−μ2 1−|μ|21−3|μ|2−μ2μ2 ¯μ2−¯μ21−3|μ|21−|μ|2 −¯μ2¯μ21−|μ|21−3|μ|2⎞ ⎟⎟⎟⎠, (19) where the bar over ¯ μ(and, later, over ¯A) indicates complex conjugation. The inhomogeneous exchange interaction H inhom , meanwhile, is given by J/(2n) times the matrix ⎛ ⎜⎜⎜⎝−2|A|2(∇−iAz)2+/Delta1·∇−|A|20 −(μ∇)2+4iμA·∇+A2 (∇−iAz)2−/Delta1·∇−|A|2−2|A|2−(μ∇)2−4iμA·∇+A20 0 −(¯μ∇)2+4i¯μ¯A·∇+¯A2−2|A|2(∇+iAz)2+/Delta1·∇−|A|2 −(¯μ∇)2−4i¯μ¯A·∇+¯A20( ∇+iAz)2−/Delta1·∇−|A|2−2|A|2⎞ ⎟⎟⎟⎠, (20) where /Delta1=2i(¯μA−μ¯A). These matrix Hamiltonians, and the Hamiltonians corresponding to any other two-site inter-action, exhibit notable structural differences when the syn-thetic AFM case is considered instead. In the SupplementalMaterial, we have provided a Mathematica notebook that automates the derivation of Hfor any free energy given in terms of m AandmB[52]. It also contains precomputed Hamiltonians for anisotropy, DMI, external fields, and so on,which we use in our applied examples later in the paper. III. NONABELIAN WA VE-PACKET THEORY In this section, motivated by the need to derive ηdynamics from Eq. ( 18) in the case of spatial inhomogeneity, we apply the machinery of nonabelian wave-packet theory [ 53]. What we call “wave-packet theory” was originally developed ina paper by Chang and Niu [ 54] to explain the Hofstadter butterfly spectrum, after which their treatment was codifiedby Ref. [ 55]. Since then, the theory has been applied in a variety of contexts, sometimes requiring extensions of thetheory to account for unique features of a particular physicalproblem [ 56–58]. The most relevant extension for our purposes, and indeed, one of the most ambitious and interesting developments inwave-packet theory, is the treatment of multiple degeneratebands [ 53,59]. In this case, the theory is called nonabelian wave-packet theory because, in dealing with a vector of mul- tiple band energies at once, the “coefficients” must becomematrix valued (and therefore, generally speaking, an elementof a nonabelian matrix representation) in order to act on themultiband wave function. In this paper, we extend the non-abelian wave-packet theory to account for both the unusualτ z⊗σzfactor in our Lagrangian and our explicitly ap r i o r i nonabelian gauge field [ 60]. A detailed derivation involving the internal workings of wave-packet theory is crucial forestablishing our main results. Since details of wave-packettheory, even in the abelian case, are not widely studied, wecarefully guide the interested reader through the derivation inAppendix D. The basic idea of abelian wave-packet theory is to con- sider a momentum-space superposition |W/angbracketright=/integraltext w qψqddq of eigenvectors, where the eigenvectors are drawn from thespectrum of the Hamiltonian evaluated at some ( xc,qc)o n a classical phase space. In nonabelian wave-packet theory,the eigenvector is expressed as a general state lying in thedegenerate subspace spanned by our right- and left-handedmodes |W(x c,kc)/angbracketright =/integraldisplay dqw(q,t)[η0(q,t)|/Psi10(q,t)/angbracketright+η1(q,t)|/Psi11(q,t)/angbracketright]. (21) The coefficient wgives the shape of the wave packet, as in Fig. 3. The vector |η/angbracketright=(η0,η1) is, again, called the isospin . We demand that the otherwise generic wave packet possess (1) a momentum space distribution localized enough to be approximated as δ(q−qc), (2) a well-defined mean position xc=/angbracketleftW|ˆx|W/angbracketright, and (3) sufficient spatial localization that the environment where the wave packet has appreciable support is approxi-mately translationally invariant. These assumptions form a set of sufficient conditions under which a wave function’s semiclassical dynamics can be for-mulated, using wave-packet theory, on a classical phase space/Gamma1/owner(x c,qc). The nonabelian version, Eq. ( D13), includes an η-valued fiber over /Gamma1. By appealing to the time-dependent variational principle, we can write the Lagrangian which generates the equation ofmotion ( 18), namely, L WP=/angbracketleft/Psi1|L|/Psi1/angbracketrightwith L=i(τz⊗σz)d dt−H−Az tσz. (22) We then assume |W/angbracketrightas the solution for |/Psi1/angbracketright. Since the wave packet is sufficiently [ 61] described by the 3-tuple ( xc,qc,η), we can reduce LWPto a Lagrangian of the phase-space variables xc,qc, andηthat specify |W/angbracketright. The result is LWP=Ldt+LH+LEM,where (23a) Ldt=/angbracketleft˜η|˙xc·ˆax+˙qc·ˆaq+ˆat+i∂t|˜η/angbracketright−˙qc·xc,(23b) LH=− /angbracketleftη|H|η/angbracketright, (23c) LEM=− ˙Az·/Gamma1q−χ/parenleftbig˙Az·xc+Az t/parenrightbig . (23d) 134450-5DANIELS, CHENG, YU, XIAO, AND XIAO PHYSICAL REVIEW B 98, 134450 (2018) FIG. 3. Under the assumptions of wave-packet theory, the magnon wave packet has its magnitude w(q,t) strongly localized in real and momentum space. Consequently, the wave function is sufficiently specified by its mean coordinates ( xc,kc) on phase space. The wave-packet theory machinery uses this assumption to resolve the wave theory (left) described by Eq. ( 18) into a particle theory (right) described by the classical Lagrangian ( 23). Not pictured is the isospin degree of freedom, which lives in an SU(2) fiber over the classical phase space. The full semiclassical dynamics described by Eqs. ( 27) occurs on the induced fiber bundle. Ldt,LH, andLEMderive from the time derivative, Hamilto- nian, and emergent field terms from Eq. ( 22), respectively. Here in the main text, we simply pause to describe the variousphysical variables in Eqs. ( 23) that fall out of the derivation. First, let us define the 4 ×2m a t r i x E=|0/angbracketright/angbracketleft/Psi1 0|+| 1/angbracketright/angbracketleft/Psi11|, (24) where |0/angbracketrightand|1/angbracketrightare understood as the basis vectors (1,0) and (0,1) for the isospin |η/angbracketright=η0|0/angbracketright+η1|1/angbracketright.Eis essentially a change of basis matrix (which chooses /Psi10and/Psi11as the canonical basis vectors), followed by a projection to theforward-time degenerate Hilbert subspace that they span. E † represents the embedding of the isospin dynamics into the full spin wave dynamics, and as such the induced isospinHamiltonian is given by H=EHE †. (25) Next, we define the various 2 ×2 matrices ˆaμ. These are the matrix-valued Berry connections in isospin space ˆaij μ=/angbracketleftbig /Psi1i q/vextendsingle/vextendsingleiσz∂μ/Psi1j q/angbracketrightbig . (26) These diagonal matrices will generate Berry curvatures (ef- fective, emergent magnetic fields) in the equations of mo-tion [ 53]. The term /Gamma1 q=/angbracketleftη|τzˆaq|η/angbracketright−/angbracketleftη|τz|η/angbracketright/angbracketleftη|ˆaq|η/angbracketrightarises uniquely due to the τz⊗σzmetric structure of our full four- dimensional Hilbert space, and is absent from existing non-abelian wave-packet theories which deal only with Euclideanspaces. It gives rise to a nonlinear potential V χ=δ/Gamma1q/δη. Finally, the tilde decoration on ˜η=Gηrefers to a gauge trans- formation G=exp[−i(τz⊗12)Az·x] discussed in Eq. ( D9). Hamilton’s principle δS=0 gives us equations of motion for the dynamical variables: ˙qc=χ(E+˙xc×B)−∂E ∂xc, (27a) ˙xc=∂E ∂qc+/angbracketleft/Omega1qq/angbracketright˙qc+/angbracketleft/Omega1qx/angbracketright˙xc+/angbracketleft/Omega1qt/angbracketright,(27b) id dtη=/bracketleftbig H−At+τzAz t+ˆVχ/bracketrightbig η, (27c) with At=˙xc·ˆax+˙qc·ˆaq+ˆat,Ethe linearly perturbed spin wave energy (as in Ref. [ 53]), and /Omega1are the various Berrycurvature terms /angbracketleftbig /Omega1αβ μν/angbracketrightbig =/angbracketleftη|/parenleftbigg∂ˆaβν ∂αμ−∂ˆaαμ ∂βν/parenrightbigg |η/angbracketright. (28) Finally, the emergent electromagnetic fields are B=∇×Az andE=∇Az t, familiar to those who have studied magnetic skyrmion physics [ 35,62]. The reduction of LWPto single-particle Lagrangian ( 23)i s quite technical, and we relegate the derivation to Appendix D. The process is illustrated schematically in Fig. 3. The equa- tions of motion ( 27), as well as their derivation, are tightly related to the results of Ref. [ 53]. The differences arise due to the non-Euclidean metric τz⊗σzin the Lagrangian. This geometry gives rise to the dynamical charge χ=/angbracketleftη|τz|η/angbracketright coupled to the Lorentz force, and also gives rise to thenonlinear potential V χ(through /Gamma1q). Although Vχcan contribute at O(A2) in perturbation the- ory in principle, it only contributes at third order or abovefor the interactions we consider concretely in this paper. Tocontribute in our formalism, it would require that ˆa qmanifest at leading order in the perturbation theory, or else that we goto higher order in the perturbation theory, as a nonabelianand non-Euclidean extension of second-order wave-packettheory [ 63,64]. If such a system could be identified, then the physics of V χ, which induces a Gross-Pitaevskii equation for the isospin, could be quite interesting. In the coupling betweena wave packet and a rigid soliton, for instance, we see thatthis term produces at leading order a force proportional to ˙ χ. Thus, a change in the spin carried by the magnon produces areal-space force on the soliton. We leave the search for sys-tems in which V χcould produce significant effects to future research. Finally, let us caution the reader that Eq. ( 27c)g i v e st h e dynamics of the isospin, which is defined with respect totheAandBsublattices, not with the laboratory frame. A right-handed mode, for instance, is by our definition alwaysdominated by the Asublattice, which means that it carries opposite spin on either side of a domain wall. To return tothe laboratory frame, one should apply the inverse rotationoperator R −1to the spin texture. To extract the laboratory- frame spin, then, lift R−1to SU(2) by the standard homomor- phism [ 65] and apply it to the isospin. The (semiclassical) spin 134450-6NONABELIAN MAGNONICS IN ANTIFERROMAGNETS PHYSICAL REVIEW B 98, 134450 (2018) z x yVG=⇒expiπτz 2j = eiζDj FIG. 4. The system under investigation in Sec. IV A . An in-plane easy-axis ( ˆz) AFM is oriented in a nanostrip geometry, perpendicular to the easy axis ( ˆx). A section of the sample is subjected to a gate voltage VGapplied normal to the sample plane, in the ˆydirection. We show in Eq. ( 32) that the resulting isospin dynamics corresponds to a rotation about σzon the Bloch sphere (Fig. 2). We define here the notation Dj=diag(1 ,eiπ/2j), and is given by this applied DMI gate up to a dynamical phase eiζ. Note, as a reference, that τz=D0. carried at time tby the magnon with isospin η(t) is then [ 66] |s(t)/angbracketright=/parenleftBigg ei(ψ+φ)cosθ 2ei(ψ−φ)sinθ 2 −e−i(ψ−φ)sinθ 2e−i(ψ+φ)cosθ 2/parenrightBigg |η(t)/angbracketright,(29) where the Euler angles defining the texture are evaluated at xc(t). The observable magnetization carried by the isospin is then mz=− /angbracketleft s|σz|s/angbracketright, the sign arising from the fact that right-handed waves |0/angbracketrightcarry negative spin. Since we are generally interested in systems with easy-axis anisotropy, thematrix transformation in Eq. ( 29) will typically result in a simple sign m z=∓ /angbracketleftη|σz|η/angbracketrightdepending on whether the local Néel order along the easy axis is pointing along ±ˆz. The key result of our wave-packet analysis, as regards the remainder of this paper, is that the isospin ηobeys an emergent Schrödinger equation, and its dynamics is therefore governedby unitary time evolution. By tailoring our Hamiltonian, wecan generate unitary rotations about multiple different axes inisospin space. We display a collection of different rotations inthe coming examples, which taken together will be sufficientto generate any generic rotation (in three Euler angles) of theisospin. IV . APPLICATION TO SELECTED MAGNONIC PRIMITIVES In the previous section, we derived a set of semiclassi- cal equations governing the isospin-coupled dynamics of amagnon wave packet. Now, we apply that formalism to twoAFM magnonic systems: a gated 1D wire and a 1D domainwall. We conclude by mentioning the effects of magneticfields and hard-axis anisotropy. A. A gated AFM nanostrip: The magnon FET In this section, we consider the application of a gate volt- age across a one-dimensional (1D) AFM nanowire (extendedalong ˆx) with in-plane easy-axis anisotropy (along ˆz). The gate voltage breaks inversion symmetry, and will thereforegenerate a nonzero DMI simply by symmetry considera-tions [ 21]. In comparison to the DMI statically generated by inversion asymmetry due to interfacial or crystal structureeffects, though, one expects the DMI produced by the gateto be tunable, and therefore a useful knob to access in amagnonic computing scheme. The system has been outlinedschematically in Fig. 4. Our motivation here is threefold. First, this gate will be extremely important in our device proposals later in the paper,so it is worthwhile to present the theoretical treatment here.Second, this simple example which does notpossess any spin texture will provide a transparent presentation to demonstratethe general solution method to the reader. Finally, solving this problem, which has already been considered in the Néelvector picture, for the special case of linearly polarized waves,by Ref. [ 11], will serve as a validation of our theoretical methods against the literature. The free energy has four parts: homogeneous and inhomo- geneous exchange, easy-axis anisotropy, and DMI. The firstt h r e eo ft h e s ea r et h es a m ea si sg i v e ni nE q s .( 1), and the DMI term is F DMI=1 2/integraldisplay D·[mA×∂xmB+mB×∂xmA]dx, (30) where D=Dˆz. From the corresponding 4 ×4 Hamiltonian, we construct the 2 ×2 isospin Hamiltonian by using the embedding E†and Eq. ( 25). Writing out H=H0+Hjσj explicitly for this problem, we find that it has an unimportant [67] constant part as well as a σzcomponent: Hz=J|D|k/epsilon1/parenleftbig 1−(k/epsilon1)2 2/parenrightbig ¯hs/radicalbig 2KJ+(Jk/epsilon1)2. (31) If we assume both that K/J is small and that kis in a regime where the distance between the split bands is constant in k, namely, well above the resonance frequency, then the denom-inator of Eq. ( 31) can be approximated merely by ¯ hsJk/epsilon1 , canceling the linear contribution in the numerator and leavingonly the constant term with a weak quadratic correction.Making these approximations in Eq. ( 31), we arrive at an isospin Hamiltonian H z=D/S. (32) How does this Hamiltonian act on the isospin state? Since we are dealing with a Schrödinger equation [Eq. ( 27c)], we need only compute the unitary time evolution operator U(t1,t0)=exp/bracketleftbiggiτz ¯hs/integraldisplayt1 t0Ddt/bracketrightbigg (33) =exp/bracketleftBigg iτz ¯hs/parenleftbigg∂ω ∂k/parenrightbigg−1/integraldisplayx1 x0Ddx/bracketrightBigg . (34) This is a rotation operator in isospin space, rotating about theˆzaxis on the Bloch sphere by a total angle proportional to Dand the length of the gate, but inversely proportional to the spin wave speed ∂kωand the spin magnitude S. The rate of rotation on the Bloch sphere works out to ∇φ=1 sD J. (35) Note that we have cited the rate of rotation on the Bloch sphere, where φis the azimuthal angle: this differs by a factor 134450-7DANIELS, CHENG, YU, XIAO, AND XIAO PHYSICAL REVIEW B 98, 134450 (2018) of 2 from the polarization angle of the staggered order. X- and Y-polarized states, which appear to be rotations of π/2a w a y from each other in the trace of a spin excitation, are actuallyπaway from each other on the Bloch sphere (Fig. 2). Because the rate of rotation scales with the DMI itself, the rotation on a single gate can be manipulated online simplyby modulating the gate voltage. We concerned ourselves inRef. [ 11] [with which our result in Eq. ( 32) agrees] mostly with a rotation between XandYpolarizations, but access to generic rotations will be crucial for a mature implementationof nonabelian magnonics. B. Domain-wall retarder Since applied AFM magnonics has become fashionable in the last decade, the AFM domain wall has undergonequite a bit of new analysis [ 12–14,16], and in decades past was a prototypical nontriviality for the AFM nonlinear sigmamodel [ 68–71]. Many such studies have concluded that spin waves passing through a domain wall experience a relativefrequency shift between the right- and left-handed compo-nents [ 14]. In systems with DMI, they can express even more pronounced shifts between linearly polarized modes, givingrise to a retarding waveplate effect [ 12]. Our formalism allows us to calculate this shift precisely, and in terms of the SU(2)isospin. In this section, we consider a Bloch-type domain wall in a synthetic AFM with easy-axis anisotropy and a bulk-typeDMI. Take the Walker solution for the 1D texture as θ(x)=− 2a r c t a n/parenleftbigg expx λ/parenrightbigg andφ(x)=−π/2,(36) withλ=√J/K=O(A−1) the domain-wall width. With this texture, we can immediately calculate the texture- induced gauge fields from Eq. ( B2) (taking ψ=0 for con- creteness): we have Az=0 and Ax=1 λsinψsechx λ(37a) and Ay=1 λcosψsechx λ(37b) ⇒A=i λ√ 2sechx λ, (37c) where we have suppressed the space-time index since there is only one [ 72]. The bulk-type DMI is written as Dij=Dˆrij, and minimization of DMI energy has been used to determineφ(x). Using spin wave Hamiltonian Hfor synthetic AFMs de- tailed in the Supplemental Material [ 52], we compute the appropriate coefficients of the semiclassical dynamics inEqs. ( 27). The resulting isospin Hamiltonian has an again unimportant 1 2component as well as a τxcomponent. The τxterm is Hx=DK(Z+2Jk2) 4/lscript√ JKsechx λ. (38) Since Hhas no other nontrivial component, we see imme- diately that it will carry out a rotation of the isospin about ˆx on the Bloch sphere, and will do so most strongly near thecenter of the domain wall due to the exponential localization provided by sech( x/λ). From there, we have E(since we have H),H(since we have HandE), and we know that the B=E=0 by inspec- tion of Az. The other Berry curvature terms are easily seen to vanish as well. We immediately construct the semiclassicalequations ( 27) and integrate them with an adaptive-step size Runge-Kutta-Fehlberg solver ( RKF45 ), using the parameters for yttrium iron garnet to define our ferromagnetic layers[73]. Our results are displayed and discussed in Fig. 5.N o t e that, deep within the domain wall, the “easy axis” is nolonger aligned with the textural slow mode, and the dispersionbecomes imaginary for modes below a critical energy. In thiscase, spin transferred to the domain wall is the dominant pro-cess, and our numerical calculations break down close to thisregime. Augmenting our theory with a collective coordinatetheory of the domain wall, effectively allowing it to absorbspin, may be used to address this problem. Here, however, wekeep the problem pedagogical by simply assuming that spinwaves are sufficiently high energy that the local Hamiltonianremains Hermitian. In our analysis of the domain-wall retarder, we note an important difference between the g-type and synthetic AFM in action. Define C=σ x⊗12, which exchanges each under- lying basis field with its conjugate (time-reversed) partner.This operation corresponds to charge conjugation .Cchanges the sign of the coupling between spin wave and the emergentelectromagnetic fields arising from spin texture and DMI.Together with time reversal (given by complex conjugation),the full chirality operator S=TC is a symmetry of the degenerate Néel-state Hamiltonian H=ˆh⊕ˆh ∗. The breaking ofSsymmetry by spin texture in the domain wall is what allows the relative amplitudes of right- and left-handed modesto change in the overall wave function. Now, define I=1 2⊗σx, which defines the sublattice interchange operation. TI is also a symmetry of the de- generate Hamiltonian. In the g-type AFM case, spin texture will break TIsymmetry in general because an infinitessimal misalignment is present in each unit cell [ 28]. In the synthetic antiferromagnetic (SAF), however, the two sublattice sites ina unit cell are never misaligned, so that TIis preserved even in the presence of spin texture. Algebraically, the TIsymmetry of the SAF restricts off- block-diagonal terms of the 4 ×4 spin wave Hamiltonian to be purely real. Since the embedding Eis itself real, it follows that the isospin Hamiltonian cannot have a nonzeroτ ycomponent. The disentangling of TIfrom Ssymmetry in SAFs should be seen as a virtue: it means that we can useSAFs to carry out rotations about precisely known axes. Bycontrast, the g-type calculation in Fig. 5shows that symmetry- unconstrained rotations can be quite complex. Not only isthe axis of rotation not about a canonical basis vector, butthe axis of rotation changes dynamically as the wave packet travels through the continuum of different local Hamiltonianspresented by the spin texture. Precise rotations appear tobe insufferably difficult to control in such an AFM, so ourprescription to experimentalists and device engineers is to usean SAF when precision is needed. However, SAFs presenttheir own challenges. Unlike pure g-type AFMs, SAFs present a shape anisotropy that may make the realization of uniaxial 134450-8NONABELIAN MAGNONICS IN ANTIFERROMAGNETS PHYSICAL REVIEW B 98, 134450 (2018) FIG. 5. Semiclassical dynamics of a single magnon passing through a Bloch-type domain wall. The horizontal axes represent time, given in picoseconds. Left: integration of Eqs. ( 27) for a wave packet, initially with right-handed polarization η=|0/angbracketright, passing through a domain wall in a synthetic AFM. The SAF material parameters were taken from YIG, and the initial frequency of the wave packet was tuned to result in aπ/2 rotation on the Bloch sphere. The top plot gives the isospin expectation values; bottom, these have been rotated to give the true spin current. Right: the same semiclassical dynamics, domain wall, and YIG parameters are simulated, but the system is assumed to be g-type AFM. We merely substitute the ferromagnetic exchange for the inhomogeneous exchange, and antiferromagnetic for homogeneous exchange. Because TIsymmetry is broken in the g-type configuration, the rotation is unavoidably more complex. Bottom: schematic illustration of a g-type versus a synthetic AFM domain wall. We have illustrated Néel-type walls for simplicity, but the calculation was done for Bloch-type walls. perpendicular magnetic anisotropy (PMA) difficult to main- tain. A possible solution would be to use a-type AFMs. These materials are magnetically ordered at the lattice level, butare AFM ordered in layers, rather than by nearest neighbors.These may present the best of both worlds: their symmetryconstraints will disentangle different rotations, as with anSAF, but they would avoid shape anisotropy issues. Furthermaterials research in this direction is warranted. We emphasize that although our wave-packet theory de- scribes a single semiclassical particle, it nonetheless applies toa global spin wave state [ 74]. Our results for both the domain wall and the magnon field effect transistor (FET) match themicromagnetic simulations of Refs. [ 12] and [ 11] to within 5% error in the driving frequency [ 75]. Formally, the global wave function can be decomposed usefully into wave packetsthrough a Gabor transformation. Standard signal analysisindicates that this use of isospin wave packets as a basis forthe spin wave signal is accurate as long as the grid spacingneeded to sample the spatially inhomogeneous texture doesnot exceed the spread of wavelengths under consideration:/Delta1x c/Delta1kc/lessorequalslant2π.C. Other gates We have carried out explicit example calculations in the previous sections because they can be immediately comparedto results in the literature, unifying these previous investiga-tions under a single formalism and allowing the reader to putour results in context. However, our formalism is far reaching and several other gates can be readily designed. From straightforward calcu- lations of Hand H, one sees that a hard-axis anisotropy will provide a rotation about σ x[76]. Note that this actually implies spin nonconservation since the magnetization (relativeto the local quantization axis) carried by a spin wave corre-sponds to the polar angle of its isospin. Such nonconservationmechanisms have been explored elsewhere [ 77]; here we merely accept that they fall out of the isospin dynamical equations. Meanwhile, an applied magnetic field parallel with the AFM order will provide a rotation about σ zsince it breaks the chiral degeneracy but not the U(1) symmetry ofthe ground state. In this way, a parallel Bfield gives the same effect as a normal Efield used to generate the DMI in Sec. IV A . 134450-9DANIELS, CHENG, YU, XIAO, AND XIAO PHYSICAL REVIEW B 98, 134450 (2018) eiπ τz τxτx τz = iτy FIG. 6. Applying different unitary gates to different branches of a spin wave signal makes the entire Lie algebra of rotational generators available from a set of two, as in the generation of σy from the known σxandσzgates in the figure. By applying σxon one branch and σzon another, one could for instance generate a Hadamard gate. Note that in such a Hadamard gate, the designer must take care to ensure that the overall dynamical phase betweenthe branches is equivalent, so as to avoid wave interference in the output channel. Since the U(1) phase is abelian, though, one need not worry about this in the iτ ygate pictured above. Using a D1=−iτz gate instead of a pure τzgate would generate the same, “extraneous” π/2 phase on both branches. A local modification of the easy-axis anisotropy can raise or lower the local AFMR frequency, and can therefore beused to adjust the relative U(1) phase between two spinwave arms of a multichannel magnonic signal. For instance,such a modification could be used to generate the blue e iπ gate in Fig. 6. There, the sign provided by the U(1) relative phase is crucial for computing the commutator, rather than theanticommutator, of σ xandσz, without the eiπgate, the loop in Fig. 6would simply produce total destructive interference, annihilating the input signal. If one could implement this ina gate-controlled, switchable fashion, then electronic controlover the e iπgate (EAA) and the σzgate (DMI) would turn Fig. 6into a switchable σx↔σygate. The presence of an eiπ gate allows multichannel schemes such as Fig. 6to explore the full Lie algebra structure of SU(2). Options for implementinga switchable e iπgate could include gate-controlled easy-axis anisotropy or a perpendicular (to n) applied Bfield. An espe- cially important use of this gate in a isospin computer would be to compensate the accidental dynamical phase accumulated during the execution of rotational gates. If one is interested in investigating the effects of inter- actions not considered here, one can simply derive the spinwave Hamiltonian in the four-dimensional basis we haveused in this paper and then project it to the operator spaceover the degenerate subspace. One immediately obtains thecorresponding isospin Hamiltonian. We have tried to cover themain classes of interactions in the Supplemental Material [ 52] but more unique interactions such as compass anisotropy [ 78] or honeycomb DMI [ 18] could provide useful interfaces to other isospin operations. V . DISCUSSION Our objective to this point has been to present the reader with a cohesive program for isospin magnonics. We startedby reviewing the idea of chirality and the isospin vectorthat parametrizes it. Our key foundational results were thesemiclassical equations ( 27) describing the isospin dynamics of an AFM magnonic wave packet. With these equations inhand, we described a collection of physical gates, with a focus on voltage gates and domain walls, that could manipulate theisospin in predictable, calculatable ways. As this paper draws to a close, let us reflect on our results and potential avenues for future research. From thecomputing standpoint, recognition of the chiral degree offreedom in AFM magnons is of paramount importance. Usingthe isospin vector as a data carrier represents a paradig-matic improvement, on multiple fronts, over the amplitude-modulating proposals that permeate FM magnonics. First,power management and energy efficiency concerns that arisewhen information is encoded in the FM spin wave powerspectrum become immaterial when the data are carried byAFM isospin. Many of the problems of architecture scal-ing, which plague FM magnonic computing, are significantlyalleviated in AFMs. Second, the isospin carries a higherdimensionality of information. We have seen that this con-siderably broadens the scope of magnon algorithmics. Forinstance, it may be possible to replicate semiclassical quantumcomputing gates in isospin logic. If one is willing to acceptthe use of 2 Nisospin signals in place of 2 Nqubits, and can map between these schemes faithfully, then perhaps one can“classically simulate” nonentangling quantum circuits on aclassical magnonic platform. To this end, a great deal of studyis needed here to properly characterize the power and scopeof isospin computing. Our key contribution to the field of magnonics is the development of a generic, unified formalism for describingthe isospin dynamics in terms of unitary time evolution, aframework with which every physicist is intimately familiar.Together with our mechanical recipe ( A10) for generating the isospin Hamiltonian from the free energy, we expect thatour theory provides a cohesive platform for future theoreticaland experimental investigations into the challenges of isospinmagnonics. Among these challenges are both extensions and applica- tions of our theoretical apparatus. The gates we investigated inSec. IVwere purely one dimensional, and from these simple components one can produce quite sophisticated computingdevices. We have taken pains, however, to keep the spatialdimensionality of our theory generic; one can apply the re-sults of this paper to 2D and 3D systems. Even in quasi-1Dmagnetic strips, two-dimensional textures such as skyrmionsor magnetic vortices could produce interesting effects. Theinteractions between such solitons and AFM spin waves inopen systems is also an open question. Our theory could beused to address these issues. There of course exist magnonic applications outside the spin wave approximation that underlie the theory in thispaper. There, our technical theory may not be a suitabletool, but we hope that our phenomenological description ofthe SU(2) isospin, a concept which relies solely on a thefact that there are two sublattice degrees of freedom with arelative phase between them, will prove useful. Recently, forinstance, AFM auto-oscillators have been proposed [ 79,80]. The dynamical differences between AFM and FM (Klein-Gordon versus Schrödinger) suggest that existing theoriesof magnetic auto-oscillation [ 81] will need to be extended for the AFM case. This has already been done in the caseof easy-plane oscillators, where the magnetization produced 134450-10NONABELIAN MAGNONICS IN ANTIFERROMAGNETS PHYSICAL REVIEW B 98, 134450 (2018) by an oscillation is relatively fixed [ 82]. Other second-order oscillator theories exist, but, especially once they becomecoupled, are often intractable [ 83,84]. They are also usually considered as phase oscillators. Whether these are the mostnatural theories for describing isospin oscillators is an openquestion. In the AFM case, for instance, will the concept of an auto- oscillation bandwidth extend to neighborhoods on the isospinBloch sphere? Such questions, which inherently depend onnonlinearity, call for an understanding of isospin beyond theharmonic spin wave regime. Along a different direction, theadventurous theorist might consider extending our theory toan AFM of more than two sublattices, attempting to derivethe dynamics of an SU( N) isospin. Practical questions remain about the classes of materials that can reliably support the sort of dynamics we have es-poused in this paper. Although our theory readily applies touniaxial AFMs such as MnF 2, AFMs with biaxial or multiax- ial anisotropy may not support low-lying circularly polarizedmodes. Circularly polarized modes may still be used as abasis, especially if the second anisotropy axis is weak, but asthe band splitting becomes greater and greater, our “nearlydegenerate” assumption breaks down. More thorough work isneeded in understanding isospin dynamics in this regime [ 77]. Finally, we note that ferrimagnets satisfy conceptual pre- requisites for an SU(2) isospin, but are usually treated [inthe yttrium iron garnet (YIG) case, at least] merely as low-damping FMs. Given the importance of ferrimagnets to mod-ern magnonics, a theoretical extension of our formalism tothese systems could be of immense interest. Although thetwo modes in ferrimagnets would not be degenerate as theyare in AFMs, and therefore would require more energy forswitching, one might still in principle be able to carry outisospin logical operations. Research into such systems couldbe critical for applied isospin computing. ACKNOWLEDGMENTS We gratefully acknowledge X. Wu, Y . Gao, J. Lan, V . Siddhu, and Z. McDargh for our insightful conversations.This work was supported by the National Science Founda-tion (NSF), Office of Emerging Frontiers in Research andInnovation, under Award No. EFRI-1433496 (M.W.D.), theNSF East Asia and Pacific Summer Institute under AwardNo. EAPSI-1515121 (M.W.D.), and the National NaturalScience Foundation of China under Grants No. 11722430 andNo. 11474065 (W.Y . and J.X.). APPENDIX A: TEMPORAL DYNAMICS FROM THE BERRY PHASE LAGRANGIAN Although we introduced spin wave dynamics via Eq. ( 4), it is possible to bypass the Landau-Lifshitz equation altogether.Instead, we can appeal directly to the Lagrangian of ourclassical field theory on α pmandβ±, given by L[α+,β+,α−,β−]=LBP−F, (A1) where Fis the magnetic free energy and LBPis the so-called Berry phase Lagrangian. The Berry phase Lagrangian is givenby LBP=S /epsilon1dA,B/summationdisplay /Gamma1/integraldisplay/Omega1/Gamma1×m/Gamma1 1−/Omega1/Gamma1·dm/Gamma1 dtddx, (A2) where /epsilon1is the lattice constant and /Omega1is the gauge-dependent orientation of the local Dirac string [ 85–87]. If one takes the variational derivative of LBPbyαandβ, we will find the left-hand side of Eq. ( 5). Even though we have already arrived at this result from the perspective of the Landau-Lifshitzequation, we repeat the derivation here using the Lagrangianpicture. We do so because the Lagrangian formalism should beof greater generality and modularity [ 88], so that others may simply add terms to the Lagrangian and repeat the process weare about to demonstrate. Define λ A=/radicalbig 1−2|α|2,λB=/radicalbig 1−2|β|2, and λm=/radicalbig 1−2|μ|2(we use the convention that |α|2=α+α−and so on). The basic idea in evaluating LBPis simply to make the substitutions RmA=ˆx√ 2[α++α−+λA(μ+μ∗)] +ˆy i√ 2[α+−α−+λA(μ−μ∗)] +ˆz(λAλm−α−μ−μ∗α+), (A3a) RmB=ˆx√ 2[β++β−+λB(μ+μ∗)] +ˆy i√ 2[β+−β−+λB(μ−μ∗)] −ˆz(λBλm−β−μ−μ∗β+) (A3b) into the Lagrangian and expand the result. The “monolithic substitutions” ( A3) are derived in Appendix C. As long as the Lagrangian is a linear operator on the spin wave fields α±and β±, we end up with a collection of terms LBP=LBP 0+LBP 1+LBP 2, (A4) where we have collected terms at zeroth, linear, and quadratic order in the spin wave fields. Linear spin wave theory, uponwhich our formalism is built, cannot support terms at cubicorder or higher, as these would constitute nonlinearities in theequations of motion. Because we are interested in taking functional derivatives with respect to the spin wave fields, we can immediatelyneglect the terms L BP 0[89]. As for LBP 1, we see that functional derivatives of this term would actually introduce inhomoge-neous terms in the equations of motion. The fastidious readerwill find in her derivations that we apparently dohave such terms in our Lagrangian, which do not vanish ap r i o r i . Such terms, if they properly belong to a physical description of thesystem, would seem to imply spontaneous emission of spinwaves since they will let ˙/Psi1take on a nonzero value even when /Psi1is everywhere zero. However, the reader is simultaneously invited to notice that we have introduced more “perturbations” than we canactually control. The problem is that A, which we treat as an independent field, encodes the ground state of the system,as predetermined by anisotropy and DMI. In fact, once the 134450-11DANIELS, CHENG, YU, XIAO, AND XIAO PHYSICAL REVIEW B 98, 134450 (2018) boundary conditions are given, Ais strictly determined by these parameters [ 90]. In equilibrium, one may compute A in principle by minimizing the free-energy functional withrespect to the textural gauge fields/braceleftBigg δF[D,K] ∂Aj μ=0/bracerightBigg μ,j⇒Aequilibrium [D,K]. (A5) Formally, these equations should be solved simultaneously with the actual spin wave equation. On physical grounds,though, we assume that these inhomogeneous terms alwaysvanish when the system under consideration is in equilibriumor else, the system would not in equilibrium, leading to acontradiction. The mathematical mechanism transmitting thisassumption is precisely the set of constraints ( A5). If the system is not in equilibrium, say, if a soliton is moving,then generally speaking it should generate spin waves inho-mogeneously. Although our formalism allows for temporalbehavior of the underlying spin texture, we assume that itis always in quasistatic equilibrium , that is, we neglect any inhomogeneous spin waves it generates. After the above considerations are implemented, we find that we need deal only with the harmonic Lagrangian L BP/mapsto→LBP 2. (A6) Keeping only the quadratic terms in the spin wave modes, keeping terms only to order O(|A|2) in our perturbative expansion, and summing over the sublattices /Gamma1∈{A,B},w e are left merely with LBP 2=S/bracketleftbigg Az tα−α++in 2(α−˙α+−α+˙α−)/bracketrightbigg −S/bracketleftbigg Az tβ−β++in 2(β−˙β+−β+˙β−)/bracketrightbigg ,(A7) where n=1+|μ|2is the effective index of refraction be- tween the local and vacuum values of the spin wave speed,as seen from the Klein-Gordon formulation (see Appendix E). One readily observes the difference of a minus sign separatingsublattices AandB, as well as a minus sign between each field and its conjugate partner. These signs are precisely ourτ z⊗σzfactor from Eq. ( 5). Defining /Psi1=(α+,β+,α−,β−), we find that setting δL/δ ¯/Psi1=0 results in i(τz⊗σz)˙/Psi1=/epsilon1d nSδF δ¯/Psi1−Az t(12⊗σz)/Psi1, (A8) where /epsilon1is the lattice constant. Since we will only keep the quadratic terms in Fby the arguments that lead to Eq. ( A6), we know that δF/δ/Psi1∗is a linear operation on /Psi1that can be written in the form i(τz⊗σz)˙/Psi1=/epsilon1d nSH/Psi1−Az t(12⊗σz)/Psi1 (A9) analogous to Eq. ( 5). In general, the spin wave Hamiltonian is given by H=⎛ ⎜⎜⎜⎜⎝/angbracketleftbigδF ∂α−/vextendsingle/vextendsingleα+/angbracketrightbig/angbracketleftbigδF ∂α−/vextendsingle/vextendsingleβ+/angbracketrightbig/angbracketleftbigδF ∂α−/vextendsingle/vextendsingleα−/angbracketrightbig/angbracketleftbigδF ∂α−/vextendsingle/vextendsingleβ−/angbracketrightbig /angbracketleftbigδF ∂β−/vextendsingle/vextendsingleα+/angbracketrightbig/angbracketleftbigδF ∂β−/vextendsingle/vextendsingleβ+/angbracketrightbig/angbracketleftbigδF ∂β−/vextendsingle/vextendsingleα−/angbracketrightbig/angbracketleftbigδF ∂β−/vextendsingle/vextendsingleβ−/angbracketrightbig /angbracketleftbigδF ∂α+/vextendsingle/vextendsingleα+/angbracketrightbig/angbracketleftbigδF ∂α+/vextendsingle/vextendsingleβ+/angbracketrightbig/angbracketleftbigδF ∂α+/vextendsingle/vextendsingleα−/angbracketrightbig/angbracketleftbigδF ∂α+/vextendsingle/vextendsingleβ−/angbracketrightbig /angbracketleftbigδF ∂β+/vextendsingle/vextendsingleα+/angbracketrightbig/angbracketleftbigδF ∂β+/vextendsingle/vextendsingleβ+/angbracketrightbig/angbracketleftbigδF ∂β+/vextendsingle/vextendsingleα−/angbracketrightbig/angbracketleftbigδF ∂β+/vextendsingle/vextendsingleβ−/angbracketrightbig⎞ ⎟⎟⎟⎟⎠, (A10)where the bra-ket notation simply indicates a functional inner product under which the basis vectors corresponding to α± andβ±are orthogonal. Since the formula we have given for His explicit and straightforward [just make the substitutions (A3) into the free energy and start taking functional deriva- tives of the quadratic sector], we will not bore the reader withpages of algebra by deriving concrete manifestations of Hin the main text. We have provided computer algebra code (in theWolfram language) that derives Hfor an assortment of useful free energies in the Supplemental Material [ 52], with a focus on those free energies needed to explore our various examplesin Sec. IV. We hope readers interested in their own systems will use the recipe described above to generate their own spinwave Hamiltonians, which project onto a Hamiltonian H governing the unitary dynamics of the isospin vector |η/angbracketrightin Sec. III. APPENDIX B: SPIN TEXTURE A principal mechanism [ 91] by which we break U(1) symmetry and mix the chiralities is through the introductionof a nonuniform ground state. To that end, we require a formalstructure for encoding information about the ground state inour dynamical equations. Much of the contemporary literature dealing with spin texture opts to assemble a local coordinate frame, generally{ˆe r,ˆeθ,ˆeφ}, so that the linearization process we used to derive Eq. ( 6) can be recycled in the {ˆeθ,ˆeφ}plane. This amounts to a passive transformation, taking the oscillatory plane of the spinwave fluctuations to align with the local texture. We instead opt to carry out the equivalent active trans- formation , rotating each spin so that its spin wave plane coincides with the global xyplane [ 34,35,44]. A thorough introduction to this technique in the ferromagnetic case isgiven by Ref. [ 34]. In ferromagnets, one simply defines a rotation matrix ˆR(x,t)b y ˆRm 0=ˆz, so that it sends the ground-state configuration m(x,t) at each point to the global ˆzaxis. This rotation matrix gives rise to a gauge field Aμ= (∂μˆR)ˆRT. Formally, ωmay be regarded as a matrix-valued [SO(3)-valued] one-form. One can show in the lattice formalism that Arepresents the infinitesimal rotation 1 +RiRT j=expAijbetween two sites, that is,Ais a generator of rotations. It can thus be decomposed into the standard basis for SO(3): Aμ=Ax μˆJx+Ay μˆJy+Az μˆJz. (B1) Defining R in terms of the Euler angles R= e−iψˆJze−iθˆJye−iφˆJz, we can express the vector fields Aj in terms of the spherical angles describing the spin texture. This is why we have chosen to include minus signs in theexponentials defining R: they show that we first “undo” the spherical angles by sending the azimuth to φ−φ=0 and then sending the polar angle to zero. Taking this conventiongives us A x μ=− sinψ∂μθ+cosφsinθ∂μφ, (B2a) Ay μ=− cosψ∂μθ−sinθsinψ∂μφ, (B2b) Az μ=− cosθ∂μφ−∂μψ. (B2c) 134450-12NONABELIAN MAGNONICS IN ANTIFERROMAGNETS PHYSICAL REVIEW B 98, 134450 (2018) Since only two angles are needed to specify the state of each spin, the third rotation by ψappears to by extraneous, though certainly permitted since it leaves invariant the spin texturenow lying along ˆz. In this sense, it represent the U(1) gauge freedom associated with the U(1) symmetry of a coherentspin. In practice, though, ψwill often notbe a gauge freedom because the U(1) symmetry will often be broken by meansother than the immediate spin texture. If the spin texture hasany misalignment with the easy axis, that is, if there is any deviation from the Néel ground state, then the anisotropyenergy will not be invariant under the rotation by ψ.D M Io r hard-axis anisotropy vectors lying perpendicular to the groundstate would also break this symmetry. The fact that we have chosen ˆzas the global axis to which the texture is rotated means that we will mostly be concernedwith the J zcomponent of the curvature form /Omega1=dA.T h e main consequence is that it is the curl of Az, rather than the curl of AxorAy, which will provide the emergent electromag- netic field, familiar to students of magnetic skyrmions [ 62], generated by a spin texture. The reader may recall that A, and therefore the 3-tuple (Ax,Ay,Az), was supposed to describe an infinitesimal ro- tation between neighboring spins. Such a rotation belongsto a two-dimensional group, and should be describable byexactly two numbers; therefore, we should seek a singleconstraint among our three vector potentials A j. By analyzing the curvature form, one can quickly show that this constraintis ∇×A z=Ax×Ay. (B3) Because ˆzis privileged, it will be convenient to keep using Az in our equations. For AxandAy, though, we define a more concise complex field via Aμ=Ax μ+iAy μ√ 2. (B4) Then, we see that we can substitute the right-hand side of Eq. ( B3)f o r[ 92] ˆz·(Ax×Ay)=Ax xAyy−Ax yAyx=2iA∗ xAy. (B5) We conclude that A∗ xAy, and, therefore, A∗ yAx, is a physically interesting quantity, as it encodes the same emergent electro-magnetic field as the curl of A z. What of the symmetric products A∗ μAμ? It turns out that these elements are also gauge-invariant physical quantities. Inthe general case, one finds A ∗ μAν=gμν+i 2Fμν, (B6) defining Qμν=AμA∗ ν, (B7) where gμν=Ax μAxν+Ay μAyνreduces in spherical angles of the texture to gμν=∂μθ∂μθ+sin2θ∂μφ∂νφ (B8) ⇒g=dθ2+sin2θd φ2. (B9) In other words, gis just the first fundamental form on the sphere. It is the differential line element ds2by which arclengths of the spin texture through spin space are measured. The matrix gis the spherical metric. Qμνis called the quantum geometric tensor . There is very little “quantum” about it in our case, but the nomenclature isalready out there [ 10,93–95]. APPENDIX C: A MONOLITHIC SUBSTITUTION FOR INTRODUCING THE SPIN WA VE FIELDS In the antiferromagnetic case, we choose the rotation ma- trix to send the staggered order to the global ˆz. Generally speaking, mAandmBare not perfectly antiparallel, so after this rotation we will still be left with in-plane components ofthe (rotated) local magnetization. We have already alluded to the fact that our two-level system does not fully describe the spin wave dynamics. This isbecause the basis fields a x+iayandbx+ibyonly represent circular modes. If we want to access modes with linearcomponents, say, fluctuations of a xwithay=0, then our Hamiltonian needs to couple to a linear combination of botha x+iayand its complex conjugate. To address this, we have introduced the fields α+,α−,β+,andβ−to represent our spin wave fluctuations on each sublattice. Now, let us fold these new variables intoour formalism. First, split each rotated field into its slow ( ˜m 0 A and ˜m0 B) and fast ( αandβ) modes, which are perpendicular by construction, and then split the slow modes into thelocal staggered order and local magnetization ( Rn=λ mˆzand ˜m=Rm, withλm=√ 1−m2) of the quasistatic equilibrium spin texture. We have introduced factors of λA=/radicalbig 1−|α|2 andλB=/radicalbig 1−|β|2in order to maintain the normalization of the slow modes ˜mAand ˜mBin the presence of spin wave fluctuations. In other words, we have RmA=λA(˜m+λmˆz)+α, (C1a) RmB=λB(˜m−λmˆz)+β. (C1b) A few notes about the quantities we have just defined. First, ˜mlies in the xyplane since ˜n=λmˆzis perfectly out of plane. Second, though we have opted out of a concern for brevitynot to decorate nandmwith any kind of indicator, keep in mind that these variables only encode the slow modes of the system. All spin wave fluctuations of these quantities havebeen restricted by construction to the excitations αandβ. Notice that we have chosen Rthrough the realignment ofnto avoid choosing a preferred sublattice. Because each m 0 Aandm0 Bis subtly misaligned from nin the presence of a texture, however, our rotated spin wave fluctuations are αand βhave small out-of-plane components. It would be convenient instead to restrict them to the xyplane, so let us now compute exactly what their out-of-plane component is. Since they areorthogonal to the sublattice slow modes by construction, wehave (on the Asublattice, for instance) 0=α·˜m 0 A=α·˜m+λmαz (C2) so that αz=−λ−1 mα·˜mandβz=λ−1 mβ·˜m. Defining aandb as the planar projections of the spin wave fields, we can thensimply write α=a−λ −1 m(a·˜m)ˆzand so on. 134450-13DANIELS, CHENG, YU, XIAO, AND XIAO PHYSICAL REVIEW B 98, 134450 (2018) Finally, we define complex variables α±=(ax±iay)/√ 2, β±=(bx±iby)/√ 2, and μ=(˜mx+i˜my)/√ 2. These four complex variables will be treated as independent; with theunderstanding that the real part must be taken before extract-ing physical quantities from this complex formalism, fourreal degrees of freedom are maintained. Taking all of thesedefinitions together, we have our two monolithic substitutions ˜m A=ˆx√ 2[α++α−+λA(μ+μ∗)] +ˆy i√ 2[α+−α−+λA(μ−μ∗)] +ˆz(λAλm−α−μ−μ∗α+), (C3a) ˜mB=ˆx√ 2[β++β−+λB(μ+μ∗)] +ˆy i√ 2[β+−β−+λB(μ−μ∗)] −ˆz(λBλm−β−μ−μ∗β+). (C3b) With these quantities in hand, the free energy can be computed explicitly, and by taking variations by αandβof the conse- quent Lagrangian, we can ultimately determine the spin waveequation of motion. APPENDIX D: A MORE DETAILED DISCUSSION OF NONABELIAN WA VE-PACKET THEORY Before computing a phase-space Lagrangian governing the semiclassical dynamics, we establish some self-consistencyproperties of the wave packet that will provide for usefulidentities during our calculation. 1. Normalization condition First, let us enforce a normalization condition on |W/angbracketright, given by /angbracketleftW|τz⊗σz|W/angbracketright=1. (D1) This leads to a normalization condition for the η, namely, that /angbracketleftW|τzσz|W/angbracketright=(−1)j/integraldisplay dqdkw∗ kwqη∗ j,kηj,q/angbracketleftbig ψj k/vextendsingle/vextendsingleσz/vextendsingle/vextendsingleψj q/angbracketrightbig (D2) =(−1)2j/integraldisplay dq|wq|2|ηj|2(D3) ⇒1=/angbracketleftη|η/angbracketright. (D4) Equation ( D4) suggests that, unlike |W/angbracketrightand|/Psi1/angbracketright,|η/angbracketrightwill be subject to a traditional, Euclidean Schrödinger dynamics.Recall that the σ zinner product in the two-level system did not provide a useful normalization condition, as a result of theinternal hyperbolic geometry. It is only here in the four-levelsystem, where the signs from internal and external geometriescancel each other, that we arrive at a normalizable spin wavedensity (rather than spin density). The calculational patterns from Eq. ( D2) detail the internal derivations of wave-packet theory. We briefly outline thelogical flow of the computation for readers unfamiliar withthe formalism. The key stages needed to reduce any of our wave-packet inner product are as follows: (1) Use the fact that the wave vectors are “block-diagonal” [in the sense of Eq. ( 11)] to reduce the τ zt oas i n g l e( −)j, and to avoid any cross terms between eigenvectors from differentbands. (2) Establish an inner product of the internal band struc- ture (e.g., /angbracketleftψj k|σz|ψj q/angbracketright). Extract the translation operators to find a factor of exp[ i(q−k)x] and use the inner product, a real-space integral over the sample, to produce a δd(q−k). (3) Carry out one of the momentum-space integrals to activate the Dirac delta function and reduce the problem toa single Brillouin zone. (4) If the inner product from step 2 was a normalization condition of the internal geometry, then it produced a ( −) j that, together with the sign from τ, cancels to give positive unity. Otherwise, there is a nontrivial inner product /angbracketleftη|ˆO|η/angbracketright that must be tracked. (5) Integrate by parts, use product rules, and use the normalization condition as necessary to manifest a factor of|w q|2in the integrand. Interpret |wq|2/mapsto→δd(q−qc) to carry out the final integral. Before evaluating the Lagrangian proper, we have one more useful identity to compute: the expectation value of theposition operator. 2. Position operator Let us consider the self-consistency condition for the wave- packet center. This means that we require the observable ˆxto be diagonal in the wave-packet basis, with eigenvalue xcfor wave packet |W(xc,qc,η,t)/angbracketright. Therefore, /angbracketleftW|(τz⊗σz)ˆx|W/angbracketright=xc/angbracketleftW|(τz⊗σz)|W/angbracketright. (D5) The bra-ket on the right then reduces to unity by the wave- packet normalization. Before we proceed, let us define the nonabelian Berry connection ˆaj μ=/parenleftbigg/angbracketleftbig /Psi10 q/vextendsingle/vextendsingleiσz∂μ/Psi10 q/angbracketrightbig 0 0 −/angbracketleftbig /Psi11 q/vextendsingle/vextendsingleiσz∂μ/Psi11 q/angbracketrightbig/parenrightbigg , (D6) wherein μis a coordinate of the phase-space dynamics. We will therefore be concerned alternatively with ˆax,ˆaq, and ˆat. Calculating the left-hand side of Eq. ( D5) using the matrix elements of the position operator from Ref. [ 96], we find [ 97] xc=/angbracketleftW|(τz⊗σz)ˆx|W/angbracketright (D7) =/angbracketleftη|ˆaq|η/angbracketright+∂γc ∂q. (D8) In deriving Eq. ( D8), we see our first example of a noncancel- lation between σzandτz. The Berry connection is not merely the normalization condition /angbracketleft/Psi1|σz|/Psi1/angbracketright, and therefore cannot produce the sign needed to cancel the ( −)jfactor. Instead, these signs have all been contained within ˆa. 3. Extracting the electromagnetic Lagrangian In Sec. II B, we introduced the collection of vector potentials Ajwhich encode the spin texture. Generally 134450-14NONABELIAN MAGNONICS IN ANTIFERROMAGNETS PHYSICAL REVIEW B 98, 134450 (2018) speaking, the introduction of spin texture breaks the contin- uous translational symmetry of the (continuum limit of the)Néel ground state. Since the A jare not necessarily gauge invariant, though, one expects that the translational proper-ties of the vector potentials need not align in general withtranslational properties of the physical system. The situationis similar to introducing an electromagnetic vector potential instandard quantum mechanics; there, the canonical momentum operator −i∂ xmust be adjusted to the mechanical momentum operator, −i∂x−ieA, where only the latter is properly con- served. Even without explicitly computing the spin wave Hamilto- nian, we expect that the kinetic energy term we explored inthe two-level system will appear to undergo a sort of Peierlssubstitution by A z. With this in mind, we will now perform a gauge transformation, removing the Azfrom the kinetic energy terms and collecting it into a new Lagrangian termwhich will completely encapsulate the emergent electromag-netic interaction. Define the matrix G=exp[−i(τ z⊗12)(Az·x)]. (D9) Then, inserting factors of G†Ginto the Lagrangian, we have /angbracketleftW|G†G/parenleftbigg iτzσzd dt−H−Az tσz/parenrightbigg G†G|W/angbracketright, (D10) where the wave packets and Hamiltonian are, at this point, still in the original gauge choice, and the brackets represent thematrix element of the operator on the diagonal in the wave-packet basis. To save space, we have removed the explicittensor product notation. We leave it to the reader to interpretτ z/mapsto→τz⊗12andσz/mapsto→12⊗σzas the context demands. The value of the transformation by Gis not only in an in- ternal simplification of H, but also in elegantly extracting the emergent electromagnetic Lagrangian early in the calculation.One readily sees after carrying out the time derivative that theLagrangian is /angbracketleft˜W|/parenleftbigg −(σ z˙Az·ˆx)−Az tσz+iτzσzd dt−˜H/parenrightbigg |˜W/angbracketright,(D11) where ˜H=GHG†. Collecting the first two terms together, this can be naturally split into three components: L=LEM+Ldt+LH. (D12) These components represent the emergent electromagnetic, dynamical, and free-energy sectors of the spin wave equation. The gauge transformation has also affected the wave packet itself. Concretely, the wave packet is now |˜W/angbracketright:=G|W/angbracketright=/integraldisplay dqw(q,t) ×[˜η0(q,t)|/Psi10(q,t)/angbracketright+˜η1(q,t)|/Psi11(q,t)/angbracketright],(D13) where |˜η/angbracketright=(˜η0,˜η1) locates the gauge-transformed wave packet within the degenerate subspace. From Eq. ( D11), we see the need to evaluate −˙Az·/angbracketleft˜W|(12⊗σz)ˆx|˜W/angbracketright−Az t/angbracketleft˜W|12⊗σz|˜W/angbracketright (D14)the first of which terms will invoke a calculation analogous to those in Appendix D2.W eh a v e /angbracketleft˜W|(12⊗σz)ˆx|˜W/angbracketright=/angbracketleftη|τzˆaq|η/angbracketright+/angbracketleftη|σz|η/angbracketright∂γc ∂q.(D15) Substituting in the self-consistency condition ( D8)o nxcfor theγcderivative, we end up with LEM=− ˙Az·/Gamma1q−χ/parenleftbig˙Az·xc+Az t/parenrightbig , (D16) where χ=/angbracketleftη|τz|η/angbracketright, and/Gamma1qis the covariance /angbracketleftη|τzˆaq|η/angbracketright− /angbracketleftη|τz|η/angbracketright/angbracketleftη|ˆaq|η/angbracketright. Note that we have simplified these terms back to η, rather than ˜η, since Gcommutes with τzand ˆaμ. Interpreting χas a charge, the second half of Eq. ( D16) is just the interaction Lagrangian for a charged particle in anelectromagnetic field [ 98]. Note that, in particle physics, there is also a sense in which the electromagnetic charge is a τ z expectation value: one can rotate the isospin of a positively charged proton, through some SU(2) “isospin” space, to theneutrally charged neutron. That we have a similar sort ofcontinuum-valued (emergent) charge is our motivation foremploying the “isospin” nomenclature in our definition of η. 4. Time derivative term Although we have already encountered a few time deriva- tives without comment in the wave-packet theory, a fewwords are certainly in order concerning the time variable. Itstreatment is one of the most delicate and subtle parts of wave-packet theory, and it is easy to make dangerous systematicerrors without a proper treatment. For the reader interested inreplicating our derivation, we have given some notes on thematter in Appendix D7. The time derivative term L dtin the Lagrangian is i/integraldisplay dqdk/angbracketleftbig /Psi1i q/vextendsingle/vextendsingle˜η∗ i,qw∗ q(τzσz)d dt/parenleftbig wk˜ηj,k/vextendsingle/vextendsingle/Psi1j k/angbracketrightbig/parenrightbig . (D17) Since our eigenvectors are themselves block diagonal, and sinceτz⊗σzas well as Gare both diagonal, we know there can be no terms connecting i/negationslash=j. The first term (on wk) in a product rule of expansion of Eq. ( D17)i ss i m p l y ∂tγc. The next term, on ˜ ηj,k, gener- ates the isospin dynamics, and the final term gives rise tomatrix-valued Berry connections. All together, these termsbecome L dt=/angbracketleft˜η|˙xc·ˆax+˙qc·ˆaq+ˆat+i∂t|˜η/angbracketright−˙qc·xc.(D18) We have used the self-consistency condition to replace the Berry phase term ∂tγcwithxc−/angbracketleftη|ˆaq|η/angbracketright. 5. Hamiltonian terms Finally, we have the terms coming from the spin wave Hamiltonian itself. These are LH=− /angbracketleftW|H|W/angbracketright (D19) =−1 ns/integraldisplay dq|wq|2[˜η∗ i˜ηj/angbracketleft/Psi1i|˜H|/Psi1j/angbracketright].(D20) Let us define the matrix ˜H=/parenleftBigg/angbracketleftbig /Psi10 c/vextendsingle/vextendsingle˜H/vextendsingle/vextendsingle/Psi10 c/angbracketrightbig/angbracketleftbig /Psi10 c/vextendsingle/vextendsingle˜H/vextendsingle/vextendsingle/Psi11 c/angbracketrightbig /angbracketleftbig /Psi11 c/vextendsingle/vextendsingle˜H/vextendsingle/vextendsingle/Psi10 c/angbracketrightbig/angbracketleftbig /Psi11 c/vextendsingle/vextendsingle˜H/vextendsingle/vextendsingle/Psi11 c/angbracketrightbig/parenrightBigg . (D21) 134450-15DANIELS, CHENG, YU, XIAO, AND XIAO PHYSICAL REVIEW B 98, 134450 (2018) One may think of Has a projection of the original Hinto the two-dimensional orthochronous degenerate subspace that weare now calling “isospin space,” the copy of SU(2) in which η resides. Defining the embedding E †=|/Psi10/angbracketright/angbracketleft0|+|/Psi11/angbracketright/angbracketleft1| (D22) which sends vectors in the isospin subspace to their represen- tation in parent 4 ×4 Hilbert space space, His merely ˜H=E˜HE†. (D23) This Hermitian matrix will govern the dynamics of |˜η/angbracketrightin the semiclassical dynamics we are about to describe. Thetotal contribution from these energy terms to the wave-packetLagrangian is, simply, L H=− /angbracketleft ˜η|˜H|˜η/angbracketright. (D24) 6. Phase-space EOM Let us take stock of our progress. We have a Lagrangian of three terms, which have been reduced to LEM=− ˙Az·/Gamma1q−χ/parenleftbig˙Az·xc+Az t/parenrightbig , (D25a) Ldt=/angbracketleft˜η|˙xc·ˆax+˙qc·ˆaq+ˆat+i∂t|˜η/angbracketright−˙qc·xc,(D25b) LH=− /angbracketleft ˜η|˜H|˜η/angbracketright. (D25c) Now, we can take variations against xc,qc, and|η/angbracketrightto derive semiclassical equations of motion (EOM). First, let us find the force equation by taking a variation against xc. For the Lorentz force term, we unsurprisingly have δLA δxμ c=−χ/bracketleftbig ∂tAz μ+∂xμ cAz t+˙xc·∂xμ cAz−(˙xc·∇)Az μ/bracketrightbig (D26) =χE+χ˙xc×B, (D27) where we define the fields E=−∇Az t−∂tAzandB=∇× Azin the obvious ways. The time derivative term meanwhile gives δLdt δxμ c=− ˙qc+/angbracketleftbig /Omega1xx μν/angbracketrightbig˙xν c+/angbracketleftbig /Omega1xq μν/angbracketrightbig˙qν c+/angbracketleftbig /Omega1xt μ/angbracketrightbig , (D28) where /angbracketleftbig /Omega1αβ μν/angbracketrightbig =/angbracketleftη|/parenleftbigg∂ˆaβν ∂αμ−∂ˆaαμ ∂βν/parenrightbigg |η/angbracketright (D29) is the η-density trace of the nonabelian Berry curvature, as discussed in Ref. [ 53]. Finally, we have a contribution from the gauged Hamilto- nian. In most cases we consider in this paper, no such termssurvive at O(|A| 2); the terms that might nominally survive are those wrapped encoded in LEM. Examples of terms that may survive and not be included in LEMcould include spatially de- pendent anisotropy or DMI, arising from, e.g., wedge-shapedlayers in magnetic heterostructures. Taking this term and theLorentz force together, the force equation is ˙q c=Tr/bracketleftbigg ˆρ/parenleftbigg τz(E+˙xc×B)−∂E ∂xc/parenrightbigg/bracketrightbigg , (D30) where Eis the energy of the unperturbed degenerate bands, and where we have defined the density operator ˆρ=ρ0|0/angbracketright/angbracketleft0|+ρ1|1/angbracketright/angbracketleft1|. (D31)Now, we turn to the velocity equation. The results are little different from what we would expect from standard non-abelian wave-packet theory, giving us the classical velocitytogether with Berry-curvature-induced transverse velocities ˙x c=Tr[ ˆρ(∂qE+/Omega1qq˙qc+/Omega1qx˙xc+/Omega1qt)]. (D32) Now, we turn to the most interesting equation of motion, generated by the variation again /angbracketleftη|. This generates terms of the form δLEM δ˜η∗=− ˙Az·δ/Gamma1q δ˜η∗−(˙Az·xc)τz˜η−Az tτz˜η, (D33) δ/Gamma1q δ˜η∗=τzˆaq˜η−τz(/angbracketleft˜η|ˆaq|˜η/angbracketright)˜η−χˆaq˜η, (D34) δLdt δ˜η∗=/bracketleftbigg ˙qc·ˆaq+i∂ ∂t/bracketrightbigg ˜η, (D35) δLH δ˜η∗=− H˜η. (D36) The final general equation of motion is i/parenleftbiggd dt+At/parenrightbigg η=/bracketleftbig H+τzAz t+ˆVχ/bracketrightbig η, (D37) where H=EHE†(note that we have removed the gauge transformation G),Ais the time-covariant connection on phase space At=˙qq·ˆaq+˙xc·ˆax+ˆat, (D38) Aij t=/angbracketleftbig ψi c/vextendsingle/vextendsingleiσzd dt/vextendsingle/vextendsingleψj c/angbracketrightbig , (D39) and ˆVχis a nonlinear term deriving from /Gamma1q.I ti sg i v e nb y ˆVχ=− ˙Az·(τzˆaq−τzˆPηˆaq−ˆaqˆPητz), (D40) where ˆPη=|η/angbracketright/angbracketleftη|is the projector onto the isospin state, and as before the dot product (with ˙Az) is taken with the subscript inˆaq. This potential is nonlinear in the sense that, through ˆPη, it depends quadratically on the current state, and the resultingterm in the Hamiltonian has the schematic form |ψ| 2|ψ/angbracketright. However, this nonlinear term balances precipitously on theedge of irrelevance. ˙A zis itself O(A2), so this term survives only if ˆaqisO(A0). Although there is no reason (to our knowledge) this could not happen in principle, none of theconcrete systems we consider later in the paper can activatethis term. What is more, the term would seem only relevant inthe case of a moving spin texture, so that ˙A zis nonzero. Such a term may be of interest for those working in the dynamicsof AFM solitons, but we leave that to future research. 7. Notes on time derivatives in wave-packet theory There are two variables, xandq, that have been float- ing around as dummy variables of integration in some ofour calculations. Several functions, such as the wave-packetenvelope a q=a(q,t) or the Bloch eigenvectors eiqxu(q,t), are functions of both q(orx) and time. For these functions, there is no difference between a total time derivative and apartial time derivative because qandxare clearly independent 134450-16NONABELIAN MAGNONICS IN ANTIFERROMAGNETS PHYSICAL REVIEW B 98, 134450 (2018) variables (merely coordinates of a space) that do not, them- selves, possess any temporal dynamics. On the other hand, the gauge field Az=Az(xc,t)w a s originally, and will always be, evaluated at xcin its spatial argument. This xcis a dynamical variable, which does de- pend on time and has dynamics. Our notation, which followsRef. [ 55], is that a partial time derivative of such a function acts only on the second argument slot, where there is anexplicit time dependence. A total time derivative, on the otherhand, would include the time dependence through x c, so that d dt=∂ ∂t+˙xc·∂ ∂xc. (D41) So far, our discussion has perhaps clarified the notation, but is by no means unusual. The delicacy of these operations inwave-packet theory occurs when evaluation of a wave-packetexpectation value promotes a function in the integrand, whereit may have possessed only an explicit time dependence, to afunction of q c(t), due to the firing of the Dirac delta function |aq|2. The question is as follows: Should the time derivative under the integrand be lifted to a total time derivative or apartial time derivative once the function acquires a new timedependence in the phase-space coordinate arguments x c(t) andqc(t)? The answer is that we must promote it to a partial time derivative. The original, physical meaning of such a timederivative in the integrand was to ask how, at any given pointin space, a function changed with time. We are concerned withthe function’s temporal behavior , not the temporal behavior of the combined wave-packet/function system. From a differentperspective, we note that we are certainly free to take the timederivative as early as possible. Suppose we “carry out” thetime derivative in the integrand by replacing ∂ tf(t,q) with its formal derivative F(t,q). Now, Fis just a function which we have determined in principle before ever introducing thephase-space path ( x c,qc), so after firing the Delta function wesimply have F(t,qc(t)). Clearly, F(t,qc(t))=∂tf(t,qc(t)), with the derivative only in the first argument. APPENDIX E: STAGGERED ORDER Suppose we changed the basis of Eq. ( 6) by a Hadamard matrix M=1 2/parenleftbigg 11 1−1/parenrightbigg , (E1) sending αandβtoδm=mx+imyandδn=nx+iny, respectively. Neglecting anisotropy for the moment, the re-sulting Schrödinger equation on ˆhis iσ xd dt/bracketleftbigg δm δn/bracketrightbigg =1 2/bracketleftbig Z+σz(Z−J∇2)/bracketrightbig/bracketleftbigg δm δn/bracketrightbigg . 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