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PhysRevB.101.144414.pdf

PHYSICAL REVIEW B 101, 144414 (2020) Higher-order exceptional points in ferromagnetic trilayers Tianlin Yu, Huanhuan Yang, Lingling Song, Peng Yan ,*and Yunshan Cao† School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China (Received 7 February 2020; revised manuscript received 17 March 2020; accepted 19 March 2020; published 10 April 2020) Magnetometers with exceptional sensitivity are highly demanded in solving a variety of physical and engineering problems, such as measuring Earth’s weak magnetic fields and prospecting mineral deposits andgeological structures. It has been shown that the non-Hermitian degeneracy at exceptional points (EPs) canprovide a new route for that purpose, because of the nonlinear response to external perturbations. One recentwork [H. Yang et al. P h y s .R e v .L e t t . 121,197201 (2018 )] has made the first step to realize the second-order magnonic EP in ferromagnetic bilayers respecting the parity-time symmetry. In this paper, we generalize the ideato higher-order cases by considering ferromagnetic trilayers consisting of a gain, a neutral, and a (balanced-)losslayer. We observe both second- and third-order magnonic EPs by tuning the interlayer coupling strength, theexternal magnetic field, and the gain-loss parameter. We show that the magnetic sensitivity can be enhanced bythree orders of magnitude comparing to the conventional magnetic tunneling junction-based sensors. Our resultspave the way for studying high-order EPs in purely magnetic system and for designing magnetic sensors withultrahigh sensitivity. DOI: 10.1103/PhysRevB.101.144414 I. INTRODUCTION The magnetometer, for measuring the intensity of mag- netic fields, was first created by Gauss in 1833 [ 1] and has achieved tremendous progress since then. It has been widelyutilized in mineral explorations [ 2,3], accelerator physics [ 4], archaeology [ 5], mobile phones [ 6], etc. A long-term goal in the community is to pursue magnetometers with ultra-high sensitivity. Conventional techniques in magnetic sensorsencompass many aspects of physics. For example, the fluxgate magnetometer works due to the nonlinear characterof soft magnetic materials when they are saturated [ 7,8]. Magnetoresistive devices typically are made of thin stripsof permalloy whose electrical resistance varies with externalmagnetic fields [ 9]. Although different magnetometric devices are designed based on different physical mechanisms, theyshare a general rule that the variation of the order parameterlinearly varies with respect to the magnetic field. Presently,ultra-high-sensitive magnetometers such as superconductingquantum interference devices can reach a magnetic sensitivityof 1 fT /Hz 1/2, but they require an extremely low working temperature and an oversized volume [ 10,11]. Seeking a solid- state, small size, room-temperature magnetometer with ultra-high sensitivity is thus one central issue. Recently, it has beendemonstrated that the peculiar non-Hermitian degeneracy inmagnetic structures [ 12–16] may provide a promising way to solve the problem. The Hamiltonian obeying the parity-time ( PT) symmetry constitutes a special non-Hermitian system, which is invariant *yan@uestc.edu.cn †yunshan.cao@uestc.edu.cnunder combined parity Pand time-reversal Toperations. It has attracted a lot of attention due to both the fundamen-tal interest in quantum theory [ 17–19] and the promising application in many fields [ 20–22], such as optics [ 23–25], tight-binding modeling [ 26,27], acoustics [ 28,29], electronics [30,31], and very recently in spintronics [ 12–16]. A PT- symmetric Hamiltonian could exhibit entirely real spectra anda spontaneous symmetry breaking accompanied by a real-to-complex spectra phase transition at the exceptional point(EP) where two or more eigenvalues and their correspondingeigenvectors coalesce simultaneously. In the vicinity of theEP, the eigenfrequency shift follows the 1 /Npower law of the external perturbation, where Nis the order of the EP. Such a feature can significantly enhance the sensitivity and has beenobserved by several experiments [ 20,32–35]. PT symmetry and EP in magnetic systems are receiving growing recent interest. In a simple bilayer structure of twomacrospins with balanced gain and loss, the second-order EP(EP2) was observed at a critical Gilbert damping constant[12]. In Ref. [ 36], it was proposed to realize the pseudo- Hermiticity in a cavity magnonics system with the third-orderEP (EP3). By taking the spin-wave excitation into account,some of the present authors reported a novel ferromagnetic-to-antiferromagnetic (AFM) phase transition at the EP thatdepends on the magnon’s wave vector [ 13]. In Ref. [ 14], an exceptional magnetic sensitivity was predicted in the vicinityof the EP3 for PT-symmetric cavity magnon polaritons. However, high-order EPs in purely magnetic /magnonic sys- tem is yet to be explored. In this work, we propose a ferromagnetic trilayer structure consisting of a gain, a neutral, and a loss layer to achieve theEP3. We show that, in the vicinity of the EP3, the separationof eigenfrequencies follow a power law /Delta1ω EP3∝/epsilon11/3. Here, 2469-9950/2020/101(14)/144414(8) 144414-1 ©2020 American Physical SocietyYU, YANG, SONG, YAN, AND CAO PHYSICAL REVIEW B 101, 144414 (2020) Gain LossNeutralGain LossNeutralm m m Oxyz(a) (b) FIG. 1. (a) Illustration of three exchange-coupled macrospins consisting of a gain (red), neutral (gray), and (balanced-)loss (blue) spin. (b) Schematic plot of a ferromagnetic heterostructure with a gain, neutral, and loss layer, denoted by red, gray, and blue colors,respectively. The magnetizations of all spins are initially along ˆ x direction. the perturbation /epsilon1comes from the disturbing magnetic field. We find mode-dependent EPs when the spin-wave excitationis allowed by including the intralayer exchange coupling.A ferromagnetic-to-antiferromagnetic phase transition is ob-served when the PTsymmetry is broken. Our results suggest a promising way to realize higher-order non-Hermitian degen-eracy in a purely magnetic system and to design magnetome-ter with ultrahigh sensitivities. The paper is organized as follows. Section IIgives the macrospin model. The condition for observing EP3 is analyti-cally derived. The one-half and one-third power law aroundEP2 and EP3 are demonstrated, respectively. The effect ofnoise on the magnetic sensitivity is analyzed as well. InSec. III, we extend the idea to ferromagnetic trilayers by allowing spin-wave excitations. Discussion and conclusionsare drawn in Sec. IV. II. MACROSPIN MODEL We first consider a ternary macrospin structure shown in Fig. 1(a). The Hamiltonian contains the Zeeman energy, magnetic anisotropy, and exchange coupling: H=−/summationdisplay nB·Mn−/summationdisplay nKn 2/parenleftbig mx n/parenrightbig2−λμ0M2·(M1+M3), (1) where Mn(mn=Mn/Mn) is the spin (unit spin) with subscript index nlabeling the nth layer ( n=1,2,3),Mnis the saturated magnetization, B=Bˆxis the external magnetic field applied on the whole structure, Kn>0 is the uniaxial anisotropy, λ> 0 is the ferromagnetic exchange-coupling strength be- tween two adjacent layers, and μ0is the vacuum permeability. The top and bottom layers are assumed to be the same materialbut with opposite Gilbert damping parameters, to guaran-tee the PT symmetry. The coupled magnetization dynamics is described by the Landau-Lifshitz-Gilbert (LLG) equation[37,38]: ∂m 1 ∂t=−γm1×Beff,1−αm1×∂m1 ∂t, (2a) ∂m2 ∂t=−γm2×Beff,2, (2b) ∂m3 ∂t=−γm3×Beff,3+αm3×∂m3 ∂t, (2c)where γis the gyromagnetic ratio and α> 0 is the Gilbert constant employed as the balanced gain-loss parameter. Theeffective magnetic fields read: B eff,1=Bˆx+K1 M1mx 1ˆx+λμ0M2m2, (3a) Beff,2=Bˆx+K2 M2mx 2ˆx+λμ0M1(m1+m3), (3b) Beff,3=Bˆx+K1 M1mx 3ˆx+λμ0M2m2. (3c) For small-amplitude spatiotemporal magnetization preces- sion, we assume mn=ˆx+my nˆy+mz nˆzwith|my,z n|/lessmuch1. By substituting Eqs. ( 3) into Eqs. ( 2), and introducing ψn=my n− imz n, we obtain (i+α)˙ψ1=ωB1ψ1−ωλ2ψ2, (4a) i˙ψ2=−ωλ1ψ1+ωB2ψ2−ωλ1ψ3, (4b) (i−α)˙ψ3=−ωλ2ψ2+ωB1ψ3, (4c) where ωB1=γ(B+K1/M1+λμ0M2),ωB2=γ(B+ K2/M2+2λμ0M1),ωλ1=γλμ 0M1, and ωλ2=γλμ 0M2. Imposing a harmonic time dependence ψn=φnexp(−iωt), we have the secular equation: ωφ=Hφ, (5) withφ=(φ1,φ2,φ3)T, and H=⎛ ⎜⎝ωB1 1−iα−ωλ2 1−iα0 −ωλ1ωB2−ωλ1 0 −ωλ2 1+iαωB1 1+iα⎞ ⎟⎠. (6) A. Eigensolutions The eigenfrequencies are determined by the zeros of the characteristic polynomial of ( 6): aω3+bω2+cω+d=0, (7) with a=−(1+α2)<0,b=2ωB1+(1+α2)ωB2,c= 2ωλ1ωλ2−ω2 B1−2ωB1ωB2, and d=ω2 B1ωB2−2ωB1ωλ1ωλ2. It is known that if and only if A=B=0, the equation has a triple real root, where A=b2−3acandB=bc−9ad.W e therefore arrive at the constraint supporting the EP3: (2ωB1+ωB2+α2ωB2)2+3(1+α2)/parenleftbig 2ωλ1ωλ2−ω2 B1 −2ωB1ωB2/parenrightbig =0, (8a) (2ωB1+ωB2+α2ωB2)/parenleftbig 2ωλ1ωλ2−ω2 B1−2ωB1ωB2/parenrightbig +9(1+α2)/parenleftbig ω2 B1ωB2−2ωB1ωλ1ωλ2/parenrightbig =0. (8b) To obtain reasonable αandB, we note that the difference between ωB2andωB1should be close to ωλ2. In the calcula- tions, we thus choose the annealed and deposited Co 40Fe40B20 [39,40] as the top- (bottom-) and the middle-layer ma- terials, with the saturation magnetization M1=1.098× 106A/m and M2=1.003×106A/m, and the anisotropy constant K1=4.36×105J/m3and K2=1.07×105J/m3, respectively. For each λ, we numerically calculate the allowed magnetic field Band gain-loss parameter α, as shown in Fig. 2(a) 144414-2HIGHER-ORDER EXCEPTIONAL POINTS IN … PHYSICAL REVIEW B 101, 144414 (2020) EP3EP2(b) (a) FIG. 2. (a) Parametric space for EP3. The gray region marks the allowed values of the external magnetic field B, the gain-loss parameter α, and the interlayer coupling strength λ. (b) Evolution of eigenvalues as the gain-loss parameter αforλ=0.18 and B= 29.2 mT. The solid and dashed curves represent the real and imagi- nary parts of eigenfrequencies, respectively. with the black and blue curves, respectively. We note that B>max{−K1/M1−λμ0M2,−K2/M2−2λμ0M1}should be satisfied to guarantee a stable ferromagnetic ground state andthe typical value of αranges from 0 to 1, leading to the reasonable parameters labeled by the gray region in Fig. 2(a). From Fig. 2(a), we can see that the critical α(magnetic field) decreases (increases) with the increasing of λ. Figure 2(b) shows a typical evolution of eigenvalues as the gain-lossparameter αforλ=0.18 and B=29.2 mT, in which both the EP2 and EP3 emerge, marked by green and red dots,respectively. Next, we discuss the magnetic sensitivity in the vicinity of EP2 and EP3. The gain-loss parameters α EP2=0.399 and αEP3=0.652 are chosen in the following calculations. B. Perturbing the top spin Supposing a perturbation /epsilon1only on the top macrospin, induced by an external magnetic field B/epsilon1, i.e.,/epsilon1=γB/epsilon1/ωλ2, we modify Eq. ( 5)t o : /Omega1φ=H/epsilon1φ, (9) with H/epsilon1=ωλ2⎛ ⎜⎜⎝ωB1ω−1 λ2+/epsilon1 1−iα−1 1−iα0 −ωλ1 ωλ2ωB2 ωλ2−ωλ1 ωλ2 0 −1 1+iαωB1ω−1 λ2 1+iα⎞ ⎟⎟⎠. (10) To highlight the key role played by the order of the EP, we first investigate the effect of the perturbation on a single-layerferromagnet with /epsilon1ranging from 10 −10to 10−2. We find that the ferromagnetic resonance (FMR) frequency varies linearlywith respect to the perturbation plotted in Figs. 3(a) and 3(b), as naturally expected. Then, we evaluate the variation of eigenvalues with respect to the perturbation near the EP2and EP3, as depicted in Fig. 3(c)and Fig. 3(e), with the mode splitting on a logarithmic scale being plotted in Fig. 3(d) and Fig. 3(f), respectively. We numerically demonstrate that the separation of frequencies scales as /epsilon1 1/2and/epsilon11/3for EP2 and EP3, respectively. To have a quantitative comparison,we choose /epsilon1=0.005 and calculate the frequency difference. We identify 0.03 GHz, 0.14 GHz, and 1.23 GHz shift forthe normal FMR, EP2, and EP3 mode, respectively. The(e) 1 2 31 2(a) (c)0(b) slope=1.0 (d) slope=0.50 (f) slope=0.33 FIG. 3. (a) The FMR frequency for a single-layer ferromagnet as a function of the perturbation /epsilon1. (b) The frequency shift /Omega1−/Omega10is depicted in logarithmic coordinates, with the slope being 1. (c) Thevariation of eigenfrequencies near the EP2 as a function of the pertur- bation. (d) Frequency splitting Re( /Omega1 1−/Omega12) on a logarithmic scale, with the one-half slope indicating the /epsilon11/2response. (e) The splitting of eigenfrequencies near EP3 vs the perturbation. Solid and dashed curves represent numerical and analytical results, respectively. (f) Frequency splitting of Re( /Omega12−/Omega13) on a logarithmic scale, with the slope approximately being 0.33, suggesting the /epsilon11/3response. sensitivity is thus enhanced by 4.7 and 41 times around EP2 and EP3 with respect to the FMR mode, respectively. In the following, we analytically derive the frequency split- ting near the EP3, by perturbatively solving the characteristicequation of H /epsilon1. Based on the Newton-Puiseux series [ 41], we obtain: /Omega1n ωλ2=c0+cn1/epsilon11 3+cn2/epsilon12 3+cn3/epsilon1, (11) with complex coefficients cni(i=1,2,3) [42] and c0=2.28. Solutions ( 11) are depicted with dashed orange curves in Fig. 3(e), showing a nice agreement with numerical results. The (real part) frequency splitting between /Omega11,/Omega12, and/Omega13is thus Re(/Omega11−/Omega12)=ωλ2(0.4/epsilon11 3−0.62/epsilon12 3−0.64/epsilon1), Re(/Omega11−/Omega13)=ωλ2(1.53/epsilon11 3−0.61/epsilon12 3−0.64/epsilon1), Re(/Omega12−/Omega13)=ωλ2(1.13/epsilon11 3+0.01/epsilon12 3),(12) with the leading terms diverging as /epsilon11/3, i.e., /Delta1/Omega1 EP3=cωλ2/epsilon11/3, (13) 144414-3YU, YANG, SONG, YAN, AND CAO PHYSICAL REVIEW B 101, 144414 (2020) FIG. 4. Evolution of the eigenfrequencies as a function of the perturbation near the (a) EP2 and (b) EP3 for /epsilon1<0. Inset: frequency splitting Re( /Omega11−/Omega12)a n dR e ( /Omega11,2−/Omega13) on a logarithmic scale, with the slopes approximately being 0.5 and 0.33, respectively. Evolution of the eigenfrequencies as a function of the perturbationnear the (c) EP2 and (d) EP3 for /epsilon1>0. Inset plots the frequency splitting on a logarithmic scale. for the separation of /Omega12and/Omega13spectral lines with c= Re(c21−c31). Supposing the frequency resolution |/Delta1/Omega1 EP3|≈κ, where κ is FMR linewidth, we can express the magnetic sensitivity as S=|δB|√κ, (14) where δB=κ3/(γc3ω2 λ2). Using the following parameters: the damping constant 0.001, the FMR frequency 5 GHz, κ≈ 0.005 GHz, and ωλ2=6.35 GHz, we estimate the sensitivity as 3×10−14T/Hz1/2, which is three orders of magnitude higher than the conventional magnetic sensor based on mag-netic tunneling junction [ 43]. C. Perturbing the whole structure In Sec. II B, we have considered perturbations only on the top spin. Because of the nonlocal nature of the magnetic field,it may affect the whole macrospin system. For such cases, werewrite the matrix H /epsilon1: H/prime /epsilon1=ωλ2⎛ ⎜⎜⎝ωB1ω−1 λ2+/epsilon1 1−iα−1 1−iα0 −ωλ1 ωλ2ωB2 ωλ2+/epsilon1−ωλ1 ωλ2 0 −1 1+iαωB1ω−1 λ2+/epsilon1 1+iα⎞ ⎟⎟⎠. (15) As shown in Fig. 4(a), the eigenfrequency near the EP2 splits into two branches for /epsilon1<0, with the inset displaying the one- half power-law behavior. The frequency near the EP3 splitsto two branches as well including two degenerate modes. Theseparation of two frequencies follows the one-third power law,as plotted in Fig. 4(b).F o r/epsilon1>0, the solutions contain a real root and a pair of complex conjugated roots. The perturbationpushes the spectrum into the exact PT phase region and thus can not remove the degeneracy of EP2, as depicted inFIG. 5. Sensitivity-diminution factor F0as a function of x0. Fig. 4(c). Figure 4(d) shows the frequency splitting in the vicinity of EP3, which is similar to that shown in Fig. 4(b). When the whole trilayer structure is perturbed for /epsilon1>0, we find the sensitivity approximately to be 10−14TH z−1/2, which is the same order of magnitude as the case studied in Sec. II B. D. Effect from statistical noise Noise is inevitable in magnetic systems, which may be caused by material imperfections or fluctuating environments.Following the method in Refs. [ 14,44], we consider a Gaus- sian distribution of the perturbation /epsilon1: P(/epsilon1−/epsilon1 0)=1√ 2πσexp/bracketleftbigg −1 2/parenleftbigg/epsilon1−/epsilon10 σ/parenrightbigg2/bracketrightbigg , (16) with the signal /epsilon10to be detected and the noise level σ.T h e ensemble-average sensitivity can be obtained by: /angbracketleft/Delta1/Omega1 EP3/angbracketright=/integraldisplay+∞ −∞cωλ23√/epsilon1P(/epsilon1−/epsilon10)d/epsilon1 =cωλ2σ1/3 √ 2π/integraldisplay+∞ −∞|x+x0|1/3e−1 2x2dx,(17) with x=(/epsilon1−/epsilon10)/σandx0=/epsilon10/σ. In the small and large signal to noise ratio limit, we obtain: /angbracketleft/Delta1/Omega1 EP3/angbracketright=⎧ ⎪⎨ ⎪⎩21/6cωλ2σ1/3 √π/Gamma1/parenleftbigg2 3/parenrightbigg ,x0/lessmuch1 cωλ2/epsilon11/3 0,x0/greatermuch1. (18) For a large signal to noise ratio, /angbracketleft/Delta1/Omega1 EP3/angbracketrightrecovers Eq. ( 13). By defining the sensitivity-diminution factor F0= c−1ω−1 λ2/epsilon1−1/3 0/angbracketleft/Delta1/Omega1 EP3/angbracketright, we can evaluate the influence of noise on the sensitivity, which is plotted in Fig. 5. It shows that the sensor performs well when x0>1. III. TRILAYER FERROMAGNETIC FILMS In this section, we extend the macrospin model to trilayer ferromagnets, which include both intralayer and interlayerexchange couplings, as shown in Fig. 1(b). The Hamiltonian 144414-4HIGHER-ORDER EXCEPTIONAL POINTS IN … PHYSICAL REVIEW B 101, 144414 (2020) of the system is then given by: H=−/summationdisplay n/summationdisplay /angbracketlefti,j/angbracketrightJnmn,i·mn,j−/summationdisplay n/summationdisplay iBn,i·Mn,i −/summationdisplay n/summationdisplay iKn 2/parenleftbig mx n,i/parenrightbig2−λμ0/summationdisplay iM2,i·(M1,i+M3,i), (19) where Mn,i(mn,i=Mn,i/Mn,i) is the spin (unit spin) at the ith site in the nth layer ( n=1,2,3) with the saturation magnetization Mn,i,Jn>0 is the intralayer exchange coupling constant, /angbracketlefti,j/angbracketrightsums over all nearest-neighbor sites in the same layer, and Bn,i=Bn,iˆxis the external magnetic field at the ith site in the nth layer. The last term in the model Hamiltonian ( 19) describes the interlayer exchange coupling between layer 1 and layer 2, and between layer 2 and layer3. In the calculations, we adopt the same material parametersas the macrospin model and consider the intralayer exchangecoupling constant J 1,2,3=J=2.44×107J/m3. A homoge- neous magnetic field is assumed to be applied over the wholesystem, i.e., B 1,i=B2,i=B3,i=B. The magnetization dynamics is described by the LLG equation ( 2) but with the following effective fields: Beff,1,i=J M1/summationdisplay /angbracketlefti,j/angbracketrightm1,j+Bˆx+K1 M1mx 1,iˆx+λμ0M2m2,i, Beff,2,i=J M2/summationdisplay /angbracketlefti,j/angbracketrightm2,j+Bˆx+K2 M2mx 2,iˆx+λμ0M1(m1,i+m3,i), Beff,3,i=J M1/summationdisplay /angbracketlefti,j/angbracketrightm3,j+Bˆx+K1 M1mx 3,iˆx+λμ0M2m2,i,(20) where J/summationtext /angbracketlefti,j/angbracketrightmn,jrepresents J[mn,(ix−1)a,iya+mn,(ix+1)a,iya+ mn,ixa,(iy−1)a+mn,ixa,(iy+1)a] with ( ixa,iya) being the coordi- nate of the ith unit spin vector, ix(y)is an integer, and ais the lattice constant. Considering a small-angle dynamics, we set mn,i=ˆx+ my n,iˆy+mz n,iˆzwith|my,z n,i|/lessmuch1. Substituting the effective field into Eqs. ( 2) and imposing the complex scalar fields ψn,i= my n,i−imz n,i, we obtain: i˙ψ1,i=γJ M1⎛ ⎝4ψ1,i−/summationdisplay /angbracketlefti,j/angbracketrightψ1,j⎞ ⎠+ωλ2(ψ1,i−ψ2,i) +γ/parenleftbigg B+K1 M1/parenrightbigg ψ1,i−α˙ψ1,i, i˙ψ2,i=γJ M2⎛ ⎝4ψ2,i−/summationdisplay /angbracketlefti,j/angbracketrightψ2,j⎞ ⎠+ωλ1(2ψ2,i−ψ1,i−ψ3,i) +γ/parenleftbigg B+K2 M2/parenrightbigg ψ2,i, i˙ψ3,i=γJ M1⎛ ⎝4ψ3,i−/summationdisplay /angbracketlefti,j/angbracketrightψ3,j⎞ ⎠+ωλ2(ψ3,i−ψ2,i) +γ/parenleftbigg B+K1 M1/parenrightbigg ψ3,i+α˙ψ3,i, (21)FIG. 6. The external magnetic field and gain-loss parameter dependence on the interlayer coupling strength λat EP3 for (a) ( kx,ky)=(π 30a,0) and (b) (π 20a,0). The gray region marks the parametric space allowing the EP3. (c) Evolution of eigenvalues with respect to the gain-loss parameter αforλ=0.175 and B= 99 mT at ( kx,ky)=(π 30a,0). (d) The real and imaginary parts of the eigenvalues as a function of the gain-loss parameter αforλ=0.158 andB=170 mT at ( kx,ky)=(π 20a,0). with the abbreviation/summationtext /angbracketlefti,j/angbracketrightψn,j=ψn,(ix−1)a,iya+ψn,(ix+1)a,iya +ψn,ixa,(iy−1)a+ψn,ixa,(iy+1)a. Expanding the spatiotemporal magnetization in terms of plane waves ψn,i=φn,iexp(ik·r−iωt), we have: ωφi=Hiφi, (22) with Hi=⎛ ⎜⎝ω/prime B1 1−iα−ωλ2 1−iα0 −ωλ1ω/prime B2−ωλ1 0 −ωλ2 1+iαω/prime B1 1+iα⎞ ⎟⎠, (23) andφi=(φ1,i,φ2,i,φ3,i)T, where ω/prime B1=˜ω1(kx,ky)+γ(B+ K1/M1+λμ0M2) and ω/prime B2=˜ω2(kx,ky)+γ(B+K2/M2+ 2λμ0M1) with ˜ ωn(kx,ky)=2γJ/Mn[2−cos(kxa)−cos(kya)]. It is straightforward to see that, for kx=ky=0, Eq. ( 22) is reduced to Eq. ( 5). We aim to search for all EPs in ferromagnetic trilayers. Following Ref. [ 13], we know that the emergence of EP3 depends on magnon’s wave vector k= (kx,ky). As two examples, we set k=(π 30a,0) and (π 20a,0) without loss of generality, to illustrate the condition support-ing the EP3, which are depicted in Fig. 6(a) and Fig. 6(b), respectively. We then explicitly demonstrate the emergenceof EP3 in Fig. 6(c) and Fig. 6(d). We observe that the EP2 appears for all spin-wave modes. At a given ( k x,ky), there exists a critical gain-loss parameter αc, beyond which the exact PTsymmetry is broken. We plot the distribution of the critical gain-loss parameter over the entire Brillouin zone inFig.7(a). The red circle marks the critical αfor the emergence of EP3. In comparison to previous work [ 13], we did not note a special region where the PT symmetry is never broken. This is due to the fact that the chiral spin-spin coupling, i.e., 144414-5YU, YANG, SONG, YAN, AND CAO PHYSICAL REVIEW B 101, 144414 (2020) FIG. 7. (a) Contour plot of the critical gain-loss parameters dependence on spin-wave modes k. The parameters are identical to the ones in Fig. 6(c). (b) FM-AFM phase diagram of the PT- symmetric trilayer and bilayer in α-λplane. The solid and dashed curves represent the phase boundary in the two cases. Dzyaloshinskii-Moriya interaction, is absent in the present model. As first predicted in Ref. [ 13], for a PT-symmetry fer- romagnetic bilayer, antiferromagnetism could emerge in thePT broken phase. As to the ferromagnetic trilayer, it can exhibit a FM-AFM phase transition as well. In Fig. 7(a),w e find that the minimum of α c(k) appears at the boundary of the Brillouin zone. We calculate the corresponding criticalgain-loss parameter at k=(± π a,±π a) for different λ, αc=/radicalbig X(k)−1/vextendsingle/vextendsinglek=(±π a,±π a), (24) where X=1 12ω/prime3 B2/parenleftbig ω/prime2 B1ω/prime B2−2ω/prime B1ωλ1ωλ2/parenrightbig/bracketleftbig c1+/parenleftbig c3−/radicalBig c2 3−c3 2/parenrightbig1/3 +/parenleftbig c3+/radicalBig c2 3−c3 2/parenrightbig1/3/bracketrightbig , (25) with c1=−27β2 1−6β1ω/prime B2(3β2+4ω/prime B1ω/prime B2)+β2 2ω/prime2 B2, c2=/bracketleftbig 27β2 1+6β1ω/prime B2(3β2+4ω/prime B1ω/prime B2)−β2 2ω/prime2 B2/bracketrightbig2 −48β1ω/prime3 B2/parenleftbig 9β1β2ω/prime B1+12β1ω/prime2 B1ω/prime B2−β3 2−β2 2ω/prime B1ω/prime B2/parenrightbig , c3=−/bracketleftbig 27β2 1+6β1ω/prime B2(3β2+4ω/prime B1ω/prime B2)−β2 2ω/prime2 B2/bracketrightbig3 +72β1ω/prime3 B2/bracketleftbig 27β2 1+6β1ω/prime B2(3β2+4ω/prime B1ω/prime B2)−β2 2ω/prime2 B2/bracketrightbig ×/parenleftbig 9β1β2ω/prime B1+12β1ω/prime2 B1ω/prime B2−β3 2−β2 2ω/prime B1ω/prime B2/parenrightbig −864β2 1ω/prime2 B1ω/prime6 B2/parenleftbig 8β1ω/prime B1−β2 2/parenrightbig , β1=ω/prime2 B1ω/prime B2−2ω/prime B1ωλ1ωλ2, β2=2ωλ1ωλ2−ω/prime2 B1−2ω/prime B1ω/prime B2, (26)as plotted by the solid black curve in Fig. 7(b), in which the blue and red regions represent the AFM and FMphases, respectively. The phase boundary for PT-symmetric bilayer is α c=λμ0M1/radicalBig 8J M1+B+K1 M1/radicalBig 8J M1+B+K1 M1+2λμ0M1(27) marked by the dashed line in Fig. 7(b), as a comparison. IV . DISCUSSION AND CONCLUSION Negative damping (gain) is the key to realize our pro- posal. In previous work [ 12,13], it has been suggested that the spin transfer torque, the parametric driving, the ferro-magnetic|ferroelectric heterostructure [ 45], and the interac- tion between magnetic system and environment [ 46–48]a r e possible mechanisms to achieve the magnetic gain. Slavinet al. analytically demonstrated that the main effect of the spin-polarized current in a free magnetic layer is a negativedamping [ 49]. In Ref. [ 50], the Slonczewski form of the spin torque is treated as a negative damping too. To achieve the EP3, FM coupling between two adjacent layers should fall into the allowed parametric space, whichcan be realized by tuning the thickness of the nonmagneticspacer between them [ 51,52]. A single-mode spin wave can be excited via the Brillouin light scattering technique [ 53,54], which is essential to observe the mode-dependent EP3. In the present model, we have assumed that the middle layer is dissipationless. However, a more realistic case is thatit suffers a positive damping α m. In this case, we expect that the mode coalescence will disappear. Indeed, we find that agap opens at the original exceptional point with the frequencysplitting following the one-third power law /Delta1ω EP3∝α1/3 m(not shown). This feature could provide a new method to determinematerials damping parameter with an ultrahigh sensitivity. In summary, we have theoretically investigated the dy- namics of PT-symmetric ternary macrospin structure and ferromagnetic trilayer. We observed both EP2 and EP3 underproper materials parameters. We demonstrated the one-halfand one-third power-law response to external perturbationsin the vicinity of EP2 and EP3, respectively. 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PhysRevA.95.013620.pdf

PHYSICAL REVIEW A 95, 013620 (2017) Geometrically frustrated coarsening dynamics in spinor Bose-Fermi mixtures Nguyen Thanh Phuc,1Tsutomu Momoi,1,2Shunsuke Furukawa,3Yuki Kawaguchi,4Takeshi Fukuhara,1and Masahito Ueda1,3 1Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan 2Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan 3Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 4Department of Applied Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan (Received 7 March 2016; revised manuscript received 4 September 2016; published 19 January 2017) Coarsening dynamics theory describes equilibration of a broad class of systems. By studying the relaxation of a periodic array of microcondensates immersed in a Fermi gas, which mediates long-range spin interactionsto simulate frustrated classical magnets, we show that coarsening dynamics can be suppressed by geometricalfrustration. The system is found to eventually approach a metastable state which is robust against random fieldnoise and characterized by finite correlation lengths together with the emergence of topologically stable Z 2 vortices. We find universal scaling laws with no thermal-equilibrium analog that relate the correlation lengthsand the number of vortices to the degree of frustration in the system. DOI: 10.1103/PhysRevA.95.013620 I. INTRODUCTION Coarsening dynamics theory [ 1,2] has been developed to describe the phase-ordering kinetics following a quench suchas a ferromagnet suddenly quenched below the Curie point,a binary alloy undergoing phase separation, and a spinorBose gas quenched across a phase transition point [ 3–5]. An important assumption of this theory is that domain structuresand correlation functions at different times in the equilibration process differ only in the overall length scale, and that this length scale grows in time according to the power lawξ(t)∝t 1/zwhere zis the dynamical critical exponent. Such a scaling has been examined numerically and experimentallyin several models of relaxation dynamics that differ in thesymmetry of the order parameter manifold and the type ofconserved quantities. Frustration, on the other hand, has long been among the most challenging issues in condensed matter physics [ 6–9]. Geometrical frustration arises, for example, in a triangularlattice with an antiferromagnetic interaction, where spinscannot align in any energetically favored antiparallel con-figuration and must instead compromise between competingconfigurations [ 7,10]. Magnetic frustration gives rise to a huge degeneracy in the classical ground-state manifold of the system, leading to exotic phases such as spin ice with a macroscopically large residual entropy at zero tempera-ture [ 11–13] and spin liquids, in which constituent spins are highly correlated yet strongly fluctuate down to abso-lute zero [ 14–16]. Frustrated spin systems also provide a platform for various emergent phenomena such as hiddenspin nematic order [ 17,18], extended criticality [ 19–21], and magnetic monopoles [ 22–24]. The presence of geometrical frustration is often diagnosed by its susceptibility fingerprintin thermodynamic measurements [ 16,25]. In the present work, by studying the relaxation of a periodic array of those microcondensates immersed in a cloud offermionic atoms which can mimic frustrated classical magnets,we show that the coarsening dynamics can be suppressedby geometrical frustration. The system then approaches ametastable state which has the same local order as the groundstate but with finite correlation lengths. It is remarkable thatthis frustration-induced metastable state is robust against botha random field noise and a small tunneling rate of atoms between microcondensates. Unlike conventional ultracoldatomic systems where the small superexchange interactionbetween two neighboring atomic spins is used [ 26,27], the present spinor Bose-Fermi mixtures can provide a platform tocreate long-range spin interactions between microcondensatesthat can extend beyond nearest-neighbor (NN) sites. Theinteractions are generated through the fermionic mediumand enhanced in strength by Bose condensation. The signand magnitude of the spin interactions can be tuned byvarying the densities of fermions and bosons, allowing for anantiferromagnetic interaction which is the needed element formagnetic frustration. Compared with the Ising [ 28],XY[29], and anisotropic XXZ [30,31] antiferromagnets, all of which have been simulated by ultracold atoms, our system realizes anisotropic Heisenberg antiferromagnetic spin model. By vary-ing the strength of next-nearest-neighbor (NNN) interactionthat lifts the macroscopic degeneracy in the highly frustratedkagome lattice, the correlation lengths of the metastable statecan be changed, allowing us to investigate its universal criticalproperties. In particular, we find new scaling laws with nothermal-equilibrium analog that relate the correlation lengthsto the degree of frustration in the system. Furthermore, we findthat the metastable state characterized by finite correlationlengths contains Z 2vortices which are topologically stable in triangular and kagome lattices, both of which have beenrealized for ultracold atoms [ 32,33]. The number of generated Z 2vortices is also related to the degree of frustration through a scaling law. The formation of Z 2vortices can be directly observed in our system with a spin-resolved measurement [ 34]. The paper is organized as follows. Section IIintroduces the system of spinor Bose-Fermi mixtures used to study thespin relaxation dynamics of frustrated classical magnets. The magnitudes of spin interactions are evaluated for a mixture of 87Rb and7Li atoms. Section IIInumerically studies the time evolution of the magnetizations of microcondensateswhich are placed on two types of lattices with geometricalfrustration: triangular and kagome lattices. In Sec. III A , the time dependencies of nearest-neighbor spin and chiralitycorrelations are investigated, from which it is evident thatthe system forms the same local order as the ground state 2469-9926/2017/95(1)/013620(12) 013620-1 ©2017 American Physical SocietyNGUYEN THANH PHUC et al. PHYSICAL REVIEW A 95, 013620 (2017) after a moderately short time. In Sec. III B , the development of long-range correlations over a long time is investigated,which shows that the system approaches a robust metastablestate characterized by finite correlation lengths. In Sec. III C , the generation of Z 2vortices, which are topologically stable excitations in triangular and kagome lattices, is investigated.The location of these vortices can be experimentally deter-mined from the spatial distributions of the three componentsof magnetization of the microcondensates. Section IVstudies the critical properties of the metastable state, in which scalinglaws relating the correlation lengths and the number of vorticesto the degree of frustration in the system are found. The criticalexponents, which are independent of the initial condition, arefound to change from one ground state to another, which canqualitatively be understood by examining the energy landscapeof the system. Section Vconcludes the paper. The detailed derivation of the effective interaction between bosons mediatedby fermions and the evaluation of the magnitudes of spininteractions are given in Appendixes AandB, respectively. II. SYSTEM Consider a two-dimensional periodic array of microconden- sates in an optical lattice immersed in a harmonically trappeddegenerate Fermi gas as illustrated in Fig. 1. We assume that the spatial variation of the harmonic trapping potentialis smooth over the length scale of the inverse Fermi wavenumber k −1 Fso that the Fermi gas can be regarded as uniform. For the sake of concreteness, we consider spin-187Rb BECs and spin-1 /26Li fermions [ 35–37]. The interaction between bosons and fermions can be decomposed as V(r1,r2)=δ(r1−r2)[g0ˆ1+g1ˆF1·ˆF2], (1) where g0=4π/planckover2pi12(2a3/2+a1/2)/(3M) andg1=8π/planckover2pi12(a3/2− a1/2)/(3M) with a3/2anda1/2being the scattering lengths for the total hyperfine spin Ftot=3/2 and 1 /2, respectively. Here M=MbMf/(Mb+Mf) is the reduced mass of bosons with massMband fermions with mass Mf;ˆ1 and ˆFdenote the identity and spin operators, respectively. The spin-exchangeinteraction governs the spin dynamics of the system. Weconsider a typical case in which the interaction energies aremuch smaller than the Fermi energy. By using the Schrieffer-Wolff transformation [ 38] to adiabatically eliminate the virtual FIG. 1. Spinor Bose-Fermi mixture consisting of an array of microcondensates immersed in a degenerate Fermi gas. The conden-sates, whose magnetizations are represented by blue solid arrows, are located at the lattice sites of an optical lattice (grid), while spin-1 /2 fermions (brown spheres with arrows) move freely (zigzag trajectories) in a harmonic trapping potential [magenta (light gray)].particle-hole excitations in the Fermi gas, we obtain the following effective interaction between microcondensates (seeAppendix Afor details): ˆV eff=−V0/integraldisplay dr/integraldisplay dr/primeλ(kF|r−r/prime|)ˆF(r)·ˆF(r/prime), (2) where V0=g2 1Mfk4 F/(64π3/planckover2pi12) and the kernel λ(x)= [sin(2x)−2xcos(2x)]/x4is the same as that of the RKKY interaction in magnetic metals [ 39–41]. For spin- 1 bosons, the spin density operator is given by ˆF(r)=/summationtext1 m,n=−1ˆψ† m(r)fmnˆψn(r), where ˆψmis the annihilation opera- tor of bosons in the Zeeman sublevel m=1,0,−1 and fmnis the matrix element of the spin-1 matrix vector. As the typical size of a microcondensate is much smaller than the spin healing length, the single-mode approximationis valid [ 42–44]. This implies that the three spin components share the same spatial distribution, and thus a microcondensateat lattice site jis characterized by an order parameter ψ j=√Nb(χ1,j,χ0,j,χ−1,j)T, where Nbis the total number of particles in a microcondensate and the spinor order parameteris normalized to unity: 1/summationdisplay m=−1|χm,j|2=1. (3) Suppose that the spatial distribution of particles in a microcon- densate is described by a wave function φ(r) localized around the corresponding lattice site, i.e., ˆψm(r)=/summationtext jφ(r−rj)ˆam,j where ˆam,jis the annihilation operator of a boson with spin statemin the microcondensate at lattice site j. Then we can express the interaction energy in terms of the spinor orderparameter as V({χ m,j})=J0/summationdisplay jS2 j+/summationdisplay (i,j)JijSi·Sj, (4) where Sj=/summationtext1 m,n=−1χ∗ m,jfmnχn,j. The coupling constants J0 andJijare given by J0=−g2 1Mfk4 FN2 b 64π3/planckover2pi12/integraldisplay d3r/integraldisplay d3r/primeλ(kF|r−r/prime|)|φ(r)φ(r/prime)|2, (5) Jij=−g2 1Mfk4 FN2 b 32π3/planckover2pi12/integraldisplay d3r/integraldisplay d3r/primeλ(kF|r−r/prime|) ×|φ(r−ri)φ(r/prime−rj)|2. (6) In addition to the above effective interaction ( 2) mediated by fermions, spin-1 bosons in each microcondensate can interactdirectly with one another through the contact interaction.This results in an interaction energy having the same formas the first term in Eq. ( 4) with the coefficient J 0re- placed by J/prime 0=(c1N2 b/2)/integraltext d3r|φ(r)|4, where c1=4π/planckover2pi12(a2− a0)/(3Mb) with a2anda0being the scattering lengths of two bosons in the total-spin Ftot=2 and Ftot=0 channels, respectively [ 45,46]. Therefore, the total interaction energy of the system is given by Eq. ( 4) with J0replaced by ˜J0= J0+J/prime 0. In the following, however, to avoid an unnecessary complication we use the notation of J0in place of the total coupling constant ˜J0. 013620-2GEOMETRICALLY FRUSTRATED COARSENING DYNAMICS . . . PHYSICAL REVIEW A 95, 013620 (2017) Each microcondensate becomes a giant spin and the spin interactions are enhanced by the Bose-Einstein condensation.Their signs and magnitudes can be tuned by varying the densityn fof fermions, the spatial extent dof a microcondensate, and the lattice constant a. For example, if we consider a mixture of87Rb and6Li with Nb/similarequal1000 and nf/similarequal5×109cm−3 in a triangular or kagome lattice with a/similarequal4.6μm and an isotropic harmonic distribution φ(r)=e−r2/(4d2)/(2πd2)3/4 withd=k−1 F/2/similarequal1μm, the onsite, NN, and NNN inter- actions are estimated to be J0//planckover2pi1/similarequal− 300 Hz ,J1//planckover2pi1/similarequal70 Hz, andJ2//planckover2pi1/similarequal− 7 Hz, respectively (see Appendix Bfor details). Long-range spin interactions beyond J2are negligibly small. These coupling constants can be made even larger by, forexample, elongating the microcondensates in the directionperpendicular to the 2D lattice. Since J 0<0 and J1>0, the microcondensates tend to be polarized locally, and interactwith one another by an antiferromagnetic NN interaction. III. FRUSTRATED SPIN DYNAMICS We now study the relaxation dynamics of the spinor micro- condensate ensemble. Since the atomic interactions are smallcompared with the critical temperature of the Bose-Einsteincondensation and the number of particles are sufficiently largein each microcondensate, the dynamics of the system canbe described by the time-dependent Gross-Pitaevskii (GP)equation [ 47]. In addition to the effective spin interaction described above, the coupling between bosons and fermionsalso leads to a spin relaxation of the microcondensatescharacterized by a nonlocal Gilbert damping term γ(|r−r /prime|) in the Landau-Lifshitz-Gilbert (LLG) equation generalized toa spatially inhomogeneous spin system: ˙m(r,t)=−m(r,t)×B eff(r,t) +m(r,t)×/integraldisplay d3r/primeγ(r,r/prime)˙m(r/prime,t). (7) Here, mis the unit vector representing the direction of the local spin density, Beffis the effective local magnetic field, and ˙mdenotes the time derivative of m. Similar to the kernel λin the RKKY interaction, the fermion-induced nonlocal Gilbert damping γ(r,r/prime) is an oscillating and rapidly decaying function of the distance |r−r/prime|[48]. Therefore, the dominant contribution to the spin relaxation of a microcondensate arisesfrom the dynamics of the condensate’s particles, leading to aneffective local LLG equation: ˙S j=−Sj×Beff j+/Gamma1Sj×˙Sj, (8) with the effective Gilbert damping of /Gamma1= Nb/integraltext d3rγ(r)|φ(r)|2∼Nbg2 1M2 fk2 F//planckover2pi14. Using the parameters of the87Rb -6Li mixture, we find /Gamma1∼0.1. On the other hand, the spin relaxation of a ferromagnetic BEC can equivalently betaken into account by adding the Gilbert damping coefficient/Gamma1to the left-hand side of the GP equation [ 49], yielding (i−/Gamma1)/planckover2pi1dχ m,j dt=⎛ ⎝2J0Sj+/summationdisplay i/negationslash=jJijSi⎞ ⎠·/parenleftBigg/summationdisplay nfmnχn,j/parenrightBigg . (9)(a) (b) (c),1iS ,2iS ,3iSiC=1a2a 1b2b FIG. 2. Triangular (a) and kagome (b) lattices. Each plaquette i contains the magnetizations Si,1,Si,2,Si,3at three vertices and the spin chirality Cidefined by Eq. ( 11). In the ground state of an antiferromagnet, the magnetizations form an angle of 120◦with one another due to frustration. The magnetizations in blue shadedplaquettes are identical in the ground state, and the winding number of Z 2vortices is calculated along the directed loops (big red triangles with arrows) connecting these plaquettes (see the main text fordetails). The vectors a 1,a2,b1,b2are coordinate axes. (c) The spin and chirality vectors form a structure similar to a tetrahedron with color-labeled vertices. We numerically solve Eq. ( 9) to find the spin relaxation dynamics of the system in two types of lattices with ge-ometrical frustration: triangular and kagome lattices (seeFig. 2). Here, we use the open boundary condition to simulate realistic experiments; the number of sites in one directionof the Bravais lattice is L=100, and the normalization of the order parameter χ m,j [Eq. ( 3)] is performed at each time step to ensure the conservation of the number ofparticles in each microcondensate. It has been numericallyjustified that the normalization of the order parameter givesthe same time evolution as that obtained by introducing aset of Lagrangian multipliers μ j’s to the energy functional: E−/summationtext jμj/summationtext m|χm,j|2. Here, the conservation of the number of particles in each microcondensate results in μj=(2J0Sj+/summationtext i/negationslash=jJijSi)·Sj. Moreover, since we concentrate here on the stationary state that the system eventually approaches, we donot include in Eq. ( 9) random fluctuations associated with the dissipation of the system. As an initial state, we first consider the case in which most of the atoms in the condensates are prepared in them F=1 hyperfine state. Since the perfect ferromagnetic state is a steady state, there would be no time evolution startingfrom such a state. However, in reality the system will bedriven away from the initial state by a small fluctuation inthe population distribution among different hyperfine stateswhich may arise from the imperfection in the preparationprocess, experimental noises, and finite-temperature effects.If one percent of particles occupy each of the m F=0 and mF=− 1 hyperfine states, the spinor order parameter at t=0 is given by χ1,j=√ 0.98,χ 0,j=0.1eiϕ0,χ −1,j=0.1eiϕ−1,(10) where the phases ϕ0andϕ−1are chosen randomly. A. Short-time evolution: local-order formation In a moderately short time, the system forms the same local order as the ground state, in which the three magnetization 013620-3NGUYEN THANH PHUC et al. PHYSICAL REVIEW A 95, 013620 (2017) FIG. 3. Time dependences of nearest-neighbor spin and chirality correlations of the system on triangular and kagome lattices. Here, the time is measured in units of /planckover2pi1/J1,w h e r e J1>0 is the nearest- neighbor antiferromagnetic spin interaction. For the kagome lattice, the next-nearest-neighbor spin interaction J2=− 0.1J1is added to lift the degeneracy in the ground-state manifold. vectors in each triangular plaquette make an angle of 120◦ with each other (see Fig. 2). Figure 3shows the time-dependent nearest-neighbor spin correlation gNN s≡/angbracketleftSi·Sj/angbracketrightand nearest- neighbor chirality correlation gNN c≡/angbracketleftCμ·Cν/angbracketrightof the system on the triangular and kagome lattices. Here, the chirality vectorC μof plaquette μis defined as Cμ≡2 3√ 3(Sμ,1×Sμ,2+Sμ,2×Sμ,3+Sμ,3×Sμ,1),(11) withSμ,1,Sμ,2,Sμ,3being the magnetizations at three vertices of plaquette μ[see Fig. 2(a)], and /angbracketleft ···/angbracketright indicates the average over all pairs of nearest neighbors. It is clear from Fig. 3that for both triangular and kagome lattices after t/similarequal50/planckover2pi1/J1,gNN s approaches −0.5 which is its value for the ground state, reflecting the fact that each magnetization makes an angle of120 ◦with its neighbors. It is also evident that gNN capproaches its value of −1 for the ground state, although in a longer time thangNN s. To provide yet another evidence of the local spin structure, we show in Fig. 4the Fourier transform gs(k)o ft h es p i n correlation function gs(r). The two peaks are clearly seen atk1,2=± (˜a1−˜a2)/3, where ˜a1and ˜a2are the reciprocal lattice vectors with respect to the basis vectors a1anda2of the Bravais lattice. The widths of these peaks are inverselyproportional to the spin correlation length. If we take thelimit of delta-function-like Bragg peaks, and make an inverseFourier transformation, we obtain the spin correlation functionas g s(r)∝eik1·r+eik2·r∝cos/bracketleftbigg2π(x1−x2) 3/bracketrightbigg , (12) where r=x1a1+x2a2.H e r ew eu s e d ˜ai·aj=2πδij(i,j= 1,2). This is consistent with the 120◦spin structure in each plaquette.FIG. 4. Fourier transform gs(k) of the spin correlation function of the system in (a) triangular and (b) kagome lattices at timeJ 1t//planckover2pi1=100. Here k1andk2are the components of momentum k along the directions given by vectors ˜a1and ˜a2, which are reciprocal to the basis vectors a1anda2of the Bravais lattice, respectively (|a1|=|a2|=a). For the kagome lattice, the next-nearest-neighbor spin interaction J2=− 0.1J1is added to lift the degeneracy in the ground-state manifold. B. Long-time evolution: robust metastable state To study the long-range correlations of the system in a long-time evolution, we have numerically evaluated thespin-correlation and chirality-correlation lengths ξ sandξcover a time scale as long as t/similarequal2000/planckover2pi1/J1. Since the system is in a nonequilibrium state and due to a finite-size effect, thespin- and chirality-correlation functions g s(r) and gc(r)d o not follow rigorous exponential functions and show a smallanisotropy. The correlation lengths, however, can be estimatedby the distances at which the magnitudes of the correlationfunctions drop to halves of their maximum values in a specificdirection. To obtain the values of ξ sandξc, we have calculated gs(r) andgc(r) along the symmetry axis of the lattices, i.e., along the direction of the unit vector b1(see Fig. 2). Due to the discrete lattice structure, a systematic error equal tohalf of the lattice constant should be added to the error barsof the correlation lengths. Moreover, since the correlationlengths are determined here by the first lattice points at whichthe magnitudes of the corresponding correlation functions drop to below halves of their maximum values, the values of correlation lengths are expected to be shifted on averageby half of the lattice constant. Those effects of the discretelattice structure have been taken into account in the numericalevaluations of ξ sandξc. Figure 5shows representative time evolutions of the spin- correlation length ξsand chirality-correlation length ξcfor the triangular and kagome lattices. It is clear from Fig. 5that the spin systems in lattices with geometrical frustration approach ametastable state with finite ξ sandξc, in contrast to the standard picture of coarsening dynamics where the correlation lengthsgrow indefinitely. Unlike typical metastable states in many-body systems which often decay to the ground state as a small random fieldis added, the frustration-induced metastable state turns out tobe robust against such a noise. To investigate the robustnessof the metastable state with finite correlation lengths against arandom noise, we add both temporally and spatially dependent 013620-4GEOMETRICALLY FRUSTRATED COARSENING DYNAMICS . . . PHYSICAL REVIEW A 95, 013620 (2017) FIG. 5. Time-dependent spin-correlation and chirality- correlation lengths, ξsandξc, for the triangular and kagome lattices. Here, time is measured in units of /planckover2pi1/J1,w h e r e J1>0i st h e nearest-neighbor antiferromagnetic interaction, and the correlation lengths are measured in units of the system’s size L. For the kagome lattice, the next-nearest-neighbor interaction J2=− 0.1J1is added to lift the degeneracy in the ground-state manifold. The stepwise feature in the evolutions of ξsandξcis a consequence of the fact that the correlation lengths are only determined by integer multiples of the lattice constant (see the text for details). The early stage of the evolution ( J1t//planckover2pi1/lessorequalslant70) is not shown here as the initial ferromagnetic spin order remains dominant and thus the antiferromagnetic spin correlation length is ill-defined. magnetic field B(r,t) with a fixed magnitude |B(r,t)|=B0 but pointing in a random direction. A representative time evolution of the spin- and chirality-correlation lengths forB 0=/planckover2pi1J1/gLμB(i.e., the Zeeman energy due to the random field has the same magnitude as the spin interaction), where μB is the Bohr magneton and gLis the Land ´eg-factor, is shown in Fig. 6. It is clear that the system approaching a metastable state with finite correlation lengths is robust against the randomnoise. Moreover, by varying the strength of the random fieldover a wide range (0 .01/planckover2pi1J 1/gLμB/lessorequalslantB0/lessorequalslant2/planckover2pi1J1/gLμB), we find that the steady-state values of the correlation lengths almost remain unchanged to within their error bars, which indicates that the noise-robust finite correlation lengths shouldresult from the frustration in the system rather than fromthe high-temperature equivalent random fluctuations. Themetastable state is also robust against the introduction ofa small tunneling of atoms between lattice sites as shownin Fig. 7. In this case, a repulsive onsite spin-independent interaction of bosons (e.g., 87Rb) is added to the generalized GP equation ( 9) to avoid the collapse of the system as all particles tend to accumulate at one single lattice site. In thepresence of atomic tunneling, the normalization of the orderparameter changes to/summationtext j/summationtext m|χm,j|2=N, where Nis the number of lattice sites. It is evident from Fig. 5that the growth of the long- range correlation in the system slows down with increasinggeometrical frustration, i.e., changing from the triangular tokagome lattices. Moreover, the growth of ξ cis always slower than that of ξs. This can be understood as a collective effectFIG. 6. Time evolutions of the spin-correlation length (black, solid) and chirality-correlation length (red, dashed) of the systemon the triangular lattice with a random magnetic field of strength B 0=/planckover2pi1J1/gLμB,w h e r e μBis the Bohr magneton. The random field varies both temporally and spatially with a fixed magnitude|B(r,t)|=B 0but pointing in a random direction. since the formation of a spin chirality vector of a triangular plaquette involves the magnetizations at three vertices. C. Topological excitations: Z 2vortices The finite correlation lengths in Sec. III B suggest that spin domains appear in the metastable state. Similar to the quench dynamics through a second-order phase transition withspontaneous symmetry breaking [ 50–52], the formation of spin domains here is expected to accompany the emergence FIG. 7. Time evolutions of the spin-correlation length (black, solid) and chirality-correlation length (red, dashed) of the system in the triangular lattice with a small tunneling rate of atoms between lattice sites. The tunneling is introduced by adding a hopping termt/summationtext m/summationtext /angbracketlefti,j/angbracketright(a† m,jam,i+a† m,iam,j)w i t h t=0.1J1to the Hamiltonian of the system. A repulsive onsite spin-independent interaction of bosons J00/summationtext mm/prime/summationtext ja† m,ja† m/prime,jam/prime,jam,jwithJ00=10J1is also added to avoid the collapse of the system. 013620-5NGUYEN THANH PHUC et al. PHYSICAL REVIEW A 95, 013620 (2017) FIG. 8. Spatial distributions of (a) the winding number of Z 2 vortices and (b)–(d) three components Sx,Sy,Szof the magnetization of the system in the triangular lattice at time J1t//planckover2pi1=100. The winding number and the magnetization are represented by the gray scale and the color gauge, respectively. Here x1andx2are the spatial coordinates of the Bravais lattice in the directions given by unit vectors b1andb2in Fig. 2. The arrows indicate the position of a representative Z2vortex. of topological defects. Specifically, for an antiferromagnet in either the triangular or the kagome lattices, the local-orderformation in Sec. III A results in the three magnetizations in each plaquette making an angle of 120 ◦with one another. The three magnetizations and the spin chirality vector of aplaquette then form a tetrahedron, whose free rotation in spaceyields the SO(3) order-parameter manifold of the system.As the first homotopy group of this manifold is given byπ 1(SO(3) )=Z2, there can exist a stable topological defect called Z 2vortex [ 53,54]. To obtain the spatial distribution of the Z 2vortices, we first calculate the winding number of the rotation of the tetrahedron along a loop (a red triangle)that connects three blue shaded plaquettes in Figs. 2(a) and2(b). In the ground state, those plaquettes have the same spin configuration. The SO(3) rotation of the tetrahedron is specified by a rotation axis nand a rotation angle θ, which can be obtained from the SO(3) rotation matrix Rby noticing thatnis the eigenvector of Rwith the eigenvalue 1 and Tr(R)=1+2 cosθ, where Tr denotes the trace of a matrix. To distinguish a 2 πrotation from no rotation, we use the SU(2) representation of the SO(3) rotation as given by thematrix U=e −iθn·σ/2.H e r e σrefers to the vector of Pauli matrices. With this representation, a Z 2vortex with a winding number n=1g i v e s U=−Iwhile a nontopological spin configuration gives U=I, where Iis the 2 ×2 identity matrix. A representative spatial distribution of the windingnumber of Z 2vortices for the triangular lattice is shown in Fig. 8(a). The local broadening of the winding number distribution at the positions of Z 2vortices should be associated with the fact that the above assumption of the local 120◦ spin configuration is not fully satisfied as the system is notin the ground state. The Z 2vortices can be experimentallyobserved by using a spin-resolved measurement [ 34]o ft h e three components Sx,Sy, andSzof the magnetization as shown in Figs. 8(b)–8(d). Here, all the spatial distributions are coarse grained over the sublattice [containing blue shaded plaquettesin Figs. 2(a) and2(b)] where the spin configuration becomes homogeneous in the ground state. IV . SCALING LAWS OF METASTABLE STATE IN KAGOME LATTICE While the ground state is uniquely determined (up to a global spin rotation) for antiferromagnets in the triangularlattice with a NN interaction J 1>0, there is a macroscopically large degeneracy in the classical ground-state manifold of thesystem in the kagome lattice due to geometrical frustration.To lift this degeneracy and to induce a long-range spin order,a NNN interaction J 2/negationslash=0 is needed. The J2dependencies of the long-time correlation lengths ξsandξcand the number of vortices Nvare shown in Fig. 9forJ2<0. Here the data are averaged over the random phases of spinor componentsin the initial state. The error bars involve both the limitedprecision in determining the correlation lengths due to thediscrete lattice structure and the statistical standard deviationdue to the random initial phases. It is evident that the corre-lation lengths ξ sandξcincrease with increasing magnitude ofJ2by which frustration is reduced, while the number of vortices Nvdecreases as the spin domains get bigger. Linear relations in logarithmic scales in Fig. 9imply the scaling laws of ξs,ξc, andNvwith respect to |J2|.U s i n gt h e least-square fitting procedure, we find ξs∼|J2|α,ξc∼|J2|β, andNv∼|J2|−γwithα=0.33±0.03,β=0.35±0.04, and γ=0.63±0.03. The relation of α/similarequalβ/similarequalγ/2 to within their error bars can be understood by the fact that the numberof vortices is approximately equal to the area of the system FIG. 9. Dependencies of the spin-correlation length ξs(black triangles), the chirality-correlation length ξc(red squares), and the number of Z 2vortices Nv(green circles) on the ratio of the next-nearest-neighbor interaction J2<0 to the nearest-neighbor one J1>0 for the system on the kagome lattice. They are evaluated at a fixed time t=200/planckover2pi1/J1in the dynamics of the system and displayed in the logarithmic scales. The correlation lengths are measured in units of the system size. The averages are taken over 10 initial states with random phases in the spinor components. Straight lines showthe least-square fittings of the corresponding numerical data. 013620-6GEOMETRICALLY FRUSTRATED COARSENING DYNAMICS . . . PHYSICAL REVIEW A 95, 013620 (2017) FIG. 10. Critical exponents α,β,a n dγfor three different condi- tions with the initial state being either ferromagnetic or paramagnetic (the polar state) and for different values of the energy dampingrate/Gamma1. divided by the area of a spin domain which is approximately given by the correlation length squared. To check the universality of the above scaling laws, we investigate the spin relaxation dynamics and the frustration-induced metastable state of the system with a varying system’sparameter (the Gilbert damping rate /Gamma1)a sw e l la sw i t h changing between the initial ferromagnetic state and the initialparamagnetic polar state where most of the particles in thecondensate occupy the m F=0 Zeeman magnetic sublevel. The result is shown in Fig. 10. It is clear that the critical exponents α,β, andγdepend on neither the magnitude of the energy damping rate nor the initial condition to withintheir error bars. This implies their universality. In obtainingthe critical exponents, ξ s,ξc, andNvare evaluated at a fixed time of t=200/planckover2pi1/J1in the dynamics of the system. By comparing the values of α,β, andγat several time points up to t=300/planckover2pi1/J1, we find that the values of those critical exponents do not change with time to within their error bars. Hence we expect that the time dependencies of the correlation lengths and the number of Z 2vortices in the kagome lattice should have the forms of ξs(t)/L=f1(tJ1//planckover2pi1)(J2/J1)α,ξc(t)/L= f2(tJ1//planckover2pi1)(J2/J1)β, andNv(t)=f3(tJ1//planckover2pi1)(J2/J1)−γ, where l is the size of the system and f1(x),f2(x), andf3(x) are dimen- sionless functions that saturate at large values ( x/greaterorsimilar1000). We also investigate the spin relaxation dynamics and the metastable state of the system for J2>0. When the NNN interaction J2changes its sign from negative to positive, the ground state of the system changes from the√ 3×√ 3 N´eel state to the q=0N ´eel state [ 55]. The numerically obtained values of the critical exponents α,β, and γfor J2>0a r es h o w ni nF i g . 11in comparison with those for J2<0. We can see that they increase as J2changes from negative to positive. It can be understood qualitatively bylooking at the energy landscape of the system. Indeed, usingthe Luttinger-Tisza method [ 56], the ground state of a classical spin system is given by the minimum of the energy functionFIG. 11. Critical exponents α, β,a n dγforJ2<0a n d J2>0 with all the other parameters and the initial condition being identical. E({Sj})=J1/summationtext /angbracketlefti,j/angbracketrightSi·Sj+J2/summationtext /angbracketleft/angbracketlefti,j/angbracketright/angbracketrightSi·Sjunder the con- straint |Sj|=1, where /angbracketlefti,j/angbracketrightand/angbracketleft/angbracketlefti,j/angbracketright/angbracketrightdenote NN and NNN pairs of lattice sites, respectively. It is obtained by introducingthe Lagrangian multipliers λ jand minimizing E({Sj})−/summationtext jλj(|Sj|2−1). Assuming λj=λfor all jand making a Fourier transformation, we get the eigenvalue equation, J(k)S(k)=λS(k), (13) where kis the wave vector, and J(k) and S(k) are the Fourier transforms of the interaction and the spin, respectively, in themomentum space. For the kagome lattice where a unit cell con-tains three lattice sites, J(k)i sa3 ×3 matrix. By numerically solving Eq. ( 13), we obtain the energy landscape as shown in Figs. 12–14. It is clear that the ground state is the k=0N´eel state for J 2>0, while it is the√ 3×√ 3N´eel state for J2<0 with wave vector klocated at either one of the two independent corners of the hexagonal Brillouin zone. In order to evaluate the“stiffness” of the ground state, we analytically solve Eq. ( 13) for momenta in the direction connecting the energy minimum FIG. 12. Energy landscape of the lowest-energy band of an antiferromagnet in the kagome lattice with a negative next-nearest-neighbor interaction J 2=− 0.1J1. 013620-7NGUYEN THANH PHUC et al. PHYSICAL REVIEW A 95, 013620 (2017) FIG. 13. Energy landscape of the first low-energy band of an antiferromagnet in the kagome lattice with a positive next-nearest- neighbor interaction J2=0.1J1. and maximum points in the energy landscape. Expanding the energy around the momentum value of the ground state, we findthe two lowest-energy bands which are degenerate at k=0: E 1=− 2(J1+J2)+3J2k2+O(k4), (14) E2=− 2(J1+J2)+J1k2+O(k4) (15) forJ2>0, and the lowest-energy band with the minimum energy at k±=± (˜a1−˜a2)/3: E1=− 2(J1−2J2)−2J2(2J1−3J2) J1−2J2|k−k±|2 +O(|k−k±|3) /similarequal− 2(J1−2J2)−4J2|k−k±|2+O(|k−k±|3) (16) forJ2<0. Here, in obtaining the second equality in Eq. ( 16) we used |J2|/lessmuchJ1. It is evident that for J2>0 there is one energy band with a stiffness, i.e., the second coefficient inthe above momentum expansion, much larger ( ∼J 1) than FIG. 14. Energy landscape of the second low-energy band of an antiferromagnet in the kagome lattice with a positive next-nearest-neighbor interaction J 2=0.1J1.the others ( ∼J2). As a result, there is a higher degeneracy and in turn larger frustration in the manifold of low-energymetastable states for J 2<0. Indeed, if we consider only the remaining energy bands with small stiffness, the area inmomentum space occupied by states with excitation energiessmaller than a given value of δEisA=πδE/ (3J 2)f o rJ2>0 andA=πδE/ (2|J2|)f o rJ2<0. The larger frustration then suppresses the growth of correlation lengths (see Sec. III B ), resulting in smaller correlation lengths for J2<0 than forJ2>0. V . CONCLUSION By studying the spin relaxation dynamics of a periodic array of microcondensates immersed in a Fermi gas, we haveshown that the coarsening dynamics can be suppressed bygeometrical frustration. Instead of decaying to the groundstate, the system is found to approach a metastable state whichhas the same local order as the ground state but with a finitecorrelation length. The frustration-induced metastable stateturns out to be robust against both random noise and a smalltunneling rate of atoms between lattice sites. This metastablestate also contains Z 2vortices, which are topologically stable in triangular and kagome lattices and can be directly observedby spin-resolved measurements. By varying the next-nearest-neighbor spin interaction in the kagome lattice, we are able tostudy the universal critical properties of the metastable state. Inparticular, we find new scaling laws that relate the correlationlengths and the number of Z 2vortices of the metastable state to the degree of frustration in the system. Althoughhere we consider a system of ultracold atoms, a similarphenomenon can be expected in any frustrated classical spinsystem. Furthermore, by using the same setup with fermionicatoms in larger-hyperfine-spin states, the dynamics of a systemwith exotic spin interactions that do not exist in conventionalcondensed matters can be explored. ACKNOWLEDGMENTS This work was supported by KAKENHI Grants No. JP23540397, No. JP25800225, No. JP26287088, No.JP15K17726, and No. JP16K05425 from the Japan Societyfor the Promotion of Science, a Grant-in-Aid for Scientific Re-search on Innovative Areas “Topological Materials Science”(KAKENHI Grants No. JP15H05855 and No. JP16H00989),the Photon Frontier Network Program from MEXT of Japan,and the ImPACT Program of Council for Science, Technologyand Innovation (Cabinet Office, Government of Japan). APPENDIX A: EFFECTIVE INTERACTION BETWEEN BOSONS MEDIATED BY FERMIONS Consider a spinor Bose-Fermi mixture in Sec. II. The part of the Hamiltonian involving fermions consists of the kinetic energy H0=/summationtext k,m/epsilon1kψf† k,mψf k,mwith/epsilon1k=/planckover2pi12k2/(2Mf) and the boson-fermion interaction [Eq. ( 1)]. Here ψf k,mdenotes the annihilation operator of a fermion with momentum /planckover2pi1kand spin statem=↑,↓. In the second quantization, the boson-fermion 013620-8GEOMETRICALLY FRUSTRATED COARSENING DYNAMICS . . . PHYSICAL REVIEW A 95, 013620 (2017) interaction is expressed in terms of ψf k,mas V=g1 2/Omega1/integraldisplay d3rFb(r)·⎛ ⎝/summationdisplay k,k/prime/summationdisplay m,n=↑,↓ei(k−k/prime)·r ×ψf† k/prime,mσmnψf k,n⎞ ⎠, (A1) where Fb(r)=/summationtext m,n=1,0,−1ψb† mfmnψb nis the spin density operator of bosons. Here σmnandfmnrefer to the matrix elements of the Pauli matrices and those of the spin-1 matrices,respectively, and /Omega1denotes the volume of the system. We assume that the characteristic energy scales given by the temperature and interactions are much smaller thanthe Fermi energy in ultracold atomic systems so that theireffects on the fermionic part of the mixture are negligible.Then we can adiabatically eliminate the virtual particle-holeexcitations in the Fermi gas to get an effective Hamiltonianfor the low-energy subspace. This is done by using theSchrieffer-Wolff transformation [ 38]. Specifically, we perform a unitary transformation given by an operator S, yielding ˜H=e iSHe−iS=H0+i[S,H 0]+V+i[S,V] −1 2[S,[S,H 0]]+O(S3,V3).(A2) By choosing such Sthat satisfies [ S,H 0]=iV, we obtain ˜H=H0+i 2[S,V]+O(S3,V3). (A3) Then the effective Hamiltonian ˜Hno longer contains terms linear in Vbecause all particle-hole excitations in the Fermi gas that are induced by the boson-fermion interaction Vhave been adiabatically eliminated. What remains is the low-energysubspace, on which the effective Hamiltonian ˜Hwould act. Using the Leibniz rule for commutators and anticommutators,[A,BC ]=[A,B ]C+B[A,C ]={A,B}C−B{A,C}, [AB,C ]=A[B,C ]+[A,C ]B=A{B,C}−{A,C}B, we find thatSshould be given by S=ig 1 2/Omega1/integraldisplay d3rFb(r)·/summationdisplay k,k/prime/summationdisplay m,n=↑,↓ei(k−k/prime)·r /epsilon1k−/epsilon1k/prime ×ψf† k/prime,mσm,nψf k,n. (A4) Substituting Eq. ( A4) into Eq. ( A3), we obtain ˜H=H0−g2 1 8/Omega12/integraldisplay d3r/integraldisplay d3r/prime/summationdisplay k,k/prime,k/prime/prime m,n,l/braceleftbiggei[(k−k/prime)·r+(k/prime/prime−k)·r/prime] /epsilon1k−/epsilon1k/prime ×[Fb(r)·σmn][Fb(r/prime)·σnl]ψf† k/prime,mψf k/prime/prime,l −ei[(k−k/prime)·r+(k/prime−k/prime/prime)·r/prime] /epsilon1k−/epsilon1k/prime[Fb(r)·σmn][Fb(r/prime)·σlm] ×ψf† k/prime/prime,lψf k,n/bracerightbigg . (A5) As mentioned above, after adiabatically eliminating virtual particle-hole excitations in the Fermi gas, we are left withthe effective Hamiltonian ˜Hacting on the low-energy Hilbertsubspace containing the Fermi gas in the ground state. Focusing on the bosonic degrees of freedom, we then obtainthe effective Hamiltonian for atoms in the microcondensatesas ˜H=H 0−g2 1 8/Omega12/integraldisplay d3r/integraldisplay d3r/primeTr{[Fb(r)·σ][Fb(r/prime)·σ]} ×/summationdisplay k,k/primeei(k−k/prime)·(r−r/prime) /epsilon1k−/epsilon1k/prime(nk/prime−nk), (A6) where nk=[e(/epsilon1k−μ)/(kBT)+1]−1is the Fermi-Dirac distribu- tion function, and Tr denotes the trace operator in the spin spaceof fermions. The trace can be evaluated directly for Pauli ma- trices, giving Tr {[F b(r)·σ][Fb(r/prime)·σ]}=2Fb(r)·Fb(r/prime). The effective interaction between bosons mediated by fermions is then given by Veff=g2 1 4/integraldisplay d3r/integraldisplay d3r/prime[Fb(r)·Fb(r/prime)]λ/prime(r−r/prime), (A7) where λ/prime(r−r/prime)=1 /Omega12/summationdisplay k,k/primeei(k−k/prime)·(r−r/prime)(nk−nk/prime) /epsilon1k−/epsilon1k/prime. (A8) Exchanging the dummy variables kandk/primein the last term, we obtain λ/prime(r−r/prime)=1 /Omega12/summationdisplay k,k/primeei(k−k/prime)·(r−r/prime)+e−i(k−k/prime)·(r−r/prime) /epsilon1k−/epsilon1k/primenk.(A9) To evaluate the right-hand side of Eq. ( A9), we use the polar coordinate representations of kandk/prime.L e tθk,θk/primeandϕk,ϕk/prime be their polar and azimuthal angles with respect to R=r−r/prime. Then, we have λ/prime(R)=1 (2π)6/integraldisplay∞ 0k2dk/integraldisplayπ 0sinθkdθk/integraldisplay2π 0ϕkdϕk ×/integraldisplay∞ 0k/prime2dk/prime/integraldisplayπ 0sinθk/primedθk/prime/integraldisplay2π 0ϕk/primedϕk/prime ×ei(kcosθk−k/primecosθk/prime)R+e−i(kcosθk−k/primecosθk/prime)R /epsilon1k−/epsilon1k/primenk =1 8π4R2/integraldisplay∞ 0k2dk/integraldisplay∞ 0k/prime2dk/prime nk kk/prime(/epsilon1k−/epsilon1k/prime) ×(eikR−e−ikR)(e−ik/primeR−eik/primeR). (A10) At zero temperature, nkis given by the Heaviside unit-step function θ(kF−k) with /planckover2pi1kFbeing the Fermi momentum. The right-hand side of Eq. ( A10) then reduces to λ/prime(R)=Mf 4π4/planckover2pi12R2/integraldisplaykF 0dk/integraldisplay∞ 0dk/primekk/prime k2−k/prime2 ×(ei(k−k/prime)R−ei(k+k/prime)R+c.c.) =Mf 4π4/planckover2pi12R2/integraldisplaykF −kFdk/integraldisplay∞ −∞dk/primekk/primeei(k−k/prime)R k2−k/prime2.(A11) Here c.c. stands for the complex conjugate. To avoid the singularity, we use the principal value of the integral/integraltext dk/prime in Eq. ( A11). The Jordan lemma in complex analysis tells us 013620-9NGUYEN THANH PHUC et al. PHYSICAL REVIEW A 95, 013620 (2017) that the integral over the infinitely large semicircle in the lower half of the complex k/primeplane vanishes. Therefore, the integral overk/primein Eq. ( 8) is equal to the negative of the sum of the integrals over two semicircles C1,C2with infinitesimal radii centered at k/prime=±kin the lower-half plane: P/integraldisplay∞ −∞dk/primek/primeei(k−k/prime)R k2−k/prime2=−/integraldisplay C1+C2dzzei(k−z)R k2−z2 =iπ(1+e2ikR) 2. (A12) Here, in deriving the last equality, we have parametrized the two semicircles C1andC2byz(C1,C2)=±k+/epsilon1eiθwith /epsilon1→0 and π<θ< 2π. We then obtain the final expression for the function λ/prime(R)a s λ/prime(R)=Mf 16π3/planckover2pi122kFRcos(2kFR)−sin(2kFR) R4. (A13) Therefore, the effective interaction of bosons mediated by fermions is given by Veff=g2 1Mf 64π3/planckover2pi12/integraldisplay d3r/integraldisplay d3r/prime[Fb(r)·Fb(r/prime)] ×2kF|r−r/prime|cos(2kF|r−r/prime|)−sin(2kF|r−r/prime|) |r−r/prime|4 =−g2 1Mfk4 F 64π3/planckover2pi12/integraldisplay d3r/integraldisplay d3r/primeλ(kF|r−r/prime|)Fb(r)·Fb(r/prime), (A14) where λ(x)≡[sin(2x)−2xcos(2x)]/x4. Thus Eq. ( 2) has been derived. APPENDIX B: SPIN INTERACTIONS If the microcondensates are confined in nearly har- monic traps, the spatial distribution of particles in eachmicrocondensate is described by a wave function φ(r)= e −r2/(4d2)/(2πd2)3/4withdcharacterizing the spatial extent of the condensate. The coupling constants of the spin interactionsthen reduce to J 0=−g2 1N2 bMfk4 F 512π6/planckover2pi12/integraldisplay d3r/integraldisplay d3r/primeλ(˜kFr/prime)e−r2+(r+r/prime)2 2 =−g2 1N2 bMfk4 Fα(˜kF) 64π4/planckover2pi12, (B1) J/prime 0=c1N2 b 16π3/2d3, (B2) Jij=−g2 1N2 bMfk4 F 256π6/planckover2pi12/integraldisplay d3r/integraldisplay d3r/primeλ(˜kFr/prime)e−r2+(r−rij+r/prime)2 2 =g2 1N2 bMfk4 Fβ(˜kF,˜rij) 64√ 2π7/2/planckover2pi12, (B3) where ˜kF=kFd,˜rij=|ri−rj|/d, and the functions α(x) and β(x,y)a r eg i v e nb y α(x)≡/integraldisplay∞ 0dr/integraldisplay∞ 0dr/primerr/primee−r2 2/bracketleftbig e−(r−r/prime)2 2−e−(r+r/prime)2 2/bracketrightbig λ(xr/prime), (B4)FIG. 15. Plot of α(x)g i v e nb yE q .( B4). β(x,y)≡1 y/integraldisplay∞ 0dr/integraldisplay∞ 0dr/primerr/primee−r2 2/braceleftbigg Erf/bracketleftbiggy−r−r/prime √ 2/bracketrightbigg −Erf/bracketleftbiggy−r+r/prime √ 2/bracketrightbigg −Erf/bracketleftbiggy+r−r/prime √ 2/bracketrightbigg +Erf/bracketleftbiggy+r+r/prime √ 2/bracketrightbigg/bracerightbigg λ(xr/prime) =−2 y/integraldisplay∞ 0dr/primer/primee−(r/prime+y)2 4/parenleftbigg er/primeyErf/bracketleftbiggr/prime−y 2/bracketrightbigg +Erf/bracketleftbiggr/prime+y 2/bracketrightbigg/parenrightbigg λ(xr/prime). (B5) Here Erf( x)=(2/√π)/integraltextx 0dt e−t2is the error function. The right-hand sides of Eqs. ( B4) and ( B5) can be evaluated numerically and the obtained results are shown in Figs. 15 and16, respectively. As shown below Eq. ( 1), the coupling constant g1be- tween bosons and fermions is proportional to the differencea 3/2−a1/2between the s-wave scattering lengths for two total-hyperfine-spin- Fchannels: F=3/2 and F=1/2. For a mixture of spin-187Rb and spin-1 /26Li, the electronic spin-singlet and spin-triplet atomic potentials as well as thecorresponding s-wave scattering lengths a sandathave been calculated numerically in Refs. [ 57] and [ 58]. By expanding the hyperfine-spin states in the electronic spin bases, we obtain FIG. 16. Plot of β(x,y)g i v e nb yE q .( B5) as a function of yfor x=0.5 which is relevant to the values of the system’s parameters in Sec. II. 013620-10GEOMETRICALLY FRUSTRATED COARSENING DYNAMICS . . . PHYSICAL REVIEW A 95, 013620 (2017) the atomic potentials VFassociated with the total-hyperfine- spin-Fchannels ( F=3/2 and 1 /2) for the87Rb -6Li mixture, from which we can evaluate the corresponding scatteringlengths. Without a strong external field the interatomic in-teraction is almost isotropic, and thus the interatomic potentialis essentially the same for all Zeeman states |F,m F/angbracketrightin a given total-hyperfine-spin- Fmanifold. Therefore, we only need to calculate the atomic potential for an arbitrary hyperfine statewithin that spin manifold. First, we expand the hyperfine-spin state |F= 3 2,mF=3 2/angbracketright in the electronic-spin basis. In terms of the hyperfine spinsF 1=1 for the boson and F2=1/2 for the fermion, the state can be written as |F1=1,mF1=1/angbracketright⊗/vextendsingle/vextendsingleF2=1 2,mF2=1 2/angbracketrightbig . (B6) Since both rubidium and lithium are alkali atoms whose valence electrons in the ground states have zero orbital angularmomentum ( l=0), their hyperfine spins are the sums of theelectronic spin sand the nucleus spin I.H e r ew eh a v e I 1= 3 2,s1=1 2andI2=1,s2=1 2for87Rb and6Li, respectively. Using the table of Clebsch-Gordan coefficients [ 59], the hyperfine states in Eq. ( B6) can be expressed as |F1=1,mF1=1/angbracketright=/radicalbigg 3 4/vextendsingle/vextendsingle/vextendsingle/vextendsinglemI1=3 2,ms1=−1 2/angbracketrightbigg −1 2/vextendsingle/vextendsingle/vextendsingle/vextendsinglem I1=1 2,ms1=1 2/angbracketrightbigg ,(B7) /vextendsingle/vextendsingle/vextendsingle/vextendsingleF2=1 2,mF2=1 2/angbracketrightbigg =/radicalbigg 2 3/vextendsingle/vextendsingle/vextendsingle/vextendsinglemI2=1,ms1=−1 2/angbracketrightbigg −/radicalbigg 1 3/vextendsingle/vextendsingle/vextendsingle/vextendsinglem I2=0,ms2=1 2/angbracketrightbigg .(B8) Substituting Eqs. ( B7) and ( B8)i nE q .( B6), we can express the hyperfine state |F=3 2,mF=3 2/angbracketrightin the electronic-spin basis as /vextendsingle/vextendsingle/vextendsingle/vextendsingleF=3 2,mF=3 2/angbracketrightbigg =/radicalbigg 1 2/vextendsingle/vextendsingle/vextendsingle/vextendsinglem s1=−1 2,ms2=−1 2/angbracketrightbigg −/radicalbigg 1 6/vextendsingle/vextendsingle/vextendsingle/vextendsinglem s1=1 2,ms2=−1 2/angbracketrightbigg −1 2/vextendsingle/vextendsingle/vextendsingle/vextendsinglem s1=−1 2,ms2=1 2/angbracketrightbigg +/radicalbigg 1 12/vextendsingle/vextendsingle/vextendsingle/vextendsinglem s1=1 2,ms2=1 2/angbracketrightbigg =/radicalbigg 1 2|s=1,ms=− 1/angbracketright−/parenleftBigg/radicalbigg 1 12+/radicalbigg 1 8/parenrightBigg |s=1,ms=0/angbracketright−/parenleftBigg/radicalbigg 1 12−/radicalbigg 1 8/parenrightBigg |s=0,ms=0/angbracketright +/radicalbigg 1 12|s=1,ms=1/angbracketright, (B9) where s=s1+s2is the total electronic spin. Therefore, the interatomic potential for the total-hyperfine-spin F=3/2 scattering channel is given in terms of the electronic spin-singlet and spin-triplet counterparts V sandVtby V3/2=(19+2√ 6)Vt+(5−2√ 6)Vs 24/similarequal0.996Vt+0.004Vs. (B10) Similarly, we expand the hyperfine state |F=1 2,mF=1 2/angbracketright in the electronic-spin basis by using the Clebsch-Gordancoefficients. The resulting interatomic potential for the total-hyperfine-spin F=1/2 scattering channel is given by V 1/2=(26+2√ 6+2√ 2)Vt+(10−2√ 6−2√ 2)Vs 36 /similarequal0.94Vt+0.06Vs. (B11)It is clear from Eqs. ( B10) and ( B11) that the atomic potential V3/2is almost equal to the spin-triplet potential Vt, while there is an approximately 5% mixing of the spin-singlet potentialV sinV1/2. However, it can be seen from the numerically calculated VsandVtfor LiRb (see Figs. 1 and 2 in Ref. [ 57]) that the energy difference between two neighboring boundstates of V tis smaller than 5% of Vs. 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PhysRevB.74.184102.pdf

Vibrational properties of Ge nanocrystals determined by EXAFS L. L. Araujo *and P. Kluth Department of Electronic Materials Engineering, Research School of Physical Sciences and Engineering, Australian National University, Canberra, Australia G. de M. Azevedo Laboratório Nacional de Luz Síncrotron, Campinas, Brazil M. C. Ridgway Department of Electronic Materials Engineering, Research School of Physical Sciences and Engineering, Australian National University, Canberra, Australia /H20849Received 20 April 2006; revised manuscript received 9 August 2006; published 1 November 2006 /H20850 Extended x-ray absorption fine structure /H20849EXAFS /H20850spectroscopy was applied to probe the vibrational pro- perties of bulk crystalline Ge /H20849c-Ge/H20850and Ge nanocrystals /H20849Ge NCs /H20850of 4.4 nm mean diameter produced by ion implantation in SiO 2followed by thermal annealing. EXAFS measurements around the Ge Kedge were carried out in the temperature range from 8 to 300 K at beam line 10-2 of the Stanford Synchrotron RadiationLaboratory /H20849SSRL /H20850. Original information about thermal and static disorder, thermal expansion, and anharmo- nicity effects have been obtained for c-Ge and Ge NCs from temperature dependent EXAFS measurements using a correlated anharmonic Einstein model and thermodynamic perturbation theory. It was observed that theGe NCs were stiffer /H20849showed a stronger bond force constant /H20850than both amorphous Ge /H20849a-Ge/H20850andc-Ge. Also, the values of the linear thermal expansion /H20849thermal evolution of the mean interatomic distance /H20850obtained for the Ge NCs were smaller than the ones obtained for c-Ge. These results were compared to the ones obtained for other nanocrystalline systems. They suggest that the increased surface to volume ratio of the nanocrystallineform and the presence of the surrounding SiO 2matrix might be responsible for the different vibrational properties of c-Ge and Ge NCs. DOI: 10.1103/PhysRevB.74.184102 PACS number /H20849s/H20850: 62.25. /H11001g, 65.80. /H11001n, 61.46.Hk, 61.10.Ht I. INTRODUCTION Nanocrystalline particles are interesting subjects for ma- terials science since their properties can deviate from thoseof bulk materials, making them attractive for a variety ofpotential applications. For example, semiconductor nano-crystals in a dielectric medium have attracted attention due to their unique optical properties which are not observed intheir bulk counterparts. 1Given that the optical properties are governed by the structural properties, characterization of thelatter for Ge nanocrystals /H20849Ge NCs /H20850has recently been per- formed with the EXAFS technique 2to attain a better under- standing of the behavior of such particles. The study of Genanocrystals is further developed in this contribution, whereEXAFS was applied to determine the short-range order vi-brational properties of Ge nanocrystals synthetized in SiO 2 by ion implantation and thermal annealing. The use of EXAFS as a vibrational probe was first sug- gested in the seventies3,4and has been explored in several ways by many research groups since that time.5–7Informa- tion about thermal and static disorder, thermal expansion andanharmonicity effects have been obtained for several mate-rial systems from temperature dependent EXAFS measure-ments. In particular, the analysis of temperature dependentdata through the cumulant expansion method 8for moderately disordered systems has proven to yield reliable informationabout the distribution of interatomic distances probed byEXAFS and its evolution with temperature. 5,6,9The first three cumulants measure the average value, the variance, andthe asymmetry of the distance distribution for a given coor-dination shell, and their variation with temperature can yield information on the linear thermal expansion, thermal disor-der, and potential anharmonicity, respectively. The secondand third cumulants can be related to the force constants of aone-dimensional effective pair potential from which the vi-brational frequency or bond strength of the atoms in thatshell can be estimated, as well as the thermal variation of thepotential asymmetry. For more disordered systems it is alsoimportant to take into account the fourth term in the cumu-lant expansion series, called the fourth cumulant. This termaccounts for symmetric deviations from a Gaussian form inthe distribution of interatomic distances. The direct extrac-tion of the thermal expansion from EXAFS data, however, isnot straightforward. Care must be taken to consider effectssuch as the spherical nature of the photoelectron wave, themean free path of the photoelectron and an effect that can beexplained as due to the influence of vibrations perpendicularto the bond direction, as defined in Refs. 6,9, and 10. Due to the latter, the variation of the first cumulant measured byEXAFS is different from the one calculated only from theasymmetry of the one-dimensional effective pair potential.This effect can be isolated by combining EXAFS and XRD/H20849x-ray diffraction /H20850results for the same system. 6,9 Bulk crystalline Ge /H20849c-Ge/H20850has been thoroughly studied by EXAFS and other techniques /H20849such as XRD /H20850.2,11–13Re- garding temperature dependent EXAFS experiments, Dalbaet al. . 6,14,15performed systematic measurements for c-Ge and amorphous Ge /H20849a-Ge/H20850at temperatures from 10 to 600 K. Data were analyzed by the ratio method and results for thePHYSICAL REVIEW B 74, 184102 /H208492006 /H20850 1098-0121/2006/74 /H2084918/H20850/184102 /H208498/H20850 ©2006 The American Physical Society 184102-1first four cumulants of the distance distribution were ob- tained. Relative values of the cumulants were first deter-mined /H20849comparing low and high temperature data /H20850and then absolute values were calculated from the relative values byfitting the high temperature data using the perturbative quan-tum approach of Frenkel and Rehr. 16Low temperature quan- tum effects in the third cumulant and the ratio between theperpendicular and radial correlation terms were evaluated forthe first shell of c-Ge. 6 Moreover, Filipponi and Di Cicco performed temperature dependent EXAFS experiments for c-Ge at temperatures from 77 to 1100 K.7Data were analyzed using the GNXAS /H20849Ref. 13/H20850approach instead of using the cumulant expansion method and absolute values for the bond lengths and Debye-Waller factors were obtained. Other EXAFS studies of theGe thermal expansion can be found cited within Refs. 6,7, 14,15, and 17. In this work we will present results for c-Ge obtained by fitting experimental data with the theoretical standards givenby the FEFF8.102 code, 18which allows us to directly obtain values for the interatomic distance distributions, their vari-ance and their asymmetry. Such values are absolute under thecondition that the FEFF standard well reproduces flawlessc-Ge. We will show that a very good agreement is found between our c-Ge results and those from the ratio method analysis for the first three cumulants of the first shell distancedistribution. This reinforces the validity of our approach,which will then be extended to the more complex system ofGe nanocrystals embedded in a SiO 2matrix. The study of nanoparticle systems is complicated by the superposition ofsurface and bulk behavior, and an EXAFS thermal expansionstudy should be, in principle, affected by the differences be-tween the vibrational properties of atoms on the surface andin the core of the nanoparticles. Our comparison of bulk andnanocrystalline data should highlight the influences of sizeeffects and the surrounding matrix on the nanocrystal vibra-tional properties. There are very few temperature dependent EXAFS stud- ies for semiconductor nanoparticle systems reported in theliterature. Results obtained for Mercaptoethanol-coated ZnSnanoparticles of diameter 3.4 nm /H20849Ref. 19/H20850indicate that they are strained and stiffer than bulk ZnS, presenting a higherEinstein temperature. Thiol-capped CdS nanoparticles from 1.3 to 4.0 nm were also observed to show higher Einsteintemperatures than bulk CdS, although the difference washigher than 5% only for the smaller nanoparticles. 20The stiffening of the CdS bonds in the nanoparticles was assignedto their increased surface-to-volume ratio. The same effectwas also observed for Thiol-capped CdTe nanocrystals. 21 In this contribution we will present original results for the short-range order vibrational properties of an elementalsemiconductor in a nanocrystalline state. The differences ob-served for such properties of the Ge NCs relative to the onesfrom bulk c-Ge reveal further differences between both states and allow us to get a deeper insight about size effectsand the influence of a surrounding matrix of SiO 2on the Ge NCs. II. EXPERIMENTAL Ge nanocrystals were formed in a 2 /H9262m thick SiO 2matrix by ion implantation, at liquid nitrogen temperature, of 1/H110031017Ge/cm2at 2 MeV. The implantation was followed by thermal annealing for 1 h of the samples at 1100 °C underforming gas /H2084995% N 2+5% H 2/H20850flow. Further details are de- scribed in Ref. 2. The Ge peak concentration of 3 at. % was verified to be centered at a depth of 1.2 /H9262m inside the SiO 2 layer by RBS /H20849Rutherford backscattering spectrometry /H20850and TEM /H20849transmission electron microscopy /H20850measurements. Polycrystalline Ge standards of thickness 200 nm sand- wiched between 2 /H9262m thick SiO 2layers were also produced as described in Ref. 2, for direct comparison with the nano- crystalline samples. This way, the fluorescence EXAFS mea-surements were carried out in similar conditions for bothc-Ge and Ge NCs samples. Cross section transmission electron microscopy /H20849XTEM /H20850 results show that the nanocrystals are spherical in shape andpresent crystallinity similar to bulk c-Ge. Figure 1presents two XTEM images of the Ge NCs as grown into the SiO 2 layer; Fig. 1/H20849a/H20850shows the nanocrystal distribution inside the SiO 2layer and Fig. 1/H20849b/H20850shows a high resolution image of one Ge NC. Small angle x-ray scattering /H20849SAXS /H20850measure- ments /H20849not shown here /H20850were employed to determine the size distribution of Ge NCs. It was observed to have a meanvalue of 4.4 nm with a full width at half maximum of1.5 nm. EXAFS measurements at the Ge Kedge /H2084911.103 keV /H20850 were performed at temperatures from 8 to 300 K, at beam line 10-2 of the Stanford Synchrotron Radiation Laboratory,USA. Fluorescence spectra were recorded with a 30 elementsolid-state Ge detector and the Si /H20849220/H20850monochromator de- tuned by 50% for harmonic rejection. The raw EXAFS data were analyzed according to the standard procedure described in Ref. 22, followed by a mul- tiple data set fit, as will be explained below. EXAFS spectrawere energy calibrated, aligned, and isolated from raw absor-bance by background subtraction via the AUTOBK algo-rithm, as implemented in the code ATHENA. 23Structural parameters were then determined using ARTEMIS /H20849Ref. 23/H20850 with photoelectron momentum kand nonphase-corrected ra- dial distance rranges of 4.8–14.8 Å−1and 1.7–2.6 Å, re- spectively. ATHENA and ARTEMIS are GUIs /H20849graphical user interfaces /H20850for the IFEFFIT code. Phases and amplitudes were calculated ab initio with the FEFF8.102 code.18The amplitude reduction factor S02and threshold energy E0were FIG. 1. XTEM images of the Ge NCs grown in SiO 2. The left frame shows the Ge NCs distributed inside the SiO 2matrix and the right frame shows a high resolution image of one Ge NC.ARAUJO et al. PHYSICAL REVIEW B 74, 184102 /H208492006 /H20850 184102-2determined from the polycrystalline standard and held con- stant thereafter at the values of 0.988 and 9.68, respectively.These are the mean values for the given temperature range,calculated from the individual values for each temperature.The coordination number was kept constant at the bulk valueof 4.0 during the c-Ge analysis and determined for the Ge NCs as 3.2 from the lowest temperature NC spectrum. It isexpected to be smaller for the nanocrystalline phase due tolower-coordinated atoms at the surface. A given data set wasfitted simultaneously with multiple kweightings of 1–4, in order to reduce correlations between the fitting parameters. III. THEORY AND DATA ANALYSIS The information extracted from experimental spectra will be written as a series of cumulants of the distancedistribution 8for the first shell of Ge. The analysis of EXAFS data via the cumulant expansion method, as well as therelationship between the cumulants and the local dynamicsin crystalline materials, has recently been reviewed byFornasini et al. 9 The EXAFS second cumulant, MSRD /H20849mean square rela- tive displacement /H20850or Debye-Waller factor /H92682is sensitive to both structural and thermal disorder. As the structural com-ponent is considered not to vary with temperature, it is pos-sible to separate both contributions by performing tempera-ture dependent EXAFS measurements and fitting theresultant Debye-Waller factors with a Debye or Einsteinmodel. 24For the /H92682of the first shell of Ge, in particular, the correlated Einstein model is considered a suitable choice.25,26 The EXAFS /H92682can yield information on the vibrational dy- namics of both crystalline and noncrystalline materials.24It contains effects of correlation between the atomic motion ofabsorber and backscatterer atoms, differing from the XRDMSD /H20849mean square displacement /H20850by the DCF /H20849displacement correlation function /H20850. 3On more general grounds, the tem- perature dependence of the /H92682provides a measure of the effective bond-stretching force constant between absorberand backscatterer atoms and can be used to study thestrength of chemical bonds. The third cumulant of the distance distribution C 3mea- sures its asymmetry. C3can be different from zero even for a harmonic crystal at very low temperatures due to the effectof zero-point atomic vibrations. 9In samples that are not flawless crystals, further asymmetry in the distribution ofdistances may be observed. But this static contribution is notsupposed to increase with temperature, so that the variationofC 3with temperature can be ascribed to asymmetry in the distance distribution generated by anharmonicity of the ef-fective interaction potential. A. Relationship between thermal expansion and EXAFS cumulants A relationship between a linear thermal expansion factor a and the EXAFS cumulants in the quantum limit was derivedby Frenkel and Rehr using a correlated anharmonic Einsteinmodel and thermodynamic perturbation theory. 16In this model, a one-dimensional anharmonic effective pair potential of the formV/H20849r−r0/H20850=ke/H20849r−r0/H208502−k3/H20849r−r0/H208503+¯ /H208491/H20850 was assumed, where r0is the minimum of the effective pair potential, keis the effective harmonic spring constant, and k3 is the cubic anharmonicity constant. The following relation- ships were then derived for the temperature dependence, toleading order in k 3, of the second cumulant /H92682, third cumu- lant C3, and linear thermal expansion factor a/H20849connected to the thermal variation of the first cumulant C1/H20850, respectively5,16 C2/H20849T/H20850=/H92682/H20849T/H20850=/H6036/H9275E 2ke1+z 1−z+/H9268static2, /H208492/H20850 C3/H20849T/H20850=k3/H20849/H6036/H9275E/H208502 2ke31+1 0 z+z2 /H208491−z/H208502+C3static, /H208493/H20850 a/H20849T/H20850=3 2/H6036k3 /H92622/H9275E31+z 1−z, /H208494/H20850 where Tis the temperature, /H9275Eis the Einstein frequency /H20849ke=/H9262/H9275E2/H20850,/H9262is the reduced mass /H20849in this case, for a Ge-Ge absorber-scatterer pair /H20850, and z/H11013exp/H20849−/H9008E/T/H20850. The Einstein temperature is given by /H9008E=/H6036/H9275E/kB, where kBis the Boltz- mann constant. /H9268static2and C3staticare the static or structural /H20849temperature independent /H20850contributions to the total disorder and asymmetry, respectively. These terms have been addedhere to the temperature dependent ones in order to accountfor the effects of static disorder, expected to be different forc-Ge and Ge NCs. 2,22 This one-dimensional model can be used as a reference to analyze the thermal behavior of the cumulants of the distancedistribution obtained from experimental EXAFS data. It canbe considered as the effective potential of the one- dimensional distribution of distances sampled by the EXAFSanalysis of a given shell of a three-dimensional crystalline/H20849or nanocrystalline /H20850material. B. EXAFS effective and realdistance distributions In an experimental measurement the EXAFS photoelec- trons /H20849with mean free path /H9261/H20850probe an effective distance distribution P/H20849r,/H9261/H20850=/H9267/H20849r/H20850*exp/H20849−2r//H9261/H20850*r−2, due to the weak- ening of the photoelectron wave with distance, the spherical nature of such a wave and the finite mean free path.8On the other hand, the instantaneous interatomic distances rare dis- tributed according to the real unidimensional distribution /H9267/H20849r/H20850. For systems with low to moderate disorder, the differ- ence between the cumulants of both distributions is consid- ered non-negligible only for the first cumulant /H20849interatomic distance /H20850,6,8and this difference must be kept in mind when a thermal expansion study is undertaken. As the analysis of EXAFS experimental data by compari- son with FEFF generated standards using the IFEFFIT codeaccounts for the effects of the weakening of the photoelec-tron wave with distance, the spherical nature of such a waveand the finite mean free path, the values obtained from suchmethod are the real cumulants of the distance distribution.VIBRATIONAL PROPERTIES OF Ge NANOCRYSTALS … PHYSICAL REVIEW B 74, 184102 /H208492006 /H20850 184102-3Thus, they can be directly applied in a thermal expansion study. C. Effective pair potential and distance distributions The use of an effective pair potential to describe the dis- tribution of distances probed by EXAFS must be appliedwith care in order to take into account some of its limita-tions. The effective pair potential can be, in principle, tem-perature dependent, both in position and shape. 27In particu- lar, a positive shift of the minimum of the effective pairpotential was assigned to the effect of vibrations perpendicu-lar to the bond direction, among other causes. 9,27This sug- gests that the thermal expansion probed by EXAFS dependsnot only on the asymmetry of the effective potential /H20849given by the second and third cumulants, /H92682andC3/H20850, as implied in Frenkel and Rehr’s model, but also on its rigid shift. Stern/H20849Ref. 10/H20850also points out the importance of the vibrations perpendicular to the bond direction and explains the rigidshift as a consequence of the difference in the minimum oftheeffective andrealpotentials, i.e., the difference between the maximum of the effective andrealdistributions of inter- atomic distances. Furthermore, for bulk AgI and CdSe, avariation of the minimum of the effective potential with tem-perature has been reported, 28,29shifting to lower values as the temperature increased; such an effect was not observedfor Ge. 9 As a result, the temperature variation of the EXAFS real first cumulant and the linear thermal expansion factor aas given above cannot be considered equivalent. In order toavoid misinterpretations due to differences in both quantities,here the quantity aas defined in Ref. 16/H20851reproduced in Eq. /H208494/H20850above /H20852will be called the anharmonic contribution to the thermal expansion, since it is related to the anharmonicity ofthe effective potential only. The variation of our first cumu-lant with temperature, /H9254C1, will be called the EXAFS ther- mal expansion and will include contributions not only due tothe anharmonicity but also due to the shift of the effectivepotential. The difference between the EXAFS /H9254C1and XRD /H9254Rthermal expansions will be connected to the shift of the effective potential /H20849perpendicular vibrations /H20850, rather than to its asymmetry. D. Data analysis Theoretical spectra were simulated by the FEFF 8.102 code /H20849Ref. 18/H20850and the values of the cumulants of the dis- tance distribution for the first shell of Ge were obtainedthrough a nonlinear best fit to experimental spectra usingARTEMIS. 23,30 The analysis of experimental spectra was carried out in two steps. In the first one, each spectrum was fitted at once,giving separate values of C 1,/H92682,C3. The presence of a fourth cumulant C4was also considered in the fits, but it was ob- served to be negligible for all measurements. This confirmedthe validity of deriving the cumulant equations from /H208494/H20850to leading order in k 3. Although the /H92682values obtained this way were insensitive to small variations in the fitting conditions and presentedsmall error bars, the same was not observed for the C 1andC3values, due to the correlation between these quantities. A similar result was observed by Fornasini et al. when analyz- ing the EXAFS data for the thermal expansion of bulk Cu.27 We then fitted our obtained /H92682results as a function of tem- perature according to Eq. /H208492/H20850, obtaining the Einstein tem- perature /H20849and consequently /H9275Eandke/H20850. In the second step, the variation of the second and third cumulants /H92682and C3with temperatures were restrained to follow Eqs. /H208492/H20850and/H208493/H20850and all spectra from 8 to 300 K were fitted simultaneously, in a similar way to the methods de-scribed in Refs. 20and31. Representing the evolution of the second cumulant by the correlated Einstein model and of thethird cumulant by Eq. /H208493/H20850is appropriate since both quantities have shown to be well described by these models in theliterature. The thermal variation of the first cumulant C 1,o n the other hand, was not restrained to follow Eq. /H208494/H20850since it is, in principle, not well represented by its variation with theasymmetry of the potential only. Thus, the first cumulant foreach temperature was simply written as r 0+drT, where r0is the value of the first cumulant for the lowest temperaturedata. During the multiple data set fit, k ewas fixed to the value obtained from the fit to the /H92682from the first step, so that k3, the structural contribution to C3,r0, and the drTfor each temperature were the only fitting parameters. By doingthis, the number of free parameters was reduced and the C 3 values and variation were linked to the ones of the second cumulant, helping to decrease the errors in its determinationand to break the correlation between C 3andC1. IV. RESULTS AND DISCUSSION Figure 2shows the Fourier transforms of a selection of the temperature dependant k3-weighted EXAFS spectra as a function of the radial distance /H20849without phase corrections /H20850 obtained in this work. Figure 2/H20849a/H20850shows spectra measured at 8, 100, and 300 K for c-Ge, while Fig. 2/H20849b/H20850shows spectra measured at the same temperatures for Ge NCs. Comparingthe data for the two systems, some characteristic features of FIG. 2. Fourier transforms of k3-weighted EXAFS spectra as a function of the nonphase-corrected radial distance measured atdifferent temperatures for /H20849a/H20850polycrystalline Ge and /H20849b/H20850Ge nanocrystals.ARAUJO et al. PHYSICAL REVIEW B 74, 184102 /H208492006 /H20850 184102-4nanocrystalline materials become readily apparent. For ex- ample, at any given temperature the magnitude of the Fouriertransforms is lower for the Ge NCs due to their higher sur-face to volume ratio, which causes a reduction of the overallcoordination number and an increase in the variance of thedistance distributions compared to the bulk c-Ge ones. Also, the second and third neighbor shells are less pronounced forthe Ge NCs since they are more sensitive to variations inbond angles, which is also a result of the increased surface tovolume ratio. Furthermore, the damping of the amplitude ofthe EXAFS signal with increasing temperature for both sys-tems can be verified from the graphs. A. Debye-Waller factors and Einstein temperatures The Debye-Waller factor values obtained for both c-Ge and Ge NCs from our experimental spectra are shown as afunction of the measurement temperatures in Fig. 3. Also plotted for comparison are the data for crystalline and amor-phous Ge published previously in Ref. 14. The lines are the respective fits with the correlated Einstein model as given byEq./H208492/H20850for each data set. The values obtained from such fits for the static contribution to the total disorder and for thethermal one /H20849given in terms of the Einstein temperatures /H20850are shown in Table I. As it can be seen, our temperature dependent Debye- Waller factor data for crystalline Ge show the same tempera-ture evolution as the data from Ref. 14, but their absolute values differ by a constant offset. This offset could corre-spond to a static disorder contribution, which in principle isnot expected for bulk c-Ge. But the offset can also be the result of experiment artifacts in fluorescence EXAFS mea-surements. In order to evaluate such effects we have consid-ered normalization, I 0chamber and self-absorption correc- tions. They were calculated using the program TkATOMS/H20849Ref. 32/H20850with the Elam tables for x-ray absorption cross sections. 33The normalization correction accounts for theenergy-dependent attenuation of the amplitude of the EX- AFS signal introduced by the edge-step normalization. It wasestimated as 0.000 05 Å 2in our experiments. The I0correc- tion accounts for the fact that the energy dependence to I0is disregarded when the absorption cross section is calculatedas a function of the incident and fluorescence emitted pho-tons. The I 0chamber used in the experiments was 15 cm long and filled with N2, yielding a correction of 0.000 27 Å2. Finally, the self-absorption correction accounts for the appar-ent amplitude reduction due to the self-absorption of thefluorescing photons by the sample before they reach the de-tector. This contribution is negligible for our samples as theyare much thinner than one absorption length for the Kedge of Ge. 2The absorption length is 9.5 /H9262m while the c-Ge samples are 0.2 /H9262m thick, which amounts to 2.1% of an absorption length. The Ge NCs samples have even smalleramounts of Ge so that this contribution is even smaller forthem. Adding up these corrections it becomes apparent thatthe offset between the crystalline Ge data corresponds toexperiment induced effects rather than to static contributionto the disorder present in the sample. When this offset issubtracted from our data, an excellent agreement is found forboth crystalline Ge datasets. Strauch et al. have used ab ini- tiophonon dynamics calculations to compute Debye-Waller factors for the first three shells of c-Ge in the harmonic approximation. 34Their results, which do not include static contributions to the second cumulant, are also shown in Fig.3. We can see a non-negligible difference between the calcu- lations and the experimental data which indicates that someanharmonicity is present even for the first shell above T /H11011150 K. The Debye-Waller factors for two a-Ge samples prepared by different techniques showed a similar trend with a slightdifference in absolute values. 14,17For clarity, only the data from Ref. 14is plotted in Fig. 3. The sample prepared by thermal evaporation exhibited higher static disorder than thesample prepared by sputter deposition, as listed in Table I. These results were compared in Ref. 17, where the effect of hydrogenation of a-Ge samples /H20849not shown here /H20850was also discussed. We will concentrate on the fact that both samplesshow higher static contribution to /H92682and lower Einstein tem- peratures than both the c-Ge and Ge NCs samples. The static contribution to the Debye-Waller factor for Ge NCs grown by ion-implantation inside a SiO 2matrix lies between that of c-Ge and a-Ge. This indicates that the nano- crystals are in a state of higher configurational energy thanthe crystalline samples, but are not in a state as disordered asTABLE I. Einstein temperatures and static components of the Debye-Waller factors obtained from best fits of the correlated Ein-stein model to the experimental temperature dependent data. SYSTEM /H9008 E/H20849K/H20850 /H92682 static /H20849X10−3Å2/H20850 c-Ge, Ref. 14 355.3±5.7 0 /H20849set/H20850 c-Ge, this work 351.1±7.2 0.34±0.05 Ge NCs, this work 391.4±11.2 1.70±0.07a-Ge/H20849sputtering /H20850, Ref. 17 344.9±3.3 1.98±0.04 a-Ge/H20849evaporation /H20850, Ref. 14 323.3±4.7 2.13±0.10 FIG. 3. Debye-Waller factor /H92682values for several Ge systems /H20849symbols, see figure legend /H20850as a function of the measurement tem- peratures, with the respective correlated Einstein model fits /H20849dashed lines /H20850. The solid line shows the ab initio harmonic calculation for the thermal contribution to /H92682for a c-Ge system.VIBRATIONAL PROPERTIES OF Ge NANOCRYSTALS … PHYSICAL REVIEW B 74, 184102 /H208492006 /H20850 184102-5the amorphous phase. The higher static disorder in the NCs when compared to c-Ge originates from both the reconstruc- tion of the NCs surface due to the presence of under-coordinated atoms and the internal strain in the crystallinecore. 19 Furthermore, the thermal evolution of the Debye-Waller factor for the nanocrystals is slower than the ones for bothc-Ge and a-Ge, as can be observed by the smaller slope of the curve best fitting the data. It must be pointed out that thesame corrections estimated for c-Ge /H20849normalization, I 0and self-absorption /H20850also apply to the case of Ge NCs. As for the Einstein temperatures, it was observed that the value best fitting the Debye-Waller factor data is higher inthec-Ge samples than in both a-Ge samples, suggesting a softening of the compression modes in the amorphous phase.On the other hand, the Einstein temperature best fitting theNC data /H20849391 K /H20850is higher than the c-Ge ones /H20849351 and 355 K /H20850, indicating that the nanocrystals are stiffer than the bulk /H20849have stiffer bonds /H20850. This finding is in agreement with the results obtained for ZnS nanoparticles, 19which were re- ported to be strained and stiffer than the bulk ZnS. The stiff-ening of ZnS nanoparticles could not be explained only bythe radial compression of the nanoparticles, nor by simplemodels such as linear strain or surface-weighted radial strain,and was assigned to inhomogeneous internal strain caused bycompeting relaxations at the surface. On the contrary, mostmetallic nanoparticles appear to behave the opposite way,showing Einstein temperatures lower than the ones observedfor the bulk, as reported in Refs. 35–41. Since the binding characteristics /H20849electronic structure /H20850of metals and semicon- ductors are fundamentally different in their bulk form, it isnot surprising to observe differences between them in thenanocrystalline form. While covalent bonds tend to be stifferand very directional, metallic bonds are softer and less direc-tional, what gives more freedom to surface atoms to move inmetallic nanocrystals, hence the differences between metallicand semiconductor nanocrystals. The harmonic spring constants k eof the effective pair potential obtained from the Einstein temperatures of Table I amount to 8.1 eV/Å2for our c-Ge, which is in good agree- ment with the value obtained for c-Ge in Ref. 6, 8.5 eV/Å2. The value of 10.1 eV/Å2obtained for our Ge NCs corre- sponds to the higher Einstein temperature obtained from thefits for this system. B. Third cumulants The results obtained in the present work for the third cu- mulant C3are shown in Fig. 4. They were obtained restrain- ing the temperature dependent C3values to follow Eq. /H208493/H20850 during the multiple data set fits. Also, the results for c-Ge from Ref. 6anda-Ge from Ref. 17are plotted for compari- son. The C3values for both crystalline samples are in very good agreement. At temperatures below 150 K they are verysmall, but different from zero due to low temperature quan-tum effects and the zero point motion. 6From 150 K on- wards, the C3values show a parabolic raise that is consistent with the classical approximation. The same trend can be observed for the a-Ge data with a constant offset which shifts the data to higher total values,probably due to a static contribution to C 3.For the Ge NCs, however, the picture is somewhat differ- ent. Even at low temperatures, the total values of C3are considerably higher than for c-Ge and a-Ge. This is caused by the higher asymmetry in the distribution of distances forthe NCs, giving rise to a static contribution of 12 /H1100310 −5Å3 toC3in all the temperature range. Depending on the ratio between surface and core atoms and the strain induced in thecrystalline core, the asymmetry in the distribution of dis-tances can be significant for the NCs, even at low tempera-tures. Thus, we ascribe this difference to static asymmetrydue to the relaxation/reconstruction of the surface atoms andthe internal strain existing in the Ge NCs. The thermal only contribution to the C 3of the NCs, also plotted in Fig. 4, evolves with temperature only at a slightly higher rate than for the crystalline sample. This indicates thatthe temperature induced asymmetry is of similar magnitudefor the nanocrystals and bulk Ge in the temperature rangeunder consideration. C. First cumulants and linear thermal expansion The mean interatomic distances obtained from the fits to the experimental spectra correspond to our first cumulant C1, whose values are shown in Fig. 5forc-Ge and Ge NCs. Their variation with temperature gives the local linear ther-mal expansion for the first shell. Comparing the EXAFSthermal expansion to the crystallographic or XRD thermalexpansion /H20849from Ref. 42/H20850forc-Ge, we can see that their difference increases with temperature. This behavior can beassigned to the effect of perpendicular vibrations, as men-tioned earlier. 6,9,10By comparing both data, it is possible to calculate the perpendicular MSRD for c-Ge, as it has been done in Ref. 6. As for the Ge NCs, the thermal increase of the mean interatomic distance was verified to evolve slower with theincrease of temperature when compared to c-Ge. The higher value of C 1for the Ge NCs at 8 K is assigned to structural differences between the crystalline and nanocrystallinephases. FIG. 4. Values of the third cumulant of the distance distributions C3forc-Ge, a-Ge, and Ge NCs. The thermal contribution to C3of the Ge NCs is also plotted individually for comparison.ARAUJO et al. PHYSICAL REVIEW B 74, 184102 /H208492006 /H20850 184102-6In order to compare our first cumulant C1obtained for c-Ge with the real first cumulant C1*obtained through the ratio method analysis in Ref. 6, we calculated the /H9004C1values shown in Fig. 6, where /H9004means the variation relative to the lowest temperature data, i.e., /H9004C1/H20849T/H20850=C1/H20849T/H20850−C1/H208498K/H20850. This is necessary because the ratio method provides only relative values for the effective first cumulant, which are taken with the lowest temperature data as the reference value. The rela- tive values of the realfirst cumulant /H9004C1*are then calculated from the effective ones, as described in Refs. 6and15.I tc a n be seen that the relative values of our first cumulant /H9004C1and the ones from the realfirst cumulant /H9004C1*/H20849calculated in Ref. 6considering /H9261=6 Å /H20850are in good agreement at lower tem- peratures, where the values are really small, but at highertemperatures there is a disagreement of about 0.002 Å. Sucha disagreement might originate from the different way oftreating the kdependence of the mean free path /H9261and of handling the conjugate variable to the distance rin both ap- proaches. While in the ratio method analysis applied in Ref.6a constant value for /H9261was used to convert effective toreal interatomic distances, here the kdependence of /H9261is calcu- lated from the imaginary part of the interaction potential dur-ing the data analysis. V. CONCLUSIONS We have verified that the thermal properties of Ge NCs differ significantly from the ones observed for both c-Ge and a-Ge. Using our approach, we were able to reproduce the thermal behavior of the EXAFS cumulants previously ob-tained for the first shell of bulk c-Ge through the ratio method 6and also obtain original results for Ge NCs. Our results for Ge NCs show that they exhibit a higher Einstein temperature than both a-Ge and c-Ge, indicating stiffer bonds. It was also verified that the linear thermal ex-pansion for Ge NCs is smaller than for c-Ge. These findings are in good agreement with existing data for other nanocrys-talline semiconductor systems. 19–21The fact that the thermal evolution of the first cumulant is lower for the NCs than forthe bulk while the thermal evolution of the third cumulant isslightly higher strengthens the argument that the variation ofthe EXAFS first cumulant should not be considered as givenonly by the quantity afrom the Frenkel-Rehr model. 16In other words, it supports the assumption that the variation ofthe third cumulant should not be used to estimate the thermalexpansion or variation with the temperature of the inter-atomic distances measured by EXAFS. In a recent work, 43it was shown that Ge NCs produced in SiO 2by ion implantation are subject to a strong compressive stress in their as-grown state. This could be one of the rea-sons for the observed damping in the thermal expansion forGe NCs when compared to the ones for c-Ge. If the interac- tion between the SiO 2matrix and the Ge atoms on the sur- face of the nanocrystals is not negligible, the matrix maysuppress the movement of such atoms, increasing their stiff-ness. Furthermore, the stronger this matrix-surface atoms in-teraction is, the higher the static disorder could be. A new study is being carried out in order to further clarify the influence of the SiO 2matrix over the vibrational proper- ties of the Ge NCs. ACKNOWLEDGMENTS L.L.A. and G.de M.A. acknowledge the Brazilian agency CNPq /H20849Conselho Nacional de Desenvolvimento Científico e Tecnológico /H20850for financial support. P.K. and M.C.R. ac- knowledge the Australian Research Council and AustralianSynchrotron Research Program for financial support. Por-tions of this research were carried out at the Stanford Syn-chrotron Radiation Laboratory, a national user facility oper-ated by Stanford University on behalf of the U.S.Department of Energy, Office of Basic Energy Sciences. FIG. 5. Thermal evolution of the interatomic distances for c-Ge and Ge NCs as given by the variation of the first cumulant or mean interatomic distance, symbols. The full line is the thermalevolution of the distance between the equilibrium positions of theatoms as given by XRD. 42 FIG. 6. Relative values of the first cumulant of the distance distribution for the first shell of Ge.VIBRATIONAL PROPERTIES OF Ge NANOCRYSTALS … PHYSICAL REVIEW B 74, 184102 /H208492006 /H20850 184102-7*Corresponding author. Electronic address: lla109@rsphysse.anu.edu.au 1L. Rebohle, J. von Borany, H. Fröb, and W. Skorupa, Appl. Phys. B71, 131 /H208492000 /H20850. 2M. C. Ridgway, G. de M. Azevedo, R. G. Elliman, C. J. Glover, D. J. Llewellyn, R. Miller, W. Wesch, G. J. Foran, J. Hansen,and A. Nylandsted-Larsen, Phys. Rev. B 71, 094107 /H208492005 /H20850. 3G. Beni and P. M. Platzman, Phys. Rev. B 14, 1514 /H208491976 /H20850. 4E. Sevillano, H. Meuth, and J. J. Rehr, Phys. Rev. B 20, 4908 /H208491979 /H20850. 5L. 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PhysRevB.76.104414.pdf

Frequency- and time-domain investigation of the dynamic properties of interlayer-exchange- coupled Ni 81Fe19/Ru/Ni 81Fe19thin films M. Belmeguenai, T. Martin, G. Woltersdorf, M. Maier, and G. Bayreuther Institut für Experimentelle und Angewandte Physik, Universität Regensburg, Universitätsstraße 31, 93040 Regensburg, Germany /H20849Received 24 February 2007; revised manuscript received 25 May 2007; published 14 September 2007 /H20850 Pulsed inductive microwave magnetometer /H20849PIMM /H20850, conventional ferromagnetic resonance /H20849FMR /H20850, and vector network analyzer FMR /H20849VNA-FMR /H20850have been used for complementary studies of the various excited modes in exchange-coupled NiFe /H2084930 nm /H20850/Ru /H20849dRu/H20850/NiFe /H2084930 nm /H20850films with variable Ru thicknesses dRu. For antiferromagnetically coupled layers, two modes, which vary in their relative intensity as a function of the biasfield, are detected. These two modes, which are observable simultaneously over a limited range of the bias fieldwith PIMM, are identified as optic and acoustic modes. The mode frequencies and the interlayer exchangecoupling are found to oscillate as a function of the Ru layer thickness with a period of 8.5 Å. The frequencyoscillations of the optic mode are coupling dependent, while those of the acoustic mode are indirectly relatedto coupling via the canting angle of the layer magnetizations below the saturation. Comparison between PIMMand VNA-FMR in terms of frequency of modes shows good agreement, but the optic mode is observed over awider field range with VNA-FMR. Furthermore, we clearly observed different behaviors of the FMR line-widths as a function of the spacer thickness for the optic and acoustic modes. In addition, perpendicularstanding spin waves have been studied as a function of coupling. The FMR linewidth of the different modesincreases with the microwave frequency and typical damping constants of /H9251=0.0073 have been measured. The effect of the pulse field amplitudes on the properties of the various excited modes has been simulated andstudied experimentally. DOI: 10.1103/PhysRevB.76.104414 PACS number /H20849s/H20850: 75.40.Gb, 76.50. /H11001g, 75.70. /H11002i, 75.30.Et I. INTRODUCTION Exchange interlayer coupling between the magnetizations M1andM2of two ferromagnetic layers separated by a non- magnetic spacer layer was demonstrated experimentally in1986. 1–3This coupling is parametrized by the bilinear J1and biquadratic J2coupling parameters defined via the phenom- enological energy density expression: E=−J1M1·M2 M1M2−J2/H20873M1·M2 M1M2/H208742 . /H208491/H20850 The nature and the strength of the coupling are described by the sign and the magnitude of J1andJ2. When J1dominates, and if it is positive, the energy is minimal when M1andM2 are parallel /H20851ferromagnetic /H20849FM /H20850coupling /H20852, while if it is negative, then the lowest energy is achieved when M1and M2are antiparallel /H20851antiferromagnetic /H20849AF /H20850coupling /H20852. If, on the other hand, J2dominates and is negative, then the mini- mum energy occurs when the magnetizations are orientedperpendicularly to each other /H2084990°-type coupling /H20850. 4 Later discoveries in such coupled structures including gi- ant magnetoresistance5,6led to an explosion in the interest on these systems. Therefore, they are on the base of the devel-opment of many components which are considered now aspotential candidates for magnetic recording devices andtoggled magnetic random access memories. 7However, the precessional dynamics at 1–10 GHz, which determines thehigh-speed response, is a fundamental limit to increasingdata rates in magnetic information storage technology. 8 Therefore, understanding the nature, the extent of exchangeinteractions, and their effect on the damping in such struc-tures at the nanosecond time scale is a technological key forsuch applications.Great attention has been given in recent years to study both experimentally and theoretically the effect of couplingon spin waves in these layered systems. 9,10Brillouin light scattering11and ferromagnetic resonance12are usually used to determine the coupling constants and to study the spin- wave modes, but only methods using pulsed excitation canreach the switching regime. At high excitation amplitudes,the motion of the magnetization becomes enharmonic, and itis best modeled by solving the Landau-Lifshitz-Gilbert/H20849LLG /H20850equation numerically. In addition, a FMR experiment is generally limited to a single frequency and high staticmagnetic fields are used, so that the amount of informationobtained from a FMR measurement is rather limited and it isdifficult to study the dynamics at low fields below the satu-ration. Therefore the aim of this paper is to use pulsed in-ductive microwave magnetometer /H20849PIMM /H20850and vector net- work analyzer FMR /H20849VNA-FMR /H20850, besides a conventional FMR, for full and complementary study of the dynamics ofinterlayer-exchange-coupled systems both in time and fre-quency domain over large static and pulse field ranges.Moreover, and in contrast to the conventional FMR, VNA-FMR and PIMM allow dynamic measurements over a largefrequency range. We focused our study particularly on lowapplied bias fields not sufficient to saturate the specimens, sothat the magnetization was antiparallel or canted at a certainangle with respect to the applied field. In such situation, weshow that not only the frequency of the optic mode dependson the interlayer exchange coupling but also that of theacoustic mode. This paper is organized as follows: we first define our macrospin model and explain how the static and dynamicsimulations are carried out /H20849Sec. II /H20850. Section III introduces the samples and the different experimental setups used forPHYSICAL REVIEW B 76, 104414 /H208492007 /H20850 1098-0121/2007/76 /H2084910/H20850/104414 /H208499/H20850 ©2007 The American Physical Society 104414-1this study. Section IV starts by summarizing the main static characteristics of the samples and then presents dynamicmeasurements, the excited mode properties, and the compari-son between time- and frequency-domain methods. This sec-tion ends by presenting the effect of the coupling on thelinewidth of the different excited spin waves. The effect ofthe pulse field amplitudes on the properties of the excitedspin-waves is studied in Sec. V, and comparison to mac-rospin simulation is presented. In Sec. VI, conclusions aredrawn. II. MACROSPIN CALCULATIONS We consider two magnetic thin films, 1 and 2, of thick- nesses d1and d2separated by a nonmagnetic spacer layer with thickness d. We study only the situation where the ex- ternal static magnetic field His applied in the plane of the films, at an arbitrary angle /H9258Hwith respect to the easy axis direction. In this case, the directions of the magnetizations ofthe two films, M 1andM2, also in the plane, are characterized by the angles /H92741and/H92742with respect to the easy axis direc- tion. The equilibrium directions of M1and M2are deter- mined by the minima of the total free energy per unit areagiven by E tot/H20849/H92741,/H92742/H20850=E1,a/H20849/H92741/H20850d1+E2,a/H20849/H92742/H20850d2+Eex/H20849/H92741,/H92742/H20850. /H208492/H20850 The volume energy density Ea, composed of the Zeeman and anisotropy energies, and the exchange energy density Eexare given by Ea/H20849/H9274/H20850=Kusin2/H9274−/H92620MsHcos /H20849/H9258H−/H9274/H20850, /H208493/H20850 Eex/H20849/H92741,/H92742/H20850=−J1cos /H20849/H92741−/H92742/H20850−J2cos2/H20849/H92741−/H92742/H20850, /H208494/H20850 where Msis the magnetization at saturation and Kuis the uniaxial anisotropy constant. The equilibrium configuration /H20849/H92741,equand/H92742,equ/H20850are de- termined numerically for each applied field, taking J1andJ2 as parameters, and the normalized static hysteresis loops are given by M/H20849H/H20850 Ms=M1cos /H20849/H92741,eq−/H9258H/H20850+M2cos /H20849/H92742,eq−/H9258H/H20850 M1+M2. /H208495/H20850 The spatiotemporal evolution of magnetization of the film iis given by the numerical solution of the equation of motion written as13 dmi/H20849t/H20850 dt=−/H92620/H20841/H9253/H20841 1+/H92512mi/H20849t/H20850/H11003Heff,i −/H9251/H92620/H20841/H9253/H20841 1+/H92512/H20853mi/H20849t/H20850/H11003 /H20851mi/H20849t/H20850/H11003Heff,i/H20852/H20854, /H208496/H20850 where /H9253is the gyromagnetic ratio, /H9251is the phenomenologi- cal damping parameter, and Heff,iis the effective field vector acting on the layer iwith a normalized magnetization vector mi. The effective field comprises the applied field H, the an- isotropy field Hani, the demagnetizing field Hdemag, the pulse field Hpulse, and the bilinear and biquadratic exchange fieldsHJ1andHJ2. These exchange fields are given by HJ1,i=J1 /H92620Msdimj/H20849t/H20850 and HJ2,i=2J2 /H92620Msdi/H20898mi,xmj,x2+mj,x/H20849mi,ymj,y+mi,zmj,z/H20850 mi,ymj,y2+mj,y/H20849mi,xmj,x+mi,zmj,z/H20850 mi,zmj,z2+mj,z/H20849mi,ymj,y+mi,xmj,x/H20850/H20899 with i/HS11005j. /H208497/H20850 For the simulations and in order to be as close as possible to the real case, the real pulse field shape /H20849as measured /H20850is used. Equation /H208496/H20850is numerically integrated using the initial equi- librium state obtained from the static simulation, a step sizeof 1 ps, and a damping parameter /H9251=0.017 which is in a good agreement with the measured one using PIMM. More-over, all the static and dynamic simulations which will bepresented below considered the case of symmetrical mag-netic layers having the same thicknesses and magnetic char-acteristics. III. SAMPLES AND EXPERIMENTAL METHODS A series of Ni 81Fe19/Ru /H20849dRu/H20850/Ni 81Fe19trilayered samples with a fixed Ni 81Fe19thickness of 300 Å and variable Ru thicknesses /H208491.6 Å/H11021dRu/H1102128 Å /H20850was sequentially deposited at room temperature by dc magnetron sputtering onto silicon substrates with Ta seed and cover layers in a commercialsputtering system at IPHT Jena. The Ru thickness gradient/H208491.6–28 Å /H20850is spread over two 6 in. wafers, on which the thickness changes from 1.6 to 9 Å and from 6 to 28 Å, re- spectively. The base pressure of the sputtering system wastypically 10 −8mbar. The deposition rates were about a frac- tion of an angstrom per second. During the growth of theNi 81Fe19layers, a magnetic field of 100 Oe was applied, which induced a uniaxial magnetic anisotropy with definedeasy axis. The easy axes are parallel for both NiFe layers. Magneto-optical Kerr effect /H20849MOKE /H20850and vibrating sample magnetometer /H20849VSM /H20850were used at room temperature to obtain the hysteresis loops for each sample, both in easyand hard axis directions. The measured hysteresis loops werethen fitted numerically by minimizing the total energy of thesystem to determine the coupling constants J 1and J2,a s described in Sec. II. For samples with FM coupling wherethe determination of the interlayer exchange coupling con-stants cannot be performed using static methods, FMR mea-surements have been used to determine the total coupling/H20849effective coupling: J eff/H20850. The maximum saturation fields both in easy and hard axes are in the range of 0.8–1.4 kOefor 4.6 Å /H11021d Ru/H110216.4 Å and they vary from 3 to 500 Oe oth- erwise. The dynamic measurements were carried out by FMR, PIMM, and VNA-FMR. For PIMM and VNA-FMR, thesamples of 1 cm 2are coupled to a coplanar waveguide and the experimental setups are described in Refs. 14and15, respectively. For both methods, the data at each bias fieldBELMEGUENAI et al. PHYSICAL REVIEW B 76, 104414 /H208492007 /H20850 104414-2require the subtraction of two measurements: one with the bias field switched on and a second measurement with a1 kOe saturating field applied in the same direction as thepulse or the rf field, which removes all magnetic responsefrom the measured quantity /H20849transmission coefficient: S 21in decibel in the case of VNA-FMR and in voltage in PIMM /H20850. By subtracting this saturated measurement from the bias fieldmeasurement, all that remains is the effect of the oscillatoryresponse created by the precessing magnetization in thesample. In our case, a maximal bias field of 1 kOe was ap-plied before each bias field measurement in order to definean initial state. This field was reduced to the target bias fieldbefore the voltage pulse or rf field. The resonance frequen-cies are obtained from the Fourier transform of the time-domain magnetic response or from the Lorentzian fit of theS 21measured by VNA. For the FMR measurements, the experimental setup is the same as described in Ref. 15. The magnetic sample is mounted inside a shorted waveguide. The microwave absorp-tion is measured by monitoring the power reflected from thesample using a mixer. The sample is swept through the reso-nance condition by means of an external field. When themagnetic sample undergoes a ferromagnetic resonance, themicrowave losses are increased and the reflected powerchanges slightly. In addition, the external magnetic field ismodulated with an amplitude of 2 Oe at a frequency of130 Hz. This modulation allows lock-in detection to be usedin order to increase the signal-to-noise ratio. The measuredFMR signal is proportional to the field derivative of theimaginary part of the rf susceptibility. The FMR experimentswere carried out using 22 and 35 GHz systems. IV . RESULTS AND DISCUSSION A. Static characterization VSM hysteresis loops for a NiFe/Ru/NiFe trilayer with a 4.9-Å-thick Ru layer are shown in Fig. 1/H20849a/H20850. The data corre- spond to the field aligned along the easy and hard axes. Thehysteresis loop in the hard axis has been shifted horizontallyby 100 Oe for clarity. Comparison of the two curves indi-cates that the anisotropies in the system are small. Note theslow approach to saturation above 0.5 kOe for both cases.This asymptotic behavior, regardless of field orientation, sug-gests that a strong coupling exists across the Ru film. Theremanence is large, between 55% and 60% of saturation de-pending on field orientation. It is possible to explain the re-manence and approach to saturation with the presence of alarge biquadratic coupling between the ferromagnetic layersacross the Ru spacer. This is in good agreement with Fig.1/H20849b/H20850, where mean values of the interlayer coupling constants J 1andJ2, determined by fitting the VSM and MOKE hyster- esis loops as indicated in Sec. II, are plotted as a function ofthe spacer thickness. For each Ru thickness, J 1/H20849J2/H20850presented here is the average between J1/H20849J2/H20850obtained from VSM and that obtained from MOKE. Positive values of J1indicate FM coupling and negative values indicate AF coupling /H20849when J2 is neglected /H20850. One clearly recognizes an oscillatory behavior ofJ1as a function of the spacer thickness, which is attenu- ated for larger dRu. The coupling is purely FM and AF fordRu/H110214.3 Å and dRu/H110226.7 Å, respectively, and a non- negligible biquadratic coupling is present for samples of Ruthicknesses between these two regimes. The oscillation pe-riod is 8.5 Å and slightly smaller than usually measured inthe Co/Ru systems /H20849about 11 Å /H20850. B. Dynamic measurements by pulsed inductive microwave magnetometer and vector network analyzer ferromagnetic resonance In analogy with coupled harmonic oscillators, the magnon modes in two magnetic films coupled via a nonmagnetic in-terlayer can be classified into acoustic and optic modes de-FIG. 1. /H20849a/H20850Easy and hard axis hysteresis loops of Si/Ta/NiFe /H2084930 nm /H20850/Ru /H208494.9 Å /H20850/NiFe /H2084930 nm /H20850/Ta obtained by vi- brating sample magnetometer /H20849VSM /H20850. Arrows indicate the magne- tization states for different applied fields. /H20849b/H20850Mean values of the bilinear /H20849J1/H20850and biquadratic /H20849J2/H20850interlayer coupling constants of Si/Ta/NiFe /H2084930 nm /H20850/Ru /H20849dRu/H20850/NiFe /H2084930 nm /H20850/Ta as a function of the Ru thickness. The coupling constants have been determined by fit-ting the VSM and MOKE hysteresis loops numerically for antifer-romagnetically coupled samples and by FMR for ferromagneticallycoupled ones /H20849effective coupling /H20850. For each Ru thickness, J 1/H20849J2/H20850 presented here is the average between J1/H20849J2/H20850obtained from VSM and that obtained from MOKE. The corresponding error bar for J1 andJ2is also given.FREQUENCY- AND TIME-DOMAIN INVESTIGATION OF … PHYSICAL REVIEW B 76, 104414 /H208492007 /H20850 104414-3pending on whether the two film magnetizations precess in phase or out of phase, respectively. This assignment isstraightforward when the film magnetizations are in parallelalignment. For the antiparallel configuration, the two magne-tizations precess in opposite directions, and hence, their rela-tive phase changes continuously. Behavior of the spin-wave frequencies as a function of applied fields provides a great deal of information about themagnitude and functional form of the coupling energy. Overthe whole range of the spacer thickness, the typical experi-mental resonance frequencies as a function of the externalin-plane bias field /H20849H/H20850, measured by PIMM and VNA-FMR, are shown in Figs. 2/H20849a/H20850and2/H20849b/H20850for two Ru thicknesses of 4.9 and 14.8 Å. For d Ru=4.9 Å, both J1andJ2are large and the AF coupling is strong, while for dRu=14.8 Å, the cou- pling is weak and mainly J1exists. Our experimental results have been fitted by the model presented in Ref. 10using the parameters indicated in the caption of Fig. 2. These two samples show qualitatively similar behavior and will be dis-cussed together. There are two different frequencies whichappear in different field regimes. The variation of the modefrequencies with the external magnetic field relates to thedifferent magnetic states of the two NiFe magnetizations.These modes are identified as the optic and acoustic spin-wave modes of the coupled ferromagnetic films. This hasbeen confirmed by our simulations by comparing the phasesof the two modes after numerical solution of the LLG equa-tion. It is also in good agreement with the model of Zivieri et al. 9which predicts that for AF coupled films and at low bias fields, the acoustic mode has the lower frequency while theoptic mode has the higher frequency. However, above a criti-cal applied field /H20849H cr/H20850, which is coupling dependent /H20851see Fig. 2/H20849c/H20850/H20852a crossover between the two mode frequencies occurs and the situation is reversed /H20849i.e., the position of the acoustic mode frequency switches with that of the optic one /H20850,a si n - dicated in Fig. 2. Thus, in AF coupled multilayers, the mag- netic ground state develops as a function of the applied fieldand the classification of “acoustic” and “optic” modes aslower and higher frequency modes, respectively, is not gen-erally valid. Therefore, the knowledge of the magnetizationstate corresponding to the applied bias field is necessarywhen identifying these modes. We note that in order to resolve the optic mode over a large field range and, in particular, at its intersection with theacoustic mode, we used a different measurement configura-tion similar to the longitudinal FMR, which is more sensitiveto the optical mode. Therefore, instead of applying the rffield perpendicular to the bias field, both fields were parallelto each other. We note that, in this case, both modes have astronger signal in VNA-FMR compared to PIMM, wherethey can only be observed over a narrow field window /H20849see Fig.2/H20850. Figure 2/H20849b/H20850shows that at very low fields /H208490/H11021H /H1102120 Oe /H20850the magnetizations align antiparallel to each other /H20851see inset of Fig. 2/H20849b/H20850/H20852. Therefore, the optic mode has the higher frequency. The discontinuity in the frequencies seenin the simulations 10at 20 Oe reflects the spin-flop transition. In this spin-flop phase /H20849H/H1102220 Oe /H20850, the angle between the magnetizations continuously decreases from 180° to 0°.FIG. 2. /H20849Color online /H20850Frequencies of the optic and acoustic modes of Si/Ta/NiFe /H2084930 nm /H20850/Ru /H20849dRu/H20850/NiFe /H2084930 nm /H20850/Ta structure as a function of the in-plane bias field and for /H20849a/H20850dRu=4.9 Å and /H20849b/H20850 dRu=14.8 Å. These frequencies are obtained by fitting the module of the transmission coefficient S 21measured by VNA-FMR to a Lorentzian and by the Fourier transform of the time-domain mag-netic response measured by PIMM. Pulse field of 8.5 Oe is used forPIMM measurements. The effective exchange field is given by thefield difference above the saturation between the correspondingresonance fields at a fixed frequency, as indicated by the dottedlines. The inset shows the VSM easy axis hysteresis loop /H20849normal- ized magnetization versus static field in Oe /H20850. The corresponding simulations are obtained from the model of Ref. 10using a uniaxial anisotropy field H ani=5 Oe with /H20849a/H20850J1=−377 /H9262J/m2and J2= −514/H9262J/m2, and /H20849b/H20850J1=−140 /H9262J/m2andJ2=−15/H9262J/m2./H20849c/H20850Ru thickness dependence of the critical field /H20849Hcr/H20850, which is the value of the bias field where the frequencies of the optic and acousticmodes are equal /H20849crossover /H20850. The corresponding frequency, called critical frequency, is also represented here as a function of d Ru.BELMEGUENAI et al. PHYSICAL REVIEW B 76, 104414 /H208492007 /H20850 104414-4Above 20 Oe, the optic mode frequency and intensity start to decrease before disappearing /H20851Fig.2/H20849b/H20850/H20852. It should have a dip when the sample saturates /H20851see simulation10in Fig. 2/H20849b/H20850/H20852. The acoustic mode frequency increases continuously andforms a kink around the saturation field. In the saturated statebeyond 140 Oe, both acoustic and optic mode frequenciesincrease with the field. Therefore, at a fixed frequency, thefield difference between the optic and acoustic modes isequal to the effective exchange field /H208492H ex/H20850. The obtained value for effective coupling is in good agreement with that obtained from the fit of the VSM and MOKE measurements. In Fig. 2/H20849a/H20850, similar behaviors to Fig. 2/H20849b/H20850are found, with the difference that the optic frequency increases slowly untilit reaches a maximum around 600 Oe, where again it startsto decrease. This behavior has also been reported by Kuanret al. for Fe/Al/Fe trilayer. 16We found this behavior for all the samples with 4.3 Å /H33355dRu/H3335511.2 Å, where the estimated J2is larger than or comparable to J1. The field, where the maximum of the optic mode frequency occurs, scales withthe coupling strength. This behavior of the optic mode fre-quency reported in Fig. 2/H20849a/H20850is a consequence of the contri- bution of bilinear and biquadratic interlayer exchanges andZeeman energy to the effective stiffness of the magnetiza-tions and can be reproduced with a simple single spinmodel 10using the mean values of J1andJ2obtained from the fit of the VSM and the MOKE hysteresis loops /H20851see Fig. 2/H20849a/H20850 for simulations /H20852. The frequency offset of the optic mode in the simulation is caused by the presence of a significanttwisting of the magnetization along the film normal in thereal sample. This additional effect can be treated by amultilayer simulation as shown by Buchmeier et al. 17 The effect of the biquadratic coupling on the mode fre- quency has been studied theoretically by Layadi.10For anti- ferromagnetic coupling, two situations can arise. When theapplied field is greater than the saturation field, the magneti-zations are parallel and, for the same parameters, the reso-nant frequency of the acoustic mode is constant while that ofthe optic one decreases almost linearly as J 2increases. On the other hand, and for the same parameters, when the mag-netizations are antiparallel, the mode behavior is different.The resonant frequencies of the optic mode and the acousticmode decrease as J 2increases, and the amplitudes of both modes are nonzero and vary with J2. Moreover, the effective coupling /H20849Jeff/H20850defined as Jeff=J1+2J2andJeff=J1−2J2in the parallel and antiparallel states, respectively,11increases /H20849decreases /H20850with increasing J2/H20849J2/H110210/H20850for antiferromagnetic coupling for parallel /H20849antiparallel /H20850states. Therefore, with in- creasing AF coupling strength, the optic mode frequencyshifts up for antiparallel alignment and down for parallelalignment because the AF coupling represents a restoringforce for the antiparallel alignment but not for the parallelalignment. 11 For a fixed bias field value, the frequencies of the optic and acoustic modes oscillate as a function of dRuwith the same period as J1/H20849Fig.3/H20850. The frequency of the optic mode strongly depends on the interaction, i.e., the interlayer cou-pling, whereas the acoustic modes /H20849above saturation /H20850are in- dependent of the coupling strength, again in analogy tocoupled harmonic oscillators. However, the acoustic modefrequency /H20851Fig. 3/H20849b/H20850/H20852depends on the alignment of the filmmagnetizations. This frequency is constant for strong FM coupled /H20849d Ru/H333553.7 Å /H20850and uncoupled NiFe /H20849dRu/H3335616.5 Å /H20850 layers, where the two magnetizations are collinear and par- allel to the applied field. Therefore, we attribute these oscil-lations of the acoustic mode as a function of d Ru, which vanish /H20851Fig.3/H20849b/H20850/H20852when the bias field is above 1 kOe /H20849field where all the samples are mostly saturated /H20850, to the canting angle of the two magnetizations which is coupling depen-dent. Similar trends have been reported in Ref. 18. C. Dynamic measurements by conventional ferromagnetic resonance The typical obtained FMR spectrum at 22 GHz is shown in Fig. 4for FM /H20849dRu=3.7 Å /H20850, AF coupled /H20849dRu=4.9 Å and dRu=14.8 Å /H20850, and uncoupled layers /H20849dRu=18.3 Å /H20850. The two higher field modes /H20849modes 3 and 4 in Fig. 4/H20850are the usual acoustic and optic modes, while the two other modes atFIG. 3. Mode frequencies of the Si/Ta/NiFe /H2084930 nm /H20850/ Ru /H20849dRu/H20850/NiFe /H2084930 nm /H20850/Ta coupled system as a function of the spacer thickness dRu./H20849a/H20850Optic mode frequencies measured by PIMM and VNA-FMR at an easy axis applied bias field of 5 Oeand /H20849b/H20850acoustic mode frequencies at the indicated easy axis applied bias fields. The acoustic mode frequencies presented here are ob-tained from VNA-FMR measurements.FREQUENCY- AND TIME-DOMAIN INVESTIGATION OF … PHYSICAL REVIEW B 76, 104414 /H208492007 /H20850 104414-5lower fields /H20849modes 1 and 2 in Fig. 4/H20850, which are observable at this frequency only for strong AF coupling /H208494.6 Å/H33355dRu /H333556.4 Å /H20850, are supposed to be the first perpendicular standing spin wave /H20849PSSW /H20850corresponding to the two NiFe layers. The acoustic mode /H20849mode 3 in Fig. 4/H20850is independent of the exchange energy and, therefore, is degenerate with the reso-nance field of uncoupled system /H20849Fig.4/H20850. In the case of AF /H20849FM /H20850coupling, the optic mode is at higher /H20849lower /H20850field with respect to the acoustic mode. With increasing coupling, itsresonance field increases /H20849decreases /H20850. In the case of thin film layer of thickness d m, assuming long in-plane wavelengths and under the approximation ofunpinned spins at the film surfaces which is well justified forNiFe due to the small anisotropies, the frequencies of thePSSWs are given by 19 fP=/H92620/H9253 2/H9266/H20873/H20875H+2A Meff/H20873P/H9266 dm/H208742/H20876/H20875H+2A Meff/H20873P/H9266 dm/H208742 +Meff/H20876/H208741/2 , /H208498/H20850 where Ais the exchange stiffness constant, His the external magnetic field, /H9253is the gyromagnetic factor /H20849/H9253/2/H9266 =29.5 GHz/T for Permalloy /H20850, and Pis the index of the PSSW mode. The three lowest modes corresponding to P=0, 1, and 2 are represented in Fig. 5for a NiFe layer of 30 nm in thick- ness for /H92620Meff=1.064 T and A=1.3/H1100310−11J/m.20There- fore, in our finite-band PIMM and VNA-FMR setups, wecannot observe PSSW modes /H20849P/H333561/H20850. The FMR measure- ments at 22 and 35 GHz show that as the AF couplingstrength decreases, the resonance field of the PSSW moves to lower values in the same manner as the optic mode and ingood agreement with the theoretical models. 21Moreover, the effect of the interlayer coupling on the PSSW can be ana-lyzed through the simple model proposed by Wigen et al. 22 This model assumes that each magnetic sublayer is resonat- ing with a nearly uniform amplitude but different phase,caused by interlayer coupling via the spacer. In other words, different sublayers resonate at different amplitudes to giverise to an overall spin wave. This model can only be appliedin the case that the interlayer coupling is much smaller thanthe intralayer one /H20849the exchange coupling Ashould be di- vided by the lattice parameter to be compared to the inter-layer coupling /H20850. Therefore, it is an extension of the spin- wave model for single layer magnetic thin films expressed byEq. /H208498/H20850, and the exchange stiffness constant Acan also be replaced by an effective coupling constant A eff. Using this model, we calculated Aeffcorresponding to the measured resonance fields of the PSSW modes at 22 GHz /H20851Fig.6/H20849a/H20850/H20852. This plot shows clearly that AF interlayer coupling reducesthe stiffness constant A eff, and we converge to that of a single layer in the case of zero coupling. Aeffoscillates with the Ru thickness in the same manner as the interlayer coupling. The PSSWs /H20849P=1 /H20850were also observed at microwave fre- quency of 35 GHz, but their amplitudes were smaller com- pared to those observed at 22 GHz due to the low signal-to-noise ratio at this frequency. Therefore, in contrast to themeasurements at 22 GHz, we observed only one PSSW/H20849similar to mode 2 in Fig. 4/H20850. One should mention that in contrast to mode 2 /H20849Fig.4/H20850, the resonance fields of mode 1 at 22 GHz /H20849observed for 4.6 Å /H33355d Ru/H333556.4 Å /H20850are below the saturation fields in this Ru thickness range, suggesting thatthis mode /H20849mode 1 /H20850is due to the canted magnetizations. Moreover, the PSSWs were not observed for the FM coupledsample /H20849d Ru/H110214Å /H20850. The excellent agreement between FMR results obtained at 22 and 35 GHz for uncoupled layers and those calculated using Eq. /H208498/H20850suggests that, as expected, the uniform /H20849acoustic /H20850mode is well fitted by this model /H20849P=0 /H20850FIG. 4. /H20849Color online /H20850 FMR spectrum of Si/Ta/NiFe /H2084930 nm /H20850/Ru /H20849dRu/H20850/NiFe /H2084930 nm /H20850/Ta measured at 22 GHz. FMR signal is proportional to the field derivative of theimaginary part of the rf susceptibility. Each derivative is a modeand is referenced by an integer number to identify /H20851/H208491/H20850and /H208492/H20850/H20852the perpendicular standing spin waves /H20849PSSW /H20850corresponding to each NiFe layer, /H208493/H20850the acoustic mode, and /H208494/H20850the optic mode. The spectra have been shifted vertically with respect to that correspond-ing to d Ru=14.8 Å for clarity. Inset shows a zoom in on the optic mode corresponding to dRu=4.9 Å.FIG. 5. /H20849Color online /H20850PSSW: P=1 and P=2, and uniform /H20849acoustic /H20850mode: P=0 corresponding to a NiFe thin film of 30 nm thickness obtained from Eq. /H208498/H20850using /H92620Meff=1.064 T, A=1.3 /H1100310−11J/m, and /H9253/2/H9266=29.5 GHz/T. Blue squares and cyan circles indicate respectively the corresponding FMR and VNA-FMR measurements of uniform mode /H20849acoustic mode /H20850:P=0 and the first PSSW: P=1 for uncoupled layers /H20851Si/Ta/NiFe /H2084930 nm /H20850/Ru /H208491.83 nm /H20850/NiFe /H2084930 nm /H20850/Ta /H20852.BELMEGUENAI et al. PHYSICAL REVIEW B 76, 104414 /H208492007 /H20850 104414-6and confirms that the lower field peaks /H20849mode 2 /H20850are the first PSSW modes. Figure 6/H20849b/H20850shows the dependence of the FMR linewidth of optic, acoustic, and PSSW modes on the spacer thicknessfor the NiFe/Ru/NiFe trilayers. These linewidths repre-sented here for FMR measurements at 22 GHz are defined asthe peak-to-peak linewidths of the derivative Lorentzian. Thekey observation in Fig. 6/H20849b/H20850is that the linewidths of the acoustic and optic modes behave significantly differently.Like its resonance field, the linewidth of the acoustic mode isalmost constant as a function of d Ru. Assuming a linear de- pendence of the FMR linewidth /H20849/H9004H/H20850on the microwave fre- quency /H20849f/H20850and fitting the measured results at 22 and 35 GHz to Eq. /H208499/H20850,23we have determined the effective Gilbert damp- ing/H9251as a function of Ru thickness dRu. The obtained results show that the damping is almost constant and fluctuatearound an average damping of /H9251=0.0073, which is in good agreement with NiFe thin films24/H208490.008 in Ref. 24/H20850. This can be explained by the fact that the magnetization vector ofeach FM layer is saturated at the resonance and the acousticmode is independent of the exchange energy and, thus, isdegenerate with the resonance mode of uncoupled system.Therefore, the fluctuation of /H9251with dRuis due to the varia-tions of the interface quality and to the inhomogeneities. This is also in agreement with the theoretical models which pre-dict a constant linewidth in symmetrical coupled trilayers, 25 /H9004H/H20849f/H20850=/H9004H/H208490/H20850+ 1.16/H92512/H9266f /H9253, /H208499/H20850 where /H9004H/H208490/H20850is the zero-frequency offset and is caused by magnetic inhomogeneities and, therefore, its origin is extrin- sic. Due to the inhomogeneity of the exchange coupling, the linewidth of the optic mode is usually larger than that of theacoustic mode. Interestingly, the linewidth and the resonancefield of the optic mode oscillate as function of d Ruin the same manner as the coupling /H20851see Figs. 1/H20849b/H20850and6/H20849b/H20850/H20852. This correlation with the coupling oscillations supports the expla-nation that the broadening of the linewidth is due to theinhomogeneous exchange interlayer coupling. However, incontrast to its resonance field, the linewidth of the PSSWdoes not oscillate with the Ru thickness d Ru. Its decrease with dRuis drastically for small thickness, but it remains higher than that of the acoustic mode and the uncoupledlayers. Moreover, this linewidth increases with the micro-wave frequency. V . EFFECT OF THE PULSE FIELD AMPLITUDES In our PIMM setup, we are able to increase the exciting pulse fields up to 150 Oe by applying voltage pulses of up to200 V and 250 ps duration to the coplanar waveguide.Therefore, large excitation angles in both layers can be ob-tained. However, at such large precession angles, the fre-quency spectrum gets rather complex, and micromagnetic orat least macrospin /H20849like in our case /H20850simulations are helpful for interpreting the obtained results. To validate our macrospin model presented in Sec. II, let us consider the case of the sample of 14.8 Å studied above indetail /H20851Fig. 2/H20849b/H20850/H20852. With increasing pulse fields, the PIMM measurements /H20849Fig.7/H20850show two significant effects. First, it can be seen that the optic mode is observable over a largerrange of the bias field, while the threshold /H20849the bias field value where its intensity becomes significant /H20850of the acoustic mode increases with higher pulse fields. Second, we observeat bias fields values around 100 Oe a higher harmonic of theacoustic mode. This behavior is well reproduced by the mac-rospin simulation /H20849Fig.7/H20850despite the fact that the agreement decreases at the highest excitations /H20849not shown here /H20850. This is an indication that at such high excitations, the macrospinapproximation is no longer valid due to inhomogeneous pre-cession. For the sample with 6.4 Å of thickness and in addition to the optic and acoustic modes, we observe a very intense thirdmode for which the frequency varies strongly with the pulsefield amplitude /H20849Fig.8/H20850. This mode, only observable in anti- parallel configuration at low bias fields, has been observedfor Ru thickness 6.1 Å /H33355d Ru/H333556.7 Å. As shown in Fig. 8, for low excitation amplitudes, only the optical mode is visible.With increasing pulse field amplitudes, a third mode appearsand the acoustical mode becomes more intense. For higherexcitation amplitudes, plenty of modes are present and onlyFIG. 6. Dependence of /H20849a/H20850the effective exchange stiffness cou- pling /H20849Aeff/H20850and /H20849b/H20850the FMR linewidth of the PSSW and the optic and acoustic modes on the spacer thickness dRu in Si/Ta/NiFe /H2084930 nm /H20850/Ru /H20849dRu/H20850/NiFe /H2084930 nm /H20850/Ta coupled systems. The measurements presented here were carried out at 22 GHz. Aeff was calculated by replacing AbyAeffin Eq. /H208497/H20850and using the measured PSSW resonance fields.FREQUENCY- AND TIME-DOMAIN INVESTIGATION OF … PHYSICAL REVIEW B 76, 104414 /H208492007 /H20850 104414-7the optic mode can be identified clearly. The pulse field de- pendence of this third mode frequency is most probably at-tributed to the large change of the direction of H J1,iand of the strength and direction of HJ2,ias the magnetizations of the two layers undergo large excitation angles during the first nanosecond of the precession /H20851compare Eq. /H208497/H20850/H20852. By this, the direction and also the strength of the effective field varystrongly in this time range. This is confirmed by the fact thatthis third mode is not visible any more in the fast Fouriertransform spectrum when omitting the first 1.5 ns of thetime-domain data, whereas the optic mode remains visible.However, this mode could not be reproduced by the mac-rospin simulation, suggesting that its origin lies beyond themacrospin model. VI. CONCLUSION The high frequency magnetization dynamics of interlayer coupled NiFe/Ru/NiFe films has been studied by three dif-ferent methods. We detected two modes that we identified asoptic and acoustic modes. The high frequency optic mode isdominant at low bias, while in higher fields, the acousticmode has the largest intensity. The oscillatory nature of theacoustic mode frequency, at low bias fields, with Ru thick- ness was attributed to the canting angle of the magnetiza-tions. Comparison between PIMM and VNA-FMR in termsof frequency of modes shows good agreement, but the opticmode is more observable with VNA-FMR. The first mode ofthe perpendicular standing spin-waves has been observedwith FMR for AF and uncoupled layers. The analysis of theobtained results via a simple model shows that the AF inter-layer coupling reduces the effective exchange stiffness.Moreover, the FMR measurements showed different behav-iors of the linewidths as a function of the spacer thickness forthe optic and acoustic modes. The FMR linewidth of thedifferent modes increases with the microwave frequencies,and typical damping constants of 0.0073 have been mea-sured. The effect of the pulse field amplitudes on the prop-erties of the different excited spin waves shows the existenceof additional modes at high pulse field amplitudes for somesamples. The macrospin simulations are in good agreementwith the measurements. ACKNOWLEDGMENTS This work was supported in part by the European com- munity’s Marie Curie actions /H20849Research Training Networks /H20850 under Contract No. MRTN-CT-2003-504462 and by Deut-sche Forschungsgemeinschaft /H20849DFG /H20850SPP1133. The authors would like to thank C. Back for discussions and for puttingat their disposal some experimental setups during this study,and M. Scheinfein for fruitful discussions regarding the mac-rospin simulations. 1P. Grünberg, R. Schreiber, Y. Pang, M. B. Brodsky, and H. Sow- ers, Phys. Rev. Lett. 57, 2442 /H208491986 /H20850. 2C. F. Majkrzak, J. W. Cable, J. Kwo, M. Hong, D. B. McWhan, Y. Yafet, J. V. Waszczak, and C. Vettier, Phys. Rev. Lett. 56, 2700 /H208491986 /H20850. 3M. B. Salamon, S. Sinha, J. J. Rhyne, J. E. Cunningham, R. W. Erwin, J. Borchers, and C. P. Flynn, Phys. Rev. Lett. 56, 259 /H208491986 /H20850.4P. Grünberg, Acta Mater. 48, 239 /H208492000 /H20850. 5M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas,Phys. Rev. Lett. 61, 2472 /H208491988 /H20850. 6G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Phys. Rev. B39, 4828 /H208491989 /H20850. 7B. N. Engel, J. Åkerman, B. Butcher, R. W. Dave, M. DeHerrera, M. Durlam, G. Grynkewich, J. Janesky, S. V. Pietambaram, N.FIG. 8. Different mode frequencies for a Si/Ta/NiFe /H2084930 nm /H20850/Ru /H208496.4 Å /H20850/NiFe /H2084930 nm /H20850/Ta sample at 8 Oe bias field where the magnetizations are in antiparallel configuration. FIG. 7. /H20849Color online /H20850Comparison of PIMM results and mac- rospin simulations corresponding to optic and acoustic mode fre-quencies of Si/Ta/NiFe /H2084930 nm /H20850/Ru /H2084914.8 Å /H20850/NiFe /H2084930 nm /H20850/Ta for 8.5 Oe pulse field /H20849upper row /H20850and 85 Oe pulse field /H20849lower row /H20850. The color scale indicates the mode intensities. The part of the graphrepresented here is hatched in Fig. 2/H20849b/H20850.BELMEGUENAI et al. 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PhysRevB.91.214430.pdf

PHYSICAL REVIEW B 91, 214430 (2015) Magnetic spheres in microwave cavities Babak Zare Rameshti,1,2Yunshan Cao,2and Gerrit E. W. Bauer2,3 1Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran 2Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 3Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai 980-8577, Japan (Received 9 March 2015; revised manuscript received 4 June 2015; published 25 June 2015) We apply Mie scattering theory to study the interaction of magnetic spheres with microwaves in cavities beyond the magnetostatic and rotating wave approximations. We demonstrate that both strong and ultrastrong coupling can be realized for stand alone magnetic spheres made from yttrium iron garnet (YIG), acting as an efficient microwave antenna. The eigenmodes of YIG spheres with radii of the order mm display distinct higherangular momentum character that has been observed in experiments. DOI: 10.1103/PhysRevB.91.214430 PACS number(s): 71 .36.+c,75.30.Ds,75.60.Ch,85.75.−d I. INTRODUCTION Light-matter interaction in the strong-coupling regime is an important subject in coherent quantum informationtransfer [ 1–3]. Spin ensembles such as nitrogen-vacancy centers may couple strongly to electromagnetic fields andhave the advantage of both long coherence times [ 4] and fast manipulation [ 5]. The “magnon” refers to the collective excitation of spin systems. In paramagnetic spin ensembles inan applied magnetic field, the spins precess coherently in thepresence of microwave radiation, creating hybridized statesreferred to as magnon-polaritons [ 6–8]. In the strong-coupling regime coherent energy exchange exceeds the dissipative lossof both subsystems. The coherent coupled systems is usuallydescribed by the Tavis-Cummings (TC) model [ 9,10], which defines a coupling constant gbetween the spin-ensemble and the electromagnetic radiation that scales with the square root ofthe number of spins. In ferro/ferrimagnets the net spin densityis exceptionally large and spontaneously ordered, which makesthose materials very attractive for strong-coupling studies.The exchange coupling of spins in magnetic materials alsostrongly modifies the excitation spectrum into a spectrum or spin wave band structure. An ubiquitous experimental technique to study ferromagnetism is ferromagnetic resonance(FMR), i.e., the absorption, transmission, or reflection spectraof microwaves. In the weak-coupling regime FMR gives directaccess to the elementary excitation spectrum of ferromagnets[11], including the standing spin waves in confined systems referred to spin wave resonance (SWR) [ 12]. The strong- coupling regime is studied less frequently, however, because the dissipative losses of the magnetization dynamics are usually quite large. An exceptional magnetic material is the manmade yttrium iron garnets (YIG), a ferrimagnetic insulator. Commerciallyproduced high-quality spherical YIG samples serve in magnet-ically tunable filters and resonators at microwave frequencies.By suitable doping becomes a versatile class of materials with low dissipation and unique microwave properties [ 13]. YIG has spin density of 2 ×10 22cm−3[14], and the Gilbert damping (re- ciprocal quality) factor of the magnetization dynamics rangesfrom 10 −5to 10−3[15–17], which facilitates strong coupling for smaller samples. Indeed, strongly coupled microwavephotons with magnons have been experimentally reported foreither YIG films with broadband coplanar waveguides (CPWs) [18–20], or YIG spheres in 3D microwave cavities [ 21–23]. Aseries of anticrossings were observed in thicker YIG films and split rings [ 19,20]. The coupling of magnons in YIG spheres with a superconducting qubit via a mircowave cavity mode inthe quantum limit has been reported [ 21]. An ultrahigh coop- erativity C=g 2/κγ > 105,where κandγare the loss rates of the cavity and spin system, and multimode strong coupling were found at room [ 22]a sw e l la st h el o w[ 23] temperatures. From a theoretical point of view, the standard TC model is too simple to describe the full range of coupling betweenmagnets and microwaves. Also the rotating-wave approxi-mation (RW A) (usually but not necessarily assumed in theTC model) is speaking applicable when the coupling ratiog/ω c/lessmuch1, where ωcis the microwave cavity mode frequency. We may define different coupling regimes [ 24,25], viz. (i) strong coupling (SC) when 0 .01<g / ω c/lessorsimilar0.1, (ii) ultrastrong coupling (USC) [ 26] when g/ωc/greaterorsimilar0.1, (iii) or even deep strong coupling (DSC) g/ωc≈1[27]. Cao et al. [8] adapted the TC model to ferromagnets by formulating a first-principlesscattering theory of the coupled cavity-ferromagnet systembased on the Maxwell and the Landau-Lifshitz-Gilbert equa-tion including the exchange interaction. A effectively one-dimensional system of a thin film with in-plane magnetizationin a planar cavity was solved exactly in the linear regime,exposing, for example, strong coupling to standing spin waves.Maksymov et al. [28] carried out a numerical study of the strong-coupling regime in all-dielectric magnetic multilayersthat resonantly enhance the microwave magnetic field. Aquantum theory of strong coupling for nanoscale magneticspheres in microwave resonators has been developed in themacrospin approximation [ 29], but this regime has not yet been reached in experiments. Here we apply our classical method [ 8] to spherically symmetric systems, i.e., a magnetic sphere in the center of a spherical cavity. This is basically again a one-dimensionalproblem that can be treated semianalytically and has otheradvantages as well, such as a homogeneous dipolar field andsimple boundary conditions. The eigenmodes of magneticspheres have been studied in the “magnetostatic” approxi-mation [ 30,31], in which the spins interact by the magnetic dipolar field, disregarding exchange as well as propagationeffects, which may be done when λ/greatermucha, where ais the radius of the sphere and λthe wavelength of the incident radiation. Arias et al. [32] treated the interaction of magnetic spheres with microwaves in the weak-coupling regime. In contrast, we 1098-0121/2015/91(21)/214430(7) 214430-1 ©2015 American Physical SocietyZARE RAMESHTI, CAO, AND BAUER PHYSICAL REVIEW B 91, 214430 (2015) address here the properties of the fully hybridized magnon- polaritons beyond the magnetostatic approximation (but dis- regard the exchange interaction), including the propagation effects (reflection and transmission) of microwaves, therebyextending the validity to λ<a . We are admittedly still one step from the “exact” solution by disregarding the exchange(as treated and discussed by Cao et al. [8]). Our calculated microwave spectra are complex but help in understanding someof the above-mentioned experiments. This manuscript is organized as follows. In Sec. II,w e introduce the details of our model and derive the scatteredintensity and efficiency factors for a strongly coupled systemof a magnetic sphere and microwaves. In Sec. III, we present and discuss our numerical results that demonstrate the effectsboth due to the dielectric as well as magnetic effects on the scat-tering properties and compare our results with experiments. InSec. IV, we conclude and summarize our findings. II. MODEL AND FORMALISM We model the coupling of the collective excitations of a magnetic sphere to microwaves in a spherical cavity by thecoupled Landau-Lifshitz-Gilbert and Maxwell equations. Weemploy Mie-type scattering theory, i.e., a rapidly convergingexpansion into spherical harmonics [ 33–35]. We model the incoming radiation as plane electromagnetic waves witharbitrary polarization and wave vector that are scattered bya cavity loaded by a magnetic sphere with gyromagneticpermeability tensor←→μ[36]. In order to understand the experiments it is not necessary to precisely model the detailsof the resonant cavity. Instead, we propose a generic modelcavity that is flexible enough to mimic any realistic situationby adjusting the parameters. We consider a thin spherical shellof a material with high dielectric constant /epsilon1 c//epsilon10/greatermuch1,radius R, and thickness δthat confines standing microwave modes with adjustable interaction with the microwave source (seeFig. 1). The spherical symmetry simplifies the mathematical treatment, while the parameters Randδallow us to freely tune the frequencies and broadenings of the cavity modes. The dynamics of the magnetization vector Mis described by the LLG equation, ∂ tM=−γM×Heff+α MsM×∂tM, (1) withαandγbeing the Gilbert damping constant and gyromagnetic ratio, respectively. The effective magnetic fieldH eff=Hext+Hxcomprises the external and (collinear) easy axis anisotropy fields Hextas well as the exchange field Hx=J∇2M, with Jbeing the exchange stiffness. Assuming that perturbing microwave magnetic field and magnetizationprecession angles are small, M(r,t)=M s+m(r,t), (2) H(r,t)=Hext+h(r,t), (3) where Msis the saturated magnetization vector and mthe small-amplitude magnetization driven by the rf magnetic fieldh,we linearize the LLG equation to ∂ tm=−γMs×/parenleftbigg H(1) eff−α γMs∂tm/parenrightbigg −γm×H(0) eff,(4) FIG. 1. (Color online) Plane wave with wave vector k0coming in at an arbitrary angle hits a large spherical cavity modeled by adielectric spherical shell of radius R, thickness δ, and permittivity /epsilon1 c. The spherical cavity is loaded with a magnetic sphere of radius a centered at the origin of the coordinate system. where H(0) eff=HextandH(1) eff=Hx+h. The response of fer- romagnetic spheres is affected by exchange when their radiiapproach the exchange length. Since the latter is typically afew nm, we hereafter disregard the exchange interaction andconcentrate on the dipolar spin waves. In the frequency domainand taking the zdirection as the equilibrium direction for the magnetization, iωm=z×(ω Mh−ωHm+iωαm), (5) withωM=γMsandωH=γH 0. We may recast Eq. ( 6)i n t o the form m=←→χ·h. The magnetic susceptibility tensor←→χis related to the magnetic permeability tensor by←→μ=μ0(←→1+←→χ). We find ←→μ=μ0⎛ ⎝1+χ−iκ 0 iκ 1+χ 0 00 1⎞ ⎠, (6) where χandκare given by χ=(ωH−iαω)ωM (ωH−iαω)2−ω2, (7) κ=ωω M (ωH−iαω)2−ω2. (8) The permeability tensor appears in the Maxwell equations for the propagation of the electromagnetic wave in a magneticmedium. Inside a spatially homogeneous medium a monochromatic wave with frequency ω, ∇×E=iωb,∇×h=−iωD, (9) ∇·D=0,∇·b=0. (10) The constitutive relation between the magnetic induction b, electric displacement D, magnetic field h, and the electric 214430-2MAGNETIC SPHERES IN MICROW A VE CA VITIES PHYSICAL REVIEW B 91, 214430 (2015) fieldEinside this medium are b=←→μ·h,D=/epsilon1spE, (11) where /epsilon1spis the scalar permittivity of the medium. It follows from Eqs. ( 10) and ( 11) that the magnetic induction bsatisfies the wave equation, ∇×∇× (μ0←→μ−1·b)−k2 spb=0, (12) withk2 sp=ω2/epsilon1spμ0. The surrounding (nonmagnetic) medium is homogeneous and isotropic with scalar magnetic permeability μ0,d i v e r - genceless magnetic field, and simplified wave equation ∇2b+ k2 spb=0. Due to the spherical symmetry it is advantageous to expand the magnetic field hinto vector spherical harmonics as [34,35,37,38] h=/summationdisplay nm¯ηnm/bracketleftbig pnmM(1) nm(k,r)+qnmN(1) nm(k,r)/bracketrightbig , (13) where nruns from 1 to ∞, andm=−n ,..., n with prefactors ¯ηnm=ηnmk0/(ωμ 0), ηnm=inE0/bracketleftbigg2n+1 n(n+1)(n−m)! (n+m)!/bracketrightbigg1/2 . (14) E0is the electric field amplitude of the incident wave. The vector spherical harmonics read [ 34,35,37,38] M(j) nm(k,r)=z(j) n(kr)Xnm(ˆr), kN(j) nm(k,r)=∇× M(j) nm(k,r). (15) z(j) n(kr) are spherical Bessel functions, Xnm(ˆr)=LYnm(ˆr)/√n(n+1) spherical harmonics, and L=−ir×∇ rthe an- gular momentum operator with ∇rthe gradient opera- tor. The electric field distribution is obtained by E= (i/ωc )∇×h. By invoking the vector spherical wave function expansion for band←→μ−1·bin the wave equation Eq. ( 12) leads to the dispersion relation fork(ω). We match the field distributions inside and outside the cavity to obtain the scattering solution for incident planemicrowaves. The field inside the spherical shell must beregular, while the scattered component has to satisfy thescattering wave boundary conditions at infinity. These con-ditions are fulfilled by adopting the first kind of spher-ical Bessel function j n(x) as the radial part for the in- ternal distribution and the first kind of spherical Hankelfunction h (1) n(x) for the scattered component outside the cavity hs=/summationdisplay nm¯ηnm/bracketleftbig cnmN(3) nm(k0,r)+dnmM(3) nm(k0,r)/bracketrightbig . (16) The unknown scattering coefficients cnmanddnmare deter- mined by the boundary conditions at the interface. We considerhere the situation in which the magnetic sphere is illuminatedby a plane wave with arbitrary direction of propagation andpolarization as indicated in Fig. 1. The incident field can be expanded as h inc=−/summationdisplay nm¯ηnm/bracketleftbig unmN(1) nm(k0,r)+vnmM(1) nm(k0,r)/bracketrightbig .(17)The expansion coefficients umnandvmn, unm=[pθ˜τnm(cosθk)−ipφ˜πnm(cosθk)]e−imφk, (18) vnm=[pθ˜πnm(cosθk)−ipφ˜τnm(cosθk)]e−imφk, (19) contain all information about the polarization vector and direction of propagation, where ˆp=(pθˆθk+pφˆφk)i st h e normalized complex polarization vector, with |ˆp|=1 and θk(φk) is the polar (azimuthal) angle of k0. Two auxiliary functions are defined by ˜πnm=tnmm sinθPm n(cosθ),˜τnm=tnmd dθPm n(cosθ),(20) withtnm=i−nηnm/E0andPm n(x) the first kind associated Legendre function. In order to solve the full scattering problem including the cavity we match the fields outside the cavity causedby the incoming plane microwave and the spacer regionseparating the magnetic particle and cavity. In the latter,spherical Bessel functions of both the first and second kindhave to be included into the expansion. At the surface ofthe magnetic sphere ( r=a) we adopt the standard boundary conditions h i×er=hmid×er, (21) Ei×er=Emid×er, (22) while at the surface of the cavity, assuming that its thickness is much smaller than the wavelength, [ 39,40] [hmid−hout]×er=−ξ[er×Eout]×er, (23) Emid×er=Eout×er. (24) The indexes midandoutindicate the regions within and outside of the cavity, respectively. The unit vector eris the outward normal to the surfaces and ξ=iω(/epsilon1c−/epsilon10)δwith permittivity of the cavity shell /epsilon1c. By matching the field distributions in the different regions the scattering coefficients are determined,from which we calculate the observables. At distances sufficiently far from the cavity, i.e., in the far field zone, the intensity of the two polarization components I 1 andI2are I1∼E2 0 k2 0r2|S1(θ,φ)|2, (25) I2∼E2 0 k2 0r2|S2(θ,φ)|2, (26) where θ(φ) is the polar (azimuthal) angle of the observer at distance r. The scattering amplitude functions are S1(θ,φ)=/summationdisplay nm[dnm˜τnm(cosθ)+cnm˜πnm(cosθ)]eimφ,(27) S2(θ,φ)=/summationdisplay nm[dnm˜πnm(cosθ)+cnm˜τnm(cosθ)]eimφ,(28) where the coefficients cnmanddnmcharacterize the scattered component of the fields outside the cavity. We may nowcompute the scattering and extinction cross sections as wellas their (dimensionless) efficiencies Q scaandQext, which are 214430-3ZARE RAMESHTI, CAO, AND BAUER PHYSICAL REVIEW B 91, 214430 (2015) the cross sections normalized by πR2, the geometrical cross section of the cavity: Qsca=4 k2 0R2/summationdisplay nm(|cnm|2+|dnm|2), (29) Qext=4 k2 0R2/summationdisplay nmRe(u∗ nmdnm+v∗ nmcnm). (30) The extinction cross section represents the ratio of (angle- integrated) emitted to incident intensity, i.e., with and withoutthe scattering cavity/particle between source and detector.This factor measures the energy loss of the incident beam byabsorption and scattering. The series expansion in Eqs. ( 27)– (30) is uniformly convergent and can be truncated at some point in numerical calculations depending on the desired accuracy.In the next section we present our results with emphasis onthe dielectric and magnetic contributions to the microwavescattering. III. RESULTS Here we present numerical results on the coupling of microwaves with a ferro- or ferrimagnet in a cavity based onour treatment of Mie scattering of the electromagnetic wavesas exposed in the preceding section. It applies to a dielec-tric/magnetic sphere centered in a (larger) spherical cavity, butboth may be of arbitrary diameter otherwise. We are mainlyinterested in the coherent coupling between the magnons andmicrowave photons in the strong or even ultrastrong couplingregimes that can be achieved by generating spectrally sharpcavity modes, by increasing the filling factor of the cavity,or simply by increasing the size of the sphere. The RW A,however, tends to break down as the coupling increases.This has led to different coupling regimes beyond the weak coupling, TC region, i.e., strong (SC) and ultrastrong (USC) coupling regimes. In the SC region coupling strength has tobe comparable or larger than all decoherence rates, while inthe USC it has to be comparable or larger than appreciablefractions of the mode frequency, g/ω c/greaterorsimilar0.1. We adopt the forward scattered intensities I1∼|S1(θ= π/2,φ=π)|2and scattering efficiency factors as convenient and observable measures of the microwave scattering bya spherical target. In order to compare results with recentexperiments, we chose parameters for YIG with gyromagneticratioγ/(2π)=28 GHz /T, saturation magnetization [ 41] μ 0Ms=175 mT, Gilbert damping constant [ 15–17]α= 3×10−4, and relative permittivity [ 42]/epsilon1//epsilon1 0=15.Without loss of generality we consider microwaves incident from thepositive xdirection ( θ k=π/2 and φk=0) and polarization (pθ,pφ)=(1,0), so its electric/magnetic components are in the−z/y directions (static magnetic field and magnetization H0/bardblz). Forward scattering is monitored by setting θ=π/2 andφ=πin Eq. ( 27). We also explore the dependence of the observables on the scattering angles. We can remove the cavitysimply by setting ξ=0. In Fig. 2the scattered intensity |S 1(θ,π)|2is depicted as a function of frequency ω/2πand scattering angle θfocusing first on a nonmagnetic sphere with radius a=1.25 mm. The angular dependence of the scattering with and without a cavity (with R=1.6 mm) is plotted in panels (a) and (b), respectively. The eigenmodes of the dielectric sphere shows-,p-, and d-wave characters in Fig. 2(a).s-wave scattering dominates as long as the wavelength (reduced by /epsilon1 sp) does not fit twice into the sphere, i.e., λ/greaterorapproxeqla/radicalbig/epsilon1sp//epsilon10. The spherical cavity, on the other hand, limits the isotropic scattering regimetoλ/greaterorapproxeqlR/radicalbig /epsilon1sp//epsilon10. 0.0 0.5 1.010203040506070 θ / πω / 2π (GHz) 0.0 0.5 1.0 θ / π34.8 46.4 58.0 69.6 81.2(c) (b) Loading rate a/R (%)(a) FIG. 2. (Color online) Scattering intensity |S1|2as function of scattering angle θand frequency ω/2πis shown for (a) a dielectric sphere of radius a=1.25 mm and relative permittivity /epsilon1//epsilon1 0=15 and for (b) the same sphere in a cavity of radius R=1.6 mm. In (c) the scattering intensity is plotted for the same cavity as function of frequency and loading rate a/R . The dashed lines are guides for the eye. 214430-4MAGNETIC SPHERES IN MICROW A VE CA VITIES PHYSICAL REVIEW B 91, 214430 (2015) 123456751015202530(a) (2,2)(2,-2) (1,1)(1,-1)(3,-3) (3,-3)(2,2)(3,3)(2,-2) (1,1) H0 / Msω / 2π (GHz) (1,-1)(b) 0510 1234567n = 2 n = 1 H0 / Ms FIG. 3. (Color online) Panel (a) shows the scattering efficiency factor Qscaas function of normalized magnetic field H0/Msand frequency ω/2πfor a YIG sphere of radius a=2 mm and relative permittivity /epsilon1//epsilon1 0=15. Panel (b) shows results for a nonmagnetic dielectric sphere. The character of the microwave modes sufficientlyfar from the anticrossing with the spin waves is labeled by the spherical harmonic indices ( n,m). In Fig. 2(c) we plot the forward scattered intensities I1as function of the load of the cavity by a dielectric sphere. Theeigenfrequencies of the cavity remain constant, while thoseconfined to the sphere shift to lower frequencies as ∼a −2.A t high loading rate the cavity modes are strongly mixed withthe modes in the sphere and all of them bend towards lowerfrequencies. Magnetism of the spheres can affect the microwave scat- tering properties strongly, but the issue of hybridization ofcavity and sphere resonant microwave modes is still present.A sufficiently large YIG sphere alone can therefore providestrong-coupling conditions to the magnetization even withoutan external resonator. To this end, the linear dimension of theYIG sphere must be of a size that allows the internal resonancesof the sphere to come into play in the microwave frequencyrange, i.e., when ka/greaterorapproxeqlπ/radicalbig /epsilon10//epsilon1sporλ/lessorapproxeql2a/radicalbig/epsilon1sp//epsilon10.W e therefore have a (narrow) regime a/radicalbig/epsilon1sp//epsilon10/lessorapproxeqlλ/lessorapproxeql2a/radicalbig/epsilon1sp//epsilon10 or 7.75 mm /lessorapproxeqlλ/lessorapproxeql15.49 mm (for Fig. 3) in which strong coupling and s-wave scattering can be realized simultaneously without a cavity. YIG spheres can typically be fabricated withhigh precision for radii in the range [ 43]a=0.9–2.5m m . In Fig. 3for aa=2 mm YIG sphere we observe a strong anticrossing between the linear spin wave modes and thesphere-confined standing microwaves. The YIG sphere istherefore an efficient microwave antenna that achieves strongand ultrastrong coupling without a cavity. It should be notedthat previous works [ 44–48], which have revealed the possi- bility to use all-dielectric as well as all-magneto-dielectricresonators without external resonator, were not consideredstrong coupling. Our results help to interpret recent experimental results on YIG spheres in microwave cavities with reported couplingstrength that are comparable with the magnon frequency [ 22], i.e., in the ultrastrong-coupling regime. In Fig. 4the scattering efficiency factor is shown as a function of H 0/Msandω/2π. Panel (a) addresses a YIG sphere of radii a=1.25 mm in a spherical microwave cavity of radii R=1.6 mm, chosen to be close to the leading dimensions of the cavity in theexperiments. Panel (b) holds for the same YIG sphere butwithout cavity. The obvious anticrossing in Fig. 4(a) is a2468 1 0 1 2353637383940 (1,-1)(2,2)(2,2) (3,-3) (3,3) (1,1)(2,-2) H0 / Msω / 2π (GHZ)(2,-2) 2468 1 0 1 2(3,3) (1,-1)(2,-2) (2,2) (1,1)(b) H0 / Ms(a) 0510 FIG. 4. (Color online) Scattering efficiency factor Qscaplotted as function of normalized magnetic field H0/Msand frequency ω/2π for a YIG sphere of radius a=1.25 mm and relative permittivity /epsilon1//epsilon1 0=15 (a) in the center of a spherical cavity of radius R=1.6m m and (b) without cavity. signature of the emergence of the hybrid excitation that we refer to as magnon-polariton . The anticrossing modes are labeled by the mode numbers ( n,m). For given nthere are twom=±nanticrossing modes with coupling strengths gn,n>gn,−n, where gn,mis the effective coupling strength of the magnon mode ( n,m) to the cavity. Figure 4(a) indicates that the ultrastrong-coupling strength is indeed approached sincea splitting of g/2π=2.5 GHz is achieved at a resonance frequency of ω/2π/similarequal37.5 GHz. Beside the main anticrossing with the (2 ,2) and (2 ,−2) cavity modes, we observe tails from other anticrossings with the (3 ,3) and (3 ,−3) modes at higher frequencies, as well as the (1 ,1) and (1 ,−1) modes at lower frequencies, which are standing electromagnetic resonancemodes confined by the YIG sphere. We may interpret theseas nearly pure spin wave modes that acquire some oscillatorstrengths by mixing from far away resonances due to theultrastrong coupling with standing microwaves. This can beverified by checking the scattering efficiency factor inthe absence of the cavity as in Fig. 4(b), which emphasizes the antenna action of the YIG sphere. Zhang et al. [22] indeed report additional, weakly coupled “higher modes,” but without explaining their nature. Theyreport ultrastrong coupling between magnons and the cavityphotons only in the frequency range of 35–40 GHz, but data atlower frequencies are not given. In Fig. 5we extend the plots in Fig. 4to a larger frequency interval. We observe that the main anticrossing in the frequency range of 35–40 GHz is causedby the n=2 modes, while hybridized modes originating from the n=1 resonance exist at the lower frequencies. The unperturbed modes between the anticrossing gaps are thereforenot only due to the higher modes, but lower modes with n=1 also contribute by the ultrastrong coupling. Two significantcurves in the left and right side of the higher unperturbedmodes originate from the anticrossing modes n=1( t h el e f t one) and n=2 (the right one) of the YIG sphere itself, as is more clear in Fig. 5(b) (the computed lines are broader because we use a relatively large κfor computational convenience). We thereby find again that the strong-coupling magnon-polaritonmay form also without cavity. We concentrated on the dipolar spin wave excitations driven by magnetic fields that are strongly inhomogeneousdue to a large dielectric constant. We disregard here exchange 214430-5ZARE RAMESHTI, CAO, AND BAUER PHYSICAL REVIEW B 91, 214430 (2015) 2468 1 0 1 2101520253035404550 (2,-2) (2,2) (1,1)(1,-1)(b) (3,-3) (3,3) (2,2) (1,1) H0 / Msω / 2π (GHZ) (1,-1)(a) 2468 1 0 1 2(2,-2)(3,-3) (3,3) (2,-2) (2,2)(2,2) (2,-2) (1,-1) (1,-1)(1,1) (1,1) H0 / Ms0510 FIG. 5. (Color online) Scattering efficiency factor Qscaas func- tion of normalized magnetic field H0/Msand frequency ω/2π for a YIG sphere of radius a=1.25 mm and relative permittivity /epsilon1//epsilon1 0=15 (a) in the center of a spherical cavity of radius R=1.6m m and (b) in the absence of the cavity. Dashed lines indicate the frequency range in Fig. 4. interactions, thereby limiting the validity of the treatment to YIG spheres much larger than the so-called exchange lengththat for YIG is only a few nanometers. In other words, wecannot properly describe all spin waves with relatively largewave number or frequencies relatively much higher relativeto the FMR frequency. Indeed, in the planar configurationspin wave resonances are observable for rather thick films [ 8]. Exchange-induced whispering gallery modes on the surface ofthe YIG might therefore be observable even in thicker spheres,but their treatment is tedious and beyond the scope of thepresent paper. IV. CONCLUSION In this paper we implement Mie scattering theory to study the interaction of dielectric as well as magnetic spheres withmicrowaves in cavities by the coupled LLG and Maxwell equations, disregarding only the exchange interaction. Weare mainly interested in the coherent coupling between themagnons and microwave cavity modes in the strong- oreven ultrastrong-coupling regimes characterized by the mode-dependent coupling strengths g n,m. We reveal that while in the presence of a spherical cavity both strong and ultrastrongcoupling can be realized by tuning the cavity modes andby increasing the filling factor of the cavity. Surprisingly,these regimes can also be achieved by removing the externalresonator, due to the strong confinement of electromagneticwaves in sufficiently large YIG spheres. In this regime, higherangular momentum eigenmodes of the dielectric sphere partic-ipate and the scattering shows s-a sw e l la s p-wave character. We thereby transcend studies that focus on dipolar spinwaves in a magnetostatic framework [ 30,31] by considering propagation effects via the full Maxwell equation. Our studymight be useful in designing optimal conditions to designcavities in which YIG spheres are coherently coupled to, e.g.,superconducting qubits, in microwave cavities for coherentquantum information transfer [ 21]. ACKNOWLEDGMENTS B.Z.R. thanks S. M. Reza Taheri and Y . M. Blanter for fruitful discussions. The research leading to these resultshas received funding from the European Union SeventhFramework Programme [FP7-People-2012-ITN] under Grantagreement No. 316657 (Spinicur). It was supported by JSPSGrants-in-Aid for Scientific Research (Grants No. 25247056,No. 25220910, and No. 26103006), FOM (Stichting voorFundamenteel Onderzoek der Materie), the ICC-IMR, EU-FET InSpin 612759, and DFG Priority Programme 1538“Spin-Caloric Transport” (BA 2954/1-2). [1] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf,Nature (London) 431,162 (2004 ). [2] Y . Kubo, F. R. Ong, P. Bertet, D. Vion, V . Jacques, D. Zheng, A. Dr ´eau, J. F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, Phys. Rev. Lett. 105, 140502 (2010 ). [3] S. Putz, D. O. Krimer, R. Ams ¨uss, A. Valookaran, T. N. ¨obauer, J. Schmiedmayer, S. Rotter, and J. Majer, Nat. Phys. 10,720 (2014 ). [4] N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, Nat. Commun. 4,1743 (2013 ). [5] L. Childress, M. V . Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. 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PhysRevB.99.014411.pdf

PHYSICAL REVIEW B 99, 014411 (2019) Excitation and control of large-amplitude standing magnetization waves L. Friedland*and A. G. Shagalov† Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel; Institute of Metal Physics, Ekaterinburg 620990, Russian Federation; and Ural Federal University, Mira 19, Ekaterinburg 620002, Russian Federation (Received 23 July 2018; revised manuscript received 29 October 2018; published 10 January 2019) A robust approach to excitation and control of large amplitude standing magnetization waves in an easy axis ferromagnetic by starting from a ground state and passage through resonances with chirped frequencymicrowave or similar alternative drives (spin torque, additional periodic anisotropy) is proposed. The formationof these waves involves two stages, where in the first stage, a spatially uniform, precessing magnetizationis created via passage through a resonance followed by a self-phase-locking (autoresonance) with a constantamplitude drive. In the second stage, the passage through an additional resonance with a spatial modulation ofthe driving amplitude yields transformation of the uniform solution into a doubly phase-locked standing wave,whose amplitude is controlled by the variation of the driving frequency. The stability of this excitation processis analyzed both numerically and via Whitham’s averaged variational principle. DOI: 10.1103/PhysRevB.99.014411 I. INTRODUCTION Because of the complexity and despite decades of studies, magnetization dynamics in ferromagnetic materials remainsof interest to basic and applied research. For example, non-linear spin waves and solitons in ferromagnetic films werestudied experimentally extensively (e.g., Refs. [ 1–4]). Magne- tostatic and boundary effects in such macroscopic films yieldcomplex dispersion of the spin waves. Depending on the signof the dispersion both bright and dark magnetic solitons wereobserved. The long wavelength approximation in this problemyields the nonlinear Schrodinger (NLS) model, providinga convenient theoretical basis for investigation. The NLSequation has well known traveling wave and soliton solutions[5], allowing interpretation of the experimentally observed magnetization dynamics. In recent years, applications in ferromagnetic nanowires opened new perspectives in studying magnetization wave-forms [ 6]. At the nanoscales a quasi-one-dimensional symme- try can be realized and magnetostatic effects can be reducedto additional contributions to the anisotropy [ 6–8], which can be conveniently modeled by the Landau-Lifshitz-Gilbert(LLG) equation. It is known that the one-dimensional (1D),dissipationless LLG equation, similar to the NLS equation,is integrable and has a multitude of exact solutions includingsolitons and spatially periodic waveforms [ 6,9,10], expected to be observed in nanowires. The simplest solitons are domainwalls, which are studied extensively [ 11–15]a sab a s i sf o rn e w memory and logic devices [ 16,17]. A different type of solitons are so-called breathers [ 6], which can be interpreted as an interacting pair of domain walls with opposite topologicalcharges (soliton-antisoliton pair). They are stable localized *lazar@mail.huji.ac.il †shagalov@imp.uran.ruobjects in easy-axis ferromagnetic when dissipation is negli-gible [ 9], which was also illustrated in numerical simulations [18]. These breather solitons correspond to the bright NLS solitons in the small amplitude approximation [ 9]. Solitons in ferromagnetic nanowires with a spin polarized current werealso discussed in Refs. [ 19,20] in the framework of a modified NLS model. In this paper, we focus on excitation of large amplitude standing LLG waves in an easy axis ferromagnetic, such thatthe projection M zof the magnetization vector Mon the easy axis is independent of time and periodic in z, while M⊥ precesses uniformly around the axis. These waves approach a soliton limit as their wavelength increases (see below). Thequestion is how to generate such waves by starting from asimple initial equilibrium and how to control their dynamics. Excitation by an impulse or localized external fields usuallyare unsuitable for generating pure large amplitude standingwaves because of significant residual perturbations. Here,we suggest a simple method of exciting these waves basedon the autoresonance approach via driving the system by asmall, chirped frequency external rotating magnetic field orsimilar alternative drives. This approach allows us to excitethe waves with a predefined amplitude and phase and stabilizethem with respect to dissipation. The autoresonance approachuses the salient property of a nonlinear system to stay inresonance with driving perturbations despite slow variation ofparameters. The idea was used in many applications startingfrom particle accelerators [ 21,22], through planetary dynam- ics [ 23], [24] and atomic physics [ 25,26], to plasmas [ 27], magnetization dynamics in single domain nanoparticles [ 28– 30], and more. Autoresonant excitation of both bright and dark solitons and spatially periodic multiphase waves within theNLS model were studied in Ref. [ 31–33], while the autores- onant control of NLS solitons is described in Refs. [ 34,35]. In all these applications, one drives the system of interestby an oscillating perturbation, captures it into a nonlinear 2469-9950/2019/99(1)/014411(10) 014411-1 ©2019 American Physical SocietyL. FRIEDLAND AND A. G. SHAGALOV PHYSICAL REVIEW B 99, 014411 (2019) resonance, while slowly varying the driving frequency (or other parameter). The resulting continuing self-phase-locking(autoresonance) yields excursion of the system in its solutionsspace, frequently leading to emergence and control of nontriv-ial solutions. In this work, motivated by the aforementionedresults in related driven-chirped NLS systems, we apply asimilar approach yielding arbitrary amplitude, standing mag-netization waves. The scope of the presentation will be as follows. In Sec. II, we introduce our autoresonant magnetization model and dis-cuss the problem of capturing the system into resonance witha chirped frequency microwave field followed by formation ofan autoresonant, spatially uniform magnetization state. In Sec.III, we study transition from the uniform state to a standing wave by spatially modulating the amplitude of the chirpedfrequency drive. In the same section, we will illustrate thisprocess in simulations and present a qualitative picture of thedynamics. Section IVwill be focused on the theory of the autoresonant standing waves and discuss their modulationalstability via Whitham’s averaged variational principle [ 36]. In Sec. V, we illustrate excitation of the standing waves via two alternative driving mechanisms involving spin torque or theaddition of spatially modulated hard axis anisotropy. Finally,Sec. VIwill present our conclusions. II. AUTORESONANT MAGNETIZATION MODEL Our starting point is the 1 DLandau-Lifshitz-Gilbert (LLG) equation for a ferromagnetic with the easy axis along /hatwideez in an external magnetic field H=H0/hatwideezand in the pres- ence of a weak rotating driving microwave field Hd= bcos(cos ϕd/hatwideex+sinϕd/hatwideey) having spatially periodic ampli- tudeb=b0+b1cos(kz)(k=2π/L, L being periodicity length) and slowly chirped frequency ωd(t)=−∂ϕd/∂t: ∂m ∂τ=h×m+λm×∂m ∂τ. (1) Hereλis the Gilbert damping parameter, h=∂2m ∂ξ2+(mz+h0)/hatwideez+ε(cosϕd/hatwideex+sinϕd/hatwideey), (2) and we use normalized magnetization m=M/M, dimensionless time τ=(γK/M )t, and coordinate ξ=z/δ, δ =√A/K (γ, A , andKbeing the gyromagnetic ratio, the exchange constant, and the anisotropyconstant, respectively). Furthermore in Eq. ( 2),h 0= MH 0/K, ε =ε0+ε1cos(κξ),ε0,1=Mb 0,1/K, ϕ d= −/integraltext /Omega1ddτ,/Omega1d(τ)=ωdM/(Kγ),κ=2π/l, andl=L/δ. The proposed approach to excitation of magnetization waves requires realization of the proper driving field.For example, consider the Permalloy parameters A= 10 −11J/m,K=105J/m3, andM=8×105A/m[15]. This yields the characteristic width δ=10 nm, the linear res- onance frequency (see below) f0=γK/ (2πM)(1+h0)= 3.5(1+h0) GHz, and the driving magnetic field amplitude b0=ε0K/M =3.75×10−4Tf o r ε0=0.003 (as in exam- ples in Fig. 3below). The periodicity length Lin our driven problem is lδ. We will use l∼10 in the examples below, which corresponds to L∼100 nm. Spatial modulation of the microwave magnetic field on this submicron scale is difficult.However, analogous autoresonant excitations of the magneti- zation wave can be obtained by introducing other componentsin the effective magnetic field hof a similar form but due to different physical effects. Two such alternatives will bediscussed in Sec. V. We seek spatially periodic solutions of Eq. ( 1) and proceed from the dissipationless version of this equation in polar coor-dinates ( m x=sinθcosϕ, m y=sinθsinϕ, m z=cosθ): θτ=/Phi1ξξsinθ+2/Phi1ξθξcosθ−εsin/Phi1, (3) /Phi1τ=/parenleftbigg −1 sinθθξξ+/Phi12 ξcosθ/parenrightbigg +cosθ−/Omega1/prime d(τ) −εcotθcos/Phi1, (4) where/Phi1=ϕ−ϕdis the phase mismatch and /Omega1/prime d=/Omega1d−h0. This system is a spatial generalization of the recently stud-ied autoresonant magnetization switching problem in single-domain nanoparticles [ 28,29], where one neglects the spatial modulation of the driving amplitude (so ε=ε 0) and the spatial derivatives in Eqs. ( 3) and ( 4) to get θτ=−εsin/Phi1, (5) /Phi1τ=cosθ−/Omega1/prime d(τ)−εcotθcos/Phi1. (6) In the 1D ferromagnetic case, Eqs. ( 5) and ( 6) describe a spatially uniform, rotating around the axis magnetizationdynamics. In the rest of this section, we discuss formation andstability of autoresonant uniform states in the dissipationlesscase but include dissipation in numerical simulations forcomparison. The autoresonance idea is based on a self-sustained phase locking of the driven nonlinear system to chirped frequencydriving perturbation. Typically this phase locking is achievedby passage through resonance with some initial equilibrium.In our case, we assume linearly chirped driving frequency/Omega1 /prime d(τ)=1−ατfor simplicity, proceed from θ≈0(mz=1) at large negative time, and slowly pass the resonance /Omega1/prime d=1 atτ=0. For small θEqs. ( 5) and ( 6) can be written as dθ dτ=−εsin/Phi1, (7) θd/Phi1 dτ=(ατ−θ2/2)θ−εcos/Phi1, (8) which can be transformed into a single complex equation for /Psi1=θei/Phi1 id/Psi1 dτ+(ατ−|/Psi1|2/2)/Psi1=ε. (9) This NLS-type equation was studied in many applications and yields efficient phase locking at /Phi1≈πafter passage through linear resonance at τ=0,provided εexceeds a threshold [ 37] εth=0.58α3/4. (10) Later (for τ> 0), the phase locking continues as the non- linear frequency shift follows that of the driving frequency,i.e.,θ 2/2≈ατ. Importantly, this continuing phase locking is characteristic of any variation of the driving frequency[thenαin (9) represents the local frequency chirp rate at the 014411-2EXCITATION AND CONTROL OF LARGE-AMPLITUDE … PHYSICAL REVIEW B 99, 014411 (2019) 0 50 100−0.500.5−1−0.500.51 T ξ/l−mz 0 50 100−0.500.5−1−0.500.51 T ξ/l−mz 0 50 10024 TΦ 0 50 10024 TΦ(a) (b) (c) (d) FIG. 1. The uniform autoresonant magnetization state. (a) zcom- ponent of magnetization −mzversus slow time T=α1/2τ; (c) phase mismatch /Phi1(0,T)=φ−φd. In both panels λ=0a n d ε=3× 10−3. Panels (b) and (d) are the same as (a) and (c), but λ=3×10−3 andε=3×10−2. initial resonance], while the system remains in an approximate nonlinear resonance mz=cosθ≈/Omega1/prime d(τ), (11) as long as the driving frequency chirp rate remains suf- ficiently small. Under these conditions, the magnetizationangles θandϕ≈ϕ d+πare efficiently controlled by simply varying the driving frequency. We illustrate this effect inFig. 1, showing the results of numerical simulations of the original system ( 1), assuming spatial periodicity of length l= 6 and linearly chirped frequency /Omega1 /prime d(τ)=1−ατ. The initial conditions θ=0.01|cos(κξ)|(κ=2π/l) represented a small spatial perturbation for studying stability of the uniform stateand we used parameters λ=0,h 0=5,ε=3×10−3, and α=5×10−4in Figs. 1(a) and 1(c), while λ=3×10−3 andε=3×10−2in Figs. 1(b) and 1(d). Our numerical scheme used an equivalent system of two coupled NLS-typeequations based on the quantum two-level analog [ 29,38] described in the Appendix. Figures 1(a) and 1(b) (without and with damping, respectively) show the evolution of −m z= −cosθversus slow time T=α1/2τ, which approximately follows the linear time dependence cos θ≈/Omega1/prime don time, while Figs. 1(c) and 1(d) represent the corresponding phase mis- match/Phi1(0,T)=ϕ(0,T)−ϕd(T) and illustrate the contin- uing azimuthal phase locking in the system at /Phi1≈π.N o t e that the uniform solution in this case is stable with respectto spatial perturbations. The dissipation changes the thresholdcondition for entering the autoresonant uniform state [ 29,39], has some effect on the phase mismatch [compare Figs. 1(c) and1(d)], and leads to the collapse of the solution to the initial equilibrium after dephasing. Nevertheless, in the phase-lockedstage the autoresonant uniform solutions are similar with andwithout damping and remains stable with respect to spatialperturbations. Note that a similar evolution can be obtainedby starting from the m z=− 1 equilibrium if one applies the external magnetic field in the −/hatwideezdirection ( h0<0). The driving field in this case must rotate in the opposite directionand the linear resonance takes place at the driving frequency/Omega1 d(τ)=1+|h0|. With this modification, Fig. 1and other FIG. 2. The instability of the uniform magnetization state. The parameters of the simulations in (a) and (b) are the same as in Figs. 1(a) and 1(b), respectively, but κ=2π/l < 1. A complex spatiotemporal magnetization profile develops beyond the point of instability. figures below illustrating mz(ξ,T) remain the same if one changes the label −mztomz. In contrast to the example in Fig. 1, one observes a spatial instability of the autoresonant uniform state in Figs. 2(a) and 2(b), showing the numerical simulations with the same parameters as in Figs. 1(a) and1(b),b u tl=8 instead of 6. One can see the destruction of the uniform state in Fig. 2and formation of a complex spatiotemporal structure of mz(ξ,T) starting T≈21 in Fig. 1(a) and somewhat earlier in Fig. 1(b). These results can be explained by a perturbation theory asdescribed below. We neglect damping for simplicity, freezethe time at τ=τ 0, and set /Phi1=π+δ/Phi1andθ=θ0+δθ, where θ0satisfies cosθ0−/Omega1/prime d(τ0)+εcotθ0=0. (12) Then, for small perturbations δ/Phi1andδθof frequency νand wave vector κ,E q s .( 3) and ( 4) become −iνδθ=− (κ2sinθ0−ε)δ/Phi1, (13) −iνδ/Phi1=/parenleftbigg −sinθ0+κ2 sinθ0−ε sin2θ0/parenrightbigg δθ, yielding ν2=1 sin2θ0(κ2sinθ0−ε)(−sin3θ0+κ2sinθ0−ε).(14) One can see that for small ε, the uniform solution is stable with respect to spatial perturbations provided κ> sinθ0. (15) The examples in Figs. 1(κ=1.047) and 2 ( κ=0.785) are consistent with this result. III. TRANSFORMATION FROM SPATIALLY UNIFORM SOLUTION TO A STANDING WA VE The formation of a uniform autoresonant solution cosθ0(τ)≈/Omega1/prime d(τ) in the spatially periodic LLG problem was demonstrated above using a constant amplitude chirped 014411-3L. FRIEDLAND AND A. G. SHAGALOV PHYSICAL REVIEW B 99, 014411 (2019) FIG. 3. The formation of autoresonant standing waves from the uniform magnetization state: (a) κ=0.78, (b) κ=0.45. The final waveform is reached as the driving frequency gradually decreases and stays constant for T> 69. frequency drive, yielding stable evolution provided the in- equality ( 15) is satisfied (see Fig. 1). If during the evolution, this inequality is violated, the spatial instability develops (seeFig. 2). However, one can avoid the instability and trans- form the uniform autoresonant solution into an autoresonantstanding wave by adding a simple spatial modulation of thedriving amplitude, i.e., uses ε=ε 0+ε1cos(κξ). We illus- trate this phenomenon via simulations in Fig. 3, where we use parameters α=5×10−4andλ=0, but, in the driving term, apply a modulated drive with ε1=ε0=3×10−3and switch on ε1atτ=0. The chirped driving frequency in this numerical example is of form /Omega1/prime d=1−/Delta1/Omega1sin(ατ//Delta1/Omega1) forτ<π/Delta1/Omega1/2αand/Omega1/prime d=1−/Delta1/Omega1 forτ>π/Delta1/Omega1/2α, and we use /Delta1/Omega1=0.98. Thus, as in previous illustrations, the frequency passes the resonance at τ=0 having chirp rate α but then gradually decreases reaching a constant. Figure 3(a) (where we use l=8) shows that the addition of the spatial modulation of the driving amplitudes prevents the spatialinstability and leads to the emergence of a growing amplitudestanding wave solution. Figure 3(b) (where l=13) shows a similar dynamics, yielding formation of larger amplitudestanding wave, which starts earlier, at T≈5 (we again use the slow time T=α 1/2τin this and the following figures). The excited standing wave is fully controlled by the variationof the driving frequency and precesses azimuthally with theangular velocity of the driving phase (due to the continuingphase locking of /Phi1≈π). Furthermore, the magnetization waveform is spatially locked to the driving perturbation, whilethe wave amplitude and form is controlled by the instan-taneous frequency of the drive. Importantly, as lincreases, the maximum and the minimum of the final solution for m z become near +1 and −1, respectively. We have also verified numerically that this solution approaches the well knownsoliton form with exponentially falling tails [see Eq. (6.21)in Ref. [ 9]]. We further illustrate the autoresonant control of the standing magnetization waves in Figs. 4(a)and4(c), where we show the results of simulations with all the parameters ofFig. 3(b), but instead of saturating the driving frequency, we allow it to vary according to the same sinusoidal formula foran additional time interval π/Delta1/Omega1/2α<τ<π /Delta1/Omega1/2α,s ot h e050100−0.500.5−1−0.500.51 T ξ/l−mz 050100−0.500.5−1−0.500.51 T ξ/l−mz 0 50 10024 TΦ 0 50 10024 TΦ(a) (b) (d)(c) FIG. 4. The control of the phase-locked standing magnetization wave by varying the driving frequency. In (a) and (c) the parametersof the simulations are the same as in Fig. 3(b), but after reaching its minimal value at T=69, the driving frequency increases back to the initial value, while the magnetization returns to the initial state.Panels (b) and (d) show −m zand phase mismatch /Phi1, respectively, versus slow time in the same case as (a) and (c), but λ=10−3and ε=10−2. frequency returns to its original value. Figures 4(b) and4(d) show the results of similar simulations with the same param-eters as in Figs. 4(a) and 4(c),b u tε=10 −2andλ=10−3. One observes the return of the magnetization to its initialuniform state, being continuously phase locked [see Figs. 4(c) and4(d)] to the drive with or without dissipation. The idea of the transformation from the uniform to standing wavesolution by passage through the spatial instability originatesfrom the similarity to the autoresonant excitations of standingwaves of the driven-chirped nonlinear Schrodinger (NLS)equation [ 31]: iψ τ+ψξξ+|ψ|2ψ+εe−i/integraltext ωddτ=0. (16) If one writes ψ=ae−iφand separates the real and imaginary parts in ( 16), one arrives at the system aτ=a/Phi1ξξ+2/Phi1ξaξ−εsin/Phi1, (17) /Phi1τ=−aξξ a+/Phi12 ξ−a2−ωd(t)−ε acos/Phi1, (18) where/Phi1=φ−/integraltext ωddτ. Similarly to our ferromagnetic prob- lem, the passage through the linear resonance in this systemyields excitation of the uniform autoresonant NLS solutionfollowed by transformation into autoresonant standing wave[31]. One notices the structural similarity between this NLS system and LLG Eqs. ( 3) and ( 4), so we proceed to the theory for the magnetization case using the driven NLS ideas. We assume that the time evolution in Eqs. ( 3) and ( 4)i s slow and interpret the solutions at a given time τ, as being a slightly perturbed solution of the same system of equationsbut with the time derivatives and the forcing terms set to zero,i.e., /Phi1 ξξsinθ+2/Phi1ξθξcosθ=0, /parenleftbigg −1 sinθθξξ+/Phi12 ξcosθ/parenrightbigg −/Omega1/prime d(τ)+cosθ=0.(19) 014411-4EXCITATION AND CONTROL OF LARGE-AMPLITUDE … PHYSICAL REVIEW B 99, 014411 (2019) We notice that this is a dynamical, two degrees of freedom problem ( ξserving as “time”) governed by Hamiltonian H=1 2/parenleftbig θ2 ξ+/Phi12 ξsin2θ/parenrightbig +V(θ), (20) where V(θ)=−/Omega1/prime d(τ) cosθ+1 4cos(2θ). (21) This fixed τproblem is integrable since it conserves the canonical momentum B=/Phi1ξsin2θand energy E=1 2θ2 ξ+Veff, (22) where Veff(θ,τ)=B2 2 sin2θ+V(θ). Next, we discuss oscillat- ing solutions of this problem and introduce the conventionalaction-angle variables ( I,/Theta1) and ( B,/Phi1), where the first pair describes pure θoscillations in the effective potential V eff, while the second pair is associated with the dynamics of /Phi1. If one returns to the original (time dependent and driven)system ( 3) and ( 4),E(τ) and B(τ) become slow functions of time. We will present a theory describing these slowparameters via Whitham’s average variational principle [ 36] in the next section and devote the remaining part of the currentsection to a simple qualitative picture of the dynamics. Ourqualitative picture is based on the assumption of almost purely θdynamics in the problem, i.e., setting B≈0,which means a continuous phase locking /Phi1≈π, simplifying the effective potential to V eff≈−/Omega1/prime d(τ) cosθ+1 4cos(2θ). As already dis- cussed above, the phase locking at πis guaranteed in the ini- tial excitation stage via temporal autoresonance with constantamplitude ε=ε 0, chirped frequency perturbation. But now our driving amplitude ε=ε0+ε1cos(κξ) has two terms, where the first leads to excitation of the uniform autoresonantsolution as discussed above, while the second term yields tran-sition to the standing wave solution. Initially, θis efficiently trapped at the minimum location θ mof the potential well Veff given by cos θm=/Omega1/prime d(τ).ToO(ε) this yields θ≈θm,s ot h i s dynamics corresponds to the uniform autoresonant solution[see Eq. ( 12)]. The second term ε 1cos(κξ) in the driving has little effect on the evolution at this stage, until the spatialfrequency κ 0=/radicalbig ∂2Veff/∂θ2mof oscillations of θaround θm passes the resonance with this driving term, i.e., when /Omega1/prime d(τ) cosθm−cos(2θm)≈sin2θm=κ2. (23) But this is exactly the location of the instability of the uniform solution [see Eq. ( 15)] without the term ε1cos(κξ)i nt h e drive. The passage through the resonance with this newdrive term excites growing amplitude oscillations of θin the effective potential. After the passage, the oscillations of θ become autoresonant as the amplitude increases to preservetheir spatial frequency near κcontinuously. These newly in- duced spatially phase-locked, growing amplitude oscillationsofθcomprise the autoresonant standing wave solution. The amplitude of these oscillations does not grow indefinitely.Indeed, when the potential V effbecomes shallower again as θmpasses π/2a t/Omega1/prime d(τ)=0, the spatial resonance cannot be sustained, and the autoresonance is expected to interrupt.We illustrate this dynamics in Fig. 5, showing the effective potential V eff(thin red lines) at 14 successive values of slow time starting T=− 20. The thick blue lines in the figure0 0.5 1 1.5 2 2.5−1−0.8−0.6−0.4−0.200.2 θVeff autoresonant uniform solution startsautoresonant standing wave startsT=57.4 T=15.7 T=−20 FIG. 5. The formation of the autoresonant standing wave mod- elled via dynamics of a quasiparticle in a slowly varying effectivepotential V eff.Veffversus θis shown for successive times (thin red lines) starting at T=− 20. The thick blue lines show spatial oscillations of θat these times, as obtained in simulations in Fig. 3(a). The excitation proceeds as the quasiparticle remains at the bottom of the potential well continuously, corresponding to the flat solution. After passage through resonance with the spatial modulation of thedriving amplitude, autoresonant oscillations of the quasiparticle in the effective potential are excited, describing the standing magneti- zation wave. show the value of the potential at θ(ξ,T) at these times, as obtained in the simulations in the example in Fig. 3(a). IV . WHITHAM’S A VERAGED VARIATIONAL ANALYSIS A. Averaged Lagrangian density The LLG problem governed by Eqs. ( 3) and ( 4) allows Lagrangian formulation with the Lagrangian density L= L0+L1where L0(θ,θξ,/Phi1τ,/Phi1ξ,τ)=1 2/parenleftbig θ2 ξ+/Phi12 ξsin2θ/parenrightbig +/Phi1τcosθ +/Omega1/prime d(τ) cosθ−1 4cos(2θ) (24) and the perturbing part L1=−εsinθcos/Phi1. (25) For studying the slow autoresonant evolution in system ( 3) and ( 4), we use Whitham’s averaged Lagrangian approach. Following Refs. [ 31,40], describing a similar NLS problem, we seek solutions of form θ=ϑ(τ)+U(/Theta1,τ),/Phi1=υ(τ)+V(/Theta1,τ), (26) where the explicit time dependence is slow, while /Theta1(ξ,τ)i s a fast variable and UandVare 2πperiodic in /Theta1. In addition, the frequencies /Theta1τ=−/Omega1(τ) andβ=υτare slow functions of time and the wave vector /Theta1ξ=κ=const (κ=2π/l, l being the periodicity length in our problem). The Whitham’saveraging [ 36] in this system proceeds from the unperturbed Lagrangian density L 0, where one freezes the slow time dependence at some τand, using /Phi1ξ=κV/Theta1, replaces /Phi1τ= β−/Omega1V/Theta1=β−(/Omega1/κ)/Phi1ξ. This yields L0=1 2/parenleftbig U2 ξ+/Phi12 ξsin2θ/parenrightbig +/parenleftbigg β+/Omega1/prime d−/Omega1 κ/Phi1ξ/parenrightbigg cosθ−1 4cos(2θ).(27) 014411-5L. FRIEDLAND AND A. G. SHAGALOV PHYSICAL REVIEW B 99, 014411 (2019) Recall that the explicit dependence on Uin (27) enters via θ=ϑ+U. This Lagrangian density describes a two degrees of freedom dynamical problem (for Uand/Phi1), where ξ plays the role of “time.” In dealing with this problem weuse Hamiltonian formulation. We define the usual canonicalmomenta P U=∂L0/∂Uξ=Uξ (28) P/Phi1=∂L0/∂/Phi1ξ=/Phi1ξsin2θ−/Omega1 κcosθ (29) and observe that /Phi1is a cyclic variable and therefore P/Phi1= Bis the integral of motion. We will unfreeze the slow time dependence later and B(τ) will becomes a slow function of time. The Lagrangian density L0yields the Hamiltonian in the time-frozen problem H0=PUUξ+P/Phi1/Phi1ξ−L0 (30) and, after some algebra, H0=H/prime 0(Pθ,θ)+V1(θ,B,/Omega1,β), (31) where H/prime 0(PU,U)=1 2(PU)2+V(θ)−V(ϑ), (32) V(θ)=−/Omega1/prime dcosθ+1 4cos(2θ), (33) and V1(B,/Omega1,β,θ )=V(ϑ)+/parenleftbig B+/Omega1 κcosθ/parenrightbig2 2s i n2θ−βcosθ. At this stage, we return to the full driven (still time-frozen) problem governed by the Hamiltonian H=H/prime 0(PU,U)+V1−L1 (34) (recall that L1=− [ε0+ε1cos(κξ)] sinθcos/Phi1) and make canonical transformation from PU,U to the action-angle (AA) variables I,/Theta1of Hamiltonian H/prime 0. The dynamics gov- erned by this Hamiltonian conserves its energy E=H/prime 0and is periodic of period 2 πin/Theta1, and, at this stage, we identify /Theta1 with the angle variable used in the definitions ( 26). The action variable in H/prime 0problem is I=1 2π/contintegraldisplay PUdU=1 2π/contintegraldisplay/radicalbig 2[E−V(θ)+V(ϑ)]dU, (35) where the time dependence enters both explicitly in Vand via ϑ. Note that ∂I ∂E=1 2π/contintegraldisplay1√2[E−V(θ)+V(ϑ)]dU=1 /tildewideκ, (36) /tildewideκ(ϑ, E ) being the (spatial) frequency of the oscillations of U governed by H/prime 0. Next, we write the full Lagrangian in our problem in terms of the new action angle variables L=d/Theta1 dξI−H=κI−E−V1(B,/Omega1,β,θ )+L1(θ,ξ,τ ), (37) where θ=ϑ+U(I,/Theta1,τ)i nV1andL1,a st h er e s u l t of the canonical transformation. The Whitham’s averagedLagrangian density /Lambda1is obtained by averaging Lin the time- frozen problem over one oscillation governed by H/prime 0: /Lambda1=1 2π/integraldisplay2π 0Ld/Theta1=κI−E−1 2π/integraldisplay2π 0(V1−L1)d/Theta1. (38) To complete the averaging, we calculate two remaining com- ponents /angbracketleftV1/angbracketright=1 2π/integraltext2π 0V1d/Theta1and/Lambda11=1 2π/integraltext2π 0L1d/Theta1in (38). /angbracketleftV1/angbracketright=V(ϑ)+1 2π/integraldisplay2π 0/bracketleftBigg/parenleftbig B+/Omega1 κcosθ/parenrightbig2 2s i n2θ−βcosθ/bracketrightBigg d/Theta1 =V(ϑ)+I1B2 2+I2B/Omega1 κ+I3/Omega12 2κ2−βI4, (39) where I1=/angbracketleft1 sin2θ/angbracketright,I2=/angbracketleftcosθ sin2θ/angbracketright,I3=/angbracketleftcos2θ sin2θ/angbracketright,I4=/angbracketleftcosθ/angbracketright, and the averages /angbracketleft.../angbracketrightare defined as /angbracketleft.../angbracketright=1 2π/integraldisplay2π 0(...)d/Theta1 =κ 2π/contintegraldisplay(...)√2[E−V(θ)+V(ϑ)]dU. (40) Finally, we calculate the averaged driving part of the La- grangian density (recall that θ=ϑ+Uand/Phi1=υ+V) /Lambda11=−1 2π/integraldisplay2π 0[ε0+ε1cos(κξ)] ×sin(ϑ+U) cos(υ+V)d/Theta1. (41) Here, we limit evaluation of this averaged object to small spatial oscillations of θaround ϑ, write U≈a(I) cos/Theta1 and replace sin( ϑ+U)≈sinϑ+a(I) cosϑcos/Theta1. Further- more, we will also assume that Vis sufficiently small to replace cos( υ+V)≈cosυ. Finally, assuming a continuous approximate double resonance in the problem, i.e., υ(τ)− π=υ/prime≈0 and/Theta1−κξ−π=μ(τ)≈0 (initial phase lock- ing of υatπwas shown in the uniform autoresonant solution stage), after averaging /Lambda11≈/parenleftbigg ε0sinϑ−ε1 2a(I) cosϑcosμ/parenrightbigg cosυ/prime. (42) Therefore, our final averaged Lagrangian becomes /Lambda1=κI−E−V(ϑ)−I1B2 2−I2B/Omega1 κ−I3/Omega12 2κ2+I4β +/bracketleftBig ε0sinϑ−ε1 2a(I) cosϑcosμ/bracketrightBig cosυ/prime. (43) We discuss the slow evolution of the full driven system next. Following Whitham, this evolution is obtained by unfreezingthe time and taking variations of /Lambda1with respect to all depen- dent variables E,ϑ,B, /Theta1, andυ. Obviously, only slow objects enter the averaged Lagrangian density. B. Evolution equations and stability analysis At this stage, we write variational evolution equations. The variation of /Lambda1with respect to Byields dμ dτ=−/Omega1=κI1 I2B, (44) 014411-6EXCITATION AND CONTROL OF LARGE-AMPLITUDE … PHYSICAL REVIEW B 99, 014411 (2019) and the variation with respect to /Theta1and use of ( 44) results in d dτ/bracketleftbigg/parenleftbig I2 2−I1I3/parenrightbigB κI2/bracketrightbigg ≈ε1 2acosϑsinμ. (45) Similarly, the variation with respect to Eandυgives I4Edυ/prime dτ=1−κ /tildewideκ+B2/parenleftbiggI1E 2−I2EI1 I2+I3EI2 1 2I2 2/parenrightbigg (46) +ε1 2aEcosϑcosμcosυ/prime and dI4 dτ≈−/parenleftbigg ε0sinϑ−ε1 2acosϑcosμ/parenrightbigg sinυ/prime. (47) Finally, the variation with respect to ϑyields I4ϑdυ/prime dτ=∂V(ϑ) ∂ϑ−κ∂I ∂ϑ+B2/parenleftbiggI1ϑ 2−I2ϑI1 I2+I3ϑI2 1 2I2 2/parenrightbigg −/parenleftbigg ε0cosϑ+ε1 2asinϑcosμ/parenrightbigg cosυ/prime. (48) Equations ( 44)–(48) comprise a complete set of slow evolu- tion equations for E,B,μ,υ/prime, andϑ. The solution of these equations proceeds by defining a quasisteady state B0=μ0= υ/prime 0=0,ϑ=ϑ0andE0given by (Vϑ−κIϑ)E0,ϑ0−ε0cosϑ0−ε1 2asinϑ0=0, (49) G(E0,ϑ0)=/parenleftBig 1−κ /tildewideκ+ε1 2aEcosϑ0/parenrightBig E0,ϑ0=0. (50) Note that in the case ε1=0 and small E,E q .( 49) nearly coincides with Eq. ( 12) describing the autoresonant uniform solution. Furthermore, for small E,toO(ε), Eq. ( 49) yields Vϑ0≈0, i.e., ϑ0remains near the location of the minimum ofV(θ) given by cos ϑ0≈/Omega1/prime d, as was suggested in the qualitative model in Sec. IVand seen in simulations. On the other hand, Eq. ( 50) clarifies the phase locking at μ≈0a s/tildewideκ approaches the resonance /tildewideκ=κfrom below. Despite the formal complexity of the averaged variational theory, it now allows us to easily find the quasisteady state ofthe magnetization versus time in this chirped-driven problemwithout solving the LLG equation numerically. We illustratesuch a calculation in Fig. 6. The dots in panel (a) in the figure represent the quasienergy Eversus time found by solving algebraic equation ( 50) in the two examples in Fig. 3.T h e solid lines in the same panel show the energy 1 l/integraltextl 0H/prime 0(ξ)dξ from our numerical simulations, where H/prime 0is defined in Eq. ( 32). Panel (b) in the figure shows by dots the magnetiza- tion waveform mz(ξ)=cosϑfound by quadratures, i.e., by solving1 2(dϑ/dξ )2+V(ϑ)=E. The solid lines in this panel show the results from the numerical simulations. One can seethat the agreement between the quasisteady state theory andsimulations is excellent. In contrast to the simplicity of findingthe quasisteady state via the variational theory, the analysis0 20 40 6000.10.20.30.40.5 TE −0.5 0 0.5−1−0.500.51 ξ/l−mz(b) (a) l=8l=13 l=8 l=13 FIG. 6. Comparison between the quasisteady state solution from the variational theory (dots) and numerical simulations (solid lines). Panel (a) presents the quasienergy Eversus time in the two examples in Fig. 3, while panel (b) shows the waveform mz(z) in the same examples at time T=70. of its stability illustrated in numerical simulations is more complex and is discussed below. For small perturbations δE,δB,δμ,δυ/prime, and δϑof the quasisteady state, we use I≈δE//tildewideκ≈δE/ sinϑ0to get the lowest order (linear) set of equations dδμ dτ=κI1 I2δB, (51) dδB dτ=−κε1aI2cosϑ0 2/parenleftbig I1I3−I2 2/parenrightbigδμ, (52) I4Edδυ/prime dτ=Gϑδϑ+GAδE, (53) I4EdδE dτ+I4ϑdδϑ dτ=−/parenleftbigg ε0sinϑ0−ε1 2acosϑ0/parenrightbigg δυ/prime,(54) I4ϑdδυ/prime dτ=Vϑϑδϑ−κRδE, (55) where we use /tildewideκ≈sinϑ0,s oR≈cosϑ0/sin2ϑ0and all coef- ficients in ( 51)–(55) are viewed as constants evaluated at the quasisteady state. Equations ( 51) and ( 52) yield d2δμ dτ2+ν2 1δμ≈0, (56) while Eqs. ( 53)–(55) reduce to d2δυ/prime dτ2+ν2 2δυ/prime≈0, (57) where the two frequencies satisfy ν2 1=ε1κ2I1acosϑ0 2/parenleftbig I1I3−I2 2/parenrightbig, ν2 2=/parenleftbig ε0sinϑ0−ε1 2acosϑ0/parenrightbig (GEVϑϑ−κRG ϑ) I4E(I4EVϑϑ−κRG ϑ)+I4ϑ(GEI4ϑ−GϑI4E). 014411-7L. FRIEDLAND AND A. G. SHAGALOV PHYSICAL REVIEW B 99, 014411 (2019) FIG. 7. The transition to instability of the autoresonant mag- netization wave. The parameters are as those in Fig. 3(a),b u ta larger final driving frequency (smaller excitation amplitude) and different ε1. Panels (a) and (b) show the excited waveforms for ε1/ε0=9 and 11, respectively, i.e., below and above the instability threshold ε1/ε0=9.8. The spatial phase locking is lost for T> 40 in panel (b). A positiveness of ν2 1,2guarantees stability of the (doubly) autoresonant ( υ/prime≈0 andμ≈0) evolution of the system. We observe that I1I3−I2 2∝/parenleftBigg/summationdisplay i1 si/parenrightBigg⎛ ⎝/summationdisplay jx2 j sj⎞ ⎠−/parenleftBigg/summationdisplay ixi si/parenrightBigg2 , where xi=cosθiandsi=sin2θi√[E−Veff(θi)]. Then I1I3−I2 2∝/summationdisplay i,j>i(xi−xj)2 sisj, (58) soν2 1is positive for ϑ0<π / 2. Then, since B=δB, andμ=δμ, they both remain small. Furthermore, for small excitations of E, to lowest order in E, κ= sinϑ0,Gϑ=cosϑ0/sinϑ0,GE=(1 sinϑ0−3 2 sin3ϑ0),I4ϑ= −sinϑ0,I4E=cosϑ0/sin2ϑ0. With these substitutions, one finds ν2 2≈ε0sinϑ0−ε1 2acosϑ0. Then condition ε0sinϑ0−ε1 2acosϑ0>0 guarantees the stability of the autoresonant evolution. We illustrate this conclusion in Fig. 7, showing the results of numerical simulations for parametersof Fig. 3(a),b u t/Delta1/Omega1=0.4, i.e., larger final driving frequency /Omega1 /prime=1−/Delta1/Omega1 and, thus, smaller excitation amplitude a. We estimate numerically that in this case ϑ0≈0.89rad anda≈0.25rad. This yields the transition to instability atε1/ε0=2s i nϑ0/(acosϑ0)≈9.8 . Panels (a) and (b) in Fig. 7show the excited magnetization waveform for ε1/ε0=9 and 11, respectively. One can see that below the instability condition ( ε1/ε0=9) the excited wave remains spatially phase locked to the drive, arriving at the finalquasisteady state at later times. In contrast, for ε 1/ε0=11 in panel (b), the initial excitation stage is similar to that in panel(a), but the spatial phase locking is lost beyond T≈40 due to the instability and the magnetization develops a complexspatiotemporal profile.V . ALTERNATIVE DRIVING SCHEMES Here we discuss two modifications of the driving compo- nent in the LLG equation ( 1), which may allow the required submicron spatial modulation of the drive. The first modifi-cation is using spin torque drive instead of the microwave (arelated autoresonant problem for single domain nanoparticleswas studied in Ref. [ 30]). The effective magnetic field associ- ated with the spin torque is h s=m×Is, (59) where Isis the dimensionless spin polarized current, which will be assumed of form Is=2εsinϕdexin the following, yielding hs=2ε(mzey−myez), and possibly using nanocon- tacts [ 41] for submicron spatial modulation of ε. The analog of system ( 3), (4) for this drive is θτ=/Phi1ξξsinθ+2/Phi1ξθξcosθ−εcosθsin/Phi1,(60) /Phi1τ=/parenleftbigg −1 sinθθξξ+/Phi12 ξcosθ/parenrightbigg −/Omega1/prime D−εcos/Phi1 sinθ,(61) where/Phi1=ϕ−ϕd+π/2. Note that for small θthe last two equations are nearly the same as Eqs. ( 3), (4)f o rt h e microwave drive. One consequence of this is that the autores-onance threshold when passing the linear resonance is thesame for both cases. Figure 8(a) illustrates the formation and control of the autoresonant standing wave via a spin torquedrive in simulations using the parameters of Fig. 3(b).O n e can see that the form of the excited solution in Figs. 3(b) and8(a) are very similar. Despite this similarity, a complete Whitham’s-type theory of the spin torque driven problem ismore complex than that for the microwave drive case, becausethe driving parts in Eqs. ( 60) and ( 61) do not allow Lagrangian description. Therefore, we leave this theory outside the scopeof the present work. The second driving alternative is using the same chirped frequency microwave drive of uniform amplitude ε 0,b u t 050100 −0.500.5−101 Tξ/l−mz 0 50 10000.51 TΦ050100 −0.500.5−101 Tξ/l−mz 0 50 10000.51 TΦ(b) (d)(a) (c) spin torquedrive anisotropymodulation FIG. 8. Formation of the autoresonant standing magnetization wave by chirped frequency spin torque drive [panel (a)] and via a combination of a uniform AC drive and a modulation of the hard axis anisotropy [panel (c)]. The parameters in the simulations are thesame as in Fig. 3(b) for the AC drive, while for the anisotropy mod- ulation case ε 2=− 2.5ε0. Panels (b) and (d) show the corresponding phase mismatch /Phi1(0,T) for the two drives, respectively. 014411-8EXCITATION AND CONTROL OF LARGE-AMPLITUDE … PHYSICAL REVIEW B 99, 014411 (2019) adding a spatially modulated [ 42] hard axis anisotropy in the system (along /hatwideex, for example). The driving component of the effective field in this case will become hd=ε0(cosϕd/hatwideex+sinϕd/hatwideey)−2ε2cos(κξ)mx/hatwideex,(62) 2ε2being the ratio between the easy and hard axis anisotropy coefficients. In the autoresonance, mx=sinθcosϕ≈ −sinθcosϕdand one can rewrite hdas hd≈[ε0+ε2sinθcos(κξ)](cos ϕd/hatwideex+sinϕd/hatwideey) (63) +ε2sinθcos(κξ)(cosϕd/hatwideex−sinϕd/hatwideey). The last term in this expression is rotating in the opposite direction and, being nonresonant, has a negligible effect.Thus, effectively, h dhas a form similar to that analyzed in our theory for the microwave drive. Figure 8(b) illustrates this idea in simulations using hdfrom Eq. ( 62) and the same parameters as in Fig. 8(a),b u tε1=0 and ε2=1.25ε0. We see that this different combination drive yields a verysimilar autoresonant magnetization wave as in Figs. 3(b) and 8(a). Furthermore, we have seen numerically that within this driving scheme, one can excite large amplitude autoresonantwaves with the modulation scale reaching L=1000 nm using smaller driving amplitudes and chirp rates. However, this maylimit some experiments, because it also requires a weakerdissipation for stable evolution. VI. CONCLUSIONS In conclusion, we have studied the problem of autores- onant excitation and control of 1D standing magnetizationwaves in an easy axis ferromagnetic in an external mag-netic field and driven by a weak circularly polarized, chirpedfrequency microwave field. We had modeled this problemby the spatially periodic time dependent LLG equation [seeEq. ( 1)]. We had discussed the excitation of the autoreso- nant solutions in this system via theory and compared theresults with numerical simulations. The excitation proceededas the driving frequency passed a resonance with the initiallyspatially uniform magnetization equilibrium in the directionof the easy axis (polar angle θ=0), yielding a driven spa- tially uniform magnetization with the azimuthal angle ϕof the magnetization locked (and therefore controlled) by thephase of the microwave. This phase locking (autoresonance)reflects a continuous self-adjustment of θ[s e eE q .( 11)], so that the resonance is preserved despite the variation of thedriving frequency. It was shown that the condition for thisautoresonant evolution is the driving amplitude εexceeding a threshold, which scales with the driving frequency chirprate as ε th∼α3/4[see Eq. ( 10)]. We had also shown that the uniform autoresonant magnetization state remained stablewith respect to spatial perturbations if sin θ<κ =2π/l, l being the periodicity length in the problem. In the case2π/l > 1, the stable uniform state reached a complete mag- netization inversion ( θ→π). In contrast, when θincreased during the autoresonant uniform state evolution and passedthe point where sin θ=2π/l, the spatial instability devel- oped, yielding a complex spatiotemporal magnetization waveform.We had shown that if instead of a constant driving ampli- tude, one introduced a spatially modulated amplitude ε 0+ ε1cos(κξ), then, instead of the instability, a standing wave is excited with the amplitude and form controlled by thefrequency of the driving wave. This emerging autoresonantsolution is doubly phase locked, i.e., its azimuthal angle ϕ is locked to the phase of the driving wave, while θperforms slowly evolving growing amplitude nonlinear spatial oscilla-tions in an effective potential, which are continuously phaselocked to the spatial modulation of the drive. Furthermore,as the periodicity length lincreases, the autoresonant wave approaches the well know soliton form [see Eq. (6.21) inRef. [ 9]]. The formation of the autoresonant standing wave is fully reversible and can be returned to its initial uniform(θ≈0) state by simply reversing the variation of the driving frequency. In addition to suggesting a qualitative descrip-tion of this autoresonant evolution (see Sec. III), we had developed a complete theory of the dynamics in the problembased on the Whitham’s averaged variational approach andstudied modulational stability of the autoresonant solutions(see Sec. IV). We had found numerically that a sufficiently weak dissipation does not affect the autoresonant evolutionsignificantly. The suggested method of excitation allows usto form steady standing waves of prescribed amplitude bysimply fixing the driving frequency at any time, while theautoresonant driving compensates the effect of dissipation.We had also discussed and illustrated in simulations formationof autoresonant standing waves when replacing the microwavedrive by a spatially modulated transverse spin torque drivingor adding a modulated hard axis anisotropy. Developing afull Whitham’s type theory in these cases and inclusion ofdissipation and thermal fluctuations in the theory seem to beimportant goals for future research. Finally, it is known thatthe undriven, dissipationless LLG problem ( 1) is integrable [9]. This means that there exist many additional, so-called multiphase solutions in this problem. Addressing the questionof excitation and control of this multitude of solutions bychirped frequency perturbations seems to comprise anotherinteresting goal for the future. ACKNOWLEDGMENTS The authors would like to thank J.M. Robbins and E.B. Sonin for stimulating discussions and important comments.This work was supported by the Israel Science FoundationGrant No. 30/14 and the Russian state program AAAA-A18-118020190095-4. APPENDIX: QUANTUM TWO-LEVEL MODEL We perform our numerical simulations to lowest significant order in λand, therefore, approximate LLG Eq. ( 1)a s ∂m ∂τ≈h×m+λm×(h×m)=h/prime×m, (A1) where h/prime=h−λh×m.Our numerical scheme for study- ing the evolution governed by Eq. ( A1) is based on the 014411-9L. FRIEDLAND AND A. G. SHAGALOV PHYSICAL REVIEW B 99, 014411 (2019) equivalent quantum two-level system (idea originated by Feynman [ 38], and recently used in studying magnetiza- tion inversion in single domain nanoparticles [ 29,30]). We solve i∂A 1 ∂τ=d0 2A1+dA 2, (A2) idA 2 ∂τ=−d0 2A2+d∗A1, (A3) where A1,2=A1,2(ξ,τ) are the wave functions of a pair of coupled quantum levels and d0=h/prime z, (A4) d=(h/prime x−ih/prime y) 2. (A5)The magnetization mind0anddin Eqs. ( A2), (A3) is related toA1,2via mx=A1A∗ 2+A∗ 1A2=2B1B2cosϕ, my=i(A1A∗ 2−A∗ 1A2)=2B1B2sinϕ, (A6) mz=|A1|2−|A2|2=B2 1−B2 2, where A1,2=B1,2exp(iϕ1,2) andϕ=ϕ2−ϕ1. Note that, as expected, the total population of our two level system remainsconstant, |A 1|2+|A2|2=|m|=1. Note also that m⊥=/radicalBig m2x+m2y=2B1B2, while ϕis the azimuthal rotation angle of the magnetization around ξ. Formally, the system ( A2), (A3) comprises a set of two coupled NLS-type equations for wave functions A1,2. The numerical approach to solving this system throughout this work used a standard pseudospectralmethod [ 43] subject to given initial and periodic boundary conditions. [1] M. M. Scott, M. P. Kostylev, B. A. Kalinikos, and C. E. Patton, P h y s .R e v .B 71,174440 (2005 ). [2] M. Wu, M. A. Kraemer, M. M. Scott, C. E. Patton, and B. A. Kalinikos, Phys. Rev. B 70,054402 (2004 ). [3] M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, Phys. Rev. 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PhysRevLett.102.177601.pdf

Observation of Ferromagnetic Resonance in SrRuO 3 by the Time-Resolved Magneto-Optical Kerr Effect M. C. Langner,1,2C. L. S. Kantner,1,2Y. H. Chu,3L. M. Martin,2P. Yu,1J. Seidel,3R. Ramesh,1,3and J. Orenstein1,2 1Department of Physics, University of California, Berkeley, California 94720, USA 2Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA (Received 3 December 2008; published 28 April 2009) We report the observation of ferromagnetic resonance (FMR) in SrRuO 3using the time-resolved magneto-optical Kerr effect. The FMR oscillations in the time-domain appear in response to a sudden,optically induced change in the direction of easy-axis anisotropy. The high FMR frequency, 250 GHz, andlarge Gilbert damping parameter, /C11/C251, are consistent with strong spin-orbit coupling. We find that the parameters associated with the magnetization dynamics, including /C11, have a nonmonotonic temperature dependence, suggestive of a link to the anomalous Hall effect. DOI: 10.1103/PhysRevLett.102.177601 PACS numbers: 76.50.+g, 75.30. /C0m, 78.20.Ls, 78.47. /C0p Understanding and eventually manipulating the elec- tron’s spin degree of freedom is a major goal of contem- porary condensed matter physics. As a means to this end, considerable attention is focused on the spin-orbit (SO)interaction, which provides a mechanism for control ofspin polarization by applied currents or electric fields [ 1]. Despite this attention, many aspects of SO coupling are notfully understood, particularly in itinerant ferromagnetswhere the same electrons are linked to both rapid currentfluctuations and slow spin dynamics. In these materials, SOcoupling is responsible for spin-wave damping [ 2,3], spin- current torque [ 4,5], the anomalous Hall effect (AHE) [ 6], and magnetocrystalline anisotropy (MCA) [ 7]. Ongoing research is aimed toward a quantitative understanding ofhow band structure, disorder, and electron-electron inter-actions interact to determine the size and temperaturedependence of these SO-driven effects. SrRuO 3(SRO) is a material well known for its dual role as a highly correlated metal and an itinerant ferromagnet with properties that reflect strong SO interaction [ 8–10]. Despite its importance as a model SO-coupled system,there are no previous reports of ferromagnetic resonance(FMR) in SRO. FMR is a powerful probe of SO coupling inferromagnets, providing a means to measure both MCAand the damping of spin waves in the small wave vectorregime [ 11]. Here we describe detection of FMR by time- resolved magnetooptic measurements performed on high- quality SRO thin films. We observe a well-defined reso- nance at a frequency /C10 FMR¼250 GHz . This resonant frequency is an order of magnitude higher than in thetransition-metal ferromagnets, which accounts for the non-observation by conventional microwave techniques. 10–200 nm thick SRO thin films were grown via pulsed laser deposition between 680–700 /C14Cin 100 mTorr oxy- gen. High-pressure reflection high-energy electron diffrac-tion (RHEED) was used to monitor the growth of the SRO filmin situ . RHEED patterns and atomic force microscopy imaging confirmed the presence of pristine surfaces con-sisting of atomically flat terraces separated by a single unit cell step (3.93 A ˚). X-ray diffraction indicated fully epitax- ial films and x-ray reflectometry was used to verify film thickness. Bulk magnetization measurements using aSQUID magnetometer indicated a Curie temperature, T C, of/C24150 K . Sensitive detection of FMR by the time-resolved mag- netooptic Kerr effect (TRMOKE) has been demonstratedpreviously [ 12–14]. TRMOKE is an all optical pump- probe technique in which the absorption of an ultrashort laser pulse perturbs the magnetization, M, of a ferromag- net. The subsequent time-evolution of Mis determined from the polarization state of a normally incident, time-delayed probe beam that is reflected from the photoexcitedregion. The rotation angle of the probe polarization causedby absorption of the pump, /C1/C2 KðtÞ, is proportional to /C1MzðtÞ, where zis the direction perpendicular to the plane of the film [ 15]. Figures 1(a)and1(b)show /C1/C2 KðtÞobtained on an SRO film of thickness 200 nm. Very similar results are obtained in films with thickness down to 10 nm. Two distinct types of dynamics are observed, depending on the temperatureregime. The curves in Fig. 1(a)were measured at tempera- tures near T C. The relatively slow dynamics agree with previous reports for this Tregime [ 16] and are consistent with critical slowing down in the neighborhood of thetransition [ 17]. The amplitude of the photoinduced change in magnetization has a local maximum near T¼115 K . Distinctly different magnetization dynamics are observed asTis reduced below about 80 K, as shown in Fig. 1(b). The TRMOKE signal increases again, and damped oscil-lations with a period of about 4 ps become clearly resolved. In order to test if these oscillations are in fact the signature of FMR, as opposed to another photoinducedperiodic phenomenon such as strain waves, we measuredthe effect of magnetic field on the TRMOKE signals. Figure 2(a) shows /C1/C2 KðtÞfor several fields up to 6 T applied normal to the film plane. The frequency clearlyPRL 102, 177601 (2009) PHYSICAL REVIEW LETTERSweek ending 1 MAY 2009 0031-9007 =09=102(17) =177601(4) 177601-1 /C2112009 The American Physical Societyincreases with increasing magnetic field, confirming that the oscillations are associated with FMR. The mechanism for the appearance of FMR in TRMOKE experiments is well understood [ 14]. Before photoexcitation, Mis oriented parallel to the magnetic anisotropy hA. Perturbation of the system by the pump pulse generates a sudden change in the direction of the easy axis. In SRO thin films, the magnetocrystalline anisotropy axis rotates further out of plane as the film is cooled [ 8]. The perturbation in hAthat we observe is consistent with a small rotation (on the order of 1/C14) caused by rapid laser- pulse induced heating. In the resulting nonequilibriumstate,Mandh Aare no longer parallel, generating a torque that induces Mto precess at the FMR frequency. In the presence of Gilbert damping, Mspirals towards the new hA, resulting in the damped oscillations of Mzthat appear in the TRMOKE signal. To analyze the FMR line shape we Fourier transform (FT) the time-domain data. The magnetization in the time-domain is given by the relation, /C1M iðtÞ¼Z1 0/C31ijð/C28Þ/C1hj Aðt/C0/C28Þd/C28; (1) where /C31ijð/C28Þis the impulse response function and /C1hAðtÞ is the change in anisotropy field. If we define a coordinate system in which z0is the easy axis and y0is in plane of rotation of hA, then /C1MzðtÞis related to /C31y0y0ðtÞ. For laser- induced precession one expects that /C1hAðtÞwill be a step function, as photoinduced local heating can be quite rapidcompared with cooling via thermal conduction from the laser-excited region. When /C1hAðtÞis proportional to the step function, /C1MzðtÞ/Rt /C01/C31y0y0ð/C28Þd/C28, and /C31y0y0ð!Þis proportional to the FT of the time derivative of the TRMOKE signal. In this case, the observable!Ref/C1/C2 Kð!Þgshould be closely related to the imaginary, or dissipative part of /C31y0y0ð!Þ. In Fig. 2(b) we plot !Ref/C1/C2 Kð!Þgfor each of the curves shown in Fig. 2(a). The spectra shown do indeed exhibit features that are expected for Im/C31y0y0ð!Þnear the FMR frequency. A well-defined resonance peak is evident, whose frequency increases with magnetic field as expected for FMR. The inset to Fig. 2(b) shows /C10FMRas a function of applied magnetic field. The solid line through the datapoints corresponds to parameters jh Aj¼7:2T(forg¼2) and easy-axis direction equal to 30/C14from the film normal at 5 K, consistent with previous determinations of hA based on equilibrium magnetization measurements [ 8,10]. Although the spectra in Fig. 2(b) are clearly associated with FMR, the sign change at low frequency is not con-sistent with Im/C31 y0y0ð!Þ, which is positive definite. We have verified that the negative component is always present in the spectra and is not associated with errors in assigningthet¼0point in the time-domain data. The origin of negative component of the FT is made clearer by referring back to the time domain. In Fig. 3(a)we show typical time- series data measured in zero field at 40 K. For comparisonwe show the response to a step-function change in the easy- (a) (b) FIG. 2 (color online). (a): Change in Kerr rotation as a func- tion of time delay following pulsed photoexcitation at T¼5K, for several values of applied magnetic field ranging up to 6 T. (b): Fourier transforms of signals shown in top panel. Inset: FMR frequency vs applied field.(a) (b) FIG. 1 (color online). Change in Kerr rotation as a function of time delay following pulsed photoexcitation, for several tem-peratures below the Curie temperature of 150 K. (a): Tempera-ture range 100 K <T< 150 K . (b): Temperature range 5K< T<80 K . Signal amplitude and oscillations grow with decreas- ingT. Inset: Polar Kerr rotation vs temperature.PRL 102, 177601 (2009) PHYSICAL REVIEW LETTERSweek ending 1 MAY 2009 177601-2axis direction predicted by the Landau-Lifshitz-Gilbert (LLG) equation [ 18]. It is clear that, if the measured and simulated responses are constrained to be equal at large delay times, the observed oscillations of /C1/C2 KðtÞare much larger than the LLG prediction at small delay. In principle, one explanation for the discrepancy would be that /C1/C2 KðtÞ results from a change in the magnitude of Mas well as its direction. Using this interpretation to fit the data requires a photoinduced increase in jMj, which is unphysical for a system in a stable FM phase. We have found that /C1/C2 KðtÞcan be readily fit by the LLG equation if, instead of introducing a variation in jMj, we relax the assumption that /C1hAðtÞis a step-function. In particular, we allow the change in easy-axis direction to ‘‘overshoot’’ at short times. The overshoot suggests that the easy-axis direction changes rapidly as the photoexcited electrons approach quasiequilibrium with the phonon and magnon degrees of freedom. The red line in Fig. 3(a)shows the best-fit obtained by modeling /C1hAðtÞbyHðtÞð/C300þ /C301e/C0t=/C28Þ, where HðtÞis the step function, /C300þ/C301is the change in easy-axis direction at t¼0, and /C28is the time constant determining the rate of approach to the asymptotic value /C300. The fit is clearly much better when the possibility of overshoot dynamics in /C1hAðtÞis included. The blue line shows the difference between measured and simulated response. With the exception of this very short pulsecentered near t¼0, the observed response is now well described by the LLG equation. In Fig. 3(b) we compare data and simulated response in the frequency domain. With the allowance for an overshoot in/C1hAðtÞthe spectrum clearly resolves into two compo- nents. The peak at 250 GHz and the sign change at lowfrequency are the both part of the magnetic response to/C1h AðtÞ. The broad peak or shoulder centered near 600 GHz is the FT of the short pulse component shownin Fig. 3(a). We have found this component is essentially linear in pump pulse intensity, and independent of mag- netic field and temperature. Its properties are consistent with a photoinduced change in reflectivity due to band-filling, which is well-known to cross-couple into theTRMOKE signal of ferromagnets [ 19]. By including overshoot dynamics in /C1h AðtÞ, we are able to distinguish stimulus from response in the observedTRMOKE signals. From the LLG equation, we can extractthe two parameters that describe the response: /C10 FMRand /C11; and the two parameters that describe the stimulus: /C301=/C300and/C28. In Fig. 4we plot all four parameters as a (a) (b) FIG. 3 (color online). Components of TRMOKE response in time (a) and frequency (b) domain. Black lines are the observedsignals. Green line in (a) is the simulated response to a step-function change in easy-axis direction. Best fits to the overshoot model described in the text are shown in red. Blue lines are the difference between the measured and best-fit response.FIG. 4 (color online). Temperature dependence of (a) FMR frequency (triangles) and damping parameter (circles), (b) over- shoot decay time, (c) ratio of overshoot amplitude to step-response amplitude ( /C30 1=/C300), and (d) /C27xy(adapted from [ 20]).PRL 102, 177601 (2009) PHYSICAL REVIEW LETTERSweek ending 1 MAY 2009 177601-3function of temperature from 5 to 80 K. The T-dependence of the FMR frequency is very weak, with /C10FMRdeviating from 250 GHz by only about 5% over the range of the measurement. The Gilbert damping parameter /C11is of order unity at all temperatures, a value that is approximately afactor 10 2larger than found in transition-metal ferromag- nets. Over the same Trange the decay of the easy-axis overshoot varies from about 2 to 4 ps. We note that thedynamical processes that characterize the response alloccur in strongly overlapping time scales. While /C10 FMRis essentially independent of T, the parame- ters/C11,/C301=/C300, and /C28exhibit structure in their Tdepen- dence near 40 K. This structure is reminiscent of the T dependence of the anomalous Hall coefficient /C27xythat has been observed in thin films of SRO [ 20–22]. For compari- son, Fig. 4(d)reproduces /C27xyðTÞreported in Ref. [ 20]. The similarity between the Tdependence of AHE and parame- ters related to FMR suggests a possible correlation betweenthe two types of response functions. Recently, Nagaosa andOnoda [ 23] have discussed the possibility of a connection between collective spin dynamics at zero wave vector(FMR) and the off-diagonal conductivity (AHE). At a basiclevel, both effects are nonzero only in the presence of bothSO coupling and time-reversal breaking. However, the possibility of a more quantitative connection is suggested by comparison of the Kubo formulas for the two corre-sponding functions. The off-diagonal conductivity can bewritten in the form [ 24], /C27 xyð!Þ¼iX m;n;kJxmnðkÞJy nmðkÞfmnðkÞ /C15mnðkÞ½/C15mnðkÞ/C0!/C0i/C13/C138; (2) where JimnðkÞis current matrix element between quasipar- ticle states with band indices n,mand wave vector k. The functions /C15mnðkÞandfmnðkÞare the energy and occupation difference, respectively, between such states, and /C13is a phenomenological quasiparticle damping rate. FMR isrelated to the dynamic susceptibility, with the correspond-ing Kubo form, /C31 ijð!Þ¼X m;n;kSimnðkÞSj nmðkÞfmnðkÞ /C15mnðkÞ/C0!/C0i/C13; (3) where Simnis the matrix element of the spin operator. In general, /C27xyð!Þand/C31xyð!Þare unrelated, as they involve current and spin matrix elements, respectively. However, it has been proposed that in several ferromagnets, including SRO, the k-space sums in Eqs. ( 2) and ( 3) are dominated by a small number of band crossings near the Fermi surface[22,25]. If the matrix elements S imnandJimnvary suffi- ciently smoothly with k, then /C27xyð!Þ,/C31xyð!Þ, and /C31yyð!Þ may all show features determined by the position of the chemical potential relative to the energy at which the bandscross. Furthermore, as Gilbert damping is related to thezero-frequency limit of /C31 yyð!Þ[26], i.e.,/C11¼/C10FMR /C31yyð0Þ@ @!lim !!0Im/C31yyð!Þ; (4) and AHE is the zero-frequency limit of /C27xyð!Þ, the band- crossing picture suggests a possible correlation between /C11ðTÞand/C27xyðTÞ. In conclusion, we have reported the observation of FMR in the metallic transition-metal oxide SrRuO 3. Both the frequency and damping coefficient are significantlylarger than observed in transition-metal ferromagnets.Correlations between FMR dynamics and the AHE coeffi-cient suggest that both may be linked to near Fermi surface band-crossings. Further study of these correlations, as Sr is replaced by Ca, or with systematic variation in residualresistance, could be a fruitful approach to understandingthe dynamics of magnetization in the presence of strongSO interaction. This research is supported by the U.S. Department of Energy, Office of Science. Y. H. C. acknowledges the sup-port of the National Science Council, R. O. C. [1] I. Zutic ´, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [2] V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769 (1972). [3] V. Kambersky ´, Can. J. Phys. 48, 2906 (1970). [4] J. C. Slonczewski, J. Magn. Magn. Mater.. 159, L1 (1996). [5] L. Berger, Phys. Rev. B 54, 9353 (1996). [6] J. M. Luttinger and R. Karplus, Phys. Rev. 95, 1154 (1954). [7] H. Brooks, Phys. Rev. 58, 909 (1940). [8] L. Klein et al. , J. Phys. Condens. Matter 8, 10 111 (1996). [9] P. Kostic et al. , Phys. Rev. Lett. 81, 2498 (1998). [10] A. F. Marshall et al. , J. Appl. Phys. 85, 4131 (1999). [11] B. Heinrich and J. F. Cochran, Adv. Phys. 42, 523 (1993). [12] W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79, 1134 (1997). [13] Y. Acremann et al. , Science 290, 492 (2000). [14] M. van Kampen et al. , Phys. Rev. Lett. 88, 227201 (2002). [15] K. Shinagawa, in Magneto-optics , edited by S. Sugano and N. Kojima (Springer-Verlag, Berlin, Germany, 2000). [16] T. Ogasawara et al. , Phys. Rev. Lett. 94, 087202 (2005). [17] T. Kise et al. , Phys. Rev. Lett. 85, 1986 (2000). [18] W. F. Brown, Micromagnetics (Krieger, New York, 1963). [19] B. Koopmans et al. , Phys. Rev. Lett. 85, 844 (2000). [20] R. Mathieu et al. , Phys. Rev. Lett. 93, 016602 (2004). [21] L. Klein et al. , Phys. Rev. B 61, R7842 (2000). [22] Z. Fang et al. , Science 302, 92 (2003). [23] M. Onoda, A. S. Mishchenko, and N. Nagaosa, J. Phys. Soc. Jpn. 77, 013702 (2007). [24] M. Onoda and N. Nagaosa, J. Phys. Soc. Jpn. 71,1 9 (2002). [25] X. Wang et al. , Phys. Rev. B 74, 195118 (2006). [26] D. L. Mills and S. M. Rezende, in Spin Dynamics in Con- fined Magnetic Structures II , edited by B. Hillebrands and K. Ounadjela (Springer-Verlag, Berlin, 2003).PRL 102, 177601 (2009) PHYSICAL REVIEW LETTERSweek ending 1 MAY 2009 177601-4

PhysRevLett.110.257204.pdf

Unconventional Magnetism via Optical Pumping of Interacting Spin Systems Tony E. Lee,1Sarang Gopalakrishnan,2and Mikhail D. Lukin2 1ITAMP , Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA 2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 19 April 2013; published 19 June 2013) We consider strongly interacting systems of effective spins, subject to dissipative spin-flip processes associated with optical pumping. We predict the existence of novel magnetic phases in the steady state of this system, which emerge due to the competition between coherent and dissipative processes. Specifically, for strongly anisotropic spin-spin interactions, we find ferromagnetic, antiferromagnetic, spin-density-wave, and staggered- XYsteady states, which are separated by nonequilibrium phase transitions meeting at a Lifshitz point. These transitions are accompanied by quantum correlations, resulting in spin squeezing.Experimental implementations in ultracold atoms and trapped ions are discussed. DOI: 10.1103/PhysRevLett.110.257204 PACS numbers: 75.10.Jm, 42.50. /C0p, 67.85. /C0d, 75.30.Kz Exotic magnetic states play a central role in the physics of quantum many-body systems and have been explored in awide variety of strongly correlated materials [ 1]. Realizing and exploring magnetic states has recently emerged as acentral goal in ultracold atomic physics [ 2,3]. Due to highly controllable and tunable interactions, ensembles of ultra-cold neutral atoms and ions may provide a unique labora- tory to study exotic quantum magnetism [ 2–9]. Among the main obstacles are relatively small energy scales associatedwith magnetic ordering (e.g., the superexchange scale inthe Hubbard model), requiring cooling atomic systemsdown to very low temperatures [ 2], and the slow time scales involved in spin thermalization [ 10–12]. Furthermore, ultracold atoms are fundamentally open, driven quantumsystems far from their absolute thermal equilibrium. This motivates the exploration of spin dynamics in the presence of driving and dissipation [ 13–30]. Recently, a number of schemes involving dissipation to create magnetic phases have been proposed. These typi-cally use engineered reservoirs involving coupling mul-tiple lattice sites [ 13–15]. At the same time, one expects single-site dissipation such as spontaneous decay to bedetrimental to realizing interesting magnetic states, result- ing, e.g., in unwanted decoherence. In this Letter, we demonstrate that optical pumping and spontaneous decaycan instead enrich the phase diagram, resulting in new phases and phase transitions that do not exist in conven-tional equilibrium systems. Significantly, these novel statescan be observed under conditions when the realization ofconventional equilibrium states is difficult. The key idea of this work can be understood by consid- ering the anisotropic spin- 1=2Heisenberg model (i.e., the XYZ model), which is governed by the Hamiltonian H¼ 1 2dX hmniðJx/C27xm/C27xnþJy/C27y m/C27ynþJz/C27zm/C27znÞ;(1) where /C27xn,/C27y n,/C27znare the Pauli matrices for an effective spin n. We assume that the spins are localized on ad-dimensional cubic lattice with nearest-neighbor interac- tions. In the presence of conventional optical pumping,this Hamiltonian is augmented with a dissipative processthat flips the spins down at some rate /C13[i.e., it corresponds to the jump operator /C27 /C0non every site, where /C27/C6n¼ð/C27xn/C6i/C27y nÞ=2]. The steady state of this open many-body system is easy to understand in the case of isotropic spin-spin interactions, namely, the XXZ model (with either ferromagnetic or antiferromagnetic couplings). For this, the Hamiltoniancan be rewritten in the form H¼ð1=2dÞP½2J xð/C27þm/C27/C0nþ /C27/C0m/C27þnÞþJz/C27zm/C27zn/C138. This Hamiltonian conserves the total number of spins in the j"istate and, therefore, does nothing to counteract the spontaneous decay. Thus, thesteady state is a trivial dark state with all spins polarized,j## /C1 /C1 /C1 #ih## /C1 /C1 /C1 #j , so the XXZ model never experiences a phase transition in the presence of dissipation, regardlessofJ xandJz. However, new types of magnetic order emerge for strongly anisotropic couplings. The crucial role of anisot- ropy can be understood as follows. Each spin experiences Jx/γJy/γ (a) PMFM AFMSDW −4−3−2−1 012344 3 2 1 0 −1 −2 −3 −4 Jx/γ(b) PM sXY −4−3−2−1 012344 3 2 1 0 −1 −2 −3 −4 FIG. 1. Mean-field phase diagrams for the dissipative XYZ model with (a) Jz=/C13¼1and (b) Jz¼0, showing the different phases: paramagnetic (PM), ferromagnetic (FM), antiferromag- netic (AFM), spin-density-wave (SDW), and staggered- XY (sXY). The white arrow points to a Lifshitz point.PRL 110, 257204 (2013) PHYSICAL REVIEW LETTERSweek ending 21 JUNE 2013 0031-9007 =13=110(25) =257204(5) 257204-1 /C2112013 American Physical Societyan effective magnetic field ( Jxh/C27xi,Jyh/C27yi,Jzh/C27zi), which depends on the direction of its neighbors [Fig. 2(a)]. It precesses about this effective field and also decays towards j#i. In order for the spin to point away from j#iin steady state, its precession must be strong enough to counteractthe decay. In the isotropic case, the spin is always parallelto the magnetic field, so there is no precession at all. On theother hand, when the couplings are sufficiently anisotropic (e.g., J x/C25/C0Jy), the spin is roughly perpendicular to the magnetic field, so the precession is strong enough to point the spin away from j#i[Fig. 2(a)]. This is in sharp contrast to the thermal equilibrium state, in which the spin tries toalign with the magnetic field rather than precess about it. This competition between precessional and dissipative dynamics gives rise to a remarkable phase diagram (Fig. 1), including ferromagnetic and antiferromagnetic phases as well as spin-density-wave and staggered- XY phases that do not exist in equilibrium. The spin-density- wave, paramagnetic, and ferromagnetic phases meet at multicritical Lifshitz points, at which the period of thespin-density wave diverges [ 31]; such Lifshitz points have been seen in equilibrium magnets with long-rangeinteractions [ 32,33], but generally do not exist in nearest- neighbor spin models. In addition, we find that a continuous symmetry emerges for certain couplings; thespontaneous breaking of this symmetry leads to a phase wecall the staggered- XYphase. Finally, we find that quantum correlations (as measured by spin squeezing) persist near the phase transitions. The model described here can be implemented in sys- tems of trapped ions or systems of ultracold atoms withanisotropic superexchange or dipolar interactions. The spin states j"iandj#icorrespond to two electronic states of the ion or atom. In the case of ions, the spin-spin interaction isobtained through virtual transitions involving motionalsidebands [ 4,34,35]. In the case of ultracold atoms, the spin-spin interaction is obtained using a two-photonresonance that excites and deexcites atoms in pairs [ 36], as explained in the Supplemental Material [ 37], or using superexchange interactions in p-band optical lattices [ 38]. In all cases, dissipation can be controllably introducedusing optical pumping. Model.— We now turn to detailed analysis of the phe- nomena outlined above. The dynamics of the many-bodysystem are given by a master equation for the densitymatrix /C26, _/C26¼/C0i½H;/C26/C138þ/C13X n/C20 /C27/C0n/C26/C27þn/C01 2ð/C27þn/C27/C0n/C26þ/C26/C27þn/C27/C0nÞ/C21 : (2) Equation ( 2) has a unique steady-state solution [ 39], and we are interested in whether the steady state exhibits aphase transition as the parameters J x,Jy,Jzchange. Note that the decay is independent for each spin, in contrast with the Dicke model [ 29,40]. Furthermore, the spins are not in equilibrium with the environmental bath. Thus, in contrast with the spin-boson model [ 41,42], the steady state is not the joint ground state of the system and environment. The master equation has a Z2symmetry ( /C27xn,/C27y n! /C0/C27xn,/C0/C27y n), which is spontaneously broken in the ordered phases. In practice, there may also be dephasing noise,leading to dissipative terms in Eq. ( 2) such as /C27 zn/C26/C27zn; since theZ2symmetry is unaffected by these terms, the phase transitions we describe are robust to dephasing, although the phase boundaries are shifted. Mean-field theory.— We begin by solving for the steady states of the model Eq. ( 2) at the level of mean-field theory. We allow the mean field to vary on each site to account forspatially inhomogeneous states [ 21]. The mean-field equa- tions, which are simply nonlinear Bloch equations, are dh/C27xni dt¼/C0/C13 2h/C27xniþ1 dX m½Jyh/C27znih/C27y mi/C0Jzh/C27y nih/C27zmi/C138; dh/C27y ni dt¼/C0/C13 2h/C27y niþ1 dX m½Jzh/C27xnih/C27zmi/C0Jxh/C27znih/C27xmi/C138; dh/C27zni dt¼/C0/C13ðh/C27zniþ1Þþ1 dX m½Jxh/C27y nih/C27xmi/C0Jyh/C27xnih/C27y mi/C138; (3) where the sum over mis taken over nearest neighbors of n. (A related model with only dephasing noise was studied inRefs. [ 43,44]. Another related model with an external field and nonlinear damping was studied using the Landau-Lifshitz-Gilbert equation [ 45,46].) Clearly, there is always a fixed-point solution, h/C27 xni¼ h/C27y ni¼ 0,h/C27zni¼/C0 1, in which all the spins are pointing down. We call this the paramagnetic (PM) phase, since it does not break the Z2symmetry of Eq. ( 2). We now consider the linear stability of the PM phase as a function h σ(a) (b) (c) θ θ FIG. 2 (color online). (a) Bloch-sphere plot, showing mean- field values of h~/C27i(solid red arrow) and effective magnetic field (dashed blue arrow) for Jx=/C13¼/C0Jy=/C13¼1,Jz¼0. The vectors are normalized to unit length. [(b) and (c)] sXY phase in the xy plane of the Bloch sphere; (b) one possible stable configuration.Black (pointing upper left) and red arrows (pointing lower right) correspond to sublattices AandB. (c) The Asublattice (black solid arrow) generates a magnetic field (gray dashed arrow)that the Bsublattice (red solid arrow) precesses around. Similarly, the Bsublattice generates a magnetic field (pink dashed arrow) that the Asublattice precesses around. The angle /C18can take any value.PRL 110, 257204 (2013) PHYSICAL REVIEW LETTERSweek ending 21 JUNE 2013 257204-2ofJx,Jy,Jz[47]. We consider d-dimensional perturbations with wave vector ~k¼ðk1;k2;...;kdÞwhere k‘¼2/C25=a ‘ anda‘is an integer. We find that the PM phase is unstable to perturbations of wave vector ~kwhen /C18Jx dXd ‘¼1cosk‘/C0Jz/C19/C18Jy dXd ‘¼1cosk‘/C0Jz/C19 </C0/C132 16:(4) This condition is satisfied only when the couplings are sufficiently anisotropic. When the PM phase is unstable, the system ends up in a time-independent steady state with h/C27xni,h/C27y ni/C2220,s oi t breaks the Z2symmetry of the master equation. There are four types of ordered phases. (i) A spatially uniform state,which we call the ferromagnetic (FM) phase, resultingfrom instability of the PM phase to k ‘¼0for all ‘. (ii) A spatially modulated state with a period of two latticesites in all directions; i.e., the system divides into twosublattices. We call this the antiferromagnetic (AFM)phase, and it results from instability to k ‘¼/C25for all ‘. (iii) A spatially modulated state with a period greater than two lattice sites in at least one direction, which we callthe spin-density-wave (SDW) phase. This results frominstability to all other k ‘. (iv) When Jz¼0, there is also a staggered- XY(sXY) phase, resulting from instability to both k‘¼0;/C25, which is discussed below. The phase diagram is shown in Fig. 1. The transitions from the PM phase are continuous, whereas the FM-AFM transition isdiscontinuous. We note two unusual features of this phase diagram. First, along the boundary between the PM and SDW phases, the ~kvalue at which the instability of the PM occurs approaches 0, meaning that the period of theSDW diverges [Fig. 3(a)]. This line culminates in a multi- critical Lifshitz point [ 31] between the PM, FM, and SDW phases. Lifshitz points occur in magnetic modelswith competing interactions [ 32,33] but are not found inequilibrium nearest-neighbor magnets; thus, their exis- tence in nearest-neighbor magnets out of equilibriumindicates that nonequilibrium phase diagrams can be quali- tatively richer than those in equilibrium. Lifshitz points show enhanced fluctuation effects relative to conventionalcritical points [ 31] and, hence, offer a rich venue for study- ing quantum fluctuations away from equilibrium. The second distinctive feature of the phase diagram is that the ordered phase breaks a continuous symmetry when J z¼0. In this case, the system divides into two sublattices as in the AFM phase. However, the angle between the twosublattices can take any value. In the specific case of J x¼/C0Jy, the spins on the AandBsublattices are at angles /C18and/C0/C18relative to the x¼yline on the Bloch sphere [Fig. 2(b)]. Any value of /C18corresponds to a stable con- figuration, since the sublattices remain perpendicular to each other’s magnetic field [Fig. 2(c)]. Upon ordering, this continuous Uð1Þsymmetry between the sublattice spin orientations is spontaneously broken, leading to aphase we call the sXY phase. This phase has vortexliketopological defects around which the relative orientationbetween A- and B-sublattice spins rotates by 2/C25. Comparison with equilibrium.— It is instructive to contrast the above results with the equilibrium case (for d> 1). The equilibrium ground state of Eq. ( 1) is ordered for any J x,Jy,Jz[48]. The magnetization axis is deter- mined by the strongest of the coupling constants, and the sign of that coupling determines whether the ordering isferromagnetic or antiferromagnetic. Evidently, the non-equilibrium phase diagram exhibits qualitatively differentbehavior from this equilibrium case. The qualitative dif-ferences between equilibrium and nonequilibrium remaineven in the limit /C13!0, although the steady state takes an increasingly long time to reach. Fluctuation effects.— We now turn from mean-field theory to an analysis of fluctuations. Such an analysiswas recently performed for driven polariton condensates[49] and suggests that the static critical properties (i.e., renormalization-group fixed points) of a driven Markoviansystem are related to finite-temperature equilibrium criticalproperties. This would indicate that the dissipative XYZ model discussed here undergoes true phase transitions in two or more dimensions. We estimate fluctuation effects and squeezing in the Gaussian approximation by mapping the spins to hard- core bosons [ 48]:/C27 þn!by n,/C27zn!2by nbn/C01. This gives a reliable approximation in the PM phase, where h/C27zni/C25/C0 1. To Gaussian order (which includes relaxing the hard-coreconstraint), the resulting Hamiltonian is H¼1 2d/C20 ðJxþJyÞX hmniðbymbnþbmbynÞþðJx/C0JyÞ /C2X hmniðby mbynþbmbnÞ/C04dJzX nby nbn/C21 ; (5)1 2 3 400.511.5 Jx/γk(a) Jx/γJy/γ(b) −4−3−2−1012344 3 2 1 0 −1 −2 −3 −4 0.50.751 FIG. 3 (color online). (a) Unstable wave vector kalong the lower boundary of the PM phase in Fig. 1(a). A Lifshitz point occurs at Jx=/C13¼1:32. For convenience, only one-dimensional wave vectors are shown. (b) Squeezing parameter /C162, calculated in the Gaussian approximation for Jz¼0. The sXY phase has been whited out, since the Gaussian approximation is notvalid there.PRL 110, 257204 (2013) PHYSICAL REVIEW LETTERSweek ending 21 JUNE 2013 257204-3and the dissipative terms in the master equation are /C13P n½bn/C26byn/C0ð1=2Þðbynbn/C26þ/C26bynbnÞ/C138. We now use stan- dard Keldysh path-integral techniques [ 50] to compute the relaxation rate, h/C27ziand the squeezing. We summarize the results here and provide details in the Supplemental Material [ 37]. (1) Relaxation rate.— The rate at which the steady state is approached can be read off from the poles of the retarded Green’s function. For notational simplicity, we assume d¼1here. In the Gaussian approximation, the lowest pole has complex frequency /C0i/C13=2/C6 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðJxcosk/C0JzÞðJycosk/C0JzÞq . A continuous phase tran- sition occurs when the frequency of this pole approaches zero; this precisely recovers Eq. ( 4). (2) Below-threshold fluctuations.— Near the transition, one expects to find nonanalytic behavior in the number of up spins,P nh/C27zni.F o r Jz¼0, this scales as h/C27zi/C24 ð/C132þ16JxJyÞðd/C02Þ=2. The divergence for d¼1renders the Gaussian approximation inconsistent and is related, as we shall show in a future work, to the absence of a phase transition in one dimension (consistent with the polariton-BEC case [ 49]). (3) Squeezing.— We find that spin squeezing, a measure of quantum correlations, persists near the transition. It can be calculated using the definition of squeezing for bosons [51]:/C162¼1þ2ðhbybi/C0j h bij2Þ/C02jhb2i/C0hbi2j. For the case of Jz¼0, as the phase boundary is approached, /C162! 1=2in the thermodynamic limit for the k¼0,/C25modes, signaling the presence of quantum correlations [Fig. 3(b)]. Comparison with numerics.— We have also simulated the Eq. ( 2) in one dimension (1D) using the method of quantum trajectories [ 52]. Although there is presumably no phase transition in 1D, the numerical results already show qualitative features predicted by mean-field theory. For example, when mean-field theory predicts FM, the corre-lation h/C27 xm/C27xniis positive for all distances [Fig. 4(a)]. When there should be AFM, the correlation alternates sign. When there should be SDW, the correlation varies with a wave- length that matches the mean-field value. When there should be sXY, h/C27xm/C27xniandh/C27y m/C27yniare both 0 for odd distances and positive for even distances [Fig. 4(b)]. Furthermore, the gap of the Liouvillian approaches 0 at the boundary of the PM phase, consistent with the Gaussian approximation (see Supplemental Material [ 37]). Experimental realization.— The dissipative XYZ model can be implemented experimentally using trapped ions. One can use171Ybþand let j#iandj"icorrespond to 2S1=2jF¼0;mF¼0iand2D3=2jF¼2;mF¼0i. In the presence of laser beams judiciously detuned from certain motional sidebands, the ions interact via Eq. ( 1)[4,34,35]. Jx,Jy,Jzcan be on the order of 1–5 kHz, and their magnitudes and signs can be varied by changing the laser detunings [ 4]. By admixing a small component ( 10/C04) of2P3=2using an off-resonant laser, one broadens thelinewidth of j"ito 2 kHz. (To make this a closed cycle, additional lasers optically pump back into j#ion a much faster time scale.) Thus, the parameter space shown in Fig.1is experimentally achievable. This setup can imple- ment an arbitrary lattice topology for a large numberof ions [ 9,53]. A variety of other realizations of the XYZ model are also possible. One approach is to use ultracold atoms coupledvia dipole-dipole interactions. The XYZ Hamiltonian is implemented by driving a two-photon resonance so that atoms are excited and deexcited in pairs, as explainedin the Supplemental Material [ 37]. This scheme can be realized using Rydberg-dressed atoms [ 54], Rydberg atoms [36,55,56], or dipolar atoms or molecules [ 57]. We show explicitly in the Supplemental Material that, for Rydberg- dressed atoms, the parameters needed for the phase tran-sitions (Fig. 1) are experimentally achievable. Finally, one can adapt a recent proposal for realizing XYZ models via superexchange in p-band optical lattices [ 38] to include dissipation, by optically pumping the atoms into the p x orbital via an intermediate excited orbital (e.g., dx2/C0y2) that does not decay into the sband. Conclusion.— In summary, we have computed the phase diagram of anisotropic spin models subject to spontaneous decay and shown that these models exhibit phases (SDWand sXY) and phase transitions (Lifshitz point) that arenot found in similar equilibrium models. The qualitativedifferences can be traced to the fact that in equilibrium, spins align with the magnetic field, whereas away from equilibrium, they precess about it. We find that quantumcorrelations, as measured by squeezing, persist near thedissipative transitions. This work paves the way for futureexplorations of critical behavior and nonequilibrium fluc- tuations near the phase transitions we have identified. A particularly intriguing question is how frustrated interac-tions (due to a triangular lattice) affect the AFM and sXYphases.0 2 4 6 8−0.2−0.100.10.2 |m−n|〈σx mσx n〉(a) 0 2 4 6 800.020.040.06 |m−n|〈σx mσx n〉(b) FIG. 4 (color online). Correlation function h/C27xm/C27xnifor 1D chain of 16 spins, from simulating the master equation.(a)J z=/C13¼1, showing remnant of FM for Jx=/C13¼2,Jy¼0 (blue circles, solid line); remnant of AFM for Jx=/C13¼/C0 2, Jy¼0(green triangles, dashed line); remnant of SDW for Jx=/C13¼4,Jy=/C13¼2(red squares, dash-dotted line). The period of the SDW matches the mean-field prediction (5.3 sites). (b)Jx=/C13¼/C0Jy=/C13¼1,Jz=/C13¼0, showing remnant of sXY phase.PRL 110, 257204 (2013) PHYSICAL REVIEW LETTERSweek ending 21 JUNE 2013 257204-4We thank Philipp Strack, Eric Kessler, Chris Laumann, Norman Yao, Hendrik Weimer, and Rajibul Islam foruseful discussions. This work was supported by NSF through a grant to ITAMP, the Harvard Quantum Optics Center, the Center for Ultracold Atoms, and DARPA. [1] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994) . [2] L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003) . [3] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008) . 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PhysRevB.83.184427.pdf

PHYSICAL REVIEW B 83, 184427 (2011) Effect of microwave irradiation on spin-torque-driven magnetization precession in nanopillars with magnetic perpendicular anisotropy N. Reckers,1,*J. Cucchiara,2O. Posth,1C. Hassel,1F. M. R ¨omer,1R. Narkowicz,3R. A. Gallardo,4P. Landeros,4H. Z ¨ahres,1 S. Mangin,2J. A. Katine,5E. E. Fullerton,6G. Dumpich,1R. Meckenstock,1J. Lindner,1and M. Farle1 1Faculty of Physics and Center for Nanointegration (CeNIDE), Lotharstr. 1, DE-47057 Duisdurg, Germany, and University of Duisburg-Essen, Lotharstr. 1, DE-47057 Duisburg, Germany 2Nancy-Universit ´e, Laboratoire de Physique des Mat ´eriaux, CNRS, Bo ˆite Postal 239, FR-54506 Vandoeuvre l ´es Nancy, France 3Department of Physics, University of Dortmund, Otto-Hahn-Str. 4, DE-44227 Dortmund, Germany 4Departamento de F ´ısica, Universidad T ´ecnica Federico Santa Mar ´ıa, Avenida Espa ˜na 1680, 2390123 Valpar ´ıso, Chile 5Hitachi Global Storage Technologies, Yerba Buena Road, San Jose, California 95135, USA 6Center of Magnetic Recording Research and Department of Electrical and Computer Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0401, USA (Received 10 January 2011; revised manuscript received 24 March 2011; published 25 May 2011) The effect of microwave irradiation on the spin-torque-driven magnetization dynamics is studied in (Co /Ni)- based nanopillar spin valves with perpendicular magnetic anisotropy. For this purpose, a setup was developed tomeasure the ac as well as the dc resistance of the nanopillar under applied fields and injected polarized currents,while irradiating microwaves with varying frequency (6–18 GHz) and power. We find that the microwaveirradiation amplifies and maintains the precessional state of the eigenresonance within a larger field range.The experiments are discussed in comparison to micromagnetic as well as macrospin simulations utilizing thenonlinearized Landau-Lifshitz-Gilbert equation. DOI: 10.1103/PhysRevB.83.184427 PACS number(s): 75 .78.−n, 75.47.−m, 85.75.−d, 72.25.Ba I. INTRODUCTION Within nanopillar spin valves consisting of a ferromagnetic hard layer, acting as a spin polarizer, and a ferromagnetic soft(free) layer separated by a nonmagnetic spacer layer, the spin-torque 1,2effect manifests itself by magnetization switching and by steady-state precessions.3,4Many investigations were conducted for systems with an easy axis of magnetization inthe film plane that demonstrate the steady-state precessionfrequencies in the microwave regime, allowing the effect to beemployed for microwave generation. 5,6A theory to understand microwave generation by spin-polarized currents is providedin Ref. 7Motivated by the possibility of dc-induced microwave generation, the influence of rf excitation on systems with aneasy axis of magnetization in the film plane was studied atlow frequencies (below 15 GHz) using rf currents 8and at high frequencies (40–60 GHz) using microwave irradiation.9 Experiments on nanopillars with perpendicular magnetic anisotropy (PMA) revealed that the current densities neededfor magnetization switching are effectively reduced 10as com- pared to in-plane systems. While a precessional elliptical statecan easily be studied for in-plane magnetized samples, it is farmore complicated in the case of PMA-based systems becausethecircular precession of the magnetization perpendicular to the film plane does not change the giant magnetoresistance(GMR) signal. 11Consequently, in investigations of PMA- switching layers, either polarizing layers with in-plane easymagnetization 12or a third in-plane magnetized ferromagnetic layer that indirectly monitors the motion of the PMA-switchinglayer were used. 11A recent theoretical work discussed the influence of microwave irradiation on (Co /Ni)-based systems with PMA and showed that microwave-assisted switchingmight be indeed beneficial. 13 In the present paper, the spin-transfer-torque effect within (Co/Ni)-based systems with PMA is experimentally studiedunder microwave irradiation in the frequency range of 5– 11 GHz, which is in the range of the magnetization eigenres-onance. The investigation focuses on the effect of microwaveirradiation on the precessional state of the system rather than on microwave-assisted switching. Our study providesunambiguous evidence that (i) the resonance frequency ofnanopillars for arbitrary orientation of the two magnetizations,including the special case of two PMA-based materials, canbe determined and (ii) the critical current density needed toexcite the steady-state precession is reduced by microwaveirradiation. II. EXPERIMENTAL DETAILS The layout of the experimental setup is schematically shown in Fig. 1. The ac resistance ( dV/dI ) and the dc resistance can be measured in the presence of a magnetic field whilemicrowaves in the gigahertz range are irradiating the sample.The setup to detect the magnetoresistance is based on amodified version of the measurement bridge described inRef. 14. The detection limit is /Delta1R R=10−5. The magnetic field can be swept up to 2 T and can be oriented parallelor perpendicular to the sample plane. To enable microwavebroadband irradiation, a coaxial semirigid microwave cable(SRMC) with a diameter of 2 mm is utilized (which isdescribed in more detail in the Appendix). The samples used for the investigations are made of (Co/Ni)-based PMA materials. The magnetically hard layer acting as a spin polarizer is given by a (Co /Pt)-(Co /Ni) multilayer while the free (switching) layer is given by a(Co/Ni) multilayer. Both layers are separated from each other by a copper spacer layer. Details of the sample preparationhave been published elsewhere. 15–17The films were patterned to form 50 ×300 nm2nanopillars. A total of eight devices 184427-1 1098-0121/2011/83(18)/184427(8) ©2011 American Physical SocietyN. RECKERS et al. PHYSICAL REVIEW B 83, 184427 (2011) ac-reference voltage generator / sine out inputmeasurement bridge ac in dc in ac out dc outdc current source multimeter microwave (mw) signal generatormagnet yoke magnet yokesample mw FIG. 1. (Color online) Schematic overview of the experimental setup. The connection of the different components with the mea- surement bridge, which is a modification of the bridge described inRef. 14, enables measurement of the magnetoresistance. The signal detected by means of the lock-in technique relies on the use of a modulated alternating current with a dc offset. The measurementbridge is an ac low-resistance bridge only allowing the measurement of resistances up to 20 /Omega1. were measured and showed a resistance of 3 .3±0.02/Omega1in the parallel alignment and a GMR ratio of/Delta1R R=0.25%. All measurements shown here were performed at roomtemperature with the magnetic field applied perpendicularlyto the film plane. III. RESULTS AND DISCUSSION Figure 2(a)shows the resistance change versus the magnetic field. We observed the magnetic switching of the free and fixedlayer by applying a small ac current (with frequency f= 997 Hz and amplitude A=40μA) without a dc current, I dc. The measurement starts in the parallel alignment of both layersat a high magnetic field (of 500 mT). The free layer switchesfrom the parallel to the antiparallel state at B=±95 mT. By increasing the magnetic field the hard layer switchesfrom antiparallel to parallel alignment at B=±410 mT. Between −500 and 500 mT we find an overall symmetric curve with respect to the zero field. The observed responsecan be explained by the GMR effect. Without any injected dccurrent the switching field of the hard (Co /Pt)-(Co /Ni) layer is about 400 mT. After saturation, the maximum magnetic fieldis limited to 250 mT to avoid any magnetization change in thehard layer when measuring the minor loop. Figure 2(b) shows the resistance change as a function ofI dc. The measurement starts in the parallel alignment of the magnetization at Idc=9 mA. At a critical dc current, I− C=−5.2 mA, the configuration switches from parallel to antiparallel alignment and at I+ C=2.6 mA the configuration switches back to the parallel alignment. Considering theelliptical shape of the sample, we calculate the critical currentdensities J − C≈1.10×107A/cm2(I− C=−5.2 mA) and J+ C≈5.52×106A/cm2(I+ C=2.6 mA). The parabolic shape is due to Joule heating. To avoid this parabolic backgroundfor all measurements shown in the following, the externalmagnetic field was varied at a constant dc current, I dc.T h e shifted zero point of the parabola is due to the Peltier effect.18 The measurement in Fig. 2(b)was performed at the remanence field of the magnet (which is 3 mT).-400 -200 0 200 400024681012 IDC=0RAC[m ] B[ m T ]RAC=I*dR/dI average -8 -6 -4 -2 0 2 4 6 8051015202530RAC=I*dR/dI IC=2 . 6m ARAC[m ] IDC[mA]B= 30 Oe IC=- 5 . 2m A(a) (b) FIG. 2. (Color online) (a) The ac resistance change as a function of the perpendicular magnetic field. (b) The ac resistance change as a function of the dc current. Figure 3shows the evolution of the ac resistance change, /Delta1R ac, as a function of the magnetic field for several injected currents. Idcis defined as positive if the electrons flow from the hard (Co /Pt)-(Co /Ni) layer (polarizer) to the free (Co /Ni) layer. Two types of ac resistance curves were observed. ForIdc<9 mA a hysteretic behavior is observed, as shown in Figs. 3(a) and 3(b). We note that for Idc=0 a square hysteresis loop shifted toward negative magnetic fields byabout 30 mT due to the dipolar field generated by the hardlayer 19is observed (although it is not shown in this work). The coercivity at Idc=0i s9 5m T[ s e eF i g . 2(a)]. AsIdcincreases, the coercivity decreases and the hysteresis loop shifts toward larger negative fields as shown in Figs. 3(a) and3(b) forIdc=6 mA. This is consistent with the analytical results of Fig. 6(a) and it is due to the spin-transfer torque as explained in detail in Ref. 6. The switching fields are now −117 and −70 mT, respectively. ForIdc>9 mA a peak in the ac resistance is observed as shown in Fig. 3(c). The maximum of the peak occurs at a magnetic field of −190 mT. Note that for Idc>9 mA a peak in the ac magnetoresistance can also be observed. By analyzingthe dc resistance changes, R dc[see Figs. 3(b) and3(d)], one can conclude that the process of magnetization movement isirreversible for low dc currents and reversible for larger ones (for which the peak is observed in the ac resistance change).For a dc current of 6 mA a hysteresis is observed, just like forthe ac resistance change, implying an irreversible process. Thisirreversibility is given by the switching of the magnetization 184427-2EFFECT OF MICROWA VE IRRADIATION ON SPIN- ... PHYSICAL REVIEW B 83, 184427 (2011) -220 -200 -180 -160 -1403.3403.3423.3453.3473.3503.352 IDC=9m A B [mT]RDCRDC[]-140 -120 -100 -80 -603.3333.3363.3393.3423.3453.348IDC=6m A B[ m T ]RDCRDC[] -140 -120 -100 -80 -60036912 IDC=6m A RAC=I*dR/dI averageRAC[m ] B [mT] -216 -189 -162020406080100RAC=I*dR/dI average IDC=9m ARAC[m ] B [mT](a) (d) (c)(b) FIG. 3. (Color online) The ac resistance change (left column) and the dc resistance (right column) as a function of the perpendicular magnetic field for different dc currents, Idc. between parallel and antiparallel alignment relative to the magnetization of the polarizer. At Idc=9 mA, however, the dc resistance change is nonhysteretic, clearly indicating areversible change of the magnetization direction. In this case,/Delta1R acis just given by the derivative of the dc resistance change producing the peaklike behavior. As discussed in detail belowand observed by other authors before, 5,6the reversible change is given by a steady-state precession of the magnetization andis caused by the competition between the external field, whichfavors an antiparallel alignment of the magnetizations, andthe spin-torque, which supports the parallel orientation of thetwo magnetizations. The effect of the microwave irradiationin the gigahertz range on /Delta1R acis shown in Fig. 4(a).T h e ac resistance curves are obtained for a constant dc currentofI dc=10 mA while sweeping the magnetic field up to −250 mT without irradiating the device. It is compared to the measurement performed under identical conditions butwith microwave irradiation of 7.7 GHz. When the sampleis not irradiated, a peak in /Delta1R acis observed at B=−188 mT. When the sample is irradiated at 7.7 GHz, the peak isonly slightly shifted and becomes broader and higher. Theincreasing amplitude of the peak is a hint that the microwaveirradiation enhances the magnetization precessional state. Thefact that the peak is broader under microwave irradiation showsthat a precession can be excited for a wider field range. This iscomparable to the behavior also observed in the mutual phaselocking of two spin-torque oscillators. 20,21 Figure 4(b) shows the evolution of the ac resistance, /Delta1R ac, for a fixed field of −190 mT and Idc=10 mA as a function of the frequency tuned from 5 to 11 GHz. A maximum ismeasured at 7.75 GHz [which is very close to the zero-frequency peak, the blue line in Fig. 4(a)]. The signal undermicrowave irradiation at f=7.75 GHz is strongly enhanced. Two smaller modes are observed at f=9.4 and 10.3 GHz. A clear enhancement of the ac resistance signal is observedfor frequencies from 7 to 9 GHz. The fact that several peaksappear in the frequency spectrum is due to several oscillationmodes; this is supported by the micromagnetic simulationsdiscussed below. The device shows a situation where the almost closed hysteresis is overlapped by the precession peak at 9 mA,as shown in Fig. 4(c), with additional microwave irradiation at a frequency of 7.9 GHz. This shows that the microwaveirradiation gives rise to magnetization precession and thatthe critical current needed to create magnetization oscillationcan be reduced by applying a microwave field tuned to theresonance frequency. IV . THEORETICAL DESCRIPTION A. Micromagnetic simulations of the precessional states To obtain a better understanding of the precessional state we performed micromagnetic simulations using the object-oriented micromagnetic framework ( OOMMF )22code. The normal modes of the elliptically shaped pillar were calculatedby using the same magnetic field of −190 mT perpendicular to the layers ( zdirection) as in the experiment [see Fig. 4(b)]. In addition, a small magnetic field of 1 mT in the xdirection (i.e., in the layer plane along the long axis of the elliptical pillar)is applied to generate a small deviation of the magnetizationaway from the film normal. After saturation, the small fieldis removed and the time evolution of the magnetizationprecession is calculated. The Fourier transformation directly 184427-3N. RECKERS et al. PHYSICAL REVIEW B 83, 184427 (2011) 6789 1 030405060 IDC=1 0m A B= -190 mTRAC[m ] f [GHz]7.75 GHz-250 -200 -15001020304050 IDC=1 0m ARAC[m ] B[ m T ]7.7 GHz 0G H z(a) (b) -200 -150 -10002468RAC=I*dR/dI average IDC=9m A f= 7.9 GHzRAC[m ] B[ m T ](c) FIG. 4. (Color online) (a) The ac resistance change over the magnetic field for Idc=10 mA without microwave irradiation (red line) and with a microwave at f=7.7 GHz (blue line). (b) The frequency-dependent measurement of the resistance change. (c) The ac resistance change over the magnetic field for a dc current of 9 mA andf=7.9 GHz. yields the normal, calculated modes as a function of frequency. Note that nospin-torque term has been included in the calculation. Figure 5shows the signal amplitude as a function of the microwave frequency. One clearly observes a main mode in thespectrum and two smaller modes located at higher frequencies.(Note that the amplitude of the smaller modes was multipliedby a factor of 10.) For every mode, we have plotted snapshotsof the magnetization state within the pillar at the respectivefrequency maxima of the amplitude [see Figs. 5(a)–5(c)]. The different colors visualize these deviations from the equilibriumdirection ( zdirection); red indicates a dynamic component along the xdirection in the film plane and blue indicates a deviation along the −xdirection. Blue and red spots, therefore, show spins precessing 180 ◦out of phase.0 2 4 6 8 10 12 1 4 0 2 4 6 8 10 12 14012345678910Amplitude [arb.units] f[ G H z ]x10 (a) (b) (c) FIG. 5. (Color online) OOMMF calculation of the amplitude as a function of frequency. Visualizations of the different magnetizationstates at the maxima of the amplitude reveal (a) a uniform mode and (b) and (c) spin-wave modes (for details see text). For the main mode [see Fig. 5(a)] all magnetic moments precess in phase. The snapshot of the first smaller mode[see Fig. 5(b)] shows two nodes, implying that there are phase shifts of the precession along the xdirection. Thus the mode can be identified with a so-called forward-volumemode, for which the propagation vector, k, is oriented in the plane along the long axis of the elliptical pillar while theequilibrium magnetization is aligned out of plane along thefilm normal. 23The name “forward” mode stems from the fact that for such modes the group velocity is positive (incontrast to the so-called backward-volume modes; see Ref. 23 for details). The snapshot of the second smaller mode [seeFig.5(c)] exhibits four nodes. The calculated spectrum corresponds to the experimental one of Fig. 4(b) when the following parameters are used: uniaxial out-of-plane anisotropy K 2⊥=2.23×105J/m3,s a t - uration magnetization Ms=617×103A/m, a gyromagnetic ratio of γ=175.87 GHz /T, and a Gilbert damping parameter α=0.1( t a k e nf r o mR e f . 13). The uniform mode appears at the same frequency as in the experiment ( f=ω/2π= 7.7 GHz) and the spin-wave excitations appear at higher frequencies.Their frequencies are 10.0 and 12.3 GHz. B. Macrospin dynamics To obtain a better understanding of the dynamics of the device we have modeled its current-field phase di-agram, obtained by solving the nonlinearized Landau-Lifshitz-Gilbert (LLG) equation with the spin-torquecontribution, 2 d/vectorM dt=−γ(/vectorM×/vectorBeff)+α Ms/vectorM×/vectorM dt −γβ Ms/vectorM×(/vectorM×/vectorp). (1) HereMsis the saturation magnetization, γ=gμB/¯his the gyromagnetic ratio (where gis the gfactor), and αis the Gilbert-damping parameter. We chose a coordinate systemsuch that the zaxis of the Cartesian coordinate system coincides with the direction normal to the layers. The firstterm on the right-hand side describes the precession of the 184427-4EFFECT OF MICROWA VE IRRADIATION ON SPIN- ... PHYSICAL REVIEW B 83, 184427 (2011) FIG. 6. (Color online) (a) Phase diagram showing the dc current, Idc, vs an external magnetic field Bfor the stable parallel alignment (P) of the (Co /Ni) layer and polarizer, the stable antiparallel alignment (AP), the steady-state precession, and a bistable state. The latter describes the hysteretic switching between P and AP alignment. (b)–(d) The field dependence of the out-of-plane equilibrium angle, θ,o ft h e( C o /Ni) magnetization for different (fixed) values of the dc current: (e) Idc=9 mA, (f) Idc=7.5 mA, and (g) Idc=6 mA. The solid and dashed lines indicate the direction of the external magnetic field sweep (see text). (e) and (f) The trajectory of the (Co /Ni) magnetization for the three dc current values. The solid and dashed lines indicate the direction of the external magnetic field sweep as indicated by the arrows. The external field values for the calculation are B=−83,−73, and −54 mT, respectively. The zdirection was chosen to be aligned parallel to the film normal. macrospin driven by the effective field, Beff. The latter is given by the external dc magnetic field, B, including the field produced by the polarizer stray field (measured at30 mT) and the effective magnetization of the sample, M eff= 2K2⊥/Ms−Nμ 0Ms. While the shape anisotropy, Nμ 0Ms, always favors an easy axis in the plane of the pillar, theintrinsic out-of-plane anisotropy field, 2 K 2⊥/Ms, with positive K2⊥overcompensates the shape anisotropy and stabilizes the out-of-plane easy axis of magnetization observed in (Co /Ni) multilayers. Details of the interplay between the two contribu-tions to M effand their dependence on the growth conditions in thin film samples have been discussed in Ref. 24.F o rt h e calculation, the same parameters as for the OOMMF simulation were chosen, only K2⊥=2.35×105J/m3was taken to be 5% larger than the one used in the OOMMF calculation. Note that the small lateral dimensions of the elliptical (Co /Ni) layer lead to a reduction of the demagnetizing factor, Nz≈0.94 (see Ref. 25), as compared to a thin film. The second term on the right-hand side of Eq. ( 1)i st h e Gilbert-damping term with the phenomenological dampingparameter α. The last term describes the contribution of thespin torque as proposed by Slonczewski 2with β=¯hgPIdc 2|e|MsdV, (2) where dis the film thickness, Jis the current density defined as positive for the case when the electrons flow from the polarizer to the (Co /Ni) layer, and gP=1/(−4+(3+mz)(1+P)3 4P3/2)i st h e polarization function. It should be noted that the Slonszweskiterm can have two effects: it may lead to additional dampingof the system or to a precessional contribution that drives themagnetization, that is, β> 0o rβ< 0. This is shown, e.g., in Ref. 26. After solving Eq. ( 1), we get the following set of nonlinear equations that describe the time evolution of the componentsof the magnetization (unit) vector, m i=Mi/Ms: ˙mx=α(my˙mz−mz˙my)−γmy(B+Meffmz)−γβm xmz, ˙my=α(mz˙mx−mx˙mz)+γmx(B+Meffmz)−γβm ymz, ˙mz=α(mx˙my−my˙mx)−γβ/parenleftbig m2 z−1/parenrightbig . (3) 184427-5N. RECKERS et al. PHYSICAL REVIEW B 83, 184427 (2011) Figure 6(a) shows the resulting phase diagram constructed by solving the above equations for different current andexternal magnetic field combinations. The parameters neededfor the calculation were the same as in the OOMMF simulation discussed above. The diagram is plotted for positive currentvalues and positive and negative field values. The resultsfor a given current-field combination are also shown inFigs. 6(e)–6(g). To obtain the plots, we have computed three magnetization components, m i, that describe the trajectory of the magnetization vector from the above equations. Notethat the zdirection was chosen to coincide with the direction perpendicular to the layers in the pillar. Consequently, thexandycomponents are oriented within the plane of the (Co/Ni) film. For I dc=6m A[ s e eF i g . 6(g)] andB=−54 mT the (Co /Ni) magnetization starts to precess from its initial orientation parallel to the polarizer (i.e., parallel to the −z direction) toward a stable antiparallel alignment. At thiscurrent value one obtains current-induced switching of the(Co/Ni) magnetization. In contrast, Figs. 6(e) and6(f)show the situation for the higher current values of I=7.5m A andB=−73 mT and I=9 mA and B=−83 mT. In this case, a steady-state precession is excited. The opening angledepends on the magnitude of the magnetic field and thedc current. The complete phase diagram is obtained by calculating the trajectories for different current-field values. Stable paralleland antiparallel orientation of the (Co /Ni) layer and the polarizer are denoted by P and AP, respectively. In the case ofhysteretic switching, one also obtains current-field values forwhich both orientations are stable depending on the history ofthe system. The boundary between this bistable area and thearea with either a stable parallel or an antiparallel alignment isgiven by the blue and yellow lines in the phase diagram, whichcan be calculated from Eq. ( 3) (with a detailed calculation being given in Appendix B). When sweeping the external mag-netic field, B,a tI dc=0, one obtains a classical field-driven hysteresis. The two coercive fields are shifted by the strayfield of the polarizer (which is about 30 mT) toward negativefield values. As can be seen in the phase diagram, this leads tocoercive fields of about −80 and +20 mT. Upon increasing the dc current the field swept hysteresis is shifted even more towardnegative fields and the coercivity is decreased. This behavioris in accordance with the experimental observation shown inFigs. 3(a)and 3(b). AtI dc=7 to 8 mA the calculation predicts a steady-state precession of the (Co /Ni) magnetization around the film normal, as seen in Figs. 3(d) and in 3(c). The region in phase space for which a steady-state precession at negativeexternal magnetic fields becomes possible increases for higherdc currents. Furthermore, Figs. 6(b)–6(d) show—for the same current values as used for calculating the trajectories—the fielddependence of the polar out-of-plane angle, θ. While for a dc current of I=6 mA a sharp change between 0 ◦(parallel alignment) and 180◦(antiparallel alignment) is revealed, for higher dc current values of Idc=7.5 and 9 mA a narrow field region exists for which the stable values of θ,in between the parallel and antiparallel orientation, are allowed correspondingto the steady-state precession. The plots together with the phasediagram also show that for 7 <I dc<12 mA the precessioncan be excited only when the magnetic field is starting from negative values [solid curves in Figs. 6(b)–6(d)]. In contrast, above 12 mA the precession can be obtained for both sweep directions of the applied field. Finally, we would like to make a remark on thermal fluctuations. They have a noticeable influence on the criticalcurrents of the phase diagram moving the boundaries ofthe hysteretic and nonhysteretic regions, so that such limitsacquire a degree of uncertainty. 27The main effects are that thermal fluctuations cause transitions between magnetic statesand, therefore, the width of the hysteretic region dependson temperature. 27Finite-temperature effects can be studied through statistical descriptions or by adding a random field tothe effective field, thus changing the theoretical analysis froma deterministic study to a statistical one. 27Such effects were ignored here. V . CONCLUSION In summary, we have shown that mircowave irradiation with a frequency close to the eigenprecession frequency affects thespin-transfer-torque-driven motion of the magnetization. Thiswas performed on perpendicularly magnetized nanopillarsfor which no conventional experimental method allows adetermination of the precession frequency. Furthermore, it wasshown that the critical current density can be strongly reducedby microwave irradiation. The method provides a powerfultool to study magnetization dynamics not only in nanopillarspin valves with perpendicular anisotropy but also, in general,in nanostructured samples. The results of the measurementswere compared to micromagnetic and macrospin calculations.They revealed that the modes observed in the resistancemeasurements, indeed, stem from a steady-state precessionof the spins in the sample. In addition to the eigenresonance,magnetostatic forward-volume modes are excited. ACKNOWLEDGMENTS We acknowledge financial support by the Deutsche Forschungsgemeinschaft (SFB 491) and National Sci-ence Foundation Award No. DMR-1008654, CONICYT,FONDECYT 11080246, the program “Financiamiento Basalpara Centros Cient ´ıficos y Tecnol ´ogicos de Excelencia” CE- DENNA FB0807, ISTRADE, and FRIENDS ANR program. APPENDIX A: MICROWA VE GENERATION To enable microwave broadband irradiation, a coaxial semirigid microwave cable (SRMC) with a diameter of 2 mmis utilized. To produce a high-frequency magnetic field in thefilm plane and perpendicular to the external field, the SRMC iselectrically shorted at its end by connecting the inner conductorand the ground shield of the SRMC. The direction of the shortis oriented perpendicular to the external magnetic field, B. The electric field of the microwave induces a current alongthe short, which results in a magnetic field around the shortand perpendicular to B, as schematically shown in Fig. 7(a). While the shorted SRMC can be employed also to detect FMR by measuring the reflected microwave power, 28in our case it is used for excitation only. The performance of the shorted 184427-6EFFECT OF MICROWA VE IRRADIATION ON SPIN- ... PHYSICAL REVIEW B 83, 184427 (2011) (a) (b) 0 100 200 300 400 5000102030405060Bx,By,Bz[µT] dz[µm]Bx By Bzxyz (Co/Pt)/(Co/Ni) (Co/Ni)B FIG. 7. (Color online) (a) Sketch of a coaxial semirigid microwave cable with generated microwave field and sample. (b) Dependence of the x,y,a n dzcomponents of the microwave field on the distance from the short in the middle of the short platelet. SRMC has been simulated using the finite-element-method simulation software HFSS (by Ansoft). Figure 7(b) shows the simulation of all components of the high-frequency magneticfield with respect to the coordinate system from Fig. 7(a) as a function of the distance between short and sample.While the xandzcomponents are almost zero, there is a strong ycomponent perpendicular to the rf current direction (ydirection) in the short. This field drives the magnetization. Due to the broadband properties of the SRMC, the frequencyof this excitation field can be varied from about 6 to 18 GHz.The shape of the short, chosen to be a thin platelet, stronglyinfluences the 1 /rdecay as a function of the distance to the sample, expected for a short with circular cross section andradius r. In the case of the platelet, the decrease is far less pronounced and even at a distance of about 0.5 mm the fieldamplitude still exhibits about half of its value at the surface ofthe short. The typical sample distance is 0.25–0.5 mm. In orderto increase the signal strength of the microwave generator, weuse an amplifier with a maximal output power of 1 W. APPENDIX B: CALCULATION OF CRITICAL REGIONS IN THE PHASE DIAGRAM By inserting the expressions for ˙mxand ˙my, given by Eq. ( 3), into the equation for ˙mz, and taking into account that/vectorm·˙/vectorm=0, we obtain the following equation: ˙mz/bracketleftbig 1−α2/parenleftbig m2 z−1/parenrightbig/bracketrightbig =α2mz(mx˙mx+my˙my) −γ/parenleftbig m2 z−1/parenrightbig [β+α(B+Meffmz)].By evaluating the derivatives d dt/parenleftbig m2 x+m2 y/parenrightbig =2(mx˙mx+my˙my) and d dt/parenleftbig m2 x+m2 y/parenrightbig =d dt/parenleftbig 1−m2 z/parenrightbig =−2mz˙mz, it follows that mx˙mx+my˙my=−mz˙mz. In this case, Eq. (4)can be written as ˙mz/bracketleftbig 1−α2/parenleftbig m2 z−1/parenrightbig/bracketrightbig =−α2m2 z˙mz −γ/parenleftbig m2 z−1/parenrightbig [β+α(B+Meffmz)] or ˙mz/parenleftbig 1+α2/parenrightbig =−γ/parenleftbig m2 z−1/parenrightbig [β+α(B+Meffmz)]. This yields the following equation for ˙mz: ˙mz=γ/parenleftbig 1−m2 z/parenrightbig [β+α(B+Meffmz)] 1+α2. From this equation we can extract the static solu- tions for d(mz)/dt=0. These are mz=±1a sw e l la s the nontrivial solution given by βpz+α(B+Meffmz). From the former equation we can obtain the criticalcurrents that separate regions in the B-I dcphase dia- gram, where the parallel or antiparallel states lose theirstability, I P dc=−2|e|MSVα hgP(1)(B+Meff), IAP dc=−2|e|MSVα hgP(−1)(B−Meff). In fact, the same critical regions were obtained by Mangin et al. [see Eqs. (2a) and (2b) in Ref. 15]. The critical currents are shown in Fig. 6(a), where IP dcis represented by the blue line and IAP dcby the yellow line. Note that these curves have different slopes due to the dependence on the spin-torque polarizationfactor, g P, on the magnetization. This fact ensures that both curves cross at a critical point ( B∗,I∗ dc), where IP dc=IAP dc, above which there is no hysteretic behavior. The critical fieldand current are given by B ∗=−Meffgp(−1)+gp(1) gp(−1)−gp(1), I∗ dc=−4|e|MSVαM eff [gp(−1)−gp(1)]. Then, for the currents above this threshold, precessional states should take place in a small field window, which increases witha current. The precession is characterized by a stable value ofm zthat must be a solution of mz=−B Meff−hgp(mz)Idc 2|e|MSVαM eff or mz=−B Meff−2gp(mz) (gp(−1)−gp(1))Idc I∗ dc. One should note that this solution is valid only for currents above I∗ dcand for fields in the precessional region. 184427-7N. RECKERS et al. PHYSICAL REVIEW B 83, 184427 (2011) *nathalie.reckers@uni-due.de 1L. Berger, P h y s .R e v .B 54, 9353 (1996). 2J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 3J. 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). 4E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867 (1999). 5O. Boulle, V . Cros, J. Grollier, L. G. Pereira, C. Deranlot, F. Petroff, G. Faini, J. Barnas, and A. Fert, Nat. Phys. 3, 492 (2007). 6S. 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). 7A. Slavin and V . Tiberkevich, IEEE Trans. Magn. 45, 1875 (2009). 8S. H. Florez, J. A. Katine, M. Carey, L. Folks, and B. D. Terris,J. Appl. Phys. 103, 07A708 (2008). 9M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, V . Tsoi, and P. Wyder, Nature (London) 406, 46 (2000). 10S. Mangin, Y . Henry, D. Ravelonsona, J. A. Katine, and E. E. Fullerton, Appl. Phys. Lett. 94, 012502 (2009). 11D .H o u s s a m e d d i n e ,U .E b e l s ,B .D e l a e t ,B .R o d m a c q ,I .F i r a s t r a u ,F . Ponthenier, M. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda,M.-C. Cytrille, O. Redon, and B. Dieny, Nat. Mater. 6, 447 (2007). 12W. Chen, J.-M. L. Beaujour, G. de Loubens, J. Z. Sun, and A. D. Kent, Appl. Phys. Lett. 92, 012507 (2008). 13M. Carpentieri, G. Finocchio, B. Azzerboni, and L. Torres, Phys. Rev. B 82, 094434 (2010). 14M. C. McGregor, J. Sci. Instrum. 43, 825 (1966). 15S. Mangin, D. Ravelonsona, J. A. Katine, M. J. Carey, B. D. Terris, and E. E. Fullerton, Nat. Mater. 5, 210 (2006).16S. Mangin, D. Ravelonsana, Y . Henry, J. A. Katine, and E. E. Fullerton, Assoc. Asia Pac. Phys. Soc. Bull. 18,6 (2008). 17D. Ravelosona, S. Mangin, Y . Lemaho, J. A. Katine, B. D. Terris,a n dE .E .F u l l e r t o n , Phys. Rev. Lett. 96, 186604 (2006). 18A. Fukushima, H. Kubota, A. Yamamoto, Y . Suzuki, and S. Yuasa, J. Appl. Phys. 99, 08H706 (2006). 19D. Ravelosona, S. Mangin, Y . Henry, Y . Lemaho, J. A. Katine, B. D. Terris, and E. E. Fullerton, J. Phys. D 40, 1253 (2007). 20S .K a k a ,M .R .P u f a l l ,W .H .R i p p a r d ,T .J .S i l v a ,S .E .R u s s e k ,a n dJ. A. Katine, Nature (London) 437, 389 (2005). 21F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Tehrani, Nature (London) 437, 393 (2005). 22[http://math.nist.gov/oommf/ ]. 23S. O. Demokritov and B. Hillebrands, in Spin Dynamics in Confined Magnetic Structures I ,Topics in Applied Physics ,e d i t e db y B. Hillebrands and K. Ounadjela, V ol. 83 (Springer Verlag, Berlin,2003), p. 65. 24O. Posth, C. Hassel, M. Spasova, S. Mangin, G. Dumpich, andJ. Lindner, J. Appl. Phys. 106, 023919 (2009). 25M. Beleggia, M. De Graef, Y . T. Millev, D. A. Goode, and G. Rowlands, J. Phys. D 38, 3333 (2005). 26R. Lehndorff, D. E. B ¨urgler, A. Kakay, R. Hertel, and C. M. Schneider, IEEE Trans. Magn. 44, 1951 (2008). 27M. D. Stiles and J. Miltat, in Spin dynamics in Confined Magnetic Structures III ,Spin Transfer Torque and Dynamics ,e d i t e db yB . Hillebrands and A. Thiaville, V ol. 101 (Springer Verlag, Berlin,Heidelberg, 2006), p. 225. 28F. M. R ¨omer et al. (to be published). 184427-8

PhysRevLett.117.227203.pdf

Curvature-Induced Asymmetric Spin-Wave Dispersion Jorge A. Otálora Departamento de Física, Universidad Técnica Federico Santa María, Avenida España 1680, Casilla 110-V, Valparaíso, Chile and Departamento de Física, CEDENNA, Universidad Santiago de Chile, USACH, 9170124 Santiago, Chile Ming Yan Department of Physics, Shanghai University, 99 Shangda Road, BaoShan District, Shanghai 200444, China Helmut Schultheiss Helmholtz-Zentrum Dresden —Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstraße 400, 01328 Dresden, Germany and Technische Universität Dresden, D-01062 Dresden, Germany Riccardo Hertel Karlsuhe Institute of Technology, Physikalisches Institut, Wolfgang-Gaede-Str. 1, D-76131 Karlsruhe, Germany and Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, CNRS, and Université de Strasbourg, 23 rue du Loess, F-67300 Strasbourg, France Attila Kákay* Helmholtz-Zentrum Dresden —Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstraße 400, 01328 Dresden, Germany (Received 5 May 2016; published 23 November 2016) In magnonics, spin waves are conceived of as electron-charge-free information carriers. Their wave behavior has established them as the key elements to achieve low power consumption, fast operative rates, and good packaging in magnon-based computational technologies. Hence, knowing alternative ways that reveal certain properties of their undulatory motion is an important task. Here, we show usingmicromagnetic simulations and analytical calculations that spin-wave propagation in ferromagnetic nanotubes is fundamentally different than in thin films. The dispersion relation is asymmetric regarding the sign of the wave vector. It is a purely curvature-induced effect and its fundamental origin is identified tobe the classical dipole-dipole interaction. The analytical expression of the dispersion relation has the same mathematical form as in thin films with the Dzyalonshiinsky-Moriya interaction. Therefore, this curvature- induced effect can be seen as a “dipole-induced Dzyalonshiinsky-Moriya-like ”effect. DOI: 10.1103/PhysRevLett.117.227203 Using the electron ’s spin degree of freedom for data processing instead of its charge is one great challenge. Thefirst success story can nowadays be seen in spintronicdevices employing various magnetoresistance effects inmagnetic sensors and storage applications. About ten years ago a new research field called magnonics emerged driven by the idea to use magnons as carrier of spin information[1–8]. Magnons, also called spin waves (SWs), are the dynamic eigenoscillations of the spin system in ferromag-nets with frequencies in the gigahertz to terahertz range andwith nanometer wavelengths. Novel materials allow for the coherent propagation of SWs over mesoscopic distances without any charge transport involved, paving the way forgreen data processing. Many concepts have been proposedtheoretically and experimentally, leading to prototypebuilding blocks of spin-wave-based logic [8–13]. The experimental discovery of novel phenomena such as the spin Hall effect, the Dzyaloshinski-Moria interaction [14,15] (DMI), the spin Seebeck effect, and others proved powerful mechanisms to excite, manipulate, and detectSWs in thin magnetic films on the nanometer scale via coupling of the magnons to charge and heat transport. Oneparticular feature of SWs in thin films is intriguing: Acertain set of SWs known as Damon-Eshbach (DE) [16] modes show a nonreciprocity regarding the inversion of the wave vector caused by dipolar interaction. When the propagation direction is reversed, these magnons switchfrom the top to the bottom surface of the thin film. Recentlyit was discovered that an asymmetric exchange interaction(DMI) in ultrathin ferromagnetic films can also cause anasymmetric SW dispersion [17], i.e., one can switch from positive to negative dispersion upon reversal of the wave vector. In this Letter we show that one can obtain a similarasymmetric SW dispersion that is purely caused by dipolarinteraction when going from thin films to three-dimensional structures with curved surfaces, in particularmagnetic nanotubes (MNTs). Such novel structures can nowadays be very well produced [18,19] , motivated by the broad range of applications for magnetoresistive devices,optical metamaterials, cell-DNA separators, and drugPRL 117, 227203 (2016) PHYSICAL REVIEW LETTERSweek ending 25 NOVEMBER 2016 0031-9007 =16=117(22) =227203(6) 227203-1 © 2016 American Physical Societydelivery vectors [20,21] . The high stability of their equi- librium state [22,23] against external perturbations and their robust domain walls propagating with velocities fasterthan the SW phase velocity [24] promote MNTs as appealing candidates for racetrack memory devices [25,26] and information processing [24,27] . In this Letter, we report the numerical simulation and full analytical description of curvature-induced asymmetric SW dispersion in nanotubes, which has the same mathematicalform [28–31]as the DMI but identifies the dipole-dipole interaction as the origin of the asymmetry. We demonstrate that the degree of asymmetry can be tuned with the tubegeometry but also with small electric currents flowing through the nanotube. Besides the tunability, contrary to thin films with the DMI, the asymmetry is present and issignificant even in the absence of external magnetic fields. Finite element micromagnetic simulations [32,33] were performed to study the propagation of SWs in MNTs. The numerical research is focused on a tube defined by an outer radius R¼30nm, a wall thickness d¼10nm, and a length L¼4μm. The MNT is assumed to be made of permalloy and the following material parameters are used: saturation magnetization μ 0Ms¼1T, exchange stiffness constant A¼1.3×10−11J=m, negligible magnetocrystal- line anisotropy ( Ku¼0), and low Gilbert damping αG¼0.01. Details of the simulations are presented in the Sec. S1 of the Supplemental Material [34]. The propagation and dispersion of SWs in MNTs are simulated for an equilibrium state in which the magneti-zation rotates around the circumference of the tube, thus forming a perfect flux closure configuration [35,36] . This state in the following is referenced as a vortex ( V) configuration. It is not a ground state for the given geometry and an external field is required to stabilize it. A circular Oersted field H 0≥Hcritinduced by a current flowing through the MNTor its core can serve this function. The critical field for the nanotube with the described geometry is μ0Hcrit¼53mT[37]. A schematic of the considered system is shown in Fig.1(a)with the tube in the Vstate together with the polar coordinate system used throughout the Letter, where ρ,φ, andzare the radial, azimuthal, and long axis coordinates. The SWs are excited with a homogeneous rf field applied inthe radial direction at the middle of the tube in a 100 nm wideregion, as indicated with the orange ring in Fig. 1(a). The SWs propagate from the middle of the nanotube toward its ends with wave vector k z. The circulation direction of the magnetization ˆφtogether with the propagation direction ˆz defines a chirality or handedness. The direction of propa-gation is shown in all figures such that SWs propagating to the right (left) with k R≡þjkzj(kL≡−jkzj) define the right- (left-)handed (RH and LH) chirality. Since the propagationdirection is perpendicular to the magnetization, similar to thin films, this excitation geometry is addressed as the Damon-Eshbach geometry.The SW excitation and propagation were simulated for several values of the circular field. For all field values, the continuous rf field exciting the SWs is applied until the steady state is reached. Figure 1(b) shows a snapshot in time of the SW profiles for the three different excitationfrequencies 8, 10, and 20 GHz for a circular field of 80 mT,well above the critical field. The color scheme representsthe radial component of the magnetization in an unrolledview. The rf-field position is illustrated with an orange bar.λ LandλRdenote the wavelength of the SWs on the left and right of the excitation region, respectively. Remarkably, the wavelength of the SWs propagating to the left differs fromthose propagating to the right. This difference in wave-length decreases with increasing excitation frequencies, butnever vanishes, according to the micromagnetic simula-tions for the considered range of frequencies. Figure 2shows the SW dispersion obtained from the micromagnetic simulations for two different values of the circular field, 80 mT and 1 T. The dispersion is asymmetricregarding the propagation direction and moreover, theminimum of the dispersion depends on the circular fieldas seen by comparing Figs. 2(a) and2(b). Despite the FIG. 1. (a) Schematic illustration of a nanotube in a vortex state and the cylindrical coordinate system. SWs are excited in themiddle with a radial rf field, as illustrated by the orange ring. TheSWs travel toward the ends of the nanotube with a wave vector k z perpendicular to the magnetization. þkzand−kzindicate the right and left propagation directions, respectively. (b) A snapshotin time of the SW profiles (radial component of the magnetizationcolor coded) for the three different excitation frequencies 8, 10and 20 GHz for a circular field of 80 mT. The orange bar indicatesthe position and width of the rf field. λ L(λR) denotes the wavelength of the waves traveling to the left (right).PRL 117, 227203 (2016) PHYSICAL REVIEW LETTERSweek ending 25 NOVEMBER 2016 227203-2geometrical similarity, our simulations show that the DE modes in nanotubes behave differently than their thin-film counterparts. Simulations suggest that for kz¼kLthere is a range of wave vectors wherein the group velocity isnegative, specific to the backward volume modes in thin films. A similar effect has been recently reported for thin films with the Dzyalonshinskii-Moriya interaction[17,28 –31]. For a deeper understanding of the origin of the asym- metry observed in the simulations, an analytical formula for the SW dispersion of nanotubes is presented. The analytical description is given under the framework of micromagneticcontinuum theory. The dispersion relation is calculated by(i) linearizing the Landau-Lifshitz-Gilbert equation, and (ii) solving the linear equation in terms of individual magnons with wave vector k zalong the nanotube axis ˆz, with an integer wave number ncharacteristic of the azimuthal symmetry along ˆφ, and with eigenfrequency ωnðkzÞ. An extensive analytical derivation presented in Ref.[38](guidelines can also be found in the Sec. S2 of the Supplemental Material [34]) leads to the following dispersion relation for the coherently distributed SWs [n¼0; SWs with planar wave mode profiles as shown in Fig. 1(b)] along the ˆφaxis: ω0ðkzÞ γ0μ0Ms¼K0ðkzÞþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A0ðkzÞB0ðkzÞp ; ð1Þ where the quantities A0andB0are defined as A0ðkzÞ¼l2ex/C18 k2z−1 b2/C19 þh0þL0ðkzÞ; B0ðkzÞ¼l2exk2zþh0þJ0ðkzÞð 2Þ with the functions J0,K0, andL0given byJ0ðkzÞ¼π SZ∞ 0dkk3 2ðk2þk2zÞ½Γ0ðkÞ/C1382; K0ðkzÞ¼π SZ∞ 0dkk2kz k2þk2zΓ0ðkÞΛ0ðkÞ; L0ðkzÞ¼π SZ∞ 0dk2kk2z k2þk2z½Λ0ðkÞ/C1382ð3Þ withΛ0ðkÞ¼RR rdρρJ0ðkρÞ,Γ0ðkÞ¼−2Λ1ðkÞ,J0ðxÞis the first kind of Bessel function of zero order, b−2¼2πlnðR=rÞ=S, and S¼πðR2−r2Þthe nanotube cross section, with Randrbeing the outer and inner radius, respectively. lex¼ffiffiffiffiffiffiffiffiffiffiffi ffi A=K dp is the exchange length, Ais the exchange stiffness constant, Kd¼ð1=2Þμ0M2sis the shape anisotropy constant, and h0is the circular field normalized to the saturation magnetization Ms. Figures 2(a)and2(b)show the dispersion calculated with Eq.(1). The solid line representing the analytical calcu- lations is in perfect agreement with the results of the simulations. Using Eq. (1), the SW dispersion is calculated for tubes with different diameters and a varying circular field. Twocases are summarized for tubes with a 10 nm film thickness and an outer radius of 30 and 150 nm in Figs. 3(a)and3(b), respectively. As shown, the minima of the dispersion isshifted towards larger k zvalues with increasing circular field, allowing for the manipulation of the asymmetry and the wave vector ranges for which the SWs have a negativegroup velocity. However, the asymmetry is decreased with increasing outer diameter since the curvature is reduced and completely vanishes for infinite diameters at the thin filmlimit. It is noteworthy that Eq. (1)allows for a systematic study of the eigenoscillations and its features ½k z;ω0ðkzÞ/C138as a function of nanotube size, material parameters, andapplied circular and/or axial fields without the need for expensive micromagnetic simulations.(a) (b) FIG. 2. SW dispersion relation obtained by micromagnetic simulations (red and blue dots) and analytical calculations (solidline) for circular fields of 80 mT (a) and 1 T (b). The blue squaresmark the frequencies for which the SW profile is shown inFig.1(b). A nearly perfect agreement between the results of the micromagnetic simulations and the analytical calculations isfound. (a) (b) FIG. 3. The dispersion of SWs is summarized for severalcircular fields as a function of wave number for nanotubes with(a) 30 nm and (b) 150 nm outer radius and a 10 nm film thickness.The minima of the dispersion are shifted towards larger k zvalues with increasing circular field for both diameters. The open dotsrepresent the minima for each circular field and the solid lineconnecting them is a guide to the eye only.PRL 117, 227203 (2016) PHYSICAL REVIEW LETTERSweek ending 25 NOVEMBER 2016 227203-3The asymmetric SW dispersion reported in this Letter cannot be explained within the classical frame of the DE dispersion known for thin films. The DE modes in nano- tubes with negative kzbehave as volume-charge-free backward volume modes in thin films. Such an effect,however, is already known for thin films [17] with anti- symmetric exchange (DMI) due to spin-orbit coupling. In fact the DMI favors a canting of the spins with a givenchirality and therefore introduces a local symmetry breakthat can lead to an asymmetric dispersion relation [28–31]. Nevertheless, for nanotubes the source of the asymmetricdispersion resides only in the dipole-dipole interaction, which is discussed in the following. Note that Eq. (1)has the same mathematical form as in thin films with an interfacial DMI or in crystals with a special symmetry ( C nv) and a bulk DMI [see Eqs. (6) –(9) in Ref.[30]and Table 1 in Ref. [28]].K0ðkzÞplays the same role in nanotubes as the well-known asymmetrical terms inthin films (crystals) with an interfacial (bulk) DMI (i.e., the term ð2γ 0=MsÞDkin the dispersion of thin films with an interfacial DMI [30], where Dis the DMI constant) but with the difference that K0ðkzÞoriginates from the dynamic volume charges created by the SWs as a result of the tubular curvature. From Eq. (3)it is easy to see that K0ðkzÞis an odd function [i.e., K0ðkzÞ¼−K0ð−kzÞ], therefore being the asymmetrical term in the dispersion relation. The term K0ðkzÞ, which can only be calculated by numerical integration of the corresponding Bessel func-tions, comprises the dynamic dipolar energy arising from the surface as well as from the volume charges [39] ρ v≡−ðMs=4πÞ~∇·~M. The negative dispersion or negative group velocity, however, should be related to small or closeto zero volume charges. With the magnetization in the vortex state for a SW with wave vector k z, wave number n¼0, and eigenfrequency ω, the volume charges averaged over the nanotube radius are hρVi¼h ρVi0eiðkzz−ωtþξÞ with hρVi0¼−M2s 4π/C181 ¯ρþkzffiffiffiffiffiffiffiffiffiffiffiffiffiffi B0ðkzÞ A0ðkzÞs/C19/C18 1þB0ðkzÞ A0ðkzÞ/C19−1 2; ð4Þ where A0ðkzÞandB0ðkzÞare defined in Eq. (2);ξis the phase constant of the radial and axial SW components. Itcan be seen that the amplitude is proportional to two terms. The first term 1=¯ρis the inverse of the nanotube average radius, which is proportional to the mean nanotube curva- ture[40]. The second term k zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½B0ðkzÞ=A0ðkzÞ/C138p depends on the propagation vector kz. Hence, the sum of the two terms depends on the sign of kz. Therefore, for opposite propagation directions the dynamic volume charges are different. In Fig. 4(a)the volume charge amplitude as a function of wave vector is shown for nanotubes with three differentradii. As expected from the previous considerations, it hasan asymmetric dependence on k z. Moreover, zero volume charges are obtained for kzvalues different from zero. Around these kzvalues the reduction in energy from the surface charges is larger than the energy increase from the volume charges; thus, the total energy decreases, leading toa negative dispersion. In Fig. 4(d) the SW profile as well as the divergence calculated with our TetraMag[32,33] code is shown for a case when the SWs propagating towards opposite ends have the same wavelength. Clearly, the resulting dynamic volume charges and thus the dipolar energies differ for the twosides. In experiments (or simulations) the excitation is done with a well defined frequency; therefore, the SW ’s should possess the same energy for the opposite travel directions.In nanotubes this can only be reached if the wavelengths differ such that the dynamic dipolar energy resulting from the surface and volume charges is the same for the twopropagation directions. As a consequence SWs propagating in opposite directions have different wavelengths and show an asymmetric dispersion. It is worth mentioning that thedipole-dipole interaction was reported to be also respon-sible for the asymmetric domain wall propagation in nanotubes [41]. The SW asymmetry defined as the frequency difference of the SWs traveling in opposite directions but with the same wave vector is also proportional to the asymmetricalterm and can be calculated analytically using Eq. (1).I t reads (a) (c) (d)(b) FIG. 4. (a) The volume charge amplitude as a function of wave vector. (b) SW asymmetry as a function of wave vector kzfor nanotubes with varying radius. (c) The wavelength λSWof the excited SWs for which the maximum asymmetry is reachedversus the nanotube radius. (d) SW profile for waves with equalwavelength but opposite travel direction and the correspondingvolume charges. The color scheme encodes the radial componentof the dynamic magnetization. The dark yellow rectangles markthe excitation region.PRL 117, 227203 (2016) PHYSICAL REVIEW LETTERSweek ending 25 NOVEMBER 2016 227203-4Δf¼γMs 2πjω0ðkzÞ−ω0ð−kzÞj ¼γMs πjK0ðkzÞj:ð5Þ The SW asymmetry can be estimated from Eq. (3)by looking at the dependence of K0ðkzÞon the value of kz. Equation (5)as a function of wave vector is plotted for nanotubes with different radii in Fig. 4(b). It can be seen that the maximum frequency difference decreases with increasing tube radius. For tubes with a small diameter thisvalue is in the range of several gigahertz; however, for tubes 500–600 nm in diameter —which are accessible experi- mentally due to the recent progress in material science [18] —the frequency difference is still in the range of several hundred megahertz. The SW wavelength for which the maximum asymmetry is reached is shown in Fig. 4(c)as a function of the nanotube outer radius. It is in perfect agreement with our simple predictions based on the volume charges; namely, the asymmetry (smallest contribution ofthe volume charges to the total energy) is largest forwavelengths comparable to the nanotube diameter. In a final step two limiting cases of the dispersion are presented: (1) k z¼0, and (2) kz≫1=R.F o r kz¼0 the dispersion has the following form ωFMR ¼ γ0μ0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðH0−HuÞðH0þMsÞp , which resembles the Kittel formula for the ferromagnetic resonance (FMR) of a thin film with the in-plane magnetization parallel to the applied field, and both oriented perpendicularly to the in-plane easyaxis of the shape anisotropy field H u. For a large radius, Hu≪H0; therefore, the well-known FMR formula [42] ωFMR≈γ0μ0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi H0ðH0þMsÞp for thin films with a homo- geneous in-plane magnetization parallel to the applied magnetic field H0is obtained. For a very small wavelength, kz≫1=R, the dispersion can be written as ω0ðkzÞ≈γ0μ0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðMsl2exk2z−HuþH0þMsÞðMsl2exk2zþH0Þq ; ð6Þ which is identical to the exchange-dominated dispersion relation of a planar thin film in the Damon-Esbach configuration with the in-plane magnetization oriented perpendicularly to the in-plane easy axis [16,43] (The derivation of the asymptotic analytical expressions is summarized in Ref. [38]). In summary, we have shown using micromagnetic sim- ulations as well as analytical calculations that SW propa- gation in nanotubes is fundamentally different than in thinfilms. The observed asymmetric dispersion is a purelycurvature-induced effect [44–46]and can be tuned with small electrical currents. We have shown that the SW asymmetry is in the megahertz to gigahertz range infrequency and depends on the nanotube radius. The ana- lytical expression of the dispersion has the same mathemati- cal form as in thin films with the Dzyalonshiinsky-Moriyainteraction. The fundamental origin of the asymmetric dispersion is the classical dipole-dipole interaction; therefore. it can be seen as a “dipole-induced DMI-like effect. ”We hope that the results presented here will encourage the experimental verification of this curvature- induced effect. Financial support by the Centers of Excellence with BASAL/CONICYT financing, CEDENNA No. FB0807, and Project FONDECYT Regular No. 1161403 is grate- fully acknowledged. A. K. would like to acknowledge helpful discussions with J. Lindner and J. Fassbender. M. Y. is supported by the National Natural Science Foundation of China (Grant No. 11374203) and the Shanghai Key Laboratory of High Temperature Superconductors (Grant No. 14DZ2260700). H. S. acknowledges financial support from the Deutsche Forschungsgemeinschaft within programme SCHU 2922/ 1-1. Also, the authors are very grateful to R. Gallardo for fruitful discussions. *a.kakay@hzdr.de [1] V. Kruglyak and R. Hicken, J. Magn. Magn. Mater. 306, 191 (2006) . [2] S. Neusser and D. Grundler, Adv. Mater. 21, 2927 (2009) . [3] A. Khitun, M. Bao, and K. L. Wang, J. Phys. D 43, 264005 (2010) . [4] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D43, 264001 (2010) . [5] D. Grundler, Nat. Phys. 11, 438 (2015) . [6] B. Lenk, H. Ulrichs, F. Garbs, and M. Münzenberg, Phys. Rep. 507, 107 (2011) . [7] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hille- brands, Nat. Phys. 11, 453 (2015) . [8] R. Hertel, W. Wulfhekel, and J. Kirschner, Phys. Rev. Lett. 93, 257202 (2004) . [9] A. V. Chumak, A. A. Serga, and B. Hillebrands, Nat. Commun. 5, 4700 (2014) . [10] K. Vogt, F. Y. Fradin, J. E. Pearson, T. Sebastian, S. D. Bader, B. Hillebrands, A. Hoffmann, and H. 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PhysRevLett.105.043001.pdf

Collective Dynamics of Bose-Einstein Condensates in Optical Cavities J. Keeling, M. J. Bhaseen, and B. D. Simons University of Cambridge, Cavendish Laboratory, Cambridge, CB3 0HE, UK (Received 2 March 2010; published 20 July 2010) Experiments on Bose-Einstein condensates in optical cavities have observed a coherent state of the matter-light system—superradiance. The nature of these experiments demands consideration of collectivedynamics. Including cavity leakage and the backreaction of the cavity field on the condensate, we find arich phase diagram including multiphase coexistence regions, and persistent optomechanical oscillations.The proximity of the phase boundaries results in a critical slowing down of the decay of many-bodyoscillations, which can be enhanced by large cavity loss. DOI: 10.1103/PhysRevLett.105.043001 PACS numbers: 37.30.+i, 42.50.Pq The huge advances in preparing Bose-Einstein conden- sates (BEC) in optical cavities have opened new frontierscombining cold atoms and quantum optics. One may nowreach the strongly coupled regime of cavity quantum elec-trodynamics (QED) [ 1,2] in which atoms exchange pho- tons many times before spontaneous emission and cavitylosses set in. The inherent cavity leakage also provides a window on these quantum many-body systems. In particu- lar, it allows in situ nondemolition measurements via opti- cal transmission [ 3,4]. The strong matter-light coupling also allows collective dynamics and backreaction effects,stimulating new directions in cavity optomechanics [ 5] and self-organized atomic ensembles [ 6–10]. Recently, these capabilities have been elevated by ob- servation of the superradiance transition in BECs [ 11,12]. The atom mediated coupling between a transverse pump and a cavity mode leads to a realization of the Dicke model[13–16], in which atomic momenta play the role of spin states; see Fig. 1. A significant merit of this approach is that the two-level system energy is small enough that the Dickesuperradiance transition can occur with light at opticalfrequencies [ 11,12]. These experiments are a landmark in the study of quantum phase transitions involving spins, and offer exciting and unique prospects to explore their static and dynamic properties. Indeed, the time-dependent natureof these experiments demands consideration of collectivedynamics. Motivated by these developments we investigate the collective dynamics of BECs in optical cavities. Our twoprimary goals are to establish the generic behavior, and tofocus on the precise experimental realization in Ref. [ 12]. We obtain a surprisingly rich phase diagram for a broad range of parameters, and find distinct regimes of dynamicalbehavior, including several regions of multiphase coexis-tence, and regions of persistent optomechanical oscilla-tions. For recent theoretical work see Ref. [ 17]. The experiments in Ref. [ 12] consist of a 87RbBEC with N/C24105atoms in an optical cavity with a transverse pumping laser; see Fig. 1. The excited atoms may re- emit photons either along or transverse to the cavity axis. This process couples the zero momentum atomic groundstate, jpx;pzi¼j 0;0i, to the symmetric superpositions j/C6k;/C6ki. This yields an effective two-level system or ‘‘spin,’’ where the splitting, !0, is twice the atomic recoil energy, !r¼k2=2m. One obtains an effective Dicke model for collective spins, S, of length N=2, coupled to radiation c[11,12] H¼!cycþ!0SzþUSzcycþgðcyS/C0þcSþÞ þg0ðcySþþcS/C0Þ; (1) where, !¼!c/C0!pþNU 0ð1þMÞ=2,!0¼2!r, U¼U0M,Mis a matrix element of order unity, and U0¼g2 0=ð!p/C0!aÞencodes the backreaction of the cav- ity light field on the BEC. The model includes both coro- tating and counter-rotating matter-light couplings, denotedgandg 0. In the experiment g¼g0¼g0/C10p=ð!p/C0!aÞ [12]. To describe the dynamics of the matter-light system ( 1) we construct the Heisenberg equations of motion _S/C0¼/C0ið!0þUcycÞS/C0þ2iðgcþg0cyÞSz; _Sz¼/C0igcSþþigcyS/C0þig0cS/C0/C0ig0cySþ; _c¼/C0 ½ /C20þið!þUSzÞ/C138c/C0igS/C0/C0ig0Sþ;(2) where S/C6/C17Sx/C6iSy,/C20is the cavity loss rate, and we neglect atom loss [ 12]. Various limits of these equations FIG. 1 (color). (a) BEC in a transversely pumped cavity [ 12] with pumping frequency !pand strength /C10p, single-atom cavity coupling g0, atomic transition frequency !a, cavity frequency !c, and cavity decay rate /C20. (b) Energy levels and pumping scheme showing the two-level splitting, !0¼2!r, in the effec- tive Dicke model, where !r¼k2=2mis the recoil energy.PRL 105, 043001 (2010) PHYSICAL REVIEW LETTERSweek ending 23 JULY 2010 0031-9007 =10=105(4) =043001(4) 043001-1 /C2112010 The American Physical Societyhave been explored in different contexts. For /C20¼g0¼0 they describe fermionic pairing, where cis the Feshbach resonant closed state molecular field [ 18]. This regime also arises for polariton condensates and phase-locking of os- cillators [ 19]. More recently, for g¼g0, they have emerged in an elegant proposal for realizing the Dickemodel [ 11]. As we will see, solutions of the more general equations strongly influence g¼g 0dynamics. In order to anchor the complete phase diagram, we start withU¼0and consider U/C2220below. Numerical solution of Eqs. ( 2), and the arguments below, yield the rich phase diagram in Fig. 2, where the phases indicate stable attrac- tors of the dynamics. Four distinct phases exist: all spinsdown and no photons ( +), all spins up and no photons ( *), a superradiant phase with photons (SR), and coexistence ofthe superradiant and down attractors; see S1–S4 in Fig. 2. The coexistence of different photon numbers will manifest itself in bistability of the output cavity light field. Bi- stability has also been seen in other matter–light systemswith cavity-axis pumping and U/C2220,g¼g 0¼0[5,20], where the onset of bistability coincides with the appear-ance of optomechanical oscillations. In spite of the cavitydecay rate, /C20, which may be large, the counter-rotating terms stabilize superradiant steady states. Indeed, droppingthe derivatives in Eq. ( 2) yields algebraic equations, and the determinantal condition for nontrivial solutions( c/C2220) yields Sz¼/C0!! 0ðg2þg02Þ/C6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð2!! 0gg0Þ2/C0!2 0/C202ðg2/C0g02Þ2q 2ðg2/C0g02Þ2: (3) The conditions for real physical solutions yield the blue phase boundaries shown in Fig. 2(a). Setting Sz¼/C0N=2 in Eq. ( 3) yields the ‘‘upper’’ boundary shown in Fig. 2(a) [21]. The square root vanishing yields the ‘‘lower’’ bound- ary,g0¼gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C11/C0=/C11þp , where /C11/C6¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2þ/C202p /C6!, delin- eating the onset of coexistence. To identify the green phaseboundary in Fig. 2(a)one must consider the stability of the steady states. We consider fluctuations about an arbitraryconfiguration, S¼S 0þ/C14S,c¼c0þ/C14c, with fre- quency /C23. Instability occurs if Imð/C23Þ>0, yielding the critical line g0¼gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi /C12þ=/C12/C0p shown in Fig. 2(a), where /C12/C6¼ð!/C6!0Þ2þ/C202. This separates the stable normal state+from the stable inverted state *. For the chosen parameters g0=g¼1:0043 , very close to unity. The dy- namics at g¼g0may thus be strongly influenced by proximity to this phase boundary. The parameters used in Fig.2follow the hierarchy !,/C20/C29gffiffiffiffi Np /C29!0, in which the photon decay rate, /C20¼8:1 MHz , is much greater than the level spacing, !0¼0:046 MHz [12]. In this limit one obtains a characteristic decay rate for the collective many-body oscillations, Imð/C23Þ¼/C0 /C20! 2 0=ð/C202þ!2Þ, as verified in Fig. 3(b). Notably, for a large cavity loss rate, /C20!1 , this results in Imð/C23Þ!0, or slow decay of the collectiveg′√N__ (MHz) g√N__ (MHz)0.00.51.01.52.0 0.0 0.5 1.0 1.5 2.0⇑ ⇓SR SR+⇓(a) S4 S3 S2 S1(b) S1S2S3 S4 FIG. 2 (color). (a) Dynamical phase diagram for parameters !¼20 MHz ,!0¼0:05 MHz ,/C20¼8:1 MHz taken from Ref. [ 12], showing the stable attractors of the dynamics for U¼ 0. The phases are +,Sz¼/C0N=2and no photons; *,Sz¼N=2 and no photons; SR, a superradiant state with c/C2220; and a coexistence region starting at a tricritical point d. The separatrix g0=g¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi /C12þ=/C12/C0p ¼1:0043 is close to but distinct from unity. (b) Small cavity loss regime ( /C20¼1 KHz ) showing evolution to the equilibrium superradiance transition at gþg0¼ffiffiffiffiffiffiffiffiffiffi!! 0p=N [13–16]. The Bloch spheres S1–S4 show the stable d, unstable /C13, and hyperbolic /C2fixed points (steady states) of Sand characteristic trajectories in each of the phases. Time evolution forg¼g0is shown in Fig. 3. |ψ|2 t (ms) 050100 60 80 100 120 140 160 180 200 220 240(b)Sz -0.500.5(a) |ψ|2 104106 200.0 200.2 200.4(c) FIG. 3 (color). (a) Time evolution of Szand (b) photon number forgffiffiffiffi Np ¼g0ffiffiffiffi Np ¼0:791 MHz andU¼0. The long time behavior shows the exponential envelope jcj2¼jc0j2/C6 AeImð/C23Þt(dashed lines) where Ais a nonuniversal amplitude dependant on the initial conditions, and the decay rate Imð/C23Þ¼ /C0/C20!2 0=ð/C202þ!2Þ. (c) For !0/C28!, the long time oscillation frequency is well described by the perturbative result Reð/C23Þ¼ !0jSj=Szþ/C14, where /C14¼4!g2S2z=jSjð/C202þ!2Þis a small cor- rection to leading term of order !0. The short and intermediate time dynamics can be strongly affected by the existence of additional stable or unstable fixed points.PRL 105, 043001 (2010) PHYSICAL REVIEW LETTERSweek ending 23 JULY 2010 043001-2oscillations. This may be understood as critical slowing down [ 22]. Further insight into this /C20!1 dynamics may be gained by adiabatic elimination of the fast photon field, c¼/C0 ½ iðgþg0ÞSxþðg/C0g0ÞSy/C138=ð/C20þi!Þ, to derive an effective equation of motion for the classical spins _S¼ fS;Hg/C0/C0S/C2ðS/C2^zÞ. Here H¼!0Sz/C0/C3þS2x/C0/C3/C0S2y is the Lipkin-Meshkov-Glick Hamiltonian [ 23,24], with /C3/C6/C17! /C202þ!2ðg/C6g0Þ2and/C0/C172/C20 /C202þ!2ðg02/C0g2Þ. The addi- tional term takes the form of damping in the Landau- Lifshitz-Gilbert equations [ 25]. Depending on the sign of /C0this tries to align Seither parallel or antiparallel to the z axis. The sign change at g¼g0is consistent with the /C20! 1limit of the phase boundary, g0¼gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi /C12þ=/C12/C0p , which separates the +and*steady states. One may contrast the emergence of integrable dynamics for g¼g0and/C20!1 , with the chaotic behavior when g¼g0and/C20¼0[16]. Moreover, for g/C222g0the dynamics is non-Hamiltonian. The above analysis explains why, despite the large value of/C20present in the experiment, long-lived dynamics can still exist near the phase boundaries; note, however, that else-where, such as the points illustrated by S1–S4 in Fig. 1, effects of decay are more pronounced. Having discussed the dynamics of the model [ 1] forU¼ 0, we now consider U/C2220. To make close contact with the experiments of Ref. [ 12] we hereon set g¼g 0. In Fig. 4(a) we present the dynamical phase diagram versus U. The entire topology may be found analytically from the steadystate solutions of Eq. ( 2). These reveal two classes of superradiant solutions incorporating both Uand/C20. The first class has photon population j cj2¼4g2 ~!2þ/C202/C18N2 4/C0S2z/C19 ; (4) where ~!/C17!þUSz, and Sz¼/C0! U/C6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2ð4!2/C0U2N2Þ/C0U! 0/C202 U2ð!0Uþ4g2Þs ;S y¼0;(5) andSxis determined by the normalization of jSj. Physical solutions require jSzj/C20N=2. When /C20; U!0we recover the results of equilibrium superradiance [ 13–16], and for U¼0they reduce to those of Ref. [ 11]. For sufficiently large negative UEq. ( 5) can develop unphysical complex roots. In this case one may satisfy Eq. ( 2) with ~!0/C17!0þ Ujcj2¼0,~!/C17!þUSz¼0, and c¼iffiffiffiffiffiffiffiffiffiffi ffi/C0!0 Ur ;S x¼/C0/C20 2gffiffiffiffiffiffiffiffiffiffi ffi/C0!0 Ur ;S z¼/C0! U;(6) where Syis determined by normalization. Physical solu- tions have S2xþS2z/C20N2=4. In general, these distinct so- lutions are connected for g/C222g0, so we do not distinguish them in Fig. 4(a). Nonetheless, it is important to keep track of them for analytical work when g¼g0. Figure 4(a) consists of three phase boundaries corresponding to insta-bility of +(blue), instability of *(red), and existence of the second-type superradiant phase (gold):g +;*¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi /C6½ð!/C7!UÞ2þ/C202/C138!0U 8!Uð!/C7!UÞs ;g /C3¼/C20 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi !0U !2/C0!2 Us ; (7) where !U/C17UN= 2. Instability of the normal state g+has also been considered for thermal clouds in a ring cavity [ 8]. The result for g/C3delimits the region, both for Eqs. ( 5) and (6), to have real, physical solutions. All three of these boundaries intersect at U¼/C02N/C01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2þ/C202p ,g¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi /C0!0U=4p , as shown in Fig. 4(a). Upon increasing gone finds a phase where two distinct first-type superradiantsolutions coexist; see Fig. 4(b). This is borne out in Figs. 4(c)–4(e), which compare the steady states with time integration of Eq. ( 2) (for 360 ms, to eliminate the transitory effects of critical slowing down discussed ear-lier), at points on the dash-dotted line in Fig. 4(a). Figure 4 also contains several regions involving coexistence of superradiant andnonsuperradiant phases. The points in Fig. 4(d), forU> 2!=N are limit cycles rather than steady states. Here S z¼/C0!=U andcis FIG. 4 (color). (a) Dynamical phase diagram of model ( 1) with g¼g0and parameters !¼20 MHz ,!0¼0:05 MHz ,/C20¼ 8:1 MHz taken from Ref. [ 12]. The blue, red, and gold critical boundaries correspond to Eq. ( 7). (b) Magnified portion showing a bistable superradiant phase (2SR) where both roots of Eq. ( 5) are physical. (c),(d) Cut along the dash-dotted line comparing steady state solutions and numerical integration of the equationsof motion at 360 ms. The region to the right of the blue asymp- tote corresponds to a limit cycle. For each value of U, we take many initial conditions with c¼1andSuniformly distributed over the Bloch sphere. (e) Magnified portion of (d) in the 2SR.PRL 105, 043001 (2010) PHYSICAL REVIEW LETTERSweek ending 23 JULY 2010 043001-3imaginary. Writing S/C0¼re/C0i/C18, where r2¼N2=4/C0 !2=U2, yields _/C18¼!0þUjcj2and _cþ/C20c¼ /C02igrcosð/C18Þ, with limit cycle behavior. For /C20/C29!0þ Ujcj2, these describe a damped driven pendulum. Having confirmed the phase diagram as a function of U, let us finally focus on the value UN¼/C040 MHz used in Ref. [ 12]. Figure 5shows the phase diagram as a function of!for this value of U. The superradiance boundary is accompanied by several regions of multiphase coexistence.It would be very interesting to study this experimentally. The inset shows the same data shifted and rescaled for comparison with Fig. 5of Ref. [ 12]. In summary, we have discussed the collective dynamics of BECs in optical cavities. We obtain a rich phase diagramwith different regimes of dynamical behavior, includingseveral regions of multiphase coexistence and the slowdecay of many-body oscillations. Amongst our findingsis a regime of persistent optomechanical oscillations de- scribed by a damped driven pendulum. Given the strong interest in cavity optomechanics [ 5] this may be a profit- able region to explore experimentally. Further directionsinclude the impact of cavity-axis pumping [ 26] and photon correlations. It would also be interesting to explorewhether such behavior may emerge at small finite tem-peratures, k BT/C28@!r, and to examine departures from the BEC regime. Experiments in which the coupling gis quenched through the phase boundaries may help explore this rich dynamics. We are grateful to K. Baumann, F. Brennecke, T. Esslinger, and M. Ko ¨hl for illuminating discussions. M. J. B. and J. K. acknowledge ETH Zu ¨rich, G. Blatter, S. Schmidt, and H. Tu ¨reci for hospitality and interactions. M. J. B. and B. D. S. acknowledge EPSRC Grant No. EP/E018130/1. J. K. acknowledges EPSRC Grant No. EP/ G004714/1.[1] F. Brennecke et al. ,Nature (London) 450, 268 (2007) . [2] Y. Colombe et al. ,Nature (London) 450, 272 (2007) . [3] I. B. Mekhov, C. Maschler, and H. Ritsch, Nature Phys. 3, 319 (2007) . [4] W. Chen, D. Meiser, and P. Meystre, Phys. Rev. A 75, 023812 (2007) . [5] F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, Science 322, 235 (2008) ; S. Ritter et al. ,Appl. Phys. B 95, 213 (2009) . [6] P. Domokos and H. Ritsch, Phys. Rev. Lett. 89, 253003 (2002) . [7] A. T. Black, H. W. Chan, and V. Vuletı ´c,Phys. Rev. Lett. 91, 203001 (2003) . [8] D. Nagy, J. K. Asbo ´th, P. Domokos, and H. Ritsch, Europhys. Lett. 74, 254 (2006) . [9] D. Nagy, G. Szirmai, and P. Domokos, Eur. Phys. J. D 48, 127 (2008) . [10] J. Larson, B. Damski, G. Morigi, and M. Lewenstein, Phys. Rev. Lett. 100, 050401 (2008) ; J. Larson, S. Fern´andez-Vidal, G. Morigi, and M. Lewenstein, New J. Phys. 10, 045002 (2008) ; S. Gopalakrishnan, B. L. Lev, and P. M. Goldbart , Nature Phys. 5, 845 (2009) . [11] F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, Phys. Rev. A 75, 013804 (2007) . [12] K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Nature (London) 464, 1301 (2010) . [13] R. H. Dicke, Phys. Rev. 93, 99 (1954) . [14] K. Hepp and E. H. Lieb, Ann. Phys. (Leipzig) 76, 360 (1973) . [15] Y. K. Wang and F. T. Hioe, Phys. Rev. A 7, 831 (1973) . [16] C. Emary and T. Brandes, Phys. Rev. E 67, 066203 (2003) . [17] D. Nagy, G. Konya, G. Szirmai, and P. Domokos, Phys. Rev. Lett. 104, 130401 (2010) . [18] A. V. Andreev, V. Gurarie, and L. Radzihovsky, Phys. Rev. Lett. 93, 130402 (2004) ; R. A. Barankov and L. S. Levitov, ibid. 93, 130403 (2004) ; E. A. Yuzbashyan, V. B. Kuznetsov, and B. L. Altshuler, Phys. Rev. B 72, 144524 (2005) . [19] P. R. Eastham, J. Phys. Condens. Matter 19, 295210 (2007) ; P. R. Eastham, M. H. Szymanska, and P. B. Littlewood, Solid State Commun. 127, 117 (2003) . [20] G. Szirmai, D. Nagy, and P. Domokos, Phys. Rev. A 81, 043639 (2010) . [21] For /C20¼0, we recover Dicke model superradiance at gþ g0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi !! 0=Np .F o r g¼g0it yields the results of Ref. [ 11]. See also Ref. [ 8]. [22] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977) . [23] H. J. Lipkin, N. Meshkov, and A. J. Glick, Nucl. Phys. 62, 188 (1965) ; N. Meshkov, A. J. Glick, and H. J. Lipkin, ibid. 62, 199 (1965) ; A. J. Glick, H. J. Lipkin, and N. Meshkov, ibid. 62, 211 (1965) . [24] S. Morrison and A. S. Parkins, Phys. Rev. A 77, 043810 (2008) . [25] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935); T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004) . [26] A. Tomadin et al. ,Phys. Rev. A 81, 061801(R) (2010) . FIG. 5 (color). Dynamical phase diagram as a function of ! for the experimental parameters used in Ref. [ 12] with UN¼ /C040 MHz . The blue, red, and gold phase boundaries are given by Eq. ( 7) and correspond to those in Fig. 4(a). The thick blue line is the boundary of stability of the +state that would be seen on increasing gas in Ref. [ 12]. Inset: replotted as a function of g2Nfor comparison with Fig. 5 of Ref. [ 12].PRL 105, 043001 (2010) PHYSICAL REVIEW LETTERSweek ending 23 JULY 2010 043001-4

PhysRevB.99.054423.pdf

PHYSICAL REVIEW B 99, 054423 (2019) Ultrafast generation and dynamics of isolated skyrmions in antiferromagnetic insulators Rohollah Khoshlahni,1Alireza Qaiumzadeh,2,1,*Anders Bergman,3and Arne Brataas2 1Institute for Advanced Studies in Basic Science, Zanjan, Iran 2Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 3Division of Materials Theory, Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden (Received 25 September 2018; revised manuscript received 7 January 2019; published 21 February 2019) Based on atomistic spin dynamics simulations, we report the ultrafast generation of single antiferromagnetic (AFM) skyrmions in a confined geometry. This process is achieved through an effective magnetic fieldinduced by the athermal inverse Faraday effect from a short laser pulse. The resulting field can nucleatean isolated skyrmion as a topologically protected metastable state in a collinear antiferromagnet with smallDzyaloshinskii-Moriya interaction. The radius of a single skyrmion is shown to increase by applying a uniformdc magnetic field and at increasing temperature. To investigate possible AFM spin-caloritronics phenomena,we investigate the skyrmion dynamics under an applied temperature gradient both analytically and numerically.The antiferromagnetic skyrmions move longitudinally toward the hotter region, but in contrast, small skyrmionsin the very low damping regime move toward the colder side, irrespective of the staggered topological chargenumber, with a speed that is much faster than that of their ferromagnetic counterparts. DOI: 10.1103/PhysRevB.99.054423 I. INTRODUCTION Antiferromagnetic (AFM) spintronics is an emerging and fast-growing subfield in spintronics that promises faster,smaller, and more energy efficient state-of-the-art memorydevices and data processors [ 1–7]. The dynamics of AFM systems are more complicated than that of their ferromagnetic(FM) counterparts and exhibit richer physics. Despite beingdiscovered as early as the 1930s [ 8,9], the absence of a net magnetization and the associated insensitivity to magneticfields [ 10] have hitherto limited the use of antiferromagnets. The only use for antiferromagnets is in passive exchange-biasstructures. With recent advances in experimental techniques,as well as novel theoretical proposals, the door to the AFMspintronics era has opened a little further [ 11]. Important ob- servations and predictions are unprecedented long-range spintransport in AFM insulators [ 12], detection and manipulation of the Néel order [ 13,14], engineering of AFM domain walls (DWs) [ 15], and AFM-DW motion [ 16–18]. The Dzyaloshinskii-Moriya interaction (DMI) is an anti- symmetric exchange interaction of a relativistic origin thatbreaks the chiral symmetry in magnetic systems [ 19,20]. Initially, the DMI was identified as the mechanism responsiblefor the weak magnetism observed in a few AFM systems,namely, the so-called weak FM systems. In general, within the continuum limit, the DMI decomposes into two parts inAFM systems: one being homogeneous and the other beinginhomogeneous. Whether these parts are finite depends on theunderlying crystallographic symmetry of the AFM system.The homogeneous DMI is responsible for weak ferromag-netism [ 19], while the finite inhomogeneous part breaks the *Corresponding author: alireza.qaiumzadeh@ntnu.nochiral symmetry and stabilizes exotic spin textures with well- defined chirality, such as chiral DWs and helimagnets [ 21,22]. Skyrmions, which are nanoscale swirling magnetic tex- tures, are topologically invariant chiral solitons. The inhomo-geneous DMI can stabilize skyrmions in magnetic systemswith broken inversion symmetry. Although these solitonswere predicted quite a long time ago, the experimental ob-servation and creation of skyrmions occurred only recently inFM systems, either as skyrmion lattices or as single skyrmions[23–30]. Single skyrmions can be utilized in encoding, trans- mitting, and processing information in spintronic devices[31–33]. Thus far, skyrmions have been observed only in FM and long-wavelength spin spiral systems. Recently, therehave been predictions that it is possible to stabilize thesetopological solitons even in AFM systems as either skyrmionlattices or isolated skyrmions [ 34–45]. To date, there have been only a few proposals for the generation and control of isolated skyrmions in AFM systems.Spin-transfer torques induced by spin (polarized) currents cancreate skyrmions [ 34,39,46], and spin (polarized) currents can be applied to move them [ 39,41–43,46]. These proposals for the creation and control of AFM skyrmions have some limi-tations and drawbacks. For example, some of them apply toonly metallic AFM systems. Furthermore, all of the proposedmethods depend on the use of heterostructured materials, andmore importantly, the incubation time for the generation of asingle AFM skyrmion is also long, a few nanoseconds [ 34]. In this paper, we propose a method for the ultrafast genera- tion of single AFM skyrmions in a confined geometry employ-ing an effective magnetic field induced by the optical inverseFaraday effect (IFE) [ 47]. We also study the AFM skyrmion motion induced by the magnonic Seebeck effect numericallyin an atomistic spin dynamic simulation and analytically byusing a collective coordinate approach. Thus, our method canbe used to generate and move isolated skyrmions in single 2469-9950/2019/99(5)/054423(8) 054423-1 ©2019 American Physical SocietyKHOSHLAHNI, QAIUMZADEH, BERGMAN, AND BRATAAS PHYSICAL REVIEW B 99, 054423 (2019) crystals of AFM insulators. We organize the remainder of this paper as follows. In Sec. II, we introduce our AFM system and the equations of motion for AFM spins. In Sec. III, we present our results for the rapid generation of single AFM skyrmions.We discuss the dynamics of isolated skyrmions in the presenceof thermal magnons in Sec. IV. Finally, we conclude the paper in Sec. Vand discuss the outlook on future work. II. AFM HAMILTONIAN AND DYNAMICS We consider a discrete bipartite two-dimensional (2D) AFM insulator with the following effective thermodynamicfree energy: F=−/summationdisplay /angbracketlefti,j/angbracketrightJijmi·mj−/summationdisplay /angbracketlefti,j/angbracketrightDij·mi×mj +K/summationdisplay i(mi·ˆz)2−μs/summationdisplay ih(t)·mi, (1) where miis the unit vector of the spin magnetic moment at site i. On the right-hand side of Eq. ( 1), the first term is the Heisenberg exchange interaction, with Jij<0 representing the nearest-neighbor AFM exchange energy; the second termis DMI, with the DMI vector D ij. The third term is the single-ion anisotropy in the zdirection, with K<0 being the uniaxial anisotropy energy, and the last term is the Zeemaninteraction between the external time-dependent magneticfieldhand the localized spins, with μ sbeing the sublattice saturation magnetization. The Heisenberg exchange interaction forces adjacent spins to become antiparallel, whereas the DMI encourages per-pendicular configurations of neighboring spin moments. Thecompetition between these two energy scales leads to variousexotic spin textures in the ground state or metastable states[31,48]. When the DMI strength is larger than a critical value, D>D c=4√ JK[49], the ground state differs from a collinear AFM state. In simple square lattices, there are twotypes of DMIs based on the DM vector alignment [ 50]. We denote DMI as bulk (interfacial) DMI when the DM vectoris parallel (perpendicular) to the bond direction. The bulkDMI is responsible for textures with Bloch-like structures innoncentrosymmetric crystals, while the interfacial DMI leadsto Néel-like structures at either the interface of heavy metalsand AFM bilayers or AFM systems with broken inversionsymmetry [ 22]. In this paper, we present the results for the bulk DMI. An extension of our results to the interfacial DMIis possible. In the free energy ( 1), we disregard the long-range dipolar interactions since they are negligible in thin films ofAFM systems. We also assume that the temperature is muchless than the Néel temperature. In this limit, we treat spins asthree-dimensional vectors with a fixed length, |m i|=1. The dynamics of atomic moments in an AFM system are described by the stochastic Landau-Lifshitz-Gilbert (sLLG)equation [ 51,52], dm i dt=− ˜γmi×/bracketleftbig/parenleftbig Hi+Hth i/parenrightbig +αGmi×/parenleftbig Hi+Hth i/parenrightbig/bracketrightbig ,(2) where ˜ γ=γ/(1+α2) is the renormalized gyromagnetic ra- tio,αGis the effective Gilbert damping parameter, Hi= −∂F/(μs∂mi) is the effective magnetic field on site i, andHth iis the stochastic magnetic field arising from the thermal fluctuations. The stochastic magnetic field describes how tem-perature effects enter the theory of atomistic spin dynamicsin a Langevin dynamics approach. Using the fluctuation-dissipation theorem, the thermal stochastic fields can be de-scribed by the following correlations that are local in bothspace and time: /angbracketleftbig H th i,α(t)Hth j,β(t/prime)/angbracketrightbig =2ξHδijδαβδ(t−t/prime), (3) /angbracketleftbig Hth i,α(t)/angbracketrightbig =0, (4) where ξH=αGkBT/(γμ s) is the noise power [ 53]. Through- out this paper, we use Latin letters for site numbers and Greek letters for the spatial components of a vector. In Eq. ( 3), the quantum effects that appear at lower temperatures have beenignored. Performing atomistic spin dynamic simulations, wesolve the sLLG equation, Eq. ( 7), using the Uppsala Atomistic Spin Dynamics ( UPPASD ) code [ 52,54]. III. ULTRAFAST GENERATION OF ISOLATED AFM SKYRMIONS Skyrmions appear either in the skyrmion crystalline phase in a stable state or as isolated skyrmions in a metastable state.Isolated skyrmions are central for data storage and processing.Hence, controlling single skyrmions is essential for practicalapplications. In this section, we propose an ultrafast methodto create single skyrmions in confined geometries. Creatinga single skyrmion in a metastable state requires transformingthe system from the ground state, i.e., the collinear state, intoa new local minimum containing a skyrmion state. Here, weshow that applying an intense and short magnetic field pulsecan create single skyrmions in AFM insulators via magnoninstability processes [ 29]. The recent discovery of ultrafast and nonthermal magne- tization dynamics triggered by intense and polarized laserpulses has attracted attention and promises a new route towardultrafast optomagnetism [ 55–57]. Although the underlying theory behind this effect is still unclear, phenomenologically,the effect of a polarized laser on magnetic systems is toproduce an effective magnetic field induced by the IFE h∝ E(t)×E ∗(t), where Eis the electric field of a laser pulse [47]. The amplitude of the magnetic field is proportional to the light intensity, its sign depends on the helicity of the pulse,and its direction is along the light propagation. There are recent reports of ultrafast optical nucleation of single skyrmions and skyrmion lattices in ferrimagneticand ferromagnetic materials using laser pulses, but the mi-croscopic origin is attributed to laser-induced transient heat-ing [ 58–60]. The possibility of the creation of skyrmions using optical vortex beams, electromagnetic waves carryingintrinsic orbital angular momentum, has theoretically beeninvestigated recently [ 61,62]. In this paper, we are interested in the nonthermal effects of circularly polarized laser pulsescaused by the IFE [ 47] in a confined AFM system with an initial collinear state, i.e., D<D c. We model the light- induced effective magnetic field or IFE by a time-dependentGaussian magnetic field pulse in the sLLG equation, h(t)= h pexp(−t2/2τ2 w)ˆz, where hpis the pulse amplitude and τwis 054423-2ULTRAFAST GENERATION AND DYNAMICS OF ISOLATED … PHYSICAL REVIEW B 99, 054423 (2019) miz 0 20 40 60 80 100 (a) t= -50 ps 0 20 40 60 80 100 -1 0 1 (b) t=0 ps (c) t=30 ps (d) t=50 ps FIG. 1. Snapshots of the time evolution of the spin configuration induced by a single 30-ps Gaussian magnetic field pulse normal toa square monolayer. (a) The initial state is an AFM ground state. (b) The maximum peak of a Gaussian magnetic field pulse arrives att=0, and a domain with M z=0 starts to form. (c) Evolution of a domain wall to create a preliminary design of the AFM skyrmion. (d) Domains shrink, and some reach the boundary and disappear. The remaining domains form a circle in the center. (e) Ultimately, onechiral skyrmion is stabilized in the center of the monolayer. (f) A magnified view of a chiral AFM skyrmion. the pulse width. The amplitude of this effective magnetic field can be a few teslas, and its effective duration is subnanosecond[63,64]. We consider a confined square lattice of 100 d×100d spins, where d=3 Å is the lattice constant. The Heisenberg exchange interaction is isotropic, J ij=J,a si st h eD M I , |Dij|=D. We choose typical material parameters in our atomistic spin dynamics simulations: the AFM exchangeenergy J=−0.5m e V /atom, K=0.1J , D=0.15 J, and α G=0.009. Using UPPASD , we find that the ground state of the system is a collinear AFM state with tilted spins atthe boundaries due to the competition between DMI andexchange energy (see Fig. 1(a)and the Supplemental Material [65]). Next, we apply a magnetic field pulse with h p=9 T and τw=30 ps normal to the sample. Magnons with different wave vectors are excited at the boundaries and propagate 3 3.5 4 4.5 5 5.5 6 6.5 7 hP (T)0510152025303540 τw (ps)AFM Single-Skyrmion FIG. 2. Phase diagram for the skyrmion nucleation by applying a magnetic field pulse on a square with a size of 100 d×100d.T h e sand color shows the AFM ground state. The green region represents the isolated skyrmion metastable state, which survives even afterturning the uniform and dc magnetic fields off. The blue color shows the isolated skyrmion metastable state, which exists only in the presence of an external uniform magnetic field. inside the system. Figure 1(b) shows that when the magnetic field pulse reaches its maximum, several skyrmion nucleiform in the middle of the system. After recombination andrepulsion of the nuclei, a single skyrmion survives at thecenter of the sample [see Fig. 1(e)]. Figure 1(f) shows that this AFM skyrmion, as expected, is of a Bloch type since theDMI is bulk type and isotropic in our square-lattice structure.We have also checked the effect of the next-nearest-neighborexchange interaction and observed a similar skyrmion nucle-ation process, as depicted in Fig. 1, but at a slightly smaller applied magnetic field with the same pulse duration. The application of a dc magnetic field normal to the sample can reduce the critical amplitude of the magnetic field pulse.The physical mechanism behind this reduction is that thebarrier between the global minimum, the AFM collinear state,and the local minimum, the isolated skyrmion state, dramat-ically decreases in AFM systems near the so-called spin-flopphase. To find the phase diagram for isolated skyrmion nucle-ation, i.e., τ wvshp, we turn on a dc magnetic field of h0=5T , which is smaller than the spin-flop field of the system ∼7T , before applying magnetic field pulses of different amplitudesand durations. After turning off the dc magnetic field, at theend of the skyrmion incubation process, we check whetherthe final skyrmions are stable (see Fig. 2). This phase diagram shows that it is possible to reduce the applied magnetic fieldby a few teslas. Within the phase diagram, there is a region,shown in blue, in which isolated skyrmions are stable onlyin the presence of a dc magnetic field and disappear byswitching off the magnetic field. Note that both thresholds of 054423-3KHOSHLAHNI, QAIUMZADEH, BERGMAN, AND BRATAAS PHYSICAL REVIEW B 99, 054423 (2019) 0 10 20 30 0 1 2 3 4 5 6R/d h0 (T)Simulation Theory 0 10 20 30 0 0.2 0.4R/d T/Tc FIG. 3. Skyrmion radius versus magnetic field at zero temper- ature. The red solid curve represents the analytical prediction, and the blue solid curve results from the atomistic simulations. The inset shows that the AFM skyrmion radius increases with temperature. pulse duration and amplitude for skyrmion nucleation are very material dependent. The Zeeman energy arising from the coupling of an exter- nal magnetic field with local magnetic moments appears tobe an effective hard-axis anisotropy term in the free energyof AFM systems expressed as a function of the Néel vector[66]. It is possible to demonstrate that the radius of AFM skyrmions in the regime D<D calways increases with an applied dc magnetic field, irrespective of the magnetic fieldsign, R/d=−πD/[K+μ 2 sB2/(16|J|)] [44,67]. This feature differs from FM systems, where the sign of the magneticfield controls the skyrmion size R/d=πD/(K+8μ sB/π2) [67,68]. Figure 3presents the variation in the AFM skyrmion radius as a function of an applied perpendicular dc magneticfield. The AFM skyrmion size increases with magnetic fieldirrespective of the direction of the magnetic field, which isdifferent from FM skyrmions [ 69]. Figure 3shows good agreement between the results of atomistic simulations andthe theory [ 44]. In the inset of Fig. 3, we show that the radius of AFM skyrmions increases with temperature, as has alreadybeen predicted theoretically [ 43]. IV . AFM SKYRMION MOTION INDUCED BY MAGNONIC SEEBECK EFFECT The application of skyrmions as data bits in racetrack memories requires their motion to be deterministic. In AFMinsulators, recent theories suggest that either coherent [ 70,71] or incoherent (thermal) magnons [ 17] drive domain wall motion. Traveling incoherent magnons can be excited byapplying a thermal gradient across the AFM system. Magnonsin AFM systems, contrary to their FM counterparts, pos-sess either left- or right-handed circular polarizations withopposite spin angular momenta. At finite temperatures, bothspecies of magnon polarizations are excited with an equalpopulation such that thermal magnons carry no net spinangular momentum.In this section, we explore the dynamics of single AFM skyrmions under a thermal gradient. First, we derive a theoryfor the motion of AFM solitons in the presence of a thermalgradient at the continuum level, and then we present ouratomistic simulations. A. Stochastic LLG equation for Néel vector dynamics We consider a two-sublattice AFM insulator in the con- tinuum limit, i.e., d→0. At low temperature, the magnetic moments in sublattices are mAandmB, where |mA|=|mB|=1. For analytic calculations, it is more convenient to introducetwo new variables: a total magnetization field inside theunit cell m=m A+mBand a staggered order parameter n= (mA−mB)/|mA−mB|, where m·n=0 and n=1. The total AFM Lagrangian, L=Lkin−F, is the difference between the kinetic energy Lkinand the thermodynamic free energy F, Lkin=/integraldisplay d2r1 2a˙n2, (5) F=/integraldisplay d2r/parenleftbiggA 2(∇n)2+D dn·(∇×n)/parenrightbigg , (6) where aand Aare the homogeneous and inhomogeneous ex- change stiffnesses, respectively, and Dis the inhomogeneous DM coefficient. It is straightforward to show how the energyparameters in the continuum model, Eqs. ( 5) and ( 6), are related to the energy parameters in the discrete model, Eq. ( 1) (e.g., see Ref. [ 72]). Minimizing the total Lagrangian in the presence of dissipation, using a Rayleigh dissipation functionR=(μ s/γ)αG/integraltext d2r˙n2/2, we obtain n×/parenleftbigg ¨n−afn+μs γaαG˙n/parenrightbigg =0, (7) where fn=−δF/δnis the effective staggered field. The inclusion of finite-temperature effects is via the Langevin dynamics by adding a stochastic Gaussian-shapedfield f thto the effective staggered field. Then, the sLLG equation becomes n×/parenleftbigg ¨n−a(fn+fth)+μs γaαG˙n/parenrightbigg =0. (8) The dissipation-fluctuation theorem relates the Langevin field to the damping constant, /angbracketleftbig fth α(r,t)fth β(r/prime,t/prime)/angbracketrightbig =2ξδαβδ(r−r/prime)δ(t−t/prime), (9) /angbracketleftfth(r,t)/angbracketright=0, (10) where ξ=αGkBT(x) is the correlation amplitude. We can introduce two length scales: one is the helix wave- length /Delta1≡d(A/D), and the other one is the thermal-magnon wavelength λT∝d√A/(kBT). Throughout our calculations, we assume /Delta1/greatermuchλT, which is valid for thermal magnons. B. Effective sLLG equation of AFM solitons To derive an effective description of the skyrmion dy- namics, we introduce fast spin fluctuations δngenerated by thermal fluctuations around a slowly varying magnetic texture 054423-4ULTRAFAST GENERATION AND DYNAMICS OF ISOLATED … PHYSICAL REVIEW B 99, 054423 (2019) n(0), n=/radicalbig 1−δn2n(0)+δn, (11) where δn·n(0)=0. Substituting Eq. ( 11) into the sLLG equation ( 8) and integrating over the fast oscillating component, we find theeffective stochastic equation of the motion, n (0)×/parenleftbigg ¨n(0)−afth+μs γaαG˙n(0)/parenrightbigg +τmagn=0, (12) where the thermomagnonic torques are given by τmagn=− aA/parenleftbig/angbracketleftbig δn×∂2 iδn/angbracketrightbig −∂i/angbracketleftδn2/angbracketrightn(0)×∂in(0)/parenrightbig =− a¯hJn·∇n(0)+aA(∂iρ)n(0)×∂in(0), (13) where the AFM magnon current is Jn i=(A/¯h)n(0)·/angbracketleftδn× ∂iδn/angbracketrightand the AFM magnon number density is ρ=/angbracketleftδn2/angbracketright/2. The adiabatic thermomagnonic torque, Eq. ( 13), in AFM systems has two contributions with opposite signs. The firstterm is a reactive torque, and the second one is a dissipativetorque [ 73–75]. C. Stochastic Thiele’s equation To find a stochastic equation for the dynamics of AFM solitons, we follow Thiele’s approach [ 76]. We use collective coordinates for describing the position of the skyrmion centeru(t)a sn (0)(r,t)=n(0)(r−u(t),t). Multiplying both sides of the effective sLLG equation, Eq. ( 12), by n(0)·∂αn(0)×,w e obtain −¨uβ∂βn(0)·∂αn(0)+˙uβ˙uγ(∂β∂γn(0))·∂αn(0) −a∂αn(0)·fth−μsγ−1aαG˙uβ∂αn(0)·∂αn(0) −a¯hJn βn(0)·∂αn(0)×∂βn(0) +aA(∂βρ)∂βn(0)·∂αn(0)=0, (14) w h e r ew eh a v eu s e d ˙n=− ˙uβ∂βnand ¨n=− ¨uβ∂βn+ ˙uβ˙uγ∂β∂γn. After integrating over the spatial coordinates, we finally find the stochastic Thiele’s equation for AFM skyrmions, Mαβ(¨uβ+αGaμsγ−1˙uβ)+Fth α+Fr α+Fd α=0. (15) This equation is similar to Newton’s equation of motion for the massive particles in a viscous medium, which istotally different from the massless dynamics of FM skyrmions[42,76–78]. In Eq. ( 15), the thermal, reactive, and dissipative forces are respectively defined as F th α=1 /Delta12/integraldisplay d2r∂αn(0)·fth, (16) Fr α=4π¯hQn /Delta12εαβJn β, (17) Fd α=−c2 /Delta12Mαβ∂βρ, (18) where Qn=(1/4π)/integraltext d2rn(0)·(∂xn(0)×∂yn(0)) is the topo- logical skyrmion number for the staggered field, Mαβ=(a/Delta12)−1/integraltext d2r∂αn(0)·∂βn(0)is the symmetric AFM mass ten- sor,εαβis the 2D Levi-Civita symbol, and c=√ aAis the effective AFM magnon velocity in an isotropic medium. Inperfectly circular skyrmions, M αβ=Mδβα. The thermal force satisfies the following relations: /angbracketleftbig Fth α(u,t)Fth β(u/prime,t/prime)/angbracketrightbig =2˜ξδαβδ(u−u/prime)δ(t−t/prime), (19) /angbracketleftbig Fth α(u,t)/angbracketrightbig =0, (20) where ˜ξ=(aM//Delta12)ξ. Here we should emphasize that in AFM systems, we can define another topological number for the magnetization fieldin each sublattice or magnetic topological charge Q m 1(2)= (1/4π)/integraltext d2rm1(2)·(∂xm1(2)×∂ym1(2)). Although the staggered topological charge Qnis finite for AFM skyrmions, the total topological number related to the magnetization fieldvanishes, Q m 1+Qm 2=0. We are interested in the steady-state limit of Eq. ( 15), ˙uα=−γ MαGaμs/parenleftbig Fth α+Fr α+Fd α/parenrightbig . (21) The AFM soliton velocity is inversely proportional to the Gilbert damping coefficient. Consequently, we expect a fastermotion compared to FM solitons since the damping coeffi-cient is small. D. Fokker-Planck equation for AFM skyrmions Equation ( 21) is stochastic, and it is difficult to solve it analytically. In this part, we find the steady-state velocity ofAFM skyrmions by solving a deterministic Fokker-Planckequation related to the stochastic equation ( 21). A generic stochastic equation of motion can be written as ˙m α=gαβ/parenleftbig Fβ+fth β/parenrightbig , (22) where gαβis the diffusion matrix; Fand fthare the deter- ministic and stochastic forces, respectively; and the forceautocorrelation function is /angbracketleftf th α(r,t)fth β(r/prime,t/prime)/angbracketright=2ξδαβδ(r− r/prime)δ(t−t/prime). Let P[m,t] be the probability of finding mat time t; then, the Fokker-Planck equation related to the above Langevin-like equation, Eq. ( 22), is given by [ 79] ∂tP=−∂α(gαβFβP)+∂α∂β(ξgαγgβγP). (23) We can now find the Fokker-Planck equation related to the stochastic Thiele’s equation ( 21). We consider a linear temperature gradient along the xdirection such that ∂yT=0, ∂2 xT=0,Jm y=0, and ∂yρ=0; meanwhile, we assume that the magnon current density is almost uniform throughoutthe sample, ∂ xJm x=0 and ∂2 xρ=0. Thus, the components of reactive and dissipative forces, Eq. ( 19), as well as the diffusion matrix become Fr x=Fd y=0, (24) Fr y=−4π¯hQn /Delta12Jn x, (25) Fd x=−c2 /Delta12M∂xρ, (26) 054423-5KHOSHLAHNI, QAIUMZADEH, BERGMAN, AND BRATAAS PHYSICAL REVIEW B 99, 054423 (2019) gαβ=−γ Mαaμsδαβ. (27) The reactive force Frhas a component perpendicular to the AFM magnon current direction, while the dissipative forceF dis along the AFM magnon current. In AFM systems, the diffusion matrix gαβis diagonal and inversely proportional to the effective mass and damping parameter, while in FMsystems, it has off-diagonal elements related to the magnetic topological number and diagonal elements proportional to theGilbert damping [ 77,80]. The deterministic Fokker-Planck equation for AFM soli- tons becomes ∂ tP=−/parenleftbig gFd x−2g2∂x˜ξ/parenrightbig ∂xP−gFr y∂yP+g2˜ξ/parenleftbig ∂2 x+∂2 y/parenrightbig P, (28) where P(r,t) is the probability of finding the skyrmion at position rand time t. We are interested in the lowest-order traveling wave solution in the Fokker-Planck equation, thusdefining P=P(r−vvvt) and expanding to first order in the velocity; finally, we obtain v x=gFd x−2g2∂x˜ξ=γc2 αGa/Delta12μs∂xρ−2γ2kB MαGa/Delta12μs∂xT ≡vn x−vB x, (29) vy=gFr y=4π¯hγQn MαGa/Delta12μsJn x≡vn y, (30) where vvvnandvvvBare the contributions from the AFM magnons and the stochastic Brownian motion, respectively.These two contributions have two opposite directions. Inthe low-damping regime, the first term is dominant in largeskyrmions, and these large skyrmions move toward the hotterside. In small skyrmions, the second term is dominant, andskyrmions move toward the colder side of the system. InAFM skyrmions, the dissipative torque is responsible forthe longitudinal velocity v n x, while in FM skyrmions, the longitudinal velocity arises from the adiabatic torque [ 77]. The transverse skyrmion velocity vyor skyrmion Hall velocity vanished in thermally driven skyrmion motion since thermalAFM magnons do not carry any net spin angular momentumJ n x=0. E. Atomistic simulation We simulate a 2D rectangular AFM system of 150 d×50d with open boundary conditions and material parameters asJ=−5.44 meV /atom, D=0.18 J, K=0.1J ,μ s=2μB, andαG=0.07. Within these material parameters a single skyrmion with a radius of R/d/similarequal6 can be created. In the presence of the skyrmion at ( X0,Y0)=(40d,24d), a lin- ear thermal gradient is applied along the xdirection, with T(x=40d)<T(x=150d), and we trace the center of the skyrmion. Figures 4(a) and 4(b) show the displacement of the skyrmion in the presence of different thermal gradientsin the absence and presence of a perpendicular and uniformmagnetic field, respectively. In the Supplemental Material[65], snapshots of the time evolution of skyrmion motion are presented.406080100120140 048 1 2 1 6∇T=0.2K/nm ∇T=0.26K/nm ∇T=0.33K/nm ∇T=0.4K/nmh0=0 406080100120140 048 1 2 1 6∇T=0.2K/nm ∇T=0.26K/nm ∇T=0.33K/nm ∇T=0.4K/nmh0=0 406080100120140 048 1 2 1 6∇T=0.2K/nm ∇T=0.26K/nm ∇T=0.33K/nm ∇T=0.4K/nmh0=0 406080100120140 048 1 2 1 6∇T=0.2K/nm ∇T=0.26K/nm ∇T=0.33K/nm ∇T=0.4K/nmh0=0 406080100120140 048 1 2 1 6∇T=0.2K/nm ∇T=0.26K/nm ∇T=0.33K/nm ∇T=0.4K/nmh0=0 406080100120140 048 1 2 1 6∇T=0.2K/nm ∇T=0.26K/nm ∇T=0.33K/nm ∇T=0.4K/nmh0=0 406080100120140 048 1 2 1 6∇T=0.2K/nm ∇T=0.26K/nm ∇T=0.33K/nm ∇T=0.4K/nmh0=0 406080100120140 048 1 2 1 6(a) (b)∇T=0.2K/nm ∇T=0.26K/nm ∇T=0.33K/nm ∇T=0.4K/nmh0=0X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J) 406080100120140 04 81 2 1 6h0=3.5T 406080100120140 04 81 2 1 6h0=3.5T 406080100120140 04 81 2 1 6h0=3.5T 406080100120140 04 81 2 1 6h0=3.5T 406080100120140 04 81 2 1 6h0=3.5T 406080100120140 04 81 2 1 6h0=3.5T 406080100120140 04 81 2 1 6h0=3.5T 406080100120140 04 81 2 1 6h0=3.5T 4080120 08 1 6FM h0=0 h0=−0.5 4080120 08 1 6FM h0=0 h0=−0.5 4080120 08 1 6FM h0=0 h0=−0.5 4080120 08 1 6FM h0=0 h0=−0.5X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J)X/d t(103μsγ J) FIG. 4. Skyrmion position as a function of time under different temperature gradients in the (a) absence and (b) presence of a uniform magnetic field. The inset shows the FM skyrmion velocityfor both h 0=0a n d h0=−0.5T. The atomistic simulations show only a longitudinal dis- placement of AFM skyrmions in the presence of thermalmagnons, as predicted by the analytical theory, v n y=0 [see Eqs. ( 29)]. Furthermore, also in good agreement with the theory, Eq. ( 29), the skyrmion velocity is proportional to the temperature gradient. Within the chosen parameters, theskyrmion is relatively large and moves toward the hotterregion, which means the velocity arising from the AFMmagnon contribution is the dominant term, v n x>vB x.T h e effective interaction between the skyrmion and tilted spins atthe boundary is repulsive [ 81]; thus, after some oscillations, the skyrmion lands at a distance from the rightmost edge(hotter side). Our atomistic simulations also show that thepresence of external magnetic fields, less than the criticalspin-flop field, has no significant effect on the AFM skyrmionvelocity. This differs with respect to the dynamics of FMskyrmions, in which applying a magnetic field reduces thelongitudinal skyrmion velocity [see the inset in Fig. 4(b)]. 054423-6ULTRAFAST GENERATION AND DYNAMICS OF ISOLATED … PHYSICAL REVIEW B 99, 054423 (2019) By tuning the DMI and anisotropy, we can also create smaller skyrmions. Smaller AFM skyrmions are very unstableat finite temperatures. But those which have survived movetoward the colder side of the system in the presence of anapplied thermal gradient, which means the Brownian contri-bution is the dominant term, v n x<vB x. In the Supplemental Material snapshots of the time evolution of skyrmion motionwith a radius of R/d/similarequal4 are presented [ 65]. Here we should notice that in our simulations, we have assumed a very low Gilbert damping. Increasing the Gilbertdamping leads to a drastic decay of thermal magnons throughthe system. In this case, there are many more magnons onone side of the skyrmion (the hotter side) than on the otherside (the colder side). Consequently, this leads to a largegradient of magnon number density and results in backwardmotion toward the hotter side even for smaller skyrmions, i.e.,v n x>vB x. V . SUMMARY AND CONCLUSION In summary, we have demonstrated a path for the ultra- fast creation of single homochiral skyrmions via an effectivemagnetic field arising from the optical inverse Faraday effect.Since laser pulses are localized, the method facilitates thecreation of skyrmions in a specific region, which makes itrelevant to applications such as skyrmion-based synaptic de- vices [ 82]. The created single skyrmions are metastable states of a finite AFM system in the presence of DMI. We have investigated the dynamic properties of AFM skyrmions via analytical calculations and classical atomisticsimulations. The methods agree well. Thermal magnons moveAFM skyrmions in a longitudinal direction; that is, the AFMskyrmion Hall angle is zero. In the low-damping regime,large skyrmions move toward the hotter region, and smallskyrmions move toward the colder side, while in the large-damping regime all skyrmions move toward the hotter side. Inaddition, the AFM skyrmion velocity is much faster than forFM skyrmions under similar conditions. Note added in proof. Recently, we became aware of another paper [ 83] that proposes a method for skyrmion motion in AFM insulators using a magnetic anisotropy gradient. ACKNOWLEDGMENTS We acknowledge fruitful discussions with J. Chico. 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PhysRevB.96.224410.pdf

PHYSICAL REVIEW B 96, 224410 (2017) Multiscale modeling of current-induced switching in magnetic tunnel junctions using ab initio spin-transfer torques Matthew O. A. Ellis, Maria Stamenova, and Stefano Sanvito School of Physics, AMBER and CRANN Institute, Trininty College, Dublin 2, Ireland (Received 29 September 2017; published 7 December 2017) There exists a significant challenge in developing efficient magnetic tunnel junctions with low write currents for nonvolatile memory devices. With the aim of analyzing potential materials for efficient current-operated magneticjunctions, we have developed a multi-scale methodology combining ab initio calculations of spin-transfer torque with large-scale time-dependent simulations using atomistic spin dynamics. In this work we introduce ourmultiscale approach, including a discussion on a number of possible schemes for mapping the ab initio spin torques into the spin dynamics. We demonstrate this methodology on a prototype Co/MgO/Co/Cu tunnel junctionshowing that the spin torques are primarily acting at the interface between the Co free layer and MgO. Using spindynamics we then calculate the reversal switching times for the free layer and the critical voltages and currentsrequired for such switching. Our work provides an efficient, accurate, and versatile framework for designingnovel current-operated magnetic devices, where all the materials details are taken into account. DOI: 10.1103/PhysRevB.96.224410 I. INTRODUCTION Magnetic tunnel junctions (MTJs), composed of two epi- taxially grown ferromagnetic (FM) metal layers separated byan insulating barrier (most often a few monolayers of MgOproviding a dramatic spin-filtering enhancement), constitutethe principle unit for a multitude of emerging technologies, inparticular, in magnetic random access memory (MRAM) andspin torque oscillators (STOs) [ 1,2]. In both these cases the magnetization dynamics of the free FM layer is driven by aspin-polarized current. When the free-layer magnetization ismisaligned with that of the polarizing layer, under current-carrying conditions, the exchange interaction between theitinerant and localized electron spins results in a spin-transfertorque (STT). For one of the two possible directions ofthe current this opposes the Gilbert damping torque andpromotes switching from an antiparallel (AP) to a parallel (P)magnetization state or vice versa when the current directionis reversed [ 3]. For MRAM applications it is a significant challenge to develop MTJs with a suitably low write currentso as to ensure energy efficiency and to prolong device lifetime[4]. It is becoming increasingly more apparent that computa- tional modeling can provide an initial analysis of the viabilityof materials for efficient MTJs. However, a current-carryingMTJ, where magnetization dynamics is excited, poses a rathermultiscale (both spatially and temporally) problem whichcannot be fully tackled from the most fundamental ab initio theory. In fact, to date only a few studies have attempted toanalyze a MTJ on multiple scales [ 5,6]. At one side, significant effort has been devoted to develop more precise ab initio models of spin-transfer torque [ 7,8]. These typically rely on ballistic quantum transport theory, which is suitable forthe spin-transport properties of MTJs formed by epitaxiallygrown thin layers with atomically sharp interfaces and veryfew defects, such as in Fe/MgO/Fe. At the larger scaleend, typical micromagnetic modeling of MTJs [ 9]e m p l o y s Slonczewski’s theory of STT [ 10]. This assumes perfectly symmetric junctions and incorporates all material-specificinformation of the electronic structure of the electrodes andtheir interfaces into a pair of polarization factors for the two FM leads, P LandPR. Such quantities are often taken as empirical parameters. Although the method may be suitablefor some FeMgO-based MTJs, the interface details may beof crucial importance for other junctions (for instance, inantiferromagnetic stacks [ 11,12]). Atomistic spin dynamics (ASD) has proved useful in modeling systems on a finerdetail than micromagnetics and has been developed to employab initio parameters to better describe the STT [ 13]. Still, there remains a significant gap in our modeling ability, sinceto date no quantitative and materials specific transport methodhas been combined with atomistic spin dynamics simulators.In practice, this means that we are not capable of performingcurrent-induced spin dynamics simulations without making a priori assumptions on the nature and type of the STT. In this work we attempt to bridge this gap and we present a multiscale approach to modeling current-inducedmagnetization dynamics in magnetic devices using STT. At themicroscopic scale, a quantum transport method is employed tocompute an ab initio atom-resolved STT, which is then mapped onto the Landau-Lifshitz-Gilbert (LLG) equation of motionfor atomistic magnetic moments to perform the magnetizationdynamics [ 14,15]. The method is general and can be applied to metallic and tunneling junctions on the same footing, includingnanoscaled objects such as point contacts or atoms on surfaces. Our paper is structured as follows: First we will introduce the computational scheme for calculating the ab initio STT and its mapping onto our atomistic spin model. We will thendemonstrate this methodology on an example Co/MgO/Co/CuMTJ stack. We will discuss the bias, current, and spatialdependence of the STT and how these features influence themagnetization switching of the free layer, both at zero andfinite temperature. II. METHODS Our multiscale methodology is built upon using an ab initio method at the microscale for the electron transport and an atomistic scale spin model to simulate the dynamics. 2469-9950/2017/96(22)/224410(8) 224410-1 ©2017 American Physical SocietyELLIS, STAMENOV A, AND SANVITO PHYSICAL REVIEW B 96, 224410 (2017) In particular, we utilize the SMEAGOL [16,17] code to model ballistic electron transport through the MTJ under a finite-bias voltage. SMEAGOL is an implementation of the Keldysh nonequilibrium Green’s function (NEGF) approach to thesteady-state open-boundary problem within the framework ofdensity functional theory (DFT), as implemented in the SIESTA code, which provides an efficient order- Nscaling core DFT algorithm [ 18]. Within this formalism the MTJ is modeled as a central scattering region (SR) connected to two semi-infiniteperiodic leads. As the electronic properties of the latter can bedetermined independently from those of the junction, theiraction on the scattering region can be described in termsof suitably chosen self-energy operators acting at the SRboundaries. This effectively reduces the original electronicstructure problem for an infinite nonperiodic system to anenergy-dependent problem for a finite atomic construct. Thebias voltage, V, is applied as a shift to the chemical potentials of either lead by ±V/2, and the nonequilibrium charge density of the SR can be determined self-consistently fromthe associated nonequilibrium Keldysh Green’s function. For our calculation of the spin-transfer torque we follow the approach proposed by Haney et al. [11]. The out-of- equilibrium spin density σ Vis assumed to be separable into an equilibrium spin density σ0and a transport correction σtr, where such correction is much smaller in magnitude thanthe equilibrium part. A transverse spin transport contribution arises from the noncollinearity in the open-boundary system, giving rise to a STT in the free layer. Further details of ourmethod are given in Ref. [ 12]. Here we adopt the magnetic moment version (as opposed to working with spin variables)of the atom-resolved STT, in which the STT acting on the ath atom is written as T a=μB 2/summationdisplay i∈a/summationdisplay j/Delta1ij×σtr ji, (1) where /Delta1ijare the matrix elements of the exchange-correlation field written over the localized atomic basis orbitals of SIESTA andμBis the Bohr magneton. Note that while the first summation is restricted to orbitals that belong to the atomicsitea(the atom for which the torque is calculated), the second one spans over all the orbitals in the SR. The transport spinis calculated from the difference between the equilibrium(V=0) and the nonequilibrium ( V/negationslash=0) density matrices ρ V ijas σtr=Tr [(ρV−ρ0)σ], (2) withσbeing the vector of Pauli matrices. The ab initio side of our multiscale approach is then completed with the evaluation of the dataset {Ta(V,θ)}of atom-resolved STTs as a function of the bias voltage Vand the angleθbetween the fixed and the free-layer magnetizations. It should be noted here that the use of a single angular parameterassumes that there is no noncollinearity within the free layer.In some cases, when the self-consistent calculation of thedensity matrix across a range of finite-bias grid points is tooinvolved computationally, we also utilize the linear responsequantity, namely, the spin-transfer torkance (STTk) τ a, that isdefined as τa≡∂Ta ∂V=1 2/summationdisplay i∈a/summationdisplay j/Delta1ij×Tr/bracketleftbigg∂ρji(V) ∂Vσ/bracketrightbigg V=0. (3) Once the spin-transfer torques, {Ta(V,θ)}, for the given junction are obtained, we can then proceed to computing thecurrent-induced magnetization dynamics using an atomisticspin model. ASD is a semiclassical model typically usinga Heisenberg spin Hamiltonian to describe a system ofconstant spin magnetic moments. These magnetic momentsare localized at atomic sites and their dynamics is calculatedfrom evolving discretized LLG-like equations of motion. TheLLG equations for atomic spins with additional STTs are oftenreferred to as LLG-Slonczewski equations, whose atomisticform reads ∂S i ∂t=−γSi×Hi+λSi×∂Si ∂t+1 μiTi(V,{Si}), (4) where Si=μi/μiis a unit vector in the direction of the spin magnetic moment of atom iof magnitude |μi|=μi. Since the ab initio torque in Eq. ( 1) is derived as the rate of change of the spin angular momentum, it is necessary to normalize thetorque to the unit vector used in the ASD. In Eq. ( 4)λis the atomistic damping parameter that corresponds to the Gilbertdamping parameter at the microscopic scale and H i(t)=−1 μi∂H ∂Si+ξi(t)( 5 ) is the effective magnetic field acting on spin i. The system is kept at a finite temperature through a stochastic time-dependent thermal field, ξ i(t). In the white noise limit this is represented as a Gaussian random number with the followingmoments: /angbracketleftξ ia(t)/angbracketright=0, (6) /angbracketleftξia(t)ξjb(t/prime)/angbracketright=2λkBT μsγδijδabδ(t−t/prime), (7) where i,jlabel the different atoms, a,b=x,y,z are the Cartesian components, and t,t/primeis the time. In order to model the dynamics of an MTJ free layer, we limit the Hamiltonianto contain only the Heisenberg exchange and a uniaxialanisotropy term as follows: H=−/summationdisplay ijJijSi·Sj−/summationdisplay iki(ˆeani·Si)2, (8) where Jijis the isotropic exchange constant and kiis the uniaxial anisotropy constant for spin ialong the axis ˆeani. In general one must also consider the demagnetizing fieldacting on the free layer and its contribution to the anisotropy.In the following we consider the intrinsic anisotropy to beout of plane ( ˆe ani=ˆz), and since our free layer is ultrathin the demagnetizing field can be represented as that of aninfinite thin platelet. Therefore, instead of calculating thedemagnetizing field directly, which can be costly since itinvolves adding long-range dipolar interaction to the spinHamiltonian, we incorporate it into the uniaxial field suchthatk i=ku−μ0(MsVa)2/2. Here kuis the intrinsic uniaxial 224410-2MULTISCALE MODELING OF CURRENT-INDUCED . . . PHYSICAL REVIEW B 96, 224410 (2017) anisotropy constant, μ0is the permeability of free space, Ms is the saturation magnetization, and Vais the atomic volume. The next step is to map the two-parameter discretized ab initio {Ta(V,θ)}dataset onto the STT term of Eq. ( 4), which is, in general, a continuous function of the angularcoordinates of the whole set of spins {S i}. Such mapping can be performed in several manners, and here we have implementedthree different strategies. The first is a full two-dimensionalinterpolation of the dataset, i.e., for each atom iin layer l ian interpolated STT value is obtained for the specified voltage V and the instantaneous angle θ=acos( Si·ˆP) between the local spin Siand the direction of the fixed layer magnetization ˆP. In order to simplify the calculation during the simulations, alinear interpolation is performed along Vwhile a cubic spline is used for θ, since the dynamics is more sensitive to the angular variation and only a limited set of angles are calculated at finitevoltage. Our second mapping uses the angular dependence of the STT derived by Slonczewski [ 10]. In this way we avoid calculating the angular dependence of the STT at each voltagefrom first principles. The torque magnitude, however, is takenfrom the ab initio calculations, i.e., the bias dependence of the torque is still from first principles, namely, it is interpolatedout of the ab initio dataset. This semifunctional mapping is given as T i(V,Si)=T||(V,li)Si×Si׈P+T⊥(V,li)Si׈P,(9) where T||andT⊥are the parallel and perpendicular torque magnitudes, which can be extracted at θ=90◦. Our final mapping utilizes the torkance instead of the finite voltage torques. In this manner a finite voltage is simulatedby assuming a linear voltage dependence and by scaling thetorkance to the desired Vas follows: T i(V,Si,li)=V∂T(θ,li) ∂V/vextendsingle/vextendsingle/vextendsingle/vextendsingle V=0. (10) We discuss the applicability of this linear dependence in the case of a Co/MgO-based MTJ in the following section.The angular dependence can again be interpolated using cubicsplines, but it is also possible to use the Slonczewski formgiven in Eq. ( 9). Although the STTs are extracted from ballistic transport at a constant bias voltage, we have developed a numericalscheme to utilise the ab initio –calculated I-Vcharacteristics, which allows us to simulate the atomistic spin dynamics alsounder constant-current conditions. As we will show in the nextsection, the conductance of a CoMgO-based MTJ is found tofollow the equation g(θ,V)=J(V,θ) V=A(V)+B(V) cos(θ). (11) Our model can then compute the current as it changes with the free-layer angle and apply the torque appropriately forthe given current and voltage. This is directly reflected in theprefactor of the Slonczewski STT equations [ 19].-0.10.00.10.20.3eT/μB (Aμm-2)(b) Ty Tz 0.00.51.01.52.02.5 5 10 15 20 25Moment ( μB) Atom No.(c) Co MgO Co Cumz (a) Co MgO Co Cu FIG. 1. The Co/MgO MTJ stack studied in this work. Panel (a) shows a schematic of the scattering region for the SMEAGOL calculation, while panels (b) and (c) present the atomic resolved ab initio STT for 90◦misalignment at 1 V and the atomic spin moments profiles, respectively. In (b) and (c) the first 4 Co and last 4 Cu atoms are omitted, since in the calculations these are replaced with the semi-infinite leads. III. RESULTS A.Ab initio STT in a Co-MgO MTJ Our computational strategy is now tested for a CoFeB- MgO based MTJ, which is probably the most studied magneticdevice today. In order to model such a system, we simplify thestructure to only comprise Co atoms in a Co/MgO(4)/Co(4)/Custack, where the numbers indicate the number of atomic planesin each layer. Note that the outermost layers are the semi-infinite leads as visualized in Fig. 1(a). In our generic Co-based MTJ, both leads share a bcc lattice with a lattice parameter of2.857 ˚A. This is the lattice constant of Fe and the intention to mimic the highly spin-polarized conventional CoFeB lead. Our DFT calculations are based on the local spin-density approximation with the Ceperley-Alder parametrization of theexchange-correlation functional as implemented in the SIESTA code [ 18]. A double- ζnumerical atomic basis set is used for all atomic species with additional polarization for sorbitals of the transition metal atoms. A Monkhorst-Pack Brillouin zonesampling is used, based on a 20 ×20 real-space grid. The magnetic moments of each layer are shown in Fig. 1(c). As expected, there is no magnetization in MgO and Cu, whilethe Co fixed layer shows moments close to the bulk value ofμ Co=1.72μB. Since the free layer is ultrathin, the moments are larger than in the bulk with a peak at the MgO interface.From the layer-resolved calculations we observe that the STTis strongly peaked at the MgO interface, as shown in Fig. 1(b) at 1 V for 90 ◦misalignment. Following the sharp decay of the STT inside the Co layer, there is a characteristic higher STT 224410-3ELLIS, STAMENOV A, AND SANVITO PHYSICAL REVIEW B 96, 224410 (2017) -0.10-0.050.000.050.100.15 0 π/4 π/2 3π/4 π eT/μB (Aμm-2) Angle θ (rad)(b) TorqueTxTyTz0.000.100.200.30J(θ) (Aμm-2) (a) Current density0.5V FIG. 2. The angular dependence of (a) the current density and (b) the total torque at 0.5 V . The solid line in (a) is a fit to the current density using J(θ)=A+Bcos(θ). The solid lines in (b) are a fit using the Slonczewski angular dependence given in Eq. ( 9). In both cases the fits agree well with the data, indicating that the empirical forms can be used. value also at the other interface with the Cu lead but with an opposite sign. The angular dependence of the current density with the misalignment of the ferromagnetic layers is shown in Fig. 2(a) atV=0.5 V . The solid line shows a fit obtained by using Eq. ( 11), which matches the data almost exactly and this behavior is consistent at higher voltages. Figure 2(b) shows the angular dependence of the total torque also at 0.5 V with thesolid lines showing a fit using Eq. ( 9). Also in this case the fit performs well and so the functional approximation discussedearlier is a suitable replacement for the interpolation of thedata. At higher voltages the perpendicular torque T ybecomes asymmetric, which would require a further parametrization.At present this asymmetry is neglected in the semifunctionalmapping, since such torque contributes little to the switchingso that its effect is minimal. We note that Slonczewski’sdescription of the tunneling, which is based on Fermi’s goldenrule, is valid for sufficiently wide barriers, eliminating thedirect overlap of the minority and majority spin states in theFM layers of symmetric MTJs [ 10]. As the STT decays very quickly from the interface and is practically contained withinthe free layer (see Fig. 1), Slonczewski’s sinusoidal angular dependence of the net free-layer STT appears to be a goodapproximation for our junction (see Fig. 2). Figure 3shows the total STT acting on the free layer in the Co-MgO MTJ as function of the applied bias voltage for afixed misalignment of the free-layer magnetization of 90 ◦.T h e asymmetry of the torque with bias arises from the asymmetryof the stack, namely, the free layer contains only four atomicplanes, while the fixed layer in our MTJ is semi-infinite. Inboth cases, however, there is an approximately linear and aquadratic relationship with voltage for the out-of-plane andin-plane torques, respectively. The slope of the in-plane STTaround zero matches well our zero bias torkance from Eq. ( 3),-0.20.00.20.40.60.81.01.21.4 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0Torque, eT/ μB (Aμm-2) Voltage (V)T|| T⊥ V∂T||/∂V FIG. 3. The voltage dependence of the in-plane (open squares) and out-of-plane (filled circle) torque, and the in-plane torkance (solid line). The in-plane torque shows a linear behavior up to approximately1.4 V . Within this range the torkance is a good approximation of the finite-bias torque. The out-of-plane torque shows a quadraticlike behavior, for which the zero-bias torkance is not sufficient to describe. and therefore the latter approximation offers a reasonable quantitative measure for the in-plane STT at low bias. Figure 4shows the current-voltage characteristics for our MTJ stack in both the parallel (P) and antiparallel (AP)configuration. The sharp increase of the in-plane STT above1.4 V in Fig. 3is due to the increase of the conductivity in the antiparallel configuration. This is in turn due to the factthat the /Delta1 1symmetry band for the minority spin carriers is approximately aligned to the /Delta11majority one at that bias voltage [ 5]. Intriguingly, while this leads to a lower tunneling magneto-resistance (TMR) at high voltages, the increasedelectron flow appears to result in a larger in-plane torque andin a reduction of the out-of-plane one, as can be seen in Fig. 3. -2.0-1.5-1.0-0.50.00.51.01.52.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0Current density, J(V) (A μm-2) Voltage (V)P AP FIG. 4. The resulting current density for an applied bias voltage in the Co/MgO/Co/Cu MTJ. The solid circles show the current densityin the antiparallel configuration, while the open squares show the parallel configuration. Up to approximately 1 V there is a significant TMR, but above this value more current flows in the antiparallel stateand the TMR drops. 224410-4MULTISCALE MODELING OF CURRENT-INDUCED . . . PHYSICAL REVIEW B 96, 224410 (2017) B. Switching dynamics at zero temperature We now move our attention to the switching dynamics based on the ab initio torques computed in the previous section. In order to construct the spin model, we require values forthe exchange constants, uniaxial anisotropy, atomistic Gilbertdamping, and magnetic moments. For the exchange we usethe tabulated bulk value [ 15] for bcc Fe, namely, J ij= 7.05×10−21J, which is assumed here to be similar to that of bcc Co, while the magnetic moments are taken directly fromthe SMEAGOL calculations. In order to explore a wide range of current-induced switching, we vary the anisotropy between0.001 and 0.5 meV , which, as discussed earlier, accountsfor both intrinsic anisotropy and demagnetizing fields. First-principles calculations by Hallal et al. [20] on Fe/MgO thin films found that the anisotropy is k u≈0.275 meV per atom for a layer thickness similar to ours. For comparison theswitching field at k=0.1m e V i s H k≈1.7 T, while to achieve a thermal stability of KV/k BTroom=60 an area of (36 nm)2is required. The Gilbert damping in thin films has been observedto vary with the layer thickness, and the presence of cappinglayers can enhance the damping through spin pumping effects.Experimental measurements for a Ta/CoFeB/MgO stack showdamping parameters of the order λ=0.01 for ultrathin FM layers [ 21], and so here we vary the damping from 0.01 to 0.1. The magnetization dynamics is computed by numericallysolving Eq. ( 4) using the stochastic Heun scheme [ 15] with a time step of 0.1 fs. This has been tested for stability inequilibrium. We start by investigating the voltage required to observe switching in the MTJ free layer without explicit thermaleffects. The lack of thermal effects allows us to simulate theswitching with only the basic unit cell and periodic boundaryconditions in the lateral directions. In order to measure theswitching we calculate the time that is required for m zto pass the mz=0 plane. We model the dynamics of each MTJ by initiating the simulation with a small deviation of thefree-layer magnetization from the −ˆzaxis at different applied bias voltages. The magnetization switching curves are shown in Fig. 5for (a) constant voltage and (b) constant current with an anisotropyofk=0.1 meV and a damping parameter of λ=0.01. When the junction is kept at a constant voltage the switching isuniform and stable. In practice, the magnetization of thefree layer remains antiparallel to that of the pinned one for a long time and then switches fast. This is expected since the torque increases as the two magnetization vectors becomenoncollinear, and it is maximized for θ=90 ◦. Furthermore, it is observed that increasing the voltage systematically shiftsthe transition to lower times. In contrast, at a constant current the torque can initially overcome the anisotropy but, as the misalignment angle between the fixed and the free layer decreases, the resistance of the junction also decreases. This causes the voltage requiredto maintain the desired current to be reduced, and as aconsequence, also the torque is reduced. The reduction ofthe torque as the magnetization vectors become noncollinearto each other has to be contrasted with an increase of theanisotropy, leading to a stable precessional state where a fine balance of the torques is achieved. As the current is increased further, the angle of this stable point becomes larger until it-1-0.5 0 0.5 1 0 2 4 6 8 10Mz/Ms Time (ns)(b)10 A μm-2 12.5 A μm-2 15 A μm-2 17.5 A μm-2 20 A μm-2-1-0.5 0 0.5 1Mz/Ms (a)0.175 V 0.185 V 0.190 V 0.195 V 0.200 V FIG. 5. Magnetization switching for a junction kept at (a) con- stant voltage and (b) constant current for λ=0.01 and k=0.1m e V . At constant voltage the switching is uniform above the criticalvoltage, while at constant current the torque has an additional angular dependence given by the variation of the conductivity (hence the voltage at constant current) with angle. reaches the maximum of the anisotropy torque at about 45◦. Then the full reversal occurs. Further increasing the currentreduces the reversal time and also the transition width. Figure 6shows the measured switching time against the voltage calculated with the different mapping strategies forthree values of the anisotropy. We find that there is nosignificant difference between the full and semi-interpolationmethods, since the angular dependence of the ab initio STT agrees well with the Slonczewski form. As such, only thefull interpolation results are compared to the torkance-based 0.01 0.1 1 10 0 0.5 1 1.5 2Switching time (ns) Voltage (V)lines - torkance points - full interp. K=0.1 meV 0.01 meV 0.001 meV FIG. 6. The switching time for a Co free layer as a function of bias voltage for three values of the anisotropy and a damping coefficient ofλ=0.1. The open points are for calculations performed with the full interpolation, while the solid lines are for the torkance method and the dotted ones are a guide to the eye. The arrow indicates the difference between the torkance and full interpolation methods forthek=0.1 meV case. 224410-5ELLIS, STAMENOV A, AND SANVITO PHYSICAL REVIEW B 96, 224410 (2017) 0.00.20.40.60.81.01.21.41.61.8 0.0 0.1 0.2 0.3 0.4 0.5Vc (V) Anisotropy (meV)λ=0.01λ=0.02λ=0.05λ=0.1 (a) torkance full semi 0.00.20.40.60.81.0 0.0 0.1 0.2 0.3 0.4 0.5Jc (Aμm-2) Anisotropy (meV)λ=0.01λ=0.02λ=0.05λ=0.1 (b)torkance full semi FIG. 7. The critical (a) voltage and (b) current required to switch the free layer for a given anisotropy and damping at T=0K . Three alternative methods for interpolating the STT are shown foreach case: torkance (solid lines), full interpolation (filled circles), and semifunctional (open circles). The dotted lines are a guide to the eye. ones. For each anisotropy there is no switching below a critical voltage and a sharp decay of the switching time above it. Sincethere is a large increase in the torque above approximately1.4 V (see Fig. 3), the switching time shows a consistent drop at this point. For an anisotropy of 0.1 meV (green triangles andline), the critical voltage lies close to this increased torque andwe find that there is a large difference between the calculationsusing the finite-voltage torques and those obtained at zerovoltage with the torkance method. The critical voltages and currents for a range anisotropy strengths and damping coefficients are shown in Fig. 7.T h e three interpolation methods discussed earlier are shown assolid lines for the torkance, filled points for full interpolation,and open points for the semifunctional method. Our resultsshow that there is no significant difference between thesemifunctional and the full interpolation method over the rangesimulated here. For the full interpolation method the loss ofnumerical accuracy may arise in some instances due to the poorinterpolation at θclose to the end points, 0 and π, if too few datapoints are available where the curvature is high. Such numeri- cal errors lead to longitudinal torques, which effectively (due tothe constrained spin length in the ASD) reduce the net torque. The nonlinear behavior of the critical voltage shown in Fig. 7(a) arises simply because of the calculated voltage dependence of the in-plane torque, while in (b) there is anadditional effect arising from the voltage dependence of thecurrent. At a lower damping the torkance matches the othermethods for a wider range of anisotropies. This is due tothe fact that the critical voltage is related to the product ofthe damping and the anisotropy. When the critical voltageis below approximately 1 V , then the torque is in the linearregime; hence, we find the torkance agrees well with thefinite-voltage-calculated torque (see Fig. 3). In high-anisotropy systems, where a large switching voltage may be required, anaccurate knowledge of the STT voltage dependence becomesimportant. C. Switching dynamics at finite temperature Finally, we consider the switching process at finite temper- ature. Now our simulation cell needs to be largely increased inorder to account for the temperature-induced noncollinearity.In this case we simulate a 32 ×32×4 spin slab corresponding to a lateral dimension of 9.2 nm and still apply periodicboundary conditions in the lateral directions. Ideally, one should consider thermal effects on the current and the STT as well, but here we only consider thermal effects in the ASDthrough the stochastic noise term introduced into the effectivefield in Eq. ( 5). The noncollinearity now requires a further decision to be made when mapping the STT to the ASD.The ab initio calculation of the torque is for a fully collinear free layer, but noncollinearity in ASD is required to achievea thermal spin distribution. One can then decide to use theangle of the total magnetization or that of each individual spinin order to determine the torque. The effects of this choicewill be discussed in what follows. Note that, in principle, onecan still calculate the torques from ab initio for a noncollinear situation. In fact, one can even calculate the torques at eachtime step in the ASD, for instance, as it is done for the forces inab initio molecular dynamics. This is, however, not practical here, since the transport calculations, in particular at finite bias,are much more demanding than the ASD ones. Figure 8shows the inverse average switching time at different temperatures for (a) k=0.1 meV and (b) 0.5 meV . The filled symbols show results obtained by using the angleof the total magnetization to calculate the STT, while the openones use the individual spin angle. From the figure we observethat results obtained with the different angle methods arealmost indistinguishable from each other except in (b) at 300 K.Here the switching time is averaged over 24 independentsimulations since it is a stochastic process. This may leadto an equivalence in the methods, since while these arefundamentally different the average switching time may besimilar. Different anisotropies present us two different situations. In Fig. 8(a) the inverse relaxation time is linear with the voltage, since the critical voltage is within the linear regime,while in Fig. 8(b) it is nonlinear. In general, however, for both anisotropy values increasing the temperature reduces the 224410-6MULTISCALE MODELING OF CURRENT-INDUCED . . . PHYSICAL REVIEW B 96, 224410 (2017) 0 1 2 3 4 0.1 0.2 0.3 0.4 0.5Inverse switching time (ns-1) Voltage (V)(a) filled - total angle open - spin angleT=0K, K=0.10meV 0K, 0.08meV 0K, 0.05meV 100K, 0.10meV 300K, 0.10meV 0 1 2 3 4 5 6 7 8 9 10 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2Inverse switching time (ns-1) Voltage (V)(b) filled - total angle open - spin angleT=0K, K=0.50meV 0K, 0.42meV 0K, 0.25meV 100K, 0.50meV 300K, 0.50meV FIG. 8. Inverse switching time with (a) k=0.1 meV and (b) 0.5 meV at T=0 K (solid blue line), 100 K (orange circles) and 300 K (green triangles). Filled and open symbols represent simulations runby using the angle calculated for the total magnetization or for each individual spin, respectively. The dashed lines indicate the inverse reversal time at T=0 K using a scaled anisotropy constant. switching time and also the critical voltage. Within a micro- magnetic picture this behavior is reproduced by introducingtemperature-dependent parameters, namely, the anisotropy, thedamping, and the magnetic moment. These reduced parame-ters then lead to a reduction in the critical switching voltage.Callen-Callen theory [ 22] predicts that at finite temperature the macroscopic uniaxial anisotropy constant K uscales as Ku(T)/Ku(0)=[M(T)/M(0)]3. From our simulations we find that at 100 K and 300 K the average magnetization isapproximately 0.94 and 0.80, respectively. This returns usexpected anisotropy constants of K u(100)≈0.83Ku(0) and Ku(300)≈0.51Ku(0). The dashed lines in Fig. 8, therefore, show the inverse switching time at 0 K obtained by usingthese scaled anisotropy values. As we can see in panel (b),the zero-temperature dynamics computed using these scaledconstants agree well with the average switching time obtainedat finite temperature, despite the lack of thermal fluctuations.The same is not true for the lower-anisotropy case of Fig. 8(a). Here there is agreement only at higher voltages for 100 K,while at 300 K the zero-temperature switching times at the rescaled anisotropies are constantly longer than those obtainedwith the finite-temperature dynamics. This has to be attributedto the actual thermal fluctuations, which are more pronouncedfor a lower anisotropy and cause the switching to occur faster. IV . CONCLUSION To summarize, we have developed a multiscale model- ing methodology combining ab initio calculations of the spin-transfer torque and large-scale finite-temperature spindynamics simulations. Using the SMEAGOL code, both the STT and the linear response STTk have been computed forvarious applied voltages and angles of misalignment betweenthe fixed and free magnetic layer in a nanoscopic junction.This is then mapped onto an atomistic spin dynamics model,which is used to calculate the switching times with and withoutthermal effects. We apply this methodology to a prototype MTJbased on Co/MgO, where we find that the STT is stronglylocalized on the Co atoms at the MgO interface and that theSTT is linear at low voltages. In contrast, above 1.4 V thereis a sharp increase in the total current in the AP configurationdriven by the minority spin component. Such current densityincrease leads to a sharp enhancement of the in-plane torqueand in a reduction of the out-of-plane one. The ab initio calculated torques are then mapped onto the ASD with different mapping types being analyzed. A full interpolation of the ab initio data set is preferred, but using the Slonczewski angular form together with the ab initio voltage dependence extracted at a fixed angle performs equally wellover a wide range of parameters. Due to the linear nature of theSTT, at low bias the 0-V linear response (torkance) is a suitablereplacement. At finite temperature the picture described abovedoes not change drastically, except for the fact that the thermalfluctuations reduce the critical voltage required for switching. The advantage of such multiscale methodology is that no empirical model of the STT is required, as this is calculatedat the atomic level from first-principles ballistic transporttheory. The atomic resolution allows systems where the typicalmicromagnetic models break down (e.g., where atomicallystaggered magnetic order is present) to be investigated. Whilecurrently some parameters, such as the exchange interactionand the anisotropy, are inferred from experiments, those canalso be taken from ab initio calculations of the actual MTJ stack with atomic resolution [ 20,23]. Computational feasibility may ultimately limit the size of treatable systems and the accessibletime scales; however, this prototypical MTJ study is still farfrom these limits, suggesting a range of realistic magneticmultilayered devices (including some accounts for disorder)to be well within the scope of the method. ACKNOWLEDGMENTS This work has been supported by the Science Foundation Ireland Principal Investigator Award (Grants No. 14/IA/2624and No. 16/US-C2C/3287). We gratefully acknowledge theDJEI/DES/SFI/HEA Irish Centre for High-End Computing(ICHEC) for the provision of computational facilities. Wealso acknowledge the Trinity Centre for High PerformanceComputing (TCHPC) for use of computational resources. 224410-7ELLIS, STAMENOV A, AND SANVITO PHYSICAL REVIEW B 96, 224410 (2017) [1] A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11,372(2012 ). [2] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320,1190 (2008 ). [3] J. Slonczewski, J. Magn. Magn. Mater. 159,L1(1996 ). [ 4 ]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. Kläui, I. L. Prejbeanu, B. Dieny, N. M. Dempsey, and B.Hillebrands, J. Phys. D 47,333001 (2014 ). [5] Y . Xie, I. Rungger, K. Munira, M. Stamenova, S. Sanvito, and A. W. Ghosh, in Nanomagnetic and Spintronic Devices for Energy- Efficient Memory and Computing (John Wiley & Sons, Ltd., Chichester, UK, 2016), pp. 91–132. [6] Y . Xie, J. Ma, S. Ganguly, and A. W. Ghosh, J. Comput. Electron. (2017 ), doi: 10.1007/s10825-017-1054-z . [7] C. Heiliger and M. D. Stiles, Phys. Rev. Lett. 100,186805 (2008 ). [8] P. M. Haney, D. Waldron, R. A. Duine, A. S. Núñez, H. Guo, and A. H. MacDonald, Phys. Rev. B 76,024404 (2007 ). [9] D. V . Berkov and J. Miltat, J. Magn. Magn. Mater. 320,1238 (2008 ). [10] J. C. Slonczewski, Phys. Rev. B 71,024411 (2005 ). [11] P. M. Haney, D. Waldron, R. A. Duine, A. S. Núñez, H. Guo, and A. H. MacDonald, Phys. Rev. B 75,174428 (2007 ). [12] M. Stamenova, R. Mohebbi, J. Seyed-Yazdi, I. Rungger, and S. Sanvito, Phys. Rev. B 95,060403 (2017 ).[13] J. Chureemart, R. Cuadrado, P. Chureemart, and R. Chantrell, J. Magn. Magn. Mater. 443,287(2017 ). [14] M. O. A. Ellis, R. F. L. Evans, T. A. Ostler, J. Barker, U. Atxitia, O. Chubykalo-Fesenko, and R. W. Chantrell, Low Temp. Phys. 41,908(2015 ). [15] R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. A. Ellis, and R. W. Chantrell, J. Phys.: Condens. Matter 26,103202 (2014 ). [16] A. R. Rocha, V . M. García-Suárez, S. Bailey, C. Lambert, J. Ferrer, and S. Sanvito, P h y s .R e v .B 73,085414 (2006 ). [17] I. Rungger, O. Mryasov, and S. Sanvito, P h y s .R e v .B 79,094414 (2009 ). [18] J. M. Soler, E. Artacho, J. D. Gale, A. García, J. Junquera, P. Ordejón, and D. Sánchez-Portal, J. Phys.: Condens. Matter 14, 2745 (2002 ). [19] J. C. Slonczewski and J. Z. Sun, J. Magn. Magn. Mater. 310, 169(2007 ). [20] A. Hallal, H. X. Yang, B. Dieny, and M. Chshiev, P h y s .R e v .B 88,184423 (2013 ). [21] S. Iihama, Y . Sasaki, H. Naganuma, and M. Oogane, J. Phys. D 49,35002 (2015 ). [22] E. R. Callen and H. B. Callen, Phys. Rev. 129,578(1963 ). [23] C. J. Aas, P. J. Hasnip, R. Cuadrado, E. M. Plotnikova, L. Szunyogh, L. Udvardi, and R. W. Chantrell, Phys. Rev. B 88, 174409 (2013 ). 224410-8

PhysRevB.74.134416.pdf

Theory of the spin-torque-driven ferromagnetic resonance in a ferromagnet/normal-metal/ferromagnet structure Joern N. Kupferschmidt, Shaffique Adam, and Piet W. Brouwer Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501, USA /H20849Received 5 July 2006; published 19 October 2006 /H20850 We present a theoretical analysis of current-driven ferromagnetic resonance in a ferromagnet/normal-metal/ ferromagnet trilayer. This method of driving ferromagnetic resonance was recently realized experimentally byTulapurkar et al. /H20851Nature 438, 339 /H208492005 /H20850/H20852and Sankey et al. /H20851Phys. Rev. Lett. 96, 227601 /H208492006 /H20850/H20852. The precessing magnetization rectifies the alternating current applied to drive the ferromagnetic resonance andleads to the generation of a dc voltage. Our analysis shows that a second mechanism to generate a dc voltage,rectification of spin currents emitted by the precessing magnetization, has a contribution to the dc voltage thatis of approximately equal size for the thin ferromagnetic films used in the experiment. DOI: 10.1103/PhysRevB.74.134416 PACS number /H20849s/H20850: 76.50. /H11001g, 72.25.Ba, 75.75. /H11001a, 85.75. /H11002d I. INTRODUCTION A decade ago, Slonczewski1and Berger2predicted that a spin-polarized current passing through a ferromagnet exerts atorque on its magnetic moment. The past decade has shownan abundance of experiments that have confirmed this theo-retical prediction. 3–7Since spin-polarized currents are easily generated by passing an electrical current through a ferro-magnet, the “spin transfer torque” opens the way for all-electrical manipulation of nanoscale magnetic devices. 8,9 Very recently, two groups have been able to use the spin torque to drive and detect ferromagnetic resonance in aferromagnet/normal-metal/ferromagnet /H20849FNF /H20850trilayer. 10,11 These experiments are designed such that the magnetization direction of one of the ferromagnets is fixed by anisotropyforces, whereas the other magnet is made of a softer ferro-magnetic material or has a more symmetric shape so that itsmagnetization can more easily respond to the applied currentor to an applied magnetic field. In both experiments, an al-ternating electrical current is used to drive the ferromagneticresonance, whereas the magnetization precession is detectedthrough the dc voltage generated by rectification of the ap-plied ac current by the time-dependent resistance of thedevice. 10,11The theoretical analysis of this experimental setup is the subject of this article. Not only does a spin-polarized current have an effect on the direction of the magnetization of a ferromagnet, a timevarying magnetization also causes the flow of spin currentsin normal metal conductors in electrical contact to the ferro-magnet. This “spin emission” was proposed by Tserkovnyak,Brataas, and Bauer as the cause of enhanced damping offerromagnetic resonance in thin ferromagnetic films in goodelectrical contact to a normal metal substrate. 12It is also the mechanism underlying Berger’s earlier prediction of the ex-citation of a dc voltage by a precessing magnetization in anunbiased FNF trilayer 13/H20849see also Refs. 14and15/H20850. Spin emission affects the experiments of Refs. 10and11 in two different ways. First, through the enhancement of thedamping spin emission broadens the ferromagnetic reso-nance. Second, the free layer’s precessing magnetizationemits alternating spin currents, which, in turn, generate a dcvoltage through the time-varying spin-dependent conduc-tance of the free layer. 13That way, spin emission provides an alternative to rectification of the applied ac current as amechanism for the generation of a dc voltage in these experi-ments. Our calculations show that both consequences of spinemission appear or disappear together. If spin emission givesa significant contribution to the damping of the ferromag-netic resonance, which is the case for the few-nanometers-thick free-layer ferromagnets used in the experiments, then italso provides a sizable contribution to the measured dc volt-age, and vice versa. In the remainder of this article we present the detailed theory of the electrical-current-driven ferromagnetic reso-nance needed to arrive at the above conclusion. In addition,our theory allows us to calculate how the ferromagnetic reso-nance frequency, the resonance width, and the asymmetry ofthe resonance line shape are affected by embedding the freeferromagnetic layer into the FNF trilayer. Our calculationproceeds in three parts. In Sec. II we derive general expres-sions for the spin-transfer torque, which we then apply to thecalculation of the magnetization motion in Sec. III. The gen-erated dc voltage is calculated in Sec. IV. We conclude inSec. V. II. SPIN-TRANSFER TORQUE A schematic drawing of the system we consider is shown in Fig. 1. It consists of a ferromagnetic source reservoir, held at electric voltage V, a thin normal-metal spacer layer, a thin ferromagnetic layer, and a normal-metal drain reservoir. Thedirection nof the magnetization in the ferromagnetic source is considered to be fixed, whereas the direction mof the magnetization in the thin layer can change under the influ-ence of an electrical current or an applied magnetic field. The nonequilibrium spin-transfer torque arises from the discontinuity of the spin current J sacross the free ferromag- netic layer,1,16,17 /H9270ne=− /H20851Js/H20849+/H20850−Js/H20849−/H20850/H20852, /H208491/H20850 where Js/H20849+/H20850andJs/H20849−/H20850are spin currents at the normal-metal- ferromagnet interfaces measured on the side of the drain res- ervoir and the spacer, respectively. We take it that the spacerPHYSICAL REVIEW B 74, 134416 /H208492006 /H20850 1098-0121/2006/74 /H2084913/H20850/134416 /H208496/H20850 ©2006 The American Physical Society 134416-1layer and the free ferromagnet are sufficiently thin so that all voltage drops occur across the ferromagnet-normal-metal in-terfaces, and spin relaxation can be neglected. 18/H20849Note that neglecting spin relaxation in the spacer layer is justified forthe 10-nm-thick Cu spacer used in the experiment of Ref. 11, which has a thickness much below the spin diffusion lengthin Cu. The spin diffusion length l sfin ferromagnets can be much smaller, however, and the experiments of Refs. 10and 11have a free layer thickness dcomparable to lsf, not d /H11270lsf. Still, we do not expect a strong effect of spin-flip scat- tering in this case, since the spin accumulation in the freelayer remains fixed collinear with the direction mof the magnetic moment, whereas the driving and detection of theferromagnetic resonance depend on the misalignment of thetwo magnetic moments in the device. 19/H20850With these assump- tions, the charge currents Jc/H20849±/H20850and the spin currents Js/H20849±/H20850 can be expressed directly in terms of the charge and spin accumulations /H9262cand/H9262sin the spacer layer. For the charge and spin currents Jc/H20849−/H20850and Js/H20849−/H20850one has two sets of equa- tions, one arising from the interface with the ferromagnetic source reservoir,8,9,20 Jc/H20849−/H20850=1 e/H208512G+/H20849/H9262c+eV /H20850+2G−/H9262s·n/H20852, Js/H20849−/H20850=−/H6036 2e2/H208512G+/H9262s·n+2G−/H20849/H9262c+eV /H20850/H20852n +/H6036 2e2/H208512G1/H20849/H9262s/H11003n/H20850/H11003n+2G2/H9262s/H11003n/H20852, /H208492/H20850 and one arising from the interface with the free ferromag- netic layer,8,9,12 Jc/H20849−/H20850=−1 e/H20851g+/H9262c+g−/H9262s·m/H20852 Js/H20849−/H20850=/H6036 2e2/H20851g−/H9262c+g+/H9262s·m/H20852m −/H6036 2e2g1/H208512/H9262s/H11003m+/H6036m˙/H20852/H11003m −/H6036 2e2g2/H208512/H9262s/H11003m+/H6036m˙/H20852. /H208493/H20850Here G±=/H20849G↑±G↓/H20850/2 and G1+iG2=G↑↓are determined by the interface conductances for majority and minority elec- trons and by the mixing conductance for the interface be-tween the ferromagnetic source and the normal-metal spacer,whereas g ±=/H20849g↑±g↓/H20850/2 and g1+ig2=g↑↓represent the equivalent quantities for the interface between the spacer layer and the free ferromagnet and for the interface betweenthe free ferromagnet and the source. Numerical values forthese conductance coefficients have been obtained for theinterfaces of various combinations of ferromagnetic andnormal-metal materials. 21 The two sets of equations are slightly different because there are two ferromagnet-normal-metal interfaces betweenthe spacer layer and the drain reservoir, whereas there is onlyone interface between the spacer layer and the source reser-voir, see Fig. 1. 22Also, in Eq. /H208492/H20850we omitted terms propor- tional to the time derivative n˙because the magnetization of the source reservoir is held fixed. Similarly, for Jc/H20849+/H20850and Js/H20849+/H20850we find Jc/H20849+/H20850=−1 e/H20851g+/H9262c+g−/H9262s·m/H20852 Js/H20849+/H20850=/H6036 2e2/H20851g−/H9262c+g+/H9262s·m/H20852m+/H60362 2e2g1m˙/H11003m+/H60362 2e2g2m˙. /H208494/H20850 Note that the charge current Jcand the component Js·mof the spin current parallel to the direction of the magnetizationof the free layer are conserved. The mixing conductances G 1+iG2and g1+ig2describe the coherent reflection of electrons with spin not collinearwith the magnetization directions nandmoff the interface with the fixed and free ferromagnetic layers, respectively. Weomitted terms that represent the coherent transmission ofelectrons with spin not collinear with nandm. The effect of coherent transmission is small for ferromagnets much thickerthan the ferromagnetic coherence length, which is usually onthe order of only a couple of atomic layers. We refer to Refs.12and16for a theory in which these processes are included. Since the imaginary parts G 2and g2of the mixing conduc- tances are numerically small for metallic junctions /H2084920% or less of G1and g1/H20850,21,23,24we set G2and g2to zero in the following calculations. At the end of Sec. IV we discuss howour results are modified for finite G 2and g2. The flow of electrical current through the FNF trilayer generates a spin-transfer torque only if the magnetization di-rections nandmare not collinear. In the experiment of Refs. 10and11this is achieved by an applied magnetic field which orients the free-layer magnetization mat a finite angle with respect to the fixed-layer magnetization direction nin the absence of a current. Following Ref. 11, we take this angle to be 90°. We choose a right-handed set of coordinate axes/H20849e 1,e2,e3/H20850such that npoints along e1andmpoints along e3 if no current is applied. The application of a current will cause mto deviate from e3. We will be interested in the linear response regime, in which the magnetization compo-nents m 1and m2are proportional to the applied current J. FIG. 1. Schematic drawing of the ferromagnet/normal-metal/ ferromagnet trilayer considered here. The left ferromagnet, withmagnetization direction n, acts as the source reservoir. The right ferromagnet is the free layer. Its magnetization direction mcan change in response to the applied current. The currents J/H20849+/H20850and J/H20849−/H20850in the text are evaluated at the right and left sides of the free ferromagnetic layer, respectively.KUPFERSCHMIDT, ADAM, AND BROUWER PHYSICAL REVIEW B 74, 134416 /H208492006 /H20850 134416-2With an alternating current bias, Jc=J/H20849t/H20850=Re J0ei/H9275t, Eqs. /H208492/H20850and /H208493/H20850give five independent equations from which one can solve for the five unknown variables, which are thecharge and spin accumulations /H9262cand/H9262sin the spacer layer and the bias voltage V. Solving these to lowest order in the applied current, we find that two relevant components of thespin-transfer torque /H208491/H20850are /H9270ne,1=−/H6036 2e/H20875JG− G1++/H6036m˙2g1 e/H208732−G+ G1+/H20874/H20876, /H208495/H20850 /H9270ne,2=/H6036 2e/H6036m˙1g1 e/H208732−g1 g1+G1/H20874, /H208496/H20850 where we abbreviated G1+=G++/H20849G+2−G−2/H20850/g1. /H208497/H20850 III. MAGNETIZATION DYNAMICS The magnetization is driven out of equilibrium by the spin-transfer torque of Eq. /H208491/H20850. In order to solve for the full time dependence of the magnetization, we use the Landau-Lifshitz-Gilbert equation, 25,26 m˙=/H9251m/H11003m˙+/H9253 Md/H20849/H9270eq+/H9270ne/H20850. /H208498/H20850 Here Mis the magnetization per unit length, dis the thick- ness of the free ferromagnetic layer, /H9253is the gyromagnetic ratio, and /H9251is the phenomenological Gilbert damping param- eter. The equilibrium torque /H9270eqis the combination of the torque applied by the external magnetic field and the aniso-tropy torque intrinsic to the ferromagnet. Since we are inter-ested in small deviations from equilibrium, we can expand /H9270eqaround the equilibrium direction m=e3, /H9270eq=−Md /H9253/H20849/H92751m1/H11032e1/H11032+/H92752m2/H11032e2/H11032/H20850/H11003m, /H208499/H20850 where the frequencies /H92751and/H92752are set by the energy cost for magnetization deviations along principal axes e1/H11032ande2/H11032 perpendicular to e3. The constants /H92751and/H92752depend on the dipolar field of the pinned layer, the demagnetization fieldand coercivity of the free layer, and the applied magneticfield. 27The geometric mean /H20849/H92751/H92752/H208501/2is the free layer’s fer- romagnetic resonance frequency in the absence of electrical contact to the normal-metal spacer layer and the drain reser-voir, whereas /H20849 /H92751//H92752/H208501/2is the ratio of semimajor and semiminor axis of the ellipsoidal magnetization precession in that case. If /H9270eqis dominated by the applied magnetic field H, one has /H92751=/H92752=/H9253H. Rotating to the coordinate system with unit vectors e1ande2, the two components of /H9270eqcan be written /H9270eq,1=Md /H9253/H20851−m2/H20849/H9275++/H9275−cos/H9278/H20850+m1/H9275−sin/H9278/H20852,/H9270eq,2=Md /H9253/H20851m1/H20849/H9275+−/H9275−cos/H9278/H20850−m2/H9275−sin/H9278/H20852, /H2084910/H20850 where /H9275±=/H20849/H92752±/H92751/H20850/2 and /H9278/2 is the rotation angle between e1/H11032ande2/H11032. With an applied ac current, J/H20849t/H20850=Re J0ei/H9275t, we can then solve for the magnetization components m1/H20849t/H20850=Re m10ei/H9275t and m2/H20849t/H20850=Re m20ei/H9275t, with the result m10=m0/H20849J0/e/H20850/H20849i/H9275+/H9275−sin/H9278/H20850 f/H20849/H9275/H20850, /H2084911/H20850 m20=m0/H20849J0/e/H20850/H20851/H9275+−/H9275−cos/H9278+i/H9275/H20849/H9251˜++/H9251˜−/H20850/H20852 f/H20849/H9275/H20850, /H2084912/H20850 where we abbreviated f/H20849/H9275/H20850=/H208491+/H9251˜+2−/H9251˜−2/H20850/H92752−2i/H9275/H20849/H9251˜+/H9275++/H9251˜−/H9275−cos/H9278/H20850+/H9275−2−/H9275+2, /H2084913/H20850 m0=/H9253/H6036G−/2dMG 1+, /H2084914/H20850 and /H9251˜+=/H9251+g1/H9253/H60362 4de2M/H208734−G+ G1+−g1 g1+G1/H20874, /H2084915/H20850 /H9251˜−=g1/H9253/H60362 4de2M/H20873G+ G1+−g1 g1+G1/H20874. /H2084916/H20850 The non-negative dimensionless numbers /H9251˜±are the effective Gilbert damping parameters.12We need two damping param- eters rather than one since the effective damping is aniso-tropic because of the presence of the second ferromagnet. IV. DC VOLTAGE Since we are interested in the dc voltage generated by the applied ac current, we need to calculate the voltage V/H20849t/H20850to second order in J/H20849t/H20850. This implies that we need to solve Eqs. /H208492/H20850and /H208493/H20850to first order in m1and m2, V=J/H208732G1+g+ G1g1++G++g1 2g1G1+/H20874−/H6036m˙2G− 2eG1+ −2/H20849g1+G1/H20850g−G−m1 G1g1g1+G1+/H20873J−em˙2/H9251˜− m0/H20874, /H2084917/H20850 where we abbreviated g1+=2g++/H20849g+2−g−2/H20850/G1. /H2084918/H20850 The two terms in the first line of Eq. /H2084917/H20850, which are proportional to Jand m˙2, give an alternating contribution to Vonly. The term proportional to Jis the dc resistance of the device, whereas the term proportional to m˙2is the magnetic contribution to the admittance. /H20849Electronic contributions to the admittance occur at higher frequencies than the ferro-magnetic resonance frequency and are not considered in ourtheory. /H20850The dc voltage follows from the subleading terms in the second line of Eq. /H2084917/H20850, which are proportional to Jm 1THEORY OF THE SPIN-TORQUE-DRIVEN … PHYSICAL REVIEW B 74, 134416 /H208492006 /H20850 134416-3and m2m1. The contribution proportional to Jm1is rectifica- tion of the applied alternating current by the time-dependentconductance of the device. The contribution proportional to m˙ 2m1follows from spin emission by the precessing magne- tization of the free ferromagnet. The two terms contributing to the dc voltage are easily calculated using the results of the previous section. UsingEqs. /H2084911/H20850and /H2084912/H20850, one calculates the averages of the prod- ucts Jm 1and m˙2m1over one period of the applied current, /H20855Jm1/H20856=m0/H20841J0/H208412 2e/H20841f/H20849/H9275/H20850/H208412/H20851/H9275Imf/H20849/H9275/H20850+/H9275−sin/H9278Ref/H20849/H9275/H20850/H20852, /H20855m˙2m1/H20856=m02/H20841J0/H208412/H92752 2e2/H20841f/H20849/H9275/H20850/H208412/H20851/H9275+−/H9275−cos/H9278−/H20849/H9251˜++/H9251˜−/H20850/H9275−sin/H9278/H20852. /H2084919/H20850 The dc voltage then follows from substitution into Eq. /H2084917/H20850, V=m0/H20841J0/H208412g−G−/H20849g1+G1/H20850 eG1g1g1+G1+/H20841f/H20849/H9275/H20850/H208412/H20853/H92752/H208492/H9275+/H9251˜++/H9275+/H9251˜−+/H9275−/H9251˜−cos/H9278/H20850 −/H9275−/H20851/H208491+/H9251˜+2+/H9251˜+/H9251˜−/H20850/H92752+/H9275−2−/H9275+2/H20852sin/H9278/H20854. /H2084920/H20850 In the limit /H9251˜±/H112701/H20849which is appropriate for most experi- ments /H20850, Eq. /H2084920/H20850simplifies to the asymmetric Lorentzian V=V0/H927502−/H20849/H9275−/H92750/H20850/H9254/H11032 /H20849/H9275−/H92750/H208502+/H92542, /H2084921/H20850 with V0=m0/H20841J0/H208412g−G−/H20849g1+G1/H20850 4/H927502eG1g1g1+G1+/H208492/H9275+/H9251˜++/H9275+/H9251˜−+/H9275−/H9251˜−cos/H9278/H20850, /H2084922/H20850 and /H927502=/H9275+2−/H9275−2, /H9254=/H9251˜+/H9275++/H9251˜−/H9275−cos/H9278, /H9254/H11032=2/H92750/H9275−sin/H9278 2/H9275+/H9251˜++/H9275+/H9251˜−+/H9275−/H9251˜−cos/H9278. /H2084923/H20850 The asymmetry of the line shape /H2084921/H20850depends on the anisotropy of the torque /H9270eqand on the angle /H9278/2 between the principal axes and the direction nof the magnetization of the fixed layer. In the experiment of Ref. 11the main contri- bution to /H9270eqcomes from the large magnetic field used to align the free layer magnetization perpendicular to n. This contribution is isotropic, which explains why no stronglyasymmetric line shapes were observed in Ref. 11. The ex- periment of Ref. 10finds a significantly asymmetric line shape if the applied magnetic field is small, the line shapesbecoming more symmetric at larger fields. Although this ob-servation appears consistent with our theory, we should notethat for Ref. 10the equilibrium torque /H9270eqarising from the applied magnetic field and shape anisotropy alone has /H9278=0 and, hence, cannot explain an asymmetric line shape. Refer-ence 10attributes the asymmetric line shape to the imaginarypart g 2of the mixing conductance which, if large enough, provides an alternative /H20849but approximately magnetic-field in- dependent /H20850mechanism for an asymmetric line shape, see the discussion below. The relative contributions of the rectification and the spin emission effects can be found by looking at the ratio of m0/H20855Jm1/H20856/eand /H20855m˙2m1/H20856/H9251˜−, cf. Eq. /H2084917/H20850. For/H9251˜±/H112701 this ratio is m0/H20855Jm1/H20856 e/H20855m˙2m1/H20856/H9251˜−=−2/H9254−/H9275−sin/H9278/H20849/H9275−/H92750/H20850//H92750 /H9251˜−/H20849/H9275+−/H9275−cos/H9278/H20850. /H2084924/H20850 Since both terms in the numerator are of order /H9254near the ferromagnetic resonance, whereas the denominator is of or- der/H9251˜−/H92750, the ratio /H2084924/H20850is of order /H9254//H9251−/H92750. This is of order unity if /H9251˜+and/H9251˜−are comparable, which happens precisely if the second term in Eq. /H2084915/H20850is not small in comparison to the first. This, in turn, is the condition that spin emissiongives a significant contribution to the total damping. Hence,we conclude that spin emission contributes significantly tothe measured dc voltage if and only if spin emission contrib- utes significantly to the damping. Since /H20855m˙ 2m1/H20856is symmetric around /H9275=/H92750, cf. Eq. /H2084919/H20850above, spin emission contributes to the symmetric part of the line shape only. The antisym-metric part is due to the rectification of the applied ac currentonly. In our calculations we have neglected the imaginary parts g 2and G2of the mixing conductance because in metallic junctions they are known to be numerically small in com-parison to the real parts g 1and G1. Inclusion of g2and G2 leads to a small modification of the resonance frequency, because g2and G2change the gyromagnetic ratio /H9253of the free ferromagnetic layer.12With corrections to first order in g2/g1only, the resonance frequency becomes /H927502=/H20849/H9275+2−/H9275−2/H20850/H208751−4/H9251˜−g2/H20849g1G1++2G1G1+−G1G+/H20850 g1/H20849G1G++g1G+−g1G1+/H20850/H20876. /H2084925/H20850 More importantly, with nonzero g2and G2, there is a finite asymmetry in the line shape even in the absence of magneticanisotropy in the free layer, 10 /H9254/H11032=2/H92750/H20851/H9275−sin/H9278−z/H20849/H9275++/H9275−cos/H9278/H20850/H20852 2/H9275+/H9251˜++/H9275+/H9251˜−+/H9275−/H9251˜−cos/H9278−2z/H9251˜−/H9275−sin/H9278, /H2084926/H20850 with z=G12g1+g2G−−2g12G1+G2g− g1G1/H20849g1+G1/H20850g1+G−. /H2084927/H20850 Again, our results are valid up to first order in g2/g1and G2/G1only. We have also analyzed the case that the equilibrium angle between the fixed layer magnetization nand the free layer magnetization mis not 90°. While this complicates the de- tailed expression for Vdc/H20849/H9275/H20850/H20849to the extent that it cannot be reported here /H20850, it does not change our qualitative conclusions that /H20849i/H20850spin emission and rectification of the applied ac cur-KUPFERSCHMIDT, ADAM, AND BROUWER PHYSICAL REVIEW B 74, 134416 /H208492006 /H20850 134416-4rent have comparable contributions to the generated dc volt- age if the free layer is thin enough that spin emission gives asizable enhancement of the damping and /H20849ii/H20850the asymmetry ofV dc/H20849/H9275/H20850around the resonance frequency /H92750is small in the ratios /H9275−//H9275+org2/g1. The former ratio is small if the ap- plied magnetic field is large enough to saturate the free fer-romagnet, whereas the latter ratio g 2/g1is known to be nu- merically small for metallic junctions /H20849of order 0.1 or less, see Refs. 21,23, and 24/H20850. V. CONCLUSION In this paper we have presented a microscopic theory for the spin-torque driven ferromagnetic resonance inferromagnet/normal-metal/ferromagnet trilayers. Our theoryis inspired by the experiments of Refs. 10and11. In these experiments, an alternating current is used to drive the ferro-magnetic resonance, while a generated dc voltage is used todetect the resonance. In addition to providing theoretical expressions for the width and asymmetry of the resonance, we are able to deter-mine the relative magnitude of two physical mechanisms thatcontribute to the dc voltage: rectification of the applied accurrent and rectification of the spin currents emitted by theprecessing ferromagnet. Both contributions are of similarmagnitude for the thin ferromagnetic films used in the ex-periments. The presence of two mechanisms to generate adirect response to periodic driving, rather than one, sets thisclass of magnetic devices apart from their semiconductor counterparts. A direct experimental probe of the two contributions to the dc voltage is to compare the dc voltage observed in spin-torque-driven ferromagnetic resonance with the dc voltage generated in conventional magnetic-field driven ferromag-netic resonance in the same device. The latter follows fromrectification of emitted spin currents only. Since spin emis-sion gives a symmetric line shape around the resonance fre-quency /H9275=/H92750, there should be a clear difference between the two methods to excite ferromagnetic resonance. A compari-son of the magnitudes of both contributions would require acalibration of the amplitude at which the magnetization pre-cesses. This can be achieved through a simultaneous mea-surement of the dc resistance of the device, which dependson the precession amplitude through the giant magnetoresis-tance effect. Note added . Recently, we learned of a work by Kovalev et al. with similar conclusions about spin-torque driven fer- romagnetic resonance. 28 ACKNOWLEDGMENTS We thank D. Ralph and J. Sankey for stimulating discus- sions. This work was supported by the Cornell Center forMaterials Research under NSF Grant No. DMR 0520404, theCornell Center for Nanoscale Systems under NSF Grant No.EEC-0117770, by the NSF under Grant No. DMR 0334499,and by the Packard Foundation. We thank Alex Kovalev forsending us a preprint of Ref. 28. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 /H208491998 /H20850. 4J. Z. Sun, J. Magn. Magn. Mater. 202, 157 /H208491999 /H20850. 5J.-E. Wegrowe, D. Kelly, Y. Jaccard, P. Guittienne, and J.-P. Ansermet, Europhys. Lett. 45, 626 /H208491999 /H20850. 6E. Myers, D. Ralph, J. Katine, R. Louie, and R. Buhrman, Sci- ence 285, 867 /H208491999 /H20850. 7J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 8Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 /H208492005 /H20850. 9A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep. 427, 157 /H208492006 /H20850. 10A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Mae- hara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and S.Yuasa, Nature /H20849London /H20850438, 339 /H208492005 /H20850. 11J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 96, 227601 /H208492006 /H20850. 12Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. 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B 66, 014407 /H208492002 /H20850. 18A finite resistance of the normal-metal spacer layer /H20849but without spin relaxation /H20850can be taken into account by making the re- placements G+→2GN/H20849GNG++G+2−G−2/H20850//H20851/H20849GN+G+/H208502−G−2/H20852,G− →2GN2G−//H20851/H20849GN+G+/H208502−G−2/H20852, G1→GN/H20849G12+G1GN+G22/H20850//H20851/H20849G1 +GN/H208502+G22/H20852, and G2→GN2G2//H20851/H20849G1+GN/H208502+G22/H20852in the equations below, where GNis the conductance of the normal metal /H20849per spin direction /H20850. A finite resistance of the free ferromagnetic layer can be taken into account by making the replacements 1/ g↑ →1/g↑+1/2 gF↑,1 / g↓→1/g↓+1/2 gF↓, with g±=/H20849g↑±g↓/H20850/2, while keeping g1and g2unchanged. Here gF↑and gF↓are the majority and minority conductances of the ferromagnet. 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PhysRevB.95.064402.pdf

PHYSICAL REVIEW B 95, 064402 (2017) Structure-dependent magnetoresistance and spin-transfer torque in antiferromagnetic Fe|MgO |FeMn |Cu tunnel junctions Xingtao Jia* School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454000, China Huimin Tang and Shizhuo Wang Department of Physics, Beijing Normal University, Beijing 100875, China Minghui Qin Institute for Advanced Materials and Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China (Received 17 September 2016; revised manuscript received 17 January 2017; published 2 February 2017) We predict large magnetoresistance (MR) and spin transfer torque (STT) in antiferromagnetic Fe|MgO|FeMn|Cu tunnel junctions based on first-principles scattering theory. MR as large as ∼100% is found in one junction. Magnetic dynamic simulations show that STT acting on the antiferromagnetic order parameterdominates the spin dynamics, and an electronic bias of order 10 −1mV and current density of order 105Acm−2can switches a junction of three-layer MgO, they are about one order smaller than that in Fe |MgO|Fe junction with the same barrier thickness, respectively. The multiple scattering in the antiferromagnetic region is considered to be responsible for the enhanced spin torque and smaller switching current density. DOI: 10.1103/PhysRevB.95.064402 I. INTRODUCTION Due to high stability to a parasitic magnetic field and high working frequency, antiferromagnet (AFM)-based spintronics[1–3] has attracted much attention both experimentally [ 4–16] and theoretically [ 17–29]. Similar to ferromagnet (FM)- based magnetic structures, AFM-based structures also havemagnetoresistances (MR) [ 8–12] to distinguish information states that can be controlled by spin transfer torque (STT)[16–27], photomagnetic pulses [ 4–7], and spin wave [ 30]. Relativistic spin-orbit torques provide a new powerful toolto control the magnetization [ 31–33]. A field-driven AFM domain-wall velocity induced by N ´eel spin-orbit torques can be two orders of magnitude larger than that of the FMdomain wall [ 34]. Sizable spin torque from enhanced spin Hall effects in a AFM |normal metal (NM) |FM spin valve (SV) has been demonstrated [ 35,36]. However, the poor size-scalability overshadows the application of the spin-orbit effect [ 2]. Anisotropic MR (AMR) in AFM-based magnetic structures such as NM |AFM [ 8–11] and FM |AFM [ 35,36] contacts is relatively small ( ∼1%). However, if a MgO barrier is inserted between NM and AMF, a large AMR of up to 100% isobserved at low temperatures [ 12]. Similar to FM-based SVs, a reference (ferromagnetic or antiferromagnetic) polarizercan be introduced to form an antiferromagnetic SV , wherea nonrelativistic structure-dependent MR dominates over therelativistic MR [ 12,17,18]. Indeed, antiferromagnetic SVs, especially perpendicular ones, show merit for large MR,electronic control, and size-scalability. Recently, an ultralow switching current density below 10 6Acm−2was predicted in the antiferromagnetic metal-based SVs [ 17,37], where the spin torque acting on the antiferromag- netic order parameter in the whole AFM region. Generally, *Corresponding author: jiaxingtao@hpu.edu.cnswitching current density is proportional to Gilbert dampingcoefficient, but inversely proportional to spin transfer effi-ciency [ 17]. Enhanced spin transfer efficiency from multiple reflection was demonstrated in a MgO-based tunnel junctions[38]. Because multiple reflection can exist naturally in the AFM, enhanced spin transfer and lower switching currentdensity are expected in the AFM-based magnetic structures. Here, we focus on the antiferromagnetic Fe |MgO|FeMn|Cu junctions, for which γ-FeMn is a well-studied AFM system with fcc crystal structure and a 3Q noncollinear spin structure.Under exchange coupling with a magnet, a collinear A |B|A|B spin structure forms [ 39]. As a good spin filter, MgO is used to enhance MR and STT in FM-based junctions, where both thecharge current and spin current are carried mainly by a smallportion of the k ||points in the two-dimensional Brillouin zone (2D BZ) [ 38,40]. The transport scheme in the AFM junctions has advantages over that in AFM |NM|AFM SVs [ 18], in which both the charge current and spin current are carried by almostall the k ||points in the 2D BZ. In this calculation based on first principles, we predict large MR and STT in Fe |MgO|FeMn|Cu junctions. Specifically, MR ∼100% is predicted in a junction. Spin-dynamic simulations show that a current density of order10 5Acm−2can switch a junction of 3-layered (3L) MgO. This paper is organized as follows. In Sec. II, we provide our calculation details based on first-principle scattering theory. In Sec. III, we present our results on Fe |MgO|FeMn|Cu junctions with ordered and disordered crystal structures. Section IV presents our summary. II. METHODS In this work, we study the spin-dependent transport of the two terminal multilayers Fe |MgO|FeMn|Cu (see Fig. 1). Two magnetic structures with different spin structures (L-type andG-type) in the FeMn layer are considered. In detail, the L-type 2469-9950/2017/95(6)/064402(6) 064402-1 ©2017 American Physical SocietyJIA, TANG, W ANG, AND QIN PHYSICAL REVIEW B 95, 064402 (2017) FIG. 1. Schematic two-terminal Fe |MgO|FeMn|Cu junction used in the calculations. Red arrow in the left lead indicates the fixed magnetization along the zaxis; blue or pink arrow in FeMn indicates the sublattice magnetization M1/2. We define the antiferromagnetic order parameter as l=M1−M2, which is free in the x-zplane with relative angle θwith respective to the zaxis. is a higher symmetric A |B|A|B structure (upper panel of Fig. 1) in which the magnetization of layer A is compensated withthat of layer B along the transport direction. The G-type is alower symmetric structure (lower panel of Fig. 1) in which the magnetization of one sublattice is compensated both in-planeand out-of-plane with that of the another sublattice. The lateralsupercell is used to match MgO with bcc-Fe and fcc-FeMn.The crystal MgO is reduced by 4% and rotated 45 ◦to match the bcc-Fe. A 5 ×5 lateral supercell of Fe |MgO matches well with the 4 ×4 lateral supercell of fcc-FeMn (the mismatch is about 0 .5%). A very small mismatch between the fcc-FeMn and fcc-Cu is neglected, and both FeMn and Cu are compressedalong the epitaxial direction to maintain a volume equal to thatof the bulk to match with Fe |MgO. The coherent potential approximation (CPA) is used for the potential of the FeMnalloys, and bulky potentials calculated self-consistently areput into a wave-function-matching (WFM) transport package[41]. During transport calculations, we consider the particle current along the (010) material growth direction, and a 40 × 40 k-mesh in the full 2D BZ is used to ensure good transportconvergence. Two kinds of imperfects, spin flip (SF) in thesite-ordered FeMn and oxygen vacancy (OV) at the interfaceclose to MgO, are considered. Over 30 configurations are usedin averaging the configuration convergence. More numericaldetails of the electronic structure and transport calculationscan be found elsewhere [ 18,38,40]. III. RESULTS AND DISCUSSIONS We next focus on the nonrelativistic structure-dependent MR, and then shift to STT in the antiferromagnetic region,and analyze the spin dynamics finally. Therein, two kinds ofdefects, OV and SF, are discussed. A. Structure-dependent magnetoresistance Figure 2gives the MR in antiferromagnetic Fe |MgO|Fe0.5 Mn 0.5(16)|Cu junctions with clean interface as a function of MgO thickness; the numbers in the bracket indicate the thick-ness in atomic layers. We define MR=[G(P)−G(AP)]/ G(AP) with conductance G=(e 2/h)/A/integraltext Tr[t(k)t◦(k)]dk summarized in the 2D BZ at the Fermi level EF. Here, P /AP denotes the parallel and/or antiparallel structure defined by therelative angle of 0 /πbetween the magnetic order parameters of Fe and FeMn, tis the transmission part of the scatteringFIG. 2. Magnetoresistance as function of MgO thickness in the L-type and G-type antiferromagnetic Fe |MgO|Fe0.5Mn 0.5(16)|Cu junctions with clean interface. The error bar is ∼5%. matrix s, and Ais the section area of the supercell. Both the site-ordered L-type and G-type junctions show notableMR, which is also sensitive to the thickness of MgO. Thepresence of the MR in the G-type junctions is related tothe reduced symmetry, where only one Oxygen atom sittingdirectly on top of the Fe(Mn) atom of FeMn layer in one cell,and the change of antiferromagnetic order parameter wouldchange the spin structure. Comparatively, we find zero MR(and negligible STT on the antiferromagnetic order parameterat relative angle of 90 ◦) in a highly symmetric site-ordered G-type Fe |MgO(3) |bcc-Fe 0.5Mn 0.5(16)|bcc-Cu junction with one oxygen atom sitting directly on top of one up-spin Fein the FeMn layer and another oxygen atom sitting on topof one down-spin Fe in one cell, where the reversal of theantiferromagnetic order parameter does not change the spinstructure. When crystal sites are disordered, an enhancedMR is found in the L-type junctions, whereas a near-to-zeroMR is obtained in the G-type junctions. With the exceptionof the site-disordered G-type junctions, MR increases asthe barrier thickness increases. In detail, for site-orderedG-type junctions, MR increases from 12% in the 3L MgOcase to 24% in the 9L MgO case. Comparatively, MR in thesite-ordered L-type junctions increases quickly from 22% inthe 3L MgO case to 71% in the 9L MgO case. If the crystalsites are disordered, an enhancement of 70% /20% is predicted in the L-type 3/9 L MgO junctions. Enhanced MR is alsofound in the L-type junctions with FeMn alloys of different TABLE I. MR in site-disordered L-type Fe |MgO(n)|FeMn(16) |Cu junctions with clean interface at θ=90◦. n 35 7 9 Fe0.25Mn 0.75 5 −26 −18 31 Fe0.5Mn 0.5 35 39 53 84 7Fe 0.75Mn 0.25 10 101 135 100 064402-2STRUCTURE-DEPENDENT MAGNETORESISTANCE AND . . . PHYSICAL REVIEW B 95, 064402 (2017) concentrations (see Table I), where a MR ∼135% is found in aF e|MgO(7) |Fe0.75Mn 0.25(16)|Cu junction. Similar to the well-studied MgO-based junctions, MR in the antiferromagnetic Fe |MgO|FeMn|Cu junctions is sensitive to OV . For the L-type junctions (including both site-orderedand site-disordered) and the site-disordered G-type junctions,about 10% OV at the interfaces near to MgO degrade the MRin Fe|MgO(3) |Fe 0.5Mn 0.5(16)|Cu by one order of magnitude compared with clean junctions (several ten percent for cleanjunctions to several percent for dirty junctions; see Table II). For thicker MgO junctions, we get similar results. Hence, toachieve larger MR, the junctions should be as clean as possible. Spin disorder, such as SF, changes the magnetization and shows up in effects on spin-dependent transport. In Table II, we list MR in Fe |MgO(3) |Fe 0.5Mn 0.5(16)|Cu with 10% SFs in FeMn alloy. The SF shows less effect on the MR in theL-type junctions than in the G-type junctions. For thicker MgOjunctions, we obtain similar results. B. Spin transfer torques In driving the magnetic order parameter [ 42,43], STT is the favored means as they can be induced by a bias voltageand thermal gradient [ 38,44]. With small bias voltages, STT is almost linear with bias voltage, especially in the MTJs [ 45], for which “torkance” ( τ) can be introduced. The magnetic structure of an AFM can be described by the sublatticemagnetizations M j(j=1,2 for the simplest case) with total magnetization m=M1+M2and antiferromagnetic order parameter l=M1−M2. Indeed, STT can also be used to drive the antiferromagnetic order parameter [ 17–27]. We denote the STT acting on the total magnetization mand antiferromagnetic order parameter lasτm(τm=τm1+τm2) andτl(τl=τm1− τm2), respectively. First, let us take a look at a simple model with a thin antiferromagnetic layer interacting with a spincurrent. For one sublattice magnetization M jin AFM, the STT applied to Mjis proportional to Mj×M/prime×Mj(in-plane) andMj×M/prime(out-of-plane) with M/primethe spin current source. For the simplest AFM, the in-plane /out-of-plane STT on M1(2)follows the same /opposite direction as that on M2(1). Hence, the out-of-plane component of τl(τ⊥ l) and the in-plane component of τm(τ|| m) would be enhanced, whereas the in-plane component of τl(τ|| l) and the out-of-plane component of τm(τ⊥ m) would vanish. The model analysis is suitable for a classical system but is invalid [ 19] in structures with quantum states dominating. In classical cases, for which τl/τm→0, the spin dynamics are dominated by τmwith ultrahigh working frequencies [ 20–27]. The spin-glass state can be considered as classical. A large deviation from the model analysis is observedin the Fe |MgO|FeMn|Cu junction, as shown in below, where the spin transport is dominated by quantum states. From the dependence of STT on MgO thickness [Fig. 3(a)] for both the site-ordered and site-disordered Fe|MgO|Fe 0.5Mn 0.5|Cu junctions with clean interface at rela- tive angle of 90◦, we find that: (1) both τlandτmexponentially decrease as the MgO thickness is larger than 3 L. The enhancedSTT from the interfacial resonance states in the ultra-thin (3L)barrier appears responsible for this deviation. The marked τ l(at relative angle of 90◦) in the site-ordered G-type junction is re- lated to the reduced symmetry in the lateral supercell structure,FIG. 3. Spin transfer torque as a function of MgO thickness (a) in site-ordered antiferromagnetic Fe |MgO|Fe0.5Mn 0.5(16)|Cu junctions with clean interface at relative angle of 90◦(Inset: similarly for site-disordered Fe |MgO(3) |Fe0.5Mn 0.5(16)|Cu junctions with clean interface). (b) Layer-dependent spin transfer torquein site-ordered G-type Fe |MgO(3) |Fe 0.5Mn 0.5(16)|Cu junctions at relative angle of 90◦(Inset: similarly for site-disordered G-type Fe|MgO(3) |Fe0.5Mn 0.5(48)|Cu junctions). as discussed in the above section, which follows a simple sine relation with respect to the relative angle rather than a sin(2 θ) relation [ 19] presented in a simple FM |NM|AFM model. (2) τl is comparable to τmfor both site-ordered and site-disordered junctions of L-type and G-type. Consequently, the dynamics ofthe antiferromagnetic-order parameter would be driven by τ l rather than τm[17,28], with working frequency ωA=γHA controlled by the anisotropic effective field HA. Note that ωAis considerably lower than that for the τm-driven case ω=√2ωAωEwithωE=γHEandHEis the exchange field. In detail, τlin the site-ordered and site-disordered L /G-type Fe|MgO(3) |Fe0.5Mn 0.5(16)|Cu junctions are 71 /382×1014τ0 (τ0≡¯h 2ek/Omega1−1m−2) and 22 /122×1014τ0, respectively; see list in Table II. Specifically, τlin the G-type junctions is several times larger than that in the L-type junctions, and is sensitive to site disorder. In comparison, τ|| mis not only stable to site disorder but also to spin configurations, as demonstrated in Table II. Furthermore, τ|| mcalculated by the WFM method is consistent well with that estimated from transmissions via afree-electron model [ 46]. Moreover, τ lis strongly dependent on the thickness of FeMn, whereas τmis almost constant. The behavior of τl(as a 064402-3JIA, TANG, W ANG, AND QIN PHYSICAL REVIEW B 95, 064402 (2017) TABLE II. In-plane spin transfer torque in site-ordered and site-disordered L-/G-type Fe |MgO(n) |Fe0.5Mn 0.5(16)|Cu junctions with clean interface with θ=90◦. The resistance area RA=1/G(EF).ηmandηlare the spin transfer efficiency of STT on the total magnetization mand antiferromagnetic order parameter l, respectively. VCandJCare the critical bias voltage and critical current density to switch the antiferromagnetic order parameter, circularly. We assess VCandJCby choosing easy uniaxial anisotropy field of 20 mTand Gilbert damping coefficient of 0.01 in the spin dynamics simulations. n(L) MR (%) RA(/Omega1μm2) τm(τ0) τl(τ0) ηm(¯h/2e) ηl(¯h/2e) VC(mV) JC(105Acm−2) Site-ordered cases 32 2 /12 0.11 /0.093 73 /93 71 /381 0.75 /0.80 0.74 /3.3 0.17 /0.03 1.5 /0.35 52 3 /16 3.6 /2.2 2.4 /4.2 2.9 /3.9 0.80 /0.85 0.96 /0.79 4.2 /3.2 1.2 /1.4 74 6 /22 39 /26 0.21 /0.37 0.25 /0.56 0.76 /0.88 0.89 /1.3 49 /22 1.3 /0.86 97 1 /24 352 /245 0.02 /0.04 0.02 /0.08 0.72 /0.89 0.68 /1.9 593 /147 1.7 /0.60 3a4/2 0.036 /0.034 92 /97 74 /20 0.31 /0.30 0.25 /0.061 0.17 /0.63 4.5 /19 3b22/3 0.11 /0.098 80 /94 20 /4 0.81 /0.85 0.20/ /0.034 0.62 /3.3 5.7 /34 Site-disordered cases 33 5 /0.56 0.13 /0.11 72 /76 22 /6.3 0.83 /0.77 0.25 //0.064 0.56 /2.0 4.5 /18 53 9 /0.71 3.7 /2.3 0.92 /1.07 0.91 /0.13 0.31 /0.23 0.31 /0.028 13.5 /93 3.7 /41 75 3 /−0.49 34 /27 0.31 /0.36 0.21 /0.23 0.99 /0.83 0.65 /0.54 60 /53 1.7 /2.1 98 4 /−1.59 313/ /239 0.035 /0.04 0.14 /0.18 0.98 /0.91 3.8 /4.1 89 /66 0.30 /0.29 314/3 0.039 /0.042 88 /74 35 /33 0.32 /0.29 0.13 /0.13 0.35 /0.36 8.9 /8.8 3227/5.7 0.12 /0.12 75 /75 37 /20 0.81 /0.75 0.39 /0.20 0.34 /0.61 2.9 /5.6 a10% OV at interfaces close to MgO. b10% SF in Fe 0.5Mn 0.5. function of FeMn thickness) is different from the recent model calculations [ 17,28], where a AFM |NM|AFM spin valve was studied with the spin torque acting on the antiferromagneticorder parameter increasing linearly with the thickness of AFMlayers. The difference seems to be the combination effect of aninterfacial effect (as shown in Fig. 3) and multiple scattering (as shown in Fig. 4). FIG. 4. (a, b) k||resolved STT and (c, d) spin transfer efficiency η=τ(k)/G(k) in units of ¯ h/2ewithG(k)=(e2/h)Tr[t(k)t†(k)] in site-ordered G-type Fe |MgO(3) |Fe0.5Mn 0.5(16)|Cu junction with clean interface at relative angle of 90◦.To check the spin-transfer behavior in the antiferromagnetic FeMn, we present the layer-dependent spin torque in thesite-ordered G-type Fe |MgO(3) |Fe 0.5Mn 0.5(16)|Cu junctions with clean interface in Fig. 3(b). Here, both τlandτmdecrease quickly with distance from the MgO |FeMn interface [the behavior is more clear for thicker FeMn layers; see insetof Fig. 3(b)], indicating that the spin torque may be an interfacial effect. The behavior is very similar to spin transferin FMs. In comparison, the spin torque in an antiferromagneticmetal-based spin valve spans the whole antiferromagneticregion [ 17,18,28]. Of equal importance as the magnitude of spin torque is the spin transfer efficiency in assessing the spin transfer process.We give the spin transfer efficiency in Table IIfor both STTs as applied to total magnetization and antiferromagneticorder parameter. The spin transfer efficiency of the STTon the total magnetization ( η m) is seen to be close to but less than one unit for most cases, which is quite stable inthe presence of defects such as OV and SF. The high-spintransfer efficiency is related to strong spin filtering of the MgObarrier. However, the spin transfer efficiency of the STT forthe antiferromagnetic order parameter ( η l) is sensitive to not only barrier thickness but also defects (Table II). Surprisingly, we find ηlis larger than one unit in several junctions. For example, ηlof 3.3 ¯h/2eis observed in the site-ordered G-type Fe|MgO(3) |Fe0.5Mn 0.5(16)|Cu junction with clean interface at the relative angle of 90◦. By comparing the STT with ηin the junctions, we find that high ηcontributes largely in enhancing STT. To check the enhanced τlandηl,w eg i v e k||-resolved τand ηin the site-ordered G-type Fe |MgO(3) |Fe0.5Mn 0.5(16)|Cu junction with clean interface at relative angle of 90◦(Fig. 4). Bothτmandτlare mainly from hot spots in the 2D BZ, and the pattern of the former is close to that of the transmission, 064402-4STRUCTURE-DEPENDENT MAGNETORESISTANCE AND . . . PHYSICAL REVIEW B 95, 064402 (2017) indicating that spin transfer is carried mainly by resonance states. Hot k||points with maximum ηm∼1.4¯h/2eandηl∼ 85¯h/2eare observed, indicating that spin transfer occurs more than once. Considering the relative angle of 90◦between the magnetization of the left lead and the order parameter of FeMn,the spin transfer efficiency contributed by the MgO |FeMn interface does not exceed one unit, and multiple spin transfermay stem from the FeMn region. Replacing Mn by Fe, a Fe |MgO|AFM-Fe |Cu junction is formed. We find τ lin both the L-type and G-type junctions are several times larger than τm. In detail, τm∼146/81× 1014τ0andτl∼486/346×1014τ0are observed in the L-/G- type Fe |MgO(3) |AFM-Fe(16) |Cu junctions at relative angle of 90◦. That is, multiple spin transfer is found in both the L-type and G-type junctions. Furthermore, OV shows less effect on τmfor both site-ordered and site-disordered junctions. τmof 97/92× 1014τ0and 88 /74×1014τ0are found in the site-ordered and site-disordered G −/L−type Fe |MgO(3) |Fe0.5Mn 0.5(16)|Cu junctions with 10% OV at Fe |MgO and FeMn |MgO interfaces, respectively; see Table II. They are slightly larger than those in the correspondingly clean junctions, respectively. Theenhancement in spin-dependent transmissions from vertexscattering in the dirty junctions may be responsible for theenhanced spin torque. In comparison, OV at interfaces wouldenhance τ lin the G-type junctions but suppress τlin the L-type junctions. Similar to OV , τl/τmis sensitive/insensitive to SF in the antiferromagnetic region. τmof 80/94×1014τ0, and 75/75×1014τ0andτlof 20/4×1014τ0and 37 /20×1014τ0 are found in the site-ordered and site-disordered L-/G-type Fe|MgO(3) |Fe0.5Mn 0.5(16)|Cu junction in the presence of 10% SF, respectively. Hence, to achieve large τl, spin disorder should be avoided. From angular-dependent STT, we can estimate the critical switching bias voltage and current by a phenomenologicalLandau-Lifshitz-Gilbert (LLG) equation [ 17]. We find that the angular dependency in STT for both L-type and G-typejunctions follows simple trigonometric functions. Consider-ing a single AFM domain with easy uniaxial anisotropicfieldH K∼20 mT along the zdirection and Gilbert damp- ing coefficient of 0.01, we estimated the critical switch-ing bias voltage V Cand critical current density JCin Fe|MgO|Fe0.5Mn 0.5(16)|Cu junctions (Table II). As barrier thickness increases, VCincreases exponentially. Also G(EF) (G=1/RA ) decreases exponentially, while JC, the product ofVCandG(EF), is of order 105Acm−2and changes less with MgO thickness. For example, VCof 0.17/0.032 mV and JC of 1.5/0.35×105Acm−2is found in site-ordered L-/G-type Fe|MgO(3) |Fe0.5Mn 0.5(16)|Cu junctions. Both VCandJCare about one order smaller than those in Fe |MgO|Fe junctions [38,47] with the same barrier thickness, respectively. The reduction in VCandJC, compared with those in Fe |MgO|Fe junctions, should be related with: (1) enhanced spin torque bymultiple spin transfers in the AFM region (Fig. 4), (2) absence of shape anisotropy, and (3) symmetric angular dependencein spin torque. Furthermore, the uniaxial anisotropy in AFMdepends on thickness. This is similar to that in FM. A smallH Kof less than 1 m Tin thin FeMn has been estimated inthe NiFe |FeMn|CoFe multilayer [ 48]. Using this parameter value, VCandJCwould be one order smaller than the numbers above. So far, we conclude that the spin dynamics in the antifer- romagnetic Fe |MgO|FeMn|Cu junctions is driven by τlwith low working frequency. The conclusion is weakly contingenton the right lead materials, and similar spin dynamic behaviorsare observed in junctions with the right Cu lead replaced bymetals such as bcc-Cr, fcc-Ag, and fcc-pt. Moreover, the spindynamics in an AFM driven by τ mis as effective as that driven byτl. For example, the spin Hall effect [ 49,50]a tt h eN M |AFM interface can induce pure τmin the absence of a particle current across the AFM, which driving the spin dynamics at very highworking frequency. However, it is hard to observe MR in FeMn-based structures experimentally, partly because the spin structureis complex [ 51–54] in FeMn. If the spin structure in FeMn is simply collinear, MR would be easily observedexperimentally. An experimental study [ 39] shows that the introduction of an exchange bias in the FeMn |Co interface can collinearly stabilize the antiferromagnetic order parameter.Moreover, the collinear spin structure is also experimen-tally demonstrated in bulky antiferromagnetic Mn 2Au [ 55] and CuMnAs [ 56]. We expect an experimentally observed nonrelativistic MR in this collinear antiferromagnetic spinstructure. IV . SUMMARY Based on first-principles scattering theory, we predict large MR and STT in antiferromagnetic Fe |MgO|FeMn|Cu junctions. A larger MR ∼100% was found in one junction. Spin torque acting on antiferromagnetic order parameters τl is the same order as that acting on the total magnetization τm. The marked MR and τlin the site-ordered G-type junctions are related to reduced symmetry in the system. Both MRand STT are sensitive to interfacial OV and SF in the FeMnregion. Spin dynamics, studied using a phenomenological LLG equation, suggest that τ lrather than τmdrives the magnetic dynamics. An electronic bias of order 10−1mV and current density of order 105Acm−2are predicted to efficiently switch a junction with a 3L MgO barrier, which are oneorder smaller than those in the Fe |MgO|Fe junction with the same barrier thickness, respectively. Multiple spin transferexisting in the antiferromagnetic region may be responsiblefor the enhanced spin torque and small switching currentdensity. ACKNOWLEDGMENTS X.J. thanks Ke Xia at BNU for the suggestion of the cal- culations. We gratefully acknowledge financial support fromNational Natural Science Foundation of China under GrantsNo. 11274094 and No. 51332007. 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RevModPhys.82.557.pdf

Axions and the strong CPproblem Jihn E. Kim * Department of Physics and Astronomy and Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea Gianpaolo Carosi† Physical Sciences Directorate, Lawrence Livermore National Laboratory, Livermore,California 94550, USA /H20849Published 4 March 2010 /H20850 Current upper bounds on the neutron electric dipole moment constrain the physically observable quantum chromodynamic /H20849QCD /H20850vacuum angle /H20841/H9258¯/H20841/H1135110−11. Since QCD explains a great deal of experimental data from the 100 MeV to the TeV scale, it is desirable to explain this smallness of /H20841/H9258¯/H20841 in the QCD framework; this is the strong CPproblem. There now exist two plausible solutions to this problem, one of which leads to the existence of a very light axion. The axion decay constant window, 109/H11351Fa/H113511012GeV for an O/H208491/H20850initial misalignment angle /H92581, has been obtained from astrophysical and cosmological data. For Fa/H114071012GeV with/H92581/H11021O/H208491/H20850, axions may constitute a significant fraction of the dark matter of the universe. The supersymmetrized axion solution of the strong CPproblem introduces its superpartner the axino, which might have affected the evolution of the Universesignificantly. The very light axion /H20849theory, supersymmetrization, and models /H20850using recent particle, astrophysical, and cosmological data, and present prospects for its discovery is reviewed here. DOI: 10.1103/RevModPhys.82.557 PACS number /H20849s/H20850: 14.80.Va, 12.38.Aw, 95.35. /H11001d, 11.30./H11002j CONTENTS I. Overview 557 II. The Strong CPProblem and Solutions 560 A. Neutron electric dipole moment 561B. Possible solutions 562 1. Calculable /H9258 562 2. Massless up quark 562 III. Axions 563 A. Axion shift symmetry and reparametrization invariance 564 1. Supersymmetrization 566 B. Axion mass 566 1. Axion mass with light quarks 5682. Comparison with old calculations 5703. Mesons without axions 5704. The /H9258=0 vacuum with axions 570 C. Axion couplings 571 1. Axion-hadron coupling 5712. Axion-photon-photon coupling 5743. Axion-lepton couplings 574 D. Old laboratory bounds on F a 575 IV . Axions from Outer Space 575 A. Axions from stars 575B. Axions in the universe 576C. Axion cosmology beyond the window 579D. Quintessential axion 580 V . Axion Detection Experiments 580 A. Solar axion search 5811. Axion helioscopes 581 2. Bragg diffraction scattering 5813. Geomagnetic conversion 581 B. Search for cosmic axions 581 1. General detector properties 5822. Microwave receiver detectors 5833. Rydberg atom detectors 584 C. Laser searches 585 1. Polarization shift of laser beams 5852. Light shining through walls 5853. Magneto-optical vacuum effects 586 VI. Theories for Very Light Axions 587 A. SM singlets without SUSY 587B. Composite axions 587C. Axions with extra dimensions 588D. SUSY-breaking scale, axion and axino 588E. The /H9262problem 588 F. Axions from superstrings 590 1. Model-independent axion 5912. Model-dependent axion 5923. Toward a plausible QCD axion from string theory 592 4. Hidden-sector confining forces, axion mixing, and approximate PQ symmetry 593 VII. Axino Cosmology 593 A. Neutralino and gravitino 594B. Axino 595 Acknowledgments 596References 596 I. OVERVIEW Strong interaction phenomena have revealed that the discrete symmetries of charge conjugation C, parity P,*jekim@ctp.snu.ac.kr †carosi2@llnl.govREVIEWS OF MODERN PHYSICS, VOLUME 82, JANUARY–MARCH 2010 0034-6861/2010/82 /H208491/H20850/557 /H2084945/H20850 ©2010 The American Physical Society 557and time reversal Tare separately good symmetries of nature. Therefore, quantum chromodynamics /H20849QCD /H20850 based on the gauge group SU /H208493/H20850c/H20849Han and Nambu, 1965 ;Bardeen, Fritszch, and Gell-Mann, 1972 /H20850must re- spect any combinations of these discrete symmetries C, P, and Tto be accepted as the theory of strong interac- tions. Among these discrete symmetries, the CPsymme- try is not necessarily respected in QCD due to the non- zero QCD vacuum angle /H9258, an issue known as the “strong CPproblem.” Since QCD is so successful phe- nomenologically, a possible solution to the strong CP problem is expected to be realized in nature. Currentlythe most attractive solution leads to the existence of avery light axion /H20849Kim, 1979 ;Shifman, Vainstein, and Za- kharov, 1980 ;Dine, Fischler, and Srednicki, 1981b ;Zhit- nitskii, 1981 /H20850. Searches for QCD axions generated from the Sun /H20849Andriamonje et al. , 2007 ;Inoue et al. , 2008 /H20850and remnant axions from the early Universe /H20849Rosenberg, 2004 ;Carosi, 2007 /H20850are presently ongoing. The story of axions started with the QCD U /H208491/H20850prob- lem /H20849Weinberg, 1975 /H20850which is now understood, having been solved by the ’t Hooft determinental interaction /H20849’t Hooft, 1976 ,1986 /H20850. The determinental interaction is shown as the left diagram of Fig. 1and the solution is shown as the shaded right diagram. The strong interac-tion causes the quark bilinears to condense with a vacuum expectation value /H20849VEV /H20850of order v/H11229260 MeV. The phase of this interaction /H9258¯originates from the QCD vacuum angle, which is known to be physical /H20849Callan, Dashen, and Gross, 1976 ;Jackiw and Rebbi, 1976 /H20850, and contributes to the neutron electric dipole moment /H20849NEDM /H20850with order /H9258¯times the neutron size, a large value. Peccei and Quinn /H20849PQ /H20850observed that there exists a way to make /H9258¯a phase by introducing a symmetry, now called U /H208491/H20850PQ; then physical amplitudes do not depend on/H9258¯, as in the massless quark case /H20849Peccei and Quinn, 1977a ,1977b /H20850. In the standard model /H20849SM /H20850, this phase is a pseudoscalar Goldstone boson called the “axion”among the multitude of Higgs fields as noted by Wein- berg /H208491978 /H20850and Wilczek /H208491978 /H20850. If the PQ idea was com- pleted with Fig. 1, this axion would be exactly massless /H20849but observable /H20850, and /H9258¯would behave “unphysically” in having to choose the freedom an appropriate axionVEV , which was the original PQ idea. However, there exist subleading terms, proportional to one power of m q, which close the quark lines with the current quark massinstead of a condensation. Then an axion potential de-velops, and the axion becomes a pseudo-Goldstone bo- son. The axion solution of the strong CP problem is cosmological in that the axion VEV chooses /H9258¯=0 at the minimum of this axion potential. The currently allowedaxion is very light and long lived. The properties of the axion /H20849denoted as a/H20850are mainly given by its decay constant F a, which sets the scale of nonrenormalizable axion interactions through a/Fa. Ini- tial axion searches placed Fafar above the electroweak scale and additional stringent bounds on Fawere ob- tained from studies of stellar evolution and cosmology/H20849Kim, 1987 /H20850. Axion astrophysics, started by Dicus, Kolb, Teplitz, and Wagoner /H208491978 ,1980 /H20850using earlier ideas from Sato and Sato /H208491975 /H20850and Sato /H208491978 /H20850now gives a stringent lower bound on the decay constant, F a/H333560.5 /H11003109GeV, from the study of SN1987A /H20849Raffelt, 1990a ; Turner, 1990 /H20850. With this large decay constant, the axion flux from the Sun is a small fraction of the solar neutrinoflux, but may still be detectable by the CERN AxionSolar Telescope /H20849CAST /H20850experiment and by the Tokyo helioscope. It is known that very light axions with F ain the 1012GeV region /H20849axion mass in the /H9262eV range /H20850might compose some part of cold dark matter /H20849CDM /H20850in the Universe /H20849Abbott and Sikivie, 1983 ;Dine and Fischler, 1983 ;Preskill, Wise, and Wilczek, 1983 /H20850. The exact amount of axion CDM depends on the initial axion mis- alignment angle /H92581at the time of axion creation when the universe temperature was around the axion decay constant, T/H11011Fa. This observation puts the very light ax- ion on the list of leading CDM candidate particles. Ifindeed these cosmic axions compose a significant frac-tion of CDM in the universe, they may be detectable bycollection of axion-converted photons in cavity-type de-tectors /H20849Sikivie, 1983 /H20850as tried by DePanfilis et al. /H208491987 /H20850 and Hagmann et al. /H208491990 /H20850and now continuing at the Axion Dark Matter experiment /H20849ADMX /H20850. Cosmology including CDM was the leading candidate for the early Universe in the 1980s /H20849Blumenthal, Faber, Primack, and Rees, 1984 ;Kolb and Turner, 1990 ;Wein- berg, 2008 /H20850. Since then this view has given way to the new cosmology with the discovery of dark energy /H20849DE /H20850 in 1998 /H20849Riess et al. , 1998 ;Perlmutter et al. , 1999 /H20850. The current view of the dominant components of the Uni- verse is/H9024 CDM /H112290.23 and/H9024/H9011/H112290.73 with only a few per- cent consisting of baryons /H20849Spergel et al. , 2007 /H20850. The most plausible dark matter candidates at present are thelightest supersymmetric /H20849SUSY /H20850particle /H20849LSP /H20850, the ax- ion, the axino, and the gravitino. Here we review theaxion and its CDM-related possibilities. The need for DM was suggested as early as the 1930s /H20849Zwicky, 1933 ;Smith, 1936 /H20850. Since then, evidence of non- luminous DM in the universe has been accumulating:examples include flat galactic rotation curves, Chandrasatellite photos, and gravitational lensing effects. If thegalactic bulge is the dominant mass in the galaxy, the rotational velocity vof a star located at rfrom the center should be v/H11011r−1/2. But the observed flat rotation curve /H20851see, for example, McGaugh et al. /H208492007 /H20850, and referencesuR¯uL dR¯dLsR¯sL ×e−i¯θ−v3 −v3 −v3×e−i¯θ FIG. 1. /H20849Color online /H20850The determinental interaction of light quarks. Chiral symmetry breaking introduces the anomalous /H9257/H11032mass term from the quark condensations.558 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010therein /H20852violates this expectation and implies an ex- tended mass in the halo varying as /H9267/H20849r/H20850/H110111/r2. The Chandra observation of x-ray and gravitational lensingimages also implies this matter profile around the bulletcluster /H20851Clowe et al. /H208492006 /H20850/H20852. Circular gravitational lens- ing images /H20851Jeeet al. /H208492007 /H20850/H20852also support the existence of DM. The DM density around the Solar system is /H9267DM/H112290.3–0.45 GeV/cm3. Current CDM candidates are either incoherent par- ticles or coherent oscillations of spin-0 fields. In thisview bosonic collective motions such as the axion can beconsidered as CDM. The popular incoherent CDM par-ticles are the weakly interacting massive particles/H20849WIMPs /H20850or decay products of WIMPs. A more fre- quently used independent distinction is between thermaland nonthermal relics, but there is no strict relation ofcorrespondence between the incoherent and coherentparticles and the thermal and nonthermal relics. WIMPsare massive particles with weak interaction cross sec-tions, first discussed in terms of a heavy neutrino, corre-sponding to the right-hand side /H20849RHS /H20850crossing point of Fig. 2/H20849a/H20850/H20849Lee and Weinberg, 1977b /H20850. The left-hand side /H20849LHS /H20850crossing point corresponds to a 10 eV neutrino /H20849Cowsik and McClelland, 1972 ;Marx and Szalay, 1972 /H20850. WIMPs, such as the LSP , are thermal relics when theirnumber density is determined by the freezeout tempera-ture and are nonthermal relics if their number density isdetermined by another mechanism such as the decay ofheavier relics /H20849Choi, Kim, Lee and Seto, 2008 /H20850. In Fig. 2/H20849b/H20850, we sketch the axion energy density in terms of the axion mass. The shape is flipped from that of Fig. 2/H20849a/H20850, because in the axion case the low- and high-mass regions contribute /H9024 afrom different physics, one from the vacuum misalignment and the other from the hot ther-mal relics. In addition to the heavy neutrino, SUSY with R-parity conservation allows the LSP to be just such a WIMPparticle. The LSP interaction is “weak” since the inter-action mediators /H20849SUSY particles /H20850are supposed to be in the 100 GeV range. For a WIMP to be a successful CDM candidate, usually the interaction cross section atthe time of decoupling needs to be /H20849Kolb and Turner, 1990 ;Spergel et al. , 2007 /H20850 /H20841/H20855 /H9268intv/H20856/H20841at decoupling /H110150.2/H1100310−26cm3s−1 with/H9024mh2/H112290.113 ± 0.009. /H208491/H20850 This is roughly the cross section for the LSP from low- energy SUSY, which is the reason why the DM commu-nity is so interested in the WIMP LSP . Some super-weakly interacting particles such as gravitinos, axinos,and wimpzillas /H20849Chung, Kolb, and Riotto, 1999 /H20850might be CDM candidates as well, but their cross sections donot fall in the range of Eq. /H208491/H20850. The CDM candidate particles are shown in the /H9268intversus mass plane in Fig. 3 taken with minor modification from Roszkowski /H208492004 /H20850. The incoherent fermions, such as the neutrino and theleft ends of the bars of the axino and gravitino, corre-spond to the left crossing points of Fig. 2/H20849a/H20850. The rest,except for the axion, correspond more or less to the right crossing points of Fig. 2/H20849a/H20850, with reheating after inflation considered if necessary. Currently, there are ex-perimental efforts to discover the LSP as predicted bySUSY models. Direct cosmological searches are also on-going /H20849Jungman, Kamionkowski, and Griest, 1996 ; Bernabei et al. , 2003 ,2008 ;Bertone, Hooper, and Silk, 2005 ;Lee et al. , 2007 ;Angle et al. , 2008 ;Behnke et al. , 2008 ;Ahmed et al. , 2009a /H20850. At the CERN Large Hadron Collider /H20849LHC /H20850, the probable LSP mass ranges for LSPs produced by neutrolino decay will be looked for. It is known that density perturbations must have be- gun growing much earlier than recombination time inorder to become large enough to form galaxies in theyoung universe. For galaxy formation, therefore, DM isneeded since proton density perturbations could notgrow before the recombination time, but DM perturba-tions could. With DM, the equality point of radiation and matter energy densities can occur much earlier thanthe recombination time since DM is not prohibited fromlog10(mν[GeV])log10(Ωνh2) −101234 −7−6−5−4−3−2−101102eV GeVDiracMajorana (a) log10(ma[eV])log10(Ωah2) −4−3−2−101 −7−6−5−4−3−2−101µeV eV (b) FIG. 2. /H20849Color online /H20850The Lee-Weinberg-type plots for /H20849a/H20850the neutrino/H9024/H9263h2/H20849Kolb and Turner, 1990 /H20850and /H20849b/H20850the axion /H9024ah2, where his the present Hubble constant in units of 100 km s−1Mpc−1. The dashed line in /H20849a/H20850is for/H9024/H9263h2=0.113. In /H20849b/H20850, it corresponds to the hadronic axion. The dashed lines correspond to the CDM and hot DM limits, respectively.559 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010collapsing by Silk damping /H20849Silk, 1968 /H20850. If the WIMP mass and interaction cross section fall in the region al-lowed by Eq. /H208491/H20850, the WIMP can be part of CDM. If the LSP were the only CDM component, then the LSP masswould give one number for the DM density, which may not be accurate. Thus, even if the LSP is contributing tothe CDM density, we may need the axion to account forthe correct amount of CDM around us. This is possiblein the anthropic scenario of very light axions because itis equally probable for the initial axion misalignment angle /H92581to take any value between 0 and /H9266/H20849Tegmark, Aguirre, Rees, and Wilczek, 2006 /H20850. Here we review the axion, which is probably the most interesting Nambu-Goldstone boson /H20849Nambu, 1960 ; Goldstone, 1961 ;Nambu and Jona-Lasinio, 1961 /H20850,a s well as related issues. In Sec. IIwe discuss the strong CP problem and its plausible solutions. In Sec. IIIwe review the most attractive solution giving the very light axionand present the axion theory in terms of possible axion couplings defined by c 1,c2, and c3used throughout this review. In Sec. IVwe present axion astrophysics and cosmology. Here we present a new number for the cos-mic axion abundance in view of recent accurate data onlight quark masses. In Sec. Vwe summarize the axion detection ideas and the ongoing axion detection experi-ments. In Sec. VIwe summarize the proposed very light axion models, including superstring axions. Finally inSec. VII we discuss cosmology with the axino, the ax- ion’s superpartner. If the axion was observed, it would mark one of the most profound elementary particle discoveries becauseit would confirm experimentally the instanton-based ar-guments of QCD. In addition, if it were shown to beconsistent with a cosmologically significant amount of axions, the CDM idea of bosonic collective motionwould also be confirmed experimentally. If SUSY is cor- rect and the axion is the solution to the strong CPprob- lem, axino must have affected the evolution of the Uni-verse as well. II. THE STRONG CPPROBLEM AND SOLUTIONS There are good reviews on the strong CP problem /H20849Kim, 1987 ;Cheng, 1988 ;Peccei, 1989 /H20850; here we outline a few key points. QCD with SU /H208493/H20850cgluons is a confining gauge theory with three light quarks below 1 GeV and /H9011QCD=380±60 MeV /H20849Groote, Körner, Schilcher, and Nasrallah, 1998 /H20850. The classical gluon field equations have the instanton solution /H20849Belavin, Polyakov, Schwartz, and Tyupkin, 1975 /H20850, G/H9262=if/H20849r/H20850g−1/H20849x/H20850/H11509/H9262g/H20849x/H20850,f/H20849r/H20850=r2 r2+/H92672, /H208492/H20850 where the gauge coupling is absorbed in the gauge field, g/H20849x/H20850is a pure gauge form with G/H9262/H9263/H110081/r4for a large r, and/H9267is the instanton size. The /H20849anti- /H20850instanton solution satisfies the /H20849anti- /H20850self-duality condition G/H9262/H9263=±G˜/H9262/H9263 which carries the integer Pontryagin index q=1 16/H92662/H20885d4xTrGG˜=1 32/H92662/H20885d4xG/H9262/H9263aG˜a/H9262/H9263, /H208493/H20850 where G˜a/H9262/H9263=1 2/H9280/H9262/H9263/H9267/H9268G/H9267/H9268a. The classical solution with q=−/H11009,...,−1,0,+1,...,+ /H11009, introduces a new real num- ber/H9258which parametrizes the /H20841/H9258/H20856vacuum, /H20841/H9258/H20856=/H20858 n=−/H11009/H11009 ein/H9258/H20841n/H20856. /H208494/H20850 Since the n’s are integers, in view of Eq. /H208493/H20850,/H9258is a peri- odic variable with period 2 /H9266. It is known that /H9258is an observable parameter /H20849Callan, Dashen, and Gross, 1976 ; Jackiw and Rebbi, 1976 /H20850.I nt h e/H9258vacuum, we must con- sider the P- and T-/H20849orCP-/H20850violating interaction param- etrized by /H9258¯=/H92580+/H9258weak,1 L=/H9258¯/H20853GG˜/H20854/H11013/H9258¯ 64/H92662/H9280/H9262/H9263/H9267/H9268G/H9262/H9263aG/H9267/H9268a, /H208495/H20850 where the curly bracket includes 1/32 /H92662,/H92580is the angle given above the electroweak scale, and /H9258weakis the value introduced by the electroweak CP violation. This ob- servable/H9258¯has led to the so-called strong CPproblem from the upper bound on the NEDM. For QCD to be- come a correct theory, this CPviolation by QCD must be sufficiently suppressed. 1With the canonical normalization of the gauge field, the RHS of Eq. /H208495/H20850is multiplied by gc2.log 10{σintcm−2} mDM[GeV]−35 fb−40 −45 −50 −55 −60 −65 −70 −75 −80 10−20µeV 10−1210−4GeV 10410121020neutrino ν axion aaxino ˜ a gravitino ˜ g3/2˜g1/2 WIMPχ wimpzilla FIG. 3. /H20849Color /H20850Some proposed particles in the plane of the interaction cross section vs the corresponding particle mass mi. The skeleton is taken from Roszkowski /H208492004 /H20850. The dashed curves represent schematic shapes of /H9024ivs the corresponding particle mass mi. The small red square box corresponds to the hot DM hadronic axion. Two small outside squares /H20849cyan and blue /H20850in the axion region are marked to show the plausible GUT and CDM axions, respectively. The abundances of theheavy axino, gravitino, and wimpzilla depend on how inflationends.560 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010A. Neutron electric dipole moment The interaction /H208495/H20850is the anomaly term /H20849Adler, 1969 ; Bell and Jackiw, 1969 /H20850which is the basis for solving /H20849’t Hooft, 1986 /H20850the old U /H208491/H20850problem of QCD /H20849Wein- berg, 1975 /H20850. The important size of instantons for physics is near the scale where QCD becomes strong. ’t Hooft /H208491976 /H20850showed that the determinental interaction of light quarks carries the same global symmetry as that of Eq./H208495/H20850, and it is customary to use this light quark determi- nental interaction rather than treating the gluon interac-tion /H208495/H20850. The early estimates of the NEDM proportional to /H9258¯from the determinental interaction are 2.7 /H1100310−16/H9258¯ecm /H20849Baluni, 1979 /H20850and 3.6/H1100310−16/H9258¯ecm /H20849Crewther, Di Vecchia, Veneziano, and Witten, 1979 /H20850. Other estimates from different methods are 11 /H1100310−16/H9258¯ecm /H20849Cea and Nardulli, 1984 /H20850, 1.2/H1100310−16/H9258¯ecm /H20849Schnitzer, 1984 /H20850,3/H1100310−16/H9258¯ecm /H20849Musakhanov and Is- railov, 1984 /H20850, and 5.5/H1100310−16/H9258¯ecm /H20849Kanaya and Koba- yashi, 1981 /H20850. Comprehensive reviews of the NEDM exist /H20849Dar, 2000 ;Pospelov and Ritz, 2005 /H20850. Recently, the NEDM has been estimated in the hard wall anti–de Sit- ter /H20849AdS /H20850QCD model with one extra dimension, 1.08 /H1100310−16/H9258¯ecm /H20849Hong, Kim, Siwach, and Yee, 2007 /H20850. The diagrams contributing to the NEDM are re- stricted. The neutron magnetic dipole moment arises atone loop in chiral perturbation theory. If we treat thisneutron magnetic dipole moment operator /H9262anomn¯/H9268/H9262/H9263nF/H9262/H9263emas a vertex, tree diagrams do not con- tribute to the NEDM, because the magnetic momentterm has the same chiral transformation property as thatof the mass term and hence by redefining an externalneutron field one can remove the phases in the neutronmass and in the dipole moment operator together. Let the U /H208491/H20850chiral transformation of quarks in the broken phase be encoded in the neutron mass term as m nn¯Lei/H20849/H92511/H11032/H9257/H11032/f/H9257/H11032−/H92518/H11032/H92660/f/H9266+/H9258¯/2/H20850nR+H.c. /H20849ei/H9251/H11032/H9258¯instead of e3i/H9251/H11032/H9258¯ because the baryon octet has spin1 2/H20850. The VEVs of /H92660 and/H9257/H11032are calculated in Sec. III.B . The CPviolation is present by a mismatch between the CP-conserving RHS vertex and the CP-violating LHS vertex as shown in Fig. 4/H20849b/H20850. The mass term of Fig. 4/H20849b/H20850and the neutron mag- netic dipole moment term of Fig. 5/H20849b/H20850have the same chiral transformation property and the phases appearing there can be simultaneously removed by redefining nR,for example. However, the phase appearing in Fig. 5/H20849a/H20850 cannot be removed by this phase redefinition and thiscontribution is physically observable. Since Fig. 5/H20849a/H20850is the physically observable NEDM, for the proton a simi-lar argument leads to the same magnitude and opposite sign for the proton electric dipole moment, i.e., d n+dp =0. Now we estimate the NEDM as dn e=g/H9266NNg/H9266NN 4/H92662mNln/H20873mN m/H9266/H20874, /H208496/H20850 where the CP-violating scalar coupling g/H9266NN /H20851the bullet of Fig. 5/H20849a/H20850/H20852is estimated by Crewther, Di Vecchia, Ven- eziano, and Witten /H208491979 /H20850as g/H9266NN=−/H9258¯2/H20849m/H9014−m/H9018/H20850mumd f/H9266/H20849mu+md/H20850/H208492ms−mu−md/H20850/H11015− 0.023/H9258¯, /H208497/H20850 where Z=mu/md/H110150.48, md/H110154.9 MeV, and ms/md /H1122920.1. From Eq. /H2084948/H20850of Sec. III.B , we estimate the CP-violating scalar coupling as g/H9266NN=−/H9258¯Z /H208491+Z/H20850/H11229−/H9258¯ 3. /H208498/H20850 Note that Eqs. /H208497/H20850and /H208498/H20850give a factor of /H1101110 differ- ence. Existing calculations vary within a factor of 10.These old calculations depend on the various approxi-mation methods used, but none of these estimated a VEV of /H92660. For example, for Eq. /H208497/H20850, Eq. /H2084911/H20850of Crewther, Di Vecchia, Veneziano, and Witten /H208491979 /H20850 uses the SU /H208493/H20850symmetric baryon octet coupling due to theCP-violating interaction. On the other hand, for Eq. /H208498/H20850the ground state vacuum of the mesonic fields has been used. After integrating out baryons, we look forthe vacuum below the chiral symmetry scale. Then, thecorrect vacuum choice adds the value /H208498/H20850to the value /H208497/H20850. But here we choose the one-order-larger value from the mesonic vacuum shift value /H208498/H20850for an order of mag- nitude estimate, not concerning ourselves about thesigns of the contributions. So we estimate the NEDM as 4.5/H1100310 −15/H9258¯ecm from Eq. /H208498/H20850. Since the recent upper bound on the NEDM is /H20841dn/H20841 /H110212.9/H1100310−26ecm /H20849Baker et al. , 2006 /H20850, we must require×× /angbracketleftπ0,η/prime/angbracketright• (a) (b) FIG. 4. Loop corrections for n¯n-meson coupling. Insertion of theCPviolation effect by VEVs of /H92660and/H9257/H11032in/H20849a/H20850. They can be transferred to one vertex shown as a bullet in /H20849b/H20850. With this bullet, CPviolation is present because of a mismatch between theCP-conserving RHS vertex and CP-violating LHS vertex.•Aµ π−π− n p nAµ• (a)( b) FIG. 5. Diagrams contributing to the NEDM with the bullet representing the CPviolation effect. /H20849a/H20850is the physically ob- servable contribution.561 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010/H20841/H9258¯/H20841/H110210.7/H1100310−11. /H208499/H20850 This extremely small upper bound on /H9258¯has led to the so-called strong CPproblem. /H20841/H9258¯/H20841/H1135110−11is perfectly al- lowed but its small value is not explained given that it could have chosen a value anywhere between 0 and /H11011/H9266. The strong CPproblem is the quest to understand more satisfactorily why /H9258¯is so unnaturally small. B. Possible solutions In the remainder of this paper, we simplify the nota- tion replacing /H9258¯by/H9258since there will not be much con- fusion. There are three explanations for the smallness of /H9258in the naturalness framework: case 1, calculable /H9258; case 2, massless up quark; case 3, axion. Here we discusscases 1 and 2, and concentrate on case 3 in subsequentsections. 1. Calculable /H9258 The naturalness of a theory with a parameter /H9252is de- fined by ’t Hooft /H208491979 /H20850: The theory is natural if the symmetry of the theory increases in the limit of vanish- ing/H9252. A frequently quoted example is the Dirac fermion mass m/H9274¯L/H9274R+H.c., where m→0 introduces a chiral symmetry /H9274→ei/H9252/H92535/H9274in the theory. Regarding the strong CP problem, the appropriate symmetry is parity PorCPsince the interaction /H208495/H20850vio- lates parity P, time reversal T, and CP, but conserves charge conjugation C. Requiring CPinvariance in the Lagrangian is equivalent to setting /H92580at zero. However, the observed weak interaction phenomena exhibit weak CPsymmetry violations in the neutral Kmeson system and B→K+/H9266−decay /H20849Amsler et al. , 2008 /H20850, and hence the needed introduction of CP violation in weak interac- tions with /H92580=0 must be achieved spontaneously. In this process one necessarily introduces a /H9258weak part in/H9258 which can be calculated and required to be sufficiently small within the bound given in Eq. /H208499/H20850. Along this line, many ideas have been proposed /H20849Bèg and Tsao, 1978 ; Mohapatra and Senjanovic, 1978 ;Barr and Langacker, 1979 ;Segre and Weldon, 1979 /H20850. This naturalness idea may be extended so as to effect only renormalizablecouplings /H20849Georgi, 1978 /H20850. In any case, the introduction of weak CP violation by spontaneous mechanisms /H20849Lee, 1973 /H20850or by soft scalar masses /H20849Georgi, 1978 /H20850must be checked against various weak phenomena. The current weak CP violation data fit nicely with Kobayashi- Maskawa-type CPviolation /H20849Kobayashi and Maskawa, 1973 /H20850, and these drastically different spontaneous weak CPviolation ideas are probably difficult to fit to the data but are not considered ruled out yet /H20849He, 2008 /H20850, even though the spontaneous CPviolation scheme /H20849Branco, 1980 /H20850in the Weinberg model /H20849Weinberg, 1976 /H20850is ruled out /H20849Chang, He, and McKellar, 2001 /H20850. It should be noted, though, that the models proposed above have difficultyin satisfying the bounds /H208499/H20850.The Nelson-Barr-type weak CP violation however, mimics, the Kobayashi-Maskawa-type CPviolation even though the fundamental reason for CPviolation is spon- taneous /H20849Barr, 1984 ;Nelson, 1984 /H20850. The scheme is de- signed such that the Yukawa couplings are real, i.e., /H92580 =0 from the CPinvariance. Next, spontaneous CPvio- lation is introduced through the singlet VEVs; this is thekey difference from the previous calculable models. Thus, the spontaneous CPviolation is required to occur much above the weak scale through the singlet VEVs,mediating it to light quarks through mixing with vector-like heavy quarks. In modern terms, the heavy quarkscan be considered as the mediation sector. Then, inte-grating out heavy fields we obtain the SM quarks with the Kobayashi-Maskawa-type weak CPviolation. To en- sure Arg Det M q=0 at tree level, specific forms for the Higgs couplings to the SM quarks and the superheavyvectorlike quarks are needed. Beyond the tree level, however, /H9258is generated at one loop, typically with the form /H20849Goffin, Segrè, and Welson, 1980 ;Bento, Branco, and Parada, 1991 /H20850, /H9258weak /H110151 16/H92662/H9004f2/H20858/H20849loop integrals /H20850, /H2084910/H20850 where/H9004f2is the product of couplings and the Feynman loop integral is of O/H208491/H20850. To satisfy the bound /H208499/H20850, the small coupling /H9004f2is needed. Some mechanism such as family symmetry may be needed to forbid /H9258weak at one loop /H20849Nelson, 1984 ;Chang and Keung, 2004 /H20850. This kind of Nelson-Barr-type calculable /H9258weak can be mimicked in many extra-dimensional models including superstring theory. Recently, for example, /H9258weak was cal- culated to be O/H2084910−12/H20850at a two-loop level in a seques- tered flavor and CP model /H20849Cheung, Fitzpatrick, and Randall, 2008 /H20850. Strictly speaking, the axion model also belongs to the class of calculable models but we separate it from the models with spontaneous CPviolation because there it is not necessary to set /H92580=0. 2. Massless up quark Suppose that we chiral transform a quark as q →ei/H92535/H9251q. Then the QCD Lagrangian changes as /H20885d4x/H20851−mqq¯q−/H9258/H20853gc2GG˜/H20854/H20852 →/H20885d4x/H20851−mqq¯e2i/H92535/H9251q−/H20849/H9258−2/H9251/H20850/H20853gc2GG˜/H20854/H20852, /H2084911/H20850 where /H20853GG˜/H20854=/H208491/64/H92662/H20850/H9280/H9262/H9263/H9267/H9268G/H9262/H9263aG/H9267/H9268a.I f mq=0, this is equivalent to changing /H9258→/H9258−2/H9251. Thus, there exists a shift symmetry /H9258→/H9258−2/H9251. It is known that the tunneling amplitude due to instanton solutions with a zero-massquark vanishes /H20849’t Hooft, 1976 /H20850, which implies that the shift symmetry is an exact symmetry. In this case, /H9258is not physical, and hence there is no strong CPproblem if the lightest quark /H20849i.e., the up quark /H20850is massless. The mass- less up quark solution must answer the question: Is the562 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010massless up quark phenomenologically viable? Wein- berg’s famous up-down quark mass ratio Z=mu/mdgave Z=5/9 /H20849Weinberg, 1977 /H20850. It is very similar to the recent compilation of the light quark masses, mu=2.6−1.1+0.9MeV, md=4.9−1.4+1.1Mev, and Z=0.48−1.3+1.2shown in Fig. 6. This compilation is convincing enough to rule out the mass-less up quark possibility /H20849Kaplan and Manohar, 1986 /H20850.I n this review, we use Z=0.48 when a number is needed though the appropriate bound may be 0.35 /H11021Z/H110210.60 /H20849Buckley and Murayama, 2007 ;Manohar and Sachrajda, 2008 /H20850. For some time the massless up quark possibility was taken seriously /H20849Kaplan and Manohar, 1986 /H20850. The reason is that, even if the Lagrangian mass for the up quark iszero, the ’t Hooft determinental interaction may gener-ate a useful up quark mass for chiral perturbation. Therewas confusion on this issue for some time /H20849Leutwyler, 1990 ;Choi, 1992 /H20850. Now, it is clear that the massless up quark possibility is ruled out, even without use of the lattice calculation of the ratio m u/md=0.410±0.036 /H20849Nel- son, Fleming, and Kilcup, 2003 /H20850. III. AXIONS The axion solution seems to be the most attractive one among three possible strong CPsolutions, in par- ticular at present when the massless up-quark possibilityis excluded and calculable solutions need one-loop sup-pression. Peccei and Quinn tried to mimic the symmetry /H9258→/H9258 −2/H9251of the massless quark case of Eq. /H2084911/H20850, by consider- ing the full electroweak theory Lagrangian /H20849Peccei and Quinn, 1977a ,1977b /H20850. They found such a symmetry if Hu is coupled only to up-type quarks and Hdcouples only to down-type quarks,L=−q¯LuRHu−q¯LdRHd−V/H20849Hu,Hd/H20850+ H.c. −/H9258/H20853GG˜/H20854. /H2084912/H20850 Certainly, if we assign the same global charge under the/H92535transformation to Huand Hd,q→ei/H92535/H9251q,Hu →ei/H9252Hu,Hd→ei/H9252Hd, the flavor-independent part changes to L→ −q¯Le−i/H92535/H9251uRei/H9252Hu−q¯Le−i/H92535/H9251dRei/H9252Hd −V/H20849ei/H9252Hu,ei/H9252Hd/H20850+ H.c. − /H20849/H9258−2/H9251/H20850/H20853GG˜/H20854. /H2084913/H20850 Since the full Lagrangian must possess global symmetry, the potential Vshould not allow the HuHdand /H20849HuHd/H208502 terms. The choice of /H9252=/H9251achieves the same kind of /H9258 shift as in the massless quark case, called PQ global sym- metry U /H208491/H20850PQ. Unlike an the massless up-quark case, here/H9258is physical. Even though the coefficient of /H20853GG˜/H20854 changes in the same way in Eqs. /H2084911/H20850and /H2084913/H20850, these two cases differ in that the tunneling amplitude vanisheswith a massless quark /H20849a detailed discussion will be pre- sented in Sec. III.B /H20850but not without a massless quark. The reason is that the Higgs fields transform under U/H208491/H20850 PQ, and one of the Higgs fields, called the axion a, has the shift symmetry a→a+const and corresponds to the Goldstone boson of the spontaneously broken U/H208491/H20850PQ/H20849Weinberg, 1978 ;Wilczek, 1978 /H20850. As a result we call the resulting axion from Eq. /H2084913/H20850the Peccei-Quinn- Weinberg-Wilczek /H20849PQWW /H20850axion. If the consequence of the determinental interaction is only Fig. 1, then of the two bosons /H9257/H11032and aonly/H9257/H11032obtains mass by the RHS diagram of Fig. 1and aremains massless. If are- mains massless, the strong CPproblem is solved as en- visioned by Peccei and Quinn /H208491977a /H20850since for any /H9258we can choose the VEV /H20855a/H20856such that the final /H9258is zero. This was Peccei and Quinn’s idea: that /H20855a/H20856has a shift symme- try mimicking that of the massless quark case. However, ahas interactions and it can be produced in the stars and Kmeson decay, which differs from the massless quark case. At the classical Lagrangian level, there seems to be no strong CPproblem. But the axion coupling to /H20853GG˜/H20854is generated at the one-loop level, which is the U/H208491/H20850PQ-QCD-QCD anomaly. The ’t Hooft determinen- tal interaction mentioned above is exactly this anoma-lous coupling. With this one-loop term, the Lagrangian is not invariant under the phase shift symmetry /H9252ora →a+const. Since it is explicitly broken at the one-loop level, the phase field /H9252of the Higgs fields or axion a does not have a flat potential, i.e., Fig. 1is not complete. Weinberg and Wilczek interpreted this phenomenon us-ing the spontaneous symmetry breaking of the global symmetry U /H208491/H20850 PQ. It is said that /H9258is made dynamical where/H9258/H11013a/Fa, but in the PQWW axion case the com- ponent was there from the beginning in the phases ofthe Higgs doublet fields. The free energy depending on −cos /H9258is the potential for the axion. Since it is propor- tional to −cos /H9258, the minimum of the potential is at /H9258 =0 in CP-conserving theories /H20849Vafa and Witten, 1984 /H20850,012345678 0 1 2 3 4 5 6 mu[MeV ]md[MeV ] •••• FIG. 6. /H20849Color /H20850The allowed mu-mdregion /H20849Manohar and Sachrajda, 2008 /H20850. The two downward sloping lines are from the bound on /H20849mu+md/H20850/2 and the two rising lines are from the bound on mu/md, determined by the masses of the meson oc- tet. The two vertical and horizontal boundaries are from theParticle Data Book bounds on m u=/H208511.5,3.3 /H20852MeV and md =/H208513.5,6.0 /H20852MeV /H20849Amsler et al. , 2008 /H20850.563 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010and thus the vacuum chooses /H9258=0. We discuss this effect below the chiral symmetry breaking scale in Sec. III.B . Thus, the axion solution of the strong CPproblem is a kind of cosmological solution. Note, however, that the weak CP violation shifts /H9258a little bit, leading to /H9258 /H11011O/H2084910−17/H20850/H20849Georgi and Randall, 1986 /H20850. The PQWW axion was ruled out quickly /H20849Donnely et al. , 1978 ;Peccei, 1979 /H20850, which was the reason for the popularity of calculable models in 1978 as discussed inSec. II.B.1 . Nowadays, cosmologically considered axions are very light, because of the phase of the SU /H208492/H20850 /H11003U/H208491/H20850singlet scalar field /H9268. The simplest case is the Kim-Shifman-Vainstein-Zakharov /H20849KSVZ /H20850axion model /H20849Kim, 1979 ;Shifman, Vainstein, and Zhakharov, 1980 /H20850 which incorporates a heavy quark Qwith the following coupling and the resulting chiral symmetry: L=−Q¯LQR/H9268+ H.c. − V/H20849/H20841/H9268/H208412/H20850−/H9258/H20853FF˜/H20854, /H2084914/H20850 L→ −Q¯Lei/H92535/H9251QRei/H9252/H9268+ H.c. − V/H20849/H20841/H9268/H208412/H20850 −/H20849/H9258−2/H9251/H20850/H20853GG˜/H20854. /H2084915/H20850 Here Higgs doublets are neutral under U /H208491/H20850PQ. By cou- pling/H9268toHuand Hd, one can introduce a PQ symmetry also, not introducing heavy quarks necessarily, and theresulting axion is called the Dine-Fischler-Srednicki-Zhitnitskii /H20849DFSZ /H20850axion /H20849Zhitnitskii, 1980 ;Dine, Fischler, and Srednicki, 1981a ,1981b /H20850. In string models, most probably both heavy quarks and Higgs doublets contribute to the /H9268field couplings. The VEV of /H9268is much above the electroweak scale and the axion is a very light axion2The SU /H208492/H20850/H11003U/H208491/H20850singlet/H9268field may mix with the Higgs doublet component by a smallamount, when in practice we can consider the axion as the phase of a singlet field /H9268,/H9268=/H20851/H20849v+/H9267/H20850//H208812/H20852eia/fSwith a /H11013a+2/H9266NDWFaand the axion period 2 /H9266NDWFa. Note that we use fSfor the VEV of /H9268or the value relevant in the field space and Fadefined from the coefficient of the anomaly term; namely, the coefficient of the anomaly /H20853GG˜/H20854defines Faas/H9258=a/Fawhile the VEV /H20849v/H20850of/H9268,/H9268 /H11008eia/v, defines fS. The periodicity 2 /H9266of/H9258implies that Fa cannot be larger than v/H11013fS, and we have Fa=fS/NDW.I t has been shown that models with NDW/HS110051 have an en- ergy crisis problem in the standard big bang cosmology /H20849Sikivie, 1982 /H20850. But models with NDW=1 do not have such a problem due to the mechanism of conversion ofthe two-dimensional axionic domain wall disks sur-rounded by axionic strings into radiation /H20849Barr, Choi, and Kim, 1987 /H20850. A. Axion shift symmetry and reparametrization invariance In the original PQWW axion model, the Lagrangian in the effective field theory language was extensively dis-cussed /H20849Donnelly et al. , 1978 ;Peccei, 1989 /H20850. Here, due to the simplicity in the formulas, we present the variant-type axion models where the PQ charges are assignedonly to the right-handed quark fields /H20849Bardeen, Peccei, and Yanagida, 1987 /H20850. This discussion will make it easier to introduce our general formulas below. The PQ cur-rent is /H20849Bardeen, Peccei, and Yanagida, 1987 /H20850 J /H9262PQ=Fa/H11509/H9262a+x/H20858 i=1Ng d¯Ri/H9253/H9262dRi+/H208491/x/H20850/H20858 i=1N u¯Ri/H9253/H9262uRi +/H20849−x/H20850/H20858 i=N+1Ng u¯Ri/H9253/H9262uRi, /H2084916/H20850 where Ngis the number of families, Nis the number of up-type quarks coupled to Hu, and x=/H20855Hu/H20856//H20855Hd/H20856. The color anomaly is nonvanishing, i.e., the divergence of J/H9262PQis /H11509/H9262J/H9262PQ=1 2N/H20873x+1 x/H20874/H9251c 4/H9266G/H9262/H9263aG˜a/H9262/H9263+muu¯/H20851i/H92535eia/H92535/Fax/H20852u +mdd¯/H20851i/H92535eia/H92535x/Fa/H20852d, /H2084917/H20850 where we considered the one-family model of uand d with N=1. If Nis zero, there is no color anomaly. For a nonvanishing N, we have to pick up the component or- thogonal to the longitudinal Z/H9262. Since the axial-vector part of the Z/H9262current is proportional to J/H926235, any axial U/H208491/H20850current orthogonal to the longitudinal Z/H9262is an SU/H208492/H20850flavor singlet current constructed in terms of right- handed quark fields. These include the currents corre- sponding to both /H9257/H11032and the PQ phase. Since /H9257/H11032is known to be heavy, we integrate out /H9257/H11032to obtain light fields below the chiral symmetry breaking scale. Thiscorresponds to picking up an anomaly-free piece, or- thogonal to the longitudinal Z /H9262.I ti s J/H9262a=J/H9262PQ−1 2N/H20873x+1 x/H208741 1+Z/H20849u¯/H9253/H9262/H92535u+Zd¯/H9253/H9262/H92535d/H20850, /H2084918/H20850 where Z=mu/md. The divergence of Eq. /H2084918/H20850is propor- tional to mumd, which must be the case for a particle orthogonal to /H9257/H11032. Below we use the typical axion model /H2084914/H20850because it is simple to assign the PQ charges whenever an ex- plicit example is needed. It has the following U /H208491/H20850PQ charges/H9003, Field /H9268 QL QR /H9003 1+1 2−1 2 In this example, the axial-vector current for U /H208491/H20850PQis J/H92625=Q¯/H9253/H9262/H92535Q+v/H11509/H9262a, where ais the phase field of /H9268 =/H20849v//H208812/H20850eia/v. The current corresponds to the charge flow which satisfies the current conservation equation if thesymmetry is exact. But the axial-vector current is in gen-eral violated at one loop by the anomaly /H20849Adler, 1969 ;2Once it was called an invisible axion /H20849Wise, Georgi, and Glashow, 1981 ;Nilles and Raby, 1982 /H20850but it is better to call it a very light axion due to the possibility of its detection.564 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010Bell and Jackiw, 1969 /H20850/H11509/H9262J/H92625=/H20849NQgc2/32/H92662/H20850G/H9262/H9263aG˜a/H9262/H9263,o r /H115092a=/H20849NQgc2/32/H92662v/H20850G/H9262/H9263aG˜a/H9262/H9263+/H20849mQ/v/H20850Q¯i/H92535Qwith the Q number NQ, which shows that the axion interaction with the SM fields is only the anomaly term /H20849plus the anoma- lous coupling with the SM gauge fields /H20850. Here and in Eq. /H2084917/H20850we explicitly write the QCD coupling gc2, but in the remainder of the paper we absorb the gauge coupling inthe gauge fields except in the experimental Sec. V. This axion is the one setting /H9258at zero; thus one needs the axion-gluon-gluon anomalous coupling for which the color anomaly of J/H92625should exist. This kind of symmetry /H9003is the PQ symmetry. The axion is introduced as the Goldstone boson de- gree of a spontaneously, broken global U /H208491/H20850PQsymmetry in renormalizable gauge models /H20849Peccei and Quinn, 1977a ;Kim, 1985 /H20850and/or as a pseudoscalar degree in a more fundamental theory where the axion interactionarises as a nonrenormalizable anomalous interaction inthe effective low energy theory. The most compellingnonrenormalizable interaction was observed in the com-pactification of ten-dimensional /H2084910D /H20850superstring mod- els /H20849Witten, 1984 /H20850. Below we treat the axion as being present as a dynamical degree at the electroweak scale,whether it arises from spontaneously broken PQ sym-metry or from a more fundamental theory with a non-renormalizable anomalous coupling, and focus on QCD interactions containing the axion degree, a= /H9258Fa. Then we collectively write the most general form of its inter- actions: the c1term is the derivative coupling respecting the PQ shift symmetry, the c2term is the phase in the quark mass matrix, and the c3term is the anomalous coupling or the determinental interaction Ldet, L/H9258=1 2fS2/H11509/H9262/H9258/H11509/H9262/H9258−1 4gc2G/H9262/H9263aGa/H9262/H9263+/H20849q¯LiD”qL+q¯RiD”qR/H20850 +c1/H20849/H11509/H9262/H9258/H20850q¯/H9253/H9262/H92535q−/H20849q¯Lmq Reic2/H9258+ H.c. /H20850 +c3/H9258 32/H92662G/H9262/H9263aG˜a/H9262/H9263/H20849orLdet/H20850+c/H9258/H9253/H9253/H9258 32/H92662Fem/H9262/H9263iF˜emi/H9262/H9263 +Lleptons,/H9258, /H2084919/H20850 where/H9258=a/fSwith the axion decay constant fSup to the domain wall number /H20849fS=NDWFa/H20850and qis the fermion matrix composed of SU /H208493/H20850ccharge-carrying fields. When the singlet scalar fields are easier to discuss, we use fS, and when the anomaly term is easier to discuss, we use Fa.Lleptons,/H9258is the axion interaction with leptons. c1,c2, and c3are pregiven coupling constants below the axion scale fSwith the mass parameter mdefined to be real and positive below the electroweak scale. Then, the de- terminental interaction can be used instead of the c3 term, Ldet=−2−1ic3/H9258/H20849−1/H20850Nfe−ic3/H9258 K3Nf−4Det /H20849qRq¯L/H20850+ H.c., /H2084920/H20850 where we multiplied the overall interaction by /H9258in the small-/H9258region and require the periodicity condition c3/H9258=c3/H9258+2/H9266. The periodicity can be accommodated au-tomatically if we replace −2−1ic3/H9258by 1, but then we must add a constant so that it vanishes at /H9258=0. The sign is chosen such that the potential is a minimum at /H9258=0 /H20849Vafa and Witten, 1984 /H20850. With the fixed phases, the c3 term is given from the QCD vacuum structure /H208494/H20850, which does not have any dimensional coupling. But the instan-ton physics necessarily introduces the instanton sizes and hence a kind of QCD scale Kfor the interaction respecting the chiral transformation property for a flavorsinglet operator L det. We use either the anomaly term or Ldet. The/H9258dependence of the form /H2084920/H20850is −c3/H9258sin/H20849c3/H9258/H20850, which has the parity symmetry /H9258→−/H9258. The Fourier ex- pansion satisfying these constraints is −2−1c3/H9258sin/H20849c3/H9258/H20850=−2−1/H208511 − cos /H20849c3/H9258/H20850/H20852 +/H20858 n=2ancos/H20849nc3/H9258/H20850, where the Fourier coefficients satisfy /H20858n=1/H11009n2ian=/H9254i0. Ne- glecting the n/H333562 terms, we use just the cos /H20849c3/H9258/H20850depen- dence. In the defining phase Eq. /H2084919/H20850, the PQWW axion is given by c1=0,c2/HS110050, and c3=0, the KSVZ axion by c1 =0,c2=0, and c3/HS110050, the model-independent axion /H20849Wit- ten, 1984 /H20850in superstring models by c1=0,c2=0, and c3 /HS110050, and the DFSZ axion by c1=0,c2/HS110050, and c3=0. In general, axion models from high energy will have c2/HS110050 and c3/HS110050, and the shift symmetry allows c1/HS110050 in a dif- ferent basis. For simplicity, we discuss Eq. /H2084919/H20850for one- flavor QCD first. For Nfflavors, both ciand/H9258are de- fined from Nf/H11003Nfmatrices in addition to the anomalous coupling and hence the axion is included in Tr /H9258, which also contains the /H9257/H11032meson part of QCD. For Nfflavors, ci/H9258must be replaced by Tr ci/H9258. For the following discus- sion, we refer to one-flavor QCD, but in Sec. III.B in the axion mass estimation we present the full Nfflavor QCD result with the chiral symmetry breaking taken into ac-count. For the case of the axion mass, the c 1,c2, and c3terms may be relevant, but only the combination c2+c3ap- pears. This Lagrangian has a shift symmetry a→a +const, which reparametrizes the couplings between c1, c2, and c3. Explicitly, the axion-field-dependent changes of the quark fields qL→ei/H9251a/H20849x/H20850qLand qR→e−i/H9251a/H20849x/H20850qRgive c1→c1−/H9251,c2→c2−2/H9251,c3→c3+2/H9251, and it must give the same physics, i.e., /H20849Georgi, Tomaras, and Pais, 1981 ; Kim, 1987 /H20850, /H90031PI/H20851a/H20849x/H20850,A/H9262a/H20849x/H20850;c1,c2,c3,m,/H9011QCD /H20852 =/H90031PI/H20851a/H20849x/H20850,A/H9262a/H20849x/H20850;c1−/H9251,c2−2/H9251,c3 +2/H9251,m,/H9011QCD /H20852. /H2084921/H20850 The reparametrization symmetry dictates the non- derivative couplings satisfying c2+c3=const, which is one reason that we use /H9258=/H9258QFD+/H9258QCD=/H92580+/H9258weak as a physical parameter in axion models. Usually, transfer of all couplings of axions to the coefficient of GG˜, the ax- ion decay constant Faand/H9258are defined. Instead, if we usefS/H20849defined to be the VEV of the singlet Higgs field565 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010/H9268/H20850, there exists the coefficient c3defined in Eq. /H2084919/H20850. The triangle diagrams may give an integer times /H9258and the instanton potential comes to the original value by a /H9258 shift of 2/H9266//H20849c2+c3/H20850, with c2+c3=NDWnot necessarily 1 in the pseudoscalar field space. Thus, this integer is called the domain wall number NDW /H20849Sikivie, 1982 /H20850, NDW=/H20841c2+c3/H20841=T r/H9003/H20849fcolored /H20850/H5129/H20849fcolored /H20850, /H2084922/H20850 where the trace is taken over all heavy and light quarks and /H5129is the index of the SU /H208493/H20850crepresentation of col- ored fermions and the PQ charge is given for the left-handed chiral representations. The height of the poten- tial is O/H20849/H9011 QCD4/H20850of the non-Abelian gauge interaction, which is shown in Fig. 7with the domain wall number NDW=3: the bullet, the square, and the triangle denote different vacua. Two important properties of axions in CP-conserving theories are /H20849i/H20850the periodic potential with the period 2 /H9266Fawhere Fais defined in /H2084919/H20850with Fa/H11013fS/NDW, and /H20849ii/H20850the minima at a=0, 2/H9266Fa,4/H9266Fa,.... This determines the cosine form of the potential. Thereexists the axion mixing with quark condensates as dis-cussed in more detail later. The derivative coupling, i.e., the c 1term, can never contribute to the PQ symmetry breaking effect, espe-cially to the axion mass. This axion gets its mass from the /H9258anomaly term which breaks the PQ symmetry. The global symmetry is not broken by the derivative term,which therefore cannot contribute to the axion mass.From the reparametrization invariance /H2084921/H20850, the combi- nation c 2+c3is the correct combination for the axion mass, as shown below. This derivation is included with amore complicated expression in the SUSY extension, but we show the c 2+c3dependence in this supergravity framework because it is the underlying symmetry inmany axion models. Some of the following discussion isderived from Choi, Kim, and Nilles /H208492007 /H20850. 1. Supersymmetrization We now discuss the reparametrization invariance with the SUSY generalization. In the N=1 SUSY models with chiral fields z, there are the superpotential W/H20849z/H20850 and the gauge kinetic function f/H20849z/H20850, both of which are holomorphic functions of z. The superpotential gives the c2term and the gauge kinetic function gives the anomaly term c3. The PQ-invariant Lagrangian, the c1part, has shift symmetry under the shift of the axion supermultip- let:A→A+i/H11003const. This derivative coupling must ap- pear from the Dterms in SUSY models, i.e., through theKähler potential. The real Kähler potential K/H20849z,z*/H20850 must respect the PQ symmetry in the form of A+A¯, K=K0/H20851A+A¯/H20852+/H20853Zq/H20851A+A¯/H20852q¯1q2+ H.c. /H20854, /H2084923/H20850 where the /H92770components of the fields are implied and theq’s denote quark supermultiplets, q=/H9272q+i/H9277/H9274q, /H2084924/H20850 with the anticommuting variable /H9277. Here we used /H9277for the anticommuting Grassmann number since /H9258in this review is reserved for the axion /H9258=a/Fa. B. Axion mass The axion mass arises from the anomaly coupling /H9258GG˜. In this section, first we show that only the c2and c3couplings are relevant for the axion mass, and then we present the axion mass in the broken phase of the chiralsymmetry. With SUSY, the discussion is a bit tricky, be-cause the axion remains massless due to the massless gluino /H20849as in the massless up-quark case with a sponta- neously broken PQ symmetry /H20850. For the axion mass, therefore SUSY must be broken and here one has tolook at how all supergravity terms contribute to the ax-ion mass. Nevertheless, we have the master formula /H2084921/H20850 for the axion, which must be valid even when SUSY isbroken. In this regard, SUSY is not special for the axionmass; the chief constraint is only the anomaly consider-ation. Thus, the following discussion applies even with-out SUSY, but we discuss the axion mass in detail withthe SUSY generalization to include the gluino effects and hence the c 1-type derivative couplings to matter /H20849quarks /H20850and gauginos /H20849gluinos /H20850. We have noted that there exists an anomaly coupling of the/H9257/H11032meson which is the mechanism solving the old U/H208491/H20850problem of QCD. In addition to /H9257/H11032, the axion ais introduced in the anomaly coupling and hence one must consider the mixing of /H9257/H11032and the axion /H20849Bardeen and Tye, 1978 ;Baluni, 1979 ;Kim and Kim, 2006 /H20850. The c3term is the anomaly coupling of the axion, and we normalize the anomaly as the coefficient of/H9280/H9251/H9252/H9253/H9254/H92551/H9251/H92552/H9252k1/H9253k2/H9252. With this normalization, from /H9280/H9251/H9252/H9253/H9254/H11509/H9251A/H9252/H11509/H9253A/H9254leading to − /H9280/H9251/H9252/H9253/H9254k1/H9251/H92551/H9252k2/H9253/H92552/H9254, the c3term anomaly is defined with A3=1. It can be shown that, using the Kähler potential /H2084923/H20850, the kinetic energy terms of fermions contain /H20849Cremmer, Ferrara, Girardello, and van Pröyen, 1983 ;Nilles, 1984 /H20850 /H20858 /H9274Zq/H20849/H9274¯i/H11509//H9274+1 6B/H9262/H9274¯/H9253/H9262/H92535/H9274+1 2Yq,/H9262/H9274¯/H9253/H9262/H92535/H9274/H20850 +/H20858 /H9261/H20849/H9261¯i/H11509//H9261−1 2B/H9262/H9261¯/H9253/H9262/H9261/H20850, /H2084925/H20850 where B/H9262and Yq,/H9262come from the auxiliary components of real K0and Zq, respectively. In terms of the real parts RandYofK0andZ/H20851redefined from Zqof Eq. /H2084923/H20850/H20852,w e obtainV a πFaΛ4 QCD ◦ /triangledownsld • /squaresolid /triangledownsld FIG. 7. The case with NDW=3 where three vacua are distin- guished.566 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010B/H9262=i 2/H20873/H11509K0 /H11509A/H11509/H9262A−/H11509K0 /H11509A¯/H11509/H9262A¯/H20874=−/H20873/H11509R /H11509A/H11509/H9262a/H20874, /H2084926/H20850 Yq,/H9262=i 2/H20873/H11509K0 /H11509A/H11509/H9262A−/H11509K0 /H11509A¯/H11509/H9262A¯/H20874−i/H20873/H11509lnZq /H11509A/H11509/H9262A −/H11509lnZq /H11509A¯/H11509/H9262A¯/H20874=2/H20873/H11509lnY /H11509A/H20874/H11509/H9262a, /H2084927/H20850 where G=−3l n/H20873−K 3/H20874+l n /H20841W/H208412,K=−e−K0/3,Zq=eZ, /H2084928/H20850 K0=R+iK0I=R,Zq=Y+iI=Y. The c3term is an anomaly term. In addition to the c3 term, the c1and c2couplings via loops of Fig. 8will also generate anomaly terms. The derivative coupling, if itever has to contribute to the axion mass, should do sovia the anomaly through loops. In Fig. 8, the couplings for the triangle diagrams are represented in terms of c 1 and c2. In supergravity models, we consider B/H9262and Yq,/H9262 couplings, which are nothing but c1. Consider a fermion with mass m. The derivative coupling through Fig. 8con- tains the anomaly coupling through the coefficient of /H9280/H9251/H9252/H9253/H9254/H92551/H9251/H92552/H9252k1/H9253k2/H9252/H20851see, for example, Georgi, Tomaras, and Pais /H208491981 /H20850/H20852, A1=/H20885 01 dx1/H20885 01−x1 dx2−4f/H20849x1,x2;q,k1,k2/H20850 m2−f/H20849x1,x2;q,k1,k2/H20850, /H2084929/H20850 where f=/H20849x1+x2/H20850/H208491−x1−x2/H20850q2+2x1/H208491−x1−x2/H20850q·k1 +x12k12. Also, the quark mass term of Fig. 8gives A2=/H20885 01 dx1/H20885 01−x1 dx22m2 m2−f/H20849x1,x2;q,k1,k2/H20850. /H2084930/H20850 From Eqs. /H2084929/H20850and /H2084930/H20850, we construct1 2A1+A2=/H20885 01 dx1/H20885 01−x1 2dx2=1 . /H2084931/H20850 When we calculate the axion mass in the real and positive quark mass basis as usual, the anomaly /H20849a/Fa/H20850/H20853GG˜/H20854coupling /H20849including the loop effect /H20850is the sole source of the axion mass. In this basis, and also inany basis due to the reparametrization-invariant combi- nation c 2+c3, we do not have to discuss the contribu- tions of the derivative couplings toward the axion mass.Even though the derivative coupling generates theanomaly, because it is derivative it does not contributeto the axion mass. For one-flavor QCD, we can check the above state- ment explicitly using Eqs. /H2084929/H20850–/H2084931/H20850. In the following two limiting cases, the integrals are easily computed as, using Eqs. /H2084929/H20850and /H2084930/H20850: case /H20849i/H20850,m/H11270/H9011 QCD: /H90031PI=1 16/H92662/H9280/H9251/H9252/H9253/H9254k1/H9253k2/H9252/H20875c3+2c1+O/H20873m2 k2/H20874/H20876, /H2084932/H20850 case /H20849ii/H20850,m/H11271/H9011 QCD: /H90031PI=1 16/H92662/H9280/H9251/H9252/H9253/H9254k1/H9253k2/H9252/H20875c3+c2+O/H20873k2 m2/H20874/H20876. /H2084933/H20850 Consider the quark mass term and the one-flavor deter- minental interaction with the quark condensation, /H20855q¯LqR/H20856/H11011/H9011QCD3ei/H9257/H11032/f. Then the potential takes the form V=m/H20855q¯LqR/H20856eic2/H9258+ H.c. + /H20849c3+c1A1+c2A2/H20850/H20853GG˜/H20854. /H2084934/H20850 For the anomaly combination c3+c1A1+c2A2, the reparametrization invariance Eq. /H2084921/H20850transforms c3 +c1A1+c2A2 to c3+2/H9251+/H20849c1−/H9251/H20850A1+/H20849c2−2/H9251/H20850A2=c3 +c1A1+c2A2where /H2084931/H20850is used, i.e., it is reparametriza- tion invariant. For case /H20849i/H20850, we consider the light quark below the scale/H9011QCD4. Thus, we have V=mv3cos/H20873/H9257/H11032 f−c2/H9258/H20874 +/H9011QCD4cos/H20873/H20849c3+2c1/H20850/H9258+/H9257/H11032 f/H20874, for which we choose c1=0. /H20849If we keep c1, we must con- sider the kinetic mixing of aand/H9257/H11032./H20850Integrating out the heavy/H9257/H11032field as/H9257/H11032/f=−c3/H20849a/fS/H20850from the /H9011QCD4term, which is the larger one, we obtain V=mv3cos/H20873/H20849c2+c3/H20850a fS/H20874, from which ma/H11011/H20881m/H9011QCD3/H20841c2+c3/H20841 fS. /H2084935/H20850 The quarks u,d, and sbelong to this category.c1γµγ5(Bµ,YQ,µ) ψ,λ k1,ε1 k2,ε2c1γµγ5(Bµ,YQ,µ ) ψ,λ k1,ε1 k2,ε2 (a)( b) FIG. 8. The Feynman diagrams for generating anomalous /H9258GG˜couplings from c1for a fermion with mass m. For c2,w e replace c1/H9253/H9262/H92535byc2m/H92535.567 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010For case /H20849ii/H20850, the heavy quark does not condense, and integrating out the heavy quark gives V=/H9011QCD4cos/H20875/H20849c3+c2/H20850a fS/H20876 from which the axion mass is given by ma/H11011/H9011QCD2/H20841c2+c3/H20841 fS. Again the axion mass depends only on the combination c2+c3. Heavy quarks above the chiral symmetry break- ing scale c,b, and tgive the c2term and vectorlike heavy quarks above the electroweak scale give the c3term when we write Eq. /H2084919/H20850just below the electroweak scale. 1. Axion mass with light quarks In the real world, there exist three light quarks whose masses are much smaller than the QCD scale /H9011QCD, and therefore the axion mass has the form anticipated in Eq./H2084935/H20850. Even though there are two light quarks the axion mass dependence has the form/H20881mas a result Fa/H11271f/H9266. This is because of the way in which the leading term ispicked up from the anomalous determinental interaction/H20849Kim and Kim, 2006 /H20850as shown in Fig. 9. In fact, this is obtained simply by noting that the in- stanton interaction is a U /H208491/H20850singlet /H20849Kim, 1987 /H20850. Sup- pose we integrate out quark fields; then the quark massparameters appear in the effective interaction as shownin the first diagram of Fig. 9. In this vacuum with a mass-less quark theory, the tunneling amplitude vanishes so that the strength of the first diagram must be propor- tional to m q. With three quarks, we can generalize it as 1//H20849mu−1+md−1+ms−1/H20850. Suppose that there are only gluons and a very light axion aat low energy. Integrating out heavy fields, we are left with the flavor-independent cou- pling aGG˜. Here we are not considering /H9257/H11032even below the quark condensation scale. If quarks are added, the flavor singlet coupling aGG˜can be split into quark mass terms with /H9251u/H11008x/mu,/H9251d/H11008x/md,/H9251s/H11008x/ms, etc., as if the quarks are not integrated over, muu¯LuRei/H9251u/H9258 +mdd¯LdRei/H9251d/H9258+¯, which shows that the flavor singlet coupling is of order O/H20849a/Fa/H20850. Then, even below the chiral symmetry breaking scale, we have the PQ charges proportional to 1/ mq. With this definition of quark charges, the axion mass comes from integrating out GG˜, and is proportional to /H9251u+/H9251d+/H9251swhich is /H11011mumdms//H20851ms/H20849mu+md/H20850+mumd/H20852first shown for the PQWW axion /H20849Baluni, 1979 /H20850. This is true even in the heavy quark KSVZ-type axion models. Even if the lightquarks do not have the same PQ charge as in some vari-ant axion models /H20849Krauss and Wilczek, 1986 ;Peccei, Wu, and Yanagida, 1986 ;Bardeen, Peccei, and Yanagida, 1987 ;Kim and Lee, 1989 ;Hindmarsh and Moulatsiotis, 1997 /H20850, the axion mass has the same final form due to the reparametrization invariance, which will be shown be-low. As a result the axion mass formula we write belowis quite general. However, there was an assumption in this statement: /H9257/H11032was integrated out. So it is necessary to include /H9257/H11032to obtain a more accurate axion mass. The light mesonsand axion interactions must appear from those of Fig. 9. In this framework, however, the flavor singlet conditionmust be invoked as a constraint. /H20849This flavor singlet con- dition is the anomaly matching without /H9257/H11032./H20850Along the way, we would like to see how the /H20881mdependence arises below the chiral symmetry breaking scale in the KSVZmodel. In the presence of vectorlike heavy quarks, the heavy fields are integrated out; their sole effect is encoded inthe low-energy effective theory as nonrenormalizable couplings suppressed by F a, e.g., in the anomalous c3-type couplings with the SM gauge bosons. It is as- sumed that the heavy quark does not condense, sincethe QCD coupling is very small due to the asymptoticfreedom at the heavy-quark Compton wavelength scaleand there does not exist a strong force to attract heavyquarks. Below the heavy quark scale, there are no mass-less mesons composed of heavy quarks. Therefore, the general form of the axion interaction, Eq. /H2084919/H20850, is valid at low energy. First, the determinental interaction has thesame chiral symmetry behavior as that of the anomalyterm, and the anomaly term is removed in favor of the determinental interaction to include /H9257/H11032explicitly. Sec- ond, we choose the basis where the uand dquark masses are real. Since the strange quark mass is knownto be below the QCD scale, we must include the strangequark with real and positive mass also in the instanton××msΛ2 ×× muΛ2×× mdΛ2ei(c2+c3)θ−v3 −v3 −v3eic3θ −v3 ×× muΛ2 −v3ei(cu 2+c3)θ−v3 −v3×× mdΛ2ei(cd 2+c3)θ ××msΛ −v3 −v3ei(cs 2+c3)θ +O(m2Λ4v3) FIG. 9. /H20849Color online /H20850The ’t Hooft determinental interaction. /L50098denotes the quark condensation and /H11003denotes the insertion of the current quark mass. The diagram highlighted predomi-nantly contributes to the /H9257/H11032mass, and O/H20849mumd/H20850is neglected.568 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010interaction. For simplicity, /H92660and/H9257/H11032, arising from quark condensations u¯uand d¯dwith decay constants f/H9266and f/H9257/H11032/H20849/H11015f/H9266/H20850/H20849Gell-Mann, Oakes, and Renner, 1968 /H20850, are considered explicitly but with the /H9257meson frozen. The effects of heavy quarks are included in the c3term. If we keep c1, the kinetic mixing of mesons and axion is present, due to the PCAC relation /H208550/H20841J5i/H9262/H20849x/H20850/H20841meson j/H20849k/H20850/H20856 =−ik/H9262fi2e−ik·x/H9254ijwhere J5i/H9262/H11011q¯/H9253/H9262/H92535Tiq. This would modify the axion mass, and hence it is easiest to calculate theaxion mass by choosing the reparametrization param- eter /H9251such that c1=0. In this basis, denoting /H92660,/H9257/H11032, and ain terms of dimensionless fields, /H9258/H9266=/H92660/f/H9266,/H9258/H9257/H11032 =/H9257/H11032/f/H9257/H11032,/H9258=a/Fa, we obtain the following effective inter- action below the chiral symmetry breaking scale: L=−mu/H20855u¯LuR/H20856ei/H20851/H20849/H9258/H9266+/H9258/H9257/H11032/H20850+c2u/H9258/H20852 −md/H20855d¯LdR/H20856ei/H20851/H20849−/H9258/H9266+/H9258/H9257/H11032/H20850+c2d/H9258/H20852+ H.c. + Ldet, /H2084936/H20850 where Ldetis given in Eq. /H2084920/H20850, Ldet=/H20849−1/H20850NfK−5/H20849/H20855u¯LuR/H20856/H20855d¯LdR/H20856/H20855s¯LsR/H20856ei/H208492/H9258/H9257/H11032−c3/H9258/H20850+¯ + flavor singlet constraint /H20850+ H.c., /H2084937/H20850 and Khas the mass dimension arising from QCD instan- ton physics. The above form is consistent with the anomaly /H2084932/H20850with c1=0. Note that the log det form in the effective Lagrangian was used by Veneziano /H208491979 /H20850; Witten /H208491979 ,1980 /H20850;Di Vecchia and Veneziano /H208491980 /H20850; and Di Vecchia et al. /H208491981 /H20850from the 1/ Ncexpansion consideration, but we use Eq. /H2084937/H20850because of its simplic- ity in the diagrammatic expansion. The sign of the firstdiagram inside the box in Fig. 9is determined to be negative without the weak CPviolation /H20849Vafa and Wit- ten, 1984 /H20850. The QCD vacuum with the flavor indepen- dence of light quarks without the determinental interac- tion chooses m q/H20855q¯q/H20856=−/H20841mq/H20841v3and we choose the sign of all quark masses to be positive so that /H20855q¯q/H20856=/H20855u¯u/H20856=/H20855d¯d/H20856 =−v3/H20849Dashen, 1971 ;Langacker and Pagels, 1973 ,1979 ; Gasser and Leutwyler, 1982 /H20850. Equation /H2084937/H20850is the instan- ton interaction of Fig. 9, which gives /H90114,mu/H90113,md/H90113,... by many ways of closing quark lines, shown in Fig. 9, but here one must invoke the flavor singlet constraint . The dominant term is the second diagram highlighted, whichis flavor singlet and is the main source for the /H9257/H11032mass. Now we restrict ourselves to the two-flavor case. For the axion, the key diagrams are those in the second lineof Fig. 9. If there is more than one QCD axion, then the O/H20849m umd/H20850diagram will be important at the next-level ax- ion mass. Integration over the instanton size includeslarge instantons, covering the chiral-symmetry-breakingrange where mesons appear as dynamical degrees,where we invoke the flavor singlet constraint. The effec- tive interaction Hamiltonian of /H9258/H9266,/H9258/H9257/H11032, and/H9258=a/fScan be written, using the reparametrization invariance /H2084921/H20850 with Nf=3 and/H9257fixed, as /H20851Huang /H208491993 /H20850and Kim and Kim /H208492006 /H20850/H20852 −V=muv3cos/H20849/H9258/H9266+/H9258/H9257/H11032/H20850+mdv3cos/H20849−/H9258/H9266+/H9258/H9257/H11032/H20850 +v9 K5cos/H208512/H9258/H9257/H11032−/H20849c2u+c2d+c3/H20850/H9258/H20852 +mu/H9011u2v6 K5cos/H20851−/H9258/H9266+/H9258/H9257/H11032−/H20849c2u+c2d+c3/H20850/H9258/H20852 +md/H9011d2v6 K5cos/H20851/H9258/H9266+/H9258/H9257/H11032−/H20849c2u+c2d+c3/H20850/H9258/H20852, /H2084938/H20850 where/H9011uand/H9011dare parameters describing the result of the Feynman and instanton size integrations. The /H20849−1/H20850Nf term is canceled by the fermion loop or /H20849−v/H20850factors. If mu=md,/H9011uand/H9011dare equal. For mu/HS11005md,/H9011uand/H9011d must be different. The instanton interaction is flavor in- dependent, which should be respected in the interaction /H2084938/H20850. The muandmdlinear terms from the determinental interaction should be flavor independent, i.e., mu/H9011u2 +md/H9011d2=flavor independent. Since it vanishes if one quark is massless, it must be a function of mumd. Thus, the instanton size integration with current quark masses must give mu/H9011u2+md/H9011d2=2mumdL˜2//H20849mu+md/H20850, which vanishes if any quark is massless. This is because the original gluon anomaly term /H20853GG˜/H20854does not distinguish flavors, and the smallness of the current quark massesenables us to expand the ’t Hooft determinental interac-tion in terms of powers of the current quark masses. Then, the 3 /H110033 mass matrix M 2ofa,/H9257/H11032, and/H92660, taking into account the chiral symmetry breaking and the solu-tion of the U /H208491/H20850problem, is given as Ma,/H9257/H11032,/H926602=/H20898c2/H20851/H9011/H9257/H110324+2/H9262/H9011inst3/H20852/F2−2c/H20851/H9011/H9257/H110324+/H9262/H9011inst3/H20852/f/H11032F 0 −2c/H20851/H9011/H9257/H110324+/H9262/H9011inst3/H20852/f/H11032F/H208514/H9011/H9257/H110324+2/H9262/H9011inst3+m+v3/H20852/f/H110322−m−v3/ff/H11032 0 −m−v3/ff/H11032 /H20849m+v3+2/H9262/H9011inst3/H20850/f2/H20899, /H2084939/H20850 where c=c2u+c2d+c3,F=fS,f=f/H9266,f/H11032=f/H9257/H11032,/H9011/H9257/H110324=v6/K/H110322,/H9011inst3=L˜2v3/K2,m+=mu+md,m−=md−mu, and569 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010/H9262=mumd /H20849mu+md/H20850. /H2084940/H20850 Certainly, Eq. /H2084939/H20850realizes the solution of the U /H208491/H20850prob- lem due to the /H9011/H9257/H110324term in the /H2084922/H20850component. In the limit f/F,f/H11032/F/H112701, we obtain m/H926602/H11229m+v3+2/H9262/H9011inst3 f/H92662, /H2084941/H20850 m/H9257/H110322/H112294/H9011/H9257/H110324+m+v3+2/H9262/H9011inst3 f/H9257/H110322, /H2084942/H20850 ma2/H11229c2 F2Z /H208491+Z/H208502f/H92662m/H926602/H208491+/H9004/H20850, /H2084943/H20850 where /H9004=m−2 m+/H9011inst3/H20849m+v3+/H9262/H9011inst3/H20850 m/H926604f/H92664. /H2084944/H20850 In this form, the /H9266mass has the standard m+v3plus the instanton contribution to the light quark mass /H20849Kaplan and Manohar, 1986 ;Choi, Kim, and Sze, 1988 /H20850. From Eqs. /H2084941/H20850and /H2084942/H20850, we estimate the parameter /H9011/H9257/H110324which is the source of the solution of the U /H208491/H20850problem: /H9011/H9257/H110324 =/H20849f/H9257/H110322m/H9257/H110322−f/H92662m/H92662/H20850/4/H11015/H20849202 MeV /H208504with f/H9257/H11032/H1122986 MeV and f/H9266/H1122993 MeV. In any axion model, this form is valid with /H20841c/H20841=NDW. Using the standard definition on the axion de- cay constant Fa=F/c, we obtain ma2/H11229Z /H208491+Z/H208502f/H92662m/H92662 Fa2/H208491+/H9004/H20850. /H2084945/H20850 Even though the instanton diagrams of Fig. 9contain the summation of linear quark mass diagrams, the diago-nalization process with mesons signals the predominantcontribution of the lightest quark. The flavor singlet con-dition discussed before chooses the following linearquark mass dependence: /H9262=/H208731 mu+1 md+¯/H20874−1 . /H2084946/H20850 Neglecting instanton contribution to the current quark masses, we obtain ma/H110150.60 eV /H20849107GeV/ Fa/H20850, for the mass ratio Z/H112290.48 as summarized by Manohar and Sachrajda /H208492008 /H20850. An earlier frequently cited Zis 5/9 /H20849Weinberg, 1977 ;Gasser and Leutwyler, 1982 /H20850. The cor- rect axion mass has to include the current quark masschange due to instantons. However, the resulting esti- mate of/H9004turns out to be small. 2. Comparison with old calculations Now we comment on the old anomaly matching con- dition. If any quark mass is zero, there exists an exact symmetry a→a+const, i.e., the axion is massless, above the chiral-symmetry-breaking scale. Below the chiral-symmetry-breaking scale, it is likely that this condition issatisfied. We denote the original current as J PQ/H9262. This cur- rent is anomalous above the chiral-symmetry-breaking scale, /H11509/H9262JPQ/H9262=/H20849NQ/32/H92662/H20850G/H9262/H9263aG˜a/H9262/H9263, where NQis the num- ber of heavy quarks with /H9003=1/2. Below the chiral- symmetry-breaking scale, we considered two pseudo- scalar mesons which have anomalous couplings: /H9257/H11032and a. The global anomaly matching condition will work if there is no chiral symmetry breaking /H20849’t Hooft, 1979 /H20850. For chiral symmetry breaking, there are no massless fer-mions and we consider only color singlet mesons belowthe chiral symmetry breaking scale. Thus, the bosoniccurrent must be anomaly-free after all heavy fields in- cluding /H9257/H11032are integrated out, i.e., we consider an anomaly-free current Ja/H9262instead of JPQ/H9262below the chiral- symmetry-breaking scale /H20849Kim, 1987 /H20850, Ja/H9262=JPQ/H9262−NQ 2/H208491+Z/H20850/H20849u¯/H9253/H9262/H92535u+Zd¯/H9253/H9262/H92535d/H20850, /H2084947/H20850 where the divergence of the second current gives a sin- glet pseudoscalar density so that the axion does not mix with/H92660. Equation /H2084945/H20850with/H9004shows that the finite /H9257/H11032 mass enters into the a-/H9257/H11032mixing. 3. Mesons without axions Even if there is no axion, we can diagonalize the mass matrix. If mu=0, one starts with an exact up-quark chiral transformation, which leads to a Goldstone boson /H9258in the vacuum, /H20855u¯u/H20856/HS110050. This Goldstone boson couples to a neutron through /H20849c1/H11509/H9262/H9258/H20850n¯/H9253/H9262/H92535n. In reality, /H9257/H11032obtains mass by the anomaly, and the symmetry remains unbro- ken: it is the phase symmetry of /H20855u¯u/H20856. Therefore, any violation of the shift symmetry must be such that it goes away in the limit /H9262→0; this is Dashen’s theorem /H20849Dashen, 1971 /H20850. Thus, from Eq. /H2084938/H20850we obtain the VEVs of/H9257/H11032and/H92660for a small /H9258¯, /H20855/H9257/H11032/H20856 f/H11032/H11229−/H9258¯ 2/H208491+Z/H20850/H9262v3 /H9011/H9257/H110324, /H2084948/H20850 /H20855/H92660/H20856 f/H11229/H9258¯/H208491+Z/H20850/H9262 m+. The VEVs of /H9257/H11032and/H92660are vanishing if /H9258¯=0 or any quark mass is zero. In addition, we can estimate the /H9257/H11032 properties from the interaction /H20849v9/K5/H20850cos/H208492/H9257/H11032/f/H9257/H11032/H20850, where f/H9257/H11032is the/H9257/H11032decay constant and Khas a mass dimension. This comes from the diagram of Fig. 9. Com- paring/H92660→2/H9253and/H9257/H11032→2/H9253decay widths, 7.74 eV and 4.3 keV, respectively /H20849Amsler et al. , 2008 /H20850, we obtain f/H9257/H110322=/H208494/3 /H20850/H20849m/H9257/H110323/m/H92663/H20850/H20851/H9003/H20849/H92660→2/H9253/H20850//H9003/H20849/H9257/H11032→2/H9253/H20850/H20852f/H92662,o r f/H9257/H11032 /H1101586 MeV. Fitting to the /H9257/H11032mass, we obtain K =/H20849v9/f/H9257/H110322m/H9257/H110322/H208501/5=240 MeV. 4. The /H9258=0 vacuum with axions We have shown above that the Lagrangian /H2084938/H20850 chooses/H9258=0 in CP-conserving theories if /H9258/H9266=0 and570 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010/H9258/H9257/H11032=0, which is determined by QCD dynamics. How- ever, if CPsymmetry is broken, the vacuum value of /H9258 is shifted from the /H9258=0 value by the presence of any linear term of /H9258/H9266,/H9258/H9257/H11032, or/and/H9258. The meson potential is invariant under CP symmetry with CP/H20849/H9266/H20850=CP/H20849/H9257/H11032/H20850 =CP/H20849a/H20850=−1. Such linear terms are generated by consid- ertion of CP-violating phases and chirality-flipping /H20849L↔R/H20850insertions. As a result linear terms of aare gen- erated by combining the ’t Hooft determinental interac- tion and CP-violating weak interactions. Linear terms of /H9266and/H9257/H11032can also be generated by considering the weak interactions alone without the determinental interaction,but the conditions of the flavor singlet, chirality-flipping /H20849L↔R/H20850, and CP-violating effects do not occur at the one-loop level. In the SM with the Kobayashi-Maskawa CP violation, Ellis and Gaillard /H208491979 /H20850showed that a finite correction occurs at the fourth order, O/H20849 /H92512/H20850, lead- ing to a small NEDM, but infinite corrections occur from O/H20849/H92517/H20850. These can give rise to a linear term of /H9266.I n the SM, the pioneering calculation with axions has been performed in chiral perturbation theory to obtain /H9258 /H3335510−17/H20849Georgi, Kaplan, and Randall, 1986 ;Georgi and Randall, 1986 /H20850. The estimated /H9258, however, is far below the current experimental limit of 10−11. C. Axion couplings The axion interactions are given in Eq. /H2084919/H20850which are shown in Fig. 10where we have not drawn the aWW˜and aZZ˜diagrams which are orthogonal to the a/H9253/H9253˜. The dia- grams of Fig. 10are complete for the low-energy axion phenomenology, where the suppression factor 1/ Faby the axion decay constant is explicitly shown. 1. Axion-hadron coupling When we discuss axion-hadron interactions, which are relevant low-energy laboratory experiments and physicsat the core of supernovae, we must integrate out gluonfields. Technically, this is achieved using the reparametri- zation invariance to remove the c 3/H9258GG˜coupling. If we keep the c3coupling, we must consider the axion-gluon-gluon interactions also, which are hard to treat accu- rately at face value but must be the same as in the c3 =0 basis. In this way, the quark interactions are changed from the original values as follows: c1→c¯1=c1+1 2c3, c2→c¯2=c2+c3, /H2084949/H20850 c3→c¯3=c3−c3=0 . In the notation with overbars, there exist only c¯1and c¯2. We discuss one family without separating c1,2into c1,2u,dfirst for an illustration, and then we discuss the cases with c1,2u,dand write down formulas for three fami- lies. We define the initial parameters c1,c2, and c3to- gether with the definition of the vacuum angle /H92580 /H11013/H9258QCD. In principle, the initial vacuum angle can be a free parameter. Here the vacuum angle /H9258QCD is defined such that c1=0. Picking up the axion-depen- dent chiral rotation charge defined below the chiralsymmetry breaking scale Eq. /H2084947/H20850, the chiral quarks in the chiral perturbation theory are transformed as q L →exp /H20849iQA/H9258/H20850qL,qR→exp /H20849−iQA/H9258/H20850qR, where QA=1 2M−1 TrM−1,M−1= diag/H208731 mu,1 md/H20874. /H2084950/H20850 The derivative interactions of the axion are obtained in this way /H20849Kaplan, 1985 ;Georgi, Kaplan, and Randall, 1986 /H20850. For the KSVZ axion, we have c1=c2=0 and c3=1, and the coefficient of the gluon anomaly term is a/Fa +/H9258QCD. Hence, redefining the axion as a+Fa/H9258QCD,w e obtain3 KSVZ axion /H20849c1=0 ,c2=0/H20850: c¯1=1 2c3=1 2, /H2084951/H20850 c¯2=c2+c3=1 . Here c¯2must be split according to the flavor singlet con- dition into c¯2u+c¯2d, Eq. /H2084947/H20850,o r /H2084950/H20850. For the DFSZ and PQWW axions, c1=0,c2/HS110050, and c3=0. If a nonvanishing /H9258QCD is introduced here, we have, using the reparametrization invariance /H2084921/H20850,c1/H11032 =−c2/2,c2/H11032=0, and c3/H11032=c2. Then the coefficient of the gluon anomaly term is c2/H20849a/fS/H20850+/H9258QCD, and hence, rede- fining the axion as a+/H20849fS/c2/H20850/H9258QCD and going back to the c¯3=0 basis, we obtain for one family, DFSZ and PQWW axions: c¯1=1 2/H20849−c2+c¯2/H20850, /H2084952/H20850 3The sign convention is stated below.cq 1γµγ51 Faaq qcq 2iγ51 Faaq qc31 Faa GG caγγ1 Faa γγ c/lscriptiγ51 Faa/lscript /lscript FIG. 10. The Feynman diagrams of axion couplings. Gand/H9253 are the gluon and photon, respectively. c3and ca/H9253/H9253couplings are anomalous.571 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010c¯2/HS110050,c¯3=0 . Again, c3/H11032must be split according to the flavor singlet condition to c¯2u+c¯2daccording to the anomaly matching condition, Eq. /H2084947/H20850. When the heavy /H9268field and heavy quark fields are integrated out, the massless /H20849at this level /H20850degree a =Fa/H9258which appears from the phase of the singlet field /H9268=/H20849/H20855/H9268/H20856+/H9267//H208812/H20850ei/H9258appears in the effective low-energy La- grangian. If there are multiple SM singlets Sicarrying PQ charges and VEVs, then the axion component is a=1 Va/H20858 i/H9003iVia/H20849Si/H20850,Va=/H20873/H20858 i/H9003i2Vi2/H208741/2, /H2084953/H20850 where a/H20849Si/H20850is the phase field of Si. The PQ charges are defined such that the smallest nonzero absolute value /H20849s/H20850 of the PQ charges is 1 so that every scalar field returns to its original value after a 2 /H9266shift of its phase. We now discuss axion couplings after integrating out the heavyfields. In the KSVZ model c 3is calculated using the triangle diagram of heavy quarks for the global anomaly. The domain wall number NDWisNDW=Tr/H9003/H20849QL/H20850l/H20849QL/H20850, with Fa=Va/NDWwhere/H9003/H20849QL/H20850/H20849defined as QL→ei/H9003/H20849QL/H20850/H9258QL under a→a+Fa/H9258/H20850is the PQ charge and l/H20849QL/H20850is the in- dex of SU /H208493/H20850crepresentation. Every field is represented in terms of left-handed fields, and the PQ charges are defined such that the SM singlet /H9268coupling to heavy quarks carries one unit of the PQ charge. If the lightquarks also carry the PQ charge, then Eq. /H2084922/H20850gives N DW, which belongs to the generic very light axion model discussed below. The anomaly calculation gives the one-loop coupling /H20849NDWa/Va/H20850/H20853GG˜/H20854, but since the vacuum angle /H9258or axion is given by the coefficient of /H20853GG˜/H20854,Fais defined by dividing Vaof Eq. /H2084953/H20850byNDW and hence c3= ±1 where the sign coincides with that of Tr/H9003/H20849QL/H20850l/H20849QL/H20850. As a convention, choose it to be c3=+1, which is choosing the effective PQ charges of heavy quarks to be positive. Transferring c3toc2, we split c3 =c2u+c2dusing the PQ charges of Eq. /H2084950/H20850, KSVZ axion: c¯1u,d=1 2c¯2u,d, /H2084954/H20850 c¯2u=1 1+Z,c¯2d=Z 1+Z, In the DFSZ model, c2uand c2dare calculated by transferring the phase of /H9268toHuand Hdwith the PQ symmetry such that /H20855Hu0/H20856=/H208812v2ei/H9003ua/V/H9268and /H20855Hd0/H20856 =/H208812v1ei/H9003da/V/H9268ifHu*Hd*/H92682defines the PQ charge of /H9268in terms of PQ charges /H9003uand/H9003dofHuand Hd. Here a =V/H9268/H9258is not the mass eigenstate and instead of V/H9268the mass eigenstate a˜uses the decay constant Fa=/H20851/H20849/H9003u +/H9003d/H208502V/H92682+/H9003u2vu2+/H9003d2vd2/H208521/2/H11229/H20849/H9003u+/H9003d/H20850V/H9268for V/H9268/H11271vu,vd, and the axion component a˜=/H20851/H20849/H9003u+/H9003d/H20850V/H9268a +/H9003uvEWa/H20849Hu/H20850+/H9003dvEWa/H20849Hd/H20850/H20852/Fa/H11229a, and Fa=V/H9268/NDW.In the DFSZ model they are given by Carena and Peccei /H208491989 /H20850:c2u=/H20841vd/H208412/vEW2,c2d=/H20841vu/H208412/vEW2, and c1u,d=c3=0. Us- ing the reparametrization invariance Eq. /H2084921/H20850,w ec a n use c1/H11032u=−c2u/2,c1/H11032u=−c2d/2,c2/H11032u=c2/H11032d=0, and c3/H11032=c2/H11032u+c2/H11032d =1. Removing c3/H11032according to the flavor singlet condi- tion, we obtain for one family, DFSZ axion for one family: c¯1u=−/H20841vd/H208412 2vEW2+1 2c¯2u,c¯1d=−/H20841vu/H208412 2vEW2+1 2c¯2d, /H2084955/H20850 c¯2u=1 1+Z,c¯2d=Z 1+Z, where vu=/H20841/H20855/H208812Hu0/H20856/H20841,vd=/H20841/H20855/H208812Hd0/H20856/H20841,vEW=/H20849vu2+vd2/H208501/2. The PQ charges c2u=/H20841vd/H208412/vEW2and c2d=/H20841vu/H208412/vEW2ofHuand Hdare obtained by considering the orthogonal compo- nent to the longitudinal mode of the Zboson. Remem- ber that the signs of c2u,dare chosen from the convention that the PQ charges of Hu,dare positive. This result is for one family. If we have Ngfamilies, we can calculate the couplings just below the electroweak scale where all quarks obtain masses. Thus, we obtain for three families c2u=c2c=c2t =/H20841vd/H208412/vEW2and c2d=c2s=c2b=/H20841vu/H208412/vEW2. Using the rep- arametrization invariance, we can calculate c1/H11032,c2/H11032, and c3/H11032, just above 1 GeV: c2/H11032=0, c1i/H11032=−1 2c2i, and c3/H11032=Ng/H20849/H20858ic2i/H20850 =Ng. Then we integrate out the heavy quarks c,b, and t to obtain the effective couplings just above 1 GeV; this does not introduce any new c2terms. Now there are three light quarks u,d, and sfor which we use the rep- arametrization invariance to remove the c3/H11032term such that the isosinglet condition is satisfied; /H11509/H9262J/H9262ais anomaly- free where J/H9262a=J/H9262PQ−/H9251uu¯/H9253/H9262/H92535u−/H9251dd¯/H9253/H9262/H92535d−/H9251ss¯/H9253/H9262/H92535sand /H11509/H9262J/H9262PQ=Ng/H20853GG˜/H20854. Thus,/H9251u+/H9251d+/H9251s=Ngis satisfied and the SU /H208493/H20850flavor singlet condition of /H11509/H9262J/H9262adetermines /H9251u=m/H9251 mu,/H9251d=m/H9251 md,/H9251s=m/H9251 ms, /H2084956/H20850 with m/H9251=Ngmumdms mumd+mums+mdms/H11229Ng/H20849mu+md/H20850Z /H208491+Z/H208502. Therefore, removing c3/H11032by the reparametrization invari- ance, we obtain DFSZ axion for Ngfamilies: c¯2u=1 1+ZNg,c¯2d=Z 1+ZNg, /H2084957/H20850 c¯1u=1 2c¯2u−vd2 vEW2,c¯1d=1 2c¯2d−vu2 vEW2. /H2084958/H20850 If heavy quarks and also Hu,dcarry PQ charges, we must consider all these. If one SM singlet /H9268houses the axion, then we obtain572 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010General very light axion: c¯2u=1 1+Z/H208491±Ng/H20850, /H2084959/H20850 c¯2d=Z 1+Z/H208491±Ng/H20850, /H2084960/H20850 c¯1u=1 2/H208491+Z/H20850/H208491±Ng/H20850/H11007/H20841vd/H208412 2vEW2/H9254Hu, /H2084961/H20850 c¯1d=Z 2/H208491+Z/H20850/H208491±Ng/H20850/H11007/H20841vu/H208412 2vEW2/H9254Hd, /H2084962/H20850 where the PQ charges /H20849/H11001or/H11002/H20850ofHuand Hddetermine the sign /H20849/H11002or/H11001/H20850in front of the DFSZ component and /H9254H=1 or 0 if the corresponding Higgs doublets carry the PQ charges or not. For the MI axion from superstringtheory, which is a hadronic axion, in principle there can exist an additional contribution to c 1as pointed out in Sec. VI.F.1 . If there are no heavy degrees carrying the PQ charges above the electroweak scale, then c2in the so-called PQWW model is given by the PQ charges of Huand Hd, PQWW axion: Same as Eqs. /H2084957/H20850and /H2084958/H20850. /H2084963/H20850 All models have c¯1and c¯2. For the original c2term, different models give different values; for example,some variant axion models /H20849Krauss and Wilczek, 1986 ; Bardeen, Peccei, and Yanagida, 1987 ;Kim and Lee, 1989 ;Hindmarsh, and Monlatshotis, 1997 /H20850have different c 2’s from those of the PQWW axion. For astrophysical application, we must keep both c¯1and c¯2. The c¯1and c¯2 terms give the axial-vector and pseudoscalar couplings, respectively. The axion operator in the flavor SU /H208493/H20850 space can be written as /H20849c¯1,2u−c¯1,2d/H20850F3+c¯1,2u+c¯1,2d /H208813F8+c¯1,2u+c¯1,2d 61, /H2084964/H20850 where F3and F8//H208813 are the third component of the iso- spin and the hypercharge operators, respectively, and 1is the identity operator. The derivative couplings with nucleons and mesons below the chiral symmetry breaking are the defined as L AVc¯1=/H11509/H9262a Fa/H20875Cappp¯/H9253/H9262/H92535p+Cannn¯/H9253/H9262/H92535n +iCa/H9266NN/H20873/H9266+ f/H9266p¯/H9253/H9262n−/H9266− f/H9266n¯/H9253/H9262p/H20874/H20876, /H2084965/H20850 La/H9266/H9266/H9266c¯1=Ca/H9266/H9266/H9266/H11509/H9262a Faf/H9266/H20849/H92660/H9266+/H11509/H9262/H9266−+/H92660/H9266−/H11509/H9262/H9266+ −2/H9266+/H9266−/H11509/H9262/H92660/H20850, /H2084966/H20850 whereCapp=c¯1uF+c¯1u−2c¯1d 3D+c¯1u+c¯1d 6S, /H2084967/H20850 /H2084967/H20850 Cann=c¯1dF+c¯1d−2c¯1u 3D+c¯1u+c¯1d 6S, Ca/H9266NN=c¯1u−c¯1d /H208812,Ca/H9266/H9266/H9266=2/H20849c¯1u−c¯1d/H20850 3. /H2084968/H20850 Here the axial-vector coupling parameters of the nucleon octet are given by F=0.47, D=0.81, and S /H112290.13±0.2 /H20849Amsler et al. , 2008 /H20850. For example, for the hadronic-axion couplings we obtain the results given byKaplan /H208491985 /H20850and Chang and Choi /H208491993 /H20850, C app=1 2/H208491+Z/H20850F+1−2 Z 6/H208491+Z/H20850D+1 6S, Cann=Z 2/H208491+Z/H20850F+Z−2 6/H208491+Z/H20850D+1 6S, /H2084969/H20850 Ca/H9266NN=1−Z 2/H208812/H208491+Z/H20850,Ca/H9266/H9266/H9266=1−Z 3/H208491+Z/H20850. For the DFSZ axion, there exist additional contributions from the extra terms in Eqs. /H2084957/H20850and /H2084958/H20850. Similar expressions might be attempted for the pseu- doscalar couplings in terms of c¯2u,dand the pseudoscalar coefficients F/H11032,D/H11032, and S/H11032. But for the axion current, corresponding to J/H9262a, there does not exist an anomaly as discussed in Eq. /H2084947/H20850and we do not write down the ax- ion pseudoscalar couplings. The anomaly carried by ax-ions above the chiral symmetry breaking scale is left over to /H9257/H11032below the chiral symmetry breaking scale and hence these pseudoscalar couplings are for the /H9257/H11032me- son. The axial vector current of /H9257/H11032to the nucleon octet N=q/H20002q/H20002qis J/H9262/H9257/H11032=f/H9257/H11032/H11509/H9262/H9257/H11032+gN5N¯/H9253/H9262/H92535T0N, /H2084970/H20850 where T0is properly normalized, Tr T02=1 2orT0 =1//H208812Nf, and gN5is determined by strong interaction dy- namics. The original global symmetry breaking term /H2084920/H20850 is transferred to N¯LNRei/H92511/H11032/H9257/H11032/f/H9257/H11032which is actually the nucleon mass term, /H9004L=−mNN¯LNRei/H92511/H11032/H9257/H11032/f/H9257/H11032+ H.c. /H2084971/H20850 For example, the SU /H208496/H20850wave function of a spin-up neu- tron is /H20841n↑/H20856=1 6/H208812/H208414d↑u↓d↑−2d↓u↑d↑−2d↑u↑d↓−2u↑d↓d↑ +4u↓d↑d↑−2u↑d↑d↓−2d↑d↓u↑−2d↓d↑u↑ +4d↑d↑u↓/H20856, /H2084972/H20850 where the quarks are now interpreted as constituent quarks below the chiral symmetry breaking. At low en-ergy, this is the only relevant symmetry for consider-573 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010ation. The octet charge /H92511/H11032is determined by strong inter- action dynamics. The ducaplet has a different U /H208491/H20850 charge/H92511/H11033. Two anomaly matching conditions, PQ-B-B and PQ-Q em-Qem, may be used but do not give useful information because of many form factors. So, the PQcharges of the current quarks being transferred to theconstituent quarks in the octet with a multiplcation fac- torg N5, we obtain the PQ charge of the neutron as the PQ charge of one constituent quark. Thus, N¯LNRhas the phase /H9251/H11032=2gN5/H208812Nf/Nf. If we guess that gN5is similar to the octet form factor gA/H112290.75 /H20849Georgi, 1984 ,p .1 0 0 /H20850, /H92511/H11032is estimated as 1.22. 2. Axion-photon-photon coupling As we calculated the c3coupling for the KSVZ axion, we can calculate the axion-photon-photon coupling bysubstituting the gluon lines by photon lines and thequark triangles by charged fermion triangles. Since weare interested in low-temperature experiments, we con-sider the energy scale below the electron mass. There- fore, considering V a=NDWFa,ca/H9253/H92530calculated from the PQ charges of charged fermions becomes ca/H9253/H92530=Tr/H9003/H20849QL/H20850Qem2 NDW. /H2084973/H20850 Below the QCD chiral-symmetry-breaking scale, we chiral-transform light quarks to obtain La/H9253/H9253=ca/H9253/H9253e2 32/H92662FaaF/H9262/H9263emF˜em/H9262/H9263, /H2084974/H20850 where ca/H9253/H9253/H11229ca/H9253/H92530−c/H9273SB, /H2084975/H20850 where the chiral-symmetry-breaking effect, including the strange quark mass effect, is c/H9273SB=2 3/H208494 + 1.05 Z/H20850 1 + 1.05 Z=/H208511.762,2.260 /H20852/H20849 76/H20850 for a 20% allowance from the tree level chiral perturba- tion theory estimation /H20849Kaplan and Manohar, 1986 /H20850. For illustration, we take c/H9273SB/H110151.98 for Z/H112290.5 /H20849Manohar and Sachrajda, 2008 /H20850. In the KSVZ model, ca/H9253/H92530is determined by the PQ charge-carrying heavy fermions. If there were only one neutral quark for this, then ca/H9253/H92530would be zero. If there is only one PQ charge-carrying heavy quark with the elec- tromagnetic charge Qem, then ca/H9253/H92530=Qem2. But, in realistic models from a fundamental theory it is more likely thatthere exist many PQ charge-carrying quarks, and thecoupling given for one PQ charge-carrying heavy quarkis presented just as an illustration. In the DFSZ model, we consider only light quarks and leptons. The PQ charges of H uand Hddetermine the PQ charges of uand dquarks. For the PQ charge of e, we have two possibilities: Hdgives mass to eand the PQ charge of eis the same as that of d,o rHugives mass to eand the PQ charge of eis opposite to that of u,ca/H9253/H92530=−2vd2 vEW2/H208732 3/H208742 /H110033−2vu2 vEW2/H20875/H20873−1 3/H208742 /H110033+ /H20849−1/H208502/H20876 =−8 3, electron mass by Hd, /H2084977/H20850 ca/H9253/H92530=−2vd2 vEW2/H20875/H208732 3/H208742 /H110033− /H20849−1/H208502/H20876−2vu2 vEW2/H20873−1 3/H208742 /H110033 =−2 3, electron mass by Hu†, /H2084978/H20850 where the PQ charges of Hu,dwere chosen to be positive before. In applying Eq. /H2084975/H20850, we must choose the PQ charges of light quarks to be positive and hence the signsof Eqs. /H2084977/H20850and /H2084978/H20850must be reversed. For the PQWW axion, the coupling is the same as those of Eqs. /H2084977/H20850and /H2084978/H20850with positive signs. The KSVZ and DFSZ axion models arise in several different ways, for which the axion-photon-photon cou-pling has been tabulated by Kim /H208491998 /H20850. In Table I,w e list axion-photon-photon couplings for several very lightaxion models. For a general light axion, the axion-photon-photon coupling depends on the ultraviolet completion of the theory. If the axion mass is lighter than 2 m e, its lifetime is /H9270/H20849a→2/H9253/H20850=28/H92663 ca/H9253/H92532/H9251em2Fa2 ma3/H112293.65/H110031024 ca/H9253/H92532/H20873eV ma/H208745 s /H112290.8/H11003107tU ca/H9253/H92532/H20873eV ma/H208745 , /H2084979/H20850 where Z/H112290.5 and the age of the Universe tU/H110154.35 /H110031017s. For ca/H9253/H9253=O/H208491/H20850, the axion with 24 eV mass has the lifetime tU/H20849Moroi and Murayama, 1998 ;Hannestad, Mirizzi, Raffelt, and Wong, 2008 /H20850. 3. Axion-lepton couplings The tree level axion-lepton /H20849l/H20850coupling arises in the DFSZ and PQWW axions where the lepton mass term difines the clof Fig. 10through the PQ charges of HdorTABLE I. ca/H9253/H9253in several field theoretic models. The left block is for the KSVZ and the right block is for the DFSZ. /H20849m,n/H20850in the KSVZ block denotes mcopies of Qem=2 3and ncopies of Qem=–1 3heavy quarks with the same PQ charge. In the DHSZ block x=tan/H9252=vu/vd. Qem ca/H9253/H9253 x one Higgs couples to ca/H9253/H9253 0 –1.95 any x /H20849dc,e/H20850 0.72 ±1 3–1.28 any x /H20849uc,e/H20850 –1.28 ±2 30.72 ±1 4.05 /H20849m,m/H20850 –0.28574 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010Hu. The removal of the c3term does not change the coupling cl, and hence we obtain the following tree-level couplings of the axion and lepton: DFSZ axion: mlvu2 NDWFavEW2l¯i/H92535la, lepton mass by Hd, /H2084980/H20850 mlvd2 NDWFavEW2l¯i/H92535la, lepton mass by Hu†, /H2084981/H20850 where the PQ charges of Hu,dare chosen to be positive. For the PQWW axion, just Fais replaced by vEW. For the KSVZ axion, the axion-lepton coupling occurs athigher order and is negligible in astrophysical applica-tions. For the generic very light axion, the couplingsgiven in Eqs. /H2084980/H20850and /H2084981/H20850are applicable. Even though the tree level coupling of the axion with an electron is absent in the KSVZ model, the axion- electron coupling is present at one loop through the c a/H9253/H9253 coupling /H20849Srednicki, 1985 /H20850 2.2/H1100310−15/H20873ma eV/H20874/H20875ca/H9253/H92530lnFa me−2 34+Z 1+Zln/H9011 me/H20876, /H2084982/H20850 where/H9011is the chiral symmetry breaking scale and NDW−1 must be multiplied in models with NDW/HS110051. On the other hand, the DFSZ axion coupling to the electron is 1.4/H1100310−11Xd/H208733 Ng/H20874/H20873ma eV/H20874, /H2084983/H20850 where Ng=NDW/2 is the number of families and Xd =sin2/H9252=vu2/vEW2for the case of Eq. /H2084980/H20850. D. Old laboratory bounds on Fa With the axion couplings discussed in Sec. III.C , one can estimate the axion production rates in various ex-periments. Null experimental results give the bounds onthe relevant axion couplings. These have been discussedin earlier reviews /H20849Kim, 1987 ;Cheng, 1988 ;Peccei, 1989 /H20850. These old laboratory bounds, immediately stud- ied after the proposal of the PQWW axion, basicallyrule out the PQWW axion, i.e., give an axion decay con- stant F agreater than O/H2084910 TeV /H20850, Fa/H11407104GeV /H20849old laboratory bound /H20850. /H2084984/H20850 IV . AXIONS FROM OUTER SPACE From Eq. /H2084979/H20850, we note that the axion lifetime is longer than tUforma/H1101124 eV, and this kind of axion is important in cosmology. For ma/H1135123 keV with ca/H9253/H9253=1, the axion lifetime is longer than 10 min, allowing solar- generated axions below this mass to reach Earth. Theseexamples illustrated the importance of studying low-mass axion effects in astrophysics and cosmology. The window for F aobtained from the astrophysical and cosmological constraints is given by0.5/H11003109/H11351Fa/H113512.5/H110031012GeV, /H2084985/H20850 where the upper bound is understood with an initial mis- alignment angle of order 1. A. Axions from stars In this section we present the key arguments leading to axion constraints from astrophysical sources. Axionshave very small masses and therefore can be emittedwithout important threshold effects from stars, in anal-ogy to neutrinos. The method to constrain axion modelsis basically the overall energy loss rate, whether usingthe individual stars /H20849e.g., Sun and SN1987A /H20850or the sta- tistical properties of stellar populations /H20849e.g., the stars in a globular cluster as a test population /H20850/H20849Kolb and Turner, 1990 ;Raffelt, 1996 /H20850. We may use the axion couplings to /H9253,p,n, and eto study the core evolution of a star. Simple bounds areobtained by comparing the energy loss rates by axionand by neutrino emission. Study of the evolutionary his-tory of a star by axion emission may give a strongerbound than the one obtained from the energy loss ratebut may not be as reliable. Since there are good reviewson axion astrophysics /H20849Raffelt, 1990a ,2008a ;Turner, 1990 ;Amsler et al. , 2008 /H20850, here we briefly comment on axion physics in stars /H20849Sun, low-mass red giants, super- novae /H20850to cite reliable F abound. With axion emission, the Sun consumes more fuel and needs an increased core temperature. From the Prima- koff process /H9253+Ze→a+Zein the hadronic axion mod- els, Schlattl, Weiss, and Raffelt /H208491999 /H20850gave the axion emission rate La/H112293.7/H1100310−2L/H17018with a 20% increase of the8B flux with the increased core temperature. The8B neutrino flux gives the best bound on the solar axion emission rate. The measured8B neutrino flux 4.94 /H11003106cm−2s−1/H20849Aharmim et al. , 2005 /H20850is consistent with the axion emission if La/H333550.04L/H17018/H20849Bahcall, Serenelli, and Basu, 2005 /H20850. This translates to an Fabound of Fa/ca/H9253/H9253/H333562.6/H11003106GeV for La/H333550.04L/H17018/H20849Schlattl, Weiss, and Raffelt, 1999 /H20850. For axion-electron coupling as in the DFSZ axion models, the axion emission from globular clusters gives a useful Fabound /H20849Raffelt and Dearborn, 1987 /H20850. Stars in a globular cluster are assumed to have identical Y/H20849helium fraction /H20850and metallicity fraction. The helium core be- fore ignition is degenerate and the bremsstrahlung emis-sion is very effective, whereas the Primakoff emission issuppressed by the large plasma frequency and the he- lium ignition does not give a useful F abound for the KSVZ axion. However, after helium ignition the coredegeneracy is lifted, the Primakoff effect becomes im-portant, and the consumption of helium fuel is acceler-ated by the axion energy loss, shortening the helium-burning lifetimes. Horizontal branch stars in severalglobular clusters confirm the expected helium-burninglifetimes, which agrees with the standard prediction and the axion losses should not exceed /H9255 a/H1102110 erg g−1s−1in the cores of horizontal branch stars /H20849Raffelt, 1990b ;575 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010Catelan, de Freista Pacheco, and Horvath, 1996 /H20850, which leads to Fa/ca/H9253/H9253/H333562/H11003107GeV, a factor of 10 improve- ment over the solar bound. Note that this globular clus-ter bound is for models with an appreciable axion-electron coupling. In the study of the axion emission in the small-mass red giants, the processes /H9253+Z→a+Z,e+Z→a+e+Z, and/H9253+e→a+ewere considered. The early studies were simple comparisons of the axion and neutrino emission/H20849Fukugita, Watamura, and Yoshimura, 1982a ,1982b ; Krauss, Moody, and Wilczek, 1984 /H20850. In the study of Dearborn, Schramm, and Steigman /H208491986 /H20850, it is summa- rized as F a/H333562.1/H11003107ca/H9253/H9253GeV if the Primakoff process /H9253+Z→a+Zdominates and Fa/H333563.7/H11003109sin2/H9252GeV if the Compton process dominates /H20851for the DFSZ axion, viz., Eq. /H2084980/H20850/H20852. The Primakoff process is present in any axion model, and hence the Primakoff process bound is almost model independent except in the region ma /H11022200 keV where ais too heavy to be produced in the core of a star. But this threshold effect is irrelevant sincethe PQWW axion region is already excluded. Note,however, that there is no confirmed observation of neu-trinos from the small-mass red giants, unlike from theSun and SN1987A, and the possibility of dominant axionemission from red giants is not excluded by observation /H20849Raffelt, 2008b /H20850. For the DFSZ axion, the region m a /H1102210−2eV is excluded due to the large axion-electron coupling. For the hadronic axion, Raffelt and Dearborn /H208491987 /H20850argued that an axion mass greater than about /H208492e V /H20850//H20851/H20849E/N−1.95 /H20850/0.72 /H20852would reduce the helium- burning time scale and is thus not allowed. For supernovae explosion, the core temperature can go much higher than the temperature in the ignitionphase of helium in the small-mass red giant cores. Forsupernovae, therefore, nuclear reactions are more im- portant and the F abound can be very strong. As a result we use the axion couplings to nucleons discussed in Sec.III.C.1 to study the core evolution of supernovae. In the beginning, the bounds on the axion decay constant wereobtained by comparing the nuclear burning rates of pro-duction of axions and neutrinos /H20849Iwamoto, 1984 ;Pant- ziris and Kang, 1986 /H20850. The discovery of SN1987A was important in that it propelled a much interest anew inthe calculation of the axion production rate /H20849Hatsuda and Yoshimura, 1988 ;Mayle, Ellis, Olive, Schramm, and Steigman, 1988 ;Raffelt and Seckel, 1988 ;Turner, 1988 /H20850. In principle, the same kind of bound on F acould be obtained from earlier supernovae studies. The studiesafter the discovery of SN1987A were performed withthe derivative coupling and quartic terms of Sec. III.C.1 and obtained a bound F a/H11407109GeV. But as pointed out byCarena and Peccei /H208491989 /H20850,Choi, Kang, and Kim /H208491989 /H20850,Turner, Kang, and Steigman /H208491989 /H20850,Kang and Pantzisis /H208491991 /H20850, and Sec. III.C.1 , a proper treatment of nucleon states must be taken into account. For axionemission from supernovae, one must constrain the en- ergy output to /H9280a/H333551/H110031019erg g−1s−1/H20849Raffelt, 1990a /H20850. The axion emission rate calculation of Raffelt /H208492008a /H20850is/H9280a= 3.0/H110031037/H20849erg g−1s−1/H20850CN2FaGeV−2Ta,30 MeV4F,/H2084986/H20850 where Fa,GeV=Fa/GeV, Ta,30 MeV =T//H2084930MeV /H20850, and F =O/H208491/H20850. In a supernovae explosion the axion emission can be comparable to neutrino emission. Such remnantaxions from all past supernovae explosions may bearound us but will be difficult to detect because of the small 1/ F a/H20849Raffelt, 2008b /H20850. For the smaller Faregion from supernovae explosions, axions can be trapped ifthe axion-nucleon interaction is strong enough. For the hadronic axion, this gives the bound on m a/H333561e V /H20849Raffelt, 1990a ;Turner, 1990 /H20850, and we have a hadronic axion window in the eV range. For the KSVZ axion and the MI superstring axion, c¯1 terms are present. For example, we can simply take c¯1u =1 3and c¯1d=1 6, corresponding to Z=0.5, and hence obtain capp=1 3F+1 12S/H112290.17 for the KSVZ axion. Using cappas CNin Eq. /H2084986/H20850, we obtain an Fabound from supernovae, Fa/H333560.5/H11003109GeV. /H2084987/H20850 The white dwarfs in the final evolutionary stage of low-mass stars /H20849M/H1102110±2 M/H17018/H20850, with the theoretical model implemented in the DFSZ model, may give a stronger bound on Fa/H20849Raffelt, 1986 /H20850for some region of the DFSZ parameter tan /H9252=/H20855Hu/H20856//H20855Hd/H20856. The recent study of the bremsstrahlung process gives the bound Fa/H333560.6 /H110031010sin2/H9252GeV , and even fits the cooling diagram nicely with Fa/H112291.2/H110031010/H11003sin2/H9252GeV for Hdgiving mass to the electron /H20849Isern, García-Berro, Torres, and Catalán, 2008 /H20850. Note that tan /H9252is known to be large /H20849/H3335630/H20850in SUSY grant unified theory /H20849GUT /H20850models, and the white dwarfs may give the strongest Fabound for some DFSZ axion models. The axion-nucleon coupling gets enhanced in a strong magnetic field. Magnetic fields as strong as B/H110221018Gi n neutron stars have been assumed in the scalar virial theorem /H20849Woltjer, 1964 /H20850. With B/H110221020G at the surface, the axion emission rate from neutron stars or white dwarfs will be enhanced by O/H208491/H20850compared to the B=0 case /H20849Hong, 1998 /H20850. In summary, axions once produced in the hot plasma of a star most probably escape the core, taking out en-ergy. This contributes to the energy loss mechanism of astar and is used to constrain axion models. From the nucleon-nucleon- acoupling, SN1987A gives the stron- gest astrophysical bound on the axion decay constant, F a/H110220.5/H11003109GeV /H20849Raffelt, 1990a ,2008a ;Turner, 1990 /H20850. B. Axions in the universe Axions with ma/H1140724 eV have a lifetime shorter than the age of the Universe. In this case, axion decay mightlead to photons that can be tested against the observedelectromagnetic background of the Universe, as in some spontaneously broken flavor symmetric models, /H9263i →/H9263ja→/H9263j/H9253/H9253 /H20849Berezhiani, Khlopov, and Khomeriki, 1990 /H20850. However, in this case the needed decay constant, 106GeV, is outside the current bound on Fa.576 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010The axion for ma/H1135124 eV has a longer lifetime than the age of the Universe and can affect its evolution. Theheavy thermal axions around the eV mass range of Fig.2/H20849b/H20850become the hot DM in the Universe. For the 3–8 eV mass range, they accumulate in galaxy clusters where their slow decay produces a sharp line that, inprinciple, can be observed by telescope searches as sug-gested by Bershady, Ressell, and Turner /H208491991 /H20850. In this case, the neutrino and axion hot DM must be considered together, which now constrains the axion mass to m a /H110211.02 eV /H20849Hannestad, Mirizzi, Raffelt, and Wong, 2008a ,2008b /H20850, almost closing the hadronic axion window of 1–20 eV of Fig. 15. But more attention is paid to axions behaving as the CDM candidate. The axion potential is almost flat asdepicted in Fig. 11. Therefore, a chosen vacuum stays there for a long time, and starts to oscillate when the Hubble time H −1is comparable to the oscillation period /H20849the inverse axion mass /H20850,3H/H11015ma. This occurs when the temperature of the Universe is about 1 GeV /H20849Abbott and Sikivie, 1983 ;Dine and Fischler, 1983 ;Preskill, Wise, and Wilczek, 1983 /H20850. There exists the domain wall problem in the standard big bang cosmology /H20849Sikivie 1982 /H20850. The axion strings and domain wall problem have been summarized by Sikivie /H208492008 /H20850. The axion cosmol- ogy is correlated to the reheating temperature TRHin the inflationary models, where one must deal with boththe inflaton and the axion. The density perturbationsproduced by perturbations of the inflaton field are adia- batic, /H9254/H9267matter //H9267matter =/H208493/4 /H20850/H9254/H9267rad//H9267rad. On the other hand, the perturbations produced by fluctuations of the axionfield have isocurvature. If the reheating temperature T RHis above the axion scale Fa, the limit on the isocur- vature of less than 30% from the large-scale structuredata can be used /H20849Beltrán, García-Bellido, and Lesgour- gues, 2007 /H20850. This will be commented on more in Sec. IV .C on the anthropic argument. In supersymmetric models, the reheating temperature is constrained to T RH/H11021109or 107GeV /H20849if the gluino is lighter than the gravitino /H20850from nucleosynthesis require- ments in models with a heavy gravitino /H20849Ellis, Kim, and Nanopoulos, 1984 ;Kawasaki, Kohri, and Moroi, 2005 /H20850. So with SUSY the domain wall is not so problematic.In this case, the problem of string-radiated axions re- quiring axion mass m a/H1102210−3eV /H20849Davis, 1985 ;Harari and Sikivie, 1987 ;Dabholkar and Quashnock, 1990 /H20850is no longer problematic. Axions are created at T/H11229Fa, but the axion vacuum /H20855a/H20856does not begin to roll until the Hubble parameter reaches the axion mass 3 H=ma, which occurs at T /H112291 GeV. From then on, the classical field /H20855a/H20856starts to oscillate. For a small misalignment angle, the energydensity behaves like that in the harmonic oscillator ma2Fa2, which is proportional to the axion mass times the number density. Thus, its behavior is like that of CDM,which is the reason that the axion DM is CDM eventhough its mass is very small and its interaction strengthis much weaker than “weak.” Even for a large misalign- ment angle, an adiabatic invariant Iexists and one can estimate the current axion energy density. The axionfield evolution with the adiabatic change of the axionmass has been considered before /H20849Chang, Hagmann, and Sikivie, 1998 ,1999 /H20850. The temperature-dependent axion mass /H20849Gross, Pisar- ski, and Yaffe, 1981 /H20850enters in the determination of the cosmic temperature T 1where 3 H/H20849T1/H20850/H11229ma/H20849T1/H20850. The new estimate of T1forFa/H112701016GeV is a bit below 1 GeV, T1/H112290.92 GeV /H20849Bae, Huh, and Kim, 2009 /H20850. QCD has two phases: the quark-gluon phase and the chiral symmetrybreaking hadronic phase. Near the critical temperature T c, these two phases are separated above and below Tc. The critical temperature is estimated as 148−31+32 /H20849172−34+40/H20850MeV for three /H20849two /H20850light quark flavors /H20849Braun and Gies, 2007 /H20850. So cosmology near Tcneeds informa- tion on the temperature-dependent axion mass. This re-gion is in the boundary of the weak and strong couplingregimes and it is very difficult to estimate the axion massaccurately. Early attempts in this direction are given inSteinhardt and Turner /H208491983 /H20850;Seckel and Turner /H208491985 /H20850; Turner /H208491986 /H20850. The ’t Hooft determinental interaction is shown in Fig. 9. In the quark-gluon phase, we have the first diagram in the box, which is parametrized as −K −5/H20849mumdms//H9267¯6/H20850cos/H20851/H20849c2+c3/H20850/H9258/H20852where/H9267¯is the effective instanton size in the instanton size integration. /H20851Gross, Pisarski, and Yaffe /H208491981 /H20850; Eq. /H208496.15 /H20850/H20852expressed the re- sult as n/H20849/H9267,0/H20850exp /H20853−1 3/H92612/H208492N+Nf/H20850 −1 2A/H20849/H9261/H20850/H208511+1 6/H20849N−Nf/H20850/H20852/H20854, /H2084988/H20850 where/H9261=/H9266/H9267T,A/H20849/H9261/H20850/H11229−1 12ln/H208491+/H92612/3/H20850+/H9251/H208491+/H9253/H9261−2/3/H20850−8 with/H9251=0.012 897 64 and /H9253=0.158 58, and the prefactor n/H20849/H9267,0/H20850is the zero-temperature density n/H20849/H9267,0/H20850=mumdmsCN/H20849/H9264/H9267/H2085031 /H92675/H208734/H92662 g2/H208491//H9267/H20850/H208742N e−8/H92662/g2/H208491//H9267/H20850. /H2084989/H20850 Here, the parameters are /H9273=1.3391 and CN=0.160 073 forN=3 with the Pauli-Villars regularization /H20849Gross, Pisarski, and Yaffe, 1981 /H20850.Faleev and Silvestrov /H208491996 /H20850 argue that the MS scheme is suitable for the study, where/H9264=1.3391 and CN=0.160 073 are presented for N=3. For the subsequent numerical illustration, we use theMS scheme values. For the QCD coupling constant, we use the three-loop result /H20849Amsler et al. , 2008 ; QCD byHinchliffe, 2008 /H20850,O(Fa) FIG. 11. /H20849Color online /H20850The almost flat axion potential. The misalignment angle is expected to be of order 1 but can also bevery small as shown by the thick arrow.577 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010/H9251c/H20849/H9262/H20850=gc2/H20849/H9262/H20850 4/H9266 /H112294/H9266 /H92520ln/H20849/H92622//H9011QCD2/H20850/H208751−2/H92521 /H925202ln/H20851ln/H20849/H92622//H9011QCD2/H20850/H20852 ln/H20849/H92622//H9011QCD2/H20850 +4/H925212 /H925204ln2/H20849/H92622//H9011QCD2/H20850/H20877/H20873ln/H20851ln/H20849/H92622//H9011QCD2/H20850/H20852−1 2/H208742 +/H92522/H92520 8/H925212−5 4/H20878/H20876, /H2084990/H20850 where/H92520=11−2 3Nf,/H92521=51−19 3Nf, and/H92522=2857−5033 9Nf +325 27Nf2.A t T=TGeV GeV /H20849from 700 MeV to 1.3 GeV /H20850, we parametrize the instanton size integration of Eq. /H2084988/H20850 as V/H20849/H9258/H20850=−C/H20849T/H20850cos/H20849/H9258/H20850, /H2084991/H20850 where/H9258=a/Faand C/H20849T/H20850is C/H20849T/H20850=/H9251instGeV4/H20849TGeV /H20850−n. /H2084992/H20850 We obtain /H9251inst=4.715/H1100310−12/H208491.515/H1100310−11,1.185 /H1100310−12/H20850, n=6.878 /H208496.789,6.967 /H20850 for/H9011QCD =380 /H20849440,320 /H20850MeV /H20849Bae, Huh, and Kim, 2008 /H20850. Equat- ing 3 H/H20849T/H20850and m/H20849T/H20850=/H20881C/H20849T/H20850/Fa2, we obtain the following T1for/H9011QCD=380 MeV /H20849Bae, Huh, and Kim, 2008 /H20850: T1,GeV = 0.931 /H20849Fa,12/H20850−0.184. /H2084993/H20850 For Fa=1012GeV, we obtain T1/H112290.93 GeV. This num- ber is smaller than those given in the 1980s because weused a smaller number for the product of current quark masses m umdmsbased on the recent compilation of light quark masses /H20849Manohar and Sachrajda, 2008 /H20850. /H208491/H20850No sudden change in m a/H20849T/H20850: Since the potential varies much more slowly than the field itself, we can usethe so-called adiabatic invariant theorem that if the po-tential is adiabatically changed, the area in the phasespace swept by the periodic motion is unchanged in oneaxion oscillation /H20849Landau and Lifshitz, 1976 /H20850. In this case, for a small misalignment angle the adiabatic invari- ant is /H9267/H20849t/H20850/m/H20849t/H20850, which can be interpreted as the conser- vation of the total axion number. For a large /H92581, the invariant is not the axion number density, but the CDMenergy density, which can be related to the axion num-ber density by a correction factor /H20849Bae, Huh, and Kim, 2008 /H20850. If we apply this until now, we obtain /H9267a/H20849T/H9253= 2.73 K /H20850=ma/H20849T/H9253/H20850na/H20849T/H9253/H20850f1/H20849/H92582/H20850 =/H20881Z 1+Zm/H9266f/H92663/H110031.66g*s/H20849T/H9253/H20850T/H92533 2/H20881g*/H20849T1/H20850MPFa T1 /H11003/H925822f1/H20849/H92582/H20850 /H9253/H20873T2 T1/H20874−3−n/2 , /H2084994/H20850 where f1/H20849/H92582/H20850is the anharmonic correction and we used Z/H11013mu/md/H112290.5, m/H9266=135.5 MeV, f/H9266=93 MeV, and g*s/H20849present /H20850=3.91./H9253is the entropy increase ratio from extra particles beyond the SM. This becomes roughly1.449/H1100310−11/H925812 /H9253/H20873Fa,Gev 1012T1,GeV/H20874F/H20849/H92581,n/H20850eV4, where/H92581is the initial misalignment angle at T1and/H92582is the angle at somewhat lower temperature T2where the adiabatic invariant Iis calculated. The total correction factor F/H20849/H92581,n/H20850takes into account the anharmonic effect and the initial overshoot of the misalignment angle, pre-sented by Bae, Huh, and Kim /H208492008 /H20850. For the critical density /H9267c=3.9784/H1100310−11/H20849h/0.701 /H208502/H20849eV/H208504and/H9011QCD =380/H1100760 MeV, the axion energy fraction, in terms of Fa only, is given by /H20849Bae, Huh, and Kim, 2008 /H20850 /H9024a= 0.3796 ABC/H20873/H925812F/H20849/H92581/H20850 /H9253/H20874/H208730.701 h/H208742 , /H2084995/H20850 where A=/H20849mumdms/3/H110036/H11003103 MeV3/H20850−0.092, B =/H20849Fa/1012GeV /H208501.184−0.010 xwith x=/H20849/H9011QCD/380 MeV /H20850−1, and C=/H20849/H9011QCD/380 MeV /H20850−0.733. /H208492/H20850Sudden change in m a/H20849T/H20850: We now try to calculate the misalignment angle below the critical temperature ofchiral symmetry breaking where a sudden phase change is experienced near the critical temperature T c. The QCD interaction for light quarks below 1 GeV can be written as L=− /H20849muu¯LuR+mdd¯LdR+mss¯LsR+ H.c. /H20850 −K−5/H20849u¯LuRd¯LdRs¯LsRe−ic¯3/H9258+ H.c. /H20850, /H2084996/H20850 where Khas the mass dimension arising from QCD in- stanton physics. The ’t Hooft determinental interaction/H20849’t Hooft, 1976 /H20850written above is equivalent to the anomaly term and has the same chiral symmetry behav- ior. For T c/H11351T/H11351Fa, quark bilinears are not developing VEVs, and the relevant determinental interaction forthe axion is the first diagram inside the box of Fig. 9. Now, the importance of the determination of T 1is how much the misalignment angle /H92581can be shrunk at Tc. In the hadronic phase below the critical temperature Tc, the axion potential is shown in Fig. 12. The value /H92581/H20849Tc/H20850is the boundary value at Tcwe use in the effective Lagrangian below Tc. Below Tc, the quark bilinears de- velop VEVs and we must consider the possibilities of q¯LqRreplaced with /H20855q¯LqR/H20856. The effective Lagrangian from the determinental interaction is shown in Fig. 9.TVZ (1+Z)2f2 πm2 π0 1GeV Tc Ti Tf FIG. 12. Phase transition near the critical temperature Tc /H11015150 MeV.578 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010In the limit Fa/H11271f/H11032, the mass eigenstates in one-flavor QCD are /H9257mass/H11032 /H11229/H208981 f/H11032/Fa 1+m/K/H11032/H20899,amass /H11229/H20898−f/H11032/Fa 1+m/K/H11032 1/H20899. /H2084997/H20850 Equation /H2084938/H20850with v3=0 has minima at /H9258=2/H9266n/H20849ninte- ger/H20850. For v3/HS110050, minima are at /H9258/H9257/H11032=2/H9266m/H20849minteger /H20850and /H9258=2/H9266n/H20849ninteger /H20850. Therefore, the /H9258direction can be taken as the approximate axion direction even below Tc. The minimum point in the direction of the axion is not changed when one goes from /H9258/H9257/H11032/HS110050t o/H9258/H9257/H11032=0, i.e., above and below the critical temperature. /H20851If the minimum of /H9258 is shifted by /H9266in going from /H9258/H9257/H11032=0 to/H9258/H9257/H11032=/H9266, the shrunker /H92581/H20849Tc/H20850atTcis near/H9266, and we must start from O/H208491/H20850misalignment angle at Tc./H20852In most regions of the phase transition space, a time scale /H9004tis needed for the sound wave of quark bilinears to propagate to a largedistance, which releases the latent heat to keep the tem-perature constant during the first-order phase transition/H20849Mukhanov, 2005 /H20850. Even if one considers supercooling toward a sudden phase transition, the parameter spacefor a sudden phase change is almost nil and the axionenergy density presented in Eq. /H2084995/H20850is reliable /H20849Bae, Huh, and Kim, 2008 /H20850. In Fig. 13, we present the exclusion plot for m u =2.55 MeV, md=5.04 MeV, and ms=104 MeV /H20849Mano- har and Sachrajda, 2008 /H20850in the Favs/H92581//H20881/H9253space, in- cluding the anharmonic effect and the WMAP value/H20849Dunkley et al. , 2009 /H20850of the CDM density combined with additional data /H20849Komatsu et al. , 2009 /H20850/H9024 DMh2 /H112290.1143±0.0034. Note that Faof order 1013GeV is not very unnatural; it results from the new smaller masses foruand d/H20849Manohar and Sachrajda, 2008 /H20850. If axions are the CDM component of the Universe, then they can be detected even though it may be verydifficult. The feeble axion coupling can be compensated by the huge number of axions, since the number density is/H11011Fa2and the cross section is /H110111/Fa2. So there is hope of detecting cosmic axions, which has been realized bySikivie’s cavity detector /H20849Sikivie, 1983 /H20850. But the Sikivie detector has technical limitations for the interesting large and median regions of the F awindow. For ex- ample, the Faregion Fa/H110221013GeV advocated in an- thropic arguments needs a too large cavity size and the supergravity mediation preferred region Fa/H110115 /H110031010GeV requires O/H208491.6 mm /H20850order cavities. For tech- nically preferred axion masses in the region 10−6eV, one needs a low-temperature cavity with dimension O/H20849/H11022104cm3/H20850and a magnetic field strength of O/H2084910 T /H20850. The current status of cosmic axion search is shown inFig. 14. C. Axion cosmology beyond the window IfFa/H112711012GeV, an O/H208491/H20850misalignment angle /H92581is ruled out by the cosmic energy density argument. How- ever, if/H92581/H112701, the axion energy density can be within the closure density. Rather than fine tuning /H92581to order 10−3 for a Planck scale Fa/H20849Pi, 1984 /H20850, the anthropic argument of Weinberg /H20849Weinberg, 1987 ;Linde, 1988 /H20850, that life forms can evolve in a universe with a sufficiently long lifetime, can be used for an allowable /H92581. The homogeneous axion field value /H20849with a→−asym- metry /H20850right after inflation can take any value between 0 and/H9266Faor/H92581=/H208510,/H9266/H20852because the height of the axion potential is negligible compared to the total energy den-sity right after inflation. So in the axion context withOver Closure 101110121013101410151016Fa/LParen1GeV /RParen1 0.00.51.01.52.02.53.0Θ1 Γ FIG. 13. /H20849Color /H20850Favs the misalignment angle /H92581//H20881/H9253as a func- tion of/H9024a. The overclosure portion is from the precision mea- surement requiring /H9024a/H110210.23 /H20849Komatsu et al. , 2009 /H20850. The green region is the region excluded by the condition /H9024a/H114070.23. The yellow band is the error bar region of /H9011QCD and the two red lines are the limits from the light quark mass bounds.maxion [eV]|αemcaγγ |/2πFa [GeV−1] 10−610−510−4RBF: blueFlorida: redADMX exp. (high resolution) Future ADMXFuture CARRACK10−1510−1410−1310−1210−11 KSVZ: e_Q=0 DFSZ: d^c unification Charge of KSVZ Q =1±1 3±4 3 A flipped-SU(5) model FIG. 14. /H20849Color /H20850The bounds on cosmic axion searches with some theoretical predictions. The coupling on the vertical axisis the coefficient of E·B. The future CARRACK and ADMX experiments are from Tegmark, Aguirre, Rees, and Wilczek, 2006 ;Imai, 2008 ; and van Bibber, 2008 .579 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010only the misalignment production of axions, the CDM density is chosen as a random number by the spontane- ous symmetry breaking of the U /H208491/H20850PQ. Even in the multicomponent CDM models including axions, the ax-ion misalignment angle can act as the random number.This singles out axion physics, as stressed in Tegmark, Aguirre, Rees, and Wilczek /H208492006 /H20850, from all other an- thropic arguments without axions in that the selection ofan axion vacuum is an unavoidable random process thatfixes the key cosmological parameter. This also distin-guishes axions from WIMPs, super-WIMPs, etc., wherethe abundance is fixed by particle physics parameters and not by a primordial random process. As a result /H9024 a may be at the required value by an appropriate initial misalignment angle in models with axions with Fa /H110221012GeV. Tegmark et al. studied the landscape sce- nario for 31 dimensionless parameters and some dimen-sionful parameters with which habitable planets are con-sidered for the assumed nuclear physics parameters/H20849Barr and Seckel, 1992 /H20850. For example, Fig. 12 of Teg- mark, Aguirre, Rees, and Wilczek /H208492006 /H20850presents the scalar fluctuation Q/H11229 /H9254/H9267//H9267versus the matter density per CMB photon /H9264, in which the anthropically chosen point is shown as the star. In models with axions, this pointresults from a random number after inflation. If a WIMPis the sole candidate for CDM, one obtains just one number for /H9254/H9267//H9267from particle physics parameters, which may not fit the observed point of that figure. Thenwe may need the CDM-favored WIMP and in addition the axion with F a/H110221012GeV, with the axion CDM frac- tion Ra=/H9024a//H9024CDM. But this large Faanthropic region has a potential conflict with the WMAP five-year data, as presented in the FavsEI/H20849=the inflation energy scale /H20850 plane of Fig. 2 of Hertzburg, Tegmark, and Wilczek /H208492008 /H20850. For Ra=1, for example, Fa/H333561014GeV is incon- sistent with the WMAP five-year data on the upper bound on the isocurvature fluctuation /H9251a/H110210.072 /H20849Ko- matsu et al. , 2009 /H20850. From the study of outer space axions, we present a cartoon for the Fabound in Fig. 15where the future CERN Axion Solar Telescope /H20849CAST /H20850and ADMX ex- perimental regions are also marked. D. Quintessential axion In light of SUSY breaking in supergravity, it is gener- ally believed that at least a hidden confining force isneeded at an intermediate scale. This hidden sector andthe observable sector couple weakly in most phenom-enological models. This scheme fits very well in the het-erotic string framework and in heterotic Mtheory. In cosmology, on the other hand, we have had the im-portant dark energy problem already for a decade /H20849Riess et al. , 1998 ;Perlmutter et al. , 1999 /H20850, which has led to much interest in quintessence models since the late1980s /H20849Wetterich, 1988 /H20850. The quintessence related to ax- ion physics is called the “quintessential axion” /H20849QA /H20850 which was suggested in Kim and Nilles /H208492003 ,2009 /H20850. There have been attempts to identify one of the MDaxions as the quintessential axion /H20849Choi, 2000 ;Kim, 2000 /H20850. To explain the dark energy in terms of a QA, one requires the VEV of the QA not to roll down until re-cently. Of course, it is required for the current vacuum energy density of the classical QA to be of order /H9261 4 /H11015/H208490.003eV /H208504. These two conditions restrict the QA de- cay constant fqand the QA mass mq. We can param- etrize the QA /H20849/H9278/H20850potential as V/H20851/H9278/H20852=/H92614U/H20849/H9264/H20850,/H9264=/H9278 fq. /H2084998/H20850 For/H9275=p//H9267/H11021−1+/H9254, we require fq/H11022/H20881/H208492−/H9254/H20850/6/H9254MP/H20841U/H11032/H20841 where U/H11032=dU/d/H9264/H20849Kim and Nilles, 2003 /H20850. Generically, one needs a Planckian scale quintessential axion decay constant fq. So the QA mass is extremely small, /H1135110−32eV. As a result, there are two problems to be resolved to achieve the QA idea: a large decay constantand an extremely shallow QA potential. It has long been believed that the MI axion has rather a robust model-independent prediction of its decay con-stant /H20849Choi and Kim, 1985a ;Svrcek and Witten, 2006 /H20850. Recently, however, it was shown that the MI axion maynot be model independent since the decay constant maydepend on the compactification scheme in warped inter- nal space, ds 2=hw2/H9257/H9262/H9263dx/H9262dx/H9263+gmn/H20849y/H20850dymdyn/H20849Dasgupta, Firouzjahi, and Gwyn, 2008 /H20850, Fa=/H208812 /H9252ms2 MP, /H2084999/H20850 where/H9252depends on the warping in the compact space y/H33528K, /H9252=/H20885d6y/H20881g/H208496/H20850e−/H9278hw−2 /H20885d6y/H20881g/H208496/H20850hw2. /H20849100 /H20850 Thus, the MI axion with a small /H9252can be a QA if the QCD axion decay constant can be in the intermediatescale. This possibility may be realizable in some compos-ite axion models, as recently suggested in Kim and Nilles /H208492009 /H20850. V . AXION DETECTION EXPERIMENTS There are currently a variety of experiments searching for axions, whether they are left over from the big bangor produced in stars or the laboratory. Though these ex-periments search for axions at a variety of mass and103104105106107108109101010111012101310141015104103102 10 1 10−110−210−310−410−510−610−710−810−9 Fa[GeV ]ma[eV] C A S TADMXSN1987A Red gts., Gl. cls.(DFSZ) Sun LabCDM Anthropic FIG. 15. /H20849Color /H20850A schematic for the Fabounds.580 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010coupling scales they all rely on the Primakoff process, for which the following coupling, ca/H9253/H9253is given in Eq. /H2084975/H20850: L=ca/H9253/H9253a Fa/H20853FemF˜em/H20854,ca/H9253/H9253/H11229c¯a/H9253/H9253− 1.98, /H20849101 /H20850 where c¯a/H9253/H9253=/H20841TrQem2/H20841E/H11271MZ. A. Solar axion search 1. Axion helioscopes Axions produced in the nuclear core of the Sun will free-stream out and can possibly be detected on Earthvia an axion helioscope, first described in 1983 /H20849Sikivie, 1983 ,1985 /H20850and developed into a practical laboratory detector in 1988 /H20849van Bibber, McIntyre, Morris, and Raffelt, 1989 /H20850. The technique relies on conversion of so- lar axions into low-energy x rays as they pass through astrong magnetic field. The flux of axions produced in theSun is expected to follow a thermal distribution with a mean energy of /H20855E/H20856=4.2 keV. The integrated flux at the Earth is expected to be /H9021 a=g1023.67/H110031011cm−2s−1with g10=/H20849/H9251em/2/H9266Fa/H20850ca/H9253/H92531010GeV /H20849Zioutas et al. , 2005 /H20850. The probability of a solar axion converting into a photon as it passes through a magnet with field strength Band length Lis given as P=/H20873/H9251emca/H9253/H9253BL 4/H9266Fa/H208742 2L21 − cos /H20849qL/H20850 /H20849qL/H208502. /H20849102 /H20850 Here ca/H9253/H9253is defined as the coupling of the axion to two photons as given in Eq. /H20849101 /H20850, while qis the momentum difference between the axion and the photon, defined as q=ma2/2E, where Eis the photon energy. To maintain maximum conversion probability the axion and photonfields need to remain in phase over the length of the magnet, thus requiring qL/H11021 /H9266/H20849van Bibber, McIntyre, Morris, and Raffelt, 1989 /H20850. For low-mass axions q→0, leading to a maximum conversion probability. Moremassive axions will begin to move out of phase with thephoton waves though this can be compensated for by theadditon of a buffer gas to the magnet volume, thus im-parting an effective mass to the conversion photon /H20849van Bibber, McIntyre, Morris and Raffelt, 1989 /H20850and bring- ing the conversion probability back to the maximum.The gas pressure can be varied to tune to various axionmasses. An initial axion helioscope was built at Brookhaven in 1992 and used a 2.2 ton iron core dipole magnet ori- ented at the Sun with a proportional chamber for x-ray detection /H20849Lazarus et al. , 1992 /H20850. It was followed by a 4 T superconducting helioscope, developed by the Univer- sity of Tokyo, which ran for 1 week with an evacuated bore in 1997 /H20849Moriyama et al. , 1998 ;Ootani et al. , 1999 /H20850 and for 1 month with a helium-filled bore in 2000 /H20849Inoue et al. , 2002 /H20850. Though both managed to set limits over a wide mass range their sensitivities were still well aboveeven the most optimistic KSVZ axion couplings. Re-cently, though, the University of Tokyo group releaseddata taken between December 2007 and April 2008, which were able to set a limit of g a/H9253/H9253/H11021/H208495.6–13.4 /H20850 /H1100310−10GeV−1for the axion in the mass range 0.84 /H11021ma/H110211.00 eV /H20849Inoue et al. , 2008 /H20850. In order to push into proposed axion model space, third-generation axion helioscopes have been developedat CERN /H20849CAST /H20850and at the University of Tokyo. Uti- lizing a prototype LHC magnet with L=9.3 m and B =9 T CAST began taking data in 2003. It utilizes a railsystem to track the Sun for 90 min a day at sunrise and sunset, and its dual magnet bore allows it to employ upto four different x-ray detectors /H20849one on each end of each magnet bore /H20850. Currently a time-projection cham- ber, a micromegas /H20849micromesh gaseous structure /H20850detec- tor and an x-ray reflective telescope with a charge-coupled device detector are all used to detect convertedx rays. Results from the combined 2003 and 2004 runsyield limits on axion-photon-photon couplings of c a/H9253/H9253/Fa/H110217.6/H1100310−8GeV−1/H20849Andriamonje et al. , 2007 /H20850. The experiment’s second phase utilizing4He and3He buffer gases is currently under way with the latter gasallowing for axion searches in proposed model space up to mass /H110111 eV. 2. Bragg diffraction scattering An alternative to axion helioscopes was proposed in 1994: use of crystal detectors which meet the Bragg con-ditions to search for x rays generated by coherent axion-to-photon conversion /H20849Paschos and Zioutas, 1994 /H20850. Vari- ous dark matter WIMP search collaborations were ableto look through their data sets and set limits on possibleinteractions from solar axions. These included germa-nium experiments such as COSME /H20849Morales et al. , 2002 /H20850 and SOLAX /H20849Avignone et al. , 1998 /H20850, CDMS /H20849Ahmed et al., 2009b /H20850, and the reactor germanium experiment TEXONO /H20849Chang et al. , 2007 /H20850, as well as the DAMA experiment /H20849Bernabei et al. , 2001 ,2003 /H20850which utilized NaI crystals. The limits from these searches can be seenin Fig. 16. One advantage of this technique is that its sensitivity is independent of axion mass, as long as onecan neglect any nuclear recoils /H20849Carosi and van Bibber, 2008 /H20850. 3. Geomagnetic conversion It has recently been pointed out /H20849Davoudiasl and Hu- ber, 2006 /H20850that solar axions might pass through the Earth and convert to x rays on the other side as they passthrough the Earth’s magnetic field. They could then bedetected by x-ray telescopes and the solar x-ray back-ground could be effectively shielded by the Earth. B. Search for cosmic axions Cosmic axions left over from the big bang may be detected utilizing microwave cavity haloscopes /H20849Sikivie, 1983 ,1985 /H20850. The strategy relies on primordial axions drifting through a microwave cavity immersed in astrong static magnetic field in which they can resonantly581 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010convert to microwave photons, see Fig. 17. The cosmic axions’ feeble interactions can be in part compensatedby their large numbers; since the number density varies as/H11011F a2while their cross section varies as /H110111/Fa2.I ft h e axion makes up the majority of CDM in the Universe, its local density is expected to be roughly 0.45 GeV/cm3 /H20849Gates, Gyuk, and Turner, 1995 /H20850, which yields a numberdensity of /H110111014axions/cm3if one assumes a 4.5 /H9262eV axion. The expected microwave signal will be a quasimo-nochromatic line beginning at the microwave frequencycorresponding to the axion mass and slightly broadenedupward due to the axion virial distribution, with ex- pected velocities of order 10 −3c, implying a spread in energies of /H9254E/E/H1101110−6. There could also be an additional signal from nonther- malized axions falling into the galaxy’s gravitational wellwhich would yield very sharp signals due to their low predicted velocity dispersion /H20849/H1102110 −7c/H20850/H20849Sikivie, 2003 /H20850. 1. General detector properties Since the Lagrangian for axions coupling to a mag- netic field goes as La/H9253/H9253=/H20873/H9251emca/H9253/H9253 2/H9266Fa/H20874aE·B, /H20849103 /H20850 the only resonant modes which can couple to axions are those that provide an axial electric field component /H20849TM modes /H20850. The expected power generated from axion-to- photon conversions in the cavity is given by /H20849Sikivie, 1985 /H20850 Pa=/H20873/H9251emca/H9253/H9253 2/H9266Fa/H208742 VB02/H9267aClmn1 mamin /H20849QL,Qa/H20850 = 0.5/H1100310−26W/H20873V 500/H5129/H20874/H20873B0 7T/H208742 Clmn/H20873ca/H9253/H9253 0.72/H208742 /H11003/H20873/H9267a 0.5/H1100310−24gc m−3/H20874 /H11003/H20873ma 2/H9266/H20849GHz /H20850/H20874min /H20849QL,Qa/H20850, /H20849104 /H20850 where Vis the cavity volume, B0is the magnetic field strength,/H9267is the local axion mass density, mais the ax- ion mass, Clmnis a form factor which describes the over- lap of the axial electric and magnetic fields of a particu- lar TM lmnmode, QLis the microwave cavity’s loaded quality factor /H20849defined as center frequency over band- width /H20850, and Qais the axion quality factor defined as the axion mass over the axion’s kinetic energy spread. Themode-dependent cavity form factor is defined as C lmn=/H20879/H20885 Vd3xE/H6023/H9275·B/H60230/H208792 B02V/H20885 Vd3x/H9280/H20841E/H6023/H9275/H208412/H20849105 /H20850 where E/H6023/H9275/H20849x/H6023/H20850ei/H9275tis the oscillating electric field of the TM lmnmode, B/H60230/H20849x/H6023/H20850is the static magnetic field, and /H9280is the dielectric constant of the cavity space. For a cylindri- cal cavity with a homogeneous longitudinal B/H6023field the T010mode yields the largest form factor with C010 /H110150.69 /H20849Bradley, 2003 /H20850. The mass range of cosmological axions is currently constrained between /H9262eV and meV scales, correspond-gaγ(GeV−1) max ion (eV)10−1210−1110−1010−910−810−7 10−510−410−310−210−1 11 0To ky o 0 8 CAST Phas e I4He3He CAST IITo kyo helioscopeLazaru s et al. AxionmodelsKSVZ(e(Q)=0) Kim-KyaeflippedSU(5)HB starsDA MASOLAX, COSM E CD MS HD M FIG. 16. /H20849Color /H20850Exclusion plot of axion-photon coupling vs axion mass /H20849Carosi et al. , 2008 /H20850. The black bold line limit is for phase 1 of the CAST experiment and results with inclusion ofbuffer gas are expected to increase the mass and reach plau-sible axion models. The field theoretic expectations are showntogether with the string theory Z 12−Imodel of Choi, Kim, and Kim /H208492007 /H20850. In the lower left apricot box, Fig. 14is located. FIG. 17. /H20849Color online /H20850Outline of the general configuration of a resonant microwave cavity detector along with the associatedsingal expected from axion-photon conversions. This includesboth the virial component and possible lines from coherentaxions.582 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010ing to converted photon frequencies between several hundred MHz and several hundred GHz. Since largermicrowave cavities correspond to lower resonant fre-quencies and lighter axions are more likely to contributeto the dark matter density, experiments have been de-signed to start searching at the low end of the frequencyrange. At these frequencies cavities can scan only a fewkilohertz at a time in order to maintain the maximumquality factor. Axial metallic and dielectric tuning rodsare utilized to tune the cavity’s resonant frequency as itscans over the possible axion mass range. The scan rateis determined by the amount of time it takes for a pos-sible axion signal to be detected over the microwavecavity’s intrinsic noise, and is governed by the Dicke ra-diometer equation /H20849Dicke, 1946 /H20850, SNR =P a P¯N/H20881Bt=Pa kBTS/H20881t B. /H20849106 /H20850 Here Pais the power generated by axion-photon conver- sions /H20851Eq. /H20849104 /H20850/H20852,PN=kBBT Sis the cavity noise power, B is the signal bandwidth, tis the integration time, kBis Boltzmann’s constant and TSis the system temperature /H20849electronic plus physical temperature /H20850. The scan rate for a given signal-to-noise ratio is given by df dt=12 GHz yr/H208734 SNR/H208742/H20873V 500l/H20874 /H11003/H20873B0 7T/H208744 C2/H20873ca/H9253/H9253 0.72/H208744/H20873/H9267a 0.45 GeV/cm3/H208742 /H11003/H208733K TS/H208742/H20873f GHz/H208742QL Qa. /H20849107 /H20850 One can see from Eq. /H20849106 /H20850that even a small expected signal power can be made detectable by increasing the signal power /H20849Pa/H11008VB02/H20850, increasing the integration time t, or minimizing the system noise temperature TS. Tech- nology and costs limit the size and strength of the exter-nal magnets and cavities and integration times are usu- ally t/H11011100 s in order to scan an appreciable bandwidth in a reasonable amount of time. As a result the majorityof development has focused on lowering the intrinsicnoise of the first-stage cyrogenic amplifiers. 2. Microwave receiver detectors Initial experiments were undertaken at Brookhaven National Laboratory /H20849DePanfilis et al. , 1987 /H20850and the University of Florida /H20849Hagmann et al. , 1990 /H20850, but their modest-sized cavities and magnet fields meant they werestill factors of 10–100 times away from plausible axionmodel space. There are currenly two active second-generation experiments under way, the Axion DarkMatter Experiment /H20849ADMX /H20850at Lawrence Livermore National Laboratory /H20849LLNL /H20850and the Cosmic Axion Re- search with Rydberg Atoms in Cavities at Kyoto /H20849CAR- RACK /H20850experiment in Japan. Both experiments utilize large microwave cavities immersed in a strong staticmagnetic field to resonantly convert axions to photonsbut they go about detecting these photons in two differ- ent ways. ADMX uses ultrasensitive microwave receiv-ers while CARRACK uses Rydberg atoms to detectsingle photons. The ADMX experiment is a collaboration of LLNL, MIT, the University of Florida, Lawrence Berkeley Na-tional Laboratory /H20849LBNL /H20850, U.C. Berkeley, University of Chicago, and Fermilab, and has been operating in vari-ous modes since February 1996. A diagram of the ex- periment is shown in Fig. 18. ADMX consists of an 8.5 T superconducting magnet, 110 cm in length with a 60 cmclear bore. A 200 liter stainless steel microwave cavity plated in ultrapure copper is suspended below a cryo- genic stage in the center of the Bfield. Power generated in the cavity is coupled to an adjustable antenna verti-cally input through the top cavity plate. Any signal isthen boosted by extremely low-noise cryogenic amplifi-ers before being sent through a double-heterodyne mix-ing stage. Here the gigahertz range signal is mixed down to an intermediate 10.7 MHz, sent through a crystal bandpass filter, and then mixed down to audio frequen- cies at 35 kHz. This audio signal is then analyzed by fast-Fourier-transform electronics which measure over a 50 kHz bandwidth centered at 35 kHz. There is also a “high-resolution” channel in which the signal is mixed down to 5 kHz and sent through a 6-kHz-wide bandpassfilter. Time traces of the voltage output, consisting of 2 20 data points with a sampling frequency of 20 kHz, arethen taken, resulting in a 52.4 s sample with 0.019 Hz resolution /H20849Duffy et al. , 2006 /H20850. Since the system noise is dominated by the first stage of amplification, great care was taken in choosing thecryogenic amplifiers. The initial ADMX data runs uti-lized heterojunction field effect transistor /H20849HFET /H20850am- FIG. 18. /H20849Color /H20850Schematic of the ADMX experiment /H20849Carosi and van Bibber, 2008 /H20850.583 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010plifiers developed by the National Radio Astronomy Observatory /H20849Daw and Bradley, 1997 /H20850. Even though they had noise temperatures of only 2 K, the quantum noise limit at 1 GHz /H20849defined as Tq=h/H9263/kB/H20850is only 50 mK. As a result a much development went into replacing theHFETs with more sensitive superconducting quantuminterference devices /H20849SQUIDs /H20850which had noise tem- peratures of only 15% of the quantum limit /H20849Bradley, 2003 /H20850. Currently data are being taken using the SQUIDs for the first stage of amplification 1920 . Results from the initial run using HFET amplifiers have already probed plausible axion model space in the axion mass range between 2.3 and 3.4 /H9262eV /H20849Bradley, 2003 /H20850. Results from a high-resolution search have probed further into coupling space over a smaller mass range, 1.98–2.17 /H9262eV /H20849Duffy et al. , 2006 /H20850. As of this re-view ADMX is scanning over the mass range corre- sponding to 800–900 MHz using SQUID amplifiers. 3. Rydberg atom detectors The CARRACK experiment has published proof of concept papers for their detection technique using Ryd-berg atoms as opposed to low-noise amplifiers /H20849Tada et al., 2006 /H20850. The experimental setup is shown in Fig. 21.I n it rubidium atoms are excited into a Rydberg state /H20849/H208410/H20856 →/H20841n/H20856/H20850, and move through a detection cavity coupled to an axion conversion cavity. The spacing between energylevels is tuned to the appropriate frequency utilizing theStark effect, and the Rydberg atoms’ large dipole tran-sition moment ensures efficient photon detection /H20849one photon per atom, /H20841ns/H20856→/H20841np/H20856/H20850. The atoms are then sub- jected to a selective field ionization allowing the atoms FIG. 19. /H20849Color online /H20850Medium-resolution limits from the ADMX experiment. The left plot shows limits to axion coupling assuming a dark matter halo density of /H9267=0.45 GeV/cm3/H20849upper region excluded /H20850. The right plot shows limits on the axion contribution to the dark matter halo density assuming the axion has either the KSVZ or DFSZ couplings /H20849Bradley, 2003 /H20850. FIG. 20. /H20849Color online /H20850High-resolution limits from the ADMX experiment. Limits given in terms of axion contribu-tion to the dark matter halo density assuming the axion haseither the KSVZ or DFSZ coupling strength. The medium-resolution plot for that mass range for the DFSZ coupling isalso given /H20849Duffy et al. , 2006 /H20850. The KSVZ axion with e Q=0 shown above gives /H9267aat Earth less than 0.16 GeV/cm3, which corresponds to /H9024a/H333550.36. So the ADMX line of Fig. 14using Eq. /H2084995/H20850crosses the eQ=0 KSVZ line and goes down to the DFSZ line. FIG. 21. /H20849Color online /H20850General schematic of CARRACK ex- periment utilizing Rydberg atoms to recover single photonsgenerated in the microwave cavity /H20849Tada et al. , 2006 ;Carosi and van Bibber, 2008 /H20850.584 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010with the higher-energy state /H20849/H20841np/H20856/H20850to be detected /H20849Carosi and van Bibber, 2008 /H20850. The advantage of this system is that Rydberg atoms act as single-photon detec-tors and thus do not suffer from quantum noise limita-tions. Though still in the development phase CARRACK has already gone through two iterations, CARRACK 1and CARRACK 2, and it has measured cavity emission at 2527 MHz down to a temperature of 67 mK, which is a factor of 2 below the quantum noise floor for that frequency. The eventual goal is 10 mK /H20849Tada et al. , 2006 /H20850. One disadvantage of this technique is that one cannot detect signals finer than the bandpass of the cav- ity, of order /H1101110 −5, which negates searches for late-infall coherent axions. C. Laser searches In addition to cosmological and solar axion searches there is also a class of laboratory axion searches that utilize laser photons /H20849/H9253laser/H20850traversing a magnetic field. Here the polarized laser photons can scatter off virtual photons /H20849/H9253v/H20850provided by the magnetic field and convert into axions /H9253laser+/H9253v→a. Currently, laser axion searches fall into two general categories. The first technique looksfor magneto-optical effects of the vacuum due to polar-ized laser photons disappearing from the beam as theyare converted into axions. The second looks for photonsconverting into axions in the presence of a magneticfield, which are then transmitted through a wall and con-verted back into photons by a magnetic field on theother side, so-called light shining through walls experi-ments. 1. Polarization shift of laser beams There can be axion-photon-photon anomalous cou- pling of the form aE·B. A laser-induced axionlike par- ticle search employing this coupling has been performedsince the early 1990s by the Rochester-Brookhaven-Fermilab-Trieste /H20849RBFT /H20850group /H20849Cameron et al. , 1991 /H20850. A few years ago, the same type of experiment by thePVLAS Collaboration was performed with an initial positive signal with F a/H11011106GeV /H20849Zavattini et al. , 2006 /H20850 as discussed earlier. This has led to some exotic modelswhere a vacuum dichroism is achieved by producing ax-ionlike particles as shown in Fig. 22/H20849a/H20850. Because of the nonrenormalizable interaction implied in Fig. 22/H20849a/H20850, one may reconcile this model with the astrophysical bound/H20849Mohapatra and Nasri, 2007 /H20850, or if light millicharged par- ticles are produced in a strong magnetic field as shownin Fig. 22/H20849b/H20850, a vacuum dichroism is achieved as dis-cussed in Gies, Jaeckel, and Ringwald /H208492006 /H20850,Masso and Redondo /H208492006 /H20850, and Kim /H208492007c /H20850. Here the polarization of the laser beam is looked for. With more data accumu-lation, there is no convincing evidence for an axionlike particle with F a/H11011106GeV at present, contrary to an ear- lier confusion /H20849Dupays et al. , 2005 ;Chen et al. , 2007 ; Yoo, 2007 ;Zavattini et al. , 2007 ;Chou et al. , 2008 /H20850. But this incident led to the current search for axionlike par-ticles at DESY /H20849Ringwald, 2008 /H20850. 2. Light shining through walls The “light shinging through walls” technique for searching for axions was first proposed in 1987 by van Bibber et al. /H208491987 /H20850and recently a model study has been presented /H20849Adler, Gamboa, Mendéz, and López-Sarrión, 2008 /H20850. The general experimental layout can be seen in Fig. 23where polarized laser photons pass through the magnetic field with E/H20648Band any converted axions /H20849or other psuedoscalar particles /H20850can continue through an absorber to be reconverted to photons on the other side. The probability for a photon to convert into an axion as it traverses the “axion source” region is given by P/H9253→a/H110081 4/H20873/H9251emca/H9253/H9253 2/H9266FaBL/H2087421 − cos /H20849qL/H20850 /H20849qL/H208502. /H20849108 /H20850 This is the same probability for an axion to convert back into a detectable photon in the “axion detector” regionon the other side of an absorber, which leaves the totalprobability for detecting a photon-axion-photon conver- sion as P /H9253→a→/H9253=P/H9253→a2/H20849ignoring photon detection effi- ciencies of course /H20850/H20849Battesti et al. , 2008 /H20850. There is a maxi- mum detectable axion mass for these laser experimentsbecause the oscillation length becomes shorter than themagnetic field length, causing a degradation of the form factor F/H20849q/H20850=1−cos /H20849qL/H20850//H20849qL/H20850 2, but this can be compen- sated for using multiple discrete dipoles. The first experiment using this technique was per- formed by the RBFT Collaboration in the early 1990s/H20849Cameron et al. , 1993 /H20850. Using two superconducting di- pole magnets /H20849L=4.4 m and B=3.7 T /H20850and a laser /H20849/H9261 =514 nm and P=3 W /H20850with an optical cavity providing /H11011200 reflections in the axion-generating region, they were able to set upper limits on axion couplings of g a/H9253/H9253/H110216.7/H1100310−7GeV /H2084995% C.L. /H20850for pseudoscalars with a maximum mass of ma/H1102110−3eV /H20849Cameron et al. , 1993 /H20850.a × ×γ B (a)×γ Bf f (b) FIG. 22. Possible processes leading to a vacuum dichroism. FIG. 23. /H20849Color /H20850Schemtic of “light shining through walls” ex- periment /H20849Battesti et al. , 2008 /H20850.585 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010Recent photon regeneration experiments include the BMV Collaboration at LULI /H20849Robilliard et al. , 2007 /H20850 which uses a short pulsed-field magnet and the Gam- meV Collaboration at Fermilab /H20849Chou et al. , 2008 /H20850which uses a Tevatron dipole magnet /H20849L=6 m and B=5 T /H20850 with an optical barrier in the middle. Both of these haveruled out the signal reported by the PVLAS /H20849see next section /H20850. Figure 24shows the current bounds from these latest regeneration experiments. Recently it has beenshown that photon regeneration experiments can beresonantly enhanced by encompassing both the produc-tion and reconversion magnets in matched Fabry-Perotoptical resonators /H20849Sikivie, Tanner, and van Bibber, 2007 /H20850. 3. Magneto-optical vacuum effects An alternative to the shining light through walls tech- nique is to look for the indirect effect of photons inpolarized laser light converting into axions as the beamtraverses a magnetic field. Figure 25shows two different ways in which axion interactions can modify a polarizedlaser beam, induced dichroism and vacuum birefrin-gence. Vacuum dichroism occurs when a polarized laserbeam passes through a dipole magnet with the electric field component Eat a nonzero angle /H9278relative to B. The photon component parallel to Bwill have a small probability to convert into axions, causing the polariza- tion vector to rotate by an angle /H9280. Vacuum birefrin- gence is due to the induced ellipticity of the beam /H20849/H9023/H20850as a result of virtual axions. It should be noted that higher-order QED diagrams, or “light-by-light scattering” dia-grams, are expected to contribute to vacuum birefrin-gence as well. Each of these effects can be estimated as /H9023/H11015NB 2L3ma2 384/H9275/H20873ca/H9253/H9253 Fa/H208742 sin/H208492/H9258/H20850, /H20849109 /H20850/H9280/H11015NB2L2 64/H20873ca/H9253/H9253 Fa/H208742 sin/H208492/H9258/H20850, /H20849110 /H20850 in the limit that ma2L/4/H9275/H112701. Here Lis the effective path length, Nis the number of paths the light travels in the magnetic field, mais the axion mass, /H9275is the photon energy, and /H9258is the photon polarization relative to the magnetic field /H20849Battesti et al. , 2008 /H20850. The initial experiment looking for magneto-optical vacuum effects was carried out by the RBFT Collabora-tion in the early 1990s /H20849Semertzidis et al. , 1990 /H20850. This experiment used a single-pass 8.8-m-long magnet with a magnetic field of B/H110112.1 T and N=500. It set a limit on the polarization rotation of /H9280/H110213.5/H1100310−10which was still three orders of magnitude higher than that expected bylight-by-light scattering and almost 15 orders of magni- tude greater than an m a/H1101110−3eV axion. Recently the early PVLAS Collaboration reported the positive detection of vacuum dichroism. This experi- ment consists of a 1-m-long 5 T superconducting magnet with a angular frequency /H9024magof the magnet rotation and a 6.4-m-long Fabry-Pérot cavity giving the pass number N=2/H9024mag//H9266/H1101144 000. It registered a polariza- tion shift of /H9280=/H208493.9 ± 0.5 /H20850/H1100310−12rad pass−1/H20849111 /H20850 which translates to an allowed mass range of a neutral pseudoscalar boson of 1 /H33355mb/H333551.5 meV and a coupling strength of 1.5 /H1100310−3/H33355ca/H9253/H9253/Fa/H333558.6/H1100310−3GeV−1 /H20849Zavattini et al. , 2006 /H20850. Though the report of this positive signal has been retracted /H20849Zavattini et al. , 2007 ,2008 /H20850, the interest it raised led to a number of more advancedexperimental searches such as some of the new laserregeneration experiments mentioned previously. FIG. 24. /H20849Color /H20850Current limits on axion coupling from the GammeV Collaboration /H20849Chou et al. , 2008 ;Yoo, 2008 /H20850. FIG. 25. The dichroism and birefringence effects. The upper plot shows the effect of dichroism as photons converting intoaxions cause a rotation of the linear beams polarization vectorby an amount /H9280. The lower plot shows virtual axions inducing birefringence in which the linear beam acquires ellipticity /H9023 /H20849Battesti et al. , 2008 /H20850.586 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010VI. THEORIES FOR VERY LIGHT AXIONS Axion couplings come in three types: the PQ- symmetry-preserving derivative coupling c1term, the PQ-symmetric c2term, and the anomalous c3term. The PQ symmetry gives a gluon anomaly and c2+c3must be nonzero. Generally, we can therefore define aas a pseu- doscalar field without potential terms except the onearising from the gluon anomaly under a particular basis /H20849for example in the c 2=0 basis /H20850, a Fa1 32/H92662G/H9262/H9263aG˜a/H9262/H9263. /H20849112 /H20850 Then we note that this kind of nonrenormalizable anomalous term can arise in several ways. The natural scales of Faare shown in Table II. In any case, the es- sence of the axion solution /H20849wherever it originates in Table II/H20850is that the axion VEV /H20855a/H20856seeks/H9258=0, whatever happened before. In this sense it is a cosmological solu-tion. The potential arising from the anomaly term afterintegrating out the gluon field is the axion potential /H20849with c 2+c3/H20850shown in Fig. 7. A. SM singlets without SUSY A complex SM singlet carrying the PQ charge can appear in many extensions of the SM: in grand unifiedtheories /H20849GUTs /H20850/H20849Wise, Georgi, and Glashow, 1981 /H20850,i n composite models /H20849Choi and Kim, 1985c ;Kim, 1985 ; Babu, Choi, Pati, and Zhang, 1994 /H20850, and in models with extra dimensions /H20849Di Lella, Pilaftsis, Raffelt, and Ziou- tas, 2000 /H20850. In the SU /H208495/H20850GUT, the axion can be embedded in a complex 24=/H9018/H20849Wise, Georgi, and Glashow, 1981 /H20850,i n which case the VEV /H20855/H9018/H20856breaking SU /H208495/H20850down to the SM and hence the axion decay constant is the GUT scaleand is outside the axion window. On the other hand, acomplex GUT singlet, whose VEV is not related to theGUT scale, can house the axion within the axion win-dow. A SUSY generalization of the SU /H208495/H20850GUT axion has been shown to be possible /H20849Nilles and Raby, 1982 /H20850. Recently, in view of the white dwarf evolution /H20849Isern, García-Berro, Torres, and Catalán, 2008 /H20850with the two- dark-matter scenario /H20849Huh, Kim, and Kyae, 2009 /H20850an electrophilic axion has been suggested in a SUSYflipped SU /H208495/H20850/H20849Bae, Huh, Kim, Kyae, and Viollier, 2009 /H20850.B. Composite axions A SM singlet for the very light axion can arise as a composite meson with an extra confining force whosescale is much larger than the electroweak scale. Thisconfining force can be the hidden sector gauge group insupergravity or just an extra gauge group. We call this extra confining gauge group “axicolor” SU /H20849N/H20850. To create the QCD axion below the axicolor scale, there must betwo classically conserved axial global symmetries /H20849Choi and Kim, 1985c ;Kim, 1985 /H20850. With only one axial symme- try, a massless meson would not result, as in the case ofone-flavor QCD there is no massless meson since the only meson /H9257/H11032becomes heavy by the instanton solution of the so-called U /H208491/H20850problem /H20849’t Hooft, 1986 /H20850. For two axial symmetries, we can consider two kinds of axiquark, QA/H9251,Q¯A/H9251,qA, and q¯Awhere Ais the SU /H20849N/H20850index and /H9251 is the SU /H208493/H20850cindex. For these vectorlike representations, /H20849N,3/H20850+/H20849N¯,3¯/H20850+/H20849N,1/H20850+/H20849N¯,1/H20850under SU /H20849N/H20850/H11003SU/H208493/H20850c, mass terms are not introduced. The axicolor vacuumangle problem is solved basically by the massless axi- quarks Qand q. Even though Qlooks like a massless QCD quark, it cannot be considered as the massless quark solution of the QCD /H9258problem. After integrating out the axicolor degrees, we obtain an effective La- grangian resulting from Qand q. The axibaryons are expected to be removed at the axicolor scale. Of two kinds of meson, one /H20849the axicolor /H9257/H11032/H20850is removed at the axicolor scale and the other remains exactly massless.However, this massless axicolor meson couples to theQCD anomaly and becomes a QCD axion through the c 3term, becoming the so-called hadronic axion. Of the two currents J¯ /H92625=Q¯/H9253/H9262/H92535Q+q¯/H9253/H9262/H92535q, /H20849113 /H20850 J/H92625=Q¯/H9253/H9262/H92535Q−3q¯/H9253/H9262/H92535q, the divergence of J/H92625corresponds to the massless meson abelow the axicolor scale, /H11509/H9262J/H92625=2N 32/H92662G/H9262/H9263/H9251G˜/H9251/H9262/H9263, /H20849114 /H20850 and hence we obtain the effective interaction /H20849112 /H20850.I n this minimal model, the domain wall number is N/H20849Choi and Kim, 1985c ,1985d /H20850. In a supergravity model of pre-TABLE II. Natural scales of Fa. For nlarge extra dimensions, the Planck mass is MP /H11229MD/H20849R/MD/H20850n/2. Axions from Order of Fa String theory String scale or Planck scale M theory String or the scale of the 11th dimensionLarge extra /H20849n/H20850dimension Combination of the fundamental mass M Dand extra dimension radius R Composite models Compositeness scaleRenormalizable theories U /H208491/H20850 PQ-global-symmetry-breaking scale587 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010ons, a similar mechanism was used to realize a compos- ite axion /H20849Babu, Choi, Pati, and Zhang, 1994 /H20850, where the role of q-type matter is replaced by the metacolor gluino /H9261/H11032, metacolor is the binding force of preons. Even if the metacolor gluino obtains a mass of O/H20849100 GeV /H20850, the QCD/H9258can be made to be within the experimental bound if Fais greater than 1011GeV. The composite axion of Chun, Kim, and Nilles /H208491992b /H20850 is a composite made of hidden-color scalars whose bilin-ears develop VEVs and break the PQ symmetry. Thisidea has been made more concrete by Kim and Nilles /H208492009 /H20850. In the gauge-mediated SUSY-breaking scenario of In- trilligator, Seiberg, and Shih /H20849ISS /H20850/H208492006 /H20850, for example, an SU /H20849N c/H20850confining group with Nfflavors satisfying Nc +1/H33355Nf/H110213 2Ncallows a SUSY-breaking local minimum. IfNf−Nc/H333563/H20849for example, Nc=7 and Nf=10 /H20850with one type of QA/H9251+Q¯A/H9251and Nf−3 flavors of the type qA+q¯A, then there can exist a suitable local minimum where thecomposite axion envisioned in Eq. /H20849113 /H20850can be realized. In this case, the SUSY-breaking scale and the compositeaxion scale are related, as first tried by Kim /H208491984 /H20850. C. Axions with extra dimensions With large extra dimensions, the axion identification involves a few parameters: the fundamental scale mass MF, the Kaluza-Klein /H20849KK /H20850radius R, and the number of extra dimensions n. In addition, there are several ways to allocate the field /H20849s/H20850containing the axion in the bulk and/or branes. The possibility of large extra dimensions has been considered for the flat and warped extra dimensions. The TeV scale for MFwas the main motivation to look for the next level of the current experimental limit onmillimeter-scale gravity /H20849Antoniadis et al. , 1998 ;Arkani- Hamed, Dimopoulos, and Dvali, 1998 /H20850. Because the ax- ion scale is considered to be at the intermediate scale, a string theory at the intermediate scale M Fhas also been considered /H20849Burgess, Ibanez, and Quevedo, 1999 /H20850. With the Randall-Sundrum-type warp factor /H20849Randall and Sundrum, 1999a ,1999b /H20850, it is possible to introduce the intermediate scale with a Planck scale MFvia the Giddings-Kachru-Polchinski stabilization mechanism/H20849Giddings, Kachru, and Polchinski, 2002 /H20850. Here we look only at the possibility of a TeV scale M F. Since the Planck mass is given by MP /H11015MF/H20849RM F/H20850n/2, we obtain n/H333562 for MF/H1122910 TeV /H20849Han- nestad and Raffelt, 2002 ;Kanti, 2009 /H20850. The Lagrangian in 4+ ndimensions /H20851/H208494+n/H20850D/H20852with a bulk field axion can be written as /H20849Chang, Tazawa, and Yamaguchi, 2000 /H20850, Leff=/H20885dny/H208771 2MFn/H20851/H11509/H9262a/H11509/H9262a+/H11509ya/H11509ya/H20852 +/H9264/H9251em /H9266a v¯PQF/H9262/H9263emF˜em,/H9262/H9263/H20878 /H20849115 /H20850 where v¯PQis the PQ-symmetry-breaking scale at the fun- damental scale order MF, anda/H20849x/H9262,y/H20850=/H20858 n=0/H11009 an/H20849x/H9262/H20850cos/H20873n·y R/H20874. /H20849116 /H20850 The four-dimensional /H208494D/H20850PQ symmetry-breaking scale isFa/H11015/H20849MP/MF/H20850A/nv¯PQwhere A=/H20841n/H20841=/H20881n12+¯+nn2/H11021n and Fafalls between v¯PQand MP. The very light axion is then=0component in Eq. /H20849116 /H20850, and the rest are the KK axions. The mass splitting of the KK axions is of order 1/ R, and the phenomenology of these KK axions foraKK→2/H9253has been studied by Di Lella, Pilaftsis, Raffelt, and Zioutas /H208492000 /H20850, from which we have 1/ R /H110111/H2084910/H20850eV for n=2/H208493/H20850forMF/H110151 TeV. The possibility of a Z2odd 5D gauge field in a warped fifth dimension has been suggested for a QCD axionunder the assumption that all unwanted PQ-symmetry-breaking effects are suppressed /H20849Choi, 2004 /H20850. One such constraint is that the bulk fields carry the vanishing PQcharge. D. SUSY-breaking scale, axion and axino The 4D supergravity interactions with the vanishing cosmological constant were obtained in 1983 /H20849Cremmer, Ferrara, Girardello, and van Pröyen, 1983 /H20850. The PQ symmetry can be embedded in the supergravity frame-work /H20849Kim, 1984 /H20850, W PQ=/H20849f/H9254A1A2−F12/H20850Z+/H20849f/H9280A1A2−F22/H20850Z/H11032 +fQA1Q¯1Q2, /H20849117 /H20850 where Z,Z/H11032,A1, and A2are gauge singlet chiral fields, Q¯1andQ2are chiral quark superfields, and f/H9254,f/H9280,F12, and F22are parameters. The superpotential /H20849117 /H20850leads to F-term SUSY breaking and PQ symmetry breaking at a common scale at order O/H20849F12,F22/H20850iff/H9254/f/H9280/HS11005F12/F22. The fQterm defines the PQ charge of the heavy quark and the resulting axion is of the KSVZ type.The PQ-symmetry-breaking scale is given by non- zero /H20855A 1A2/H20856/H11229/H20849F2//H9261/H20850cos/H20849/H9252−/H9251/H20850and the SUSY-breaking scale is given by nonzero ZF=−F2sin/H9251sin/H20849/H9252−/H9251/H20850and ZF/H11032=F2sin/H9251sin/H20849/H9252−/H9251/H20850, where /H9261=/H20881f/H92542+f/H92802,F2=/H20881F14+F24, tan/H9251=f/H9280/f/H9254, and tan /H9252=F22/F12. The axino does not ob- tain mass at this level, but obtains a mass at order of thesoft SUSY-breaking scale /H20849Chun, Kim, and Nilles, 1992a ; Chun and Lukas, 1995 /H20850. Note that /H20855Z /H11032/Z/H20856/H11229−cot/H9251with /H20855Z,Z/H11032/H20856=O/H20849F2/M/H20850. Early discussions of the axino can be found in Frère and Gérard /H208491983 /H20850. E. The /H9262problem If Higgs doublets Hu,dcarry vanishing U /H208491/H20850charges beyond the MSSM gauge charges, then the superpoten- tial can contain a W/H9262=−/H9262HuHdterm where /H9262can be of the order of the fundamental scale since it is a supersym-metric term. This is problematic for the TeV scale elec- troweak symmetry breaking; this is the so-called /H9262 problem /H20849Kim and Nilles, 1984 /H20850. This/H9262term is a super- symmetric Higgsino mass term and can be forbidden in588 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010Wby introducing some symmetry, continuous or dis- crete. The widely discussed ones are the PQ and Rsym- metries. In the supergravity framework, if the Higgsdoublets carry one unit of PQ charge then nonrenormal- izable interactions of the form S 2HuHd/MPcan be present in WifS2and HuHdcarry opposite PQ charges. Then the resulting /H9262is of order Fa2/MP, which can be of order of the TeV scale /H20849Kim and Nilles, 1984 /H20850. With an intermediate hidden sector with hidden-sector squarks Q1and Q¯2, one may have a nonrenormalizable interac- tion of the form Q1Q¯2HuHd/MP. In this case the hidden- sector squark condensation at the intermediate mass scale can also generate a TeV scale /H9262/H20849Chun, Kim, and Nilles, 1992b /H20850./H20851For the B/H9262term, one may consider /H20849Q1Q2*/MP2/H20850HuHdin the Kähler potential. /H20852It is better for the superpotential to possess this kind of PQ symmetry and/or Rsymmetry /H20849Hall and Randall, 1991 ;Dine and MacIntire, 1992 ;Casas and Muñoz, 1993 ;Kim and Nilles, 1994 ;Kim, 1999a /H20850. If so, even if the nonrenormal- izable interactions are not considered, the gravity me- diation scenario can generate a TeV scale /H9262via the Giudice-Masiero mechanism /H20849Giudice and Masiero, 1988 /H20850. In supergravity, the Higgsino mass term is present in the chiral fermion mass matrix given by Cremmer, Ferrara, Girardello, and van Pröyen /H208491983 /H20850and Nilles /H208491984 /H20850: e−G/H20851Gij−GiGj−Gl/H20849G−1/H20850lkGkij/H20852/H9273Li/H9273Lj, /H20849118 /H20850 where G=K/H20849/H9278,/H9278*/H20850−ln /H20841W/H208412. The term e−GGijgives/H9262 /H11011m3/2ifKcontains HuHd/H20849Giudice and Masiero, 1988 /H20850 and/H9262/H11011S2/MPifWcontains S2HuHd/MP/H20849Kim and Nilles, 1984 /H20850. In the next-to-MSSM /H20849NMMSM /H20850models with W =SH uHd, the/H9262term can be generated by the singlet VEV /H20855S/H20856at the electroweak scale /H20849Cerdeño, Hugonie, López-Fogliani, Muñoz, and Teixeira, 2004 ;López- Fogliani and Muñoz, 2006 /H20850.I na Z/H11032-added MSSM /H20849Z/H11032MSSM /H20850, the/H9262term can also be successfully gener- ated /H20849Langacker, Paz, Wang, and Yavin, 2008 /H20850. Extending the MSSM gauge group which can be bro- ken down to the MSSM at a high energy scale, one can generate a reasonable /H9262. For example, there exists an interesting solution to the problem of why there is onlyone pair of Higgsino doublets at low energy in the ex- tended SU /H208493/H20850 W/H11003U/H208491/H20850electroweak model /H20849Lee and Weinberg, 1977a /H20850. This is dictated by the extended gauge symmetry. This one-pair problem is elegantly solved inthe SUSY Lee-Weinberg-type model due to the anti- symmetric /H20851under SU /H208493/H20850 W/H20852Higgsino mass matrix /H20849Kim, 2007b /H20850, reminiscent of the “color” introduction used to put low-lying baryons in the completely symmetric rep-resentation 56 in the old flavor-spin SU /H208496/H20850/H20849Han and Nambu, 1965 /H20850. Thus, explicit steps toward a successful /H9262in the grav- ity mediation scenario can be constructed in extra-singlet models, in SUSY-GUT models, through thesuperpotential, through the Kähler potential, and incomposite models.The loop effects are important sources of SUSY breaking in the gauge-mediated SUSY-breaking/H20849GMSB /H20850scenario /H20849Dimopoulos and Raby, 1981 ;Dine, Fischler, and Srednicki, 1981a ;Dine and Fischler, 1983 ; Dine and Nelson, 1993 ;Dine, Nelson, and Shirman, 1995 /H20850, in anomaly mediation SUSY breaking /H20849AMSB /H20850 /H20849Giudice et al. , 1998 ;Randall and Sundrum, 1999a /H20850, and even in the mirage mediation scenario /H20849Choi, Jeong, and Okumura, 2005 ;Loaiza-Brito, Martin, Nilles, and Ratz, 2005 /H20850. GMSB has been suggested to solve the flavor problem /H20849Gabbiani, Gabrielli, Masiero, and Silvestrini, 1996 /H20850present in the gravity mediation scenario. In this GMSB or any other loop-generated SUSY-breaking sce- nario, the soft terms generated by the supergravity effectare required to be subdominant compared to those aris-ing from the loops, or at best comparable to them. If theloop terms are subdominant as in the GMSB or AMSB,then there are some problems. First, the generation of /H9262is difficult because /H9262term generation via the Giudice-Masiero mechanism is sub- dominant at the TeV scale. One has to generate /H9262by employing the PQ and/or Rsymmetries; this method, however, does not belong to generating all TeV scaleparameters dynamically. In this regard, another confin-ing group around TeV scale has been proposed /H20849Choi and Kim, 2000 /H20850, and the model presented there is the type of composite SU /H208492/H20850 Waxion discussed in Sec. VI.B , which was saved by introducing singlets and relevantcouplings /H20849Luty, Terning, and Grant, 2001 /H20850. Then again it does not succeed in generating all TeV scale parametersdynamically. Second, in the loop SUSY-breaking scenarios for gen- erating all TeV scale electroweak parameters by loops there exists the B /H9262//H9262problem /H20849Dvali, Giudice, and Pomarol, 1996 /H20850. Since it occurs at loop orders, we con- sider /H20848d4/H9258HuHdX†for/H9262and /H20848d4/H9258HuHdXX†forB/H9262 where the auxiliary component of Xdevelops a VEV . From this observation, one generically obtains B/H9262/H11011/H9262/H9011 where/H9011/H11011/H9262/f2can be greater than /H9262; this was remedied by making B/H9262appear at two-loop order /H20849Dvali, Giudice, and Pomarol, 1996 /H20850. This B/H9262//H9262problem occurs essen- tially because of the difference of the engineering di- mensions of the B/H9262and/H9262terms. Both generically ap- pear at one-loop order with the coefficient g2/16/H92662, and hence in describing the electroweak scale the B/H9262term lacks one power of g2/16/H92662. Recently, a better solution employing a Kähler potential HuHd/H20849lnX+lnX†/H20850has been suggested /H20849Giudice, Kim, and Rattazzi, 2008 /H20850, which can be compared to the original Giudice-Masiero Kähler potential HuHdX†+¯. There exist several more ideas about the B/H9262//H9262problem /H20849Cohen, Roy, and Schmaltz, 2007 ;Cho, 2008 ;Murayama, Nomura, and Po- land, 2008 ;Roy and Schmaltz, 2008 /H20850. Perhaps nonrenormalizable interactions are the easy solution of the /H9262andB/H9262//H9262problems even in the GMSB. Here, however, one introduces another scale. Without adetailed knowledge of the ultraviolet completion of theMSSM, the nonrenormalizable interactions are usually assumed to be suppressed by the Planck mass M P. But,589 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010there might be some heavy mass scale M, which can be somewhat smaller than the Planck mass MP, for the see- saw mass of the required nonrenormalizable interac-tions. In string compactifications, it is known explicitly that Mcan be different from M P/H20849Choi and Kim, 2006 ; Kim, Kim, and Kyae, 2007 ;Choi and Kobayashi, 2008 /H20850. A simple diagram giving an Mdependence is shown in Fig. 26where the SU /H208492/H20850doublets H1and H2form a vectorlike superheavy pair. This is a kind of seesawmechanism of Higgsino doublet pairs. For this scenario,a superpotential possessing the PQ symmetry can beconstructed: W= 1 2mX2+fXX3+1 2S2T+XH 1H2−f1SH 1Hu −f2SH 2Hd+¯, /H20849119 /H20850 where /H20855X/H20856=Mand the HuHdterm is forbidden by the PQ symmetry. Then the /H9262term for Huand Hd/H20849which give mass to up- and down-type quarks, respectively /H20850is given by/H9262=/H20855f1f2S2/M/H20856/H20849Kim and Nilles, 1984 /H20850.I f /H20855S/H20856is lowered to the hidden-sector confining scale of order /H110111010–12GeV in the GMSB, the Higgsino mass can be made to be around the TeV scale by adjusting f1f2/M. One may construct models with appropriate Fterms such that B/H9262and msoft2are of the same order in the GMSB, e.g., through the PQ-symmetry-preserving termin the Kähler potential /H20885d4/H9277f1f2T* XHuHd+ H.c., /H20849120 /H20850 which also gives a /H9262term. In the so-called mixed mediation /H20849M-mediation /H20850sce- nario, with comparable moduli, anomaly, and gauge me-diations, which includes in its parameter space theGMSB, the AMSB, the mirage mediation, and the de-flected mirage mediation /H20849Everett, Kim, Ouyang, and Zurek, 2008 /H20850, the loop-generated /H9262term in general has a severe B/H9262//H9262problem. It seems that the model presented in an AMSB scenario /H20849Pomarol and Rattazzi, 1999 ;Rat- tazzi, Strumia, and Wells, 2000 /H20850has the basic ingredient for the solution of B/H9262//H9262problem according to a PQ symmetry as stressed by Giudice, Kim, and Rattazzi /H208492008 /H20850. This can also be gleaned from the axion shift symmetry in the mirage mediation scenario /H20849Nakamura, Okumura, and Yamaguchi, 2008 /H20850. F. Axions from superstrings The most interesting theory housing axions is super- string theory. Axions from strings are described by effec-tive field theory below the compactification scale. If the axion arises from the spontaneous symmetry breaking ofa tree-level global symmetry as discussed, the answer issimple: There is no such axion since string theory wouldnot allow any global symmetry. If the compactificationprocess leads to the SM, the renormalizable terms in this effective theory respect the gauge symmetry SU /H208493/H20850 c /H11003SU/H208492/H20850/H11003U/H208491/H20850Yand the global symmetries of the bary- ion number U /H208491/H20850Band the separate lepton numbers U/H208491/H20850Li. On the other hand, if the nonrenormalizable terms are allowed, one can write, for example, qLlLuRdR, breaking both the baryon number and the lepton num-ber symmetries. If the nonrenormalizable terms are in-cluded, the SM does not respect the baryon and lepton numbers symmetries. Similarly, there is no PQ globalsymmetry if we are allowed to write all nonrenormaliz-able terms. For the PQ symmetry, the situation is more severe. Suppose the singlet carrying the PQ charge is /H9268. Then/H9268*/H9268respects the PQ symmetry but /H92682and/H9268*2do not, which has led to a discussion of gravitational effectson the axion /H20849Barr and Seckel, 1992 ;Ghigna, Lusignoli, and Roncadelli, 1992 ;Holman, Hsu, Kephart, Kolb, Watkins, and Widrow, 1992 ;Kamionkowski and March- Russell, 1992 ;Dobrescu, 1997 /H20850. Therefore, the PQ sym- metry cannot be discussed in general in terms of matterfields only, when we include gravity in the discussion asin string theory. Thus, in string compactification one must consider the gravity multiplet also. Here the gauge singlet bosonic degrees in the gravity multiplet are the graviton g MN /H20849M,N=0,1,...,9 /H20850, the antisymmetric tensor BMN, and the dilaton /H9021. In ten dimensions, gMNand BMNare gauge fields. A 4D action on Minkowski space x/H9262/H20849/H9262 =0,1,2,3 /H20850is obtained by compactifying six internal spaces yi/H20849i=4,...,9 /H20850with the compact volume VZ. Some bosonic degrees from the ten-dimensional /H2084910D /H20850anti- symmetric tensor field behave like pseudoscalars in the4D effective theory. Thus, the axion candidates, if they do not arise from the matter multiplets, must be in B MN. The pseudoscalar fields in BMNare like phase fields in axion models in field theory. Because there is no globalsymmetry in string theory, there must be no massless B MN, otherwise the shift symmetry of BMNwould have worked as a global symmetry. From the tree-level equa- tions of motion all pseudoscalar BMNfields are not mass- less. For example, if a shift symmetry of BMNis related to an anomaly as in the PQ current case, we considerthat the shift symmetry is already broken. In other words, there is no shift symmetry of pseudoscalar B MN unless it is anomalous. One must deal with these bosonic degrees in string compactification to see whether these components leadto terms in the potential /H20849or the superpotential in SUSY models /H20850, which is a technical and model-dependent pro- cedure. Here we discuss axions from strings and com-ment on their phenomenological viability. Some relevantrecent reviews describing details can be found in Conlon /H208492006 /H20850and Svrcek and Witten /H208492006 /H20850. The M-theory dis-M ˜H1˜H2˜HuS ˜HdS FIG. 26. The generation of the /H9262term by a seesaw mechanism.590 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010cussion was presented in Choi /H208491997 /H20850and Svrcek and Witten /H208492006 /H20850. The pseudoscalar fields in BMNcome in two catego- ries, one the tangential component B/H9262/H9263and the other Bij.B/H9262/H9263can be discussed in any string compactification and hence is called “model independent” /H20849MI/H20850while Bij depends on the compactification scheme as its internal coordinates iand jimply and is called “model depen- dent” /H20849MD /H20850. After presenting the string formulas con- taining BMN, we discuss the MI axion in Sec. VI.F.1 and then the MD axions present in much more speculativemodels in Sec. VI.F.2 . Now, there exists a standard formula for the string action /H20851Polchinski /H208491988 /H20850, Eq. /H2084913.3.22 /H20850/H20852, which was lack- ing in the early days of string axions /H20849Choi and Kim, 1985a ;Witten, 1985a /H20850. The type-II dilaton /H9278IIand cou- pling gII=e/H9278IIare related to the 10D gravitational cou- pling/H926010byM108=1//H9260102=4/H9266/gII2/H5129s8, where/H9251/H11032=/H5129s2//H208492/H9266/H208502 /H20849Polchinski, 1988 /H20850. For the type-II string, there are NS-NS and R-Rfluxes which can give anomalous cou- plings. These complicated systems housing pseudosca-lars are reviewed in Conlon /H208492006 /H20850with the tentative that it is difficult to realize a QCD axion in string modelswith a workable moduli stabilization /H20849Kachru, Kallosh, Linde, and Trivedi, 2003 /H20850. In heterotic string models, there does not exist a reasonable moduli stabilizationmechanism, even though an ambitious attempt has beenproposed /H20849Becker, Becker, Fu, Tseng, and Yau, 2006 /H20850. However, we discuss heterotic string axions below be-cause string axions were found first in heterotic stringmodels and key couplings in axion phenomenologymight be similarly discussed also for the type-II string.As in the type-II string, the heterotic string coupling is related to the dilaton /H9021asg h=e/H9021. The kinetic energy terms of gMN,BMN, and AMare /H20851Polchinski /H208491988 /H20850, Eqs. /H2084912.1.39 /H20850and /H2084912.3.36 /H20850/H20852, LKE=/H20881−g10e−2/H9021/H20877M108 2R−M108 4/H20879dB 2−/H92753 M102gh2/H208792 −M108/H9251/H11032 8gh2Trv/H20841F2/H208412/H20878 =/H20881−g10/H208772/H9266 gh2/H5129s8R−/H9266 gh2/H5129s8/H20879dB 2−/H92753 M102gh2/H208792 −1 4/H208492/H9266/H20850gh2/H5129s6Trv/H20841F2/H208412/H20878, /H20849121 /H20850 where Trvis the trace over vector representation and the Chern-Simons three-form is /H92753=T rv/H20849A1∧dA 1+2 3A1∧A1∧A1/H20850. /H20849122 /H20850 For E 8/H11003E8, there is the adjoint representation and we use1 30Train place of Trv. For the compact internal vol- ume VZ, the Planck mass is MP=4/H9266VZ/gs2/H5129s8and the 4D gauge coupling constant is gYM2=4/H9266gs2/H5129s6/VZor/H9251YM =gs2/H5129s6/VZ. In most compactifications, the SM gauge fields arise from the level k=1 embedding and the cou- pling/H9251YMis the coupling strength at the compactifica-tion scale. If the SM gauge fields are embedded in the level k, the SM gauge coupling at the compactification scale will be smaller by the factor k. For interactions of BMN, we consider the Bianchi identity, the gauge- invariant couplings of the gaugino /H9273/H20849Derendinger, Ibanez, and Nilles, 1985 ;Dine, Rohm, Seiberg, and Wit- ten, 1985 /H20850, and the Green-Schwarz terms /H20849Green and Schwarz, 1984 /H20850, dH=1 16/H92662/H20849trR∧R−t rF∧F/H20850,HMNP/H9273¯/H9003MNP/H9273, B∧trF∧F∧F∧F+¯, /H20849123 /H20850 where HMNP is the field strength of BMN,Fis the field strength of the gauge field A, the gauge-invariant fer- mion coupling is the SUSY counterpart of the relevantterms of Eq. /H20849121 /H20850, and the ellipsis denotes more Green- Schwarz terms. It was argued that the H MNP coupling to the gaugino must be a perfect square /H20849Dine, Rohm, Seiberg, and Witten, 1985 /H20850, which gives a vanishing cos- mological constant even for a nonvanishing gaugino con- densation with nonzero /H20855HMNP /H20856/H20849Derendinger, Ibanez, and Nilles, 1985 ;Dine, Rohm, Seiberg, and Witten, 1985 /H20850. 1. Model-independent axion B/H9262/H9263with/H9262and/H9263tangent to the 4D Minkowski space- time is the MI axion present in all string compactifica-tions /H20849Witten, 1984 /H20850. Because it is a 4D gauge boson, one cannot write potential terms in terms of B /H9262/H9263and it is massless if one neglects the anomaly term. The number of transverse degrees in B/H9262/H9263is 1, and it can be expressed as a pseudoscalar aby dualizing it, H/H9262/H9263/H9267/H11008Fa/H9280/H9262/H9263/H9267/H9268/H11509/H9268a. Even though it is massless at this level, the Bianchi iden- tity of Eq. /H20849123 /H20850gives an equation of motion of aas /H115092a=/H208491/32/H92662Fa/H20850G/H9262/H9263aG˜a/H9262/H9263, which hints that amight be an axion. For it to be really a QCD axion, c2+c3should be nonzero as discussed in Sec. III.B . It is known that c3 =1 /H20849Witten, 1985b /H20850with c3defined in Eq. /H2084919/H20850. The other possible couplings are given by the second term of Eq./H20849123 /H20850, F a M102/H9280/H9262/H9263/H9267/H9268/H11509/H9268aMI/H9273¯/H9003/H9262/H9263/H9267/H9273=Fa M102/H9273¯/H9253/H9268/H92535/H9273/H11509/H9268aMI, /H20849124 /H20850 which is the c1term defined in Eq. /H2084919/H20850. There is no c2 term and c2+c3=1, and hence H/H9262/H9263/H9267is really an axion and is model independent.4This is a hadronic axion. This MI hadronic axion can have a nonvanishing c1and hence its phenomenology might be different from that of theKSVZ hadronic axion. In Eqs. /H2084961/H20850and /H2084962/H20850for the MI hadronic axion, one has to add the relevant c 1term from Eq. /H20849124 /H20850. The domain wall number of the MI axion has been shown to be NDW=1 by considering the coupling of the MI axion to a string XM/H20849/H9268,/H9270/H20850on the world sheet 4Nevertheless, its properties may depend on models in warped space /H20849Dasgupta, Firouzjahi, and Gwyn, 2008 /H20850.591 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010/H20848d2/H9268/H9280/H9251/H9252B/H9262/H9263/H11509/H9251X/H9262/H11509/H9252X/H9263/H20849Witten, 1985a /H20850.Fais about 10−3 times the Planck mass /H20849Choi and Kim, 1985a /H20850, and the correct relation, obtained from Eq. /H20849121 /H20850,i s Fa/k =/H9251cMP/23/2/H9266/H110111016GeV, where kis the level of the SM embedding and /H9251cis the QCD coupling constant /H20849Svrcek and Witten, 2006 /H20850. But the value Fa/H110111016GeV most probably overcloses the universe. An idea for lowering the MI axion decay constant may be the following. In some compactification schemes, an anomalous U /H208491/H20850angauge symmetry results, where the U/H208491/H20850angauge boson eats the MI axion so that the U /H208491/H20850an gauge boson becomes heavy. This applies to the MI ax- ion since the coupling /H11509/H9262aMIA/H9262anis present by the Green- Schwarz term /H20849Witten, 1984 ;Chun, Kim, and Nilles, 1992b /H20850. In fact, even before considering this anomalous U/H208491/H20850angauge boson, the possibility was pointed out by Barr /H208491986 /H20850; the theory became consistent after the anomalous U /H208491/H20850anfrom string compactification was dis- covered /H20849Atick, Dixon, and Sen, 1987 ;Dine, Seiberg, and Witten, 1987 ;Dine, Ichinose, and Seiberg, 1987 /H20850. Then a global symmetry survives down the anomalous U/H208491/H20850angauge boson scale. A detailed scenario is the fol- lowing. The anomalous U /H208491/H20850anwith the gauge transfor- mation,/H9258an→/H9258an+const is obtained by calculating U/H208491/H20850ancharges of fermions. Thus, we have a nonvanish- ingc2in Eq. /H2084919/H20850asm/H9274¯L/H9274Rexp /H20849ic2/H9258an/H20850and c3of the MI axion as c3/H9258MI/H20853FF˜/H20854. For all gauge group factors, the anomaly units are calculated and they are shown to beidentical /H20849Casas, Katehou, and Muñoz, 1988 ;Kim, 1988 /H20850. For the MI axion to be part of a gauge boson, it must bea true Goldstone boson without an anomaly, i.e., it should be exactly massless; so we transform away the c 3 term by a phase redefinition of fermions such that c¯2 =c2−c3/H20855/H9258MI/H20856//H9258anand c¯3=0 can occur for all gauge fields, i.e.,aMIcoupling to the anomalies vanishes for all gauge groups. Because the longitudinal gauge boson aMI is removed, we are left with the c¯2term only, m/H9274¯L/H9274Rexp /H20849ic¯2/H9258an/H20850+H.c., without the need to consider the gauge symmetry U /H208491/H20850an. At low energy, however, the term m/H9274¯L/H9274Rexp /H20849ic¯2/H9258an/H20850has a global symmetry, /H9258an →/H9258an+const, with /H9258annot depending on x/H9262. Thus, the interaction m/H9274¯L/H9274Rexp /H20849ic¯2/H9258an/H20850+H.c. explicitly shows a global U /H208491/H20850axial symmetry or PQ symmetry below the U/H208491/H20850angauge boson mass scale: /H9274L→/H9274Le−i/H9258an/2and/H9274R →/H9274Rei/H9258an/2. This global PQ symmetry can be broken in the axion window as in the field theoretic axion models.However, this idea about the decay constant does notwork necessarily, because most fields, including those re- moved at the GUT scale, carry the U /H208491/H20850 ancharge. 2. Model-dependent axion In 4D, BMNcontains more pseudoscalars Bijwith iand jtangent to the compact space VZ. If they are axions, these are MD axions. The number of massless Bijmodes at the KK mass level is the second Betti number of thecompact space /H20851Green, Schwarz, and Witten /H208491987 /H20850, Eq. /H2084914.3.10 /H20850/H20852, which was discussed in the early days in Wit-ten /H208491984 ,1985a /H20850and Choi and Kim /H208491985b /H20850. The string propagation on M 4/H11003VZcan be described by a suitable nonlinear /H9268model. In this /H9268model description, when a closed string topologically wraps VZnontrivially then there are world-sheet instantons due to the map S1 →U/H208491/H20850. It is known that the world-sheet instantons are present precisely if the second Betti number is nonzero/H20849Green, Schwarz, and Witten, 1987 /H20850, and hence the MD axions are expected to receive non-negligible massesnonperturbatively /H20849Dine, Seiberg, Wen, and Witten, 1986 ,1987 ;Wen and Witten, 1986 /H20850, but this may be a model-dependent statement /H20849Polchinski, 2006 /H20850.I faM D axion is known to have no potential term except the anomaly terms, then one should check the c 2and c3cou- plings to confirm that it is really an axion. There hasbeen no example presented yet in this way for a MDaxion. If a MD axion is present, its decay constant isexpected to be near the string scale as explicitly given by F MD=/H9251C1/3MP/23/2/H9266k1/3gs2/3from the anomaly term alone inSvrcek and Witten /H208492006 /H20850. The Green-Schwarz term integrated over VZleads to this kind of decay constant for the MD axion /H20849Choi and Kim, 1985b /H20850. However, as discussed, one has to calculate the corresponding c2term also to pinpoint the MD axion decay constant FMD. 3. Toward a plausible QCD axion from string theory A key problem in string axion models is to find a method obtaining a QCD axion at the axion window /H20849109/H33355Fa/H333551012GeV /H20850but an attractive model in this di- rection is still lacking. Thus, the most pressing issue isthe problem of introducing a detectable QCD axionfrom superstring theory. It includes the search for an approximate PQ symmetry and a detectable QCD ax-ion. The conditions for compactified manifolds in warped space needed to lower the MI axion decay constant havebeen discussed by Dasgupta, Firouzjahi, and Gwyn /H208492008 /H20850, but its realization seems nontrivial. The idea of localizing MD axions at fixed points in order to lower the decay constant has been proposed byI. W. Kim and J. E. Kim /H208492006 /H20850. It uses the warp factor idea and one needs a so-called Giddings-Kachru-Polchinski throat /H20849Giddings, Kachru and Polchinski, 2002 /H20850in the type-II string, but in the heterotic string a non-Kähler V Zis needed /H20849Becker, Becker, Fu, Tseng, and Yau, 2006 /H20850. Indeed, a warp factor is obtained in this way, but it has power law behavior. Intermediate scale string models can introduce the ax- ion window as the ultraviolet completion scale /H20849Burgess, Ibañez, and Quevedo, 1999 /H20850. On the other hand, in this case the large radius used to generate the Planck mass isthe scale needing explanation. We note that a method of obtaining F ain the axion window is through the composite axion from super-strings as discussed in Sec. VI.B . However, the compos- ite axion has not been obtained so far from string con-struction.592 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010Even if Fais lowered, we must consider the hidden sector also in estimating the axion masses and decayconstants as discussed below. 4. Hidden-sector confining forces, axion mixing, and approximate PQ symmetry With the hidden-sector confining forces, we need at least two /H20849QCD and one hidden-sector /H20850/H9258’s which have to be settled to zero, and hence we need at least twoaxions. For definiteness, consider only one more confin-ing force at an intermediate scale, which may be thesource of gravity mediation or GMSB. In this case, atleast one MD axion is assumed to be present, and axionmixing must be considered. We assume that one decayconstant is in the intermediate scale. Here there is animportant /H20849almost /H20850theorem: the cross theorem on decay constants and condensation scales. Suppose that there are two axions a 1with F1and a2with F2/H20849F1/H11270F2/H20850which couple to axion potentials with scales /H90111and/H90112/H20849/H90111 /H11270/H9011 2/H20850. The theorem states the following /H20849Kim, 1999b , 2000 ;Kim and Kim, 2006 /H20850: according to the diagonaliza- tion process in most cases with generic couplings, the larger potential scale /H90112chooses the smaller decay con- stant F1, and the smaller potential scale /H90111chooses the larger decay constant F2. So it is not enough just to ob- tain a small decay constant. The hidden sector may stealthe smaller decay constant; the QCD axion is probablyleft with the larger decay constant. We can turn thisaround such that the hidden sector instanton potential isshallower than the QCD instanton potential since theinstanton potential is proportional to the light quarkmass as discussed in Sec. III.B . If the hidden-sector quark mass is extremely small, then the QCD axion canobtain the smaller decay constant, and the other axion isan extremely light axion which can be used to fit theobserved dark energy /H20849Riess et al. , 1998 ;Perlmutter et al. , 1999 ;Komatsu et al. , 2009 /H20850. This is named the quintessential axion /H20849Kim and Nilles, 2003 ,2009 /H20850.I tc a n be easily realized if some hidden-sector squark conden-sations are very small, as Fig. 27can generate hidden- sector quark masses /H20849Kim and Kim, 2006 /H20850. Since it is difficult to obtain a reasonable light MD axion, attempts have been made to find an approximatePQ symmetry from string compactification. Only onereference exists using a realistic string compactificationbecause of the difficulty of calculating all approximatePQ charges of quarks /H20849Choi, Kim, and Kim, 2007 /H20850. After all, the topologically attractive B ijmay not be the QCDaxion we want. In this regard, we note that there already exists a field theoretic work regarding an approximate PQ symmetry, starting with a discrete Z9symmetry /H20849Laz- arides, Panagiotakopoulos, and Shafi, 1986 /H20850. Later, gravitational nonperturbative effects such as wormholesand black holes were phenomenologically studied inview of any global symmetries /H20849Giddings and Strominger, 1988 ;Lee, 1988 ;Gilbert, 1989 /H20850. It is known that the PQ-symmetry-breaking operators in the super-potential must be forbidden up to dimension 8 /H20849Barr and Seckel, 1992 ;Ghigna, Lusignoli, and Roncadelli, 1992 ; Holman, Hsu, Kephart, Kolb, Watkins, and Widrow,1992 ;Kamionkowski and March-Russell, 1992 ;Do- brescu, 1997 /H20850. If we introduce an approximate PQ sym- metry, it is better to forbid the PQ-symmetry-breakingoperators up to dimension 8 in the superpotential; pos-sibly up to dimension 7 with reasonably small couplingssomewhere. In this spirit, it is worthwhile to check approximate PQ symmetries in string-derived models. The MSSMspresented in Kim /H208492007a ,2007b /H20850,Kim, Kim, and Kyae /H208492007 /H20850, and Kim and Kyae /H208492007 /H20850, satisfy most phenom- enological constraints and one can check approximateglobal symmetries. But it is tedious work, and so far anapproximate PQ symmetry has been checked out onlyfor the flipped SU /H208495/H20850model of Kim and Kyae /H208492007 /H20850.I n searching for an approximate global symmetry in astring-derived model, there are so many Yukawa cou-plings to be considered that a complete study up to allorders is almost impossible. For example /H20849Choi, Kim, and Kim, 2007 /H20850presented O/H2084910 4/H20850d=7 superpotential terms, and it is not a trivial task to find an approximatePQ symmetry direction, considering all these terms. Upto dimension-7 terms, there exists an approximate PQsymmetry which is spontaneously broken. The resultingaxion coupling with photons has been calculated byChoi, Kim, and Kim /H208492007 /H20850and is shown in Fig. 16to- gether with the CAST and Tokyo axion search bounds/H20849Andriamonje et al. , 2007 ;Inoue et al. , 2008 /H20850. But the axion decay constant is not lowered. This is because theneeded singlet VEVs, leading to the low energy MSSM,carry PQ charges. This is a generic problem for observ-able axions from superstrings. In comparison to the MI axion case with the anomalous U /H208491/H20850 an, it may be easier to realize the observable axion with an approximate PQsymmetry. VII. AXINO COSMOLOGY Supersymmetrization of axion models includes the fermionic superpartner axino a˜and the scalar superpart- ner saxion as discussed in Sec. VI.D . Both saxion and axino masses are split from the almost vanishing axionmass if SUSY is broken. The precise value of the axinomass depends on the model, specified by the SUSY-breaking sector and the mediation sector to the axionsupermultiplet /H20849Nilles, 1984 /H20850. Most probably the saxion mass is around the soft mass scale M SUSY . The axino mass should also be near this scale as well. But the axinomass can also be much smaller /H20849Frere and Gerard, 1983 ;/angbracketleft˜¯qh˜qh/angbracketright qh ¯qh Λ3 h/M2 P• FIG. 27. /H20849Color online /H20850The hidden-sector squark condensa- tion breaks chiral symmetry and generates hidden-sectorquark masses.593 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010Kim, Masiero, and Nanopoulos, 1984 ;Chun, Kim, and Nilles, 1992a /H20850or much larger than MSUSY /H20849Chun and Lukas, 1995 /H20850. Therefore, we take the axino mass as a free parameter here. The decoupling temperature of the axino supermul- tiplet is of order /H20849Rajagopal, Turner, and Wilczek, 1991 /H20850 Ta˜dcp=/H208491011GeV /H20850/H20873Fa 1012GeV/H208742/H208730.1 /H9251c/H208743 /H20849125 /H20850 where/H9251cis the QCD coupling constant. Saxion cosmology is a simple extension of the stan- dard cosmology with saxion mass around the SUSY-breaking scale /H20849Kim, 1991 ;Chang and Kim, 1996 ;Asaka and Yamaguchi, 1999 /H20850, but its effect is not so dramatic as the effect of the axino. Therefore, here we focus on theaxino cosmology /H20849Rajagopal, Turner, and Wilczek, 1991 ; Covi, Kim, and Roszkowski, 1999 ;Covi, Kim, Kim, and Roszkowski, 2001 ;Choi, Kim, Lee, and Seto, 2008 /H20850.I n the moduli stabilization scenario of Kachru et al. /H208492003 /H20850, the saxion VEV has been estimated by Choi and Jeong /H208492007 /H20850. The axino cosmology depends crucially on the nature ofRparity. If Rparity is conserved and the axino is lighter than the neutralino, then most probably the axino or gravitino /H20849in the case of GMSB /H20850is the LSP . If R parity is not conserved, the neutralino can decay to or-dinary SM particles, as discussed by Allanach, Dedes, and Dreiner /H208492004 /H20850. Now we focus on R-parity conservation. The neu- tralino, if it is the LSP , is a natural candidate for darkmatter. Due to TeV scale sparticle interactions, the ther-mal history of neutralinos allows them to be dark matter. But if a solution of the strong CPproblem via the axion is imposed, the thermal history involves contributionsfrom the axion sector, notably by the axino. Since axinocosmology depends on neutralino and gravitino numberdensities, we comment on the neutralino and gravitinocosmologies before discussing the effect of the axino.The neutralino cosmology depends on the neutralinofreezeout temperature /H20849Lee and Weinberg, 1977b ;Drees and Nojiri, 1993 /H20850and the gravitino cosmology depends on the reheating temperature after inflation /H20849Weinberg, 1982 /H20850. Here we list several relevant temperatures in the axino cosmology, T a˜dcp, axino decoupling temperature; TR, reheating temperature after inflation; Tfr, neutralino freezeout temperature; Ta˜-rad, axino-radiation equality temperature; TD, radiation temperature right after a˜decay. /H20849126 /H20850 Here we are interested in the axino domination of the dark matter density. In the evolution history of coldaxino dark matter, either a heavy axino has decayed al-ready or it has not decayed yet. If the axino has notdecayed yet, the current axino CDM can be estimated using T a˜dcporTR. If it has decayed already, the past cold axino dark matter requires the existence of TRminat some earlier time, 4 3ma˜Ya˜/H20849TRmin/H20850=TD /H20849127 /H20850 so that Ya˜/H20849TR/H20850=na˜/s/H33356Ya˜/H20849TRmin/H20850at the time of reheating after inflation, where TRminis the temperature above which axinos dominate the universe before they decay. A. Neutralino and gravitino The neutralino LSP seems the most attractive candi- date for CDM simply because the TeV order SUSY-breaking scale introduces the LSP as a WIMP /H20849Gold- berg, 1983 ;Ellis, Hagelin, Nanopoulos, Olive, and Srednicki, 1984 /H20850. The neutralino, which was in thermal equilibrium in the early universe, decouples and freezesout when the annihilation rate becomes smaller than the Hubble parameter. The freezeout temperature T fris nor- mally given by m/H9273/25 /H20849Lee and Weinberg, 1977b ;Kolb and Turner, 1990 /H20850, e.g., 4 GeV for a 100 GeV neutralino. Obviously, the neutralino relic density is not affected by the axino: TD/H11022Tfrsince neutralinos were in thermal equilibrium after the axino decay. This is the standardneutralino dark matter. With the introduction of the axino, therefore, we study the case T D/H11021Tfr. Gravitinos in the universe are important if they domi- nate the dark matter fraction now or affected the resultof nucleosynthesis. Thermal gravitinos produced at the Planckian time are important if m 3/2/H110111 keV /H20849Pagels and Primack, 1982 /H20850. However, in the inflationary scenario these Planckian-time gravitinos are not important now.It was observed that heavy gravitino decay affects nu-cleosynthesis /H20849Weinberg, 1982 /H20850; this problem was sug- gested to be solved by inflation /H20849Krauss, 1983 ;Khlopov and Linde, 1984 /H20850. Then the gravitino number density is roughly estimated in terms of the reheating temperature after inflation, n 3/2/H11008TR. To estimate the cosmological bound on TRrather accurately, a full supergravity inter- action /H20849Cremmer, Ferrara, Girardello, and van Pröyen, 1983 /H20850has been used and applied to the dissociation problem of rare light elements such as deuterium, etc., resulting in TR/H11021109GeV /H20849Ellis, Kim, and Nanopoulos, 1984 /H20850. A recent calculation of TRhas been performed using the nucleosynthesis code to look for7Li destruc- tion and/or6Li overproduction /H20849Kawasaki, Kohri, and Moroi, 2005 ;Kawasaki, Kohri, Moroi, and Yotsuyanagi, 2008 /H20850, following the earlier work of Cyburt, Ellis, Fields, and Olive /H208492003 /H20850, which led to a stronger bound, TR /H11021108GeV if the gravitino is lighter than the gluino and TR/H11021107GeV if the gravitino is heavier than the gluino. This gravitino problem is absent if the gravitino is the next LSP /H20849NLSP /H20850,ma˜/H11021m3/2/H11021m/H9273, since a thermally pro- duced gravitino would decay into an axino and an axion,which would not affect the BBN-produced light ele-ments /H20849Asaka and Yanagida, 2000 /H20850. If the gravitino is the LSP with the stau or neutralino as the NLSP , the gravitino can be the CDM even in the594 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010constrained MSSM /H20849or mSUGRA /H20850for some parameter space, avoiding the BBN and b→s/H9253constraints /H20849Boehm, Djouadi, and Drees, 2000 ;Ellis, Olive, Santoso, and Spanos, 2004 ;Roszkowski, Ruiz de Austri, and Choi, 2005 ;Cerdeño, Choi, Jedamzik, Roszkowski, and Ruiz de Austri, 2006 /H20850. B. Axino Thus, in SUSY theories we must consider a relatively small reheating temperature 107−8GeV. Axino cosmol- ogy must also be considered with this low reheating tem-perature. In principle, the axion supermultiplet is independent of the observable sector, in which case we may take theaxino mass as a free parameter from the keV scale to avalue much larger than the gravitino mass /H20849Chun, Kim, and Nilles, 1992a ;Chun and Lukas, 1995 /H20850. Light axinos /H20849m a˜/H11351100 GeV /H20850can be a dark matter candidate and have been studied extensively as a warm dark mattercandidate /H20849Rajagopal, Turner, and Wilczek, 1991 /H20850with the reheating temperature given by Brandenburg and Steffen /H208492004 /H20850, or a CDM candidate /H20849Covi, Kim, and Roszkowski, 1999 ;Asaka and Yanagida, 2000 ;Covi, Kim, Kim, and Roszkowski, 2001 ;Roszkowski and Seto, 2007 ;Seto and Yamaguchi, 2007 /H20850. Heavy axinos, how- ever, cannot be the LSP; they can decay to the LSP pluslight particles. This heavy axino decay to neutralinos hasalready been considered /H20849Chun and Lukas, 1995 /H20850. The heavy axino possibility was considered in studying cos-mological effects of the saxion by Kawasaki and Na- kayama /H208492008 /H20850and Kawasaki, Nakayama, and Senami /H208492008 /H20850. A more complete cosmological analysis of the heavy axino has been discussed by Choi, Kim, Lee, and Seto /H208492008 /H20850. Since the CDM fraction of the universe is roughly 0.23 /H20849Komatsu et al. , 2009 /H20850, we focus on the possibility of the axino or axino-related neutralino being the CDM. Forthe axino to be the LSP , it must be lighter than the light-est neutralino and gravitino. In this case, we do not have T Dof Eq. /H20849126 /H20850. If the lightest neutralino is the NLSP , ma˜/H11021m/H9273/H11021m3/2, the thermal production /H20849TP/H20850mechanism gives the aforementioned bound on the reheating tem-perature after inflation. At a high reheating tempera-ture, TP is dominant in axino production /H20849Covi, Kim, and Roszkowski, 1999 /H20850. If the reheating temperature is below the critical energy density line, there exists an-other axino CDM possibility from nonthermally pro-duced /H20849NTP /H20850axinos which result from neutralino decay /H20849Covi, Kim, Kim, and Roszkowski, 2001 /H20850. This situation is shown in Fig. 28. We note that with R-parity conser- vation the double production of low-mass axinos is neg-ligible in supernovae, and hence there is no useful exclu-sion region from SN1987A in the low-mass region. Since the final axino energy fraction is reduced by the mass ratio /H9024 a˜h2=/H20849ma˜/m/H9273/H20850/H9024/H9273h2forma˜/H11021m/H9273/H11021m3/2, the stringent cosmologically constrained MSSM parameter space for m/H9273can be expanded. As shown in Fig. 28, the NTP axinos can be CDM for a relatively low reheatingtemperature /H20849/H1102110 TeV /H20850for 10 MeV /H11021ma˜/H11021m/H9273. In Fig. 28the thin dashed yellow corner on the RHS corre- sponds to MSSM models with /H9024/H9273h2/H11021104, and a small axino mass gives the possibility of the axino forming23% of the closure density. If all SUSY mass parameters are below 1 TeV, then probably /H9024 /H9273h2/H11021100 /H20849the thick solid corner on the RHS /H20850but a sufficient axino energy density requires ma˜/H110221 GeV. Thus, if the LHC does not detect the neutralino needed for closing of the universe,axino closing is a possibility /H20849Covi, Roszkowski, and Small, 2002 ;Covi, Roszkowski, Ruiz de Austri, and Small 2004 ;Choi and Roszkowski, 2006 ;Choi, Rosz- kowski, and Ruiz de Austri, 2008 /H20850. If the NLSP is a stau with axino or gravitino LSP , the previously forbiddenstau LSP region is erased. In this case, the CDM axino issimilar to the bino LSP case, but because of the chargeon the stau it is easier to detect the stau signal at theLHC /H20849Brandenburg, Covi, Hamaguchi, Roszkowski, and Steffen, 2005 /H20850. However, it may be difficult to detect axi- nos /H20849Kim and Kim, 2002 /H20850. In the GMSB scenario, the gravitino mass is generally smaller than the neutralino mass and possibly smallerthan the axino mass. The cosmological effect for this case has been studied by Chun, Kim, and Kim /H208491994 /H20850 and Kim and Kim /H208491995 /H20850. For a heavy axino decaying to a neutralino, we present a T Rvsma˜plot for Fa=1011GeV in Fig. 29. The region TR/H11022Ta˜dcpis above the dashed blue line. An axino lifetime greater than 0.1 s is denoted by the red shaded region on the LHS. The blue shaded region onthe RHS is where the axino decays before the neutralino decouples /H20849T D/H11022Tfr/H20850. The magenta lines /H20849horizontal /H20850are the contours of the entropy increase due to the axino decay, r/H11013Sf/S0. Above the r=1 line axinos dominate the universe before they decay. The green lines /H20849vertical /H20850de- note /H20855/H9268annvrel/H20856, where/H9268annis the neutralino annihilation cross section, in units of GeV−2, which are used to giveTRH[GeV] m˜a[GeV]101102103104105106107108109 10−810−610−410−2 1 102Hot Warm Cold TFΩNTP ˜a h2≈1EXCLUDED (ΩTP ˜ah2>1) (ΩTP ˜ah2=1 ) FIG. 28. /H20849Color /H20850Constraints of the reheating temperature as a function of the axino mass. The solid line is the upper boundfrom TP . The yellow region is the region where NTP can give cosmologically interesting results /H20849/H9024 a˜NTPh2/H112291/H20850. The freezeout temperature is Tfr/H11015m/H9273/20.595 Jihn E. Kim and Gianpaolo Carosi: Axions and the strong CPproblem Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010the right amount of neutralino relic density. In Fig. 29 we use the neutralino and gluino masses m/H9273=100 GeV and mg˜=2 TeV, respectively. For a larger Faand a heavier neutralino mass, the green lines move to theright /H20849Choi, Kim, Lee, and Seto, 2008 /H20850. ACKNOWLEDGMENTS We have benefited from discussions with K.-J. Bae, S. M. Barr, Kiwoon Choi, Ki-Young Choi, D. K. Hong,J.-H. Huh, K. Imai, H. D. Kim, I.-W. Kim, A. Melissinos,C. Muñoz, H. P . Nilles, S. Park, S. Raby, G. G. Raffelt,A. Ringwald, K. van Bibber, and K. Zioutas. This workwas supported in part by the Korea Research Founda-tion, Grant No. KRF-2005-084-C00001. REFERENCES Abbott, L. F., and P . Sikivie, 1983, Phys. Lett. B 120, 133. Adler, S. C., et al. , 1996, Phys. Rev. Lett. 76, 1421. Adler, S. L., 1969, Phys. Rev. 177, 2426. Adler, S. L., J. Gamboa, F. Mendéz, and J. López-Sarrión, 2008, Ann. Phys. 323, 2851. Aharmin, B., et al. /H20849SNO Collaboration /H20850, 2005, Phys. Rev. C 72, 055502. Ahmad, Q. 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PhysRevB.95.104432.pdf

PHYSICAL REVIEW B 95, 104432 (2017) Thermal spin torques in magnetic insulators H. Yu,1,2,*S. D. Brechet,2P. Che,1,2F. A. Vetro,2M. Collet,3S. Tu,1,2Y . G. Zhang,1Y . Zhang,1T. Stueckler,1L. Wang,1,4 H. Cui,4D. Wang,4C. Zhao,4P. Bortolotti,3A. Anane,3J-Ph. Ansermet,2,†and W. Zhao1 1Fert Beijing Research Institute, School of Electrical and Information Engineering, BDBC, Beihang University, China 2Institute of Physics, Station 3, Ecole Polytechnique F ´ed´erale de Lausanne, 1015 Lausanne-EPFL, Switzerland 3Unit´e Mixte de Physique CNRS, Thales, Univ. Paris Sud, Universit ´e Paris-Saclay, 91767 Palaiseau, France 4Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China (Received 15 December 2016; revised manuscript received 3 March 2017; published 23 March 2017) The damping of spin waves transmitted through a two-port magnonic device implemented on a yttrium iron garnet thin film is shown to be proportional to the temperature gradient imposed on the device. The sign of thedamping depends on the relative orientation of the magnetic field, the wave vector, and the temperature gradient.The observations are accounted for qualitatively and quantitatively by using an extension of the variationalprinciple that leads to the Landau-Lifshitz equation. All parameters of the model can be obtained by independentmeasurements. DOI: 10.1103/PhysRevB.95.104432 The discovery of giant magnetoresistance (GMR) rev- olutionized information storage technology [ 1,2] and the spin-transfer torque (STT), predicted two decades ago bySlonczewski [ 3] and Berger [ 4], may reshape once again the magnetic memory industry [ 5]. The concept of a heat-driven spin torque, or thermal spin-transfer torque (TST), has beensuggested [ 6–8] and opened the world of spin caloritronics. Magnetic insulators are ideal for studying the fundamentals ofspin caloritronics, because they are free of the effect of heaton charge transport. Here, we demonstrate that a spin torquecan be induced in magnetic insulators by applying a thermalgradient. The effect is not linked to spin-dependent transportat interfaces since we observe a heat-driven contribution todamping of magnetization waves on a millimeter scale. Weshow that by adding to M(r) the bound magnetic current (∇×M) as state variable, the variational principle that yields the Landau-Lifshitz equation predicts the presence ofa thermal spin torque, from which we derive an expressionfor spin currents in insulators. Our experiments verify the keypredictions of this model. Thermodynamics can predict a linkbetween heat and magnetization, but cannot determine thestrength of the effect [ 9]. Spin caloritronics studies the interplay of spin, charge, and heat transport [ 10]. As the spin dependence of the electrical conductivity proved to be important since it gives rise to GMR,the spin dependence of other transport parameters has beeninvestigated, such as that of the Seebeck [ 11] and Peltier coefficients [ 12]. The combination of heat with spin and charge transport gained widespread attention owing to studies of thespin Seebeck effect [ 13,14]. The STT effect which uses a spin- polarized electrical current has shown promising applications,e.g., in magnetic memories (STT-MRAM). It was alreadyestablished that heat flowing through a ferromagnetic metalcan generate a diffusive spin current [ 15] which induces a spin torque when flowing through a magnetic nanostructure [ 6]. Experimentally, this effect was studied in Co /Cu/Co spin valve nanowires by observing the change in the switching *haiming.yu@buaa.edu.cn †jean-philippe.ansermet@epfl.chfield of magnetization due to a local thermal gradient [ 7]. It was later shown that heat couples to magnetization dynamics[16–18]. The effect of heat on magnetization was also found in magnetic tunnel junctions [ 19] and metallic spin valves [ 20]. Slonczewski predicted that a spin-transfer torque inducedby thermal magnons could be more efficient than the usualelectrically induced spin torques [ 8]. Combining TST and STT might further decrease the write-current magnitude ofMRAMs [ 21]. A 20-nm-thick yittrium iron garnet (YIG) film was grown on gadolinium gallium garnet (GGG) substrate using pulsedlaser deposition. Details of the growth condition and magneticproperties of the thin YIG layer can be found in Ref. [ 22]. Figure 1shows the experimental principle of the mea- surement. Using inductively coupled plasma etching andphotolithography, a YIG strip 100 μm wide and 4.8 mm long was prepared. The ends were designed with a 45 ◦angle in order to avoid spin-wave reflection. Following the etching process,a 10-nm-thick copper or platinum bar was deposited on topof the YIG strip by electron-beam evaporation. This bar isconnected to two large Au electrodes. These electrodes aredesigned for contact with a ground-signal-ground microprobe.The magnetic field is applied along the YIG strip, andspin waves are excited by one microprobe and detected byanother. Alternatively, a microcoil [ 23] was used for excitation. Excitation and detection are 800 μm apart. The results were obtained with contacts made of Pt with a Ta seed layer. Theresonance frequency could be tuned from 4 GHz up to 10 GHz.Lock-in detection with field modulation was used. The thermalgradient was generated by two Peltier elements and defined as∇T=(T B−TA)/lwithl=5 mm being the distance between the Peltier elements. Using an infrared camera, we verifiedthat the temperature changed linearly at the location of thesample. As shown in Fig. 2, the linewidth changes linearly with temperature gradient. Furthermore, the slope changes signwhen the field is reversed or when the propagation direction isreversed. For the latter case, we had to move the sample andthis caused a change in the linewidth of 0.03 mT when thesample was at a uniform temperature. In Fig. 2, we translated all data points by this amount when the sample was flipped. 2469-9950/2017/95(10)/104432(4) 104432-1 ©2017 American Physical SocietyH. YU et al. PHYSICAL REVIEW B 95, 104432 (2017) x z yB0w t GGGYIGA B T FIG. 1. Spin-wave propagation under a thermal gradient. 4.8- mm-long YIG strip fabricated on GGG substrate, width w=100μm, thickness t=20 nm, 10-nm-thick Cu contact connected to Au electrodes, microprobes for both excitation and detection, Peltierelements AandBheat sunk by copper blocks (not shown). We can account for the observed effect of a temperature gradient on spin-wave transmission by a model based on anextension of the variation principle which yields the well-known Landau-Lifshitz-Gilbert (LLG) equation [ 24]. In the presence of an applied thermal gradient ∇T, the LLG equation FIG. 2. Linewidth of the ferromagnetic resonance spectra at 4.2 GHz, as a function of temperature gradient. The slope changes sign upon flipping the field (top) or flipping the direction of propaga- tion at fixed field orientation (bottom). A→Bdata are translated by 0.03 mT.for the time evolution of the magnetization Mcontains a thermal spin torque term, i.e., ˙M=γM×Beff+α MSM×˙M+γτTST, (1) where γ< 0 is the gyromagnetic ratio, αis the magnetic damping parameter, and MSis the saturation magnetization. The effective magnetic field Beffis composed of the external fieldB0, the demagnetizing field Bdem, the anisotropy field Bani, and the microwave excitation field binduced by the microwave antenna. The torque τTSTcan be expressed as τTST=αTSTω |γ|M M2s×(M×mk), (2) where the effective thermal spin torque damping coefficient αTSTcan be written as αTST=−ωM ωkT k. (3) Here, ωcorresponds to the microwave frequency and mkis the out-of-equilibrium component of the magnetization for amode of wave number k. In this work, we provide a quantitative expression for the thermal wave vector k Twith no adjustable parameter: kT=ω−ω0 ωM/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 MSdM S dT/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇T, (4) where ω 0=−γB 0andωM=−γM S. The lengthy derivations of the above equations are given in the Supplemental Mate-rial [ 25]. The effective damping parameter α effis the sum of the Gilbert damping parameter αand the thermal spin torque damping parameter αTST. The observed spin-wave spectral linewidth is therefore given by [ 25] /Delta1B=/Delta1B 0+2√ 3α/vextendsingle/vextendsingle/vextendsingle/vextendsingleω K γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle−2 √ 3/vextendsingle/vextendsingle/vextendsingle/vextendsingleω K−ω0 γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 MSdM S dT/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 k∇T, (5) where ωKis the resonance frequency, given by the Kittel formula [ 26] and the first two terms are the usual ones [ 27]. Thus, our model predicts that the thermal spin torque changes sign under reversal of either the temperature gradient,the propagation direction, or the applied magnetic field (Fig. 2). Initially, we varied the applied thermal gradient and observeda linear change in the spin-wave spectral linewidth for oneorientation of the field. This linear dependence is consistentwith Eq. ( 5). Clearly, when the thermal gradient changes sign, the linewidth changes from a broadening to a narrowing withrespect to its value in the isothermal condition. It must benoted that the temperature has hardly any influence on thelinewidth [ 25]. The dependence of linewidth with thermal gradient changes sign when the magnetic field is reversed(Fig. 2, top). This can be understood as follows. If ωchanges sign because Bis reversed, then kmust change sign also if we want propagation to be maintained in the same orientation [ 25]. Therefore, according to Eq. ( 5), the slope of the linewidth plotted vs temperature gradient must change sign when themagnetic field is reversed, as confirmed by Fig. 2(top). Furthermore, if we swap the excitation and the detection,i.e., we reverse the spin-wave vector k, then we observe that the thermal spin torque effect is also reversed, as shown in 104432-2THERMAL SPIN TORQUES IN MAGNETIC INSULATORS PHYSICAL REVIEW B 95, 104432 (2017) k = 100 rad/cm k = 35 rad/cm FIG. 3. Linewidth as a function of frequency at a set temperature gradient, using microprobe (top), or metal contacts (bottom) for excitation. Wave vector based on HFSS calculation. The appliedtemperature gradients are indicated in the figure. Top: black line yields α=3.15×10 −4; red and blue lines using Eq. ( 5). Bottom: black line yields α=6.30×10−4; red line using Eq. ( 5). The error bars indicate the noise level. Fig. 2(bottom), which is consistent with the linewidth being proportional to 1 /k[Eq. ( 5)]. We now investigate the frequency dependence of linewidth variation. The upper part of Fig. 3shows the linewidth changes with frequencies from 4.7 GHz up to 9.7 GHz using a micro-probe for excitation. We ran a high frequency electromagneticfield simulation (HFSS) taking into account the dimensionsof the microprobe and acquired the field distribution at theinjection area. We then used Fourier transformation to obtainthekspace distribution [ 25]. Thus, we found that the most prominent excitation has a wave vector around 100 rad /cm, and that there are some higher order modes with much lowerintensities. The lower part of Fig. 3shows the frequency dependence of linewidth measured using the microcoil forexcitation. According to the results from HFSS, we found thatthe dominant wave vector kof excitation is much smaller, namely, 35 rad /cm. The slope of the frequency dependence is proportional to the effective damping parameter. We canobserve that the change of the slope is more significant formicrocoil excitation than that for microprobe excitation. Thiscan be understood from Eq. ( 3) where the thermal spin torque induced damping parameter is inversely proportional to thespin-wave wave vector. We can account for the data usingthekvalues deduced from the HFSS calculation. We take the temperature dependence of the saturation magnetization to be| 1 MSdM S dT|=3.8×10−3K−1based on Ref. [ 16] and confirmed by isothermal measurements of saturation magnetization [ 25]. In the lower part of Fig. 3, we fit the data based on Eq. ( 5), using the damping parameter α=6.30×10−4deduced from the data taken without any thermal gradient. This smallervalue could be due to the fact that when using the microcoilexcitation, the detection was done using a Pt bar, whereas a Cubar was used when taking data with the microprobe excitation.According to Ref. [ 18], the growth of Pt on YIG may introduce an increase of damping. In summary, the various data presentedin Fig. 3can be accounted for quantitatively with parameters that are all determined by independent measurements. Finally, we note that the thermal spin torque [Eqs. ( 2) and ( 3)] can be expressed in terms of a spin current. To first order in the linear response, the thermal spin torque is givenby [25] τ TST=kT·js, (6) where the dot stands for the tensor contraction and the thermal spin current tensor jsis defined by js=−μ0MS×∇−1mk. (7) The spin current density tensor jshas physical dimensions (J/m2in SI units) that correspond to the product of a spin density and a phase velocity. Expression ( 7) has the same geometry to first order as the spin-wave spin current tensorderived by Saitoh and Ando [ 28]. However, the physical origin of this spin current tensor is different since here, it is obtainedspecifically for the case of a spin current induced by a thermalgradient. Very recently, self-oscillation based on spin-orbit torque was found in YIG /Pt pillars [ 29] and in permalloy /Pt nanowires [ 30]. By analogy, we may expect self-oscillation driven by a thermal spin torque as well. In conclusion, we have prepared thin-film YIG microstrips and found that the linewidth of transmission spectra can bebroadened or narrowed by applying a thermal gradient. Theseobservations are accounted for by an effective damping thatis due to a thermal spin torque. A comprehensive theoreticalanalysis provides an explicit expression for this torque, whichis derived from an extension of the variational principle onwhich the Landau-Lifshitz equation is based. This study pointsto the possibility of damping control in magnonic devices usinga local thermal gradient. We wish to acknowledge the support by NSF China under Grants No. 11674020 and No. 11444005, for S.T. by the Sino-Swiss Science and Technology Cooperation SSSTC GrantNo. EG 01-032015, for F.A.V and P.C. by the Polish-SwissResearch Program NANOSPIN PSRP-045/2010, for H.Y . bythe Deutsche Forschungsgemeinschaft SPP 1538 (SpinCat)Grant No. AN762/1, and by the International Collaboration111 Project B16001 from the Ministries of Education andForeign Experts. The authors thank Vincent Cros for commentson the manuscript. 104432-3H. YU et al. PHYSICAL REVIEW B 95, 104432 (2017) [1] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas,Phys. Rev. Lett. 61,2472 (1988 ). [2] G. Binasch, P. Gr ¨unberg, F. Saurenbach, and W. Zinn, Phys. Rev. B39,4828(R) (1989 ). [3] J. C. Slonczewski, J. Magn. Magn. Mater. 159,L1(1996 ). [4] L. Berger, Phys. Rev. B 54,9353 (1996 ). [5] A. D. Kent and D. C. Worledge, Nat. 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PhysRevB.79.144404.pdf

Theory of ferromagnetic resonance in perpendicularly magnetized nanodisks: Excitation by the Oersted field R. E. Arias1and D. L. Mills2 1Departamento de Física, FCFM Universidad de Chile, Casilla 487-3 Santiago, Chile 2Department of Physics and Astronomy, University of California–Irvine, Irvine, California 92697, USA /H20849Received 30 October 2008; published 3 April 2009 /H20850 We present theoretical studies of ferromagnetic resonance in perpendicularly magnetized nanodisks, wherein spin waves are excited through the ac modulation of the dc transport current injected into the disk. We havenanopillars in mind in our analysis, where spin-polarized current is injected from a metallic ferromagnetelsewhere in the structure. We argue that in a limit described, the modulation of the Oersted field generated bythe transport current is the dominant spin-wave excitation mechanism, and our studies explore this limit. Wecalculate the critical current above which the nominal ferromagnetic state becomes unstable through studies ofthe linewidth of the lowest spin-wave mode, which vanishes when the critical current is reached. We find thatas the applied Zeeman field H 0is decreased from values above 4 /H9266MS, the critical current has a minimum when H0/H110114/H9266MSto increase for values of the external field below this value. DOI: 10.1103/PhysRevB.79.144404 PACS number /H20849s/H20850: 75.75. /H11001a, 75.30.Ds, 76.50. /H11001g I. INTRODUCTION In recent years, there has been great interest in the study of spin dynamics in objects called nanopillars, which arenanoscale structures that typically consist of two metallicferromagnets: one with magnetization pinned or fixed bylarge anisotropy and other with magnetization that is quitefree to precess in response to stimuli. The two ferromagnetsare separated by a conducting nonmagnetic layer. Spin-polarized current may pass from the pinned magnet into thefree layer, thus exciting the spins in the free layer; the mag-netization can be excited into large amplitude highly nonlin-ear motions. 1It is also the case that through point-contact injection of spin-polarized current into an ultrathin film, one may excite large amplitude spin motions in the film, in theregion just under the contact. 2–4In the case of nanopillars, large amplitude motions of the magnetization produce micro-wave radiation. It is possible to phase lock the emission froman array of nanopillars, with the consequence that the outputof the array is very much larger than that of a singlenanopillar. 5–7 It is of considerable interest to understand the nature of the spin-wave normal modes in nanopillars, since they con-trol the response of the system in the small amplitude linear-response regime. It is also the case that the eigenvectors ofthe spin-wave modes may be utilized to describe aspects ofthe magnetization motions as one enters the nonlinear regimeas well. 8Of interest is the novel ferromagnetic resonance /H20849FMR /H20850experiment reported by Sankey et al.9These authors inject dc current into the free layer of a nanolayer whosestrength is below the threshold to excite spontaneous oscil-lations of the magnetization. They then excite spin waves bysuperimposing a small amplitude ac current onto the dc cur-rent. For the reasons discussed in Ref. 9, spin-wave excita- tion by this means leaves a signature on the dc magnetore-sistance, in the form of a peak when the modulationfrequency sweeps through a spin-wave resonance of the freelayer. In nearly all experiments on nanopillars to date, the mag- netization lies in the plane of the free layer. This is true, forexample, in the ferromagnetic resonance studies reported in Ref. 9. Analyses of the nature of the spin-wave modes of in-plane-magnetized disks, along with Brillouin light-scattering studies of the modes, were reported by Gubbiottiet al. 10,11We remark that the analysis of the case where the magnetization lies in plane is difficult to approach with ana-lytic methods, so the spin dynamics is explored in these pa-pers through the use of a micromagnetic methodology theseauthors developed. Included in their studies was the interest-ing regime wherein a vortex resides within the disk. A reviewarticle that discusses this area has appeared recently. 12While this method provides both eigenfrequencies of the modes andeigenvectors, in fact the eigenvectors that emerge are notnormalized. Because of this, it is difficult to make directcontact with experimentally measured spectra through suchcalculations. In the case where both exchange and dipolarinteractions enter importantly into the description of thespin-wave modes, the procedure for normalizing the eigen-vectors is not obvious; it should be remarked. One of us hasdeveloped a scheme by which this may be done, for a sampleof arbitrary shape, for the case where the ground-state mag-netization is spatially uniform. 13 We have recently explored the nature of the ground state and also the nature of the spin-wave normal modes, for aperpendicularly magnetized disk into which a transport cur-rent is injected perpendicular to its surfaces. 14The issue ex- amined is the influence of the Oersted field associated withthe transport current on the ground state and the spin-wavemodes. We demonstrated that in the presence of the Oerstedfield, the ground state may be viewed as a vortex state whosephysical origin and character is very different than that en-countered in disks magnetized in plane. In our vortex state,the magnetization is always perpendicular to the disk sur-faces right at the disk center and at its outer edges it is cantedand acquires a nonzero azimuthal component parallel to theplane of the disk. As the external Zeeman field H 0is de- creased from large values to the vicinity of 4 /H9266MSor below, the canting angle near the edge of the disk becomes large toapproach /H9266/2 for fields well below 4 /H9266MSand the vortexPHYSICAL REVIEW B 79, 144404 /H208492009 /H20850 1098-0121/2009/79 /H2084914/H20850/144404 /H208499/H20850 ©2009 The American Physical Society 144404-1core becomes concentrated at the disk center in a region whose spatial extent is roughly the exchange length. Thespin-wave analysis shows the vortex state to be locally stablewith respect to small perturbations down to zero applied Zee-man field. In this paper, we present an analysis of the ferromagnetic resonance spectrum of such a perpendicularly magnetizeddisk, where as in Ref. 9the spin waves are excited by im- posing an ac component onto the dc transport current. Weargue that in a parameter regime outlined below, it is themodulation of the Oersted field that is the dominant sourceof spin-wave excitation, when compared to theSlonczewski 15spin torque term. The calculations we present apply to this regime, where our previous description of theOersted field-induced vortex state is valid. We shall see thatthe modes excited have a distinctly different character thanencountered in classical ferromagnetic resonance, where spinwaves are excited by an external microwave field. In ourstudy of the Oersted field-induced ferromagnetic resonance,the Green’s function method we develop properly incorpo-rates the normalization of the eigenvectors, so the relativeintensity of the various modes in our calculated spectra isrendered correctly. Through the study of the linewidth of thelowest FMR mode as the dc current is increased, we maycalculate the critical current of our disk as a function of thestrength of the applied Zeeman field. We find that the criticalcurrent assumes a minimum value when H 0is near 4 /H9266MS and increases substantially as the applied field is reduced. We now turn to our analysis. In Sec. IIwe discuss the formalism we have developed, Sec. IIIpresents our numeri- cal studies, and final comments are included in Sec. IV. II. ANALYSIS We consider a disk of radius Rand thickness dmagne- tized perpendicular to its surfaces; the saturation magnetiza- tion is then zˆMSin the quiescent state, when transport current is absent. An external Zeeman field H0is also applied paral- lel to the zaxis. We then impose a spin-polarized transport current I/H20849t/H20850=I0+/H9254I/H20849t/H20850, with the current density J/H6023 =zˆ/H20851I/H20849t/H20850//H9266R2/H20852uniformly distributed over the disk. We assume that the disk is sufficiently thin that all magnetization com-ponents are independent of the coordinate z—the direction normal to the film surfaces—and depend only on /H9267/H6023—the co- ordinate in the plane. Our interest is in the equation of mo- tion for the magnetization M/H6109/H20849/H9267/H6023,t/H20850, which we write in the form dM/H6023/H20849/H9267/H6023,t/H20850 dt=/H9253/H20851H/H6023/H20849/H9267/H6023,t/H20850/H11003M/H6023/H20849/H9267/H6023,t/H20850/H20852+/H9251M/H6023/H20849/H9267/H6023,t/H20850 MS/H11003/H11509M/H6023/H20849/H9267/H6023,t/H20850 /H11509t +/H9253H/H6023ST/H20849/H9267/H6023,t/H20850/H11003M/H6023/H20849/H9267/H6023,t/H20850. /H208491/H20850 In the first term, H/H6023/H20849/H9267/H6023,t/H20850=H/H6023/H20849T/H20850/H20849/H9267,t/H20850+h/H6023d/H20849/H9267/H6023,t/H20850+D MS/H116122M/H6023/H20849/H9267/H6023,t/H20850. The third term in H/H6023/H20849/H9267/H6023,t/H20850is the exchange effective field, with Das the exchange stiffness, h/H6023d/H20849/H9267/H6023,t/H20850is the dipolar field set up by the time-dependent motion of the magnetization, and H/H6023/H20849T/H20850/H20849/H9267/H6023,t/H20850is the vector sum of the applied Zeeman field zˆH0and the Oersted field generated by the transport current which has the form /H9272ˆ/H208512I/H20849t/H20850/H9267/cR2/H20852, and then there is the de- magnetizing field we write here as −4 /H9266Mz/H20849/H9267/H20850zˆ. In our previ- ous paper,14we discussed the influence of corrections to this local approximation to the demagnetizing field on the spin-wave spectrum of the disk. These corrections referred to asgradient corrections led to rather small quantitative correc-tions to the spin-wave frequencies. We thus set the gradientcorrections aside in the present paper, in the interest of sim-plicity. The second term on the right-hand side of Eq. /H208491/H20850is the Gilbert form of the phenomenological damping term andthe third term is the spin torque term, with the origin in thefact that the transport current injected into the film is spinpolarized. 15,16We follow Rezende et al.17by writing the spin torque effective field as H/H6023ST/H20849/H9267/H6023,t/H20850=/H20851/H9255/H6036I/H20849t/H20850/2/H9266MS2R2de/H20852 /H11003/H20851M/H6023/H20849/H9267/H6023,t/H20850/H11003nˆ/H20852. Here /H9255is the degree of spin polarization in the transport current injected into the “active” disk, eis the magnitude of the electron charge, and the vector nˆis in the direction of the injected spin current. Again we follow theauthors of Ref. 17by choosing /H9255=0.2 in our numerical esti- mates and studies, and we write nˆ=cos /H92580xˆ+sin/H92580zˆ. The spin polarization of the transport current will in general beout of the plane of the disk by virtue of the Zeeman field thatis perpendicular to the film surfaces. Our final results do notdepend sensitively on the value of /H92580, however. We proceed, as in Ref. 14, to linearize Eq. /H208491/H20850about the ground state in the presence of the transport current. One proceeds by writing the magnetization components M/H9251/H20849/H9267/H6023,t/H20850 =M/H9251/H20849E/H20850/H20849/H9267/H6023/H20850+m/H9251/H20849/H9267/H6023,t/H20850. Here M/H9251/H20849E/H20850/H20849/H9267/H6023/H20850is the equilibrium magne- tization, which is tilted away from the zdirection by the Oersted field associated with the injected current, and as weshall discuss shortly by the spin torque term as well. Then m /H9251/H20849/H9267/H6023,t/H20850is the amplitude of the spin wave, which we assume to be small. Thus one proceeds by linearizing Eq. /H208491/H20850with respect to the small amplitude spin-wave motion described bym/H9251/H20849/H9267/H6023,t/H20850. As one proceeds with this process, one encoun- ters terms zero order in m/H9251/H20849/H9267/H6023,t/H20850. These are set to zero, and when this is done one has in hand a set of differential equa-tions that determine the form of the ground-state magnetiza- tion as described by M /H9251/H20849E/H20850/H20849/H9267/H6023/H20850. There is an issue that must be discussed before we turn to a summary of the details of thisprocedure. In our previous publication, 14we explored the influence of the Oersted field on the ground state of the perpendicu-larly magnetized disk. This led us to the vortex state whosecharacter was discussed in Sec. I. The magnetization in the ground state is here invariant under rotation about the zaxis. That is the static magnetization in the ground state M /H6023/H20849E/H20850/H20849/H9267/H6023/H20850 depends only on the distance from the origin /H9267=/H20841/H9267/H6023/H20841. In fact M/H6023/H20849E/H20850has only a zˆcomponent and a /H9272ˆcomponent, so we wrote the ground-state magnetization in the form M/H6023/H20849E/H20850/H20849/H9267/H20850 =MS/H20851/H9272ˆsin/H9274/H20849/H9267/H20850+zˆcos/H9274/H20849/H9267/H20850/H20852. A differential equation that may be solved for /H9274/H20849/H9267/H20850was derived in Ref. 14. We then explored spin-wave excitations out of the vortex ground statethrough the use of the linearized Landau-Lifschitz equation,with both Gilbert and spin torque “antidamping” set aside. Inthis framework, the spin-wave normal modes are character-ized by an azimuthal quantum number m, and in the eigen-R. E. ARIAS AND D. L. MILLS PHYSICAL REVIEW B 79, 144404 /H208492009 /H20850 144404-2vectors one encounters the factor exp /H20849im/H9272/H20850found in analyses of geometries with rotational symmetry about z. The picture in the previous paragraph is rendered more complex when the spin torque term is included in the equa-tion of motion. When one linearizes the equation of motion,the zero-order equation contains a contribution proportional toM /H6023/H20849E/H20850/H20849/H9267/H6023/H20850/H11003/H20851nˆ/H11003M/H6023/H20849E/H20850/H20849/H9267/H6023/H20850/H20852. This breaks the cylindrical sym- metry of the problem. If the spin torque magnetic field H/H6023ST/H20849/H9267/H6023,t/H20850is comparable in magnitude to the Oersted field, the ground state will be complex in nature and thus influencedimportantly by the direction of the spin polarization of theinjected current. Under such circumstances, one can makefew general statements regarding either the ground state orthe nature of the spin dynamics in nanodisks such as thoseconsidered in this paper. For this reason, the analysis and numerical calculations reported here confine their attention to the circumstancewhere the effective spin torque field is modest in magnitudecompared to the Oersted field. When the equations of motionare reduced to dimensionless form, the ratio of the spintorque effective field to the Oersted field is controlled by thedimensionless parameter /H9257=/H9255/H6036c/4/H9266MSRde. We thus confine our attention to the case where this parameter is small com-pared to unity. The spin torque “antidamping” term will becomparable to the Gilbert damping term, but our interestresides in the case where the terms in the equation of motioncontributed by the Oersted field are considerably larger thanthese two. The authors of Ref. 4introduce a spin torque term iden- tical to that in our Eq. /H208491/H20850, in their discussion of the response of a perpendicularly magnetized film to a spatially localizedsource of spin-polarized current injected into the film. Theseauthors assume that at all times the magnetization has rota-tional symmetry about the zaxis, which is located at the center of the circular disk into which the current is injected.This assumption, unfortunately, is incompatible with thesymmetry of the original Landau-Lifschitz equation that isthe starting point of their analysis, after the spin torque termis introduced. In the limit that the dimensionless parameter /H9257is small compared to unity then to good approximation the ground-state configuration of the nanodisk is well approximated bythe vortex state presented in Ref. 14. The ground-state mag- netization has the form M S/H20851sin/H9274/H20849/H9267/H20850xˆ+cos/H9274/H20849/H9267/H20850zˆ/H20852, where the angle /H9274/H20849/H9267/H20850is found by solving the equation derived in Ref. 14, D /H9267/H11509 /H11509/H9267/H20873/H9267/H11509/H9274 /H11509/H9267/H20874=D 2/H92672sin/H208492/H9274/H20850+H/H20849I/H20850/H20849/H9267/H20850sin/H9274/H20849/H9267/H20850 −H/H20849Oe/H20850/H20849/H9267/H20850cos/H9274/H20849/H9267/H20850, /H208492/H20850 where H/H20849I/H20850/H20849/H9267/H20850=H0−4/H9266MScos/H9274/H20849/H9267/H20850and the Oersted field is H/H20849Oe/H20850/H20849/H9267/H20850=2I/H9267/cR2. The boundary conditions are /H9274/H208490/H20850=0 and for reasons discussed in Ref. 14/H11509/H9274//H11509/H9267/H20841R=0. We may then proceed with the linearization process. We proceed very much as in Ref. 14, with the addition of the Gilbert damping term, and the spin torque term. We erect alocal coordinate system at each point in the disk, with oneaxis parallel to the local magnetization that lies in the planeformed by the unit vectors zˆand /H9272ˆ, a second axis in the radial direction /H9267ˆ, and the third—designated by appending the subscript tto vector components parallel to it—also lies in the plane formed by zˆand/H9272ˆ. One finds two linearized equations in the variables m/H9267andmt. The Gilbert damping term and the spin torque term then add two terms to theright-hand side of Eqs. /H208496a/H20850and /H208496b/H20850of Ref. 14after seeking solutions where the magnetization components have the timedependence exp /H20849−i/H9024t/H20850. We denote these terms with the sym- bols m˙ /H9267/H20841dampandm˙t/H20841damp, m˙/H9267/H20841damp=+i/H9251/H9024mt+HST/H20851sin/H92580cos/H9274/H20849/H9267/H20850 − cos/H92580sin/H9274/H20849/H9267/H20850sin/H9272/H20852m/H9267, /H208493a/H20850 m˙t/H20841damp=−i/H9251/H9024m/H9267+HST/H20851sin/H92580cos/H9274/H20849/H9267/H20850 − cos/H92580sin/H9274/H20849/H9267/H20850sin/H9272/H20852mt. /H208493b/H20850 Here HST=/H9255/H6036I/2/H9266MSR2de. When /H9257/H112701 the current Iin the spin torque term is replaced by its dc value. One should note the terms in Eq. /H208493/H20850that depend on the azimuthal angle /H9272. These are an illustration once again that the spin torque term breaks the radial symmetry in the prob-lem, save for the very special case that the injected currenthas its spin polarization perpendicular to the film surfaces/H20849 /H92580=/H9266/2/H20850. In what follows, we refer the reader to Eq. /H208496/H20850of Ref. 14, along with the definitions of the various quantities that enter.The driving term that excites the magnetization is the oscil-latory component of the Oersted field mentioned earlier, which has the form /H9272ˆ/H208512/H9254I/H9267/R2c/H20852exp /H20849−i/H9024t/H20850. Its role in driv- ing the magnetization may be incorporated into Eq. /H208496/H20850of Ref. 14by replacing the quantity h/H9272/H20849d/H20850in Eq. /H208496a/H20850by 2/H9254I/H9267/R2c. If we then define the two quantities HST/H20849a/H20850/H20849/H9267/H20850 =HSTsin/H92580cos/H9274/H20849/H9267/H20850andHST/H20849b/H20850/H20849/H9267/H20850=HSTcos/H92580sin/H9274/H20849/H9267/H20850, then one finds /H20875i/H9024+HST/H20849a/H20850/H20849/H9267/H20850+2D /H92672cos/H9274/H20849/H9267/H20850/H11509 /H11509/H9272/H20876m/H9267−HST/H20849b/H20850/H20849/H9267/H20850sin/H20849/H9272/H20850m/H9267 +4/H9266MSsin2/H9274/H20849/H9267/H20850mt−/H20875H/H20849T/H20850/H20849/H9267/H20850−i/H9251/H9024 −D/H20873/H116122−/H20877/H20851cos/H9274/H20849/H9267/H20850/H208522 /H92672+/H20875/H11509/H9274/H20849/H9267/H20850 /H11509/H9267/H208762/H20878/H20874/H20876mt =−2MS/H9254I cR2/H9267cos/H9274/H20849/H9267/H20850/H20849 4a/H20850 and /H20875H/H20849T/H20850/H20849/H9267/H20850−i/H9251/H9024−D/H20873/H116122−1 /H92672/H20874/H20876m/H9267+/H20875i/H9024+HST/H20849a/H20850/H20849/H9267/H20850 +2D /H92672cos/H9274/H20849/H9267/H20850/H11509 /H11509/H9272/H20876mt−HST/H20849b/H20850/H20849/H9267/H20850sin/H20849/H9272/H20850mt=0 . /H208494b/H20850 The next step is to write m/H9267,t=/H20858mm/H9267,t/H20849m/H20850/H20849/H9267/H20850exp /H20849im/H9272/H20850,s ow e haveTHEORY OF FERROMAGNETIC RESONANCE IN … PHYSICAL REVIEW B 79, 144404 /H208492009 /H20850 144404-3L1/H20849m/H20850/H20849/H9267/H20850m/H9267/H20849m/H20850/H20849/H9267/H20850+i 2HST/H20849b/H20850/H20849/H9267/H20850 /H11003/H20851m/H9267/H20849m−1/H20850/H20849/H9267/H20850−m/H9267/H20849m+1/H20850/H20849/H9267/H20850/H20852+L2a/H20849m/H20850/H20849/H9267/H20850mt/H20849m/H20850 =−2MS/H9254I cR2/H9267cos/H9274/H20849/H9267/H20850/H9254m,0 /H208495a/H20850 and L2b/H20849m/H20850/H20849/H9267/H20850m/H9267/H20849m/H20850/H20849/H9267/H20850+L1/H20849m/H20850/H20849/H9267/H20850mt/H20849m/H20850/H20849/H9267/H20850+i 2HST/H20849b/H20850/H20849/H9267/H20850 /H11003/H20851mt/H20849m−1/H20850/H20849/H9267/H20850−mt/H20849m+1/H20850/H20849/H9267/H20850/H20852=0 . /H208495b/H20850 where we have the differential operators L1/H20849m/H20850/H20849/H9267/H20850=i/H9024+HST/H20849a/H20850/H20849/H9267/H20850+2iDm /H9267cos/H9274/H20849/H9267/H20850, /H208496a/H20850 L2a/H20849m/H20850/H20849/H9267/H20850=−H/H20849T/H20850/H20849/H9267/H20850+i/H9251/H9024 +D⌊1 /H9267/H11509 /H11509/H9267/H9267/H11509 /H11509/H9267−1 /H92672/H20853m2+/H20851cos/H9274/H20849/H9267/H20850/H208522/H20854−/H20873/H11509/H9274/H20849/H9267/H20850 /H11509/H9267/H208742 ⌋ +4/H9266Mssin2/H9274/H20849/H9267/H20850, /H208496b/H20850 and L2b/H20849m/H20850=H/H20849T/H20850/H20849/H9267/H20850−i/H9251/H9024−D/H208751 /H9267/H11509 /H11509/H9267/H9267/H11509 /H11509/H9267−m2+1 /H92672/H20876. /H208496c/H20850 Of interest to us is the m=0 channel, since the Oersted field excites these modes by virtue of its rotational symmetry.Thus, the modes excited in this picture are very different innature than those excited in classical microwave ferromag-netic resonance, where the microwave field will couple toonly the m=/H110061 modes in the limit where the microwave exciting field may be viewed as spatially uniform. Now wealso see that in the presence of the spin torque terms in theequation of motion, the spin-wave eigenmodes, strictlyspeaking, are not characterized by a single azimuthal quan-tum number. Indeed, we have here an infinite hierarchy ofcoupled equations that describe the spin-wave modes. How-ever, in the limit explored in this paper, where the spintorque field is viewed as small compared to the other mag-netic fields in the problem, the admixture of m=/H110061 modes into the m=0 modes will be a small effect. In the limit that the amplitude of the spin torque field is small, it is straight-forward to develop a perturbation theoretic description ofthis admixture. One writes out the equations which describethem=/H110061 amplitudes, and the term which involves the m =0 amplitude acts as a driving term which excites the m =/H110061 modes. When the amplitudes of the m=/H110061 modes are fed back into the equation for the m=0 amplitude, we have terms proportional to /H20849H ST/H20849b/H20850/H208502. In this paper, where we explore the limit where the spin torque fields are small, we mayconfine our attention to terms linear in the amplitude of thespin torque field. Thus, we proceed by ignoring the role of the terms in H STb. In this limit, it remains the case that the spin-wave eigenmodes may be described to very good ap-proximation as modes characterized by the azimuthal quan-tum number mand, as mentioned earlier, the time-dependentcomponent of the Oersted field excites only the manifold of m=0 modes. We then address a structure that we may write in the form /H20873L1/H208490/H20850/H20849/H9267/H20850L2a/H208490/H20850/H20849/H9267/H20850 L2b/H208490/H20850/H20849/H9267/H20850L1/H208490/H20850/H20849/H9267/H20850/H20874/H20873m/H9267/H208490/H20850/H20849/H9267/H20850 mt/H208490/H20850/H20849/H9267/H20850/H20874=/H20873f/H20849/H9267/H20850 0/H20874, /H208497/H20850 where f/H20849/H9267/H20850=−2MS/H9254I cR2/H9267cos/H9274/H20849/H9267/H20850. Such a structure may be solved by presenting a pair of Green’s functions that satisfy /H20873L1/H208490/H20850/H20849/H9267/H20850L2a/H208490/H20850/H20849/H9267/H20850 L2b/H208490/H20850/H20849/H9267/H20850L1/H208490/H20850/H20849/H9267/H20850/H20874/H20873G/H9267/H208490/H20850/H20849/H9267,/H9267/H11032/H20850 Gt/H208490/H20850/H20849/H9267,/H9267/H11032/H20850/H20874=/H20873/H9254/H20849/H9267−/H9267/H11032/H20850 0/H20874. /H208498/H20850 Once the Green’s functions are constructed, we have m/H9267/H208490/H20850/H20849/H9267/H20850=−2MS/H9254I cR2/H20885 0R G/H9267/H208490/H20850/H20849/H9267,/H9267/H11032/H20850/H9267/H11032cos/H9274/H20849/H9267/H11032/H20850d/H9267/H11032 /H208499a/H20850 and mt/H208490/H20850/H20849/H9267/H20850=−2MS/H9254I cR2/H20885 0R Gt/H208490/H20850/H20849/H9267,/H9267/H11032/H20850/H9267/H11032cos/H9274/H20849/H9267/H11032/H20850d/H9267/H11032./H208499b/H20850 In Ref. 14, we argued that appropriate boundary condi- tions for such a nanodisk are that m/H9267/H208490/H20850/H20849R/H20850=0 and /H11509mt/H208490/H20850//H11509/H9267/H20841R=0. These boundary conditions assume strong surface anisotropy at the edge of the disk, with the aniso-tropy axis normal to the edge of the disk. Until a theory suchas that developed here can be brought into direct contactwith data, the actual form of the boundary condition at thedisk edge is not known; we regard this choice as reasonablefrom the physical point of view. We remark that our principalconclusions are not affected sensitively by the choice ofboundary condition at the edge of the disk. These boundary conditions will be imposed if we require G /H9267/H208490/H20850/H20849R,/H9267/H11032/H20850=0 and /H11509Gt/H208490/H20850/H20849/H9267,/H9267/H11032/H20850//H11509/H9267/H20841/H9267=R=0. Analysis of the leading behavior of Eq. /H208498/H20850close to the origin implies that the two Green’s func- tions vanish at /H9267=0 for fixed /H9267/H11032. For this class of problem, the construction of the Green’s functions is not straightfor-ward. We develop a means by which this may be done inAppendix A. III. STUDIES OF THE FMR SPECTRUM OF A PERPENDICULARLY MAGNETIZED DISK In this section, we present our numerical studies of the FMR spectrum of a uniformly magnetized disk, along withrelated issues. As noted in Sec. II, the theoretical treatment outlined above is applicable to disks for which the dimen-sionless parameter /H9257is small compared to unity. In this sec- tion, we shall present numerical studies of a model Permal-loy disk with a radius of 150 nm and a thickness of 10 nm.If, following the authors of Ref. 17we take the spin transfer efficiency /H9255to be 0.2, then the parameter /H9257=0.16. The ex- change stiffness Dhas been chosen such that the exchange length lex=/H20849D/4/H9266MS/H208501/2is equal to 6 nm, which is appropri- ate to Permalloy. The Zeeman field, of course, is perpendicu-lar to the disk, and this will tip the magnetization of thepinned layer in a nanopillar out of plane. To simulate the roleof the out-of-plane component of the spin torque field, weR. E. ARIAS AND D. L. MILLS PHYSICAL REVIEW B 79, 144404 /H208492009 /H20850 144404-4have taken the angle /H92580=30°. Strictly speaking, of course, /H92580 will vary with the applied magnetic field. To include this effect in our calculations, we need quantitative informationon magnetic properties of the pinned layer. We do not havesuch information at present for any actual sample; it will bea straightforward matter to improve this aspect of our simu-lation when data on real samples are available. In our calcu-lations, we have scaled the various magnetic fields to 4 /H9266MS. We have taken the dimensionless measure /H9251in the Landau- Lifschitz equation to be 0.02. In Fig. 1, we show a model FMR spectrum for the disk just described, for two values of the dimensionless Zeemanfield h 0=H0/4/H9266MS. We display results for h0=1.2 and h0 =0.7. We see three modes clearly, with a very small feature that is the fourth mode. Qualitatively the spectra bear a re-semblance to those published in Ref. 9, but of course no meaningful comparison is possible since the samples used inthis experiment are elliptical in shape and magnetized inplane, as opposed to the geometry analyzed in the presentpaper. We remark that we use as a measure of the FMRresponse of the disk the quantity /H20855/H208414 /H9266m−/H208412/H20856 =/H20853/H208480Rd/H9267/H9267/H208414/H9266m−/H208412/H20854/R2, where m−=m/H9267−imt. This is the domi- nant contribution to the spin-wave eigenvector, and in theabsence of mixing between m −andm+=m/H9267+imtprovided by the spin torque term, it is the only component of the eigen-vector. Its average over the disk thus serves as a sensiblemeasure of the amplitude of the FMR signal. Much information on the character of the spin waves in the disk may be extracted from the Wronskian contained inthe Green’s function /H20851the quantity W/H20849R/H20850described in Appen- dix A /H20852. In the absence of damping, the Wronskian has zeros on the real axis at each of the spin-wave frequencies withazimuthal quantum number m=0. Of course, one may easily use the method outlined in Appendix A for calculating theWronskian associated with any value of the azimuthal quan-tum number m, and the zeros of each of these provide one with the spin-wave frequencies for the appropriate azimuthalquantum number. Thus, if one wishes to generate the fre-quencies of the spin-wave normal modes rather than addressthe eigenvalue problem directly through the solution of thedifferential equations for the spin-wave modes as we did inRef. 14, one may recover the same information through the use of the Wronskian. We remark that from the computa-tional point of view, the use of the Wronskian in this matteris less demanding. When damping is added to the problem, the zeros of the Wronskian move off the real axis of the /H9024plane into the lower half plane. In our studies, in the presence of damping,we study the width of the spin-wave modes by plotting thequantity F=1 //H20841W/H20849R/H20850/H20841as a function of frequency. This func- tion has peaks centered at the spin-wave frequencies, and thefull width of the peaks at half maximum provides us with ameasure of the linewidth of the mode. We proceed by fittingthese peaks to a Lorentzian, very much as experimentalistsdo, and from this we extract the linewidth of each mode. Asthe dc current is turned on in our calculations and the spintorque effects assert themselves, the spin-wave linewidthsnarrow. The poles in the Wronskian migrate toward the realaxis in the complex /H9024plane as the dc current is increased. For any given spin-wave mode, at a certain critical current,the linewidth collapses to zero; the zero in the Wronskian lieson the real axis at this point. As the current is increasedabove the critical current, the zero of the Wronskian migratesinto the upper half plane. Once this happens, the vortex stateis unstable. The linearized equations of motion for m /H9267andmt admit solutions which increase exponentially in time in this current regime. Application of the dc current will set themagnetization into oscillatory motion when the current ex-ceeds the critical value. Of course, our theory breaks down atthis point. But through the method just described we canoutline the current regime in which the vortex state is stable,and we can calculate the critical current above which spon-taneous oscillations set in. In Fig. 2below, for the two applied magnetic fields em- ployed in Figs. 1/H20849a/H20850and1/H20849b/H20850, we show the linewidth of the first two spin-wave modes as a function of dc current for ourmodel disk. We see that the lower of the two modes goesunstable first. Then in Fig. 3we provide the critical current above which the vortex state is unstable as a function ofapplied magnetic field. What is striking is the minimum incritical current very near the applied field of H 0=4/H9266MS.I n Ref. 14, we found that when appreciable dc current is present, the spin-wave frequencies display a minimum at orvery near this field. As the Zeeman field is lowered below4 /H9266MS, the modes stiffen and this suggests that the vortex state becomes more stable as the field is lowered. We can generate eigenvectors for the spin-wave modes through the use of Eq. /H208499/H20850. The eigenvectors for the various modes may be calculated from these relations upon setting FIG. 1. /H20849Color online /H20850We show the ferromagnetic resonance spectrum of the model disk described in the text for two values ofthe Zeeman field. The quantity h 0=H0/4/H9266MSwith H0the Zeeman field applied perpendicular to the surface of the disk. In /H20849a/H20850, the dc current has been taken to be 15 mA, and in /H20849b/H20850the dc current has been taken to be 10 mA /H20849notice that the first peak is off scale, its value is on the order of 25 000 /H20850.THEORY OF FERROMAGNETIC RESONANCE IN … PHYSICAL REVIEW B 79, 144404 /H208492009 /H20850 144404-5the frequency equal to that of the peaks in the FMR response displayed in Fig. 1. In Figs. 4and5we illustrate the nature of the eigenvectors of the first two m=0 modes. As expected, the low-frequency mode is nodeless, while the second modehas a single node. We see that as the Zeeman field is low-ered, the peaks in the eigenvectors are drawn in toward thecenter of the disk, where the vortex core resides. As theapplied field is lowered, the effective field near the center ofthe disk becomes weaker; the central region acts as a poten-tial well which deepens as the field is lowered. The lowestfrequency mode is drawn inward as the well deepens, and thefirst maximum of the second mode behaves similarly. We donot show the eigenvector of the third mode. While this modeappears in our calculated spectra, its behavior is qualitativelysimilar to the first two low-lying modes just illustratedexcept—of course—it has two nodes.IV . RESULTS AND DISCUSSION We have developed a theoretical structure that allows us to discuss the spin dynamics and ferromagnetic resonancespectrum in perpendicularly polarized nanodisks, whereinthe excitation of the spin waves has its origin in ac modula-tion of the transport current injected into the disk. The ex-plicit calculations we present focus on samples in which themodulation of the Oersted field is the dominant source ofexcitation. As we pointed out, under circumstances where theeffective magnetic field associated with the spin torque termis comparable to the Oersted field, the presence of spin po-larization in the injected current breaks the rotational sym-metry of the disk; while in principle our method will apply tothis situation, it will be necessary to solve numerically ahierarchy of radial equations. As we noted in Sec. II, the authors of Ref. 4have overlooked this complication, so far as we can see. It is our view that the methodology set forthhere can be readily adapted to a full discussion of the generalproblem. We plan to turn to this in the future. Our method allows us to calculate the critical current by exploring the spin dynamics in the disk in the low currentstable vortex state presented in Ref. 14and then increasing the current until the linewidth of the lowest-lying spin-wavemode vanishes. We find the striking dependence of the criti-cal current on applied magnetic field displayed in Fig. 3, where the critical current is not a monotonic function of ap-plied magnetic field but rather has a minimum for appliedfields H 0in the near vicinity of 4 /H9266MS. We will be most FIG. 2. /H20849Color online /H20850We show the linewidth of /H20849a/H20850the lowest m=0 and /H20849b/H20850the second m=0 spin-wave mode as a function of dc current, for the two applied Zeeman fields used in the calculationsof the spin-wave spectrum in Fig. 1. We give the ratio of the full width at half maximum to the spin-wave frequency in these figures. FIG. 3. /H20849Color online /H20850The critical current as a function of ap- plied magnetic field, for the model disk discussed in the text. Theinstability is controlled by the m=0 spin wave, whose linewidth is the first to go to zero. FIG. 4. /H20849Color online /H20850The radial variation in the dominant con- tribution to the eigenvector of the lowest frequency spin-waveeigenmode in the m=0 manifold, for the two magnetic fields used in the calculation of the FMR spectrum in Fig. 1. As in Fig. 1, the dc current assumes the value of 15 mA for the lower of the twoapplied fields and 10 mA for the higher applied field.R. E. ARIAS AND D. L. MILLS PHYSICAL REVIEW B 79, 144404 /H208492009 /H20850 144404-6interested to see the experimental study of samples such as those explored here. ACKNOWLEDGMENTS We have enjoyed several stimulating discussions with I. Krivorotov, regarding studies of the spin-wave spectrum innanodisks through modulation of the spin torque current.This research was supported by the U. S. Army through Con-tract No. CS000128. R.E.A. also acknowledges support fromFONDECYT under Contract No. 1085028 /H20849Chile /H20850and from the Millennium Science Nucleus “Basic and Applied Magne-tism” under Contract No. P06-022-F /H20849Chile /H20850. APPENDIX A: CONSTRUCTION OF THE GREEN’S FUNCTIONS In what follows, the two functions G/H9267/H208490/H20850/H20849/H9267,/H9267/H11032/H20850and Gt/H208490/H20850/H20849/H9267,/H9267/H11032/H20850will be considered to be the functions of the vari- able/H9267with/H9267/H11032held fixed. We will make use of solutions of the homogeneous equations /H20873L1/H208490/H20850/H20849/H9267/H20850L2a/H208490/H20850/H20849/H9267/H20850 L2b/H208490/H20850/H20849/H9267/H20850L1/H208490/H20850/H20849/H9267/H20850/H20874/H20873m˜/H9267/H208490/H20850/H20849/H9267/H20850 m˜t/H208490/H20850/H20849/H9267/H20850/H20874=/H208730 0/H20874, /H20849A1 /H20850 where we have added the tilde above the mto distinguish these special functions from the physical magnetization ofthe sample. We may construct four independent solutions/H20853m ˜/H9267i/H208490/H20850/H20849/H9267/H20850,m˜ti/H208490/H20850/H20849/H9267/H20850/H20854through the use of “one-sided” boundary conditions as follows: m˜/H92671/H208490/H20850/H208490/H20850=m˜t1/H208490/H20850/H208490/H20850=0 , m˜/H92672/H208490/H20850=m˜t2/H208490/H20850=0 , /H20849A2a /H20850 /H20879/H11509m˜/H92671/H208490/H20850 /H11509/H9267/H20879 0=1 ,/H20879/H11509m˜t1/H208490/H20850 /H11509/H9267/H20879 0=0 ,/H20879/H11509m˜/H92672/H208490/H20850 /H11509/H9267/H20879 0=0 ,/H20879/H11509m˜t2/H208490/H20850 /H11509/H9267/H20879 0=1 , /H20849A2b /H20850 m˜/H92673/H208490/H20850/H20849R/H20850=m˜t3/H208490/H20850/H20849R/H20850=0 , m˜/H92674/H208490/H20850/H20849R/H20850=0 ,m˜t4/H208490/H20850/H20849R/H20850=1 , /H20849A2c /H20850 /H20879/H11509m˜/H92673/H208490/H20850 /H11509/H9267/H20879 R=1 ,/H20879/H11509m˜t3/H208490/H20850 /H11509/H9267/H20879 R=0 ,/H20879/H11509m˜p4 /H11509/H9267/H20879 R=/H20879/H11509m˜t4 /H11509/H9267/H20879 R=0 . /H20849A2d /H20850 Such solutions of the homogeneous equation exist for any value of the frequency The Green’s functions we seek may then be written in the form G/H9267,t/H208490/H20850/H20849/H9267,/H9267/H11032/H20850=g/H9267,t/H11021/H20849/H9267,/H9267/H11032/H20850/H9258/H20849/H9267/H11032−/H9267/H20850+g/H9267,t/H11022/H20849/H9267,/H9267/H11032/H20850/H9258/H20849/H9267−/H9267/H11032/H20850, /H20849A3 /H20850 where /H9258/H20849x/H20850is the Heaviside step function. We assert that the functions g/H11022/H20849/H9267,/H9267/H11032/H20850andg/H11021/H20849/H9267,/H9267/H11032/H20850can be written in the form g/H9267,t/H11021/H20849/H9267,/H9267/H11032/H20850=m˜/H9267,t1/H208490/H20850/H20849/H9267/H20850a1/H20849/H9267/H11032/H20850+m˜/H9267,t2/H208490/H20850/H20849/H9267/H20850a2/H20849/H9267/H11032/H20850/H20849 A4a /H20850 and g/H9267,t/H11022/H20849/H9267,/H9267/H11032/H20850=m˜/H9267,t3/H208490/H20850/H20849/H9267/H20850a3/H20849/H9267/H11032/H20850+m˜/H9267,t4/H208490/H20850/H20849/H9267/H20850a4/H20849/H9267/H11032/H20850. /H20849A4b /H20850 If these forms are inserted into Eq. /H208499/H20850and the conditions in Eq. /H20849A2 /H20850are noted then the boundary conditions for m/H9267,t/H208490/H20850/H20849/H9267/H20850stated in Sec. IIare automatically obeyed. We next turn to an argument from which the four functions ai/H20849/H9267/H11032/H20850 may be determined. We may determine the functions a/H9267,ti/H20849/H9267/H11032/H20850by matching g/H9267,t/H11022/H20849/H9267,/H9267/H11032/H20850tog/H9267,t/H11021/H20849/H9267,/H9267/H11032/H20850at the point /H9267=/H9267/H11032and by also utilizing the jump condition on the derivatives, which reads /H11509gt/H11022/H20849/H9267,/H9267/H11032/H20850//H11509/H9267/H20841/H9267/H11032−/H11509gt/H11021/H20849/H9267,/H9267/H11032/H20850//H11509/H9267/H20841/H9267/H11032=+1 /D while /H11509g/H9267/H11022/H20849/H9267,/H9267/H11032/H20850//H11509/H9267/H20841/H9267/H11032−/H11509g/H9267/H11021/H20849/H9267,/H9267/H11032/H20850//H11509/H9267/H20841/H9267/H11032=0. These conditions lead to four inhomogeneous linear equations which—when inverted—allow the functions ai/H20849/H9267/H11032/H20850to be expressed in terms of the known functions m˜/H9267,ti/H208490/H20850/H20849/H9267/H11032/H20850. These equations may be written in the form M/H20849/H9267/H11032/H20850·A/H20849/H9267/H11032/H20850=B, /H20849A5 /H20850 where M/H20849/H9267/H11032/H20850i sa4/H110034 matrix whose elements are the func- tions m˜/H9267,ti/H208490/H20850/H20849/H9267/H11032/H20850and the derivatives /H11509m˜/H9267,ti/H208490/H20850//H11509/H9267/H20841/H9267/H11032, M/H20849/H9267/H11032/H20850=/H20898m˜t1/H208490/H20850/H20849/H9267/H11032/H20850,m˜t2/H208490/H20850/H20849/H9267/H11032/H20850,−m˜t3/H208490/H20850/H20849/H9267/H11032/H20850,−m˜t4/H208490/H20850/H20849/H9267/H11032/H20850 m˜/H92671/H208490/H20850/H20849/H9267/H11032/H20850,m˜/H92672/H208490/H20850/H20849/H9267/H11032/H20850,−m˜/H92673/H208490/H20850/H20849/H9267/H11032/H20850,−m˜/H92674/H208490/H20850/H20849/H9267/H11032/H20850 m˜t1/H11032/H208490/H20850/H20849/H9267/H11032/H20850,m˜t2/H11032/H208490/H20850/H20849/H9267/H11032/H20850,−m˜t3/H11032/H208490/H20850/H20849/H9267/H11032/H20850,−m˜t4/H11032/H208490/H20850/H20849/H9267/H11032/H20850 m˜/H92671/H11032/H208490/H20850/H20849/H9267/H11032/H20850,m˜/H92672/H11032/H208490/H20850/H20849/H9267/H11032/H20850,−m˜/H92673/H11032/H208490/H20850/H20849/H9267/H11032/H20850,−m˜/H92674/H11032/H208490/H20850/H20849/H9267/H11032/H20850/H20899, /H20849A6 /H20850 and FIG. 5. /H20849Color online /H20850The radial variation in the dominant con- tribution to the eigenvector of the second spin-wave eigenmode inthem=0 manifold, for the two magnetic fields used in the calcula- tion of the FMR spectrum in Fig. 1. As in Fig. 1, the dc current assumes the value of 15 mA for the lower applied field and 10 mAfor the higher field.THEORY OF FERROMAGNETIC RESONANCE IN … PHYSICAL REVIEW B 79, 144404 /H208492009 /H20850 144404-7A/H20849/H9267/H11032/H20850=/H20898a1/H20849/H9267/H11032/H20850 a2/H20849/H9267/H11032/H20850 a3/H20849/H9267/H11032/H20850 a4/H20849/H9267/H11032/H20850/H20899 with B=/H208980 0 −1 /D 0/H20899. The functions ai/H20849/H9267/H11032/H20850are found by inverting the matrix in Eq. /H20849A5 /H20850. When this is done, of course one encounters the deter- minant of the matrix M/H20849/H9267/H11032/H20850,W/H20849/H9267/H11032/H20850=Det /H20851M/H20849/H9267/H11032/H20850/H20852, which we call the Wronskian since it is a generalization of the Wronsk-ian encountered in classical Sturm-Liouville theory. We referto this quantity as the Wronskian in the text, and in whatfollows here. We conclude with the statement of a usefultheorem which serves as a generalization of the standardtextbook theorem regarding the behavior of the Wronskianencountered in the construction of the Green’s function instandard Sturm-Liouville theory. Suppose we have a set of N functions /H20853F n/H20849/H9267/H20850/H20854which satisfy differential equations of the form dFn/H20849/H9267/H20850 d/H9267=/H20858 l=1N pnl/H20849/H9267/H20850Fl/H20849/H9267/H20850. /H20849A7 /H20850 Equation /H20849A7 /H20850will admit Nsets of linearly independent solutions and the jth set will be labeled /H20853Fn/H20849j/H20850/H20849/H9267/H20850/H20854. Let Q/H20849/H9267/H20850be the matrix Q/H20849/H9267/H20850=/H20898F1/H208491/H20850/H20849/H9267/H20850F1/H208492/H20850/H20849/H9267/H20850¯F1/H20849N/H20850/H20849/H9267/H20850 F2/H208491/H20850/H20849/H9267/H20850F2/H208492/H20850/H20849/H9267/H20850¯F2/H20849N/H20850/H20849/H9267/H20850 ] ]] F N/H208491/H20850/H20849/H9267/H20850FN/H208492/H20850/H20849/H9267/H20850¯FN/H20849N/H20850/H20849/H9267/H20850/H20899. /H20849A8 /H20850 We then have the following theorem: d d/H9267DetQ/H20849/H9267/H20850=T r P/H20849/H9267/H20850DetQ/H20849/H9267/H20850, /H20849A9 /H20850 where Tr P/H20849/H9267/H20850=/H20858l=1Npll/H20849/H9267/H20850. Now, we may apply this theorem by making the identifi- cation F1/H20849i/H20850/H20849/H9267/H20850=m˜/H9267i/H208490/H20850/H20849/H9267/H20850,F2/H20849i/H20850/H20849/H9267/H20850=m˜ti/H208490/H20850/H20849/H9267/H20850,F3/H20849i/H20850/H20849/H9267/H20850=/H11509m˜/H9267i/H208490/H20850//H11509/H9267, F4/H20849i/H20850/H20849/H9267/H20850=/H11509m˜ti/H208490/H20850//H11509/H9267/H20849the negative signs on the third and fourth columns of Eq. /H20849A6 /H20850are innocuous as far as the determinant is concerned /H20850. When Eq. /H20849A1 /H20850is written in terms of the set /H20853Fn/H20849i/H20850/H20849/H9267/H20850/H20854it has a form compatible with those in Eq. /H20849A7 /H20850, and the matrix M/H20849/H9267/H20850has the form of Q/H20849/H9267/H20850in Eq. /H20849A8 /H20850. One finds that Tr P/H20849/H9267/H20850=−2 //H9267so that Det /H20851M/H20849/H9267/H20850/H20852=W/H20849/H9267/H20850=CR2//H92672. The constant C=C/H20849/H9024/H20850may be evaluated by computing W/H20849R/H20850so we haveW/H20849/H9267/H20850=/H20873R /H9267/H208742 W/H20849R/H20850. /H20849A10 /H20850 The structure of C/H20849/H9024/H20850=W/H20849R/H20850as a function of frequency is used in this paper in order to obtain the frequencies andlinewidths of the modes. The proof of the theorem in Eq. /H20849A9 /H20850is sketched in Ap- pendix B. APPENDIX B: PROOF OF EQ. ( A9) In what follows we use the summation convention wherein one sums over repeated indices. For simplicity, wealso confine our attention to the case where there are four functions in our set /H20853F n/H20849j/H20850/H20854. The extension of the proof to the case where we have Nfunctions in the set is straightforward. Thus, Eq. /H20849A7 /H20850is written /H11509Fi/H20849j/H20850 /H11509/H9267=pilFl/H20849j/H20850. /H20849B1/H20850 Then after we form the matrix Q/H20849/H9267/H20850defined in Appendix A, we have DetQ=/H9255ijknFi/H208491/H20850Fj/H208492/H20850Fk/H208493/H20850Fn/H208494/H20850. /H20849B2/H20850 where /H9255ijknis the Levi-Civita tensor of rank four, which van- ishes when any two indices are equal, and which equals +1or −1 depending on whether ijkn is an even or odd permu- tation of 1234. Clearly, if D=Det /H20849Q/H20850, dD d/H9267=/H9255ijknpilFl/H208491/H20850Fj/H208492/H20850Fk/H208493/H20850Fn/H208494/H20850+/H9255ijknpjlFi/H208491/H20850Fl/H208492/H20850Fk/H208493/H20850Fn/H208494/H20850+ ... /H20849B3/H20850 In the terms which involve pilwith l/HS11005i,lhas to be equal to j,k,o rnand of course i,j,k, and nmust all be different by virtue of /H9255ijkn. A similar statement applies to the term which involves pjl. Suppose in the first term on the right-hand side of Eq. /H20849B3/H20850, we consider the term with l=j, which has the form/H9255ijknpijFj/H208491/H20850Fj/H208492/H20850Fk/H208493/H20850Fn/H208494/H20850. We compare this with the contri- bution from the second term for which l=i. This has the form /H9255ijknpjiFi/H208491/H20850Fi/H208492/H20850Fk/H208493/H20850Fn/H208494/H20850. Upon interchanging the summation indices iand jin the last-mentioned term, it becomes /H9255jiknpijFj/H208491/H20850Fj/H208492/H20850Fk/H208493/H20850Fn/H208494/H20850, and we see that it exactly cancels the first-mentioned term. Similarly, all other couples of this typecancel each other. Thus the only terms in Eq. /H20849B3/H20850which survive are the terms which come from the diagonal elements of p il. Thus, Eq. /H20849B3/H20850becomes dD d/H9267=/H9255ijknpiiFi/H208491/H20850Fj/H208492/H20850Fk/H208493/H20850Fn/H208494/H20850+/H9255ijknpjjFi/H208491/H20850Fj/H208492/H20850Fk/H208493/H20850Fn/H208494/H20850+ ... =/H20849pii+pjj+pkk+pnn/H20850/H9255ijknFi/H208491/H20850Fj/H208492/H20850Fk/H208493/H20850Fn/H208494/H20850=T r /H20849P/H20850D, /H20849B4/H20850 where as in Appendix A, Tr /H20849P/H20850=/H20858i=14pii.R. E. ARIAS AND D. L. MILLS PHYSICAL REVIEW B 79, 144404 /H208492009 /H20850 144404-81See, for instance, I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, R. C. Ralph, and R. A. Burhman, Science 307, 228 /H208492005 /H20850. 2M. Tsoi, A. G. M. Jansen, J. Bass, W. C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 /H208491998 /H20850. 3W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. 92, 027201 /H208492004 /H20850. 4M. A. Hoefer, M. J. Ablowitz, B. Ilan, M. R. Pufall, and T. J. Silva, Phys. Rev. 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Car- lotti, T. Okuno, T. Shinjo, and M. Grimsditch, Phys. Rev. B 70, 172404 /H208492004 /H20850. 12C. H. Back, D. Pescia, and M. Buess, in Spin Dynamics in Con- fined Strutures III: Topics in Applied Physics , edited by B. Hill- ebrands and A. Thiaville /H20849Springer-Verlag, Heidelberg, 2006 /H20850, Vol. 101, p. 137. 13D. L. Mills, J. Magn. Magn. Mater. 306,1 6 /H208492006 /H20850. 14R. E. Arias and D. L. Mills, Phys. Rev. B 75, 214404 /H208492007 /H20850. 15J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 16L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 17S. M. Rezende, F. M. de Aguiar, and A. Azevedo, Phys. Rev. Lett. 94, 037202 /H208492005 /H20850.THEORY OF FERROMAGNETIC RESONANCE IN … PHYSICAL REVIEW B 79, 144404 /H208492009 /H20850 144404-9

PhysRevMaterials.5.064411.pdf

PHYSICAL REVIEW MATERIALS 5, 064411 (2021) Spin-wave localization and guiding by magnon band structure engineering in yttrium iron garnet Rouven Dreyer, Niklas Liebing, Eric R. J. Edwards,*Andreas Müller, and Georg Woltersdorf† Institute of Physics, Martin Luther University Halle-Wittenberg, 06120 Halle, Germany (Received 2 March 2021; accepted 1 June 2021; published 21 June 2021) In spintronics, the propagation of spin-wave excitations in magnetically ordered materials can also be used to transport and process information. One of the most popular materials in this regard is the ferrimagnetic insulatoryttrium iron garnet due its exceptionally small spin-wave damping parameter. While the small relaxation rateallows for large propagation length of magnetic excitations, it also leads to nonlocality of the magnetic properties.By imaging spin waves, their band structure is mapped with high-frequency resolution using a magneto-opticsuper-Nyquist sampling technique. In doing so, wave-vector selection is shown to suppress dispersion effectsto a large extent, allowing for local measurements of spin relaxation. Moreover, we demonstrate even highercontrol of magnon propagation by employing the wave-vector selectivity near an avoided crossing of differentspin-wave modes where the group velocity approaches zero. Here the local engineering of the dispersion allowsus to construct magnonic waveguides, and at the same time it reveals the local relaxation properties. DOI: 10.1103/PhysRevMaterials.5.064411 In recent years, spin-wave propagation and its control have been an intensely studied topic [ 1,2]. In parallel, it has been demonstrated that spin waves may be used to transport heat[3] and angular momentum [ 4]. In many of these experiments, yttrium iron garnet (YIG) has proven to be a valuable material.The insulating properties of YIG were used in YIG/metal hy-brid structures to demonstrate a flurry of magnetoresistive andmagnetothermal phenomena, which are explained by the exci-tation or annihilation of spin waves in YIG [ 5–9]. At the same time, the exceptionally small Gilbert damping constant of onlyα=5×10 –5of YIG enables spin transport on the millimeter length scale [ 3,10]. In most cases, the presence of magnon ex- citations in YIG can be probed on the nanoscale by the inversespin Hall effect [ 6,11,12]. However, this approach is not sen- sitive to the properties of the spin wave that is converted into asignal, i.e., its wavelength and propagation direction. Magne-tization dynamics at the micro- and nanoscale can be studiedinductively [ 13–17] or by optical methods. Four distinct op- tical approaches are typically used: (i) microfocus Brillouinlight scattering ( μBLS) [ 18–23], (ii) time-resolved scanning transmission x-ray microscopy (TR-STXM) [ 24–26], (iii) time-resolved magneto-optic Kerr microscopy (TR-MOKE)[27–33], and (iv) diamond nitrogen-vacancy (NV) center res- onance imaging [ 34–37]. The long spin-wave relaxation times in YIG complicate the analysis since extrinsic effects suchas sample inhomogeneity, magnon-magnon scattering [ 38], or instrumental effects related to the excitation of multi-ple spin-wave modes or limited frequency resolution usuallydominate the measured linewidth [ 39]. On the other hand, the large spin-wave propagation length allows to investigate *Present address: IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA. †georg.woltersdorf@physik.uni-halle.decoupling phenomena such as avoided crossings in the spin- wave dispersion in single YIG films [ 40,41] or YIG-based heterostructures [ 42–44]. In such systems, the strong-coupling regime between different magnon modes is accessible [ 45], paving the way for a novel playground for coherent infor-mation processing based on magnons [ 46]. Due to the large propagation lengths in YIG, it is difficult to confine spinwaves. Usually, confinement is only achieved by physicallypatterning (e.g., dry-etching) the YIG material. Unfortunately,this approach introduces defects and modifies the magneto-static properties. Therefore, for a number of experiments itwould be highly desirable to have a method at hand allowingus to control the spin-wave properties locally without the needfor patterning of the YIG structures. In this article, we study the properties of spin waves in thin YIG layers by phase-resolved magneto-optic imagingof coherently excited spin waves. To reach the required fre-quency resolution and sensitivity for experiments with YIG,a modified version of the TR-MOKE method is introduced.In addition to the direct measurement of the spin-wave dis-persion and avoided crossings of different spin-wave modes,we demonstrate that a truly local measurement of spin-waverelaxation properties becomes possible. In addition, by ex-tracting group velocities and relaxation times of the excitedspin waves near an avoided spin-wave mode crossing, weobtain an independent estimate for the local Gilbert dampingparameter. Finally, we engineer the spin-wave dispersion lo-cally to construct a soft magnonic waveguide, which allows usto study spin-wave propagation inside the spin-wave band gapopened by an avoided crossing of different spin-wave modes. In our experiments, we perform TR-MOKE experiments on a 200-nm-thick YIG layer. The magnetization is excitedcoherently using the rf-field generated by a coplanar waveg-uide (CPW) patterned on top of the YIG layer [as shownin Fig. 1(a)]. The wave-vector spectrum of the excited spin waves is determined by the static in-plane magnetic field as 2475-9953/2021/5(6)/064411(6) 064411-1 ©2021 American Physical SocietyROUVEN DREYER et al. PHYSICAL REVIEW MATERIALS 5, 064411 (2021) FIG. 1. (a) Geometry of the experiments. Spatial resolved images of spin waves in DE and BV configurations with their corresponding wavelength are shown in the gap. (b) The frequency comb generated by a femtosecond laser is given by multiples of the laser repetitionrate f rep. The excitation frequency f rfaliases back to the Nyquist frequencies (red dotted lines). The lowest aliasing frequency cor- responds to the difference frequency εbetween f rfand the nearest comb line. well as the frequency and spatial distribution of the excitation field. We distinguish between the Damon-Eshbach (DE) andbackward volume (BV) configurations as limiting cases ofthe in-plane dispersion [ 40], cf. supplemental Fig. S1 [ 47]. As a light source, a femtosecond laser operating at 510 nmwith a repetition rate of f rep=80 MHz is used to sample the magnetization dynamics via the polar magneto-opticalKerr effect (MOKE), as shown in supplemental Fig. S2 [ 47]. Due to the small spin relaxation, studying spin waves inYIG requires a method with a frequency resolution on theorder of 1 MHz. To meet this requirement, we introduce ameasurement scheme that we term super-Nyquist samplingMOKE (SNS-MOKE). The MOKE effect allows for mix-ing of the excitation frequency f rfand the nth harmonic of the laser pulse-repetition frequency yielding an intermediatefrequency f rf=nfrep+ε. Demodulating the Kerr signal at frequency εdirectly yields real and imaginary components of the magnetic rf-susceptibility, and in doing so it providesphase-resolved measurements of the spin precession (see thesupplemental material [ 47] for details). Clearly, the advantage of the SNS-MOKE technique is that it allows for tuning ofthe rf-frequency in arbitrary steps. This is an enormous im-provement over conventional pump-probe microscopy, wherethe frequency resolution is given by the laser repetition rate(i.e., 80 MHz in our case). Previously, we have used theSNS-MOKE technique only at fixed frequencies to imagespin-wave modes [ 31,48,49]. Typical data of the spatially resolved, complex suscepti- bility are shown in Fig. 1(a) and recorded in DE and BV configurations, respectively. Here the external magnetic field FIG. 2. Extracted spin-wave dispersion for a fixed external field of 79 mT. The dispersion curves were computed with the recipe by Kalinikos and Slavin [ 40,58] as discussed in the supplemental material. The red solid lines indicate the DE and BV dispersion. The dotted red line shows an angular orientation of nearly flat dispersion. The black solid line depicts the dispersion branch for the first PSSW.The top left inset shows the spin-wave profile in the vicinity of the avoided mode crossing recorded in the gap of the CPW. The top right inset shows a magnified view of the avoided mode crossing with thecalculated mode repulsion [ 40,58]. is fixed at 142 mT and the rf-frequency is varied from im- age to image. The excited wave vector is determined bythe maximum of the product of the k-dependent rf-magnetic field and rf-magnetic susceptibility h(k)×χ(ω,k)[39], and it can be determined by counting the number of maximanobserved over distance Land calculating the wave num- ber as |k|=2πn/L. In the following, we use the spin-wave wavelengths λ=|k|/2πdetermined from spatially resolved images [Fig. 1(a)] or line scans to map out the dispersion of BV and DE modes by extracting the wave vectors as afunction of frequency for a fixed magnetic field, as shown inFig.2. Using the SNS-MOKE method, we are able to map out the spin-wave dispersion with a pronounced avoided crossing[24,50–53] using a step size of only 2 MHz, as shown in the inset of Fig. 2. The avoided crossing of the first-order perpendicular standing spin-wave mode (PSSW) and the DE mode hasa much larger frequency splitting than the linewidth ofindividual spin-wave modes involved, indicating strong cou-pling. Following the formalism introduced by Kalinikos andSlavin [ 40,54], we determine the mode repulsion of DE and first-order PSSW mode and the corresponding size of the fre-quency splitting (see supplemental Fig. S3 [ 47]). In particular, we obtain a coupling constant g=f splitting/(/Delta1f1+/Delta1f2)o f 220 at 4 GHz from the experiment. Note that values largerthan g=1 are referred to as strong coupling [ 55], and they allow for coherent exchange of information between the twomodes. The solid lines in Fig. 2show the calculated dispersion for DE, BV (red), and PSSW (black) modes. In Fig. 2,t h e left inset shows the observed spatial profile of one of thehybridized modes in the vicinity of the avoided crossing. Herethe propagation of the spin wave is strongly suppressed in 064411-2SPIN-WA VE LOCALIZATION AND GUIDING BY MAGNON … PHYSICAL REVIEW MATERIALS 5, 064411 (2021) FIG. 3. (a) Angular dependence of the local field swept measure- ments of the rf-susceptibility measured with an excitation frequencyof 4.0 GHz. A clear minimum of the dispersion around a magnetic field direction of 55 degrees is visible. The inset shows the nearly uniform spatial distribution of the magnetic excitation across the gaprecorded at the dispersion minimum. (b)–(e) Detailed measurement of the spin-wave dispersion minimum for frequencies between 3 and 6 GHz. The inset in panel (e) depicts the extracted linewidth of the 5.6 GHz measurement as a function of in-plane field orientation clearly showing a minimum at nearly flat dispersion. comparison to the maps shown in Fig. 1(a). We attribute this effect to the nearly flat dispersion and therefore small groupvelocity close to the anticrossing. Specifically, the spin-wavegroup velocity is reduced from 200 m/s to values of about10 m/s near the mode repulsion (indicated by the purple arrow in the right inset of Fig. 2). The Gilbert damping parameter is usually determined from the linewidth by sweeping the magnetic field at a fixed fre-quency across the ferromagnetic resonance (FMR). In Fig. 3 we record the SNS-MOKE signal in two-dimensional plots asa function of in-plane magnetic field magnitude and in-planeorientation. As one can see, e.g., by following the signalalong the dotted line, locally measured field sweeps are not FIG. 4. (a) Frequency dependence of the linewidth for FMR measurements (black cross) and at the dispersion minimum (red dots). The blue data points show the calculated linewidth in thevicinity of the avoided crossing. Solid lines are fits to extract the Gilbert damping. (b) Gilbert damping for localized spin-wave modes for different frequency. The dotted black line indicates the averagedamping parameter, while the gray area is the standard deviation. The orange star marks the data point taken within the magnonic waveguide presented in Fig. 5(d). suitable to determine the Gilbert damping (cf. supplemental Fig. S1 [ 47]). To evaluate these spectra, it would be necessary to take the wave-vector distribution of the excitation field,the dispersion, and the spin-wave propagation effects prop-erly into account [ 56]. One might overcome this problem by identifying the magnetization direction where the dispersionis nearly flat and the spin waves cannot propagate. A nearlyflat dispersion is expected for an intermediate angle of thefield orientation between BV and DE configurations wherethe different dipolar contributions compensate each other (reddotted line in Fig. 2). The actual angle where the dispersion becomes nearly flat is indicated by black circles in Fig. 3and depends on the rf-frequency (Fig. S4 [ 47]). A flat dispersion results in the simultaneous excitation of spin waves of allexcited wave vectors, causing destructive interference of allspin-wave modes except for the uniform mode ( k=0). This behavior is indeed observed in the inset of Fig. 3(a), where mostly a uniform precession of the magnetization occurs. At 4GHz, a flat dispersion is expected for an angle between kand Mof 55 degrees as indicated in Fig. 2. In addition, because the dispersion is flat, the excited spin waves have a nearlyvanishing group velocity v g=∂ω/∂ kand cannot propa- gate. As demonstrated in the inset of Fig. 3(e), a pronounced minimum of the resonance linewidth is indeed observed forthese conditions. By measuring the susceptibility at the anglesof minimal dispersion, we extract Lorentzian resonance lineshapes from the local SNS-MOKE spectra that can be easilyinterpreted in terms of their linewidth. Figure 4(a) shows the frequency-dependent linewidth determined from a seriesof such spectra obtained in the point of minimal dispersion.The Gilbert damping determined from the red data points inFig.4(a) corresponds to a value of α=(5.41±1.07)×10 –5 with a very small zero-frequency linewidth offset /Delta1H0of only 064411-3ROUVEN DREYER et al. PHYSICAL REVIEW MATERIALS 5, 064411 (2021) 16μT and hence three times smaller compared to conven- tional FMR measurements we performed on the same sample.The remaining zero-frequency linewidth offset is most likelycaused by two-magnon scattering processes, which are facili-tated by a flat dispersion extending to large k-vectors. An even better localization of the probed properties may be achieved by controlling the magnon band structure suchthat degenerate magnon modes can be avoided completely. Inthe following, we will look in detail at the presented avoidedcrossing, and in doing so study spin waves with tunable prop-agation length with the aim to localize them. The associatedchange in curvature of the dispersion offers a handle to controlthe magnon propagation length since it is given by the productof spin-wave lifetime and group velocity [ 39]:λ prop=τvg, where the lifetime [ 57] is related to the Gilbert damping pa- rameter by τ=2 α1 γμ 0(M0+H). (1) Obviously, in the case of a flat dispersion, the probed signal has a truly local character and provides access to intrinsiclocal properties such as internal fields and the Gilbert dampingparameter. In the region of mode repulsion, the excited spin-wave modes are localized close to the edges of the conductorsof the CPW (left inset in Fig. 2). Using different frequen- cies and a nearly flat dispersion found in the vicinity of theavoided mode crossing (shown in supplemental Fig. S3 [ 47]) we extract the propagation length λ propand the group velocity vgfor these modes and calculate an average Gilbert damp- ing parameter of α=(4.08±0.97)×10–5for frequencies between 1 and 8 GHz using the equation above. The narrowlinewidths for different frequencies [Fig. 4(a)] and the corre- sponding Gilbert damping parameters [Fig. 4(b)] calculated from the obtained data are a consequence of the local charac-ter of the damping measurement under nearly flat dispersionconditions. In the following, we will demonstrate that the magnon band gap at small wave vectors created by the avoided crossingallows us to realize magnon guiding along a track definedby tiny local fields. One can set the external field and theexcitation frequency such that no spin waves can be excitedand propagate [gray shaded area in, e.g., Fig. 5(a)]. To lo- cally guide spin waves along a predefined track, it is requiredto locally shift this band gap. For this we define magneticstructures on top of the YIG layer in order to provide a localbias field in their gap, as shown in Figs. 5(c) and5(d).T h e local in-plane stray field amounts to ∼0.5 mT for 10-nm-thick Permalloy structures with a 5 μm gap, and it induces a shift of the magnon dispersion of about 100 MHz [Fig. 5(b)]. Now one can select a frequency where the spin-wave propagationwithin the magnonic waveguide is still allowed while it isforbidden due to the magnon band gap in the surroundingmagnetic material. This situation is demonstrated in Fig. 5(d) for a frequency of 4.45 GHz and an applied external field of79 mT. The spin-wave mode within the magnonic waveguidecan propagate over several 10 μm while the propagation on the left and right side of the waveguide is evanescent. Bydetermining the decay length of this specific spin-wave modeand the corresponding group velocity from the dispersion, wefind a Gilbert damping parameter of α=(6.4±1.3)×10 –5 FIG. 5. Parts (a) and (b) show the magnonic band gap for an external field of 79 mT at positions next to and within the magnonic waveguide, respectively. In (b) the dispersion is slightly shifted to-ward larger frequencies. Here we find a regime where in contrast to the band state in (c), the propagation of DE spin-wave modes is only possible within the waveguide (gap state) as observed in (d). (and we calculate a linewidth of 10.3 ±2.1μT) (both indi- cated by orange stars in Fig. 4). In summary, we demonstrate that by selecting the orien- tation of the wave vector, it is possible to avoid spin-wavedispersion to a large extent. Even more interestingly, strongmodification of the dispersion in the vicinity of the avoidedcrossing allows us to select arbitrarily low values of thespin-wave velocity and hence to address the Gilbert dampinglocally. Using the exceptional frequency resolution of theSNS-MOKE technique introduced here, we show that thespin-wave interaction in the vicinity of the mode crossingrepresents a case of strong coupling, allowing us to study co-operative phenomena such as Rabi oscillations for spin-waveexcitations as well in the future. Finally, we demonstrated thata forbidden magnon band can be created by the avoided cross-ing. By local modification of the dispersion using the strayfield of magnetic microstructures, we have designed a “soft”magnonic waveguide. This waveguide supports spin waves inthe vicinity of the avoided crossing, while the propagation inthe surrounding material is forbidden. 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PhysRevB.87.134403.pdf

PHYSICAL REVIEW B 87, 134403 (2013) Magnetoelectric resonances and predicted microwave diode effect of the skyrmion crystal in a multiferroic chiral-lattice magnet Masahito Mochizuki1,2,*and Shinichiro Seki3 1Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan 2Institute of Theoretical Physics, University of Cologne, D-50937 Cologne, Germany 3Department of Applied Physics and Quantum Phase Electronics Center, The University of Tokyo, Tokyo 113-8656, Japan (Received 20 September 2012; published 3 April 2013) We theoretically discover that unique eigenmodes of skyrmion crystal (SkX) are not only magnetically active to an ac magnetic field ( Hω) but also electrically active to an ac electric field ( Eω) in a multiferroic chiral-lattice magnet Cu 2OSeO 3, which amplifies the dynamical magnetoelectric coupling between Eωand the spin texture. The resulting intense interference between their electric and their magnetic activation processes can lead to anunprecedentedly large diode effect on the microwave, i.e., its absorption by SkX changes up to ∼20% when the incident direction is reversed. Our results demonstrate that the skyrmion could be a promising building block formicrowave devices. DOI: 10.1103/PhysRevB.87.134403 PACS number(s): 76 .50.+g, 75.10.Hk, 75 .70.Ak, 75 .78.−n Skyrmion, a topological vortex-like swirling spin texture,1 is now attracting a great deal of interest. It was predicted that the skyrmion and its crystallized form, so-called skyrmioncrystal (SkX), are realized in chiral-lattice magnets without in-version symmetry through competition between ferromagnetic(FM) and Dzyaloshinskii-Moriya (DM) interactions undera magnetic field H. 2,3Quite recently, the SkX phase was indeed observed in metallic B20 alloys such as MnSi,4–7 Fe1−xCoxSi,8,9and FeGe,10as well as in the insulating magnet Cu2OSeO 3,11–13by small-angle neutron-scattering (SANS) experiments and Lorentz transmission electron microscopy(LTEM). Since then, several experiments have been performed and have reported intriguing transport properties in SkX 14–19and electric control of skyrmions with spin-polarized current20,21in metallic systems. The emergence of spin-driven ferroelectric polarization Phas been observed in the insulating SkX phase of Cu 2OSeO 3,11,22,23and the electric-field control of this mul- tiferroic skyrmion texture was experimentally demonstrated.24 There the research interest, more or less, comes from the possible application to next-generation spintronics devices.However, a lot of researchers are presaging further potentialityin skyrmions. Nevertheless, the sorts of experimental work arequite limited, i.e., observations by means of LTEM or SANSand transport measurements only. In this article, we theoretically propose a brand new direction for the research on skyrmions from the viewpointsof microwave functionalities and dynamical phenomena atgigahertz frequencies. We discover that collective rotationaland breathing motions of skyrmions in SkX can be resonantlyactivated not only by an ac magnetic field ( H ω) but also by ac electric field ( Eω) as unique eigenmodes of SkX in multiferroic chiral-lattice magnets. These resonances amplify the dynam-ical coupling of underlying spin texture with E ωandHω, and the resulting intense interference between the electric andthe magnetic activation processes can lead to unprecedentedlylarge directional dichroism of the electromagnetic (EM) wavein Cu 2OSeO 3; i.e., its absorption by SkX changes up to ∼20%, depending on the sign of its incident direction. This effect isenhanced especially at eigenfrequencies of the aforementionedskyrmion resonances ( ∼GHz) and can work as an efficient microwave diode. Currently, most microwave-device functionsare achieved using designed combinations of waveguides,circuits, and elements made of ferrites with ferrimagneticorder. 25Our finding provides a guideline for designing new microwave devices such as a magnetically tunable isolator.Our work will be a trigger for a broad-based quest for novelfunctions of skyrmions and related spin textures. The magnetic structure of Cu 2OSeO 3is composed of tetrahedra of four Cu2+(S=1/2) ions as shown in Fig. 1(a). Recent powder neutron diffraction26and NMR27experiments suggested that a three-up- and one-down ( uuud )-type collinear spin arrangement is realized on each tetrahedron belowT c∼58 K. We regard this four-spin assembly as a magnetic unit and treat it as a classical vector spin miwhose norm mis unity. We employ a classical Heisenberg model on a cubic lattice28–30to describe the magnetism in a thin specimen of Cu 2OSeO 3, which contains the FM-exchange interaction, the DM interaction,31and the Zeeman coupling to the external Hnormal to the plane. The Hamiltonian is given by H0=−J/summationdisplay /angbracketlefti,j/angbracketrightmi·mj−D/summationdisplay i,ˆγmi×mi+ˆγ·ˆγ −gμ Bμ0Hz/summationdisplay imiz, (1) where g=2, and ˆγruns over ˆxand ˆy. We set the ratio D/J=0.09, for which the periodicity in the SkX phase becomes ∼99 sites. Since the distance between adjacent Cu-ion tetrahedra is ∼5˚A, this periodicity corresponds to a skyrmion diameter of ∼50 nm, in agreement with the LTEM observation.11All the spin textures considered here are slowly varying, which can be described by a continuum spin model.It justifies our treatment based on a lattice spin model afterdivision of the space into square meshes and coarse graining ofmagnetizations. We first analyze the model, (1), using the replica-exchange Monte Carlo technique and obtain the phase diagrams at lowtemperature ( T) shown in Fig. 1(b). The SkX phase emerges 134403-1 1098-0121/2013/87(13)/134403(5) ©2013 American Physical SocietyMASAHITO MOCHIZUKI AND SHINICHIRO SEKI PHYSICAL REVIEW B 87, 134403 (2013) 62’ abc 21||<100> 3||<111>(e)mzi 21’P||[001] (||y) M||H||[110] (||z)(d) zxyH||z(a) gμBμ0H[110] /J 1.875x10-3 6.3x10-3(b) HL FM SkX -0.5 0.5 0pzi /λ xy100 sites 100 sites z(f)(c) FIG. 1. (Color) (a) Spin structure of Cu 2OSeO 3, composed of tetrahedra of four Cu2+ions (S=1/2) with three-up and one-down spins. (b) Phase diagram of the spin model, (1). Here HL, SkX, and FM denote the helical, skyrmion-crystal, and ferromagnetic phases, respectively. (c) Spin structure in the SkX phase, which possessesa sixfold rotation axis, 6, and six twofold rotation axes, followed by time reversal, 2 /prime. Arrows represent in-plane spin components. (d) Under H/bardbl[110], the system becomes polar along [001] and the emergence of ferroelectric polarization P/bardbl[001] is allowed. (e) Symmetry axes in a Cu 2OSeO 3crystal, which belongs to the P213 space group: threefold rotation axes, 3, along /angbracketleft111/angbracketrightand twofold screw axes, 2 1, along /angbracketleft100/angbracketright. (f) Real-space configurations of local electric polarizations piin the skyrmion under H/bardbl[110]. in the range 1 .875×10−3<|gμ Bμ0Hz/J|<6.3×10−3, sandwiched by the helical and FM phases, in agreementwith the experiment for thin-plate samples. 11Skyrmions are crystallized into a triangular lattice and magnetic moments mi directly antiparallel (parallel) to Hat the center (periphery) of each skyrmion as shown in Fig. 1(c). For the SkX state formed under H/bardbl[110] as shown in Fig. 1(d), the emergence of P/bardbl[001] perpendicular to the net magnetization M/bardblHis expected from symmetry considerations. As shown in Fig. 1(e), the crystal structure of Cu 2OSeO 3, which belongs to a nonpolar space group, P213, possesses threefold rotation axes, 3, along /angbracketleft111/angbracketrightand 21-screw axes along /angbracketleft100/angbracketright. The spin texture in the SkX phase is also nonpolar, with a sixfold rotation axis, 6, along Hand twofold rotation axes followed by time reversal, 2/prime, normal toHas shown in Fig. 1(c). When the SkX sets in under H/bardbl[110] on the Cu 2OSeO 3crystal, most of the symmetries should be broken, and only the 2/prime 1axis (/bardbl[001]) normal to Hsurvives, as shown in Fig. 1(d). Consequently, the system becomes polar along [001]. Indeed the emergence of P/bardbl[001] under H/bardbl[110] was experimentally observed.23Here we define the Cartesian coordinates, x/bardbl[¯110], y/bardblP/bardbl[001], andz/bardblM/bardbl[110], shown in Fig. 1(d), for convenience of the following formulations.The net magnetization Mand the ferroelectric polarization Pare given by sums of the local contributions as M= gμB NV/summationtextN i=1miandP=1 NV/summationtextN i=1pi, respectively, where the index iruns over the Cu-ion tetrahedra with uuud spinsN is the number of tetrahedra, and V(=1.76×10−28m3)i s the volume per tetrahedron. Because of the cubic symmetry,the local polarization p ifrom the ith tetrahedron is given using the spin components mia,mib, and micin the P213 setting as pi=(pia,pib,pic)=λ(mibmic,micmia,miamib).(2) We can easily evaluate the local contributions piandmi from each tetrahedron in the ferrimagnetic phase where all the tetrahedra give uniform contributions. Then the couplingconstant λis evaluated as λ=5.64×10 −27μCm from the experimentally measured P[001]=16μC/m2in the ferrimag- netic phase under H/bardbl[111] at 5 K.11 Because of this strong coupling between magnetism and electricity, collective oscillations of this SkX can be activatednot only magnetically by an ac magnetic field H ωbut also electrically by ac electric field Eω. As demonstrated below, with the special configuration of P⊥M, both the Hωand the Eωcomponents of an EM wave propagating along P×Mcan activate common oscillation modes. To see this, we calculatedynamical magnetic and dielectric susceptibilities, χ mm αβ(ω)=Mω α μ0Hω β,χee αβ(ω)=Pω α /epsilon10Eω β, (3) by numerically solving the Landau-Lifshitz-Gilbert equation using the fourth-order Runge-Kutta method. The equation isgiven by dm i dt=−mi×Heff i+αG mmi×dmi dt, (4) where αG(=0.04) is the Gilbert-damping coefficient. The effective field Heff iis calculated from the Hamiltonian H= H0+H/prime(t)a sHeff i=−∂H/∂mi. Here the first term H0is the model Hamiltonian, (1), while the perturbation term H/prime(t) represents a short rectangular pulse of a magnetic field orelectric field. After applying the pulse at t=0, we calculate M(t) andP(t) and obtain their Fourier transforms M ω αandPω α. Calculations are performed using a system of N=288×288 sites with the periodic boundary condition. In Fig. 2(a), we display imaginary parts of the calculated dy- namical magnetic susceptibilities, Im χmm yy(ω) and Im χmm zz(ω), forHω/bardblyandHω/bardblz. We also plot the imaginary parts of the calculated dielectric susceptibilities, Im χee zz(ω) and Im χee yy(ω), forEω/bardblzandEω/bardblyin Fig. 2(b).I nI m χmm yy, we find a strong resonance active to Hω/bardblyatωR/J=6.12×10−3, which was ascribed to the counterclockwise rotation mode,where all the skyrmion cores in the SkX uniformly rotatein the counterclockwise fashion. 32,33This rotation mode can also be seen in the spectrum of Im χee zzas a peak at the same frequency, indicating its simultaneous electric activitytoE ω/bardblz. The spectrum of Im χmm yyhas one more resonance at a higher frequency, ωR=1.135×10−2J, which was ascribed to another rotation mode with opposite rotational sense, i.e.,the clockwise rotation mode. 32,33We can see a very tiny peak in Imχee zzat the corresponding frequency, indicating its weak 134403-2MAGNETOELECTRIC RESONANCES AND PREDICTED ... PHYSICAL REVIEW B 87, 134403 (2013) (a) (b)0.02 0.04 0 00.020.040412ImχαβeeImχαβmmImχzzmm Imχyymm Imχzzee Imχyyee8ω/J CCWrotation CWrotationbreathing 6.12x10-37.76x10-31.135x10-2 FIG. 2. (Color online) (a) Imaginary parts of the dynamical magnetic susceptibilities, Im χmm yyand Im χmm zz, as functions of the frequency ωatgμ Bμ0Hz/J=3.75×10−3.I nR e f . 32, the strong (weak) resonance in Im χmm yywas ascribed to the counterclockwise (clockwise) rotation mode, while the resonance in Im χmm zzwas ascribed to the breathing mode. (b) Imaginary parts of the dynamical dielectric susceptibilities, Im χee zzand Im χee yy, as functions of ω.T h e three above-mentioned magnetically active resonances can be seen in the spectra of dielectric susceptibilities at the same frequencies, indicating their simultaneous electric activities. electric activity. On the other hand, the spectrum of Im χmm zzhas a single resonance active to Hω/bardblzatωR/J=7.76×10−3, which was ascribed to the breathing mode, where areas of allthe skyrmions in the SkX oscillatory expand and shrink in auniform way. 32,33Again, this mode is simultaneously active to Eω/bardbly, and the corresponding peak can be seen in Im χee yyat the same frequency. A recent microwave experiment found clearabsorptions at these spin-wave resonances, while absorptionsat off-resonant frequencies turn out to be negligibly small. 34 The presence of collective modes active to both Eω andHωis nothing but the source of interesting microwave activity. From Maxwell’s equations, we can derive the relation Hω/bardblKω×Eωfor the EM wave. This relation indicates that the relative directions of HωandEωare determined by the propagation vector Kω, and their relationship should be reversed upon the sign reversal of Kω. When a lineally polar- ized EM wave with Eω/bardblzandHω/bardblypropagates parallel (antiparallel) to ( P×M)/bardblxa ss h o w ni nF i g . 3(a), where sgn[Re Kω]=+ 1(sgn[Re Kω]=− 1) with Kω=Kωˆx,t h e oscillation of Pinduced by Eωand that of MbyHω contributes in a subtractive (an additive) way to the collective oscillation, which results in weaker (stronger) absorption ofthe EM wave. Such a nonreciprocal absorption of the EMwave is expected also for E ω/bardblyandHω/bardblz,a ss h o w ni n Fig. 3(b). To study microwave absorption and nonreciprocal direc- tional dichroism (NDD) quantitatively, we start with thefollowing Fourier-formed Maxwell’s equations for materialswithMandP: 35 ωBω=Kω×Eω,−ωDω=Kω×Hω, (5) M KHω EωK HωEω abc zxysgn(ReKω)=+1 Eω ΔMΔP P M Hωadditive subtractive EEωωEEEEEω ΔMΔP P M HωEω ΔMΔP P M HωEEEωωEEEE Eω ΔMΔP P M HωωωMK HωEω KHω Eω abc zxy Eω ΔMΔP P MP MHω P MP MEω Eω Eω Hω HωHωadditive subtractive 21’P21’P sgn(ReKω)=+1sgn(ReKω)= -1 sgn(ReKω)= -1(a) Hω||y, Eω||z (b) Hω||z, Eω||y FIG. 3. (Color online) Configurations of the microwave Hωand Eωcomponents, for which nonreciprocal directional dichroism is expected when P/bardblyandM/bardblzwithP⊥MandKω/bardblP×M/bardbl x:( a )Eω/bardblzandHω/bardbly,a n d( b ) Eω/bardblyandHω/bardblz. Cartesian coordinates x/bardbl[¯110], y/bardbl[001], and z/bardbl[110] are defined as shown, where the zaxis is parallel to H. where Bω=μ0(ˆμ∞Hω+Mω),Dω=/epsilon10ˆ/epsilon1∞Eω+Pω,(6) Mω=μ0ˆχmm(ω)Hω+ˆχme(ω)/radicalbigg/epsilon10 μ0Eω, (7) Pω=/epsilon10ˆχee(ω)Eω+ˆχem(ω)√/epsilon10μ0Hω. (8) As discussed above, NDD of microwaves with Kω=Kωˆx is expected for the following configurations of HωandEω: (i)Hω/bardblyandEω/bardblz, and (ii) Hω/bardblzandEω/bardbly.W e introduce the complex refractive index N(ω)=n(ω)+iκ(ω), which is related to KωasKω=ω cN(ω). We solve Eqs. (5) and obtain N(ω)∼/radicalBig/bracketleftbig /epsilon1zz∞+χeezz(ω)/bracketrightbig/bracketleftbig μyy∞+χmmyy(ω)/bracketrightbig −sgn(Re Kω)/bracketleftbig χme yz(ω)+χem zy(ω)/bracketrightbig/slashbig 2( 9 ) and N(ω)∼/radicalBig [/epsilon1yy∞+χeeyy(ω)][μzz∞+χmmzz(ω)] +sgn(Re Kω)[χme zy(ω)+χem yz(ω)]/2 (10) for cases (i) and (ii), respectively. These expressions contain the sign of Re Kωindicating the direction dependence of the microwave absorption because the absorption coefficient α(ω) 134403-3MASAHITO MOCHIZUKI AND SHINICHIRO SEKI PHYSICAL REVIEW B 87, 134403 (2013) is related to N(ω)a s α(ω)=2ωκ(ω) c∝ωImN(ω), and the absorption intensity is given by I(ω)= I0exp[−α(ω)l], where lis the sample thickness. The nonreciprocal absorption /Delta1α(ω)=α+(ω)−α−(ω) represents the magnitude of NDD, where α+andα−are absorption coefficients for microwaves propagating in the positive andnegative directions, respectively. In order to evaluate N(ω) andα ±(ω) quantitatively, we need to calculate not only the dielectric and magnetic susceptibilitiesbut also the following dynamical ME susceptibilities: χ em αβ(ω)=Pω α√/epsilon10μ0Hω β,χme αβ(ω)=/radicalbiggμ0 /epsilon10Mω α Eω β. (11) For the values of /epsilon1∞ zzand/epsilon1∞ yyin Eqs. (9)and(10), we assume an isotropic dielectric tensor, i.e., /epsilon1∞ zz=/epsilon1∞ yy=/epsilon1∞, for simplicity and set /epsilon1∞=8 according to the dielectric-measurement data.22,36In turn, we take μ∞ zz=μ∞ yy=1 for permeability. The value of Jis set to be J=1 meV so as to reproduce the experimental Tcfor the SkX-paramagnetic transition. In Figs. 4(a) and 4(b), we display the calculated ω dependence of the absorption coefficients, α+(ω) andα−(ω), at several values of HzforEω/bardblzandHω/bardblyforEω/bardbl yandHω/bardblz, respectively. The out-of-plane Eω/bardblzand in-plane Hω/bardblyactivate two rotation modes with opposite senses,32,33i.e., lower-lying counterclockwise and higher- lying clockwise modes, which give two spectral peaks inFig. 4(a). We find that, for the lower-lying resonance, the /Delta1α(ω) increases as H zincreases and reaches more than 0.25 cm−1, which corresponds to a relative change /Delta1α/α ave= 2(α+−α−)/(α++α−)∼20% at maximum. On the other hand, both the in-plane Eω/bardblyand the out-of-plane Hω/bardblz activate a breathing mode,32,33which gives a single spectral peak in Fig. 4(b).A g a i n ,t h e /Delta1α(ω) increases with increasing Hzand reaches approximately 0.14 cm−1, corresponding to a/Delta1α/αaveof 10%. These values do not depend on the value ofαG. Such a huge directional dichroism is quite rare for any frequency range35,37–39and has never been realized at gigahertz frequencies. This is because most of the multiferroics withsimple magnetic orders have resonant frequencies much higherthan microwave frequencies due to the large spin gaps. In turn,the long-period skyrmion textures have small spin gaps andtheir nontrivial collective modes with gigahertz frequenciesenable us to achieve interesting microwave functions. In summary, we have theoretically predicted that the SkX phase of the chiral-lattice insulator Cu 2OSeO 3shows an enhanced diode effect on linearly polarized microwaves00.40.81.21.6 α−α+6.25 X10-3α+, α− (cm-1)(a) Hω//y, Eω//z 3.75 X10-3 0.060.12 13 ω (GHz)(b) Hω//z, Eω//y α−α+ 01 234 56 7 8 ω (GHz)6.25 X10-33.75 X10-3 1.875 X10-3 6.25 X10-33.75 X10-31.875 X10-31.875 X10-3 0.40.81.21.6α+, α− (cm-1) 0 Δα=α+−α− (cm-1) Δα=α+−α− (cm-1) -0.2-0.11.875 X10-3 6.25 X10-33.75 X10-3 13 ω (GHz)gμμ0Hz/J= gμμ0Hz/J=gμμ0Hz/J=gμμ0Hz/J= FIG. 4. (Color online) Calculated absorption coefficients, α+(ω) andα−(ω), for microwaves with sgn(Re Kω)=+ 1 and sgn(Re Kω)= −1, respectively, at several values of Hzin the cases of (a) Hω/bardbly andEω/bardblzand (b) Hω/bardblzandEω/bardbly. as a consequence of interference between magnetic and electric responses from the multiferroic skyrmion texture. 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PhysRevLett.118.097201.pdf

Magnetic Domain Wall Floating on a Spin Superfluid Pramey Upadhyaya, Se Kwon Kim, and Yaroslav Tserkovnyak Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA (Received 4 August 2016; revised manuscript received 9 November 2016; published 27 February 2017) We theoretically investigate the transfer of angular momentum between a spin superfluid and a domain wall in an exchange coupled easy-axis and easy-plane magnetic insulator system. A domain wall in the easy-axis magnet absorbs spin angular momentum via disrupting the flow of a superfluid spin current in the easy-plane magnet. Focusing on an open geometry, where the spin current is injected electrically via a nonequilibrium spin accumulation, we derive analytical expressions for the resultant superfluid-mediated motion of thedomain wall. The analytical results are supported by micromagnetic simulations. The proposed phenomenon extends the regime of magnon-driven domain-wall motion to the case where the magnons are condensed and exhibit superfluidity. Furthermore, by controlling the pinning of the domain wall, we propose a realization of areconfigurable spin transistor. The long-distance dissipationless character of spin superfluids can thus be exploited for manipulating soliton-based memory and logic devices. DOI: 10.1103/PhysRevLett.118.097201 Introduction. —Spin currents carried by a collective excitation of magnets, in lieu of charge currents, have recently attracted vibrant experimental and theoreticalactivities, opening a subfield of spintronics dubbed mag- nonics [1]. This is motivated in part by the prospects of constructing low-dissipation spintronic devices. Apart fromallowing for the Joule heating-free transfer of spin signals,magnons also offer the possibility of imparting their spinangular momentum to topological solitons [2]. These solitons [3], such as domain walls and Skyrmions, are robust against fluctuations and are thus considered ideal candidates for encoding nonvolatile information [4]. Recent experimental demonstrations of thermal magnon-induced domain-wall [5]and Skyrmion motion [6]could thus provide a basis for all-magnonic nonvolatile memory(such as the racetrack register [4]) and logic devices [7]. On another front, these magnons offer a unique pos- sibility to form coherent condensates at room temperature, as demonstrated experimentally by parametric (microwave)pumping in a magnetic insulator [8]. Such condensates present an exciting opportunity for magnonics by support-ing a long-distance coherent superfluidlike transport of thespin current [9], as opposed to the exponentially decaying spin currents carried by the incoherent thermal magnons. In addition to the pumped systems, such spin superfluidity isalso supported by easy-plane magnets having a Uð1Þorder parameter [10]. More recently, these spin superfluids have gained increased attention with proposals of realizing themin various easy-plane systems [11,12] . The superfluid nature of spin currents results in an algebraically decayingtransport of spin [12], magnetic analogues of the Josephson effect [11,13] , dissipation via phase slips [14], and macro- scopic qubit functionality [15]. While these proposals establish the feasibility of an efficient transport of thespin information, the possibility of transferring angularmomentum by these superfluidlike spin currents to topo- logical solitons remains unexplored. In this Letter, we fillthis gap by proposing a scheme for coupling spin currents carried by superfluids to magnetic solitons. The main idea is to form an exchange coupled bilayer of an easy-plane and an easy-axis magnetic insulator. The bilayer is invariant under global spin rotations around an axis of symmetry, which coincides with the easy axis andthe normal to the easy plane. See Fig. 1for a schematic (where zis the symmetry axis). The easy-plane magnet plays the role of a spin superfluid and the easy-axis magnet harbors a domain wall. When a spin current polarized alongthe symmetry axis is injected into the bilayer, it is trans- ported coherently by the gradient of the azimuthal angle ( φ) of the spin density in the easy-plane magnet [10]. A static domain wall blocks the flow of this spin current by pinning φunderneath the domain wall. The pinning occurs due to the finite exchange coupling between the spin order incoherent spin source coherent spin-transport channeleasy-axis magnet easy-plane magnet FIG. 1. A bilayer of an easy- z-axis magnet exchange coupled (with the coupling strength g) to an easy- x-y-plane magnet. A z-polarized spin current is injected from an incoherent spin source and propagates as a superfluid spin current through the easy-plane magnet. This spin current is ∝∇φ, where φis the azimuthal angle of the spin order parameter within the x-yplane. This spin current is interrupted and absorbed by a domain wall in the easy-axis magnet, where it is converted into its sliding motion atspeed v.PRL 118, 097201 (2017) PHYSICAL REVIEW LETTERSweek ending 3 MARCH 2017 0031-9007 =17=118(9) =097201(5) 097201-1 © 2017 American Physical Societyparameters in the easy-axis and easy-plane magnets. However, the Uð1Þsymmetry of the combined system demands conservation of the total angular momentum along the symmetry axis. Consequently, the coherently trans-ported spin current in the easy-plane magnet is absorbedby the domain wall and converted into its motion. Theproblem of deriving analytical expressions for this spintransfer-induced domain-wall motion and using it to proposea spin transistor are the main focuses of this Letter. Our proposal extends the concept of magnon-induced torques (due to the exponentially decaying incoherent magnoncurrents [16]) to the more efficient case, where the magnons are condensed and exhibit superfluidity. Model.—We focus on a quasi-one-dimensional model with a bilayer strip extended along the xaxis and discuss two possible routes for forming the proposed system. That is, when an easy-axis ferromagnet is exchange coupled to a spin superfluid formed by (1) an easy-plane ferromagnet(referred to as FM-FM), or (2) a Heisenberg antiferromag-net (referred to as AFM-FM). For clarity, in the remainderof the main text, we focus upon the FM-FM case. Similarresults apply mutatis mutandis to the AFM-FM case [17]. In the FM-FM case, the free-energy density (per unit area in the x-yplane) of the system can be written as F¼F isþFsfþU, with Fis¼¯At∂xm2=2−¯Ktm2z=2; Fsf¼At∂xn2=2þKtn2z=2; ð1Þ andU¼−gm·n. Here, A(A),¯K(K),t(t), and m(n) represent the magnetic stiffness, the anisotropy, the thick-ness, and the unit vector oriented along the order parameterin the easy-axis (easy-plane) magnet, while gparametrizes the strength of the exchange coupling between the easy-axis and easy-plane magnets. The easy-axis and easy-plane characters are enforced by having ¯K> 0and K> 0. Within the easy-axis magnet, the equilibrium configurationof interest is that of a single domain wall (referred to asregion II) connecting magnetic domains (referred to asregions I and III for malong þzand−z, respectively). See Fig.2for a schematic. Furthermore, we focus on the small exchange coupling regime, where g≪Ktandg≪¯K t. Within this regime, the equilibrium out-of-plane canting ofnand the deviation of maway from the zaxis (within regions I and III) are small. The proposed bilayer can be realized by using perpendicular racetrack material (such ascobalt iron boron [20]or cobalt and nickel multilayers [21]) for the easy-axis magnet, and magnetic insulators (such asyttrium iron garnet) for the easy-plane magnet. Theexchange coupling between the easy-axis and the easy-plane system can be controlled via insertion of a non- magnetic layer, such as copper [22]. Coupled spin hydrodynamics. —We begin by outlining a hydrodynamic theory for describing the proposed spinsuperfluid-mediated domain-wall motion. The central ideais to write down the continuity equation for the flow of the z component of the spin current in the bilayer. In regions I and III, this spin current is transported within the easy- plane magnet. In the strong anisotropy and the long-wavelength limit of the spin dynamics, the transport is described by [12] st_n z¼−∂xJs−αst_φ; ð2Þ with Js≡−At∂xφ, and sbeing the magnitude of the saturated spin density in the easy-plane magnet. The firstterm on the right-hand side describes the flow of a super-fluid spin current (per unit length along the yaxis), and the second term describes the transfer of the spin current to the atomic lattice due to a finite Gilbert damping, α, within the easy-plane magnet. In region II, an additional spin current, J Φ, is absorbed by the domain wall. Using the collective coordinate approach [23], the resultant domain- wall dynamics can be written as ¯s_Φ−¯α¯s_X=λ¼0; ð3aÞ 2¯st_Xþ2¯α¯stλ_Φ¼JΦ; ð3bÞ where the so-called soft modes XandΦrepresent the location and the spin azimuthal angle at the center of the domain wall, where the zcomponent of the spin density vanishes. Here, λis the domain-wall width and ¯sis the spin Hall injector FIG. 2. The model of a domain wall of width λcoupled to a spin superfluid. The domain wall divides the bilayer into three regions:(I) up domain, (III) down domain, and (II) the domain wall. A spin current, J shs, is injected on the left by converting a charge current, j, into a spin accumulation via the spin Hall effect. Upon reaching thedomain-wall region, a portion of this spin current, J Φ, is absorbed from the easy-plane magnet by the domain wall. The resultantdynamics of the domain wall is characterized by the generalizedcoordinates XandΦ, parametrizing its position and the associated azimuthal angle. The dynamics of the spin superfluid pumps a spin current, J p s, back to the contact. (Bottom panel) The corresponding superfluid spin current flowing in the easy-plane magnet, asobtained by plotting −At∂ xϕ.PRL 118, 097201 (2017) PHYSICAL REVIEW LETTERSweek ending 3 MARCH 2017 097201-2magnitude of the saturated spin density in the easy-axis magnet. Equation (3b)describes the flow of the spin current within the domain-wall region. Namely, the spin currentabsorbed by the domain wall, J Φ, is converted into its motion, giving rise to the term proportional to _X.I n addition, a portion of the absorbed spin current is trans- ferred to the atomic lattice in the easy-axis magnet, resulting in the term proportional to ¯α. In the spirit of the long-wavelength spin dynamics, throughout this Letter, we consider the domain wall as apointlike object satisfying λ∼ffiffiffiffiffiffiffiffiffi ffi¯A=¯Kp ≪1=∂ xφ. In this case, the width of region II can be neglected, and the discontinuous jump in the spin current flowing in the easy-plane magnet at x¼X(see the bottom panel of Fig. 2) should be the same as J Φ. Consequently, using Eq. (3),w e have J−−Jþ¼JΦ¼2stð1þα2Þ_X: ð4Þ Here, J−andJþare the spin current flowing in the easy- plane magnet just before and after the domain wall (i.e., region II), respectively. Equipped with this boundarycondition, at x¼X, we are now ready to discuss the motion of the domain wall in response to a spin current injected from the left of the bilayer. Specifically, we consider the open geometry proposed in Ref. [12]. See the top panel of Fig. 2for a schematic. A charge current flowing along the yaxis at the metal and easy-plane magnet interface, with a density j(per unit thickness), is converted into a spin current via the spin Hall effect [24]. Within the spin Hall phenomenology [25], the corresponding spin current injected into the bilayer can be written in terms of the so-called spin Hall angle θ[26], the charge of an electron e, and the length (along the xaxis) of the metallic contact l,a sJ shs¼tℏjtanθ=2el. In addition, the dynamics induced via such an injection pumps a portion of the spin current, Jp s¼tℏg↑↓n×_n=4π[27], back into the metallic contact resulting in the following boundary con- dition at the left end: Jsjx¼0¼ϑj−tℏg↑↓n×_n=4π: ð5Þ Here, g↑↓parametrizes the real part of the spin mixing conductance, and we have defined ϑ≡ℏtanθ=2el. Finally, for the right interface, we assume the usual exchangeboundary condition: J sjx¼L¼0: ð6Þ Linear regime. —We proceed to look for dynamic sol- utions of the form _Φ¼Ω,φðx; TÞ¼fðxÞþΩT, and _nz¼0, where Tdenotes time. Physically, such an ansatz represents the following dynamic state. The spins in the easy-plane magnet rotate globally about the zaxis with a linearly decaying spin current in regions I and III [12], anda steady-state motion of the domain wall with _X¼v.W e highlight that within this ansatz, the domain-wall angle ispreccessing at the same frequency as the underlying spinsuperfluid and refer to this dynamic regime as the “locked ” phase. Furthermore, in the presence of a moving domain wall, the assumption of having a position independent Ωis not self-evident. We justify and discuss its validity a posteriori [17]. Balancing the flow of spin current, via substitution of the ansatz in Eqs. (2)and (3)and the boundary condition equations (4)–(6), yields v¼ ϑjt 2stð1þα2Þþαtðγ↑↓þγαÞ=λ: ð7Þ Here, we have used n×_n¼Ωzand defined γα≡αsL, γ↑↓≡ℏg↑↓=4π. This is one of the central results of the Letter. In the absence of Gilbert damping, all of the injectedspin current is absorbed by the domain wall giving a velocity obtained by the conservation of the angular momentum, i.e., v¼ϑjt=2 st, while the loss of the spin current results in a reduction of the velocity from thisperfect absorption case. This loss of spin current occurs attwo sources: (a) the interface to the metal (due to spin pumping), giving rise to the term proportional to γ ↑↓, and (b) the bulk, giving an algebraically decaying velocity withthe length of the bilayer. Nonlinear regime. —At a critical strength of the external drive, the velocity of the domain wall can no longer increase linearly with the injected spin current. Thisphenomenon is referred to as the Walker breakdown[28] and is observed for both external field and current- induced domain-wall motion [29]. In this section we focus on the analogue of the Walker breakdown phenomenon for the superfluid-mediated spin transfer. For this purpose, wederive an analytical expression of J Φwithin the Landau- Lifshitz phenomenology. The zcomponent of the torque applied on the easy-axis magnet, due to the coupling to theeasy-plane magnet, reads τ z¼−z·m×δmU=t. The spin current absorbed by the domain wall is then given by integrating the torque over the domain-wall region, i.e., JΦ¼tR λτzdx. Substituting the following parametrization of the Cartesian components of the unit vector fields, m≡ ðsinθcosϕ;sinθsinϕ;cosθÞandn≡ðcosφ;sinφ;nzÞ, intoU, we get JΦ¼πgλsinðφjX−ΦÞ; ð8Þ where φjXis the value of φatX. For a given coupling g, there exists a maximum value of the absorbed spin current Jc Φ, i.e., when φjX−Φ¼π=2. This results in a corresponding critical value for the injected spin current, Jsc, and a critical domain-wall velocity [from Eq.(3b)],vc¼Jsc=2stð1þα2Þ, above which the locked phase can no longer exist. Namely, ΦandφjXprecess at different frequencies, resulting in an oscillatory exchangeof the spin current between the domain wall and the spinPRL 118, 097201 (2017) PHYSICAL REVIEW LETTERSweek ending 3 MARCH 2017 097201-3superfluid (corresponding to the jump j−−jþ, shown in the bottom panel of Fig. 2, oscillating between a positive and a negative value). We refer to this transition as a lockedto unlocked breakdown. Consequently, as in the case of the Walker breakdown, the domain wall is expected to drift in an oscillatory fashion, with hvi<v c. Substituting the value of critical velocity into Eq. (7), we obtain, for the break- down spin current, Jsc¼ϑjct¼πgλ/C20 1þαðγ↑↓þγαÞ 2sð1þα2Þλ/C21 : ð9Þ This is the second main result of the model, predicting a linear dependence of the breakdown spin current on λ. Here, we note that the locked to unlocked transition is analogous to the transition of superconducting Josephsonjunctions from the zero-voltage state to the finite-voltage state [30]. In Fig. 3(a), we compare the analytical results with micromagnetic simulations [17]. As predicted by the model, two regimes are observed in the simulations:(a) linearly increasing domain-wall velocity below a critical value of the injected spin current ( J cs), and (b) oscillatory drift of the domain wall with a reduced average velocityabove J cs. Moreover, both the velocity in the linear regime and the value of the critical current for the locked to unlocked breakdown agrees well with the simulations. Spin transistor. —We propose utilizing the domain-wall width dependence of the locked to unlocked breakdown in conjunction with the voltage control of the magnetic anisotropy (VCMA) [31] to construct a spin transistor. For this purpose we consider the case of a strongly pinneddomain wall, i.e., with _X¼_Φ¼0. The pinning of Φcould be achieved by fabricating a nanowire geometry for theeasy-axis magnet. In this case, the dipolar interaction forces the domain-wall magnetization to be oriented along the long axis of the nanowire. The domain-wall position can bepinned by engineering “notches, ”which create a local energy minima with respect to X[4]. For an injected spin current J ins≡ϑjt < Jc Φ, a static solution results for the spin superfluid with the domain wall absorbing all of the spin current injected at the left contact. See the off schematic in the inset of Fig. 3(b). On the other hand, forJins>Jc Φlocked to unlocked breakdown occurs, result- ing in a precessing solution for the superfluid. Since JΦ∝sinðφX−ΦÞ, the spin current absorbed by the domain wall averages to zero. Utilizing the inverse spin Hall effect[32], the spin current beyond the domain wall can be detected by adding a right metal contact. See the on schematic in the inset of Fig. 3(b). Focusing on the case where the interfaces dominate over the bulk, i.e., γ ↑↓≫γα, half of the spin current is pumped back to the left contact and the other half is detected by the right contact, i.e., Jouts¼Jins=2. Here, the interfaces are assumed to be symmetric, parametrized by the same γ↑↓. The λdepend- ence of Jc Φthen translates into the following transistorlike action [plotted in Fig. 3(b)]. The off(on) state of the device is defined as Joutsbeing zero (nonzero). In the absence of the gate voltage, Vg, the device is biased to be below the locked to unlocked breakdown, and hence in the offstate. Application of a gate voltage changes λ(by changing Kvia VCMA) and turns the device onabruptly, via inducing locked to unlocked breakdown. The proposed spin 00.01 0.02 0.03 0.04 0.05 0.06 0.07 0.0800.010.020.03 0.06 0.01 0.006 0.014 0.09 0 1 2 3 4 5 6 7 800.511.522.533.5 9(a) (b) FIG. 3. (a) For a given exchange coupling ~g≡g=Kt , two regimes for domain-wall motion are obtained. A steady-state regime with linearly increasing velocity ( ~v) and oscillatory motion above a critical value of the injected spin current ~Js. The broken line plots the analytical result from Eq. (7). (Inset) The critical ~Jsincreases linearly with ~g. The broken line shows the analytical result from Eq. (9). (b) The spin current detected at the right end of the bilayer ( Jouts) exhibits nonlinear behavior in the presence of a pinned domain wall. When the injected spin current, Jins, is below (above) a critical breakdown current, Jouts¼0(Jouts≠0). Solid and broken curves plot the nonlinear characteristics for λ¼10and 5 nm, respectively. The nonlinearity can be used to construct a transistor, as indicated by the vertical dashed-dotted line. Fixing Jinsand changing λby an external gate switches the device from an off(Jouts¼0)t oa n on(Jouts≠0) state. These offandonstates are depicted schematically in the insets.PRL 118, 097201 (2017) PHYSICAL REVIEW LETTERSweek ending 3 MARCH 2017 097201-4transistor has an added advantage; i.e., the domain wall can be moved to a desired location by applying a magneticfield, making the device reconfigurable. This work was supported by FAME (a SRC STARnet center sponsored by MARCO and DARPA) and, in part, bythe Army Research Office under Contract No. W911NF-14-1-0016. [1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015) . [2] D. Hinzke and U. Nowak, Phys. Rev. Lett. 107, 027205 (2011) ; P. Yan, X. S. Wang, and X. R. Wang, Phys. Rev. Lett. 107, 177207 (2011) ; A. A. Kovalev and Y. Tserkovnyak, Europhys. Lett. 97, 67002 (2012) ; L. Kong and J. Zang, Phys. Rev. Lett. 111, 067203 (2013) ; J. Iwasaki, A. J. Beekman, and N. Nagaosa, Phys. Rev. B 89, 064412 (2014) ; A. A. Kovalev, Phys. Rev. B 89, 241101 (2014) ; C. Schütte and M. Garst, Phys. Rev. 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PhysRevB.104.014425.pdf

PHYSICAL REVIEW B 104, 014425 (2021) Dynamic detection of current-induced spin-orbit magnetic fields L. Chen ,1,2,*R. Islinger,2J. Stigloher,2M. M. Decker,2M. Kronseder,2D. Schuh,2D. Bougeard,2 D. Weiss ,2and C. H. Back1 1Department of Physics, Technical University of Munich, Garching bei Munich, Germany 2Institute of Experimental and Applied Physics, University of Regensburg, Regensburg, Germany (Received 26 November 2020; revised 17 June 2021; accepted 6 July 2021; published 23 July 2021) Current-induced spin-orbit torques (SOTs) in ferromagnet/nonmagnetic metal heterostructures open vast possibilities to design spintronic devices to store, process, and transmit information in a simple architecture.It is a central task to search for efficient SOT devices, and to quantify the magnitude as well as the symmetryof current-induced spin-orbit magnetic fields (SOFs). Here, we report an approach to determine the SOFs basedon magnetization dynamics by means of time-resolved magneto-optic Kerr microscopy. A microwave currentin a narrow Fe/GaAs (001) stripe generates an Oersted field as well as SOFs due to the reduced symmetry atthe Fe/GaAs interface, and excites standing spin wave (SSW) modes because of the lateral confinement. Due totheir different symmetries, the SOFs and the Oersted field generate distinctly different mode patterns. Thus, it ispossible to determine the magnitude of the SOFs from an analysis of the shape of the SSW patterns. Specifically,this method, which is conceptually different from previous approaches based on line shape analysis, is phaseindependent and self-calibrated. It can be used to measure the current-induced SOFs in other material systems,e.g., ferromagnetic metal/nonmagnetic metal heterostructures. DOI: 10.1103/PhysRevB.104.014425 I. INTRODUCTION The investigation of the mutual conversion between charge and spin currents has witnessed significant attention in recentyears due to its possible technological impact for spintronicdevices [ 1,2]. In ferromagnet (FM)/nonmagnetic metal (NM) heterostructures, a charge current flowing in the NM alongthe xaxis will generate a transverse spin accumulation σ along the ydirection at the interface via the spin Hall ef- fect and/or the inverse spin galvanic effect [ 1]. The resulting spin accumulation acts on the ferromagnetic layer via field-like (τ FL) and dampinglike ( τDL) spin-orbit torques (SOTs), which can be written as τFL=−γμ 0hFLm×yandτDL= −γμ 0hDLm×m×y, where γis the gyromagnetic ratio, μ0the magnetic constant, mthe magnetization unit vector, and hFL(hDL) the corresponding effective fieldlike (damping- like) spin-orbit magnetic field hSOF. These torques modify the magnetization’s equation of motion, i.e., the Landau-Lifshitz-Gilbert (LLG) equation, and are responsible for anumber of spin-orbit related functionalities including mag-netization switching [ 3,4], domain wall motion [ 5–7], and auto-oscillations of the magnetization [ 8,9]. II. HISTORIC DEVELOPMENT OF THE SPIN-TORQUE FERROMAGNETIC RESONANCE METHOD To optimize material parameters leading to efficient SOTs, the magnitude of the SOFs must be determined accurately.One frequently used approach is the spin-transfer-torque *lin0.chen@tum.deferromagnetic resonance (STT FMR) method [ 10], which is based on a line shape analysis of the rectified dc voltageinduced by FMR. It is generally assumed that the symmetriccomponent of the dc voltage, V sym, corresponds to the out of plane hDLwhile the antisymmetric component, Va-sym, corre- sponds to the in-plane Oersted field generated by the currentflowing in NM. This method is so-called self-calibrated sincethe spin Hall angle in the NM (related to h DL) is determined by the ratio of Vsym/Va-sym. Initially, the importance of hFL, which also generates Va-sym, has not been properly taken into consideration. Pai et al. further modified this method and extracted hDLand hFLby measuring the dependence of Vsym/Va-sym on the FM layer thickness tFM, assuming that hFL is independent of tFM[11,12]. However, this does not hold since magneto-optical [ 13] and magnetotransport [ 14] meth- ods show that hFLstrongly depends on tFM, which possibly leads to a wrong estimation of hFLand hDL. A second well- established technique based on FMR is the spin-orbit-torqueFMR (SOT FMR) method, which has been utilized to charac-terize the SOFs in single-crystalline ferromagnetic materialswith broken inversion symmetry [ 15]. In contrast to bilayer systems, there is no in-plane Oersted field since only onelayer is involved, and the single-crystalline ferromagnet actsboth as spin current generator and detector (see Appendixes A andBfor the differences between STT FMR and SOT FMR and details concerning these two methods). Up to now, STTFMR and SOT FMR have been used to study spin-orbitrelated phenomena in a large variety of materials (see thelarge number of references which cite Refs. [ 10,15]), includ- ing nonmagnetic metals [ 4,16], topological materials [ 17–20], magnetic semiconductors [ 21], antiferromagnets [ 22–24], and transition-metal dichalcogenides [ 25–28]. It should be noted 2469-9950/2021/104(1)/014425(11) 014425-1 ©2021 American Physical SocietyL. CHEN et al. PHYSICAL REVIEW B 104, 014425 (2021) (c) (d)(b) (a) (arb. units) FIG. 1. Schematic of device and driving fields. (a) Schematic of the device used for the detection of magnetization dynamics driven by electric current. The out of plane component of the dynamic magnetization mz(t) is detected by time-resolved magneto-optical Kerr (TRMOKE) microscopy. A microwave current jFMwith a frequency fof 12 GHz is fed into the Fe stripe deposited on a semi-insulating GaAs(001) substrate, and excites m(t) by the combination of spin-orbit field hSOFand Oersted field hFM,z rf. The external magnetic field His applied parallel to jFM. (b) Phase relations in the TRMOKE setup. /Phi1inis the phase of input microwave current jin, which can be adjusted by the time of laser pulse (green arrow). /Phi1mis the assumed phase shift of jFM(red dashed line). Since jFMinduces hSOFand hFM,z rfwhich drive m(t); thus m(t) is of the same phase /Phi1masjFM. The phase difference between the laser pulse and mz(t),/Phi1l−m, is thus the sum of /Phi1inand /Phi1m; i.e.,/Phi1l−m=/Phi1in+/Phi1m. (c) Current-orientation dependence of hSOFinduced by Bychkov-Rashba-like (red arrow) and Dresselhaus-like (green arrow) spin-orbit interaction. Since His parallel to jFM, only the transverse components of hSOFexcite magnetization dynamics. (d) Lateral distributions of hSOF(red dashed line) and hFM,z rf (black solid line). hSOFis symmetric across y, while hFM,z rf is antisymmetric. The different symmetries of the excitations lead to distinctive standing spin wave patterns; i.e., the symmetric hSOFexcites odd spin wave modes ( n=1,3···), while the antisymmetric hFM,z rfexcites even modes ( n=2,4···). that for both STT FMR and SOT FMR, the out-of-plane Oersted field hFM,z rfgenerated by the current flowing in the ferromagnetic material itself contributes no net effect to themeasured dc voltage since it is antisymmetrically distributed(see Appendix A). This, however, becomes the key ingredient for the present study. For electric-current driven FMR, it is generally believed that the phase /Phi1 mof microwave driving current in the spin- current source materials suffers no phase shift, i.e., /Phi1m= 0 always holds, and /Phi1mis expected to show no position dependence along the current direction of the device. Onlyrecently, it has been noticed by spatially resolved ferromag-netic resonance phase imaging that a possible phase differencetransverse to a CoFeB/Pt stripe exists [ 29], and affects the de- termination of the magnitude of h DL. Note that for the sample studied in Ref. [ 29], a part of the microwave current flows also in the CoFeB layer, and the generated hFM,z rfcan influence the line shape and subsequently the phase. Therefore, the openquestions, which have not been properly addressed so far, areas follows: Does the assumption /Phi1 m=0 always hold at differ- ent positions for any spin-current source material, irrespectiveof the details of the material/device? If not, is it still pos- sible to quantitatively determine SOFs by magnetizationdynamics? III. EXPERIMENTAL RESULTS A. Evidence of phase shift for electric-current driven FMR Here, we use Fe/GaAs (001) bilayers as a model material system to investigate a possible variation of /Phi1mat different positions of the device; see Fig. 1(a). The advantages of the single-crystalline system Fe/GaAs are (i) presence of sizableinterfacial SOFs having the same symmetry as FM/NM bilay-ers [30]; (ii) low Gilbert damping constant; (iii) the electrical current j FMflows solely in Fe and thus a complex analy- sis can be avoided; (iv) tunable resistivity of Fe simply bychanging the Fe layer thickness t Fe. The measurements are carried out by phase-sensitive time-resolved magneto-opticalKerr effect (TRMOKE) microscopy [ 31] (see Supplemental Material [ 32]; also see [ 13,30,33]). As shown in Fig. 1(b),a t certain position x, the phase difference between the pulse laser 014425-2DYNAMIC DETECTION OF CURRENT-INDUCED … PHYSICAL REVIEW B 104, 014425 (2021) (a) (c)(b) FIG. 2. Determination of the position dependence /Phi1mfor electric-current driven magnetization dynamics. Position-dependent Kerr voltage VKerrmeasured at the center of the stripe ( y=0, where hFM,z rf=0) for (a) Fe thickness tFe=3.5n ma n d( b ) tFe=0.8 nm. For both devices, /Phi1in is set to 50 ° and the device dimensions are 6.4 μm×100μm. One can see that both the magnitude and the line shape for tFe=3.5 nm remain unchanged, but change dramatically for tFe=0.8 nm. The solid lines in (a,b) are fits to Eq. ( 3), which give the magnitude of ϕ. The bump at about 66 mT for tFe=3.5 nm is due to the formation of a standing spin wave; see Sections B and C. (c) Position dependence of /Phi1mobtained from Eq. ( 4), displaying a clear variation of /Phi1mfortFe=0.8n m . and the dynamic magnetization m(t),/Phi1 l−m(x), can be written as /Phi1l−m(x)=/Phi1in+/Phi1m(x), (1) where /Phi1inis the controlled phase between pulse laser and input microwave current jin, and /Phi1m(x) the assumed x- dependent phase shift of microwave current jFMin Fe. The polar Kerr signal at a certain /Phi1inand x,VKerr(/Phi1in,x), is pro- portional to the real part of the out of plane component of thedynamic magnetization m z, which can be obtained from the complex dynamic susceptibility [ 30]: VKerr(/Phi1in,x)∼/bracketleftbig Re(χo)ho−Im/parenleftbig χi a/parenrightbig hi/bracketrightbig cos/Phi1l−m(x) −/bracketleftbig Im(χo)ho+Re/parenleftbig χi a/parenrightbig hi/bracketrightbig sin/Phi1l−m(x).(2) Here Re( χo)[Im(χo)] is the real (imaginary) part of the diagonal dynamic susceptibility due to out of plane exci-tation h o, and Re( χi a)[Im(χi a)] is the real (imaginary) part of the off-diagonal dynamic susceptibility due to in-planeexcitation h i. For Fe/GaAs studied here, hicontains only the position-independent hFLalong the ydirection hy FL; i.e., hi=hy FL. Note that hFLalong the xdirection does not excite magnetization dynamics since the external magnetic field Hisapplied parallel to the xaxis. In contrast, hocontains both the y-dependent Oersted field hFM,z rfand the y-independent hDL; i.e., ho=hFM,z rf(y)+hDL. It is worth mentioning that Im( χi a) and Im( χo)[ R e (χi a) and Re( χo)] show a symmetric (antisym- metric) line shape with respect to H, and their magnitudes can be calculated by solving the LLG equation [ 30]. Figures 2(a)and2(b) show the position dependence of the Kerr voltage VKerrmeasured at the center of the stripe ( y=0 and hFM,z rf=0) under /Phi1in=50◦fortFe=3.5 and 0.8 nm. Both devices have the same dimensions of 6.4 μm×100μm but show an opposite temperature coefficient in the temper-ature dependence of the resistivity (see Appendix C). For t Fe=3.5 nm, the line shape as well as the magnitude of VKerr remain the same along the stripe from x1tox5, while they change dramatically for tFe=0.8 nm. To extract /Phi1m, the char- acteristic VKerrspectra can be fitted by VKerr=Acosϕ/Delta1H2+sinϕ/Delta1H(H−HR) 4(H−HR)2+/Delta1H2. (3) Here Ais an apparatus-dependent coefficient, HRthe mag- netic field at FMR, /Delta1Hthe full width at half maximum, and ϕ is the phase factor which determines the line shape of VKerr(H). 014425-3L. CHEN et al. PHYSICAL REVIEW B 104, 014425 (2021) From Eqs. ( 1)–(3), the magnitude of /Phi1mcan be derived as /Phi1m=tan−1Re(χo)hDLcosϕ+Im/parenleftbig χi a/parenrightbig hy FLsinϕ Re/parenleftbig χia/parenrightbig hy FLcosϕ−Im(χo)hDLsinϕ−/Phi1in,(4) which provides a measure of the phase shift of the driv- ing microwave current in the spin-current source materialvia time-resolved magneto-optical Kerr (TRMOKE) spec-tra. Using the corresponding dynamic susceptibilities as wellasϕvalues for both devices, and considering that h DL∼ hy FL, the magnitude of /Phi1mcan be obtained from Eq. ( 4). Figure 2(c)summarizes /Phi1mas a function of position for both devices. One can see that /Phi1mshows no significant change within experimental error for tFe=3.5 nm. However, a sizable variation of /Phi1mis observed for tFe=0.8 nm. The variation of /Phi1mcould be due to the fact that the rf characteristics of Fe change from a good conductor to a dielectric upon decreasing tFe(see Appendix C). Besides ultrathin Fe, we show in the Supplemental Material [ 32] that sizable phase variation is also found in Py /Bi2Se3bilayers, where a more resistive Bi 2Se3 also changes the phase of microwave current. The phase vari- ation at certain positions will not influence the line shape ofthe rectified dc voltage induced by the coupling of the mi-crowave current and magnetization dynamics (Appendix B). However, the spatial variation of /Phi1 mcould lead to the for- mation of a spin wave spin current traveling along the x direction [ 34] and subsequent conversion to a symmetric dc voltage through the spin galvanic effect if a Dresselhaus-type spin-orbit interaction is present [ 15,21,28]. We propose that this should be carefully examined and possibly excludedif the line shape analysis method is used to quantify theSOFs. Therefore, it is of vital importance to establish a phase- independent technique to reliably determine the SOFs basedon magnetization dynamics. Here, we report a self-calibratedand phase-independent approach to measure current-inducedSOFs by analyzing the shape of the standing spin wave(SSW) mode patterns, i.e., a method which is distinctly dif-ferent from previous electrical methods based on line shapeanalysis. B. Formation of standing spin waves in a laterally confined Fe/GaAs stripe Formation of SSWs is a prerequisite for this work. Figure 3(a) shows the calculated SSW eigenmodes for a 2.8μm wide, 3.5 nm thick Fe stripe with Happlied along the [110] direction of the GaAs substrate, which corresponds tothe Damon-Eshbach geometry [ 35,36]. In the calculation, the following parameters determined by separate magnetizationand FMR measurements are used: saturation magnetiza-tionμ 0MS=2.1 T, effective demagnetization field μ0HK= 1.75 T, and Landé gfactor g=2.12. The intersection at a frequency fof 12 GHz specifies HRof each mode, which is expected to be observed in the experiment. The lateralconfinement leads to a mode separation of 4 mT (i.e., 8 mTbetween odd modes), which is comparable to the magni-tude of /Delta1H. The normalized profiles of m zfor the first five modes ( n=1–5), i.e., mzas a function of space coordinate y, are displayed in Fig. 3(b). One can see that the odd (even)(a) (b) FIG. 3. Eigenmodes and distribution of confined SSWs. (a) Cal- culated eigenmodes for a laterally confined Fe/GaAs stripe with tFe=3.5n m a n d w=2.8μm. The external magnetic field His applied along the [110] direction of GaAs, and the intersection de- fines the required HRfor each standing spin wave (SSW) mode at f=12 GHz. (b) The lateral distribution of SSW modes for n=1–5. The symmetric modes ( n=1, 3, and 5) can be excited by symmetric excitations; antisymmetric modes ( n=2 and 4) can be excited by antisymmetric excitations. modes are symmetrically (antisymmetrically) distributed with respect to the center of the stripe. Consequently, the odd(even) modes can be excited by symmetrically (antisymmet-rically) distributed driving fields due to symmetry reasons[35,36]. We first analyze the eigenmodes of the Fe/GaAs stripe under homogeneous (symmetric) excitation. The stripe, whichis 2.8 μm in width and 20 μm in length with the long side along the [110] direction of the GaAs substrate, is integratedin the gap of a coplanar waveguide (CPW) by using electron-beam lithography, as shown in the inset of Fig. 4(a). Here, the Fe stripe is exposed to homogeneous excitation by anout of plane Oersted field h CPW,z rf, which is generated by microwave current flowing in the signal and ground line ofthe CPW. According to Eq. ( 2), the detected Kerr signal can be simplified as V Kerr(/Phi1in)∼Re(χo)hCPW,z rfcos(/Phi1in+/Phi1m)− Im(χo)hCPW,z rfsin(/Phi1in+/Phi1m). Figure 4(a) shows the normal- ized VKerr(H,y) image measured at /Phi1in=90◦. As expected, only the odd modes with n=1, 3, and 5 appear. Figure 4(b) presents the micromagnetic simulation [ 33] of the SSW modes using the same parameters as those used in Fig. 3, which reproduces the experimental results well (see Supple-mental Material [ 32]). To have a closer look at the obtained modes, we perform a horizontal scan for the data in Fig. 4(a), i.e., by placing the laser at the center of the stripe and sweep-ingH. As shown in Fig. 4(c), the cut shows only symmetric line shapes, which can be fitted using the corresponding cutof the simulation data in Fig. 4(b). The locations of the first, third, and fifth modes are marked by solid points, and the 014425-4DYNAMIC DETECTION OF CURRENT-INDUCED … PHYSICAL REVIEW B 104, 014425 (2021) FIG. 4. SSWs driven by a symmetric excitation. (a) SSWs detected in a Fe (3.5 nm)/GaAs stripe by TRMOKE microscopy, where the magnetization dynamics is excited by a homogeneous (symmetric) out of plane Oersted field through a coplanar waveguide (CPW). Only symmetric odd modes ( n=1, 3, and 5) can be observed. The inset shows the schematic of the CPW device, where the Fe stripe is integrated into the gap of the CPW, and His applied along the long axis of the stripe, i.e., along the [110] direction of GaAs. (b) Micromagnetic simulation of the SSW modes using MUMAX 3, which reproduces the experimentally observed modes well. In the simulation, we use the same material parameters as for the calculation of the eigenmode and we convolve the simulation with a Gaussian beam profile. (c) Horizontal line cut of the Kerr signal at the center of the stripe ( y=0). The three peaks can be fitted by symmetric Lorentzians, and the positions of the first, third, and fifth modes are indicated by red, green, and blue circles, respectively. (d) Vertical cut of modes for n=1, 3, and 5. All the modes show symmetric profiles and can be well fitted by MUMAX simulations. mode spacing coincides well with the eigenmode calcula- tion shown in Fig. 3(a). Note that the mode position differs between Figs. 3(a) and4(c); this is because the in-plane bi- axial and uniaxial magnetic anisotropies are not included inthe eigenmode calculation. Since only purely symmetric lineshapes are observed, one can infer that /Phi1 m=0◦. Otherwise an antisymmetric component in the VKerrtrace originating from Re(χo) is expected. This is not surprising since the microwave current and hCPW,z rfare intrinsically in phase due to the fact that the CPW is impedance matched with the rf network.These results also prove the validity of the proposed phaseanalysis presented above. Figure 4(d) shows the first, third, and fifth modes as a function of lateral space coordinate y. All the modes show symmetric profiles with the peak wave-amplitude ratio of ∼10:2:1, which can also be well fitted by micromagnetic simulations.C. Determination of SOFs by the shape of the standing spin wave pattern Next, measurements are performed on a 2.8 μm wide, 100μm long stripe orientated along the [110] direction of GaAs using an electric-current excitation as shown in Fig.1(a). A rf-current density jFM=1.0×1011Am−2is ap- plied to the device, and His set parallel to jFM. The magnitude ofjFMis calibrated by the Joule heating induced resistance increase [ 15]. As shown in Fig. 1(d), the driving fields here contain both symmetric hSOFand antisymmetric hFM,z rfcompo- nents. In addition to the odd modes driven by the symmetric SOFs, even modes excited by the antisymmetric hFM,z rfare ex- pected. Figure 5(a)shows the SSW pattern measured at /Phi1in= 90◦. In contrast to Fig. 4(a) where only the symmetric odd modes are observed, for the case of electric-current excitation, 014425-5L. CHEN et al. PHYSICAL REVIEW B 104, 014425 (2021) (a) (c) (d)(b)(arb. units) (arb. units)First Expt.Second Third FIG. 5. SSWs driven by electric current for tFe=3.5 nm. (a) Image of the TRMOKE signal measured at /Phi1in=90◦and jFM/bardblH/bardbl[110]. (b) Profiles of the first three modes, i.e., vertical cuts along the dashed lines in (a). The position of the maxima (minima) of n=1(n=3) shifts away from the center of the stripe by /Delta1∼0.4μm due to the interference with the second mode, as indicated by the dashed lines. (c) Corresponding image of V/Phi1m−free Kerr for a [110] device obtained by square and root operation of VKerr(0 °) and VKerr(90 °); i.e., V/Phi1m−free Kerr =/radicalbig [VKerr(0◦)]2+[VKerr(90◦)]2. (d) Horizontal cut of V/Phi1m−free Kerr aty=0, which can be fitted by a symmetric Lorentzian. both the first and third modes are not located at the center of the stripe anymore. This indicates the emergence of the antisymmetric second mode. Because the mode spacing is ofthe same magnitude as the FMR linewidth, the nearest modes merge, and the shape of the SSW pattern is dramatically altered and shifted. For example, the second mode increases V Kerrof the first mode on the lower part of the stripe while reducing it on the upper side. A clearer shift of the patterns can be seen from the profile (vertical cut) of each mode. Asshown in Fig. 5(b), the maximum (minimum) position of the first (third) mode shifts away from center to the lower part of stripe by an absolute value of /Delta1∼0.4μm. If the phase term /Phi1mis unknown, it is impossible to extract the magnitude of SOF from Fig. 5(a). However, it is possi- ble to eliminate /Phi1mthrough square and root operations of VKerr(/Phi1in) measured at two phases with 90 ° phase shift. Based on Eq. ( 2), the/Phi1m-independent Kerr voltage V/Phi1m-free Kerrcan be obtained as V/Phi1m-free Kerr=/radicalBig [VKerr(/Phi1in)]2+[VKerr(/Phi1in+90◦)]2, ∼Im/parenleftbig χi a/parenrightbig hy FL/radicaltp/radicalvertex/radicalvertex/radicalbt/bracketleftbigg 1−Re(χo) Im/parenleftbig χia/parenrightbighDL+hFM,z rf hy FL/bracketrightbigg2 +/bracketleftbiggRe/parenleftbig χia/parenrightbig Im/parenleftbig χia/parenrightbig+Im(χo) Im/parenleftbig χia/parenrightbighDL+hFM,z rf hy FL/bracketrightbigg2 . (5) The corresponding V/Phi1m-free Kerrimage for the [110]-oriented device is shown in Fig. 5(c). For the present sample with μ0HKof 1.75 T, the magnitude of the susceptibility under out of plane excitation is much smaller than the in-plane one, and the ratios ofthe dynamic susceptibilities under the square root are determined as [ 30]R e (χ o)/Im(χi a)=0.1, Re( χi a)/Im(χi a)=−0.5, and 014425-6DYNAMIC DETECTION OF CURRENT-INDUCED … PHYSICAL REVIEW B 104, 014425 (2021) (a) (b) (c)(d) (e) (f) FIG. 6. Determination of SOF by the shape of SSW pattern for tFe=3.5n m . I m a g e o f V/Phi1m-free Kerr (H,y) signal for ajFM/bardblH/bardbl[110], (b) jFM/bardblH/bardbl[¯110] and (c) jFM/bardblH/bardbl[010]. In the plots, jFMhas been normalized to 1 ×1011Am−2. The configurations of the SOFs induced by Bychkov-Rashba-like hRand Dresselhaus-like hDare also presented in the insets. hRand hDare constructive for [110]-oriented devices, but destructive for [ ¯110]-oriented devices. For [010] orientation, only hRis detected. Corresponding images obtained by micromagnetic simulations for devices oriented along (d) [110], (e) [ ¯110], and (f) [010]. Im(χo)/Im(χi a)=−0.2. At the center of the stripe ( hFM,z rf=0 and hDL=hy FL), Eq. ( 5) can be further simplified to V/Phi1m-free Kerr≈Im/parenleftbig χi a/parenrightbig hy FL/radicaltp/radicalvertex/radicalvertex/radicalbt 1+/bracketleftbiggRe/parenleftbig χo/parenrightbig Im/parenleftbig χia/parenrightbig/bracketrightbigg2 +2/bracketleftbiggRe/parenleftbig χia/parenrightbig Im/parenleftbig χia/parenrightbigIm(χo) Im/parenleftbig χia/parenrightbig−Re(χo) Im/parenleftbig χia/parenrightbig/bracketrightbigghDL hy FL=Im/parenleftbig χi a/parenrightbig hy FL/radicalBigg 1+/bracketleftbiggRe(χo) Im/parenleftbig χia/parenrightbig/bracketrightbigg2 . This means only hy FLcontributes to V/Phi1m-free Kerr, and the effect of h DLcan be neglected in the analysis due to the large effective demagnetization field. Equation ( 5) also suggests that, at the center of the stripe, the line shape of the V/Phi1m-free Kerr trace is symmetric with respect to H, which is confirmed by the horizontal cut shown in Fig. 5(d). However, when the laser is moved away from the center of the stripe, theabove assumption becomes invalid, since h FM,z rf>hy FLholds. The appearance of even modes excited by hFM,z rfcan alter the shape of the odd mode pattern, which, therefore, pro-vides a phase-independent way to determine the magnitude ofh y FL. Figures 6(a)–6(c) present the images of V/Phi1m-free Kerr(H,y)f o r devices structured along the [110], [ ¯110], and [010] orien- tations. To compare the amplitudes of V/Phi1m-free Kerr, all imagesare normalized to the current density jFM=1×1011Am−2. As shown in the images, the coalescence of the first threemode patterns leads to the formation of three main regionsas indicated by the closed dashed lines. The odd and even modes merge and become indistinguishable after the treat- ment of square and root operations. All the V /Phi1m-free Kerr(H,y) images show similar patterns indicating similar excitations for each device. However, the maximum intensity of the Kerr signal, Vmax, differs significantly for different crystallographic directions with V[110] max=1.2V[010] max=1.7V[¯110] max. This implies a dependence of hy FLon the current direction due to interfer- ence of Bychkov-Rashba-like and Dresselhaus-like spin-orbit interactions. As sketched in the insets of Figs. 6(a)–6(c), con- structively aligned Dresselhaus hDand Bychkov-Rashba hR SOFs are detected (hy FL=hR+hD) for the [110] orientation, 014425-7L. CHEN et al. PHYSICAL REVIEW B 104, 014425 (2021) while for the [ ¯110] orientation, hDand hRadd destructively (hy FL=hR−hD). For the [010] orientation only hRcan be detected (hy FL=hR). To quantify hDand hR, we repeat the micromagnetic simulations, similar to the case where the magnetization dynamics is only driven by a homogeneoush CPW,z rforiginating from the CPW, but now including both y-independent hy FLand y-dependent hFM,z rfcalculated from the Biot-Savart law. A least square algorithm is carried out tominimize the difference between images obtained by exper-iment and simulations (see Supplemental Material [ 32]). As shown by Figs. 6(d)–6(f), the corresponding simulation im- ages reproduce the experiments reasonably well. For the [110]device, the magnitude of the extracted SOF is μ 0hR+μ0hD= 0.28 mT; and for the [ ¯110] device, μ0hR−μ0hD=0.13 mT, which gives μ0hR=0.21 mT and μ0hD=0.07 mT. The mag- nitude μ0hRin turn is consistent with the value determined from a [010] device with μ0hR=0.22 mT. Moreover, we compare the magnitude of SOF obtained by SSW and dc voltage detection for the same device of tFe=3.5n m i n t h e Supplemental Material [ 32], and the results show quantitative agreement between our method and dc voltage detection for samples with no phase variation. All these results indicate the validity of our method. Although the present experiment only determines the mag- nitude of the fieldlike torque (corresponding to hy FL) due to the relatively large HKvalue, we propose in the Supplemental Material [ 32] that it is also possible to determine the magni- tude of fieldlike and dampinglike torques in FM/NM bilayerswith a reduced H K, showing the completeness of this method. It should be noted that to perform the SSW pattern method inFM/NM bilayers, the prerequisite is that the effective dampingconstant of the FM should be low ( ∼0.005), which leads to sizable mode spacing comparable to the FMR linewidth. This,however, is not the case for most FM/NM bilayers [also for t Fe=0.8 nm shown in Fig. 2(a)] due to the large effective damping caused by extrinsic effects, such as spin pumpingand/or inhomogeneous broadening. A possible solution forthis problem could be using metallic ferromagnets with ul-tralow damping, such as CoFe [ 37]. IV . CONCLUSIONS We have demonstrated by TRMOKE measurements that a possible phase variation of the driving microwave cur-rent can be detected when using electric-current excitation.We have proposed a phase-independent and self-calibratedway to quantify the spin-orbit fields by using the shift ofstanding spin wave patterns excited by the combined actionof current-induced spin-orbit fields and Oersted field. Thisunique approach goes beyond the standard electrical measure-ments based on line shape analysis and solves a long-standingproblem in the determination of SOFs based on magneti-zation dynamics. Our method is not specific to Fe/GaAs,but can also be used for other systems, e.g., ferromagneticmetal/nonmagnetic metal bilayers. ACKNOWLEDGMENT This work is supported by the German Science Foundation (DFG) via SFB 1277.(a) (b) FIG. 7. Schematics of the driving fields for (a) STT FMR in a FM/NM bilayer and (b) SOT FMR in a single-crystalline Fe/GaAs heterostructure. jNMis the microwave current flowing in NM, and jFMis the microwave current in FM. hNM rfand hFM rfare the Oersted fields generated by jNMand jFM, respectively. In FM/NM bilayers, hDLand hFLare induced by jNMdue to the spin Hall effect and/or the inverse spin galvanic effect, while in Fe/GaAs, hDLand hFLare induced by jFMdue to the inverse spin galvanic effect. APPENDIX A: DIFFERENCES BETWEEN STT FMR AND SOT FMR Figures 7(a) and7(b) present the schematics of the excita- tion fields for STT FMR in FM/NM bilayers and SOT FMR insingle-crystalline FMs with reduced symmetry, respectively.For STT FMR, microwave current flows both in NM( j NM) and FM ( jFM).jNMinduces the Oersted field hNM rfas well as hDLand hFLdue to the spin Hall effect and/or the inverse spin galvanic effect. The net in-plane component of hNM rf, hNM,y rf, is symmetrically distributed across y, and is parallel or antiparallel to hFLdepending on the sign of hFL, while the out of plane component of hNM rf,hNM,z rf, is antisymmetrically distributed across y. If the magnetization dynamics is probed by electrical measurements, hNM,z rfcontributes no net effect to the detected dc voltage. Similarly, the Oersted field hFM rf generated by jFMis also antisymmetrically distributed, and contributes no net effect to the measured dc voltage for bothSOT FMR and STT FMR. The symmetry of all the drivingfields is summarized in Table I. APPENDIX B: ELECTRICAL AND OPTICAL DETECTION OF MAGNETIZATION DYNAMICS Figure 8(a) shows the setup for the detection of magnetization dynamics by dc voltage for STT FMR andSOT FMR. The dc voltage V dcis measured by sweeping the external magnetic field at fixed microwave frequency;a typical V dctrace is presented in Fig. 8(b).T ofi tt h e TABLE I. Symmetry of the driving fields for STT FMR and SOT FMR. hNM,y rf(hFM,y rf)a n dhNM,z rf(hFM,z rf) are the in-plane and out of plane components of hNM rf(hFM rf). Sy(Ay) stands for symmetrically (antisymmetrically) distributed with respect to the yaxis. Azrepre- sents antisymmetrically distributed with respect to the zaxis. For dc voltage detection of magnetization dynamics, all the antisymmetric excitations contribute no net effect to the measured dc voltage. hFL hDL hNM,y rf hNM,z rf hFM,y rf hFM,z rf STT FMR SySySyAyAzAy SOT FMR SySyN.A. N.A. AzAy 014425-8DYNAMIC DETECTION OF CURRENT-INDUCED … PHYSICAL REVIEW B 104, 014425 (2021) (a) (b) Expt. FIG. 8. (a) Depiction of a scheme for dc voltage detection of magnetization dynamics for STT FMR and SOT FMR. Here ϕMis the angle between jrfandM. (b) Typical spectrum of the dc voltage Vfor STT FMR and SOT FMR around the resonance field of FM, which can be decomposed into symmetric and antisymmetric parts. characteristic line shape, we introduce a symmetric ( Lsym= /Delta1H2/[4(H−HR)2+/Delta1H2]) and an antisymmetric Lorentzian ( La-sym=−4/Delta1H(H−HR)/[4(H−HR)2+/Delta1H2]). Vdcis fitted by a combination of Lsym and La-sym, VsymLsym+Va-sym La-sym, with Vsym(Va-sym) the magnitude of the symmetric (antisymmetric) component of the dcvoltage. By fitting, we obtain values for H R,/Delta1H,Vsym, and Va-sym.HRand/Delta1Hare related to the magnetic properties of FM, while Vsymand Va-sym are related to the current-induced driving fields including SOFs and/or Oersted field. Being different from VKerr, which is proportional to the real part of out of plane dynamic magnetization Re( mz), Vdcprobes the real part of in-plane dynamic magnetization Re(my) through the anisotropic magnetoresistance effect of FM. The total detected Vdcis obtained by summing up dVdc for all positions of the device, i.e., V dc=∫l 0dVdc(x), with dVdc(x)=−/Delta1ρn(x)[jFM(x)·n(x)]|xdx, (B1) where lis the length of the device, /Delta1ρis the magnitude of the anisotropic magnetoresistance of FM, and n(x)i s the unit dynamic magnetization at position x. In the mea- surement coordinate system ( x,y,z), the microwave current density jFMflows along the xdirection and the dc voltage is also detected along this direction. In the coordinate systemlabeled ( x /prime,y/prime,z/prime),n(x) and jFM(x) can be respectively writ- ten as n(x)=M-1(M,myei[ωt−/Phi1m(x)],mzeiωt), and jFM(x)= jFMei[ωt−/Phi1m(x)](cosϕM,−sinϕM,0), where my(mz) is the dy- namic magnetization along the y(z) direction, ωis the angular frequency of magnetization precession, and ϕMis the magne- tization angle as defined in Fig. 8(a). At each position xin FM, the microwave current and the induced SOFs/Oersted field arecoherently coupled (the phase difference between these twodynamic quantities is 0). Thus, /Phi1 m(x) cancels out, and dVdc(x) can be obtained as dVdc(x)=−/Delta1ρjFM 2Msin 2ϕMRe(my)dx. (B2) Re(my) is obtained through the complex dynamic suscep- tibility as [ 30,38] /parenleftbigg my mz/parenrightbigg =/parenleftbigg χi−iχo a iχi aχo/parenrightbigg/parenleftbigg hicosφM ho/parenrightbigg , (B3) where χi(χi a) is the complex diagonal (off-diagonal) dynamic magnetic susceptibility due to the in-plane excitation hi, andχo(χo a) is the complex diagonal (off-diagonal) dynamic mag- netic susceptibility due to the out of plane excitation ho. Each component of the susceptibility χhas both real and imaginary parts, χ=Re(χ)+iIm(χ) and can be calculated numeri- cally. From Eq. ( B3), the position dependence Re[ my(x)] can be written as Re[my(x)]=Re(χi)hicosϕM+Im/parenleftbig χo a/parenrightbig ho. (B4) For most of the ST FMR measurements, Mlies in plane, and hiand hocan be expressed as hi=hFL+hNM,y rfand ho= hDL,hFL(/bardbly) and hDL(∼m×y) represent the fieldlike and dampinglike SOFs, and hNM,y rfthe rf current-induced Oersted field in NM (for the case of SOT FMR detecting a single layer of single-crystalline FM with reduced symmetry, hNM,y rf=0). Based on Eqs. ( B2)–(B4),VsymandVa-sym can be, respectively, expressed as Vsym=−/Delta1ρjFMl 2MIm/parenleftbig χo a/parenrightbig hDLsin2φM, Va-sym=−/Delta1ρjFMl 2MRe(χi)/parenleftbig hFL+hNM,y rf/parenrightbig cosϕM. (B5) The magnitude of hDLand h FL+hNM,y rfcan be respectively determined by Vsymand Va-sym through Eq. ( B5). However, optical detection directly probes the real part of out of plane dynamic magnetization mz, and thus the phase variation must be included in Eq. ( B3)a s /parenleftbigg my mz/parenrightbigg =/parenleftbigg χi−iχo a iχi aχo/parenrightbigg/parenleftbigg hicosφM ho/parenrightbigg ei/Phi1l−m(x), (B6) (a) (b) FIG. 9. (a) Temperature dependence of the resistivity of Fe/GaAs with Fe thickness tFeof 3.5 and 0.8 nm, which shows a metal- insulator transition upon decreasing tFe. (b) Equivalent transmission line circuit for tFe=0.8 nm. Because the device length lis larger than the microwave guide wavelength λg, the transmission line can be treated as a series of Ninfinitesimal segments. Each segment in length /Delta1lcontains a RLC circuit, where Lnis the inductance per length, Rnthe resistance per length, and Cnthe capacitance. /Phi1inis the initial phase of the input microwave current. At position xn,t h e phase of the microwave current changes to /Phi1ndue to dielectric loss. 014425-9L. CHEN et al. PHYSICAL REVIEW B 104, 014425 (2021) where /Phi1l−m(x)[=/Phi1in+/Phi1m(x)] is the phase difference be- tween the laser pulse and the dynamic magnetization atposition x. APPENDIX C: DISCUSSION OF THE MECHANISM RESPONSIBLE FOR THE V ARIATION OF /Phi1m Figure 9(a) shows the temperature dependence of the re- sistivity for tFe=3.5 and 0.8 nm. One can see that the temperature coefficient changes from a metal- to a semicon- ductorlike behavior with decreasing tFe. This indicates that, fortFe=0.8 nm, the Fe film is not a good conductor anymore, but behaves like a dielectric, which can be understood fromthe mixing of metallic Fe and semiconducting GaAs states atthe interface [ 39,40]. The microwave guide wavelength λ gcan be calculated by [ 41] λg=λ0√μrεr, (C1)where λ0is the microwave wavelength in free space, μrthe relative permeability, and εrthe relative permittivity. For tFe= 0.8n m ,λ0=2.5 cm (at 12 GHz), μr∼1×105,εr=13 (the dielectric constant of GaAs is approximately adopted), and λg is estimated to be 2.5 μm, which is smaller than the length of the device l. Since λg/lessmuchl, the equivalent transmission circuit can be treated as a series of Ninfinitesimal segments as shown in Fig. 9(b) [41]. Each segment in length /Delta1lcontains aRLC circuit, where Lnis the inductance per length, Rnthe resistance per length, and Cnthe capacitance to ground. Since the capacitor and inductor give a phase shift of 90◦,i ti s expected that the phase of jFM rfis position dependent. This is the possible origin of position-dependent /Phi1mas shown in Fig. 2(c) of the main text, and this could be similar to the case of detection of magnetization dynamics by dc voltage ina CPW, where the phase shift between inductive current anddriving Oersted field may not necessarily be the same [ 42]. 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PhysRevLett.56.1756.pdf

VOLUME56, NUMBER 16 PHYSICAL REVIEW LETTERS 21 APRIL 1986 Comment on "Landau-Lifshitz Equation of Fer- romagnetism: Exact Treatment of the Gilbert Damping" In a recent Letter1 Lakshmanan and Nakamura re ported "that the effect of Landau-Lifshitz-Gilbert damping is just a rescaling of the time variable t by a complex constant, so that for every given solution of the undamped Landau-Lifshitz (LL) equation in any dimension the exact solution of the fully damped ver sion can be given straightforwardly.1' Our aim in this Comment is to point out that, unfor tunately, this result is based on an inadmissable as sumption, and is in general invalid, except in the trivi al case of a single spin in a constant magnetic field. The authors1 transform the LL equation for a uniax ial ferromagnet into an equation of the form (l-/X)-16a>/df-/(co,a>*) (1) and its complex conjugate, for the complex functions a>^(Sx+iSy)/(l+Sz)f (2) co*==(Sx--/^)/(l+Sz). [Compare Eqs. (13) and (4a) of Ref. 1.] By a rescaling of the time variable according to i —, T = (i — /x)r the damping parameter X seemingly disappears, and Eq. (1) takes the same form in the complex time r as its undamped version in the real time /. The authors then conclude that "For every solution in the X = 0 case, we have the corresponding solution in the damped (X^O) case just by the rescal ing / —• T of the time parameter. The corresponding damped spin field S(r,/) can then be constructed sim ply from Eq. (4)" [corresponding to the above Eq. (2)]. However, it must be noted that if a> depends on T, then <D* depends on r*. Thus, the assumption that o> depends on r only is inconsistent with Eq. (1), except in the special case that j (<o,a>*) is independent of a>*. In other words, a> will in general depend on both r and r*, and thus nothing is gained by the rescaling. If, in addition, time-dependent magnetic fields are present, the function /(co,a>*) becomes explicitly dependent on /, and therefore on X after the rescaling, which invalidates the procedure even in the single-spin case. We illustrate these general considerations by dis cussing the motion of a single spin in a uniaxial aniso- tropy field Feff=Szez. For X = 0 the spin precesses around the anisotropy axis Sz according to S=(a xcoss0/> —a sins0f, s0) where s0 and a are constants (a2 + SQ = 1). Hence co(t) = a(\ +s0)~1exp(- is0t), and the rescaling procedure would yield for the damped spin motion a>(r) = a(l+So)"1exp[-/50(l-/X)r]. (3) It is easy to check, however, that this expression does not satisfy Eq. (1), which in the present case [as ob tained from Eq. (13) of Ref. 1 with Vw = 0 and A = — -1 ] takes the form /(l+a>ai*)9oi/9T-co(l-aiQi*) = 0. (4) In the case of a single spin in an applied field BU) = (0,0,£(r)), on the other hand, Eq. (1) be comes idw/BT = fjLB(t)a)f (5) which shows that /(o>, eo*) is independent of cu*. Thus the rescaling procedure does give the correct result in this case if the field B is constant. Finally, an inspection of Eq. (13) of Ref. 1 (com pleted with the terms due to an external field1) shows immediately that the single spin in a constant magnetic field is the only case in which 9cu/9/ becomes indepen dent of (o*. In conclusion, the validity of the rescaling procedure proposed by Lakshmanan and Nakamura1 is strictly limited to the trivial case of a single spin in a constant magnetic field. E. Magyari, H. Thomas, and R. Weber Instttut fur Physik der Universitat Basel CH-4056 Basel, Switzerland Received 2 May 1985 PACS numbers: 75.30.Ds, 03.40.Kf, 75.10.Hk *M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53, 2497 (1984). 1756 © 1986 The American Physical Society

PhysRevB.92.024105.pdf

PHYSICAL REVIEW B 92, 024105 (2015) Theoretical investigation of the magnetic and structural transitions of Ni-Co-Mn-Sn metamagnetic shape-memory alloys Chun-Mei Li,1,2,*Qing-Miao Hu,2Rui Yang,2B¨orje Johansson,3,4,5and Levente Vitos3,4,6 1College of Physical Science and Technology, Shenyang Normal University, Shenyang 110034, China 2Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China 3Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm SE-100 44, Sweden 4Condensed Matter Theory Group, Physics Department, Uppsala University, Post Office Box 516, SE-75120 Uppsala, Sweden 5School of Physics and Optoelectronic Technology and College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China 6Research Institute for Solid State Physics and Optics, Post Office Box 49, Budapest H-1525, Hungary (Received 10 December 2013; revised manuscript received 11 June 2015; published 13 July 2015) The composition-dependent crystal structure, elastic modulus, phase stability, and magnetic property of Ni2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50) are studied by using first-principles calculations in combination with atomistic spin dynamics method. It is shown that the present lattice parameters and Curie temperature ( TC) are in agreement with the available experimental data. The martensitic phase transformation (MPT) occurs forx< 0.43, where the austenite is in the ferromagnetic (FM) state whereas the martensite is in the antiferromagnetic (AFM) one at 0 K. The xdependence of the lattice parameter, elastic modulus, and energy difference between the FM austenite and the AFM martensite well accounts for the decrease of the MPT temperature ( T M) with the Co addition. With increasing x, the increase of the magnetic excitation energy between the paramagnetic and FM austenite of these alloys is in line with the TC∼x.T h eN i3 das well as the Co 3 delectronic states near the Fermi level are confirmed mainly dominating the phase stability of the studied alloys. DOI: 10.1103/PhysRevB.92.024105 PACS number(s): 31 .15.A−,61.66.Dk,62.20.de,75.10.−b I. INTRODUCTION A new class of metamagnetic shape-memory alloys [ 1] (e.g., Ni-Co-Mn-In and Ni-Co-Mn-Sn) has attracted muchattention in recent years. Due to the magnetic-field-inducedmartensitic phase transformation (MPT) [ 1–3], they display high output stress level and relatively large magnetic shape-memory effect (MSME) in comparison with the traditionalNi-Mn-Ga-based alloys [ 4–6]. Experimentally, a 1.0% one- way and a 0.3% two-way MSME have been observed inNi 1.72Co0.28Mn 1.52Sn0.44[7]. The Ni-Co-Mn-Sn group [ 7–9], containing no expensive element and with considerableMSME, is even more promising for magnetic actuationapplications, such as magnetic refrigeration and as a magneticsensor [ 10,11]. The magnetic and martensitic transitions of Ni-Co-Mn- Sn are highly composition dependent [ 12–20]. Different compositions may result in different combination of the Curietemperature ( T C) and the MPT temperature ( TM). Lowering the Sn content relative to that of Mn increases the TMbut decreases theTC[17,19]. On the other hand, adding more Co content relative to that of Ni increases the magnetization but suppressesthe MPT [ 12,15]. For example, in Ni 1.72Co0.28Mn 2−xSnx[17], withxdecreasing from 0.40 to 0.28, TMgoes up from 423 to 561 K whereas TCgoes down from 593 to 393 K. In Ni 2−xCoxMn 1.56Sn0.44, with xrising up to 0.40, the TC increases to 450 K [ 12], while the MPT cannot occur there even in very low temperature because of the decrease of the TMwith increasing x. The different combinations of TMandTCresult in different properties and also various technological significance *Corresponding author: cmli@imr.ac.cnof the alloys. In Ni 2−xCoxMn 1.60Sn0.40withx≈0.20 [20], theTMandTCare close to each other and consequently the structural and magnetic transitions may couple to eachother. This coupling induces some attractive properties suchas giant magnetocaloric effect, magnetostriction, and mag-netoresistance, which are important for the applications ofthe magnetic shape memory, energy conversion, or solid-staterefrigeration [ 5,10,21]. To build the connection between the composition and T Mas well as TC, and to understand their origin and the underlying physics, are critical for designingnew Ni-Co-Mn-Sn with desirable properties. From the atomic scale of investigations, in the present paper we will explore systematically the composition-dependentmagnetic and structural transitions of Ni 2−xCoxMn 1.60Sn0.40 (0/lessorequalslantx/lessorequalslant0.50). It is known that these studied X 2MnZ types of shape-memory alloys generally possess cubic L21structure in the austenite but a tetragonal one in the martensite [ 22]. Based on first-principles calculations, we first study in detailthe composition dependence of the crystal structure, lattice pa-rameters, elastic constants, and free-energy difference betweenthe two phases, and examine their connection with the T M∼x. Furthermore, in combination with atomistic spin dynamicscalculations, the T C∼xis estimated, and its correlation with thexdependence of the magnetic excitation energy between the paramagnetic (PM) and ferromagnetic (FM) states of thesealloys is investigated. Finally, the electronic origin of thephase stability is presented in combination with the Jahn-Tellertheory. The rest of the paper is arranged as follows: in Sec. II,w e describe the first-principles and the atomistic spin dynamicsmethods we used and the calculation details; in Sec. III,t h e composition-dependent crystal structure, lattice parameters,elastic constants, phase stability, magnetic property, and 1098-0121/2015/92(2)/024105(9) 024105-1 ©2015 American Physical SocietyLI, HU, YANG, JOHANSSON, AND VITOS PHYSICAL REVIEW B 92, 024105 (2015) electronic origin are presented. Finally, we summarize the main results of this work in Sec. IV. II. METHODS AND CALCULATION DETAILS A. Calculation of the total energy To carry out the electronic structure and total-energy calculations, the first-principles exact muffin-tin orbitals(EMTO) method [ 23–27] is used in the present work. Within this program, the Kohn-Sham potential is represented bylarge overlapping potential spheres, which are optimized byminimizing the deviation between the exact and overlappingpotentials. Thus, one describes more accurately the exactcrystal potential compared to the conventional muffin-tin ornonoverlapping methods. Another important trait is that theEMTO tool can conveniently incorporate coherent potentialapproximation (CPA) method [ 24,27], which is one of the few possible approaches to deal with both the compositional andmagnetic disorder at the first-principles level. In a number ofprevious works, the EMTO-CPA method has been shown tobe suitable and accurate enough to compute the anisotropiclattice distortions, and thus the elastic constants of randomalloys [ 23,25,27,28]. For the present application, the exchange-correlation po- tential is described within the Perdew-Burke-Ernzerhof gener-alized gradient approximation. The EMTO basis sets includes,p,d, andfcomponents, and the scalar-relativistic and soft- core approximation are employed. The overlapping potentialsphere radius ( R Ni mt) and the atomic radius ( RNi WS)o nt h e Ni sublattice are optimized by RNi mt=0.95RWSandRNi WS= 1.10RWS, respectively, where RWSis the average Wigner-Seitz radius. For the other two sublattices ( X=Mn and Sn), the usual setups RX mt=RWSandRX WS=RWSare adopted. The Brillouin zone is sampled by a 13 ×13×13 uniform k-point mesh without any smearing technique. The equilibrium lattice parameters, bulk modulus, and magnetic moments are determined by fitting the total energiesversus volume (nine data points) to a Morse function [ 29]. The elastic constants are calculated with the mathematical formuladescribed in our previous paper [ 30]. The Debye temperature is obtained by means of the Hill average [ 31] with Eqs. (6.27) in Ref. [ 24]. The magnetic ordering is described by three kinds of configurations: (a) the FM state with parallel alignmentbetween Mn on the Mn sublattice (Mn 1) and Mn on the Sn sublattice (Mn 2); (b) the antiferromagnetic (AFM) state with antiparallel alignment between Mn 1and Mn 2; and (c) the PM state described by the disordered local magnetic model [ 32]. The number of valence electrons per atom ( e/a) is calculated with Ni 3 d84s2,C o3d74s2,M n3 d54s2, and Sn 4 d105s2p2. B. Calculation of the magnetism The temperature dependence of the magnetic property is evaluated with the Uppsala Atomistic Spin Dynamics(UppASD) program [ 33–36]. Within this method, the itinerant electron system is mapped to an effective classical Heisenbergmodel: H=−1 2/summationdisplay i/negationslash=jJijmi·mj, (1)where Jijare the interatomic exchange interactions; the indices iandjare 1, 2, and 3, representing the Mn, Ni, and Co atoms. The miis the magnetic moment of atom i, the motion of which is described using the Landau-Lifshitz-Gilbert equation [ 33,34]: ∂m i ∂t=−γmi×[Bi+bi(t)] −γα mmi{mi×[Bi+bi(t)]}. (2) In this expression, Bi=−∂H ∂mi, is the so-called effective field experienced by each atom i.γis the gyromagnetic ratio. bi(t) is a stochastic magnetic field with a Gaussian distribution withrespect to temperature ( T), and its magnitude is related to the damping parameter α, which eventually brings the system into thermal equilibrium. With the solved m iin the given T,t h e magnetization ( M) and magnetic susceptibility ( χ) are then calculated from M=1 N/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftBigg/summationdisplay imx,i/parenrightBigg2 +/parenleftBigg/summationdisplay imy,i/parenrightBigg2 +/parenleftBigg/summationdisplay imz,i/parenrightBigg2 (3) and χ=1 kBT2[/angbracketleftM2/angbracketright−/angbracketleftM/angbracketright2], (4) respectively, where N(=3) means the three types of magnetic atoms (Mn, Ni, and Co) and kBis the Boltzmann constant. In our calculations, the cubic periodic box size is kept to 15×15×15 unit cells. The time step for solving the above differential equations ( 2)i s1 0−16s. The number of the time steps used is 10 000. The αis set at 0.01. Including the interactions between the atoms within the tenth-nearestneighbors, the 0-K J ijare calculated using the magnetic force theorem [ 37] implemented in the EMTO-CPA program [ 24]. III. RESULTS AND DISCUSSION A. Crystal structure Figure 1shows the total electronic energies for the FM, AFM, and PM states of Ni 2−xCoxMn 1.60Sn0.40withx=0.10, as functions of the tetragonal lattice ratio ( c/a) and the Wigner-Seitz radius ( rWS). For the FM state [Fig. 1(a)], we get only one energy minimum around RWS=2.757 bohrs andc/a=1, corresponding to the L21phase. Nevertheless, for both the AFM and PM states [Figs. 1(b) and 1(c), respectively], the energy shows two minima: one is at c/a=1, meaning the cubic austenite, and another one is aroundc/a=1.20–1.30, corresponding to the tetragonal martensite. In comparison, in Figs. 1(a)–1(c), the austenite with the lowest energy is in the FM state, whereas the martensitewith relative lower energy tends to be in the AFM one.For Ni 1.90Co0.10Mn 1.60Sn0.40, the electronic energy prefers the austenite in the FM state and the martensite in the AFM one. The xdependence of the relative electronic en- ergy of the AFM and PM austenite [ /Delta1EAus AFM(x) and /Delta1EAus PM(x)] and martensite [ /Delta1EMar AFM(x) and /Delta1EMar PM(x)] of Ni2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50) are listed in Table I. The reference in each xis the electronic energy of the FM austenite [ EAus FM(x)]. For the austenite, the /Delta1EAus AFM(x)i s 024105-2THEORETICAL INVESTIGATION OF THE MAGNETIC AND . . . PHYSICAL REVIEW B 92, 024105 (2015)(bohrs) FIG. 1. (Color online) Total electronic energy contours (in Ry) for the (a) FM, (b) AFM, and (c) PM Ni 1.90Co0.10Mn 1.60Sn0.40alloys as a function of the tetragonal lattice ratio ( c/a) and the Wigner-Seitz radius ( rWS). smallest for x=0 whereas for 0 .10/lessorequalslantx/lessorequalslant0.50 the FM state tends to be lowest in the energy because of the positive valuesof both /Delta1E Aus AFM(x) and /Delta1EAus PM(x). For the martensite, since /Delta1EMar AFM(x) is always much smaller than /Delta1EMar PM(x) in each x, the AFM state is energetically stabilized for all of these alloysat 0 K. In Fig. 2, the equilibrium lattice parameters [ a(x)] of the L2 1phase with FM, AFM, and PM states, respectively, are TABLE I. Relative electronic energy (in mRy) of the AFM [/Delta1EAus AFM(x)] and PM [ /Delta1EAus PM(x)] austenite and the AFM [ /Delta1EMar AFM(x)] and PM [ /Delta1EMar PM(x)] martensite to that of the FM austenite of Ni2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50) alloys. x/Delta1 EAus FM(x)/Delta1EAus AFM(x)/Delta1EAus PM(x)/Delta1EMar AFM(x)/Delta1EMar PM(x) 0.00 0.00 −0.28 1.57 −2.16 1.49 0.10 0.00 0.10 1.99 −1.69 1.80 0.20 0.00 0.46 2.40 −1.21 2.08 0.30 0.00 0.82 2.82 −0.71 2.39 0.40 0.00 1.20 3.24 −0.18 2.73 0.50 0.00 1.58 3.65 0.36 3.09FIG. 2. (Color online) Composition ( x) dependence of the equi- librium lattice parameter a(x)o ft h e L21-Ni 2−xCoxMn 1.60Sn0.40 (0/lessorequalslantx/lessorequalslant0.50) with FM, AFM, and PM states, respectively. The solid points denote our present a(x) values. The open squares mean the experimental a(x)f o rx=0.20 and 0.24, which are from Refs. [ 20] and [ 15], respectively. shown against x. In any of the three magnetic states, the a(x) decreases linearly with increasing x. In each composition, thea(x) is always biggest in the FM state but smallest in the AFM one. The open squares in the figure denote theexperimental data for x=0.20 and 0.24, respectively [ 15,20]. It is clear that our present a(x) in the FM sate is in much better agreement with them than those in the AFM and PMstates. Since these experimental a(x) are measured above room temperature [ 15,20], they are shown a little larger than our values in the FM state due to the thermal expansion. Thexdependences of the lattice parameters [ a(x) and c/a(x)] of the tetragonal martensite are depicted in Fig. 3. With increasing x,t h ea(x) in both the AFM and PM states decrease linearly as well. Nevertheless, the c/a(x) decreases in the AFM state but keeps almost constant around 1.20 in the PM state.It is noted that in the AFM state the c/a(x) values are around 1.25–1.31, which are comparable to the data (1.31) calculated in Ni 2Mn 1.50Sn0.50[38]. In addition, following the relationship ofTM(x)∼c/a(x) found in the NiMn-based alloys [ 39,40], a larger c/a(x) corresponds to a higher TM(x); the present decrease of c/a(x) in the AFM state happens to correspond to the decrease of the experimental TM(x) with Co addition in Ni2−xCoxMn 1.60Sn0.40[15,20,41]. B. Elastic property In Table II, the calculated bulk modulus B(x), elastic constants Cij(x), and Debye temperature /Theta1(x)o ft h eF M , AFM, and PM L21-Ni 2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50) alloys are listed. According to the dynamical or mechanicalstability condition of a lattice, the stability criteria forcubic crystals requires that C 11>|C12|,C11+2C12>0, and C44>0. From our calculations, these Cij(x) of the FM and PM states satisfy all of the above conditions. However, inthe AFM state, the C 11(x)i ss m a l l e rt h a n |C12(x)|when 024105-3LI, HU, YANG, JOHANSSON, AND VITOS PHYSICAL REVIEW B 92, 024105 (2015) FIG. 3. (Color online) Composition ( x) dependence of the equi- librium lattice parameters [ a(x) in (a) and c/a(x)i n( b ) ]o ft h e tetragonal Ni 2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50) with AFM and PM states, respectively. 0/lessorequalslantx/lessorequalslant0.40, and for x=0.50 they are almost comparable because C/prime(x){=1 2[C11(x)−C12(x)]}is merely about 0.9 GPa in this composition. It is supposed that the Cij(x)i nt h eA F M state do not follow the requirement of C11(x)>|C12(x)|. Neglecting the temperature effect on the Cij(x), the AFM TABLE II. Composition ( x) dependence of the theoretical bulk modulus [ B(x), in GPa], elastic constants [ Cij(x), in GPa], and Debye temperature [ /Theta1(x), in K] of the FM, AFM, and PM L21- Ni2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50). The tetragonal shear elastic constant C/prime(x)=1 2[C11(x)−C12(x)]. States xB (x)C11(x)C12(x)C44(x)C/prime(x)/Theta1(x) FM 0.00 144.6 160.6 136.6 114.8 12.0 332.2 0.10 145.8 163.5 136.9 116.2 13.3 340.9 0.20 147.0 167.1 136.9 117.7 15.1 350.8 0.30 148.2 171.0 136.8 119.2 17.1 361.70.40 149.4 175.5 136.3 120.4 19.6 373.5 0.50 150.8 180.4 136.0 122.2 22.2 385.1 AFM 0.00 143.2 141.5 144.1 122.7 −1.3 0.10 143.9 141.6 145.0 124.2 −1.7 0.20 144.7 142.4 145.8 125.7 −1.7 0.30 145.5 143.9 146.3 127.3 −1.2 0.40 146.3 145.9 146.5 128.7 −0.3 0.50 147.2 148.4 146.6 130.3 0.9 PM 0.00 142.4 152.4 137.4 119.3 7.5 300.9 0.10 143.4 151.4 139.4 120.6 6.0 286.4 0.20 144.3 151.0 141.0 121.9 5.0 274.3 0.30 145.3 151.0 142.4 123.1 4.3 265.20.40 146.4 151.6 143.8 124.5 3.9 259.8 0.50 147.5 152.4 145.0 125.8 3.7 257.1L21-Ni 2−xCoxMn 1.60Sn0.40therefore is mechanically unsta- ble in low temperature. This may be the reason whyNi 2Mn 1.60Sn0.40is measured with the FM state but not the AFM one at 4.2 K [ 41,42], in spite of the latter one being confirmed energetically favorable from above calculations. Inthe present work, all these studied austenitic alloys are con-firmed both thermodynamically and mechanically stabilizedby the FM state at 0 K, which is in good agreement with theexperimental measurements in low temperatures [ 7,43]. In Table III, the calculated B(x),C ij(x), and /Theta1(x)i nt h e AFM and PM martensite are listed. The dynamical or mechan-ical stability criteria for tetragonal crystals requires that C 11> |C12|,C33>0,C44>0,C66>0, (C11+C33−2C13)>0, and (2 C11+C33+2C12+4C13)>0. Our present Cij(x)i n both the AFM and PM states follow these conditions. Since theAFM martensite is relatively lower in energy than the PM one,the martensite of all these alloys is both thermodynamicallyand mechanically stabilized by the AFM ordering betweenMn 1and Mn 2at 0 K. The antiparallel alignment between Mn 1 and Mn 2has been confirmed in the tetragonal structure of Ni2Mn 1+xSn1−xternary alloys by means of both first-principle calculations [ 38,44] and neutron-diffraction experiment [ 45]. Although around room temperature, several different magneticstates have been reported in the martensitic NiCoMnSn qua-ternary alloys [ 7,12,14,20,43], such as antiferromagneticlike, ferrimagnetic, paramagnetic, superparamagnetic, and super-spin-glass states. Almost all of these nonferromagnetic statesindicate the existence of the AFM coupling between Mn 1and Mn 2in the phase. Therefore, in the present work it is seen as reasonable that with less than 25% Ni replaced with Co inNi 2−xCoxMn 1.60Sn0.40the martensite still remains in the AFM state at 0 K. In Tables IIand III, the tetragonal shear elastic mod- ulus of the austenite, C/prime(x), and that of the martensite {Cs(x)[=C11(x)+C12(x)+2C33(x)−4C13(x)]}are espe- cially shown for comparison. It is found that in the samecomposition xtheC /prime(x) is very small whereas the Cs(x)i s relatively quite large. The particularly low value of C/prime(x) indicates a strong negative contribution of the entropy ( −TS) to the free energy ( F) of the austenite, which ultimately stabilizes the phase against the martensite with increasingT. The FM and AFM couplings between the Mn 1and Mn 2in Ni 2−xCoxMn 1.60Sn0.40correspond to the ground-state magnetic ordering of the austenite and martensite, respectively.With increasing x(or decreasing e/a), theC /prime(x) in the FM state increases whereas the Cs(x) in the AFM state decreases. The Co doping tends to mechanically stabilize the cubic relativeto the tetragonal structure in low temperature, which results inlower experimental T M(x) of this type of alloys [ 15,20,41]. In the PM state, the C/prime(x) decreases but the Cs(x) increases with increasing x(or decreasing e/a), preferring the stability of the martensite relative to the austenite. Then, the oppositetrend of experimental T M(x)∼xis estimated. It means that the 0-K C/prime(x)∼xandCs(x)∼xin the PM state fail to account for the experimental trend of TM(x)∼x. This failure could be ascribed to the fact that, in the high-temperature PMstate, the temperature effects on the elastic constants, such aselectronic entropy, phonon smearing, thermal expansion, andmagnetism [ 46], might be significant and thus could not be ignored. 024105-4THEORETICAL INVESTIGATION OF THE MAGNETIC AND . . . PHYSICAL REVIEW B 92, 024105 (2015) TABLE III. Composition ( x) dependence of the theoretical bulk modulus [ B(x), in GPa], elastic constants [ Cij(x), in GPa], and Debye temperature [ /Theta1(x), in K] of the AFM and PM tetragonal Ni 2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50). The tetragonal shear elastic constant Cs(x)= C11(x)+C12(x)+2C33(x)−4C13(x). States xB (x) C11(x) C12(x) C13(x) C33(x) C44(x) C66(x) Cs(x) /Theta1(x) AFM 0.00 145.8 177.5 118.3 128.8 188.2 99.9 88.1 157.0 377.7 0.10 146.4 182.3 114.7 129.6 188.4 104.4 89.4 155.4 385.7 0.20 147.0 184.5 113.7 130.2 189.1 108.8 90.9 155.7 389.90.30 147.8 189.5 109.7 133.4 183.8 110.8 92.5 133.0 388.7 0.40 148.5 195.0 105.6 134.0 184.8 116.0 93.1 134.2 396.4 0.50 149.3 199.1 102.9 135.7 183.3 119.9 94.0 125.6 398.1 PM 0.00 141.4 198.6 86.0 134.1 159.6 117.9 58.9 67.3 356.0 0.10 142.6 197.9 89.5 134.0 164.1 117.6 65.1 79.4 365.7 0.20 144.0 199.8 90.6 134.2 168.4 122.1 69.3 90.3 377.00.30 145.3 201.0 92.2 134.8 171.4 119.9 73.9 96.7 388.1 0.40 146.6 202.6 93.2 135.9 173.4 121.0 77.4 99.2 383.7 0.50 147.9 205.3 93.3 137.0 175.2 122.8 79.7 101.1 388.0 C. Phase stability In NiMn-based shape-memory alloys [ 47,48], the large free-energy difference between the austenite and martensite(/Delta1F AM) generally means the big driving force of the MPT, and then the high critical temperature TM. Here, we cal- culate the /Delta1FAM(x) with the approximation, /Delta1FAM(x)≈ /Delta1EAM(x)+/Delta1FAM ph(x), where the /Delta1EAM(x) is the electronic energy difference between the austenite and martensite andthe/Delta1F AM ph(x) is that of the phonon vibrational free-energy difference. The /Delta1FAM ph(x) may be calculated with Eq. (5) in a previous paper [ 49], which is nevertheless very time consum- ing because of the temperature-dependent Debye temperatureterm [ 46]. For the sake of simplicity, the present 0-K /Delta1F AM ph(x) is evaluated from its zero-point expression, /Delta1FAM ph(x)≈ 9 8kB[/Theta1A(x)−/Theta1M(x)], with the /Theta1A(x) and/Theta1M(x) being the Debye temperature in the austenite and martensite, respec-tively [ 30]. In finite temperature, the /Delta1F AM ph(x)i ss i m p l y estimated from its high-temperature expansion, /Delta1FAM ph(x)≈ 3kBT/Theta1A(x)−/Theta1M(x) /Theta1A(x)[50]. Listed in Tables IIandIII, in addition to the fact that the C/prime(x) is much smaller than the Cs(x), the/Theta1A(x) is always smaller than /Theta1M(x) in each x. This means that the /Delta1FAM ph(x) provides a negative contribution to the /Delta1FAM(x). With the obtained ground-state magnetic ordering of the two phases, we calculate the /Delta1FAM(x)o f Ni2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50) in different tempera- tures from 0 to 500 K with intervals of 100 K. In Fig. 4, the estimated /Delta1FAM(x)∼xis shown in each temperature, in comparison with the available experimental TM(x)∼x. It is clear that in each xthe/Delta1FAM(x) indeed decreases with increasing Tbecause of the negative contribution of the /Delta1FAM ph(x), which prefers the relative stability of the parent phase. With increasing x(or decreasing e/a), the/Delta1FAM(x) decreases in each temperature, corresponding to the TM(x) decreasing with the Co doping [ 15,20,41]. Positive /Delta1FAM(x) means that the AFM martensite is lower in energy and then more stable than the FM austenite, whereasnegative /Delta1F AM(x) means that the latter one is more stable. In Fig. 4, the 0-K /Delta1FAM(x) is close to zero around x=0.43, reflecting that the austenite and martensite of the alloy areenergetically comparable at 0 K. For alloys with x< 0.43, the AFM martensite is relative more stable in low temperature be-cause/Delta1F AM(x) tends to be positive, whereas above x=0.43 due to /Delta1FAM(x)<0 the FM austenite is always stabilized at ambient temperature. It suggests that even in very lowtemperature the alloys with x> 0.43 would not undergo MPT and instead they are stabilized with the FM cubic structure. Thepredicted critical composition ( x=0.43) of whether the MPT can occur or not in Ni 2−xCoxMn 1.60Sn0.40is comparable to that (around 0.32–0.36) measured in Ni 2−xCoxMn 1.56Sn0.44[12]. D. Magnetic property In Table IV, the 0-K local magnetic moments of Ni, Co, two types of Mn, and Sn atoms, together with the total magneticmoments of Ni 2−xCoxMn 1.60Sn0.40alloys are summarized. FIG. 4. (Color online) Free-energy difference between the FM austenite and the AFM martensite [ /Delta1FAM(x)], together with the available experimental TM(x)o fN i 2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant 0.50) with respect to xand the number of valence electrons per atom ( e/a). The /Delta1FAM(x) is calculated in different temperatures from 0 to 500 K with intervals of 100 K. The TM(x) are cited from Refs. [ 15,20,41]. 024105-5LI, HU, YANG, JOHANSSON, AND VITOS PHYSICAL REVIEW B 92, 024105 (2015) TABLE IV . Local magnetic moments (in μB)o fN i ,C o ,a n dM n on Mn(Mn 1)a n dS ns i t e s( M n 2), and Sn atoms, together with the total magnetic moments (in μB) of the FM, AFM, and PM states of Ni2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50). States Ni Co Mn 1 Mn 2 Sn Tot FM 0.55 1.34 3.48 3.49 −0.05 6 .70–6.98 AFM 0.13 0.65 3.40 −3.45 −0.05 1 .55–1.77 PM 0 0 3.44( −3.44) 3.45( −3.45) 0 0 The concentration of Co as well as the crystal structure do not influence these local magnetic moments significantly, andtherefore, in the table we show them only as a function ofthe magnetic ordering for the alloy with x=0.1. It is found that, in all the FM, AFM, and PM states, the Mn 1and Mn 2 atoms are spin polarized, and in absolute value their magneticmoments (around 3.45 μ B) are almost the same. The Ni and Co atoms are spin polarized only in the FM and AFM states.Their magnetic moments are always parallel to those of theMn atoms on the Mn sublattice, and the values in the AFMstate (0.13 μ Bfor Ni, 0.65 μBfor Co) turn out to be lower than their correspondents in the FM one (0.55 μBfor Ni, 1.34 μB for Co). The Sn atoms are almost non-spin-polarized in all the three magnetic states. In a result, the 0-K total magneticmoment is around 6.70 μ B∼6.98μBin the FM state, and 1.55μB∼1.77μBin the AFM state. It reveals that with xincreasing from 0 to 0.50 the total magnetic moments show an increase of less than 0.30 μBin both the FM and AFM states of alloys. In order to explore the magnetic transition from the FM state to the PM one of the austenitic alloys in finite temperature,we calculate both the Mandχof these alloys at different temperatures from 0 to 700 K with intervals of 25 K, bymeans of EMTO-CPA in combination with UppASD method.In Fig. 5, the obtained temperature dependence of the χas well as the normalized magnetization ( M/M 0, with Mand M0being the magnetization at Tand 0 K, respectively) are shown for each x, together with the TC(x) estimated from the temperature corresponding to the maximum of χ. It is found that with increasing xfrom 0 to 0.50 our theoretical TC(x) monotonically goes up from 317 to 424 K, which is in line with the available experimental data [ TExp. C(x)] [15,20,41]s h o w ni n Fig. 6. Similar to Fe-doped NiMn-based alloys [ 51], the Co addition increases the TC(x) and then enhances the saturated magnetization under the magnetic field, which is consequentlyhoped to improve the output stress of these alloys during theMPT. Seen from the energy calculations in Fig. 6,i ti ss h o w n that the electronic energy difference between the PM and theFM austenite [ /Delta1E PF(x)] linearly increases with x, which is consistent with the trend of TC(x)∼x. It indicates that the relationship of TC(x)∼xshould be originated from the trend of/Delta1E PF(x)∼x, i.e., the Co addition increases the magnetic FIG. 5. (Color online) Normalized magnetic moment ( M/M 0, with MandM0being the magnetization at Tand 0 K, respectively) as well as susceptibility ( χ) with respect to temperature ( T), together with the estimated TC(x)o fN i 2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50). 024105-6THEORETICAL INVESTIGATION OF THE MAGNETIC AND . . . PHYSICAL REVIEW B 92, 024105 (2015) FIG. 6. (Color online) Total electronic energy difference be- tween the PM and FM austenite [ /Delta1E PF(x)] as well as the TC(x) ofL21-Ni 2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50) with respect to x and the number of valence electrons per atom ( e/a). The Tthe C(x) denotes the present theoretical values whereas the Texp. C(x) means the experimental data from Refs. [ 15,20,41]. excitation energy and therefore enhances the driving force of the magnetic transition from the FM state to the PM one inNi 2−xCoxMn 1.60Sn0.40. E. Electronic structure The stability of the parent phase in NiMn-based alloys has been demonstrated to be closely related to the minority (spin-down) density of states (DOS) around Fermi level [ 52–54]. In order to explore the electronic origin of the composition-dependent MPT, we calculate and compare the DOS of FML2 1-Ni 2−xCoxMn 1.60Sn0.40(x=0, 0.20, and 0.40) alloys with 0 and 5% tetragonal distortion used to calculate C/prime(x), as shown in Fig. 7.F o rN i 2Mn 1.60Sn0.40, there exits a pseudogap in the total DOS both with and without lattice distortionat about −0.05 Ry, which was shown to be the covalent bonding characters between Sn pand Ni 3 das well as Co 3delectrons [ 55]. However, for the total DOS without lattice distortion [in Fig. 7(a)], a small peak appears at about −0.02 Ry below the Fermi level, resulting in the Jahn-Teller instabilityof the cubic phase [ 52,54,56–58]o fN i 2Mn 1.60Sn0.40. Upon tetragonal distortion [in Fig. 7(b)], this peak splits and then the DOS near the Fermi level reduces, leading to a more stabletetragonal phase of the alloy. Shown in Fig. 7, with increasing xthe pseudogap is gradually filled and then becomes more and more shallow. Itdeserves to be noted that with Co doping [in Fig. 7(a)] the small peak gradually disappears. Upon 5% tetragonal distortion [inFig. 7(b)], its splitting is thus less and less significant with increasing x. This means that the Co doping reduces the Jahn-Teller instability and then depresses the tetragonal latticedistortion, which corresponds to the increase of C /prime(x)b u tt h e decrease of the TM(x) with the Co addition. In comparison to the DOS of Ni as well as Co atoms shown in Fig. 8, it is found that the pseudogap is formed by the Ni 3dand Co 3 dstates. Both of them include triple-degeneratedFIG. 7. (Color online) Total minority (spin-down) density of states (DOS) of FM L21-Ni 2−xCoxMn 1.60Sn0.40(x=0, 0.20, and 0.40) with no lattice distortion (a) and 5% tetragonal distortion used to calculate C/prime(x) (b). The vertical lines indicate the Fermi level. T2gand double-degenerated Egbands. In Figs. 8(a) and8(c), the pseudogap formed by Ni 3 dT 2gand Ni 3 dE gis around −0.05 Ry. In Figs. 8(b) and8(d), the pseudogap formed by Co 3dT 2gand Co 3 dE gis in a relative higher energy level, which is almost right on the Fermi level. Therefore, withNi replacing with Co, the hybridization between Ni 3 dand Co 3delectrons around −0.05 Ry becomes more and more strong, and the pseudogap in the place is gradually filled forNi 2−xCoxMn 1.60Sn0.40. In Figs. 8(a) and8(b), the small peak in the total DOS of Ni 2Mn 1.60Sn0.40is shown to be mainly contributed by Ni 3dE g, whereas the Ni 3 dT 2gas well as the whole Co 3 d states seem to have no connection with the peak. In Fig. 8(c), upon 5% tetragonal distortion, the Ni 3 dE gof the alloy splits into two levels: one is on a little higher energy side withthex 2−y2orbital, whereas the other one is on the relative lower energy side with the 3 z2−r2orbital. With increasing x,t h eN i3 dE gstates reduce and the peak is weakened [in Fig.8(a)]. Upon tetragonal distortion [in Fig. 8(c)], its splitting is therefore less and less with the Co addition, meaning thatthe Jahn-Teller instability in the total DOS reduces and theFM cubic structure gets relatively more and more stable withincreasing xin Ni 2−xCoxMn 1.60Sn0.40. IV . CONCLUSION Using first-principles EMTO-CPA in combination with UppASD method, we have systematically investigated thecomposition-dependent crystal structure, lattice parameters,elastic property, phase stability, Curie temperature, andelectronic structure of Ni 2−xCoxMn 1.60Sn0.40(0/lessorequalslantx/lessorequalslant0.50) quaternary shape-memory alloys. The main results are sum-marized as follows. (1) The present lattice parameters a(x) andc/a(x)o ft h e FM austenite and the AFM martensite decrease with increasing 024105-7LI, HU, YANG, JOHANSSON, AND VITOS PHYSICAL REVIEW B 92, 024105 (2015) FIG. 8. (Color online) Minority (spin-down) density of states (DOS) of Ni 3 d,N i3dE g,a n dN i3 dT 2gas well as Co 3 d,C o3dE g,a n d Co 3dT 2gin FM L21-Ni 2−xCoxMn 1.60Sn0.40(x=0, 0.20, and 0.40) with no lattice distortion (upper panel) and 5% tetragonal distortion used to calculate C/prime(x) (lower panel). The figure illustrates how T2gandEgbands of Ni 3 das well as Co 3 dare split by the lattice distortion. The vertical lines indicate the Fermi level. x, which are in good agreement with the available theoretical and experimental data. (2) The MPT is found occurring below x=0.43. Above the composition, the alloys are stabilized by the FM cubicL2 1phase even in very low temperature. For x< 0.43, the austenite is stabilized by the FM coupling between Mn 1and Mn 2, whereas the martensite is with the AFM ordering at 0 K. (3) With increasing x(or decreasing e/a), the c/a(x)o f the AFM martensite decreases, the shear elastic modulus ofthe FM austenite C /prime(x) increases whereas that of the AFM martensite Cs(x) decreases, and the free-energy difference between the two phases /Delta1FAM(x) decreases, which all well account for the decrease of the experimental TM(x) with increasing x. (4) The estimated TC(x)∼xis in line with the available experimental data. With the Co addition, the magnetic excita-tion energy /Delta1E PF(x) increases, which therefore enhances thedriving force of the magnetic transition from the FM state to the PM one. (5) The calculated electronic structure indicates that the Ni 3das well as the Co 3 dstates near the Fermi level mainly dominate the phase stability of these studied alloys. 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PhysRevB.86.144415.pdf

PHYSICAL REVIEW B 86, 144415 (2012) Effect of disorder on transverse domain wall dynamics in magnetic nanostrips Ben Van de Wiele,1Lasse Laurson,2and Gianfranco Durin3,4 1Department of Electrical Energy, Systems and Automation, Ghent University, B-9000 Ghent, Belgium 2COMP Centre of Excellence, Department of Applied Physics, Aalto University, P .O. Box 14100, FIN-00076 Aalto, Finland 3ISI Foundation, Via Alassio 11/c, I-10126 Torino, Italy 4Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, I-10135 Torino, Italy (Received 29 June 2012; published 24 October 2012) We study the effect of disorder on the dynamics of a transverse domain wall in ferromagnetic nanostrips, driven either by magnetic fields or spin-polarized currents, by performing a large ensemble of graphics processingunit-accelerated micromagnetic simulations. Disorder is modeled by including small, randomly distributednonmagnetic voids in the system. Studying the domain wall velocity as a function of the applied field andcurrent density reveals fundamental differences in the domain wall dynamics induced by these two modes ofdriving: For the field-driven case, we identify two different domain wall pinning mechanisms, operating belowand above the Walker breakdown, respectively, whereas for the current-driven case pinning is absent abovethe Walker breakdown. Increasing the disorder strength induces a larger Walker breakdown field and current,and leads to decreased and increased domain wall velocities at the breakdown field and current, respectively.Furthermore, for adiabatic spin-transfer torque, the intrinsic pinning mechanism is found to be suppressed bydisorder. We explain these findings within the one-dimensional model in terms of an effective damping parameterα ∗increasing with the disorder strength. DOI: 10.1103/PhysRevB.86.144415 PACS number(s): 75 .78.Fg, 72 .25.Ba, 75 .78.Cd Domain wall (DW) dynamics in nanoscale ferromagnetic wires and strips driven by magnetic fields or spin-polarizedcurrents is a subject of major technological importance for theoperation of potential future nanoscale magnetic memory 1,2 and logic3devices. In these devices information is typically stored as magnetic domains along a nanostrip or wire and isprocessed by DW motion. For the reliable operation of suchdevices it is of fundamental importance to understand andcontrol the effect of imperfections or disorder on the DWdynamics, necessarily present in any realistic samples, e.g.,in the form of thickness fluctuations and grain structure ofthe sample, or various impurities and defects in the material.At the same time, such systems constitute a low-dimensionallimit of the general problem of driven elastic manifolds in arandom potential. 4 While the crucial importance of disorder for the dynamics of higher-dimensional DWs is well established, resultingin phenomena such as the Barkhausen effect, 5a majority of studies of DW motion in systems with nanostrip orwire geometry neglect disorder effects. This applies to boththeoretical studies and interpretations of experimental results.Some exceptions include studies demonstrating enhancedDW propagation due to the roughness of the edges of thestrip. 6,7Recently also the effect of spatially varying saturation magnetization Mson the dynamics of vortex walls was studied, resulting in an effective damping increasing withthe disorder strength. 8Similar spatially distributed disorder has also been studied in a simplified, line-based model of atransverse DW. 9,10Experimental studies of DW dynamics in wires have revealed its stochastic nature in the case of shortcurrent pulses, 11and has been attributed to the presence of disorder in the samples, in combination with thermal effects.For longer current pulses, the resulting average DW velocitieshave been shown to be quite low, 12likely due to pinning effectsinduced by structural disorder. Dynamical pinning effects have also been observed in experiments of field-driven vortex walldynamics. 13,14However, despite these advances, many details of the disorder effects on DW dynamics in nanostructuresremain to be clarified. In this paper, we consider by micromagnetic simulations the effect of disorder on the field and current-driven dynamics of atransverse DW in a narrow and thin permalloy strip. Disorder ismodeled by including randomly positioned small nonmagneticregions (voids) in the system. Our results show that thefield- and current-driven DW dynamics exhibit remarkabledifferences which are only revealed in the presence of disorder.In particular, we identify two fundamentally different DWpinning mechanisms acting in a field-driven system, operatingbelow and above the Walker breakdown field, respectively,with the latter mechanism being absent in the current-drivencase. Also the Walker breakdown itself is affected by thepresence of disorder, such that it is shifted to larger fieldand current values with increasing disorder strength. At thesame time, the DW velocities at the breakdown field andcurrent get smaller and larger, respectively. Furthermore, foradiabatic spin-transfer torque, the intrinsic pinning mecha-nism is found to be suppressed by disorder. These findingsemphasize the importance of understanding the interplaybetween disorder, the DW structure, and the properties ofthe external driving force, and are shown to be related to aneffective damping parameter α ∗increasing with the disorder strength. We perform a large ensemble of micromagnetic simulations with the graphics processing unit (GPU)-based micromagneticsimulator MuMax, 15making it possible to obtain large statis- tics for averaging over the disorder realizations. To study thetime evolution of the magnetization M(r,t) with an amplitude M s, we solve the Landau-Lifshitz (LL) equation with the 144415-1 1098-0121/2012/86(14)/144415(5) ©2012 American Physical SocietyBEN V AN DE WIELE, LASSE LAURSON, AND GIANFRANCO DURIN PHYSICAL REVIEW B 86, 144415 (2012) spin-transfer torque terms,16 ∂M ∂t=−γ 1+α2M×Heff−αγ Ms(1+α2)M×(M×Heff) −bj M2s(1+α2)M×[M×(j·∇)M] −bj Ms(1+α2)(ξ−α)M×(j·∇)M, (1) where Heffis the effective magnetic field (with contributions from the external, exchange, and demagnetization fields), γis the gyromagnetic ratio, αis the Gilbert damping constant, ξ is the degree of nonadiabaticity, jis the current density, and bj=PμB/[eMs(1+ξ2)], with Pthe polarization, μBthe Bohr magneton, and ethe electron charge. We consider permalloy strips of width w=100 nm and thickness 10 nm, such that the stable DW structure is a head-to-headV-shaped symmetric transverse wall, separating in-plane domains pointing along the strip axis. 17The used material parameters are those of permalloy, i.e., Ms=860×103A/m andα=0.02, and no anisotropy fields are included in Eq. (1). To clearly see the effect of quenched disorder on the DWdynamics, we set the temperature T=0. The system is discretized by considering Ncells of size 3 .125×3.125× 10 nm 3. Upon application of an external magnetic field Hext=Hextˆxalong the strip axis in the absence of disorder, the DW is displaced along the strip. If the field is below the Walkerbreakdown field H W, the DW essentially keeps its equilibrium structure during the propagation, with a small out-of-planecomponent close to the tip of the Vshape, and a velocity roughly linearly proportional to the applied field. Above H W, an antivortex is nucleated at the tip of the Vshape. It then propagates across the strip width, reversing the polarity of theDW magnetization. This process is repeated such that the DWpolarity oscillates back and forth, dramatically decreasing theaverage DW velocity. 18 With disorder included in the form of randomly positioned nonmagnetic voids of linear size 3.125 nm with varyingdensities σwithin a strip of length L=3.2μm, the DW can get pinned even for nonzero applied fields. 19This makes measurement and even definition of the DW velocity anontrivial task. Thus, in what follows we consider both the“conditional velocities” v mof the moving DWs, conditioned on the fact that the DWs will not get pinned during the timeinterval /Delta1t=20 ns we consider in the simulations (i.e., the DW will either reach the end of the strip or it is still movingafter/Delta1t=20 ns), 20and the probability Ppinfor the DW to get pinned during /Delta1t. These are computed by averaging over 50 disorder realizations for each Hextandσ. Notice that here we consider a T=0 system, such that a pinned DW cannot depin. An alternative measure of the DW velocity (which is likely tobe closer to typical experimental measurements where T> 0) is given by v exp=(1−Ppin)vm. In general, Ppinwill increase with the observation (time and length) scale, thus making alsov expa scale-dependent quantity. Figure 1shows the resulting average velocities vmof the moving DWs as a function of Hextandσ. The presence of voids induces a finite depinning field Hdep(σ) increasing with σ.F o r Hext>H dep(σ),vmfirst increases until a maximum velocity is reached at Hext=HW(σ), and then starts to decrease0100200300400500vm [m/s]σ = 0 σ = 3125 μm-2 σ = 6250 μm-2 σ = 9375 μm-2 σ = 12500 μm-2 0 5 10 15 Hext [mT]00.51Ppin0 2.5 5 7.5 Hext [mT]0200400vexp [m/s] FIG. 1. (Color online) The average velocity vmof the moving DWs (main figure) and vexp=(1−Ppin)vm(inset) as a function of Hextandσ. Error bars correspond to the standard deviation of vm.T h e pinning probabilities Ppinduring the 20 ns simulation (bottom panel) exhibit large values for large Hextdue to the core pinning mechanism. again. The position HW(σ) of this maximum, corresponding to the Walker breakdown, is shifted towards larger fieldvalues as σis increased, and the corresponding maximum velocity v m[HW(σ)] decreases with σ. The error bars in Fig. 1 correspond to the standard deviation of vm, and indicate that the dynamics of moving DWs has a stochastic nature dueto the random disorder. Notice in particular that the pinningprobability P pinexhibits a nonmonotonic dependence on Hext, with strong pinning for both small and large Hext, while for intermediate applied fields (corresponding to large values ofv m) pinning is less likely. The maximum value of vexp(inset of Fig. 1) exhibits a strong dependence on σ, and depends also on the observation scale via Ppin(not shown). For large Hext,Ppinis close to 1 for /Delta1t=20 ns, and consequently vexp is essentially zero. Similar pinning effects for large applied fields have been observed experimentally for vortex walls.13,14 To gain insight on the mechanisms behind this behavior, we consider snapshots of the DW configurations and the variouscontributions to ∂M/∂t in Eq. (1). For small H ext, we find that the overall DW structure is preserved, with the disorderinducing only minor distortions. If the DW gets pinned,this happens by a collective action of several voids. Thismechanism is known as collective pinning , and it is responsible for the nonzero depinning field H dep<H W(σ). Remarkably, we identify a fundamentally different pinning mechanism forlarge fields, H ext>H W(σ): In this regime, an antivortex is able to propagate to the interior of the strip, resulting inpinned DW configurations (occurring with probability P pin) with the antivortex core positioned exactly on top of a voidor a local void structure. We refer to this mechanism ascore pinning , and attribute it to the fact that the energy of the system can be significantly lower when the antivortexcore or part of it—involving large magnetization gradientsand out-of-plane magnetization—is placed in a nonmagneticregion (or more generally, in a region with low M s). In the field-driven case the DW is susceptible to get pinned by thismechanism because the Zeeman torque is relatively small in 144415-2EFFECT OF DISORDER ON TRANSVERSE DOMAIN W ALL ... PHYSICAL REVIEW B 86, 144415 (2012) FIG. 2. (Color online) Examples of the spatial distribution of the contributions of the applied field Hext=5 mT (top) and current density jext=20×1012A/m2withξ=0 (middle) to ∂M/∂t in Eq.(1), corresponding to the magnetization configuration shown in the bottom panel, exhibiting an antivortex in the middle of the strip. ∂M/∂tis given in units of Ms/s. The randomly positioned voids with σ=3125μm−2are shown as gray dots. magnitude and does not directly displace the DW (top panel of Fig. 2); instead, the small out-of-plane magnetization due to the Zeeman torque induces demagnetizing fields, which act tomove the DW. Such an indirect driving mechanism is sensitiveto the perturbations due to disorder, leading to several effects,including σ-dependent H depandHW, and in particular the core pinning mechanism for high Hext. We proceed to contrast these results with the current-driven case by applying a current density j=−jextˆxwithP=0.5 along strips of length L=6.4μm. We first consider perfect adiabaticity ( ξ=0, top panel of Fig. 3) .D u et oi n t r i n s i c pinning,21there is a nonzero depinning current jdep,intin the absence of disorder, above which DW motion involvesrepeated polarity transformations mediated by antivortexpropagation across the strip width. Adding disorder with thesame procedure as above reveals two intriguing observations:First, it appears that the DW is able to move even for currentsslightly below j dep,int. This surprising finding can be explained by noticing that the intrinsic pinning mechanism is due to theability of the DW to deform in such a way that the torques dueto interactions within the DW (i.e., the effective field) exactlycounterbalance the adiabatic spin-transfer torque. 21However, the presence of disorder induces additional DW deformationsand imposes constraints on the ability of the DW to counteractthe current-induced torques, leading to nonzero values for both050010001500vm [m/s] σ = 0 σ = 3125 μm-2 σ = 6250 μm-2 σ = 9375 μm-2 σ = 12500 μm-2 0 5 10 15 20 25 30 jext [1012 A/m2]00.51Ppin0 5000 10000 σ [μm-2]0.020.030.04α*ξ = 0.02 ξ = 0.03 ξ = 0.040200400600800vm [m/s] σ = 0 σ = 3125 μm-2 σ = 6250 μm-2 σ = 9375 μm-2 σ = 12500 μm-2 0 5 10 15 20 25 30 jext [1012 A/m2]00.51Ppin10 15 20 jext [1012 A/m2]0100200300vexp [m/s] FIG. 3. (Color online) The average velocity vmof the moving D W sa saf u n c t i o no f jextandσ,f o rξ=0 (top) and ξ=0.04 (bottom). Error bars correspond to the standard deviation of vm.T h e pinning probabilities Ppinduring the 20 ns simulation highlight the absence of core pinning for large current densities. The insets show vexp=(1−Ppin)vmforξ=0 (top panel), and the effective α∗(σ)f o r various ξ(bottom panel), respectively. vmand 1−Ppinforjextsomewhat below jdep,int. Notice that whilevexp(inset of the top panel in Fig. 3) exhibits nonlinear field dependence reminiscent of typical creep motion for smallfields, we are considering here a T=0 system in which a pinned DW cannot depin due to the absence of thermalfluctuations. 22 The second observation is that for larger jext, core pinning is absent. Even if for jext>jW(σ) the antivortex core is constantly moving back and forth across the strip width, itnever gets pinned by the voids, strongly contrasting with thefield-driven case. To explain this observation, we considerthe spatial distribution of the current-induced contribution to∂M/∂t (middle panel of Fig. 2), and find that the current acts directly (in contrast to the indirect mechanism in thefield-driven case) and strongly on the antivortex core where themagnetization gradients are large, facilitating its propagationalong the strip across the energy barriers due to the voids. Thisis also directly visible in the the LL equation [Eq. (1)], where the current acts on the gradient of Mrather than on Mitself. 144415-3BEN V AN DE WIELE, LASSE LAURSON, AND GIANFRANCO DURIN PHYSICAL REVIEW B 86, 144415 (2012) TABLE I. Predictions for jdep,int,HW,a n dvm(HW) from the one-dimensional model in terms of σ-dependent effective α∗andC∗≡ (/Delta1M2 s|Ny−Nx|)∗, compared with the simulated values. C∗is estimated by fitting the expression for jW(see text) to the data in the bottom panel of Fig. 3. σ(μm−2) α∗C∗(A2/m) jpred dep,int(A/m2)jsim dep,int(A/m2)Hpred W(mT) Hsim W(mT) vpred m(HW)( m / s ) vsim m(HW)( m / s ) 0 0.0200 2 .92×10−1014×101215×10122.75 2.75 457 457 1562.5 0.0221 2 .52×10−1012.1×101213×10123.05 3.0 398 437 3125 0.0238 2 .45×10−1011.7×101212×10123.25 3.25 389 419 4687.5 0.0258 2 .36×10−1011.3×101211.5×10123.52 3.25 377 403 6250 0.0283 2 .28×10−1010.9×101211×10123.78 3.5 368 387 Finally we consider the role of the nonadiabatic spin- transfer torque (bottom panel of Fig. 3, where the ξ=0.04 case is shown) on the DW dynamics. For ξ> 0 and σ=0, there is no intrinsic pinning, and the DW propagates, preserv-ing its internal structure with a finite velocity linearly propor-tional to the current density j extup to a Walker breakdown current jW.F o rjext>jW, an antivortex is again nucleated and propagates across the strip width, reversing the polarityof the DW magnetization, and decreasing the average DWvelocity. For larger j ext, the velocity again increases with jext. Adding disorder induces a finite depinning threshold jdep(σ), and pushes the local maximum of vmor the Walker breakdown to higher jext.A tt h es a m et i m e , vmatjW(σ) increases with σ. Thus, the voids are able to inhibit the antivortex entering thestrip, enhancing the DW propagation and structural stability forintermediate current densities, j W(σ=0)<j ext<jW(σ> 0). This effect arises as the antivortex core is pushed across thestrip width by the effective field terms in Eq. (1)(notice that the effect of the current is symmetric such that no antivortexdisplacement along the ydirection arises directly due to the current—see the middle panel of Fig. 2), a mechanism sensitive to the disturbances due to disorder. Again, there isno core pinning for j ext>jW(σ), for the same reason as in the adiabatic ( ξ=0) case. Forjdep(σ)<j ext<jW(σ),vmdepends linearly on jext, and by extrapolating linear fits to the data to jext=0 all the lines cross at vm=0 (not shown). Thus, we estimate effective values of the damping parameter from the slopes of these linearfits, 8as within one-dimensional models23vm∝(β/α)jextfor jext<jW, with β=ξ/(1+ξ2). Our simulations (inset of the lower panel of Fig. 3) with different ξindicate that the data can be interpreted in terms of an effective α∗increasing with σ.8 Also an effective M∗ s=(1−σLw/N )Msemerges naturally. Thus we can explain our results with the one-dimensionalmodel in terms of σ-dependent effective parameters: For in- stance, j W(σ)=4πγ(M2 s/Delta1|Ny−Nx|)∗α∗/(gμBP|β−α∗|), with/Delta1the DW width and NxandNythe demagnetizing factors, and jdep,int(σ)≡jW(σ,ξ=0).23Using the expression forjWand the values of α∗to estimate C∗≡(/Delta1M2 s|Ny− Nx|)∗, the scaling of jdep,intwithσcan be reproduced remarkably well—see Table I. A similar analysis in thefield-driven case, with HW=2πα∗(Ms|Ny−Nx|)∗and vm(HW)=(γ/Delta1∗/α∗)HW,23reproduces the observed scaling of both HWandvm(HW) with σ(Table I). Notice that in our casevm(HW) depends on σthrough the σ-dependent effective parameters, while for systems with only edge roughnessv m(HW) is independent of the amount of edge roughness.6 To summarize, we have presented a detailed analysis of the effect of disorder on the field- and current-driven transverseDW dynamics in a narrow and thin permalloy nanostrip. Wehave identified two fundamentally different pinning mecha-nisms, acting in different regimes of the DW propagation. Theobservation that there is no core pinning in the current-drivencase whereas it dominates the field-driven dynamics for largefields highlights the different nature of the field and currentdrive in a way that can be observed only in the presence ofdisorder. In general, we have seen that the pinning mechanismsoperating will depend on the details of the DW structure,and thus we expect that the core pinning mechanism isabsent for systems with high perpendicular magnetocrystallineanisotropy as there is no (anti)vortex core that could getpinned, but it could play a role in the dynamics of vortex wallsoccurring in wider soft strips, 8possibly also for small applied fields. If only edge roughness is present, no core pinning shouldoccur. Experiments should be performed to systematicallystudy the scale dependence of P pinandvexp. Finally, we point out that the observation that disorder tends to stabilize the DWinternal structure and increase the maximum DW velocity bysuppressing the Walker breakdown in the current-driven casesuggests that it could be desirable to deliberately engineerdisorder in the system, for instance, to replace notches to pinthe DW in various technological applications. 24 Stefano Zapperi is thanked for numerous interesting dis- cussions on DW dynamics and disorder, and Mikko J. Alavafor useful comments on the manuscript. We thank Luc Dupr ´e and Dani ¨el De Zutter for supporting this research. L.L. has been supported by the Academy of Finland througha Postdoctoral Researcher’s Project (Project No. 139132)and through the Centres of Excellence Program (ProjectNo. 251748). B.V .d.W. has been supported by the FlandersResearch Foundation FWO. 1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 2M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin,Science 320, 290 (2008).3D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, a n dR .P .C o w b u r n , Science 309, 1688 (2005). 4V . Lecomte, S. E. Barnes, J.-P. Eckmann, and T. Giamarchi, Phys. Rev. B 80, 054413 (2009). 144415-4EFFECT OF DISORDER ON TRANSVERSE DOMAIN W ALL ... PHYSICAL REVIEW B 86, 144415 (2012) 5G. Durin and S. Zapperi, in The Science of Hysteresis ,e d i t e db y G. Bertotti and I. Mayergoyz, V ol. II (Academic, Amsterdam, 2006),pp. 181–267. 6Y . Nakatani, A. Thiaville, and J. Miltat, Nat. Mater. 2, 521 (2003). 7E. Martinez, J. Phys.: Condens. Matter 24, 024206 (2012). 8H. Min, R. D. McMichael, M. J. Donahue, J. Miltat, and M. D. Stiles, P h y s .R e v .L e t t . 104, 217201 (2010). 9L. Laurson, A. Mughal, G. Durin, and S. Zapperi, IEEE Trans. Magn. 46, 262 (2010). 10L. Laurson, C. Serpico, G. Durin, and S. Zapperi, J. Appl. Phys. 109, 07D345 (2011). 11G. Meier, M. Bolte, R. Eiselt, B. Kr ¨uger, D.-H. Kim, and P. Fischer, P h y s .R e v .L e t t . 98, 187202 (2007). 12M. Kl ¨a u i ,P . - O .J u b e r t ,R .A l l e n s p a c h ,A .B i s c h o f ,J .A .C .B l a n d , G. Faini, U. R ¨udiger, C. A. F. Vaz, L. Vila, and C. V ouille, Phys. Rev. Lett. 95, 026601 (2005). 13H. Tanigawa, T. Koyama, M. Bartkowiak, S. Kasai, K. Kobayashi, T. Ono, and Y . Nakatani, P h y s .R e v .L e t t . 101, 207203 (2008). 14X. Jiang, L. Thomas, R. Moriya, M. Hayashi, B. Bergman, C. Rettner, and S. S. P. Parkin, Nat. Commun. 1, 25 (2010). 15A. Vansteenkiste and B. Van de Wiele, J. Magn. Magn. Mater. 323, 2585 (2011). 16S. Zhang and Z. Li, P h y s .R e v .L e t t . 93, 127204 (2004).17Y . Nakatani, A. Thiaville, and J. Miltat, J. Magn. Magn. Mater. 290-291 , 750 (2005). 18A. Thiaville and Y . Nakatani, in Spin Dynamics in Confined Magnetic Structures III , edited by B. Hillebrands and A. Thiaville, Topics in Applied Physics, V ol. 101 (Springer, Berlin, 2006),p. 161. 19We have checked that our main conclusions remain the same ifinstead of voids one considers small areas with half the saturationmagnetization M s, suggesting that our results are not limited to the specific kind of disorder we study here. 20To avoid any effect related to the initial acceleration and of thedemagnetizing fields at the end of the wire, we actually calculatedthe average speed between a point at 0.5 μm after the initial position and at 0.5 μm before the end of the wire. 21Z. Li and S. Zhang, Phys. Rev. B 70, 024417 (2004). 22In fact, studying the true thermally activated creep motion by micromagnetic simulations is very challenging, due to the longtime scales needed to observe several repeated pinning-depinningevents, a requirement for reliable estimation of the DW velocitiesin the creep regime. 23A. Mougin, M. Cormier, J. P. Adam, P. J. Metaxas, and J. Ferre,Europhys. Lett. 78, 57007 (2007). 24M. A. Basith, S. McVitie, D. McGrouther, and J. N. Chapman, Appl. Phys. Lett. 100, 232402 (2012). 144415-5

PhysRevLett.102.127202.pdf

Physical Origin and Generic Control of Magnonic Band Gaps of Dipole-Exchange Spin Waves in Width-Modulated Nanostrip Waveguides Ki-Suk Lee, Dong-Soo Han, and Sang-Koog Kim * Research Center for Spin Dynamics & Spin Wave Devices and Nanospinics Laboratory, Department of Materials Science and Engineering, Seoul National University, Seoul 151-744, Republic of Korea (Received 7 November 2008; revised manuscript received 31 January 2009; published 25 March 2009) We report, for the first time, on a novel planar structure of magnonic-crystal waveguides, made of a single magnetic material, in which the allowed and forbidden bands of propagating dipole-exchange spin waves can be manipulated by the periodic modulation of different widths in thin-film nanostrips. The origin of the presence of several magnonic wide band gaps and the crucial parameters for controlling thoseband gaps of the order of /C2410 GHz are found by micromagnetic numerical and analytical calculations. This work can offer a route to the potential application to broadband spin wave filters in the gigahertzfrequency range. DOI: 10.1103/PhysRevLett.102.127202 PACS numbers: 75.40.Gb, 75.30.Ds, 75.40.Mg The engineering of electronic band gaps in a periodic atomic structure has played a crucial role in the develop- ment of currently advanced semiconductor technologies. Reliable manipulations of the propagations of electrons in atomic-scale periodic structures as well as electromagnetic waves (photons) in submicron- or larger-scale structures are one of the long-standing fundamental issues in the field of condensed matter physics. Controlling the propagation of photons in a variety of artificially fabricated periodic structures known as photonic crystals is a good example [1]. Owing to various applications of the photonic crystal to optical nanodevices such as photonic waveguides [ 2] and integrated circuits [ 3], the photonic crystals have been given considerable attention. Meanwhile, in the areas of nanomagnetism and magnetization ( M) dynamics, the magnetic counterpart of the photonic crystals, the so-called magnonic crystal (MC), is a subject of growing interest, owing to its applications to spin wave (SW) waveguides and filters [ 4]. In recent years, many theoretical and ex- perimental studies have been conducted on not only vari- ous types of MCs including one-dimensional (1D) structures, such as periodic multilayers [ 5,6], periodic arrays of nanostrips [ 7], corrugated films [ 8,9], and comb- like [ 10] or serial loop structures [ 11], but also 2D or 3D structures [ 12–14]. In such structures, the allowed and forbidden SW modes (called magnons) are controllable by periodic structures artificially fabricated with different magnetic material parameters [ 5,6,15], shapes [ 8–11], and exchange-bias fields [ 16]. Despite recent advances in fundamental understandings of those MCs as well as the wave properties of excited SW modes, few studies have focused on MC waveguides composed of simple structures for its practical applica- tion to broadband SW filters [ 9]. For future SW-based signal processing devices [ 17,18], it is necessary to find micrometer-size (or smaller) MC waveguides having sim-ple planar structures, with controllable wide band gaps (of a few gigahertz) of dipole-exchange spin waves (DESWs). In this Letter, we report, for the first time, on a new type of simple planar-patterned thin-film nanostrip waveguidesin which the DESWs’ magnonic bands along with theirwide band gaps on the order of 10 GHz can be manipulatedby periodic modulations of different widths (of a few tens of nanometers). The physical origin of the presence of magnonic wide band gaps in such width-modulated nano-strips and the relations of the allowed DESW modes andband gaps to geometric variation of the proposed MCswere found by micromagnetic numerical and analyticalcalculations. We performed micromagnetic simulations on DESW propagations in magnetic thin-film nanostrips. We used, as a model system, 10 nm-thick Permalloy (Py) nanostrips of different widths (24 and 30 nm here, for example)modulating with a periodicity Pranging from 12 to 42 nm [the light-gray area in Fig. 1(a)], which were con- FIG. 1 (color online). (a) Geometry and dimensions of pro- posed nanostrip MCs with periodic modulation of different strip widths. The initial M’s point in the /C0xdirection, as indicated by the black arrow. The dark-brown and yellow areas indicate the SW generation and waveguide component, respectively. (b) The unit period of P¼P1þP2, where P1andP2are the segment lengths of 24 and 30 nm widths, respectively. (c) Temporal evolution of spatial Mz=Msdistribution excited by a sinc func- tion field with H0¼1:0Tapplied along the yaxis to only the dark-brown area.PRL 102, 127202 (2009) PHYSICAL REVIEW LETTERSweek ending 27 MARCH 2009 0031-9007 =09=102(12) =127202(4) 127202-1 /C2112009 The American Physical Societynected directly to a segment of the Py nanostrip of 10 nm thickness and 30 nm width [the yellow area in Fig. 1(a)]. The unit period of the nanostrips consists of the samePy segments with the different widths of 24 and 30 nm and with the corresponding lengths P 1andP2, respec- tively, as illustrated in Fig. 1(b). The OOMMF code (ver- sion 1.2a4) [ 19] was used to numerically calculate the dynamics of the M’s of individual unit cells (size: 1:5/C2 1:5/C210 nm3[20]) interacting through exchange and di- polar forces, which code uses the Landau-Lifshitz-Gilbertequation of motion [ 21]. The chosen material parameters corresponding to Py are as follows: the saturation magne-tization M s¼8:6/C2105A=m, the exchange stiffness Aex¼1:3/C210/C011J=m, the damping constant /C11¼0:01, the gyromagnetic ratio /C13¼2:21/C2105m=As, and with zero magnetocrystalline anisotropy. For the local homoge-neous excitation and subsequent propagation of the lowest-mode DESWs, along the length direction, with frequenciesf SW, ranging from 0 to 100 GHz, we applied a ‘‘sine cardinal (sinc)’’ function HyðtÞ¼H0sin½2/C25/C23Hðt/C0t0Þ/C138 2/C25/C23Hðt/C0t0Þ, with H0¼1:0Tand the field frequency /C23H¼100 GHz , only to a local area of 1:5/C230 nm2indicated by the dark- brown color shown in Fig. 1(a). The results obtained by the fast Fourier transform (FFT) of the temporal Mz=Msevolution for DESW propagations along the xaxis at y¼15 nm are plotted in Fig. 2[18]. The frequency spectra clearly reveal the allowed and/orforbidden bands of DESWs propagating through the nano-strips: The allowed bands are indicated by the coloredregion, and the forbidden bands by the white region. Forcomparison, the fundamental mode DESWs propagating insingle-width (24 and 30 nm) nanostrips are shown inFigs. 2(a)and2(b). Obviously, there is no forbidden band except for below the intrinsic potential barrier ( <14 GHz ), owing to the quantization of the lowest mode of DESWs due to the geometric confinement of the nanostrip’s narrowwidth [ 18,22–25]. For the different-width-modulated nano- strips, by contrast, there are several wide forbidden bandsof the order of /C2410 GHz [see Figs. 2(c)and2(d)]. More- over, the number of forbidden bands as well as the bands’position and gap width differ according to not only Pbutalso the motif (represented by P 1=P). More specifically, for P¼18 nm with½P1;P2/C138¼½ 9n m;9n m/C138, two wide band gaps (11 and 16 GHz) appear in the DESW modes ranging from 14 to 100 GHz, whereas, for P¼30 nm with ½P1;P2/C138¼½ 15 nm ;15 nm /C138, five forbidden bands with smaller gap widths ( 3:8–8:6 GHz ) exist (for more data, see supplementary Fig. 1 [ 26]). To comprehensively understand such striking band-gap features, we plotted the dispersion curves of the DESWmodes in the 24 and 30 nm-wide nanostrips and in thenanostrip of ½P 1;P2/C138¼½ 9n m;9n m/C138[27] as an example. Because of the pinning of DESWs at the longer (length direction) edges of the nanostrips, there exist certain widthmodes having quantized k yvalues [ 23]. Generally, in single-width nanostrips, it is expected that the several width modes are excited [ 22–25], and, thus, several con- cave branches appear in the dispersion curves [ 18]. In the present simulation, however, there was a single parabolicdispersion curve, as shown in Fig. 3(a), because homoge- neous DESW excitations along the width direction em-ployed in this study led to only the lowest mode hav- ing the smallest k yvalue [see Fig. 1(c)]. Accordingly, one would expect the dispersion curves for the width- modulated nanostrips to be folded and have band gaps at the Brillouin zone (BZ) boundaries, similar to those found typically in a 1D periodic system [ 2,28]. However, the dispersion curves for ½P1;P2/C138¼½ 9n m;9n m/C138show rather more complicated band features [Fig. 3(b)]: The band gaps occur not only at the BZ boundaries, kx¼n/C25=P with integers n(black dashed lines), but also at certain kvalues, kx¼½ ð2nþ1Þ/C25/C61:44/C138=P(red dotted lines). The former can be explained by a periodic translation symmetry asso- ciated with the different width modulation along the DESW propagation direction, but the latter cannot beunderstood by such a 1D approach. FIG. 2 (color online). Frequency spectra obtained from FFTs ofMz=Msoscillation along the xaxis at y¼15 nm , for single- width nanostrips (24 and 30 nm) and for MCs of different [ P1 andP2] values noted. The vertical dashed orange lines indicate the boundary between the single-width nanostrip waveguide [the yellow area in Fig. 1(a)] and the MC of the width-modulated nanostrip [the light-gray area in Fig. 1(a)]. FIG. 3 (color online). (a) Dispersion curves for DESWs prop- agating through single-width nanostrips of 24 and 30 nm. (b) Dispersion curves of DESWs existing in the MC of ½P1;P2/C138¼½ 9n m;9n m/C138within the nanostrip area only, from x¼501to 1500 nm, obtained from FFTs of temporal Mz=Ms oscillations along the xaxis at y¼15 nm . The black dashed lines indicate the Brillouin zone boundaries k¼n/C25=P , where n¼0;/C61;/C62;..., and the red dotted lines denote certain k values, kx¼½ ð2nþ1Þ/C25/C61:44/C138=Pat which the forbidden band gaps occur.PRL 102, 127202 (2009) PHYSICAL REVIEW LETTERSweek ending 27 MARCH 2009 127202-2In order to quantitatively elucidate the physical origin of such different band gaps varying with different widthmodulation, we compared magnonic band diagrams [the thick black lines in Fig. 4(a)] obtained numerically from micromagnetic simulations for the case of ½P 1;P2/C138¼ ½9n m;9n m/C138, combined with the analytical calculation of the band structure of a single-width (27 nm) nanostrip: Note that this width is just the average of the 24 and 30 nm widths employed in the width-modulated nanostrips. Forthe 27 nm-wide nanostrip, the dispersion relation wasanalytically derived and expressed in terms of a quantized in-plane wave vector /C20 2m¼k2xþk2y;m, with integers m¼ 1;2;3, etc. [ 18,23]. The kxandkycorrespond to the longi- tudinal and transverse components of /C20m, respectively. The ky;mvalue can be obtained by considering the ‘‘effective’’ pinning [ 23], for example, ky;m¼m/C20:072 nm/C01for the 27 nm-wide nanostrip [ 29]. In Fig. 4(a), the solid red line indicates the dispersion curves of the DESW mode with m¼1, the lowest mode excited. Owing to the periodicity of the width modulation, the dispersion curves are folded atthe first BZ boundary (the dashed vertical line), as shown inFig.4(a), and thus these folded branches intersect with the original one at the BZ boundaries. Such a crossing of the dispersion curves indicates the ‘‘diagonal’’ coupling be-tween the two identical modes having opposite propaga-tion vectors [ 30]. This diagonal coupling represents interference between the initially propagating forward mode and its backward mode reflected at the BZ boundary,resulting in the standing wave pattern of DESWs with k x¼ n/C25=P in the MC of P, and a split in the energy band (a band gap) [see the thick black lines in Fig. 4(a)][28]. Next, Fig.4(b) shows calculations of the spatial distributions ofthe FFT powers of the local Mz=Msoscillations for the indicated specific frequencies selected at the top and bot-tom of the magnonic band for the case of ½P 1;P2/C138¼ ½9n m;9n m/C138. It is evident that the origin of the first band gap [ 31] is the diagonal coupling between the two identical but oppositely propagating lowest-mode ( m¼1) DESWs, as explained above. In addition to the first band gap at the BZ boundaries associated with the diagonal coupling, the dispersionbranch of a higher-quantized width mode ( m¼3), noted by the dotted orange lines in Fig. 4(a), intersects with that of the lowest mode ( m¼1)a t k x¼½0:5/C60:2/C1382/C25=P (away from the BZ boundaries) indicated by the bluecircles and the arrows in Fig. 4(a). To understand these band gaps, we consider a 2D scattering of the lowest- mode ( m¼1) DESWs from the edge steps between the narrower- and wider-width strip segments [see Fig. 1(b)]. Such edge steps periodically arranged in the width-modulated nanostrips play a crucial role as new sourcesfor excitations of the higher width modes ( m¼3;5;7;...) [32]. In general, those DESWs scattered from the edge steps propagate in wide angles on the x-yplane, so that they interfere destructively with themselves. However,for the phase-matching condition of the scattered DESWs, they can interfere constructively with themselves; in other words, other higher-width-mode ( m¼3) DESWs being propagated in the opposite direction are excitedand interact with the initial propagating lowest mode.Consequently, the interactions between the initial lowest-mode ( m¼1) and the excited higher width-mode ( m¼3) DESWs lead to quite complicated 2D standing wavepatterns of DESWs. For f SW¼66:8 GHz , the nodes ap- pear in both the width and length direction [see Figs. 4(b) and4(c)], subsequently leading to complex 2D normal modes in thin-film nanostrips of the lateral confinements [33]. This gives rise to the anticrossing of dispersion curves as well as band gaps [ 30] [see the thick black lines for the second and the third bands and the diagonal-line-patternedblue region between them in Fig. 4(a)]. Such strong coupling between the initially propagating mode and the newly excited higher mode and the resultingband gaps are known as the mini stop bands of electro-magnetic waves in photonic crystal waveguides [ 2,30,34]. It is worthwhile to note that the lateral geometric confine- ments in the width-modulated nanostrips can also yieldsignificant internal field inhomogeneities, as reported for aDaemon-Eschbach geometry in Refs. [ 24,25]. Simulation results for the dynamic Mprofiles of the normal modes between the uniform and nonuniform internal field (seesupplementary Fig. 2 [ 26]) reveal that the dynamic M profiles in the width-modulated nanostrips could not beexplained simply in terms of the inhomogeneity of the internal field. The complex modes of DESWs and associ- ated band gaps in the width-modulated MCs are the resultof the cooperative phenomena of the diffraction, reflectionof the DESWs scattered at the edge steps of the width-modulated nanostrips, and their interference with the ini- FIG. 4 (color online). (a) Comparison of magnonic band struc- ture (black thick curves) of a nanostrip-type MC of ½P1;P2/C138¼ ½9n m;9n m/C138obtained from micromagnetic simulations and that of a single-width nanostrip of 27 nm width, for two differentmodes of m¼1(solid red curves) and m¼3(dotted orange curves) obtained from the analytical form of Eq. (1) in Ref. [ 18]. (b) Perspective view of the FFT power distributions of local M z=Msoscillations for specific frequencies for the top and bottom of allowed bands, as indicated by orange horizontal lines in the dispersion curves shown in Fig. 3(b). (c) Cross-sectional FFT power profiles of standing wave modes in the width direc-tion ( ydirection). The red (green) line indicates the standing wave profile in the width direction at the center of the 30 nm- (24 nm-) wide segment.PRL 102, 127202 (2009) PHYSICAL REVIEW LETTERSweek ending 27 MARCH 2009 127202-3tially propagating lowest-mode DESWs [ 26]. On the basis of such novel DESW band structures, it was found thatmagnonic band gaps vary sensitively according to both the periodicity and the motif in width-modulated nanostrips. From an application perspective, this novel property can beimplemented as an effective means of manipulating theallowed DESW modes in their propagations through suchwidth-modulated nanostrips, a new type of SW waveguidesthat pass DESWs in a chosen narrow-band frequencyregion but filter out most DESWs having other frequencies.Moreover, these results can resolve the bottleneck of spin wave devices–the trade-off among their speed, miniatur- ization, and controllability by applied magnetic fields [ 35]. In conclusion, we found that complex DESW band structures and wide band gaps originate from the diagonalcoupling between the identical lowest modes, as well as thecoupling between the initially propagating lowest modeand the higher-quantized width mode newly excitedthrough the DESWs scattering at the edge steps of different-width-modulated nanostrips. Moreover, we found that the magnonic band-gap width, the position, and thenumber of band gaps are controllable by the periodicityand the motif of the different width modulation. We express our thanks to B. Hillebrands and A. Slavin for their careful reading of this manuscript. This work wassupported by Creative Research Initiatives (the ResearchCenter for Spin Dynamics and Spin Wave Devices) of MEST/KOSEF. *Corresponding author. sangkoog@snu.ac.kr [1] E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987). [2] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, Princeton, NJ, 2008), 2nd ed. [3] N. Engheta, Science 317, 1698 (2007). [4] R. L. Carter et al. , J. Appl. Phys. 53, 2655 (1982). [5] D. S. Deng, X. F. Jin, and R. Tao, Phys. Rev. B 66, 104435 (2002). [6] S. A. Nikitov, Ph. Tailhades, and C. S. Tsai, J. Magn. Magn. Mater. 236, 320 (2001). [7] M. P. Kostylev, A. A. Stashkevich, and N. A. Sergeeva, Phys. Rev. B 69, 064408 (2004); M. Kostylev et al. , Appl. Phys. Lett. 92132504 (2008). [8] C. G. Sykes, J. D. Adam, and J. H. Collins, Appl. Phys. Lett. 29, 388 (1976); P. A. Kolodin and B. Hillebrands, J. Magn. Magn. Mater. 161, 199 (1996). [9] A. V. Chumak et al. , Appl. Phys. Lett. 93, 022508 (2008). [10] H. Al-Wahsh et al. , Phys. Rev. B 59, 8709 (1999). [11] A. Mir et al. , Phys. Rev. B 64, 224403 (2001). [12] J. O. Vasseur et al. , Phys. Rev. B 54, 1043 (1996). [13] Yu. V. Gulyaev et al. , JETP Lett. 77, 567 (2003). [14] M. Krawczyk and H. Puszkarski, Phys. Rev. B 77, 054437 (2008).[15] V. V. Kruglyak, and R. J. Hickena, J. Magn. Magn. Mater. 306, 191 (2006); V. V. Kruglyak et al. , J. Appl. Phys. 98, 014304 (2005). [16] C. Bayer, M. P. Kostylev, and B. Hillebrands, Appl. Phys. Lett. 88, 112504 (2006). [17] R. Hertel, W. Wulfhekel, and J. Kirschner, Phys. Rev. Lett. 93, 257202 (2004); T. Schneider et al. , Appl. Phys. Lett. 92, 022505 (2008); K.-S. Lee and S.-K. Kim, J. Appl. Phys. 104, 053909 (2008). [18] S. Choi, K.-S. Lee, K. Y. Guslienko, and S.-K. Kim, Phys. Rev. Lett. 98, 087205 (2007). [19] A version of the OOMMF code used is 1.2a4. See http:// math.nist.gov/oommf. [20] Simulation results using 1:5/C21:5/C22:5n m3are in good agreement with those using 1:5/C21:5/C210 nm3. [21] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935); T. L. Gilbert, Phys. Rev. 100, 1243 (1955). [22] T. W. O’Keeffe and R. W. Patterson, J. Appl. Phys. 49, 4886 (1978); R. Arias and D. L. Mills, Phys. Rev. B 70, 094414 (2004); V. E. Demidov and S. O. Demokritov, Phys. Rev. B 77, 064406 (2008). [23] K. Yu. Guslienko et al. , Phys. Rev. B 66, 132402 (2002); K. Yu. Guslienko and A. N. Slavin, Phys. Rev. B 72, 014463 (2005). [24] M. P. Kostylev et al. , Phys. Rev. B 76, 054422 (2007). [25] V. E. Demidov et al. , Appl. Phys. Lett. 92, 232503 (2008). [26] See EPAPS Document No. E-PRLTAO-102-023915 for two figure files and five movie files. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html. [27] Dispersion curves were obtained from the 2D FFTs of the temporal Mz=Msoscillations along the xaxis ( x¼ 501–1500 nm )a t y¼15 nm (the dashed line on the nanostrip MC in Fig. 1). [28] C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996), 7th ed. [29] In the calculation of the ky;mvalues for the 27 nm-width nanostrip having a homogeneous internal field distribution across the strip width, the effective pinning parameterexpressed by Eq. (5) in Ref. [ 23] was used. The exact values of k y;mcan be obtained using Eq. (6) in Ref. [ 24]. However, the approximate analytical calculation describedin Ref. [ 23] is in good agreement with the result obtained with Eq. (6) in Ref. [ 24] for the single-width nanostrip of longitudinally saturated magnetizations. [30] J.-M. Lourtioz et al. ,Photonic Crystals: Towards Nano- scale Photonic Devices (Springer, Berlin, 2005). [31] The top of the first allowed band ( f SW¼25:6 GHz ) and the bottom of the second allowed band ( fSW¼36:4 GHz ) appear at the first BZ boundary kx¼/C25=P. [32] Since the edge steps are symmetric with respect to the xaxis (mirror plane), higher-quantized width modes have a mirror symmetry in the profiles along the width axis (yaxis); in other words, they have odd-number mvalues (i.e.,m¼3;5;7;...). [33] M. Grimsditch et al. , Phys. Rev. B 69, 174428 (2004); M. Grimsditch et al. , Phys. Rev. B 70, 054409 (2004). [34] S. Olivier et al. , Phys. Rev. B 63, 113311 (2001). [35] S. V. 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PhysRevB.100.094417.pdf

PHYSICAL REVIEW B 100, 094417 (2019) Creep of chiral domain walls Dion M. F. Hartmann ,1,*Rembert A. Duine,1,2Mariëlle J. Meijer,2Henk J. M. Swagten,2and Reinoud Lavrijsen2 1Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, NL-3584 CE Utrecht, The Netherlands 2Department of Applied Physics, Eindhoven University of Technology, P .O. Box 513, 5600 MB Eindhoven, The Netherlands (Received 7 January 2019; published 11 September 2019) Recent experimental studies of magnetic domain expansion under easy-axis drive fields in materials with a perpendicular magnetic anisotropy have shown that the domain wall velocity is asymmetric as a functionof an external in-plane magnetic field. This is understood as a consequence of the inversion asymmetry ofthe system, yielding a finite chiral Dzyaloshinskii-Moriya interaction. Numerous attempts have been made toexplain these observations using creep theory, but, in doing so, these have not included all contributions tothe domain wall energy or have introduced additional free parameters. In this article we present a theory forcreep motion of chiral domain walls in the creep regime that includes the most important contributions to thedomain-wall energy and does not introduce new free parameters beyond the usual parameters that are includedin the micromagnetic energy. Furthermore, we present experimental measurements of domain wall velocities asa function of in-plane field that are well described by our model, and from which material properties such as thestrength of the Dzyaloshinskii-Moriya interaction and the demagnetization field are extracted. DOI: 10.1103/PhysRevB.100.094417 I. INTRODUCTION The interest in nanomagnetic materials has grown steadily since magnetic storage devices, such as the racetrack mem-ory, were proposed as a new tool to meet the ever increas-ing demand for computer storage capacity [ 1–4]. For such applications the domain wall (DW) chirality is an impor-tant parameter as it affects the speed and direction of DWmotion. The interfacial Dzyaloshinskii-Moriya-interaction(DMI) [ 5,6] arises from perpendicular inversion asymmetry in the system and affects the DW chirality. Hence it is ofparamount importance to be able to measure the magnitude ofthe DMI using a simple experimental method. The interfacialDMI is modeled as an effective field that lies in-plane (IP)and is always perpendicular to the domain wall (DW) nor-mal, hence preferring a Néel wall [ 7]. Superpositioning the DMI field with an externally applied IP magnetic field couldprovide means of measuring it. This has lead to a boom ofexperimental studies on DW dynamics under the influence ofan IP magnetic field [ 8–17]. There are several regimes of DW dynamics, determined by the strength of the DW driving force compared to thepinning force. In the flow regime the driving force is signif-icantly higher than the pinning force and in this regime IPmagnetic fields and DMI is successfully modeled by means ofthe Landau-Lifschitz-Gilbert equation [ 18–20]. In the creep regime however, the DW is considered to be mostly pinnedand in local equilibrium and has a net displacement becausethe bias is assisted by thermal fluctuations. The creep model was successfully implemented to interpret magnetic domain growth driven by an external magneticfield H zin the direction of the magnetization of one of the *d.m.f.hartmann@uu.nldomains, resulting in the famous universal creep law for the DW velocity v:l n (v)∝H−1/4 z [21]. When introducing a magnetic field perpendicular to the magnetization direction ofthe domains, a modification to this creep law was proposed:ln(v)∝(E el/Hz)1/4, where Eelis the elasticity of the DW [ 8]. This modification turned out to describe experimental findingswell for small IP magnetic fields, but is not able to describe thehigh-field region [ 10]. Recent attempts to improve the theoret- ical model exposed the dispersive nature of the elasticity butcompromised on universality as extra free parameters wereintroduced [ 14]. Chiral damping was proposed to explain the asymmetric component of the velocity profiles [ 11,15,16,22]. We contend however that in the quasistatic creep regimedynamic effects such as chiral damping should not play asignificant role. In this paper we construct a theory for motion of chiral domain walls in the creep regime which does not involve thefree parameters introduced in Ref. [ 14]. We use it to interpret our experimental data on the DW velocity as a function ofthe IP magnetic field. We show that our model allows forquantitative determination of the strength of the interfacialDMI from field-driven DW creep measurements. II. MODEL In Fig. 1(a) the deformation of a DW due to a thermal fluctuation in the presence of an easy-axis driving field Hzis il- lustrated. The deformation size Lis determined by the balance between the gained Zeeman energy from the driving field andthe elastic energy cost. The deformations can be seen as nucle-ations whose chance of survival is determined by L. For such a nucleation process, Arrhenius’ law tells us that the rate atwhich these surviving deformations will occur is determinedby the height of the energy barrier F b(i.e., the free energy at the tipping point): ln( v)∝−Fb/(kBT)[23,24]. Jeudy et al. 2469-9950/2019/100(9)/094417(5) 094417-1 ©2019 American Physical SocietyDION M. F. HARTMANN et al. PHYSICAL REVIEW B 100, 094417 (2019) FIG. 1. (a) Top view of a DW (blue lines) that gets deformed over a length Land displaced over a distance udue to a thermal fluctuation. The DW can be tilted over an angle α. The magnetization is indicated by the red vectors, which at the DW are characterized by the IP angle ϕ. Note that the IP magnetization changes due to the displacement, affecting the elasticity. The IP magnetic field Hx (green) as well as the effective DMI field HD(yellow) and effective Bloch field HB(purple) are indicated locally. (b) Model to describe the deformation. (c) When an IP magnetic field is applied to a sample with PMA, the magnetization inside a domain tilts towards the IP magnetic field by an angle θtdetermined by the balance of PMA and IP magnetic field β=MSHxcos(ϕ)/KP=sin(θt) (orange) compared to the β=0 case (red). have shown that defining Fb=Td[(Hd/Hz)1/4−1], in terms of the depinning field Hdand an effective disorder temperature Td, can describe the DW motion accurately in both the creep and depinning regime [ 25,26]. A recent study also used this form to capture the in-plane magnetic field effects into thedepinning field [ 27]. Instead of postulating a form of F bwe will determine it from micromagnetics well inside the creepregime; F b=max LF(L). We capture the complexity of the asymmetric DW dynamics in F(L). As a consequence the optimization required to determine Fbis semianalytical. Here we introduce an insightful numerical procedure as opposedto a full analytical treatment as has been done extensively inliterature [ 8,14,21,25]. This numerical approach allows us to address a plethora of effects in the underlying physics of DWdynamics in the creep regime. F(L) is composed of the elastic energy cost and the Zee- man energy gain, which depend not only on L, but also on the DW displacement u:F(u,L)=E el(u,L)+EZeeman (u,L). To express uin terms of Lwe use u(L)=uc(L/Lc)2/3[21,28,29], where Lcis the Larkin length scale determined by minimizing the sum of the elastic and pinning energy density for u=ξ, anducis a proportionality constant. Hence the next step is to determine the elastic energy to be able to compute Lcand express u, and thereby F, in terms of L.The elastic energy is defined as the difference in internal, i.e., excluding pinning and driving, energy between the do-main wall before and after the deformation. Due to the appli-cation of the external IP magnetic field the DW energy densityitself depends on the orientation of the DW with respect to thisapplied field. Furthermore, the IP magnetization of the sampleat the DW is affected by the exchange interaction. Following Blatter et al. we model the deformation as an angular shape for simplicity, see Fig. 1(b) [30]. Other shapes are possible, but this is the lowest order approximation.Note that Pellegren et al. chose an arc shape [ 14], but did not implement the exchange energy cost due to the kink inthe connection with the straight DW segments, resulting inunphysical divergences (as demonstrated in the SupplementalMaterial [ 31]) that do not occur in our theory. We have approximated the IP magnetization of each seg- ment to be constant and implement a nearest neighbor ex-change interaction at the bending points. The energy of thesystem is then minimized (numerically) over the IP magneti-zation angle of the two segments. We compute the energy density of the domain wall by inserting the domain-wall solution into the micromagneticenergy functional E(α,ϕ)=2/radicalbig 1−β2J λ+MSπλ{g(β)HBcos2(ϕ−α) −f(β)[Hxcos(ϕ)+HDcos(ϕ−α)]}. (1) For more details see [ 31]. The first term is the exchange interaction Joverλ, the DW thickness. The second term is the demagnetization energy, expressed in terms of theeffective Bloch field H B(this energy favors a Bloch DW, hence the nomenclature), the angle αbetween the DW normal and the xaxis and the angle ϕthe IP magnetization at the DW with the direction of the IP magnetic field, see Fig. 1(a).T h e third term is the Zeeman energy due to the applied IP magneticfield H xand the fourth is the DMI expressed in terms of an effective field HDfavoring a Néel type DW. The prefactors involving βincorporate the tilting in the xdirection of the magnetic domains due to the external IP magnetic field [seeFig.1(c)]. The functions fandgare given in the Supplemental Material [ 31]. Similarly, we obtain the Zeeman energy from the driving field H z, EZeeman (u,L)=MSHztuL/radicalbig 1−β2. (2) Again, the factor/radicalbig 1−β2comes from the tilted domains as illustrated in Fig. 1(c). By dividing out D(=MSλHD)i n Eqs. ( 1) and ( 2), the relevant dimensionless parameters be- come ˜J≡Jλ−1D−1,˜HB≡2HB/HD,˜Hx≡Hx/HD, and ˜Hz≡ Hz/HD. Using Eq. ( 1), we compute the optimal orientation angle of the undeformed DW α0and the corresponding internal magnetization IP angle ϕ0by minimizing E(α,ϕ)/cos(α). The factor 1 /cos(α) arises because we allow the DW to orient itself with respect to the IP magnetic field at the costof elongating. For example, a mixed Bloch-Néel DW tilts itsnormal to better align with the external IP magnetic field. Thistilting however would induce a stretching factor of 1 /cos(α), increasing the energy cost. This effect is illustrated in Fig. 1(a) 094417-2CREEP OF CHIRAL DOMAIN WALLS PHYSICAL REVIEW B 100, 094417 (2019) FIG. 2. α0as a function of the applied IP magnetic field (a) and the corresponding minimized azimuthal angle of the internal mag- netization ϕ0(b). The green curve shows the solution for ϕ0when αis fixed at 8◦, which switches sign at Hx=0. The corresponding energy density however, remains continuous and smooth. Note that the green curve does not saturate in but converges to the Néel wall. and the optimal angle α0and corresponding minimized angle ϕ0are shown as a function of Hxin Fig. 2. The energy of the unperturbed DW is then given by LtE(α0,ϕ0). The profile of ϕ0shown in Fig. 2(b) exhibits sharp kinks for both α(Hx)=α0(Hx) andα(Hx)=0. This feature arises because in the energy density of Eq. ( 1) we neglected higher order anisotropy terms proportional to cosn(ϕ−α)f o r n> 2 which are allowed by symmetry. As a consequence, thissimplified energy density yields a sharp transition in DWtype from mixed Bloch-Néel to pure Néel at f(β)(˜H x−1)= g(β)˜HBas demonstrated in Fig. 2where ϕ0saturates to 0o rπ. To effectively include for the higher order terms in the energy density, we adjust the value of αto some nonzero value, e.g., α=8◦as done by Pellegren et al. [14]. This removes the symmetry between the two deformed segments and prevents the saturation of ϕ. With this mod- ification, ϕis smooth around ϕ=0o rϕ=πas demon- strated by the green curve in Fig. 2(b). We will assume D>0 in this paper, the results for D<0 are obtained by ˜Hx→− ˜Hx. A kink between two DW segments, as illustrated in Fig.1(b), gives an energy cost Eben(ϕ1,ϕ2)=Jλ a[1−cos(ϕ1−ϕ2)], (3) withϕ1andϕ2the IP angles of the internal magnetization of the segments. Here ais the distance between neighboring atoms in the magnetic layer. Due to variations in the latticestructure and to account for non-nearest neighbor interactions,an effective value of a∼1 nm is used. The effect of aon the DW dynamics is investigated in the Supplemental Material[31].The elastic energy is computed by minimizing over ϕ 1 andϕ2: Eel t=min ϕ1,ϕ2/bracketleftbiggL 2/radicalBigg 1+/parenleftbigg2u L/parenrightbigg2 [E(α1,ϕ1)+E(α2,ϕ2)] +Jλ a[3−cos(ϕ0−ϕ1)−cos(ϕ0−ϕ2) −cos(ϕ1−ϕ2)]/bracketrightbigg −LE(α0,ϕ0). (4) The first term is the length of each of the two segments of the deformed DW multiplied by their respective energy densities.α 1andα2are the orientations of the respective segments. The second term is the bending energy for the three corners, seeFig.1(b). The third term is the energy of the unperturbed DW. With this expression we compute L c, express uin terms of L, and thereby obtain F(L)=F[u(L),L] from which the DW velocity is found as ln( v)∝−Fb/(kBT). For more detail, see the Supplemental Material [ 31] (and Refs. [ 32–36] therein). In summary, the derivation of the DW velocity involves multipleoptimization steps to determine α 0,ϕ0,ϕ1,ϕ2,Lc, and finally Fb. Due to the complexity of the elastic energy, our results are obtained numerically. Approximating the elasticity to be proportional to u2/L does allow for analytic solutions, but these are not able tofully explain recent experimental observations. For example,Jeet al. approximated Eq. ( 4) by setting α 0=0,ϕ0=ϕ1=ϕ2 and neglecting the ±arctan(2 u/L) in the first two terms [ 8]. Because Pellegren et al. have chosen a different DW profile, we cannot directly compare the expression in Eq. ( 1) with their results [ 14]. They do, however, treat Las a free parameter and do not find it by optimization. Moreover, they do notaccount for the bending costs that we model by the termsinvolving ϕ 0−ϕ1andϕ0−ϕ1in Eq. ( 1). III. RESULTS In Fig. 3(a) the modeled DW velocity as a function of the applied IP magnetic field Hxis shown for different values of˜HB(a). Figure 3(b) shows the asymmetric component A=ln[v(↑↓)/v(↓↑)] for ˜HB=0.5. The kinks in the solid lines at ˜Hx=1±˜HBmark the saturation of internal DW magnetization angle into a Néel wall perpendicular to the IPmagnetic field. These are expected from the form of Eq. ( 1) where we neglected terms O[cos 4(ϕ)]. The dashed curves are the result of setting α=8◦fixed to compensate for the simplified energy density. In the high IP magnetic field regime, i.e., |˜Hx|>˜HB,t h e profile straightens out. In this regime the azimuthal angle ofthe internal magnetization is saturated to align with the IPmagnetic field, yielding a Néel DW. Due to this saturation,the orientation dependence of the elasticity no longer varieswith further increasing |H x|. As a result, the elasticity becomes isotropic and the logarithmic increase in velocity is solely dueto the gained Zeeman energy. Note that the demonstrated asymmetry of the profile com- pares well with experiments [ 10,11,14–17,37]. Furthermore, the minimal velocity is not attained at ˜H x=1 as in the model of Je et al. [8]. 094417-3DION M. F. HARTMANN et al. PHYSICAL REVIEW B 100, 094417 (2019) FIG. 3. Dependence of the IP magnetic field ˜Hxof the DW velocity (a) and the asymmetric component of the velocity Afor ˜HB=0.5 (b). The profiles in (a) are given a vertical offset for clarity. The dashed lines represent the result for fixing α=8◦.F o rt h i s calculation Hz=10 mT. Note moreover that the asymmetric velocity component switches sign as |˜Hx|increases. This feature has been observed experimentally and explained by chiral damping [ 11,13,15]. In our model there are no chiral damping effects, showing thatthis feature need not be an indication for chiral damping. Finally, we compared and fitted our model to experimental data. The results are shown in Fig. 4showing good quantita- tive agreement in a broad variety of samples over a wide range FIG. 4. Fitted DW velocity curve (dashed line) to experimental data (dots) of three different samples. The data shown in (b) and (c) are obtained for this paper. The data in (a) are from Ref. [ 10]. The obtained fit parameters are shown in the table. (d) The HD(1/tfilm) trend for a cobalt film thickness sample study in Pt /Co(tfilm)/Gd (orange) and Pt /Co(tfilm)/Ir (blue) stacks. The corresponding data and fits can be found in the Supplemental Material [ 31].of IP magnetic fields. The asymmetric behavior is clearly demonstrated in the experiment. We performed measurements on two different samples stacks, see Figs. 4(b) and4(c). The samples are grown via Ar DC magnetron sputter deposition in a sputter chamber with abase pressure of ∼3×10 −9mbar. The detailed composition of the samples is Sample a SiO 2/Ta (4)/Pt(4)/Co(0.6)/Pt(4); Sample b SiO 2/Ta (4)/Pt(4)/Co(0.8)/Gd(3)/Pt(2); Sample c SiO 2/Ta (4)/Pt(4)/Co(0.9)/Ir(4). The number in parentheses indicates the thickness of the layer in nanometers. These samples are representatives ofthe variety of the velocity profiles observed in the literatureof asymmetric domain expansion experiments [ 8,10,12,13]. We image the magnetic domains and the expansion of thosedomains with a Kerr microscope setup. We use an OOP pulsemagnet with a pulse length of 0.8–400 ms and strength up to±33 mT, and an IP magnet with a strength up to ±300 mT. Furthermore, we also interpret data from previous research ofRef. [ 10]i nF i g . 4(a).I nF i g . 4(d) the obtained values for H D are plotted as a function of the film thickness tand confirm our expectation that HDshould decrease as a function of t [20,31,38]. IV . CONCLUSION The DMI and IP magnetic field complexify DW dynamics significantly due to the orientation dependence of elasticity.To grasp and expose this complexity, we defined a modelfollowing creep theory and solving the dynamics semianalyt-ically. The model has a profound sensitivity to DMI and de-magnetization. As a result, the model provides a quantitativeinterpretation of experimental data of DWs that demonstrateasymmetric velocity profiles as a function of H x. Experimental studies that do not exhibit a kink at Hx= HD±2HBare often fitted with the constant elasticity model proposed by Je et al. [8]. In these studies the measurement range of Hxmight not be large enough to expose these kinks. Figure 4(b) demonstrates that our model resembles results from the constant elasticity model of Je et al. [8], but yields a different value of the DMI: at Hx=HD, the velocity is not minimized. The parameter αhas been set to a fixed value to account for the omission of higher order anisotropy terms in the energydensity. As a result the angle ϕwill not saturate for large H x. Previous research used αas a fitting parameter to account for roughness [ 14]. If roughness forces the DW to tilt, the tilting angle is not fixed to one value. Hence a fixed value of αshould not be interpreted as a physical tilting of the DW. We remark that assuming ϕto be constant along an axis normal to the DW is only a first approximation. For a mixedBloch-Néel DW, ϕwill adjust so that the magnetization aligns with the IP magnetic field well inside the domains, but doesnot at the DW. As ϕplays a key role in the DW dynamics, future research could focus on the exact behavior of ϕ. In recent publications the asymmetric shape of the DW velocity profile as a function of H xis used as an argument for significant effect of chiral damping on the DW dynam-ics [11,15,16]. However, our model demonstrates a similar 094417-4CREEP OF CHIRAL DOMAIN WALLS PHYSICAL REVIEW B 100, 094417 (2019) asymmetry without chiral damping. Furthermore, in the qua- sistatic creep regime dynamic effects such as chiral dampingshould not affect creep motion. The comparison experimental data demonstrates the broad applicability of our model. Future research could apply ourmodel to an extensive sample study to investigate the effectsof sample growth parameters and layer thickness on param-eters of the model such as the effective lattice spacing a. Furthermore, measurements over a broader range in H zcould be performed to test the universality.ACKNOWLEDGMENTS R.A.D. is member of the D-ITP consortium, a program of the Dutch Organisation for Scientific Research (NWO)that is funded by the Dutch Ministry of Education, Cultureand Science (OCW). This work is funded by the EuropeanResearch Council (ERC). This work is part of the researchprogramme of the Foundation for Fundamental Research onMatter (FOM), which is part of the Dutch Organisation forScientific Research (NWO). [1] S. S. Parkin, M. Hayashi, and L. Thomas, Science 320,190 (2008 ). [2] K.-W. Moon, D.-H. Kim, S.-C. Yoo, S.-G. Je, B. S. Chun, W. Kim, B.-C. Min, C. Hwang, and S.-B. Choe, Sci. Rep. 5,9166 (2015 ). [3] D. A. Allwood, G. Xiong, C. Faulkner, D. Atkinson, D. 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PhysRevApplied.9.054011.pdf

Magnetization Switching of a Co=PtMultilayered Perpendicular Nanomagnet Assisted by a Microwave Field with Time-Varying Frequency Hirofumi Suto,*Taro Kanao, Tazumi Nagasawa, Koichi Mizushima, and Rie Sato Corporate Research and Development Center, Toshiba Corporation, 1 Komukai-Toshiba-cho, Saiwai-ku, Kawasaki 212-8582, Japan (Received 12 September 2017; revised manuscript received 27 March 2018; published 8 May 2018) Microwave-assisted magnetization switching (MAS) is attracting attention as a method for reversing nanomagnets with a high magnetic anisotropy by using a small-amplitude magnetic field. We experimentally study MAS of a perpendicularly magnetized nanomagnet by applying a microwave magnetic field with a time-varying frequency. Because the microwave field frequency can follow thenonlinear decrease of the resonance frequency, larger magnetization excitation than that in a constant- frequency microwave field is induced, which enhances the MAS effect. The switching field decreases almost linearly as the start value of the time-varying microwave field frequency increases, and it becomessmaller than the minimum switching field in a constant-frequency microwave field. To obtain this enhancement of the MAS effect, the end value of the time-varying microwave field frequency needs to be almost the same as or lower than the critical frequency for MAS in a constant-frequency microwave field. Inaddition, the frequency change typically needs to take 1 ns or longer to make the rate of change slow enough for the magnetization to follow the frequency change. This switching behavior is qualitatively explained by the theory based on the macrospin model. DOI: 10.1103/PhysRevApplied.9.054011 I. INTRODUCTION Magnetization switching that utilizes ferromagnetic resonance (FMR) excitation has attracted attention as a write method in next-generation magnetic recording [1–20]. To date, experimental studies on this kind of magnetization switching have employed a microwave field with a time-constant frequency and have shown that the switching field of a nanomagnet substantially decreases byapplying a microwave field with a frequency on the order ofthe FMR frequency of the nanomagnet [5]. This switching method is called microwave-assisted magnetization switch- ing (MAS). Furthermore, magnetization switching inducedsolely by a circularly polarized microwave field has been proposed and experimentally demonstrated [10,11] .F M R is a nonlinear phenomenon and the resonance frequency atwhich the magnetization excitation becomes largest depends on the amplitude of the magnetization excitation. Because the FMR-based magnetization switching methodsutilize large magnetization excitation, nonlinearity plays an important role and needs to be taken into account to explain the switching behavior [12–14]. At the same time, the nonlinearity of FMR suggests that a microwave field with a time-varying frequency can induce larger magnetization excitation than a microwave field with a time-constantfrequency. Regarding nanomagnets with perpendicularmagnetic anisotropy, which are of interest in high-density magnetic recording applications, the resonance frequency decreases as the amplitude of the magnetization excitationevolves. Therefore, by gradually decreasing the frequencyof the microwave field, the nonlinear decrease of theresonance frequency can be followed and the magnetizationexcitation can be enhanced. Theoretical and micromagnetic simulation studies have reported that this kind of micro- wave field can efficiently induce magnetization switching[15–19]. However, no experimental studies have yet been reported. In magnetic recording applications, it has been proposed that a spin-torque oscillator (STO) can be used as a micro- wave field source [21,22] . One way to realize a varying- frequency microwave field is to change the current injectedinto the STO [17]. Furthermore, it has been reported that, in a certain geometry, the STO spontaneously changes thefrequency during the switching process because of theinteraction with the media magnetization [19]. In this paper, we experimentally study magnetization switching of a Co =Pt nanomagnet with perpendicular magnetic anisotropy by applying a microwave field witha time-varying frequency and compare the switchingbehavior to that in a microwave field with a constant frequency. For convenience, we introduce the following symbols for microwave field frequency ðf rfÞ. In constant- frequency MAS (CF MAS), the microwave field frequencyis constant and is referred to as f const rf. In varying-frequency*hirofumi.suto@toshiba.co.jpPHYSICAL REVIEW APPLIED 9,054011 (2018) 2331-7019 =18=9(5)=054011(10) 054011-1 © 2018 American Physical SocietyMAS (VF MAS), the start and end values of the time- varying microwave field frequency are referred to as fstart rf andfend rf, respectively. In CF MAS, the switching field of the nanomagnet decreases almost linearly with an increas- ingfconst rf value, reaches a minimum at the critical fre- quency, and then increases abruptly. In VF MAS, the switching field similarly decreases linearly with increasinganf start rfvalue but continues to decrease even when fstart rf becomes higher than the critical frequency for CF MAS. The switching field thus becomes lower than the minimumswitching field for CF MAS, showing that the MAS effect is enhanced by a microwave field with varying frequency. To obtain this enhancement of the MAS effect, f end rfneeds to be almost the same as or lower than the critical frequency for CF MAS. In addition, the frfchange needs to take approximately 1 ns or longer when fend rfis set to half of fstart rf to make the rate of frfchange sufficiently slow. The switching behavior of VF MAS can be explained qualita- tively by the theory that describes the magnetizationdynamics of a single spin in a microwave magnetic field. II. SAMPLE STRUCTURE AND EXPERIMENTAL SETUP Figure 1(a) shows the sample structure and the meas- urement setup. A Ta bottom layer and Co =Pt multilayer magnetic film are deposited on a sapphire substrate byusing a magnetron sputtering system. The film structure from bottom to top is Ta 200=Pt50=ðCo13.6=Pt5Þ× 5=Pt50=Ta50(thicknesses are given in angstroms). Figure 1(b) shows magnetization as a function of the z-direction magnetic field ðH zÞmeasured by using a vibrating sample magnetometer. The average saturationmagnetization ðM sÞof the Co =Pt film is estimated to be 1200 emu=cm3, assuming that the total thickness of the magnetic layer is from the bottommost Co layer to thetopmost Co layer. Figure 1(c) shows vector network analyzer (VNA)-FMR spectra of the film sample as a function of H z. At around Hz¼0Oe, the FMR peak disappears because of the formation of reversed magnetic domains, and the FMR frequency at Hz¼0Oe is esti- mated to be 5 GHz by linear extrapolation. This FMR frequency indicates that the effective perpendicular anisotropy field ðHeff aniÞof the film, including the demag- netizing field, is approximately 1.8 kOe, and the perpendicular anisotropy field ðHaniÞis estimated to be 16.9 kOe from Hani¼Heff aniþ4πMs. Figure 1(d)shows the linewidth of the FMR absorption peak ðΔfFMRÞas a function of the resonance frequency ðfFMRÞobtained in theHzrange from −2.4to−0.8kOe. This relationship can be expressed as ΔfFMR¼2αfFMRþΔf0, where Δf0 denotes linewidth broadening due to the film inhomoge- neity. From the fitting, the damping parameter ( α) of the film is estimated to be 0.035.The Co =Pt film is then patterned into a nanomagnet with a diameter of 50 nm by electron-beam lithography and Ar ion milling. The Ta bottom layer is patterned into a Hall cross to detect switching of the nanomagnet by the anomalous Halleffect. After that, insulating layers of 20-nm-thick SiN and 80-nm-thick SiO 2are sputter deposited, and a coplanar waveguide (CPW) made of 100-nm-thick Cu with thinadhesion layers is fabricated. The signal line of the CPWpasses above the nanomagnet with a separation of 100 nm. The widths of the CPW signal ( S) line, ground ( G) lines, and gaps are 1, 2, and 2μm above the nanomagnet and gradually expand to 80, 80, and 45μm at the contact pads. These dimensions are chosen to make the characteristic impedance roughly 50Ω, as calculated by using AppCADsoftware. Note that the CPW dimensions are the designvalues, and the actual (a) (b) (c) (d) (g)(f)(e) FIG. 1. (a) Sample structure and experimental setup. (b) M-Hz loop obtained for the film sample having an area of 1cm2. (c) VNA-FMR spectra versus Hzobtained for the film sample. (d) Linewidth of the FMR absorption peak versus FMR fre- quency. The dotted line depicts the linear fit. (e) Microwavetransmission between the cable end connected to the AWG andthe cable end connected to the CPW input. (f) Microwavetransmission between the input and output of the CPW. Halfof the measured value is employed as microwave transmissionbetween the CPW input and above the nanomagnet. (g) Micro-wave reflection at the CPW input.SUTO, KANAO, NAGASAWA, MIZUSHIMA, and SATO PHYS. REV. APPLIED 9,054011 (2018) 054011-2dimensions deviate from them and have a tapered cross section. The length of the CPW is 750μm. This length is approximately one tenth of the wavelength of microwavesignal traveling through the CPW at 16 GHz, which is the highest in the studied frequency range. A transmission- electron-microscopy image of the nanomagnet that has aslightly different Co thickness but is fabricated using thesame process is provided in Ref. [9]. We study switching of the nanomagnet by applying H z and an in-plane microwave magnetic field from the CPW. When no microwave field is applied, the switching z-direction dc magnetic field ðHswÞof the nanomagnet is 5.7 kOe. Hereafter, this value is referred to as the intrinsic Hsw. To generate a microwave field, a microwave signal is generated from a Keysight M8195A arbitrary waveformgenerator (AWG) with a 64-GHz sampling rate, amplified by an RF-LAMBDA RFLUPA00G22GAwide-band ampli- fier, and introduced to the CPW. The amplifier has abandwidth of 0.02 –22 GHz, an average gain of þ30dB, and a gain flatness of /C62.5dB. Figure 1(e) shows the microwave transmission between the cable end connectedto the AWG and the cable end connected to the input of theCPW, as measured by the VNA. The variation from þ22to þ28dB is due to the frequency dependence of the gain of the amplifier and the attenuation of the cables. The micro-wave property of the CPW is also evaluated. Figure 1(f) shows the microwave transmission between the input and the output of the CPW. The attenuation ranges from −2to −3dB. The microwave reflection at the CPW input [Fig. 1(g)] is approximately −14dB, showing that there is no severe reflection point, such as a large impedancemismatch. Considering that the phase of the reflection is close to 0° (data not shown), the characteristic impedance of the CPW is estimated to be approximately 75Ω. In this estimation, multiple reflection in the CPW is neglected. Inaddition, the CPW is shorter than one tenth of the wave- length; thus, impedance mismatch has a small effect on the microwave transmission. We employ the sum of themicrowave transmission from the AWG to the CPW input and half of the microwave transmission of the CPW as the microwave transmission from the AWG to above thenanomagnet because the nanomagnet is located below the middle of the CPW. This calculated microwave trans- mission is used to construct the waveform of the AWG. The microwave field amplitude ðH rfÞgenerated from the CPW is estimated by using the Biot-Savart formulaassuming a uniform current in the signal line. In this estimation, the tapered cross section of the CPW observed by transmission electron microscopy is consid-ered. When a microwave signal with a voltage ðV rfÞof 1 V travels above the nanomagnet, a microwave field with Hrf¼85Oe is applied. The microwave signal is modu- lated into pulses of nanosecond-order duration to avoidheating the sample and is emitted repeatedly from the AWG at 122 kHz.Figure 2(a)shows the waveform of a constant-frequency microwave signal that has a 5-ns rise and fall time, a 10-ns plateau time, V rf¼1.0V, and fconst rf¼12GHz. The rise and fall times are fixed to 5 ns throughout the experiments,and the plateau time is 10 ns except when the plateau timedependence is measured in Sec. III D. The measured voltage is larger than 1 V because this signal waveformis measured by disconnecting the cable end from the CPWinput and connecting it to an oscilloscope with an 80-GHzsampling rate. The signal amplitude, therefore, furtherattenuates by half of the microwave transmission inFig.1(f)and becomes almost 1 V above the nanomagnet. Figure 2(d)shows the instantaneous f rfestimated from the zero-cross intervals of the waveform, which confirms thatf rfis actually 12 GHz. Similarly, a varying-frequency microwave signal is generated that takes into account thefrequency dependence of the microwave transmission.Figure 2(b) shows the waveform of a varying-frequency microwave signal with V rf¼1.0V,fstart rf¼12GHz, and fend rf¼0.02GHz. During the rise and fall time, frfis constant and, during the plateau time, frfdecreases. The signal amplitude is almost the same as that with the(a) (d) (b) (c) (f)(e) FIG. 2. (a) –(c) Waveforms of signals for the following parameter sets:ðVrf¼1.0V;fconst rf¼12GHzÞ,ðVrf¼1.0V;fstart rf¼12GHz; fend rf¼0.02GHzÞ, and ðVrf¼1.5V;fconst rf¼12GHzÞ. These waveforms are measured at the cable end connected to the CPW input, and the amplitude further attenuates by half ofthe microwave transmission in Fig. 1(f)when above the nano- magnet. Because this attenuation is from −1to−1.5dB, the signal amplitude becomes 89% to 84% of the measured voltage. (d)–(f) Instantaneous f rfvalues estimated from the zero-cross intervals of the waveforms. Each dot corresponds to one zero-crossinterval. The dots overlap the designed f rfdepicted by dashed lines.MAGNETIZATION SWITCHING OF A Co/Pt … PHYS. REV. APPLIED 9,054011 (2018) 054011-3constant frequency [Fig. 2(a)], although it fluctuates because the frequency-dependent microwave transmission is not perfectly compensated for. Regarding the frfchange, we think that the rate of frfchange should be faster when frfis higher because it takes a shorter time for a microwave field to induce magnetization excitation when frfis higher. To realize such an frfchange, we employ the following function: frfðtÞ¼fstart rfexp/C18 −t tplateaulnfstart rf fendrf/C19 ; ð1Þ where tplateau denotes the plateau time. This function is derived from dfrf=dt∝frf, where the rate of frfchange is proportional to frf. This function is one example of frf change, and further study is necessary to optimize frf change for efficient MAS. Note that Ref. [15] reported a function in which frfalways matches the resonance frequency when αis small. Figure 2(e)shows the estimated instantaneous frfof the waveform in Fig. 2(b), which confirms that frfchanges as designed. Figures 2(c)and2(f) show a constant-frequency signal waveform with Vrf¼ 1.5V and fconst rf¼12GHz and the instantaneous frf, which we mention later in the next section. III. EXPERIMENTAL RESULTS A. Comparison between constant-frequency MAS and varying-frequency MAS, and effect of the start frequency on varying-frequency MAS In this section, we first study switching of the nano- magnet in a constant-frequency microwave field, thenapply a varying-frequency microwave field to study the effect of f start rf. Figure 3(a)shows the dependence of Hswonfconst rffor CF MAS. These data and this kind of frequency depend- ence of Hsw, as shown later in this paper, are measured as follows: at each frequency, the nanomagnet is initialized tothe–zdirection, and H zis increased in steps of 10 Oe per 0.3 s until magnetization switching is detected. During this Hzincrease, a pulsed microwave field is applied repeatedly. Each curve in Fig. 3(a)is obtained by setting Vrfto 0.5 – 2.0 V, which generates Hrfof 43 –170 Oe. As fconst rf increases, Hswdecreases almost linearly until Hswtakes a minimum at the critical frequency, then Hswincreases abruptly to the intrinsic Hswof 5.7 kOe. This kind of switching behavior has been reported by previous exper-imental studies on MAS [5], which can be understood as follows. When H zis applied in the opposite direction to the magnetization, the FMR frequency decreases. Therefore,the resonance condition in which the FMR frequency is near f const rfresults in a Hsw−fconst rfcurve with a negative slope. However, when fconst rf becomes higher than the critical frequency, matching Hzbecomes so small that a microwave field cannot induce magnetization excitationlarge enough to induce magnetization switching. As Hrf increases, the critical frequency increases and the corre- sponding Hswdecreases, showing that a larger MAS effect is obtained by applying a microwave field with a largerH rfvalue. Figure 3(b)shows the dependence of Hswonfstart rffor VF MAS. To focus on the effect of fstart rf,fend rfis set as low as 0.02 GHz. When fstart rfis smaller than the critical frequency of CF MAS, the Hswversus fstart rfcurves almost coincide with the Hswversus fconst rf curves for CF MAS. This coincidence indicates that magnetization switching in the varying-frequency microwave field occurs in the same manner as CF MAS because the frequencies of the twokinds of microwave field are almost the same at the beginning of the f rfchange. When fstart rfbecomes higher than the critical frequency, Hswcontinues to decrease and becomes smaller than the minimum value for CF MAS, showing that the MAS effect is enhanced. We now confirm that this enhancement of the MAS effect actually originatesfrom the varying-frequency microwave field by examining the waveform of the microwave signals. In VF MAS, H sw forVrf¼1.0V decreases to 3.05 kOe at fstart rf¼12GHz. In CF MAS, no MAS effect is obtained for Vrf¼1.0Va t fconst rf¼12GHz because fconst rf is above the critical fre- quency, and the MAS effect is obtained by increasing Vrfto 1.5 V, where the critical frequency is at 12.5 GHz. Waveforms of these signals are shown in Figs. 2(a), 2(b), and 2(c). The amplitude of the varying-frequency microwave signal for Vrf¼1.0V [Fig. 2(b)] is clearly(a) (b) FIG. 3. (a) Hswversus fconst rffor CF MAS. (b) Hswversus fstart rf for VF MAS obtained by setting fend rf¼0.02GHz. Squares show the corresponding critical frequency and Hswfor CF MAS.SUTO, KANAO, NAGASAWA, MIZUSHIMA, and SATO PHYS. REV. APPLIED 9,054011 (2018) 054011-4smaller than that of the constant-frequency microwave signal for Vrf¼1.5V [Fig. 2(c)]. The fact that these two signals achieve almost the same MAS effect isevidence that the enhancement in the MAS effect is dueto the varying-frequency microwave field. Asf start rfincreases, the Hswcurves for Vrf¼0.5, 1.0, and 1.5 V take the minimum and increase abruptly. ForV rf¼2.0V, such an abrupt Hswincrease does not appear, probably because its frequency is above 16 GHz. Thisabrupt increase in H swcannot be explained by considering only frf. For example, HswforVrf¼1.0V and fstart rf¼ 12GHz (below the abrupt increase) is smaller than that for Vrf¼1.0V and fstart rf¼12.5GHz (above the abrupt increase), which is inconsistent with the fact that the frf change for fstart rf¼12.5GHz passes through 12GHz and decreases to fend rf¼0.02GHz. The fact that the VF-MAS result cannot be explained by considering only frfindicates that the rate of frfchange needs to be taken into account as follows. When fstart rfis too high, the rate of frfchange becomes too fast for the magnetization excitation to follow. As a result, the varying-frequency microwave field can nolonger enhance the MAS effect. Above the abrupt increase, H swbecomes approximately the same as Hswat the critical frequency for CF MAS. This result indicates that magneti-zation switching occurs in the same manner as CF MASwhen f rfdecreases and matches the critical frequency for CF MAS. B. Effect of the end frequency on varying-frequency MAS We next examine the effect of fend rfon VF MAS. As shown in the previous section, enhancement of the MASeffect is not apparent for V rf¼2.0V because the critical frequency is already close to the upper limit of the studied frequency range. Thus, we show the results for Vrf¼0.5, 1.0, and 1.5 V. We fix fstart rfto 8, 12, and 15 GHz, respectively, for Vrf¼0.5, 1.0, and 1.5 V, at which Hsw takes the minimum in Fig. 3(b), and fend rfis varied. Figures 4(a)–4(d) show the waveforms and estimated instantaneous frfvalues of signals with Vrf¼1.0V, fstart rf¼12GHz, and fend rf¼1and 11 GHz, respectively, which confirms that the signals have the designed ampli- tude and frequency regardless of the amount of frfchange. Figure 4(e) shows the dependence of Hswonfend rf.A s already shown in the previous section, Hswbecomes smaller than Hswat the critical frequency for CF MAS when fend rf¼0.02GHz because the varying-frequency microwave field enhances the MAS effect. As fend rf increases, Hswis first constant and then abruptly increases to the intrinsic Hsw, showing that the MAS effect dis- appears when fend rfis too high. Waveforms of the signals for fend rfbelow and above the abrupt Hswincrease [Figs. 4(a) and4(b)] confirm that this drastic change of the switching behavior originates from the different fend rfvalue. Thefrequency at which Hswincreases is almost the same as the corresponding critical frequency. This result shows that fend rfneeds to be approximately the same as or lower than the critical frequency for CF MAS to enhance the MAS effect by applying a varying-frequency microwave field. C. Minimizing the switching field by applying a varying-frequency microwave field In Sec. III A, the Hswcurves for VF MAS exhibit the abrupt increase because the rate of frfchange becomes too fast. This result suggests that Hswcan be even smaller when the rate of frfchange is sufficiently slow. To determine the minimum Hswthat can be achieved by VF MAS, we again measure the dependence of Hswonfstart rf. In Sec. III B,i ti s found that fend rfneeds to be almost the same as or lower than the critical frequency for CF MAS. Based on this finding, fend rfis set to the critical frequencies of 7, 9.5, and 12.5 GHz, respectively, for Vrf¼0.5, 1.0, and 1.5 V. As shown in Fig. 5,Hswgradually decreases with an increas- ingfstart rfvalue, then Hswbecomes almost constant with no abrupt increase. This constant Hswvalue means that magnetization switching occurs in the same manner in this(a) (c) (d) (b) (e) FIG. 4. (a),(b) Waveforms of signals for the following param- eter sets: ( Vrf¼1.0V,fstart rf¼12GHz, fend rf¼1GHz) and (Vrf¼1.0V,fstart rf¼12GHz, fend rf¼11GHz). (c),(d) Instanta- neous frfvalues estimated from the zero-cross intervals of the waveforms. (e) Hswversus fend rffor VF MAS obtained by setting fstart rf¼8, 12, and 15 GHz, respectively, for Vrf¼0.5, 1.0, and 1.5 V. Squares show the corresponding critical frequency and Hsw for CF MAS.MAGNETIZATION SWITCHING OF A Co/Pt … PHYS. REV. APPLIED 9,054011 (2018) 054011-5fstart rfrange because the frfchange for a certain fstart rfpasses through the frfchange for a lower fstart rfvalue. The constant Hswalso means that the rate of frfchange is sufficiently slow. Therefore, the obtained Hswvalue is considered to be the minimum that can be achieved by VF MAS. Thedifference between the minimum H swvalue for CF and VF MAS is the largest for Vrf¼1.0V, followed by 1.5 and 0.5 V, showing that a varying-frequency microwave fieldcan enhance the MAS effect most efficiently for a certain H rf. This Hrfdependence is theoretically discussed in Sec. IV. D. Effect of rate of change in microwave field frequency In this section, we study the effect of the rate of frf change by varying tplateau . We set fstart rfto 8, 12, and 15 GHz, respectively, for Vrf¼0.5, 1.0, and 1.5 V, which are used for measuring the fend rfdependence [Fig. 4(e)], and we set fend rfto half of fstart rf. Figures 6(a) and 6(b) show the waveform and estimated instantaneous frfvalues of a signal with Vrf¼1.0V,fstart rf¼12GHz, fend rf¼6GHz, andtplateau ¼2ns, which confirms that the signal has the designed amplitude and frequency even for the short tplateau . Figure 6(c)shows the dependence of Hswontplateau . When tplateau is 2 ns, Hswis the same as that for the slow frf change [Fig. 5], showing that the rate of frfchange is sufficiently slow at 2 ns. As tplateau decreases, Hswis first constant and then increases abruptly. This increase appears attplateau ¼1.0–1.2ns, depending on Vrf. Below these tplateau values, the rate of frfchange becomes too fast and the enhancement of the MAS effect disappears. Immediately after this abrupt increase, Hswbecomes almost the same as Hswat the critical frequency for CF MAS. In this condition, magnetization switching occurs in the same manner as CF MAS when frfdecreases to the critical frequency, as already discussed in Sec. III A .A s tplateau further decreases, Hswgradually increases. This tplateau dependence is explained as follows. As the rate of frf change becomes faster, the time duration in which frfisnear the critical frequency becomes shorter. Because the magnetization excitation is still developing on this time- scale, the MAS effect weakens, and thus Hswincreases. IV. THEORY OF MAGNETIZATION SWITCHING IN A VARYING-FREQUENCY MICROWAVE FIELD BASED ON THE MACROSPIN MODEL The magnetization dynamics of a single spin in a rotating microwave field can be described by the Landau-Lifshitz- Gilbert (LLG) equation formulated in a rotating frame, and the switching condition can be derived by examining thestability of the steady-state solutions of the LLG equation[12]. Although issues such as spatially nonuniform mag- netization excitation [6], a quasiperiodic magnetization motion [12], and thermally activated magnetization switch- ing[14] are not accounted for, it is known that the switching behavior of CF MAS is qualitatively reproducedby this approach. In this section, we explain the switching behavior of VF MAS using this approach. We employ the following normalization, which is applicable to magneti-zation with uniaxial anisotropy, regardless of the strength ofthe anisotropy field. The rotating microwave field andz-direction dc magnetic field are normalized in units of the anisotropy field ðH aniÞ:hrf¼Hrot rf=Hani,hz¼Hz=Hani. Note that Hrot rfhere means the amplitude of the rotating microwave field, whereas Hrfin the experiments means the amplitude of the microwave field alternating in onedirection. It has been reported that a rotating microwave field induces the same MAS effect at half the microwave field amplitude compared to an alternating microwavemagnetic field [8,9]. This is the case because an alternatingFIG. 5. Hswversus fstart rffor VF MAS obtained by setting fend rf¼7, 9.5, and 12.5 GHz, respectively, for Vrf¼0.5, 1.0, and 1.5 V. Squares show the corresponding critical frequency and Hsw for CF MAS.(a) (c)(b) FIG. 6. (a),(b) Waveform and instantaneous frfof a signal with Vrf¼1.0V,fstart rf¼12GHz, fend rf¼6GHz, and tplateau ¼2ns. (c)Hswversus tplateau for VF MAS obtained by setting fend rf¼ fstart rf=2.SUTO, KANAO, NAGASAWA, MIZUSHIMA, and SATO PHYS. REV. APPLIED 9,054011 (2018) 054011-6microwave field is decomposed into two rotating micro- wave fields that rotate in the opposite direction and have half the amplitude, and only the rotating microwave fieldthat rotates in the same direction as the FMR precession induces magnetization excitation. The microwave field frequency is normalized in units of FMR frequency:ω rf¼ð2πfrfÞ=ðγHaniÞ, where γdenotes the gyromagnetic ratio. Similarly, time is normalized as τ¼tðγHaniÞ. The LLG equation that describes the dynamics of the magnetization direction ˜min the rotating frame ð˜x;˜y;˜zÞis given by [14] d˜m dτ¼−˜m×½hrfe˜xþð−hzþ ˜m˜z−ωrfÞe˜z þαωrf˜m×e˜z/C138þα˜m×d˜m dτ: ð2Þ Note that the damping constant αis the only remaining parameter as a result of the normalization. Figures 7(a)and7(b)show the cone angle of the steady- state solutions obtained by setting d˜m=dτ¼0and analyti- cally solving Eq. (2). The stability of the solution —stable, saddle, and unstable —evaluated by introducing a small deviation is also shown. Parameters α¼0.17andhrf¼ 0.05are chosen to reproduce the experimentally obtained CF- and VF-MAS results for Vrf¼1.0V, which we discuss later in detail. For clarity, the cone angle is shownup to 90°, and there is always a stable state near 180°, which corresponds to the switched state. Here, stable statemeans that the magnetization can stay in the state and rotates in synchronization with the microwave field. The magnetization cannot stay in unstable and saddle states andmove to the stable state. Figure 7(a) corresponds to the critical frequency ðω rf¼0.32Þfor CF MAS. As hz increases from zero, the magnetization follows the line of the stable state and the cone angle gradually increases because hzapproaches the resonance condition. At around hz¼0.5(the dashed line), the stable state disappears, which means that the induced magnetization excitation overcomes the barrier for switching. Thus, the magnetiza-tion moves to the other stable state near 180°, and MAS occurs. The solution shows hysteresis like a protrusion toward the lower-right direction. In CF MAS, however,this hysteresis is saddle or unstable and has no effect onmagnetization switching. Figure 7(b) shows calculation results for ω rfvalues higher than the critical frequency. At ωrf¼0.58, a peak appears in the cone angle due to FMR. As ωrfdecreases to 0.5, hysteresis appears and two stable states exist in anarrow h zrange near 0.3. These two stable states are referred to as a lower-angle branch and higher-angle branch. As shown in the inset of Fig. 7(b), the magneti- zation follows the higher-angle branch in the downward hz sweep, and the cone angle abruptly decreases at the edge of the higher-angle branch. Similarly, the magnetization follows the lower-angle branch in the upward hzsweep, and the cone angle abruptly increases at the edge of thelower-angle branch. This is called the fold-over effect [23]. In the experiments, h zis swept only in the upward direction. Thus, in CF MAS where ωrfis fixed, the magnetization is always on the lower-angle branch. At ωrf¼0.45, this hysteresis becomes more obvious. In these three conditions, MAS does not occur because one or morestable states exist. At ω rf¼0.39, an unstable state appears around the edge of the higher-angle branch. In CF MAS, this condition is still higher than the critical frequency, andmagnetization switching does not occur because of one or more stable states. As seen in Fig. 7(a), this unstable state expands as ω rffurther decreases. Now we apply a varying- frequency microwave field. In the experiments, frfis changed on the nanosecond timescale, while Hzis changed on the second timescale. Therefore, we consider that themagnetization moves on the curves for different ω rfvalues at constant hz. As the magnetization moves on the curves from a higher ωrfto a lower one, the magnetization is able to stay on the higher-angle branch, which is in contrast to CF MAS, where the magnetization is always on the lower- angle branch. When the higher-angle branch becomesunstable at ω rf¼0.39, the magnetization can move to the stable state near 180° instead of the stable lower-angle branch, which results in the enhanced MAS by a varying-frequency microwave field. The unstable state in the higher-angle branch appears when ω rfis slightly higher than the critical frequency,(a) (c)(b) FIG. 7. (a) Cone angle and stability of the magnetization excitation for hz¼0.05andα¼0.17at the critical frequency ðωrf¼0.32Þ, (b) at higher ωrfvalues, and (c) at lower ωrfvalues. In (b), ωrfis 0.58, 0.5, 045, and 0.39 from the curve with the smallest cone angle to the one with the largest cone angle. (Inset)An enlarged view of the data for ω rf¼0.5. In (c), ωrfis 0.28 and 0.24 from the curve with the smaller cone angle to the one withthe larger cone angle.MAGNETIZATION SWITCHING OF A Co/Pt … PHYS. REV. APPLIED 9,054011 (2018) 054011-7which indicates that ωrfneeds to decrease to a slightly higher value than the critical frequency to induce VF MAS.This result explains the experimentally obtained depend- ence on f end rf[Fig. 4(e)], in which the MAS effect appears when fend rfis almost the same as or lower than the critical frequency. The dependence on fstart rfcan be explained by using Figs. 7(a)and7(b). The switching condition for CF MAS is determined by the edge of the lower-angle branch,where the stable state disappears. In VF MAS, the cone angle first increases at the edge of the lower-angle branch. Asf rfdecreases, the magnetization stays on the higher- angle branch until it becomes unstable. In other words, bothCF and VF MAS are initiated by the transition of themagnetization excitation at the edge of the lower-angle branch. Because this edge shows an almost linear relation- ship with respect to ω rf, the Hswcurves for CF and VF MAS show the same linear relationship with respect tof const rfandfstart rf, regardless of the fact that magnetization switching occurs in a different manner. The dependence on the rate of frfchange can be understood as follows. When ωrfchanges fast, the cone angle of the magnetization becomes smaller than the calculated value because the calculated value is a steady- state solution and frfchanges faster than the relaxation time of the magnetization. When the magnetization cannot keepstaying on the higher angle branch and falls to the lower- angle branch, MAS cannot be enhanced. As shown in Fig.6(c), even when the rate of f rfchange becomes so fast that the enhancement of MAS disappears, Hswis still smaller than the intrinsic Hsw. In this condition, MAS occurs when frfdecreases below the critical frequency. This result indicates that this kind of MAS in which frf changes below the critical frequency occurs for faster frf change relative to the enhancement of MAS in which frf changes above the critical frequency. This difference can be explained as follows. Figure 7(c)shows calculation results forωrfvalues slightly lower than the critical frequency. At around hz¼0.6(the dashed line) there is no stable state for either ωrf¼0.28or 0.24. When ωrfchanges in this range, the magnetization moves one way to the switched state. In contrast, when ωrfis higher than the critical frequency, the magnetization can fall from the higher-angle branch tothe lower-angle branch during the ω rfchange. Because the magnetization moves one way to the switched state during theωrfchange, this kind of MAS occurs when the rate of frfchange is relatively fast. The minimum switching hzobtained for VF MAS is 0.33, as indicated by the dashed line in Fig. 7(b). This hz corresponds to the limit where the higher-angle branch always exists during the frfchange, which is necessary for the cone angle to gradually increase until switching occurs. We compare the experimental results with the calcula- tion, and for this purpose, we estimate the Hswvalue without microwave fields as follows. The intrinsic Hswof 5.7 kOe reflects thermally activated magnetizationswitching. Although MAS is also thermally activated, the thermal effect acts effectively only during the micro- wave field application, which has a duty ratio of approx- imately 0.001 (10-ns plateau time and 122-kHz repetition).Owing to the difference in the timescale of thermal effect,MAS above the intrinsic H swis screened, and the Hsw versus fconst rf curves change a slope at around fconst rf¼ 3GHz in Fig. 3(a). If the thermal effect were reduced to that of the timescale of MAS, the Hswversus fconst rf curves would have a constant slope and the intercept at fconst rf¼0Hz would be Hswin the static in-plane field with an amplitude of Hrf. Thus, the intercept of the extrapolated Hswversus fconst rfcurve for Vrf¼0.5V, which is approx- imately 7 kOe, is employed as the Hswwith no microwave field under the reduced thermal effect. The ratios of the minimum HswforVrf¼1.0V obtained by CF and VF MAS [3.8 kOe in Fig. 3(b) and 2.6 kOe in Fig. 5]t o thisHswvalue are 0.54 and 0.37, respectively, which approximately coincides with the calculation results of0.5 and 0.33. We would like to comment on αand the microwave field amplitude. The damping parameter α¼0.17is larger than the value estimated from the VNA-FMR measurementof the film. This deviation may originate from the factthat MAS involves large-amplitude magnetization excita- tion, whereas the VNA-FMR measurement uses small- amplitude magnetization precession. Increase of αby a factor of 5 in large-amplitude magnetization excitation hasbeen reported [24]. In addition, αis affected by the fact that the nanomagnet has a nonuniform demagnetizing field,whereas the film has a uniform demagnetizing field. Whenwe use the H swof 7 kOe without a microwave field as Hanifor a rough estimation, hrf¼0.05corresponds to Hrot rf¼350Oe for a rotating microwave field and Hrf¼ 700Oe for an alternating microwave field, which is much larger than Hrf¼85Oe in the experiments. This disagree- ment is because of the fact that the calculation does notinclude thermal activation and spatially nonuniform mag-netization excitation. According to the study using amacrospin model with thermal activation [14], thermal effects alone cannot explain the disagreement, and spatially nonuniform magnetization excitation may make a largecontribution. The issue of nonuniform magnetization exci-tation is presented in Ref. [6], which discusses a compari- son of experimental results, macrospin simulations, andmicromagnetic simulations. Figures 8(a) and8(b) show the calculation results for h rf¼0.075. Similar to the case of hrf¼0.05, enhancement of the MAS effect by a varying-frequency microwave fieldappears. According to this approach, the enhancement ofthe MAS effect becomes larger as h rfincreases, which cannot explain the experimental result in which the enhancement of the MAS effect is the largest forV rf¼1.0V. Because αis the only parameter in Eq. (2), the experimental result can be understood as increasedSUTO, KANAO, NAGASAWA, MIZUSHIMA, and SATO PHYS. REV. APPLIED 9,054011 (2018) 054011-8damping in large magnetization excitation. As shown in Figs. 8(c)and8(d), the hysteresis becomes less evident as α is increased to 0.22. Because CF MAS does not utilize thehysteresis, the MAS effect of CF MAS is almostunchanged. However, because VF MAS utilizes the hys- teresis, the enhancement of the MAS effect by VF MAS decreases as αincreases. This αdependence is consistent with previous theoretical and simulation studies [16,20] . This result implies that the effective damping of the nanomagnet may increase as the magnetization excitationbecomes larger, which reduces the enhancement ofthe MAS effect in a varying-frequency microwave field. The effective damping includes the intrinsic damping of the material, the spatial inhomogeneity of the magneticanisotropy and the demagnetizing field, the spatially nonuniform magnetization excitation, and the spin pumping. V. SUMMARY In this paper, we study the switching of a perpendicularly magnetized nanomagnet in a microwave field with time- varying frequency and explain the switching behavior by using the theory based on the macrospin model. When thefrequency of the microwave field gradually decreases, a larger MAS effect than that in a constant-frequency micro- wave field is obtained because the microwave field fre-quency follows the nonlinear decrease of the resonancefrequency and induces larger magnetization excitation. The switching field decreases almost linearly as the start frequency of the microwave field increases up to a certain frequency, beyond which further increases in the start frequency do not change the switching field. To obtain enhancement of the MAS effect, the end frequency of the microwave field needs to be approximately the same asor lower than the critical frequency for constant-frequency MAS. In addition, frequency change of a microwave field needs to take approximately 1 ns to make the rate of change sufficiently slow that the magnetization excitation can follow the varying-frequency microwave field. ACKNOWLEDGMENTS We thank Canon ANELVA Corp. for the technical support. This work was supported by Strategic Promotion of Innovative Research and Development from the Japan Science and Technology Agency, JST. [1] C. Thirion, W. Wernsdorfer, and D. Mailly, Switching of magnetization by non-linear resonance studied in singlenanoparticles, Nat. Mater. 2, 524 (2003) . [2] J.-G. Zhu, X. Zhu, and Y. Tang, Microwave assisted magnetic recording, IEEE Trans. Magn. 44, 125 (2008) . [3] J.-G. Zhu and Y. Wang, Microwave assisted magnetic recording utilizing perpendicular spin torque oscillator withswitchable perpendicular electrodes, IEEE Trans. Magn. 46, 751 (2010) . [4] I. Tagawa, M. Shiimoto, M. Matsubara, S. Nosaki, Y. Urakami, and J. Aoyama, Advantage of MAMR read-write performance, IEEE Trans. Magn. 52, 3101104 (2016) . [5] S. Okamoto, N. Kikuchi, M. Furuta, O. Kitakami, and T. Shimatsu, Microwave assisted magnetic recording technol-ogies and related physics, J. Phys. D 48, 353001 (2015) . [6] M. Furuta, S. Okamoto, N. Kikuchi, O. Kitakami, and T. Shimatsu, Size dependence of magnetization switching andits dispersion of Co =Pt nanodots under the assistance of radio frequency fields, J. Appl. Phys. 115, 133914 (2014) . [7] H. Suto, T. Nagasawa, K. Kudo, T. Kanao, K. Mizushima, and R. Sato, Layer-Selective Switching of a Double-LayerPerpendicular Magnetic Nanodot Using Microwave Assis-tance, Phys. Rev. Applied 5, 014003 (2016) . [8] S. Okamoto, N. Kikuchi, and O. Kitakami, Magnetization switching behavior with microwave assistance, Appl. Phys. Lett. 93, 102506 (2008) . [9] H. Suto, T. Kanao, T. Nagasawa, K. Kudo, K. Mizushima, and R. Sato, Subnanosecond microwave-assisted magneti-zation switching in a circularly polarized microwave mag-netic field, Appl. Phys. Lett. 110, 262403 (2017) . [10] T. Taniguchi, D. Saida, Y. Nakatani, and H. Kubota, Magnetization switching by current and microwaves, Phys. Rev. B 93, 014430 (2016) . [11] H. Suto, T. kanao, T. Nagasawa, K. Mizushima, and R. Sato, Zero-dc-field rotation-direction-dependent magnetizationswitching induced by a circularly polarized microwavemagnetic field, Sci. Rep. 7, 13804 (2017) .(a) (b) (c) (d) FIG. 8. (a),(b) Cone angle and stability of the magnetization for hz¼0.075andα¼0.17at the critical frequency ðωrf¼0.44Þ and at higher ωrfvalues. In (b), ωrfis 0.77, 0.7, 0.6, and 0.54 from the curve with the smallest cone angle to the one with the largestcone angle. (c),(d) Cone angle and stability of the magnetizationforh z¼0.075andα¼0.22at the critical frequency ðωrf¼0.4Þ and at higher ωrfvalues. In (d), ωrfis 0.59, 0.55, 0.5, and 0.44 from the curve with the smallest cone angle to the one with the largest cone angle. The dashed lines show the minimum switch-ingh zfor CF and VF MAS.MAGNETIZATION SWITCHING OF A Co/Pt … PHYS. REV. APPLIED 9,054011 (2018) 054011-9[12] G. Bertotti, C. Serpico, and I. D. Mayergoyz, Nonlinear Magnetization Dynamics under Circularly Polarized Field,Phys. Rev. Lett. 86, 724 (2001) . [13] T. Taniguchi, Magnetization reversal condition for a nano- magnet within a rotating magnetic field, Phys. Rev. B 90, 024424 (2014) . [14] H. Suto, K. Kudo, T. Nagasawa, T. Kanao, K. Mizushima, R. Sato, S. Okamoto, N. Kikuchi, and O. Kitakami,Theoretical study of thermally activated magnetizationswitching under microwave assistance: Switching pathsand barrier height, Phys. Rev. B 91, 094401 (2015) . [15] K. Rivkin and J. B. Ketterson, Magnetization reversal in the anisotropy-dominated regime using time-dependent mag-netic fields, Appl. Phys. Lett. 89, 252507 (2006) . 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PhysRevB.98.224426.pdf

PHYSICAL REVIEW B 98, 224426 (2018) Localized spin waves in isolated kπskyrmions Levente Rózsa,*Julian Hagemeister, Elena Y . Vedmedenko, and Roland Wiesendanger Department of Physics, University of Hamburg, D-20355 Hamburg, Germany (Received 11 October 2018; revised manuscript received 30 November 2018; published 28 December 2018) The localized magnon modes of isolated kπskyrmions on a field-polarized background are analyzed based on the Landau-Lifshitz-Gilbert equation within the terms of an atomistic classical spin model, with systemparameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion order ka higher number of excitation modes are found, including modes with nodes in the radial eigenfunctions. It is shown that at low fields2πand 3πskyrmions are destroyed via a burst instability connected to a breathing mode, while 1 πskyrmions undergo an elliptic instability. At high fields all kπskyrmions collapse due to the instability of a breathing mode. The effective damping parameters of the spin waves are calculated in the low Gilbert damping limit, and they arefound to diverge in the case of the lowest-lying modes at the burst and collapse instabilities but not at the ellipticinstability. It is shown that the breathing modes of kπskyrmions may become overdamped at higher Gilbert damping values. DOI: 10.1103/PhysRevB.98.224426 I. INTRODUCTION Magnetic skyrmions are localized particlelike spin config- urations [ 1], which have become the focus of intense research activities over the last years due to their promising applica-tions in spintronic devices [ 2–5]. While their particlelike prop- erties make them suitable to be used as bits of information, thecollective excitations of the spins constituting the magneticskyrmion, known as spin waves or magnons, open possibleapplications in the field of magnonics [ 6]. These spin wave modes were first investigated theoretically [7–10] and experimentally [ 11–13] in skyrmion lattice phases, where the interactions between the skyrmions lead to theformation of magnon bands. If a skyrmion is confined in afinite-sized nanoelement, it will possess discrete excitationfrequencies [ 14–17]. Although such geometries have also been successfully applied to the time-resolved imaging ofthe dynamical motion of magnetic bubble domains [ 18,19], in such a case it is not possible to distinguish between theexcitations of the particlelike object itself and spin wavesforming at the edges of the sample [ 14]. In order to rule out boundary effects, the excitations of isolated skyrmionshave to be investigated, as was performed theoretically inRefs. [ 20–23]. It was suggested recently [ 24] that the experi- mentally determined excitation frequencies in the Ir/Fe/Co/Ptmultilayer system may be identified as spin wave modes ofisolated skyrmions, rather than as magnons stemming froman ordered skyrmion lattice. In most investigations skyrmions correspond to simple domains with the magnetization in their core pointing op-posite to the collinear background. However, it was shownalready in Ref. [ 25] that the Dzyaloshinsky-Moriya interac- tion [ 26,27] responsible for their stabilization may also lead to the formation of structures where the direction of the *rozsa.levente@physnet.uni-hamburg.demagnetization rotates multiple times between the center ofthe structure and the collinear region. Such target states orkπskyrmions, where kis the number of sign changes of the out-of-plane magnetization when moving along the radialdirection, have also been investigated in constricted geome-tries [ 28–32]. The experimental observation of localized spin structures with multiple rotations has been mainly restrictedto systems with negligible Dzyaloshinsky-Moriya interactionso far [ 19,33,34], where the formation of domain structures is attributed to the magnetostatic dipolar interaction. The collapse of isolated kπskyrmions and their creation in nanodots by switching the external field direction wasrecently investigated in Ref. [ 35]. It was found that during the creation process the skyrmions display significant sizeoscillations resembling breathing eigenmodes. In Ref. [ 25], the stability of kπskyrmions was studied in a system with a ferromagnetic ground state, and it was found that applying theexternal field opposite to the background magnetization leadsto a divergence of the skyrmion radius at a critical field value,a so-called burst instability. This instability can be attributedto a sign change of one of the eigenvalues of the energyfunctional expanded around the kπskyrmion configuration, intrinsically related to the dynamics of the system. However,the spin wave frequencies of isolated kπskyrmions remain unexplored. Besides the excitation frequencies themselves, the lifetime of spin waves is also of crucial importance in magnonicsapplications. This is primarily influenced by the Gilbert damp-ing parameter α[36], the value of which can be determined experimentally based on resonance lineshapes measured in thecollinear state [ 11,19,24]. It was demonstrated recently [ 23] that the noncollinear spin structure drastically influences theeffective damping parameter acting on the spin waves, leadingto mode-dependent and enhanced values compared to theGilbert damping parameter. This effect was discussed throughthe example of the 1 πskyrmion in Ref. [ 23] ,b u ti ti sa l s oe x - pected to be observable for kπskyrmions with higher order k. 2469-9950/2018/98(22)/224426(10) 224426-1 ©2018 American Physical SocietyLEVENTE RÓZSA et al. PHYSICAL REVIEW B 98, 224426 (2018) Here the localized spin wave frequencies of isolated kπskyrmions are investigated in a classical atomistic spin model. The parameters in the Hamiltonian represent thePd/Fe/Ir(111) model-type system, where the properties ofskyrmions have been studied in detail both from the experi-mental [ 37,38] and from the theoretical [ 35,39–41] side. The paper is organized as follows. The classical atomistic spinHamiltonian and the method of calculating the eigenmodesis introduced in Sec. II A, while the angular momentum and nodal quantum numbers characterizing the excitationsare defined in Sec. II B within the framework of the corre- sponding micromagnetic model. Eigenfrequencies equal toor approaching zero are discussed in Sec. II C, and the ef- fective damping parameters are introduced in Sec. II D.T h e eigenmodes of kπskyrmions with k=1,2,3 are compared in Sec. III A , the instabilities occurring at low and high field values are discussed in connection to magnons with vanishingfrequencies in Sec. III B, and the effective damping parame- ters of the different modes are calculated for vanishing andhigher values of the Gilbert damping in Secs. III C andIII D , respectively. A summary is given in Sec. IV. II. METHODS A. Atomistic model The system is described by the classical atomistic model Hamiltonian H=−1 2/summationdisplay /angbracketlefti,j/angbracketrightJSiSj−1 2/summationdisplay /angbracketlefti,j/angbracketrightDij(Si×Sj) −/summationdisplay iK/parenleftbig Sz i/parenrightbig2−/summationdisplay iμsBSi, (1) with the Siunit vectors representing the spins in a single-layer triangular lattice; J,Dij, and Kdenoting nearest-neighbor Heisenberg and Dzyaloshinsky-Moriya exchange interactionsand on-site magnetocrystalline anisotropy, respectively, whileμ sandBstand for the spin magnetic moment and the external magnetic field. The Pd/Fe/Ir(111) system selected for the in-vestigations presented here belongs to the C 3vsymmetry class due to the fcc stacking of the atomic layers and the breakingof inversion symmetry at the surface. Following the symmetryrules established by Moriya [ 42], the D ijvectors must lie in the mirror plane perpendicular to the nearest-neighbor bondson the lattice. Only the in-plane components of these vectorswill be considered here, being sufficient for explaining theformation of kπskyrmions, while the out-of-plane compo- nents only appear as higher-order terms in the correspondingmicromagnetic energy functional [ 43]. The numerical val- ues of the parameters are taken from Ref. [ 35], being J= 5.72 meV ,D=|D ij|=1.52 meV ,K=0.4 meV, and μs= 3μB. These were determined based on measuring the field dependence of 1 πskyrmion profiles in the system by spin- polarized scanning tunneling microscopy in Ref. [ 38]. During the calculations the external field Bis oriented along the out-of-plane zdirection. The equilibrium kπ skyrmion structures are determined from a reasonable initialconfiguration by iteratively rotating the spins S itowards the direction of the effective magnetic field Beff i=−1 μs∂H ∂Si.T h e iteration is performed until the torque acting on the spins,Ti=−Si×Beff i, becomes smaller at every lattice site than a predefined value, generally chosen to be 10−8meV/μB.T h e calculations are performed on a lattice with periodic boundaryconditions, with system sizes up to 256 ×256 for the largest kπskyrmions in order to avoid edge effects and enable the accurate modeling of isolated skyrmions. Once the equilibrium configuration S(0) iis determined, the spins are rotated to a local coordinate system ˜Si=RiSiusing the rotation matrices Ri. In the local coordinate system the equilibrium spin directions are pointing along the local zaxis, ˜S(0) i=(0,0,1). The Hamiltonian in Eq. ( 1) is expanded up to second-order terms in the small variables ˜Sx i,˜Sy ias (cf. Ref. [ 23]) H≈H0+1 2(˜S⊥)THSW˜S⊥ =H0+1 2[˜Sx˜Sy]/bracketleftbiggA1A2 A† 2A3/bracketrightbigg/bracketleftbigg˜Sx ˜Sy/bracketrightbigg . (2) The matrix products are understood to run over lattice site indices i, with the matrix components reading A1,ij=− ˜Jxx ij+δij/parenleftBigg/summationdisplay k˜Jzz ik−2˜Kxx i+2˜Kzz i+μs˜Bz i/parenrightBigg ,(3) A2,ij=− ˜Jxy ij−δij2˜Kxy i, (4) A3,ij=− ˜Jyy ij+δij/parenleftBigg/summationdisplay k˜Jzz ik−2˜Kyy i+2˜Kzz i+μs˜Bz i/parenrightBigg .(5) The energy terms in the Hamiltonian are rotated to the lo- cal coordinate system via ˜Jij=Ri[JI−Dij×]RT j,˜Ki= RiKRT j,and˜Bi=RiB, where Iis the 3 ×3 identity matrix, Dij×is the matrix describing the vector product with Dij, andKis the anisotropy matrix with the only nonzero element beingKzz=K. The spin wave frequencies are obtained from the linearized Landau-Lifshitz-Gilbert equation [ 36,44] ∂t˜S⊥=γ/prime μs(−iσy−α)HSW˜S⊥=DSW˜S⊥, (6) with σy=/bracketleftbig0 −iIs iIs 0/bracketrightbig the Pauli matrix in Cartesian com- ponents and acting as the identity matrix Isin the lattice site summations. The symbol γ/primedenotes the gyromagnetic ratioγ=ge 2mdivided by a factor of 1 +α2, with gthe electron gfactor, ethe elementary charge, mthe electron’s mass, and αthe Gilbert damping parameter. Equation ( 6)i s rewritten as an eigenvalue equation by assuming the time de- pendence ˜S⊥(t)=e−iωqt˜S⊥ qand performing the replacement ∂t→− iωq. Since the kπskyrmions represent local energy minima, HSWin Eq. ( 2) is a positive semidefinite matrix. For α=0 theωqfrequencies of DSWare real and they always occur in±ωqpairs on the subspace where HSWis strictly positive, for details see, e.g., Ref. [ 23]. In the following, we will only treat the solutions with Re ωq>0, but their Re ωq<0 pairs are also necessary for constructing real-valued eigenvectors ofEq. ( 6). The zero eigenvalues are discussed in Sec. II C. As is known from previous calculations for 1 πskyrmions [21–23], the localized excitation modes of kπskyrmions are 224426-2LOCALIZED SPIN WA VES IN ISOLATED kπSKYRMIONS PHYSICAL REVIEW B 98, 224426 (2018) found below the ferromagnetic resonance frequency ωFMR= γ μs(2K+μsB). During the numerical solution of Eq. ( 6) these lowest-lying eigenmodes of the sparse matrix DSWare determined, as implemented in the MONTECRYSTAL atomistic spin simulation program [ 45]. B. Micromagnetic model The atomistic model described in the previous section enables the treatment of noncollinear spin structures where thedirection of the spins significantly differs between neighbor-ing lattice sites. This is especially important when discussingthe collapse of kπskyrmions on the lattice as was performed in Ref. [ 35]. Here we will discuss the micromagnetic model which on the one hand is applicable only if the characteristiclength scale of noncollinear structures is significantly largerthan the lattice constant, but on the other hand enables asimple classification of the spin wave modes. The free energy functional of the micromagnetic model is defined as H=/integraldisplay A/summationdisplay α=x,y,z(∇Sα)2+K(Sz)2−MBSz +D(Sz∂xSx−Sx∂xSz+Sz∂ySy−Sy∂ySz)dr,(7) where for the Pd/Fe/Ir(111) system the following parameter values were used: A=2.0p J/m is the exchange stiffness, D=−3.9m J/m2is the Dzyaloshinsky-Moriya interaction describing right-handed rotation [ 39],K=−2.5M J/m3is the easy-axis anisotropy, and M=1.1M A/mi st h es a t u r a - tion magnetization. As discussed in, e.g., Refs. [ 23,25,46], Eq. ( 7) is char- acterized by two independent dimensionless parameters, theanisotropy K dl=KA D2and the magnetic field ( MB)dl=MBA D2. Two systems display identical properties at the same values ofthese dimensionless parameters after an appropriate rescalingof the length and time units, thereby enabling a comparisonof the excitation frequencies between different materials. Thedimensionless anisotropy takes the value −K dl=0.33 for the Pd/Fe/Ir(111) system, and a qualitatively similar behaviorof isolated kπskyrmions is expected for easy-axis systems with a spin spiral ground state in the absence of an external magnetic field, 0 /lessorequalslant−K dl/lessorequalslantπ2 16≈0.62 [25]. The equilibrium spin structure S(0)=(sin/Theta10 cos/Phi10,sin/Theta10sin/Phi10,cos/Theta10)o f kπ skyrmions will be cylindrically symmetric, given by /Phi10(r, ϕ)=ϕ+πdue to the right-handed rotational sense and /Theta10(r, ϕ)=/Theta10(r), which is the solution of the Euler-Lagrange equation A/parenleftbigg ∂2 r/Theta10+1 r∂r/Theta10−1 r2sin/Theta10cos/Theta10/parenrightbigg +|D|1 rsin2/Theta10 +Ksin/Theta10cos/Theta10−1 2MBsin/Theta10=0. (8) The skyrmion order kis encapsulated in the boundary conditions /Theta10(0)=kπ,/Theta10(∞)=0. Equation ( 8)i ss o l v e d numerically in a finite interval r∈[0,R] significantly larger than the equilibrium kπskyrmion size. A first approximation to the spin structure is constructed based on the correspondinginitial value problem using the shooting method [ 25], theniteratively optimizing the structure using a finite-difference discretization. The spin wave Hamiltonian may be determined anal- ogously to Eq. ( 2), by using the local coordinate sys- tem/Theta1=/Theta1 0+˜Sx,/Phi1=/Phi10+(sin/Theta10)−1˜Sy. The matrices in Eqs. ( 3)–(5) are replaced by the operators A1=−2A/parenleftbigg ∇2−1 r2cos 2/Theta10/parenrightbigg −2|D|1 rsin 2/Theta10 −2Kcos 2/Theta10+MBcos/Theta10, (9) A2=4A1 r2cos/Theta10∂ϕ−2|D|1 rsin/Theta10∂ϕ, (10) A3=−2A/braceleftbigg ∇2+/bracketleftbigg (∂r/Theta10)2−1 r2cos2/Theta10/bracketrightbigg/bracerightbigg −2|D|/parenleftbigg ∂r/Theta10+1 rsin/Theta10cos/Theta10/parenrightbigg −2Kcos2/Theta10+MBcos/Theta10. (11) Due to the cylindrical symmetry of the structure, the solutions of Eq. ( 6) are sought in the form ˜S⊥(r, ϕ, t )= e−iωn,mteimϕ˜S⊥ n,m(r), performing the replacements ∂t→ −iωn,mand∂ϕ→im. For each angular momentum quantum number m, an infinite number of solutions indexed by n may be found, but only a few of these are located belowω FMR=γ M(−2K+MB), hence representing localized spin wave modes of the kπskyrmions. The different nquantum numbers typically denote solutions with different numbersof nodes along the radial direction, analogously to thequantum-mechanical eigenstates of a particle in a box. Because of the property H SW(m)=H∗ SW(−m) and HSW being self-adjoint, the eigenvalues of HSW(m) and HSW(−m) coincide, leading to a double degeneracy apart from the m= 0 modes. The ±ωqeigenvalue pairs of DSWdiscussed in Sec. II A for the atomistic model at α=0 in this case can be written as ωn,m=−ωn,−m. The operator A2in Eq. ( 10), appearing due to the non- collinear structure of kπskyrmions, depends on the sign of m or−i∂ϕ. Considering only the modes with Re ωn,m>0, this leads to ωn,m/negationslash=ωn,−mindicating nonreciprocity or an energy difference between clockwise ( m< 0) and counterclockwise (m> 0) rotating modes [ 17,23]. For finding the eigenvectors and eigenvalues of the micro- magnetic model, Eq. ( 6) is solved using a finite-difference method on the r∈[0,R] interval. For treating the Laplacian ∇2in Eqs. ( 9) and ( 11) the improved discretization scheme suggested in Ref. [ 47] was applied, which enables a more accurate treatment of modes with eigenvalues converging tozero in the infinite and continuous micromagnetic limit. Thespin wave modes of the atomistic model discussed in Sec. II A were assigned the ( n,m) quantum numbers, which are strictly speaking only applicable in the micromagnetic limit withperfect cylindrical symmetry, by visualizing the real-spacestructure of the numerically obtained eigenvectors. 224426-3LEVENTE RÓZSA et al. PHYSICAL REVIEW B 98, 224426 (2018) C. Goldstone modes and instabilities The translations of kπskyrmions on the collinear back- ground in the two-dimensional plane along the xory directions represent continuous symmetries of the system,which are spontaneously broken by the presence of the kπ skyrmions. The Goldstone modes appearing due to this sym-metry breaking are represented by two eigenvectors of thespin wave Hamiltonian H SWbelonging to zero eigenvalue. Within the micromagnetic description of Sec. II B, these may be expressed analytically as [ 21–23] (˜Sx,˜Sy)=e−iϕ/parenleftbigg −∂r/Theta10,i1 rsin/Theta10/parenrightbigg , (12) (˜Sx,˜Sy)=eiϕ/parenleftbigg −∂r/Theta10,−i1 rsin/Theta10/parenrightbigg . (13) Equations ( 12) and ( 13) represent eigenvectors of the dy- namical matrix DSWas well. From Eqs. ( 2) and ( 6) it follows that the eigenvectors of HSWandDSWbelonging to zero eigenvalue must coincide, HSW˜S⊥=0⇔DSW˜S⊥=0, be- cause ( −iσy−α)i nE q .( 6) is an invertible matrix. Since we will only keep half of the solutions of the equation of motion(6), namely the ones satisfying Re ω n,m>0, the eigenvectors from Eqs. ( 12) and ( 13) will be denoted as the single spin wave mode ω0,−1=0. Since the eigenvectors and eigenvalues are determined numerically in a finite system by using a discretization pro-cedure, the Goldstone modes will possess a small finite fre-quency. However, these will not be presented in Sec. III A together with the other frequencies since they represent anumerical artifact. For the 1 πand 3πskyrmions the ω 0,1 eigenmode has a positive frequency and an eigenvector clearly distinguishable from that of the ω0,−1translational mode. However, for the 2 πskyrmion both the ω0,−1and the ω0,1 eigenfrequencies of DSWare very close to zero, and the cor- responding eigenvectors converge to Eqs. ( 12) and ( 13)a st h e discretization is refined and the system size is increased. Thiscan occur because D SWis not self-adjoint and its eigenvectors are generally not orthogonal. In contrast, the eigenvectors of HSWremain orthogonal, with only a single pair of them taking the form of Eqs. ( 12) and ( 13). In contrast to the Goldstone modes with always zero en- ergy, the sign change of another eigenvalue of HSWindicates that the isolated kπskyrmion is transformed from a stable local energy minimum into an unstable saddle point, leadingto its disappearance from the system. Such instabilities weredetermined by calculating the lowest-lying eigenvalues of H SWin Eq. ( 2). Due to the connection between the HSW andDSWmatrices expressed in Eq. ( 6), at least one of the precession frequencies ωqwill also approach zero at such an instability point. D. Effective damping parameters For finite values of the Gilbert damping α,t h es p i nw a v e s in the system will decay over time as the system relaxes to theequilibrium state during the time evolution described by theLandau-Lifshitz-Gilbert equation. The speed of the relaxationcan be characterized by the effective damping parameter,which for a given mode qis defined as α q,eff=/vextendsingle/vextendsingle/vextendsingle/vextendsingleImω q Reωq/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (14) As discussed in detail in Ref. [ 23],α q,effis mode dependent and can be significantly higher than the Gilbert dampingparameter αdue to the elliptic polarization of spin waves, which can primarily be attributed to the noncollinear spinstructure of the kπskyrmions. For α/lessmuch1,α q,effmay be expressed as αq,eff α=/summationtext i/vextendsingle/vextendsingle˜S(0),x q,i/vextendsingle/vextendsingle2+/vextendsingle/vextendsingle˜S(0),y q,i/vextendsingle/vextendsingle2 /summationtext i2I m/bracketleftbig/parenleftbig˜S(0),x q,i/parenrightbig∗˜S(0),y q,i/bracketrightbig, (15) where the eigenvectors in Eq. ( 15) are calculated at α=0 from Eq. ( 6). Equation ( 15) may also be expressed by the axes of the polarization ellipse of the spins in mode q,s e eR e f .[ 23] for details. For higher values of α, the complex frequencies ωqhave to be determined from Eq. ( 6), while the effective damping parameters can be calculated from Eq. ( 14). Also for finite values of αfor each frequency with Re ωq>0 there exists a pair with Re ωq/prime<0 such that ωq/prime=−ω∗ q[23]. The spin waves will be circularly polarized if A1=A3andA† 2=−A2 in Eq. ( 2), in which case the dependence of ωqonαmay simply be expressed by the undamped frequency ω(0) qas Reωq(α)=1 1+α2ω(0) q, (16) /vextendsingle/vextendsingleImωq(α)/vextendsingle/vextendsingle=α 1+α2ω(0) q. (17) These relations are known for uniaxial ferromagnets; see, e.g., Ref. [ 48]. In the elliptically polarized modes of non- collinear structures, such as kπskyrmions, a deviation from Eqs. ( 16) and ( 17) is expected. III. RESULTS A. Eigenmodes The frequencies of the localized spin wave modes of the 1π,2π, and 3 πskyrmion, calculated from the atomistic model for α=0 as described in Sec. II A,a r es h o w ni nF i g . 1. For the 1 πskyrmion six localized modes can be observed below the FMR frequency of the field-polarized backgroundin Fig. 1(a), four of which are clockwise rotating modes (m< 0), one is a gyration mode rotating counterclockwise (m=1), while the final one is a breathing mode ( m=0). The excitation frequencies show good quantitative agreementwith the ones calculated from the micromagnetic model forthe same system in Ref. [ 23]. Compared to Ref. [ 21], the additional appearance of the eigenmodes with m=1,−4,−5 can be attributed to the finite value of the anisotropy parameterKin the present case. Increasing the anisotropy value makes it possible to stabilize the skyrmions at lower field values,down to zero field at the critical value in the micromagnetic model |K c|=π2D2 16A, where the transition from the spin spiral to the ferromagnetic ground state occurs at zero external field[46]. Since the excitation frequencies decrease at lower field values as shown in Fig. 1(a), this favors the appearance of 224426-4LOCALIZED SPIN WA VES IN ISOLATED kπSKYRMIONS PHYSICAL REVIEW B 98, 224426 (2018) 0.7 0.8 0.9 1.0 1.1 1.20255075100125150175(a) 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.150255075100125150175(b) 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.150255075100125150175(c) FIG. 1. Frequencies of localized spin wave modes at α=0f o r (a) the 1 π,( b )t h e2 π, and (c) the 3 πskyrmion. Selected spin wave modes are visualized in contour plots of the out-of-plane spincomponent and denoted by open symbols connected by lines in the figure, the remaining modes are denoted by connected dots.0 1 02 03 04 05 0-2-023 -0.100.000.10 FIG. 2. Comparison between the 3 πskyrmion profile (left verti- cal axis) and the eigenvectors of the breathing modes ( m=0) with different numbers of nodes n=0,1,2 (right vertical axis). The cal- culations were performed using the micromagnetic model describedin Sec. II BatB=1 T, the lattice constant is a=0.271 nm. Double arrows between vertical dashed lines indicate the extensions of the domain walls in the structure. further modes. Simultaneously, the FMR frequency increases withK, meaning that modes with higher frequencies become observable for larger uniaxial anisotropy. For each angularmomentum quantum number m, only a single mode ( n=0) appears. In the case of the 2 πskyrmion an increased number of eigenmodes may be seen in Fig. 1(b). This can mainly be attributed to the appearance of spin waves with higherangular momentum quantum numbers both for clockwise (uptom=−17) and counterclockwise (up to m=12) rotational directions. Furthermore, in this case modes with n=1 node in the eigenfunction can be observed as well. The same trendcontinues in the case of 3 πskyrmions in Fig. 1(c),t h el a r g e number of internal eigenmodes can be attributed to angularmomentum quantum numbers ranging from m=−22 tom= 16, as well as to spin wave eigenvectors with up to n=2 nodes. The different rotational directions and numbers ofnodes are illustrated in Supplemental Videos 1–4 [ 49] via the square-shaped modes ( n=0,1;m=±4) of the 3 πskyrmion atB=0.825 T. The increase of possible angular momentum quantum numbers for higher skyrmion order kas well as for decreas- ing magnetic field Bmay be qualitatively explained by an increase in the skyrmion size. Modes with a given value of m indicate a total of |m|modulation periods along the perimeter of the skyrmion; for larger skyrmion sizes this corresponds toa modulation on a longer length scale, which has a smallercost in exchange energy. The breathing modes of the 3 πskyrmion with different numbers of nodes are visualized in Fig. 2atB=1T . T h e results shown in Fig. 2are obtained from the micromagnetic model in Sec. II B, which is in good quantitative agreement with the atomistic calculations at the given field. All theeigenmodes display three peaks of various heights, while theydecay exponentially outside the 3 πskyrmion. As can be seen in Fig. 2, the peaks are localized roughly around the regions 224426-5LEVENTE RÓZSA et al. PHYSICAL REVIEW B 98, 224426 (2018) where the spins are lying in-plane, indicated by the domain walls (DW) between pairs of dashed lines. The widths ofthe domain walls were determined by approximating the 3 π skyrmion profile with linear functions close to the inflectionpoints r j,/Theta10,j,j=1,2,3 where the spins are lying in-plane, and calculating where these linear functions intersect integermultiples of πin/Theta1 0. Thus, the domain walls are located be- tween the inner Rin,j=rj+[∂r/Theta10(rj)]−1[(4−j)π−/Theta10,j] and outer Rout,j=rj+[∂r/Theta10(rj)]−1[(3−j)π−/Theta10,j] radii. Such a description was used to calculate the skyrmion radiusin, e.g., Ref. [ 46], and it was also applied for calculating the widths of planar domain walls [ 50]. The nodes of the eigenmodes are located roughly be- tween these domain walls, meaning that typically excita-tion modes with n=0,...,k −1 nodes may be observed inkπskyrmions, in agreement with the results in Fig. 1. A higher number of nodes would require splitting a singlepeak into multiple peaks, the energy cost of which gen-erally exceeds the FMR frequency, thereby making thesemodes unobservable. The sign changes in the ˜S x n,meigen- vectors mean that the different modes can be imagined asthe domain walls breathing in the same phase or in opposite phase, as can be seen in Supplemental Videos 5–7 [ 49]. Note that eigenmodes with higher nquantum numbers may also be observed for skyrmions confined in nanodots [ 14–16] where the peaks of the eigenmodes may also be localized atthe edge of the sample, in contrast to the present case whereisolated kπskyrmions are discussed on an infinite collinear background. It is also worth noting that the lowest-lying nonzero gy- ration mode is n=0,m=1f o rt h e1 πand 3πskyrmions, while it is n=1,m=1f o rt h e2 πskyrmion, see Fig. 1.A s already mentioned in Sec. II C, numerical calculations for the 2πskyrmion indicate both in the atomistic and the micro- magnetic case that by increasing the system size or refiningthe discretization the eigenvectors of both the n=0,m=−1 and the n=0,m=1 modes of D SWin Eq. ( 6) converge to the same eigenvectors in Eqs. ( 12) and ( 13) and zero eigenvalue, which correspond to the translational Goldstone mode in theinfinite system. This difference can probably be attributedto the deviation in the value of the topological charge, be-ing finite for 1 πand 3πskyrmions but zero for the 2 π skyrmion [ 35]. B. Instabilities Skyrmions with different order kdeviate in their low- field behavior. Since the considered Pd/Fe/Ir(111) systemhas a spin spiral ground state [ 38], decreasing the mag- netic field value will make the formation of domain wallsenergetically preferable in the system. In the case of the1πskyrmion this means that the lowest-lying eigenmode of H SWin Eq. ( 2), which is an elliptic mode with m=±2, changes sign from positive to negative, occurring betweenB=0.650 T and B=0.625 T in the present system. This is indicated in Fig. 1(a) by the fact that the frequency of then=0,m=−2 eigenmode of D SWin Eq. ( 6) converges to zero. This leads to an elongation of the skyrmion into aspin spiral segment which gradually fills the ferromagneticbackground, a so-called strip-out or elliptic instability already4.45 4.46 4.47 4.48 4.49 4.50020406080100 FIG. 3. Frequency of the breathing mode n=0,m=0o ft h e 1πskyrmion close to the collapse field. Calculation data are shown by open symbols, red line denotes the power-law fit f0,0= Af(Bc,1π−B)βf. discussed in previous publications [ 21,46]. In contrast, for the 2πand 3πskyrmions the lowest-lying eigenmode of HSWis a breathing mode with m=0, which tends to zero between B=0.800 T and B=0.775 T for both skyrmions. This is indicated by the lowest-lying n=0,m=0 mode of DSWin Fig. 1(b) for the 2 πskyrmion, which is the second lowest after the n=0,m=1 mode for the 3 πskyrmion in Fig. 1(c). This means that the radius of the outer two rings of 2 π and 3πskyrmions diverges at a finite field value, effectively decreasing the skyrmion order kby 2 and leading to a burst instability. A similar type of instability was already shownto occur in Ref. [ 25] in the case of a ferromagnetic ground state at negative field values, in which case it also affects 1 π skyrmions. At the burst instability, modes with n=0 and all angular momentum quantum numbers mappear to approach zero be- cause of the drastic increase in skyrmion radius decreasing thefrequency of these modes as discussed in Sec. III A . A similar effect was observed for the 1 πskyrmion in Ref. [ 22] when the critical value of the Dzyaloshinsky-Moriya interaction,|D c|=4 π√A|K|, was approached at zero external field from the direction of the ferromagnetic ground state. In contrast,the elliptic instability only seems to affect the n=0,m= −2 mode, while other mvalues and the nonreciprocity are apparently weakly influenced. In the atomistic model, skyrmions collapse when their characteristic size becomes comparable to the lattice con-stant. The collapse affects only the innermost rings of kπ skyrmions, thereby decreasing the skyrmion order kby 1. It was investigated in Ref. [ 35] that for the 1 π,2π, and 3 π skyrmions the collapse of the innermost ring occurs at B c,1π≈ 4.495 T, Bc,2π≈1.175 T, and Bc,3π≈1.155 T, respectively. As can be seen in Figs. 1(b),1(c), and 3, this instability is again signaled by the n=0,m=0 eigenfrequency going to zero, but in contrast to the burst instability, the otherexcitation frequencies keep increasing with the field in thisregime. Figure 3demonstrates that close to the collapse field the excitation frequency may be well approximated by the 224426-6LOCALIZED SPIN WA VES IN ISOLATED kπSKYRMIONS PHYSICAL REVIEW B 98, 224426 (2018) 0.7 0.8 0.9 1.0 1.1 1.21.01.52.02.5 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15125102050100 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15125102050100 FIG. 4. Effective damping parameters calculated according to Eq. ( 15) for the eigenmodes of the (a) 1 π,( b )2 π,a n d( c )3 π skyrmions, plotted on a logarithmic scale. The corresponding excita-tion frequencies are shown in Fig. 1. power law f0,0=Af(Bc,1π−B)βf, with Af=175.6GHz Tβf, Bc,1π=4.4957 T, and βf=0.23. C. Effective damping parameters in the limit of low α The effective damping parameters αn,m, effwere first cal- culated from the eigenvectors obtained at α=0 following4.45 4.46 4.47 4.48 4.49 4.50024681012 FIG. 5. Effective damping parameter α0,0,effof the breathing mode n=0,m=0o ft h e1 πskyrmion close to the collapse field. The corresponding excitation frequencies are shown in Fig. 3. Calcu- lation data are shown by open symbols, red line denotes the power- law fit α0,0,eff/α=Aα(Bc,1π−B)−βα. Eq. ( 15). The results for the 1 π,2π, and 3 πskyrmions are summarized in Fig. 4.A sd i s c u s s e di nR e f .[ 23], theαn,m, eff values are always larger than the Gilbert damping α, and they tend to decrease with increasing angular momentum quantumnumber |m|and magnetic field B. The spin wave possessing the highest effective damping is the n=0,m=0 breathing mode both for the 1 πand 2πskyrmion, but it is the n= 0,m=1 gyration mode for the 3 πskyrmion for a large part of the external field range where the structure is stable. Exci-tation pairs with quantum numbers n,±mtend to decay with similar α n,m, effvalues to each other, with αn,|m|,eff<αn,−|m|,eff, where clockwise modes ( m< 0) have lower frequencies and higher effective damping due to the nonreciprocity. The effective damping parameters drastically increase and for the lowest-lying modes apparently diverge close to theburst instability, while no such sign of nonanalytical behaviorcan be observed in the case of the 1 πskyrmion with the elliptic instability. For the same n,m mode, the effective damping parameter tends to increase with skyrmion order k away from the critical field regimes; for example, for the n= 0,m=0 mode at B=1.00 T one finds α 0,0,eff,1π/α=2.04, α0,0,eff,2π/α=5.87, and α0,0,eff,3π/α=10.09. Close to the collapse field, the effective damping param- eter of the n=0,m=0 breathing mode tends to diverge as shown in Figs. 4(b),4(c), and 5for the 2 π,3π, and 1πskyrmions, respectively. Similarly to the eigenfrequency converging to zero in Fig. 3, the critical behavior of the effective damping may be approximated by a power-law fitα 0,0,eff/α=Aα(Bc,1π−B)−βαas shown in Fig. 5, this time with a negative exponent due to the divergence. The fittingyields the parameters A α=0.96 Tβα,Bc,1π=4.4957 T, and βα=0.23. Naturally, the critical field values agree between the two fits, but interestingly one also finds βf=βαup to two digits precision. Rearranging Eq. ( 14) yields α0,0,eff αReω0,0=1 α|Imω0,0|, (18) 224426-7LEVENTE RÓZSA et al. PHYSICAL REVIEW B 98, 224426 (2018) 0.0 0.2 0.4 0.6 0.8 1.00102030405060 0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.6 FIG. 6. (a) Frequency f0,0=Reω0,0/2πand (b) inverse lifetime |Imω0,0|of the n=0,m=0 breathing mode of the 1 πskyrmion atB=1 T as a function of the Gilbert damping parameter α.T h e solutions of Eq. ( 6) for the elliptically polarized eigenmode of the 1 π skyrmion are compared to Eqs. ( 16)a n d( 17) which are only valid for circularly polarized modes. where the left-hand side is proportional to ( Bc,1π−B)βf−βα which is approximately constant due to the exponents can- celing. This indicates that while Re ω0,0diverges close to the collapse field, |Imω0,0|/αremains almost constant at low α values. D. Damping for higher αvalues Due to the divergences of the effective damping param- eters found at the burst instability and collapse fields, it isworthwhile to investigate the consequences of using a finiteαvalue in Eq. ( 6), in contrast to relying on Eq. ( 15) which is determined from the eigenvectors at α=0. The αdependence of the real and imaginary parts of the ω 0,0breathing mode frequency of the 1 πskyrmion is displayed in Fig. 6,a ta field value of B=1 T far from the elliptic and collapse instabilities. As shown in Fig. 6(a), unlike circularly polarized modes described by Eq. ( 16) where Re ωqdecreases smoothly and equals half of the undamped value at α=1, the Re ω0,0 value for the elliptically polarized eigenmode displays a much0.85 0.90 0.95 1.00 1.05 1.10 1.1505101520 0.85 0.90 0.95 1.00 1.05 1.10 1.150.000.020.040.060.080.10 FIG. 7. (a) Frequency f0,0=Reω0,0/2πand (b) inverse lifetime |Imω0,0|of the n=0,m=0 breathing mode of the 2 πskyrmion at α=0.1 as a function of the external magnetic field B. The solutions of Eq. ( 6) for the elliptically polarized eigenmode of the 2 πskyrmion are compared to Eqs. ( 16)a n d( 17) which are only valid for circularly polarized modes. faster decay and reaches exactly zero at around α≈0.58. According to Eq. ( 14), this indicates that the corresponding effective damping parameter α0,0,effdiverges at this point. Since the real part of the frequency disappears, the ωq/prime= −ω∗ qrelation connecting Re ωq>0 and Re ωq/prime<0 solutions of Eq. ( 6) discussed in Sec. II D no longer holds, and two different purely imaginary eigenfrequencies are found in thisregime as shown in Fig. 6(b). This is analogous to overdamp- ing in a classical linear harmonic oscillator, meaning that thepurely precessional first-order differential equation describingcircularly polarized modes is transformed into two coupledfirst-order differential equations [ 23] with an effective mass term for the breathing mode of kπskyrmions. This implies that when performing spin dynamics simulations based onthe Landau-Lifshitz-Gilbert equation, the value of the Gilbertdamping parameter has to be chosen carefully if the fastestrelaxation to the equilibrium spin structure is required. Thehigh effective damping of the breathing mode in the α/lessmuch1 limit [cf. Fig. 4(a)] ensures that the inverse lifetime of the elliptically polarized excitations remains larger for a widerange of αvalues in Fig. 6(b) than what would be expected 224426-8LOCALIZED SPIN WA VES IN ISOLATED kπSKYRMIONS PHYSICAL REVIEW B 98, 224426 (2018) for circularly polarized modes based on Eq. ( 17). Note that contrary to Sec. III B,R eω0,0becoming zero in Fig. 6(a) does not indicate an instability of the system, since stabilityis determined by the eigenvalues of the matrix H SWin Eq. ( 2) which are independent of α. Since the disappearance of Re ω0,0and the bifurcation of |Imω0,0|occurs as the excitation frequency becomes smaller, it is expected that such an effect may also be observed atafi x e d αvalue as the external field is decreased. This is illustrated for the n=0,m=0 breathing mode of the 2 π skyrmion in Fig. 7atα=0.1. For this intermediate value of the damping, the breathing mode becomes overdampedaround B=0.875 T, which is significantly higher than the burst instability between B=0.775 T and B=0.800 T [cf. Fig. 1(b) and the circularly polarized approximation in Fig. 7(a)]. This means that the lowest-lying breathing mode of the 2 πskyrmion cannot be excited below this external field value. In Fig. 7(b) it can be observed that contrary to the cir- cularly polarized approximation Eq. ( 17) following the field dependence of the frequency, for the actual elliptically polar-ized eigenmode |Imω 0,0|is almost constant for all field values above the bifurcation point. Although a similar observationwas made at the end of Sec. III C as the system approached the collapse field at α=0, it is to be emphasized again that no instability occurs where Re ω 0,0disappears in Fig. 7(a). IV . CONCLUSION In summary, the localized spin wave modes of kπ skyrmions were investigated in an atomistic spin model,with parameters based on the Pd/Fe/Ir(111) system. It wasfound that the number of observable modes increases withskyrmion order k, firstly because of excitations with higher angular momentum quantum numbers mforming along the larger perimeter of the skyrmion, secondly because of nodesappearing between the multiple domain walls. It was foundthat the 2 πand 3πskyrmions undergo a burst instability at low fields, in contrast to the elliptic instability of the 1 π skyrmion. At high field values the innermost ring of thestructure collapses in all cases, connected to an instability ofa breathing mode. The effective damping parameters of the excitation modes were determined, and it was found that for the same n,m mode they tend to increase with skyrmion order k.T h e effective damping parameter of the n=0,m=0 breathing mode diverges at the burst and collapse instabilities, but nosuch effect was observed in case of the elliptic instability. Forhigher values of the Gilbert damping parameter αa deviation from the behavior of circularly polarized modes has beenfound, with the breathing modes becoming overdamped. Itwas demonstrated that such an overdamping may be observ-able in 2 πand 3πskyrmions for intermediate values of the damping significantly above the burst instability field wherethe structures themselves disappear from the system. The results presented here are expected to hold qualita- tively for all systems where kπskyrmions may be stabilized, as long as the ground state is a spin spiral in the absenceof an external magnetic field. Therefore, they may motivatefurther experimental and theoretical studies on kπskyrmions, offering a wider selection of localized excitations comparedto the 1 πskyrmion, thereby opening further possibilities in magnonics applications. ACKNOWLEDGMENTS The authors would like to thank A. Siemens for fruitful dis- cussions. Financial support for this work from the Alexandervon Humboldt Foundation, from the Deutsche Forschungs-gemeinschaft via SFB 668, and from the National Research,Development and Innovation Office of Hungary under ProjectNo. K115575 is gratefully acknowledged. [1] A. N. Bogdanov and D. A. Yablonski ˘ı, Sov. Phys. JETP 68, 101 (1989). [2] A. Fert, V . Cros, and J. Sampaio, Nat. Nanotechnol. 8,152 (2013 ). [3] W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. B. 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PhysRevB.86.134433.pdf

PHYSICAL REVIEW B 86, 134433 (2012) Dynamics of vortex nucleation in nanomagnets with broken symmetry Jaroslav T ´obik,1,*Vladim ´ır Cambel,1and Goran Karapetrov1,2 1Institute of Electrical Engineering, Slovak Academy of Sciences, D ´ubravsk ´a cesta 9, SK-841 04 Bratislava, Slovakia 2Department of Physics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104, USA (Received 19 March 2012; revised manuscript received 14 October 2012; published 31 October 2012) We investigate the dynamics of magnetic vortex nucleation in sub-100-nm mesoscopic magnets with the aim of establishing an independent control of vortex polarity and chirality. We consider the dynamic behavior of thevortex spin structure in an object with broken symmetry—a Pacman-like nanomagnet shape—proposing a modelbased on classical electrodynamics and providing a proof by conducting micromagnetic calculations. The modelprovides evidence that the desired vortex chirality and polarity could be established by applying solely quasistaticin-plane magnetic field along specific directions with respect to the structure’s asymmetry. We identify the modesof vortex nucleation that are robust against external magnetic field noise. These vortex nucleation modes arecommon among a wide range of sub-100-nm magnets with broken rotational symmetry. The results could leadto the practical realization of high density magnetic memories based on magnetic vortices. DOI: 10.1103/PhysRevB.86.134433 PACS number(s): 75 .75.Fk, 75 .60.Jk, 75.78.Cd Confinement leads to fundamental changes in the physical behavior of materials due to the increased role of the surface.In mesoscopic magnetic materials such changes in the energylandscape could lead to novel magnetic spin configurationssuch as vortices. The equilibrium properties of these topo-logical states are governed by both the properties of themagnetic material and the geometry of the object. On theother hand, confinement also leads to the distinct dynamicbehavior of these topological states since the available energylevels are very much limited. The transition probabilitiesbetween different states can thus be controlled by carefulengineering of the geometry of the mesoscopic object. Herewe show that by tailoring the geometry of the mesoscopicmagnet one can produce deterministic dynamic switchingbetween well-defined degenerate topological states using onlyin-plane magnetic fields. We present an analytical modelthat explains the mechanism of the vortex nucleation andorigin of robustness of the vortex polarization. Confirmationof the model is accomplished by conducting micromagneticsimulations. The findings could lead to practical realization ofa two-bit magnetic memory cell based on a controlled settingand the readout of the polarity and chirality of the magneticvortex. Controlled manipulation of magnetic domains in ferro- magnet nanostructures have recently opened opportunitiesfor novel fast, high-density, and low-power memories withnovel architectures. 1–3Any perspective magnetic memory architecture, such as submicron nonvolatile magnetic memory,has to have (1) a well-defined switching field used to set thememory bits, and (2) a reproducible switching behavior usinga simple sequence of external magnetic field pulses. Therefore,the dynamics of the switching between different ground stateshas to be understood in detail. Recent advances in fabrication technology at nanoscale have enabled studies of magnetic systems that are welldefined in all three dimensions on a nanometer length scale(<100 nm). 4,5The size reduction of nanomagnets leads to novel spin topological states such as the vortex state, C state, Sstate, flower state, and so on,6and to simplified transitions between these states in the external magnetic field.The transition between ground states in such nanomagnets is of fundamental importance.7,8It is governed by competition between the magnetostatic energy and exchange energy, and itis influenced by the magnetic material used and by the choiceof the nanomagnet shape. For example, the magnetization ofdisks in a zero field can be oriented in-plane, out-of-plane, or avortex state can be created depending on the disk diameter andthickness. 4,9In the disk the flux-closure magnetic state reduces the long-range stray fields (i.e., reduces the magnetostaticinteraction between neighboring disks). Therefore, such diskmagnetic systems have a potential for high-density magneticstorage elements, with bits represented by the chirality andpolarity of a basic vortex state. In submicron-sized disks four possible states have to be con- trolled. It was shown experimentally that in the nanomagnetswith broken rotational symmetry, chirality can be controlledeasily by the in-plane field of a selected direction. 10–13At the same time the polarity of the vortex core, which representsthe second bit, can be controlled by an out-of-plane magneticfield, 13spin-polarized current,8high-frequency in-plane mag- netic field,14or by an in-plane magnetic pulse of precisely defined amplitude and duration.15 In our previous work we have proposed a prospective shape of a nanomagnet with broken symmetry which permits thecontrol of chirality and polarity bits by the application of thein-plane field only. 16In this paper we analyze the mechanisms that establish specific chirality and polarity values in such aPacman-like (PL) nanomagnet by taking a closer look at theenergies and dynamics that govern the switching processes.Using an analytical model we show that the polarity and the chirality of the vortex core nucleated in the decreasing in-planemagnetic field is implicitly defined by the direction of themagnetic field with respect to the missing sector of the PLnanomagnet. We consider a magnetic dot of cylindrical shape. To construct a PL structure it is necessary to remove an outersector that is 45 ◦wide, and has 1 /3 of the disk radius (see Fig.1). Due to a symmetry analysis that will be presented later, we choose the orientation of the x,zaxes such that they define the mirror symmetry plane σy, which leaves the PL object 134433-1 1098-0121/2012/86(13)/134433(5) ©2012 American Physical SocietyT´OBIK, CAMBEL, AND KARAPETROV PHYSICAL REVIEW B 86, 134433 (2012) σyR R cutxy FIG. 1. Geometry of Pacman-like nanomagnet. The structure is symmetric with respect to reflection plane σy. invariant. Another symmetry operation is mirroring through the plane x,ynoted in the following text by σz. First, we define polarity /vectorπ[/vectorf] and chirality /vectorχ[/vectorf] vectors as functionals of an arbitrary vector field /vectorf: /vectorπ[/vectorf]=1 /Omega1/integraldisplay /vectorf(/vectorr)d/Omega1, (1) /vectorχ[/vectorf]=/integraldisplay /vectorr×(/vectorf(/vectorr)−/vectorπ)d/Omega1. Polarity is just th simple average value of the field, while chirality resembles the definition of the momentum of quantity /vectorfin classical mechanics. The subtraction of polarity in the expression for chirality is necessary due to chirality invariancewith respect to the origin coordinate system choice. Mostlywe are interested in the zcomponent of polarity and chirality. The integration domain /Omega1is over the volume of the PL nanomagnet. Let us consider that the nanomagnet is placed in a strong in-plane magnetic field that has an angle ϕwith the xaxis. To emulate the magnetic response by the missing sector, wecan consider the PL nanomagnet as a superposition of a fulldisk and a set of microscopic magnetic moments in a removedsector. These additional moments have to have the same valueand to be oriented in the opposite direction to the magnetic Hcut mcutBext xϕ FIG. 2. (Color online) (a) The magnetization of the Pacman-like nanomagnet is a superposition of the uniform magnetization of a full disk (red arrows) and the magnetization of the missing sector that is equal and opposite to the one of the full disk. (b) The sumof the compensation moments in the sector creates a dipole m cut which asymmetrically interacts with the local magnetization in the nanomagnet.moments in the disk (Fig. 2). In the first approximation all microscopic moments are parallel. Neglecting higher thandipolar moments, the missing sector behaves as a dipole with amoment /vectorm cutpositioned in the center of mass of the sector /vectorrT: /vectormcut=−/integraldisplay /vectorMd/Omega1/prime/vectorrT=1 /Omega1/prime/integraldisplay /vectorrd/Omega1/prime. (2) The minus sign reflects that dipoles of opposite orientation have to be added to eliminate the dipoles in the missing sector(see Fig. 2). The integration is over the volume of the missing sector /Omega1 /prime. To calculate the magnetic field of the PL nanomagnet, consider first the full disk. In magnetic fields exceeding the saturation field the magnetic polarization /vectorMis parallel to the direction of the applied field /vectorHextthroughout the disk volume. The internal magnetic field /vectorHis also uniformly oriented in parallel with /vectorM. As a small piece of material is removed, all the fields change slightly. To correct the internal magneticfield, the field of magnetic moment /vectorm cutgiven by Eq. (2)has to be added to the originally homogeneous internal field of thefull disk. The removed part thus creates a dipole which inducesmagnetic field with nonzero chirality χ[/vectorH], ifϕ/negationslash=0 ◦,180◦. Since dipoles partially follow the field orientation, the nonzero chirality of /vectorMis expected. Next, we explain qualitatively the mechanism which determines vortex core polarity. Taking into consideration magnetostatics only, the state with positive polarity π[/vectorM] is energetically equivalent to the state with negative polarity −π[/vectorM], if no external field in the zdirection is applied. That can be easily seen by writing the total energy functional E=μ0 4π/integraldisplay/integraldisplay/parenleftBigg/vectorM·/vectorM/prime |/vectorr−/vectorr/prime|3−3/vectorM·(/vectorr−/vectorr/prime)/vectorM/prime·(/vectorr−/vectorr/prime) |/vectorr−/vectorr/prime|5/parenrightBigg d/Omega1d/Omega1/prime +A/integraldisplay (∇/vectorM)2d/Omega1−/integraldisplay /vectorHext·/vectorMd/Omega1. (3) Changing the sign of Mzdoes not change the first two integrals of Eq. (3)since these two parts of the energy functional are quadratic in Mzand its derivative, respectively. The only linear part in Mzis in the third integral. But a change of sign of Mzwould not influence the total energy because Hexthas nozcomponent. The integration domain /Omega1is not changed by reversing zbecause the symmetry operation σz—the reflection from the plane xytransforms the PL nanomagnet to itself. The final state of the vortex core polarization is determined by magnetization dynamics. The time evolution of magneti-zation is described by the phenomenological Landau-Lifshitz-Gilbert (LLG) equation ∂/vectorM ∂t=−γ/vectorM×/vectorHeff+α/parenleftbigg /vectorM×∂/vectorM ∂t/parenrightbigg . (4) Here we show that this equation itself contains a polarity symmetry-breaking mechanism. First, we apply a strongexternal in-plane field in the direction that has an angle ϕwith thexaxis (Fig. 2). Then we slowly (adiabatically) decrease the external field amplitude to the level that is just above the vortexnucleation field. By adiabatic field change we mean that thechange of the external field with time is so slow that the energydissipation keeps the system very close to the local minimum at 134433-2DYNAMICS OF VORTEX NUCLEATION IN NANOMAGNETS ... PHYSICAL REVIEW B 86, 134433 (2012) all times. Then∂/vectorM ∂t≈0 everywhere. Any dynamics means also the dissipation of energy due to the term that is proportional to αin Eq. (4). Therefore, at local minima also /vectorHeffis parallel to magnetization /vectorM, otherwise it is not possible to satisfy Eq. (4) with vanishing left side. We note this effective field /vectorH/bardbl eff.N o w , having /vectorHextjust above the nucleation field, we decrease the external field by a small value /Delta1/vectorH. To the first approximation the effective field is /vectorHeff=/vectorH/bardbl eff+/Delta1/vectorH. When looking at the dynamics of local magnetic moments shortly after the externalfield is decreased, we can neglect the damping term in theLLG equation since α/lessmuch1. The torque on the zcomponent of magnetization is then given by the equation ∂M z ∂t=−γ(Mx/Delta1Hy−My/Delta1Hx). (5) The right-hand side of Eq. (5)is not zero locally, nor it is on average, due to the asymmetry of the PL nanomagnet withrespect to the direction of the applied field (if ϕ/negationslash=0 ◦,180◦). If the actual value of the external field is lower than the nucleationfield, the nonzero polarity of the nanomagnet starts to evolve.From pure energy considerations, the two vortex states withopposite polarity are energetically equivalent. Looking at thetime-evolution equation (4), especially at its first-order approx- imation (5), one sees that the magnetization is driven towards the well-defined direction by the geometric asymmetry of thePL nanomagnet. The direction of the initial polarity evolutionis obtained by the volume integration of Eq. (5). The confirmation of the above model approximation can be obtained by numerical simulations. We have performednumerical simulations of the PL nanomagnet using the OOMMF software package.17The parameters used in the simulation are as follows: outer radius R=35 nm, thickness (in the z direction, not shown) h=40 nm. The material used in the calculations is Permalloy Ni 80Fe20(Py), with the following material parameters: exchange constant A=13×10−12J/m, saturated magnetization Ms=8.6×105A/m, and Gilbert damping parameter α=0.5. Numerical simulations were done on a rectangular mesh of size 1 nm. To check thepossible influence of the discretization, calculations with aPL nanomagnet rotated against the discretization mesh by 10 ◦ and 22.5◦were performed. The same results for Binwithin an error of 2 mT were achieved. In Fig. 3we show the dependence of the nucleation-field amplitude on the applied field direction. The nucleation fieldis defined as the applied magnetic field at which the nonzero zcomponent of the magnetization polarity π(/vectorM) appears. The external field is adiabatically decreasing from 150 mTto zero in the selected direction. By adiabatic change we meanthe repeated process of decreasing the field b ya2m Ts t e p , followed by full system relaxation. We would like to note the symmetry properties of the PL nanomagnet. In Fig. 3(top) each quadrant corresponds to a vortex ground state of the nanomagnet with the specificchirality and polarity shown in corresponding corners. Thenanomagnet relaxes into that remanent state from a uniformmagnetization along an angle within the specific quadrant.As can be seen in Fig. 3, the angular dependence of the nucleation field can be reconstructed from the dependenceforϕ∈[0; π 2] by inversion and reflection through the xzplane0306090 120 150 180 210 240 27030033060 30 mT ϕ=75°B= 1T B=26mT B=24mTϕ=15°B=1T B=20mT B =0T B=22mT B=0TU US CV VI I FIG. 3. (Color online) Top: Angular dependence of vortex nucle- ation field. Direction of the nucleated vortex polarity and chirality are also indicated for each quadrant. Bottom: Two different processesof vortex nucleation depending on the initial magnetization direction with respect to PL’s symmetry plane ( ϕ=15 ◦andϕ=75◦). From the uniform magnetization state ( U) the magnetization transitions to S-shape (dots) or C-shape (stars) configurations and equilibrates to a vortex state ( V) with specific polarity and chirality. The snapshot of the intermediate state ( I) shows the position of vortex core nucleation. σy. The inversion symmetry of the graph shown in Fig. 3is the consequence of the time-reversal symmetry. The reflectionsymmetry with respect to the xzplane shown in Fig. 3is related to the reflection from σ y—the geometrical operation that transforms the PL nanomagnet to itself. The simulation results show the existence of two distinct vortex core nucleation regimes [Fig. 3(bottom)]. For large angles ( ϕ∈[50◦;9 0◦]) the vortex nucleates from the C-state magnetization pattern. This form of nucleation is not robustin the sense that even the small out-of-plane field B z/similarequal1m T is sufficient to alter the resulting polarity of the nucleatedvortex along the direction of the applied field B z(Fig. 4). Instead, for small angles ( ϕ∈[0◦;4 8◦]) the vortex nucleation path is different. Just above the vortex nucleation field, themagnetization of the PL nanomagnet forms an Sstate. This configuration consists of two regions with opposite signs ofthe curvature of field lines. 18Meanwhile, there is only one curvature of field lines just below the vortex nucleation field. 134433-3T´OBIK, CAMBEL, AND KARAPETROV PHYSICAL REVIEW B 86, 134433 (2012) 0 30 60 90 120 150 1800102030405060B ( mT ) Polar angle ( deg ) BZ Bnuc FIG. 4. (Color online) Angular dependence of the in-plane vortex nucleation field Bnucand threshold out-of-plane field Bznecessary to reverse the polarity of the entering vortex. The process of vortex core nucleation in this case involves thereversal of magnetic moments in a part of the nanomagnet. This reversal process proceeds through an out-of-plane motionof the local magnetic moments, resulting in robust vortex corepolarization despite the presence of small external fields in thezdirection. In Fig. 4we show the angular dependence of the maximum external field B zfor which the PL nanomagnet is able to sustain nucleation of vortex polarity opposite to thedirection of the applied external field. TheCandSshapes of magnetization can be explained by the position of perturbing dipole /vectorm cut. As can be seen from Fig. 2, the PL nanomagnet is divided into two domains with the opposite sign of the magnetic field circulation generatedby/vectorm cut. The sizes of these two domains are determined by the orientation of /vectormcut. However, the exchange interaction tends to align local moments in parallel, thus there exists a criticalangle (around 48 ◦in our geometry), beyond which the region with minor curvature does not exist. A detailed energy balancebetween the exchange and cavity (sector) demagnetizationdetermines the scenario of vortex nucleation and its eventualrobustness with respect to external perturbation. We would like to point out that the adiabatic case discussed above describes the magnetization dynamics on the order ofseveral nanoseconds. We also performed OOMMF simulations for the nonadiabatic case, when the change of the externalmagnetic field is faster than the local magnetization dynamics.We find that the final vortex state has the same angulardependence on the initial applied in-plane magnetic field asin the adiabatic case (Fig. 4). Moreover, the robustness of the transition is increased, as measured by the magnitude of thethreshold out-of-plane B znecessary to alter the final polarity. An intuitive understanding for the increased robustness can beinferred from Eq. (5): The initial impulse given to the system is proportional to the amplitude of the in-plane magnetic fieldstep during which vortex nucleation occurs. This basicallymeans that the PL nanomagnet is more stable when operatedat higher switching speeds. The analysis provided above has been performed without taking into account finite temperature effects. Preliminaryestimates indicate that the results could be altered by thermaleffects when the thermal energy is comparable to the energycushion estimated by the opposite B znecessary to alter the final polarity of the vortex state (Fig. 4). According to simulations, at room temperature the final polarity and chirality for appliedfields at angles −45 ◦/lessorequalslantϕ/lessorequalslant45◦do not change. A more complicated situation arises for other angles of the in-planeapplied field, but the full analysis is beyond the scope of thiswork and will be addressed in the near future. To summarize, in this work we provide simple arguments that elucidate the origin of driving mechanisms for thenucleation of magnetic vortex with controlled chirality andpolarity. We also find the regime of the PL nanomagnetoperation in which the final vortex state is independent ona weak disturbing external field. This is a promising findingto consider if using the PL nanomagnet as a memory elementin bit-patterned media or as a generator of magnetic vorticesof desired polarity and chirality for microwave applications.Weak interaction among the elements as well as robustness tosmall external field perturbations makes the PL nanomagnetvery suitable for operation. Finally, we note, that the sub-100-nm PL nanomagnet is not the only unique design offering control of chiralityand polarity by in-plane magnetic field. Qualitatively similarresults are obtained in simulations for different sizes andshapes of the missing sector. According to our model, thenecessary ingredients are the symmetry of the object, thedemagnetization field strength of the removed part, and shapeanisotropy induced by the removed part. The robustness ofvortex polarity against B zis based on the vortex–antivortex annihilation during vortex core nucleation. This work has been supported by the project CENTE II, Research & Development Operational Program funded by theERDF, ITMS 26240120019, and by VEGA 2/0037/12. *jaroslav.tobik@savba.sk 1S. S. Parkin, H. Hayashi, and L. Thomas, Science 320, 190 (2008). 2K. S. Buchanan, P. E. Roy, M. Grimsditch, F. Y . Fradin, K. Yu.Guslienko, S. D. Bader, and V . Novosad, Nat. Phys. 1, 172 (2005). 3C. A. Ross, H. I. Smith, T. Savas, M. Schattenburg, M. Farhoud, M. Hwang, M. Walsh, M. C. Abraham, and R. J. Ram, J. Vac. Sci. Technol. B 17, 3168 (1999).4S. H. Chung, R. D. McMichael, D. T. Pierce, and J. Unguris, Phys. Rev. B 81, 024410 (2010). 5J. W. Lau and J. M. Shaw, J. Phys. D 44, 303001 (2011). 6R. P. Cowburn, J. Phys. D 33, 1R (2000). 7W. F. Brown, J. Appl. Phys. 39, 993 (1968); K. Yu. Guslienko, K. S. Lee, and S. K. Kim, P h y s .R e v .L e t t . 100, 027203 (2008); M. Bolte, G. Meier, B. Kr ¨uger, A. Drews, R. Eiselt, L. Bocklage, S. Bohlens, 134433-4DYNAMICS OF VORTEX NUCLEATION IN NANOMAGNETS ... PHYSICAL REVIEW B 86, 134433 (2012) T. Tyliszczak, A. Vansteenkiste, B. Van Waeyenberge, K. W. Chou, A. Puzic, and H. Stoll, ibid. 100, 176601 (2008). 8K. Yamada, S. Kasai, Y . Nakatani, K. Kobayashi, H. Kohno, A. Thiaville, and T. Ono, Nat. Mater. 6, 269 (2007). 9J. d’Albuquerque e Castro, D. Altbir, J. C. Retamal, and P. Vargas,P h y s .R e v .L e t t . 88, 237202 (2002). 10M. Schneider, H. Hoffmann, and J. Zweck, Appl. Phys. Lett. 79, 3113 (2001). 11T. Taniuchi, M. Oshima, H. Akinaga, and K. Ono, J. Appl. Phys. 97, 10J904 (2005). 12P. Vavassori, R. Bovolenta, V . Metlushko, and B. Ili ´c,J. Appl. Phys. 99, 053902 (2006).13M. Jaafar, R. Yanes, D. Perez de Lara, O. Chubykalo-Fesenko, A. Asenjo, E. M. Gonzalez, J. V . Anguita, M. Vazquez, and J. L.Vicent, P h y s .R e v .B 81, 054439 (2010). 14K. S. Lee, K. Y . Guslienko, J. Y . Lee, and S. K. Kim, Phys. Rev. B 76, 174410 (2007). 15R. Antos and Y . Otani, P h y s .R e v .B 80, 140404 (2009). 16V . Cambel and G. Karapetrov, Phys. Rev. B 84, 014424 (2011). 17M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, Technical Report No. NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD (1999). 18To be precise, we mean the curvature of field lines projected ontothexyplane since the concept of curvature with sign is meaningful for curves in two dimensions. 134433-5

PhysRevB.82.144405.pdf

Relating Gilbert damping and ultrafast laser-induced demagnetization Manfred Fähnle, *Jonas Seib, and Christian Illg Max-Planck-Institut für Metallforschung, Heisenbergstr. 3, 70569 Stuttgart, Germany /H20849Received 2 July 2010; revised manuscript received 3 September 2010; published 4 October 2010 /H20850 Based on the breathing Fermi-surface model of Gilbert damping and on the Elliott-Yafet relation for the spin-relaxation time, a relation is established between the conductivitylike contribution to the Gilbert damping /H9251at low temperatures and the demagnetization time /H9270Mfor ultrafast laser-induced demagnetization at low laser fluences. Thereby it is assumed that, respectively, the same types of spin-dependent electron-scattering pro-cesses are relevant for /H9251and/H9270M. The relation contains information on the properties of single-electron states which are calculated by the ab initio electron theory. The predicted value for /H9251//H9270Mis in good agreement with the experimental value. DOI: 10.1103/PhysRevB.82.144405 PACS number /H20849s/H20850: 75.78.Jp, 76.20. /H11001q I. INTRODUCTION: GILBERT DAMPING AND LASER- INDUCED DEMAGNETIZATION Recently, there has been an extensive research activity on short-time magnetization dynamics for two reasons. First,there is an enormous importance for magnetic devices, and,second, the microscopic mechanisms which determine thedissipative magnetization dynamics are not well understood.The dynamics /H20849e.g., magnetization switching /H20850can be driven by external magnetic fields or by spin-polarized electricalcurrents on a time scale of nanoseconds to several picosec-onds. Since the pioneering paper of Beaurepaire et al. 1it is known that the magnetization can be modulated even on apicosecond or subpicosecond time scale when exposing athin film of Ni, Fe, or Co, e.g., to an optical femtosecondlaser pulse. In the following we denote the dynamics onthese two scales as fast and ultrafast magnetization dynam-ics, respectively. For the theoretical modeling of the fast dynamics of the magnetization Mthe Gilbert equation 2is commonly used dM dt=−/H9253/H20849M/H11003Heff/H20850+1 M/H20873M/H11003/H9251dM dt/H20874. /H208491/H20850 Here the first term /H20849/H9253is the gyromagnetic ratio /H20850describes the precession of Maround the effective field Heff, and the sec- ond term with the damping scalar /H9251represents the damping. It has been shown /H20849for a review, see Ref. 3/H20850that for a general situation the Gilbert scalar /H9251has to be replaced by a damping matrix /H9251, which depends itself on the magnetization configu- ration of the whole system but for the present purpose it suffices to consider Eq. /H208491/H20850. For the modeling of the ultrafast dynamics a phenomenological three-temperature model isused 1which describes the interaction of the electron, spin, and lattice subsystems, or its recently developed microscopicversion /H20849see, e.g., Ref. 4/H20850. The quantity of interest is the demagnetization time /H9270M/H20849which for Ni can be below 100 fs for low laser fluences4/H20850describing the rate of magnetization loss of the film after laser excitation. Both damping of the fast magnetization dynamics, char- acterized by /H9251, and ultrafast demagnetization after laser ex- citation, characterized by /H9270M, require a transfer of angular momentum from the electronic system to the lattice via elec-tronic spin-flip scattering. Assuming that the dominant mi-croscopic channels for the angular momentum transfer are the same for both situations, it is desirable to find a relationbetween /H9251and/H9270M. II. SUMMARY OF A FORMER UNIFIED THEORY A first unified theory of fast and ultrafast magnetization dynamics has been presented by Koopmans et al.5who com- pared the fast precessional dynamics of a homogeneouslymagnetized system in the homogeneous effective field H eff =H/H20849composed of the external field, the anisotropy field, and the demagnetization field /H20850with the ultrafast demagnetization after laser excitation. According to Eq. /H208491/H20850, the precession damps out on the time scale /H9270P=/H6036 g/H9262BH/H9251/H208492/H20850 with the Landé factor g/H110152 and the Bohr magneton /H9262B.I n Ref. 5the physics behind the drastically different time scales /H9270Mand/H9270Pcould be explained by a simple hand-waving ar- gument. To do this, the authors of Ref. 5calculated the trans- versal spin-relaxation time /H9270M,tfor laser excited electrons, thereby describing the damped precessional motion of thesingle electrons again by the Gilbert Eq. /H208491/H20850with the same damping constant /H9251as for the fast precession but with the effective field Heff=Hreplaced by the exchange field Hex, which an individual electron spin feels that is not alignedwith the sea of other electrons. Assuming that the longitudi-nal relaxation time /H9270Mis equal to the transverse relaxation time/H9270M,teven for the extremely fast precession in the Stoner exchange field /H20849preconditions for the validity of this assump- tion are discussed in Ref. 6/H20850yields7,8 /H9270M=/H6036 g/H9262BHex/H9251. /H208493/H20850 From Eqs. /H208492/H20850and /H208493/H20850it becomes obvious that the reason for the drastically different time scales of fast and ultrafast dy-namics is given by the different effective fields. The indi-vidual spins of the ultrafast dynamics feel the exchange field,and their precession is extremely fast. In contrast, for the fastdynamics the Stoner exchange field does not appear explic-itly /H20849it just functions to guarantee the constant modulus ofPHYSICAL REVIEW B 82, 144405 /H208492010 /H20850 1098-0121/2010/82 /H2084914/H20850/144405 /H208494/H20850 ©2010 The American Physical Society 144405-1the magnetization during the precession /H20850, and the precession in the effective field is much slower. In Ref. 5the relation in Eq. /H208493/H20850between /H9270Mand/H9251has been rederived from quantum-mechanical principles. In bothsituations an equation of motion is determined which de-scribes the relaxation of the magnetic moment m/H20849t/H20850of the sample from an initial nonequilibrium situation toward equi-librium. On the microscopic level this relaxation results froman imbalance of spin-up and spin-down electron-lattice scat-tering events. Thereby it is again assumed that the sametypes of spin-flip electron scattering /H20849described by a model matrix element /H20850are relevant for /H9270Mand/H9251. For the femtosec- ond demagnetization the nonequilibrium situation arises be-cause after the action of the laser pulse /H20849which primarily raises the electronic temperature /H20850the electron, spin, and lat- tice temperatures of the above-mentioned three-temperaturemodel are different. The increased temperature of the heatbath for the individual spins /H20849provided by the electronic sub- system /H20850causes a repopulation of the spin-up and spin-down levels /H20849defined with respect to the orientation of the ex- change field H ex/H20850via spin-flip scattering events which change the energy by /H11006g/H9262BHex. For the calculation of /H9251a homogeneously magnetized sample is considered, where theindividual spins are coupled by H exto a macrospin of fixed quantum number Sbut variable magnetic quantum number mS/H20849now defined with respect to the orientation of the effec- tive field H/H20850. The initial nonequilibrium situation where the macrospin is not parallel to His again relaxed by spin scat- tering processes whereby—however—the restriction has tobe fulfilled that Sis conserved while m Scan be changed. Therefore the change in energy due to an individual spin flipis/H11006g /H9262BHrather than /H11006g/H9262BHex. From the equations of motion derived for the respective situation the quantities /H9270M and/H9251can be determined, yielding Eq. /H208493/H20850. It should be recalled that in the quantum-mechanical treat- ment all the electronic properties of the system are describedby just one effective parameter /H20849the spin-flip scattering ma- trix element /H20850. Therefore it cannot be expected that Eq. /H208493/H20850 gives a highly accurate description for all systems, albeit ityielded a correct prediction on the order of magnitude of /H9270M/H9251from a quantum-mechanical treatment. In fact, Eq. /H208493/H20850 could not be confirmed quantitatively when manipulating /H9270M and/H9251by transition-metal or rare-earth doping /H20849see, e.g., Ref. 9/H20850, either because of the oversimplified treatment of the elec- tronic and scattering properties, or because different relax-ation channels are relevant for /H9270Mand/H9251/H20849in contrast to the basic assumption of the calculation /H20850. III. DESCRIPTION OF THE PRESENT UNIFIED THEORY In the present paper we derive a relation between /H9251and /H9270Mby a completely different approach than in Ref. 5. The advantage of the present theory is that it takes into account ina much more detailed manner the specific electronic proper-ties of a material. The disadvantage is that the relation be-tween /H9251and/H9270Mis much more complicated and does not contain just one parameter /H20851Hexin Eq. /H208493/H20850/H20852but the properties of all individual electronic states which have to be calculatedby the ab initio electron theory for a comparison with theexperiment. It will be shown that the value of /H9251//H9270Mpre- dicted by the theory agrees well with the corresponding ex-perimental value. It is well known /H20849see, e.g., Refs. 10–15/H20850that there are often two contributions to /H9251, one which is proportional to the conductivity of the material and which dominates at lowtemperatures, and one which is proportional to the resistivity.We want to derive a relation between the low-temperaturedamping parameter and the demagnetization time /H9270Mafter laser excitation at low temperatures and such low laser flu-ences that the electron, spin, and lattice temperatures riseonly slightly. It has been shown /H20849see, e.g., Ref. 13/H20850that the Gilbert damping in metallic ferromagnets results predominantlyfrom the fact that the magnetization dynamics itself gener-ates pairs of excited electrons and holes which then experi-ence spin-dependent scattering at the lattice, thereby trans-ferring angular momentum from the electronic system to the lattice. We can distinguish between pairs for which the ex-cited electrons and holes appear in the, respectively, sameband, and pairs which are generated by exciting the electronsto other bands than those for which the holes appear. Therelaxation of these two types of electron-hole pairs leads tothe above discussed two contributions to /H9251/H20849see Refs. 14and 15/H20850. The intraband pairs are generated because the spin-orbit energy changes when the orientation e/H20849t/H20850of the, for example, homogeneous magnetization M/H20849t/H20850=Me/H20849t/H20850changes with time t, i.e., the single-electron energies /H9255jk/H20849jandkdenote the band index and the electronic wave vector /H20850change with time. Some states which are just below the Fermi surface forone orientation eget pushed above the Fermi surface for an orientation eat another time whereas other states which were originally above are pushed below. This means that excitedelectrons and holes are generated in the same band when weconsider the respective preceding orientation as reference.This means, that, e.g., for a precessional dynamics of M/H20849t/H20850 the Fermi surface “breathes.” The relaxation of the intrabandelectrons and holes leads to the conductivitylike “breathingFermi surface” contribution 13–16to/H9251. The interband pairs are generated because the system of electrons feels a time-dependent perturbation due to thechanging spin-orbit interaction /H20849see, e.g., Ref. 14/H20850, and this leads to electronic transitions between states /H9023 jkand/H9023j/H11032k. These excitations are pictured as14“bubbling” of individual electrons at the Fermi surface. The relaxation of these inter-band electrons and holes leads 14to the resistivitylike “bub- bling Fermi surface” contribution to /H9251. It has been shown in Ref. 14that the breathing and the bubbling Fermi-surface contributions are incorporated inKamberský’s 12torque correlation model. For the conductivi- tylike contribution itself another type of theory yielded thebreathing Fermi-surface model. 3,16Because we concentrate on the low-temperature damping, we will consider thebreathing Fermi-surface model. In the theory of Ref. 5the fast dynamics of the system is described by a statistical approach on the macrospin level,and the macrospin relaxation is driven by the fact that tran-sitions /H20841S,m S/H20856→/H20841S,mS+1/H20856lower the energy by g/H9262BH. Such transitions are realized by the spin-dependent electron-latticeFÄHNLE, SEIB, AND ILLG PHYSICAL REVIEW B 82, 144405 /H208492010 /H20850 144405-2scattering events which are characterized by the effective scattering matrix element /H20849see above /H20850but the detailed dy- namics on the level of single-electron states does not enterexplicitly the statistical approach for the macrospin. In con-trast, in the breathing Fermi-surface model the system is de-scribed by a statistical approach on the level of single-electron states. As described above, intraband electron-holepairs are generated, for instance, by a precessional dynamicsofM/H20849t/H20850due to a breathing Fermi surface. The electron-hole pairs generated by the precession survive for some lifetime /H9270 before they relax by electron-lattice scattering, thereby trans-ferring angular momentum to the lattice. Because of the fi-nite lifetime the real occupation numbers n jk/H20849t/H20850deviate from the equilibrium Fermi-Dirac occupation numbers fjk/H20851/H9255jk/H20849t/H20850/H20852, and the differences between these two occupation numbersrepresent the driving forces for a statistical treatment of therelaxation on the level of single-electron states. Altogether,the breathing Fermi-surface model yields 17 /H9251=/H9253/H9270 MFel, /H208494/H20850 where the quantity Fel=−/H20858 jk/H11509fjk /H11509/H9255jk/H20873/H11509/H9255jk /H11509e/H208742 /H208495/H20850 contains the derivatives of the single-electron energies with respect to the orientation eof the magnetization M=Me. The quantities /H11509/H9255jk//H11509ecan be calculated in the ab initio electron theory from the single-electron energies /H9255jkcalculated for two close orientations eof the homogeneous magnetization which are stabilized by the action of constraining fields.3 We now describe the calculation of the demagnetization time/H9270Mafter laser excitations at low temperatures and low laser fluences. For low fluences the laser excitation drives thesystem only slightly out of thermal equilibrium. In Ref. 5the quantity /H9270Mhas been calculated within the microscopic three-temperature model described above where the relax-ation is driven by the different temperatures of the electron,spin, and lattice subsystems. For low fluences we can use thetheory of Yafet 6,18in which a weak nonequilibrium situation for the electronic spin states is modeled by prescribing ini-tially two different chemical potentials for electrons with twospin characters, and this difference is the driving force forthe proceeding relaxation which is achieved by spin-dependent electron-lattice scattering. Whereas in Ref. 5the spin-dependent scattering is described from the very begin-ning by one effective matrix element, the theory of Yafetcontains the real matrix elements for the scattering betweendifferent electronic states /H9023 jkand/H9023j/H11032k/H11032. The key point of the theory is the fact that in a system with spin-orbit coupling the wave functions /H9023jkare always mixtures of the two spin states /H20841↑/H20856and /H20841↓/H20856with probability pjksto find an electron in the spin state s. The degree of spin mixing is described by the parameter bjk2= min /H20849pjk↑,pjk↓/H20850, /H208496/H20850 whereby for most states bjk2is much smaller than one, i.e., most states have a dominant spin character. In a simplifiedversion of Yafet’s theory the spin-flip matrix elements are not calculated explicitly but estimated by simple physical argu-ments. Within this simplified version the so-called Elliot-Yafet relation 6,19for/H9270Mis derived /H9270M=1 pb2/H9270c, /H208497/H20850 where b2is an average of bjk2over all states involved in the relaxation, pis a material-specific parameter which should be close to 4 /H20849and which should not be mixed up with the above defined probability pjksto find a single electron in the spin state s/H20850, and the quantity /H9270cis the relaxation time enter- ing Drude’s theory of electrical conductivity. Because in the breathing Fermi-surface model the lifetime /H9270is generally assumed to be identical to the Drude relaxation time/H9270c, we can derive from Eqs. /H208494/H20850and /H208497/H20850the relation /H9270M=M /H9253Felpb2/H9251, /H208498/H20850 which is the central result of the present paper. Please note two fundamental differences between Eq. /H208493/H20850, which is the central result of Ref. 5and Eq. /H208498/H20850. First, in Eq. /H208493/H20850/H9270Mis proportional to 1 //H9251whereas it is proportional to /H9251 in Eq. /H208498/H20850. The proportionality to /H9251is related to the fact that we considered the conductivitylike contribution which domi-nates the damping at long lifetimes /H9270/H20849respectively, low tem- peratures /H20850. The resistivitylike contribution depends also on the lifetime /H9270, however, in a more complicated manner. It increases monotonically with increasing /H9270−1, and for small /H9270−1/H20849where the conductivitylike contribution dominates /H20850it is proportional to /H9270−1,/H9251=F˜el//H9270. Thereby, F˜elis again a quantity which is determined by the properties of the electronic statesbut it is different from the quantity F elappearing in Eqs. /H208494/H20850 and /H208495/H20850. Whereas Felcan be expressed by matrix elements which are formed with, respectively, the same single-electron wave functions /H9023jk, the quantity F˜elcontains matrix ele- ments formed by two different wave functions /H9023jkand/H9023j/H11032k, respectively, see, e.g., Ref. 14. This procedure yields the re- lation /H9270M=F˜el pb21 /H9251/H208499/H20850 between the demagnetization time and the resistivitylike con- tribution to /H9251, and this relation has the same form as Eq. /H208493/H20850 given by Ref. 5. Altogether, it becomes clear that Eq. /H208493/H20850is not valid for all situations, it is certainly not valid for verylong lifetimes /H20849respectively, low temperatures /H20850where the conductivitylike contribution to /H9251dominates. This may be a further reason why relation /H208493/H20850could not be confirmed quan- titatively in the experiments. /H20849Please note that for Fe, Co, and Ni probably both contributions to /H9251are relevant at room temperature.10,13/H20850 The second essential difference is that in Eq. /H208493/H20850just one material parameter /H20849Hex/H20850appears which has nothing to do with spin-orbit coupling, whereas the quantities Feland F˜el of Eqs. /H208498/H20850and /H208499/H20850are determined by the sensitivity of the electronic states on changes in the spin-orbit coupling. OnRELATING GILBERT DAMPING AND ULTRAFAST LASER- … PHYSICAL REVIEW B 82, 144405 /H208492010 /H20850 144405-3the first sight it therefore looks as if Eqs. /H208493/H20850and /H208499/H20850were, in principle, not compatible with each other. Therefore we must conclude that /H9251/H9270M=F˜el/pb2should depend only weakly on the strength /H9264of the spin-orbit coupling /H9264/H20849L·S/H20850between the spin angular momentum Sand the orbital angular momen- tumLof an electron. Indeed, it has been shown14that F˜el /H11011/H92642, and because in first-order perturbation theory b2is also proportional to /H92642, the quantity /H9251/H9270Mdoes not depend on /H9264. Finally, we want to test our relation /H208498/H20850against experi- mental results. We take the case of Ni because in this mate-rial the damping is definitely dominated by the conductivity-like contribution at low temperatures, and Ref. 10gives a value of /H9261= /H9253M/H9251=1.07 /H11003108/s for that contribution at room temperature, where /H9261is the Landau-Lifshitz damping parameter. Furthermore, the value of /H9270Mas fitted from M/H20849t/H20850/M/H20849t=0/H20850at low fluences is approximately 100 fs /H20851Fig. 3d of Ref. 4/H20852. This yields the experimental value of /H9251//H9270M=1.2/H110031011/s. To calculate the theoretical value of /H9251//H9270Mby Eq. /H208498/H20850, we take p=4 and b2=0.025. This value of b2has been calculated by the ab initio electron theory20un- der the assumption that the dominant contribution to the de-magnetization arises from thermally excited electrons and holes.4The value of Felcalculated by the ab initio electron theory is taken from Fig. 2 of Ref. 21/H20849for a precession around /H20851111/H20852/H20850. Altogether, Eq. /H208498/H20850then yields /H9251//H9270M=0.6/H110031011/s, a value which agrees astonishingly well with the experimental value. To conclude, we have calculated by a purely microscopic approach a relation between the Gilbert damping parameter /H9251at low temperatures and the demagnetization time /H9270Mfor ultrafast laser-induced demagnetization at low fluences. Thepredicted value for /H9251//H9270Mis in good agreement with the ex- perimental value. The theory therefore provides a link be-tween the magnetization dynamics on the fast /H20849nanoseconds to several picoseconds /H20850and the ultrafast /H20849approximately 100 fs/H20850time scale. ACKNOWLEDGMENT The authors are indebted to Bert Koopmans for many helpful discussions. *faehnle@mf.mpg.de 1E. Beaurepaire, J. C. Merle, A. Daunois, and J. Y . Bigot, Phys. Rev. Lett. 76, 4250 /H208491996 /H20850. 2T. L. Gilbert, Ph.D. thesis, Illinois Institute of Technology, 1956. 3M. Fähnle, D. Steiauf, and J. Seib, J. Phys. D: Appl. Phys. 41, 164014 /H208492008 /H20850. 4B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fähnle, T. Roth, M. Cinchetti, and M. Aeschlimann, NatureMater. 9, 259 /H208492010 /H20850. 5B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and W. J. M. de Jonge, Phys. Rev. Lett. 95, 267207 /H208492005 /H20850. 6Y . Yafet, in Solid State Physics , edited by F. Seitz and D. Turn- bull /H20849Academic, New York, 1963 /H20850, V ol. 14, pp. 1–98. 7In Ref. 5the Stoner exchange field Hexis related to the critical temperature TC, which—however—is the Stoner critical tem- perature rather than the in general much smaller experimentalcritical temperature. 8It should be noted that an equation of the form of Eq. /H208493/H20850relating the single-spin dephasing time /H9270M,tto/H9251has been derived by completely different methods also by H. J. Skadsem, Y . Tserk-ovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 75, 094416 /H208492007 /H20850; and by L. Berger, ibid. 80, 144427 /H208492009 /H20850. 9J. Walowski, G. Müller, M. Djordjevic, M. Münzenberg, M.Kläui, C. A. F. Vaz, and J. A. C. Bland, Phys. Rev. Lett. 101, 237401 /H208492008 /H20850. 10B. Heinrich, D. J. Meredith, and J. F. Cochran, J. Appl. Phys. 50, 7726 /H208491979 /H20850. 11B. Heinrich, in Ultrathin Magnetic Structures III , edited by J. Bland and B. Heinrich /H20849Springer, Berlin, 2005 /H20850. 12V . Kamberský, Czech. J. Phys., Sect. B 26, 1366 /H208491976 /H20850. 13K. Gilmore, Y . U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 /H208492007 /H20850. 14K. Gilmore, Y . U. Idzerda, and M. D. Stiles, J. Appl. Phys. 103, 07D303 /H208492008 /H20850. 15K. Gilmore, M. D. Stiles, J. Seib, D. Steiauf, and M. Fähnle, Phys. Rev. B 81, 174414 /H208492010 /H20850. 16V . Kamberský, Can. J. Phys. 48, 2906 /H208491970 /H20850. 17For the present purpose it suffices to consider the damping for a momentary orientation of the magnetization along a high-symmetry direction in the crystal for which the damping matrixcan be replaced by a damping scalar /H20849Ref. 3/H20850. 18D. Steiauf, C. Illg, and M. Fähnle, J. Magn. Magn. Mater. 322, L5/H208492010 /H20850. 19R. J. Elliott, Phys. Rev. 96, 266 /H208491954 /H20850. 20D. Steiauf and M. Fähnle, Phys. Rev. B 79, 140401 /H20849R/H20850/H208492009 /H20850. 21D. Steiauf and M. Fähnle, Phys. Rev. B 72, 064450 /H208492005 /H20850.FÄHNLE, SEIB, AND ILLG PHYSICAL REVIEW B 82, 144405 /H208492010 /H20850 144405-4

PhysRevLett.127.067201.pdf

Skyrmion Qubits: A New Class of Quantum Logic Elements Based on Nanoscale Magnetization Christina Psaroudaki1,2,*and Christos Panagopoulos3,† 1Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA 2Institute for Theoretical Physics, University of Cologne, D-50937 Cologne, Germany 3Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link 637371, Singapore (Received 30 March 2021; accepted 30 June 2021; published 4 August 2021) We introduce a new class of primitive building blocks for realizing quantum logic elements based on nanoscale magnetization textures called skyrmions. In a skyrmion qubit, information is stored in thequantum degree of helicity, and the logical states can be adjusted by electric and magnetic fields, offering a rich operation regime with high anharmonicity. By exploring a large parameter space, we propose two skyrmion qubit variants depending on their quantized state. We discuss appropriate microwave pulsesrequired to generate single-qubit gates for quantum computing, and skyrmion multiqubit schemes for a scalable architecture with tailored couplings. Scalability, controllability by microwave fields, operation time scales, and readout by nonvolatile techniques converge to make the skyrmion qubit highly attractive asa logical element of a quantum processor. DOI: 10.1103/PhysRevLett.127.067201 Quantum computing promises to dramatically improve computational power by harnessing the intrinsic propertiesof quantum mechanics. Its core is a quantum bit (qubit) of information made from a very small particle such as an atom, ion, or electron. Proposed qubit systems includetrapped atoms, quantum dots, and photons [1–3]. Among them, superconducting circuits, currently one of the leadingplatforms for noisy intermediate-scale quantum computingprotocols [4], are macroscopic in size but with well- established quantum properties [5]. Nevertheless, despite tremendous progress, significant challenges remain, inparticular with respect to control and scalability [6]. Here we propose an alternative macroscopic qubit design based on magnetic skyrmions, topologically protectednanoscale magnetization textures, which have emerged as potential information carriers for future spintronic devices [7]. We focus on frustrated magnets, in which skyrmions and antiskyrmions have a new internal degree offreedom associated with the rotation of helicity [8–12].I n these systems, the noncollinear spin texture induces electricpolarization, allowing for electric-field modulation of theskyrmion helicity [13,14] . Along with magnetic field gradients [15] (MFGs) and microwave fields [16,17] , electric fields emerge as a new, powerful tool for acurrent-free control of skyrmion dynamics [18]. Skyrmions of a few lattice sites [19] inspired theoretical studies on their quantum properties [20,21] . Similar to Josephson junctions [22,23] , their macroscopic quantum tunneling and energy-level quantization are indicative ofquantum behavior. In sufficiently small magnets, an analo-gous quantum behavior in terms of macroscopic quantumtunneling of the magnetic moment has been experimentally verified in mesoscopic magnetic systems [24–26], while the quantum depinning of a magnetic skyrmion has beentheoretically proposed [27]. We formulate a theoretical framework of skyrmion quantization and construct skyrmion qubits based on the energy-level quantization of the helicity degree of freedom. The ability to control the energy-level spectra with externalparameters, including electric and magnetic fields, offers a rich parameter space of possible qubit variants with high anharmonicity and tailored characteristics. We proposemicrowave MFGs for skyrmion qubit manipulation and gate operation, and consider skyrmion multiqubit schemes for a scalable architecture. A skyrmion qubit has amoderately high coherence time in the microsecondregime, while nonvolatile readout techniques can be employed for a reliable qubit state readout. Finally, we discuss how scale-up multiqubit challenges can beaddressed by leveraging state-of-the-art skyrmion technol- ogy and show that skyrmion qubits are suitable for quantum computing technology. Skyrmion field quantization. —We begin by considering the inversion-symmetric Heisenberg model with competinginteractions [10], F¼− J1 2ð∇mÞ2þJ2a2 2ð∇2mÞ2−H a2mzþK a2m2z; ð1Þ where HandKare the Zeeman and anisotropy coupling, respectively, while J1andJ2denote the strength of the competing interactions and athe lattice spacing. A numberPHYSICAL REVIEW LETTERS 127, 067201 (2021) Editors' Suggestion Featured in Physics 0031-9007 =21=127(6) =067201(6) 067201-1 © 2021 American Physical Societyof geometrically frustrated magnets are good candidates to host complex spin textures [8], including the triangular- lattice magnet Gd 2PdSi 3, known to support skyrmion phases [28]. Using m¼½sinΘcosΦ;sinΘsinΦ;cosΘ/C138, we describe classical skyrmions by ΦðrÞ¼−Qϕand Θ¼ΘðρÞ, with ρ,ϕpolar coordinates. This class of solutions is characterized by an integer-valued topological charge Q¼ð1=4πÞR rm·ð∂xm×∂ymÞ, with Q¼1 (Q¼−1) for a skyrmion (antiskyrmion). The skyrmion size is defined as λ≡2a=Re½γ/C6/C138, with γ/C6¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi−1/C6˜γp=ffiffiffi 2p and ˜γ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1−4ðH=J 1þ2K=J 1Þp . The model of Eq. (1) has an unbroken global symmetry, Φ→Φþφ0, with φ0 the collective coordinate of the skyrmion helicity. By considering a skyrmion stabilized in a nanodisk (see Fig.1), we exclude the translational coordinate of position [21] and focus exclusively on the dynamics of φ0. To investigate quantum effects, we utilize a method of collective coordinate quantization. Here φ0and its con- jugate momentum Szare introduced by performing a canonical transformation in the phase space path integral[29,30] (see Supplemental Material [31]). This is achieved by ensuring momentum is conserved, S z¼P, with P¼R rð1−cosΘÞ∂ϕΦthe infinitesimal generator of rotations satisfying fP;Φg¼−∂ϕΦ. Using standard equivalence between path integral and canonical quantization, we introduce operators ˆφ0and ˆSzwith ½ˆφ0;ˆSz/C138¼i=¯S, and ¯Sthe effective spin. The classical limit is associated with ¯S≫1. Eigenstates of ˆSzare labeled by an integer charge s with ˆSzjsi¼s=¯Sjsi, and states ˆφ0jφ0i¼φ0jφ0ihave a circular topology jφ0i¼j φ0þ2πi. The relation between physical and dimensionless parameters is summarized in Table I. We construct skyrmion qubits based on textures withQ¼1. Antiskyrmion qubits follow directly from our present analysis. Fundamental skyrmion qubit types. —We now seek to construct a skyrmion qubit based on the energy-level quantization of the helicity degree of freedom. A promising qubit candidate needs to satisfy several criteria including scalability, ability to initialize to a simple fiducial state, long decoherence times, a universal set of quantum gates, and the ability to perform qubit-specific measure- ments [32]. TheSzqubit: The ability to control the energy-level spectra with external parameters, offers a rich parameter space of possible qubit variants with tailored character- istics. We introduce the Sz-qubit Hamiltonian, HSz¼κðˆSz−h=κÞ2−Ezcos ˆφ0; ð2Þ which resembles the circuit Hamiltonian of a supercon- ducting charge qubit [33]. Here κand hdenote the anisotropy and magnetic field coupling, respectively, in dimensionless units. The noncollinear spin texture gives rise to an electric polarization which couples to an electric fieldEzapplied across the nanodisk to control φ0[14](see Fig.1for a schematic illustration of the setup). The Szqubit is designed in the Ez≪κregime, such that logical qubits are spin states jsi, representing deviations of the mz component from equilibrium. The solution of the Schrödinger equation HSzΨsðφ0Þ¼EsΨsðφ0Þ, with Ψsðφ0Þ¼h φ0jsi, can be calculated exactly in the form of special functions (see Supplemental Material [31]). In Fig.2(b)we plot the potential landscape and the first three levels using κ¼0.1,h¼0.47, and Ez¼0.02. Two requirements are essential for a reliable qubit operation; nonequidistance of the energy spectrum to uniquely address each transition and suppressed sponta-neous thermal excitations to higher energy levels k BT≪ℏω12,ℏω02. The remarkable feature of skyrmion qubits is that these conditions can be met by tuning the relevant external parameters. In Fig. 2(a) we present the range of parameters ¯h¼h¯S=κandEzfor which a relatively large anharmonicity is present, jω12−ω01j>20%ω01 andjω02−ω01j>20%ω01. FIG. 1. Skyrmion qubit concept. (a) A quantum state jΨias an arbitrary superposition of skyrmion configurations with distinct helicities φ0. (b) Bloch sphere representation of jΨi¼αj0iþβj1i, with j0iand j1idenoting the two lowest energy levels of the quantum operator ˆφ0. (c) A bilayer of magnetic materials as a platform for the skyrmion qubit couplingscheme. The qubit coupling is tuned by a nonmagnetic spacer(blue), and logical states are adjusted by electric fields (yellowplates). TABLE I. Relation between physical and dimensionless parameters. We use J1¼1meV, a¼5Å,¯S¼10,J2¼J1,K¼0.4J1, Kx¼0.05J1, and PE¼20μC=cm2. MFG stands for magnetic field gradient. Length Time Frequency Temperature Magnetic field Electric field Static MFG r×0.5nmt×6.610−13sω×1519 GHz T×11.6KH=gμB¼h×0.86TE¼Ez×215V=mH⊥=gμB¼h⊥×1.72T=nmPHYSICAL REVIEW LETTERS 127, 067201 (2021) 067201-2For¯h¼1=2, the two lowest spin states j0iandj1iare degenerate, and a small Ezlifts the degeneracy creating a tight two-level system. Truncating the full Hilbert space to qubit subspace, the reduced Hamiltonian is Hq¼H0 2ˆσz−Xc 2ˆσx; ð3Þ with H0¼κð1−2¯hÞ=¯S,Xc¼Ez, and ωq¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H2 0þX2cp the corresponding qubit level spacing. The universal level repulsion diagram is shown in Fig. 2(c), with a minimum energy splitting Ez. The Sz-qubit operation regime in physical units is given in Table II. We note that the proposed qubit platform has large anharmonicity, and thevoltage bias for qubit manipulation is several orders of magnitude smaller compared to those required for the electric-field skyrmion creation and annihilation [18]. The helicity qubit: Inspired by the superconducting flux qubit and proposals on magnetic domain walls [34],w e seek to construct a double-well potential landscape for the helicity φ 0, in order to define the qubit logical space using the two well minima. This is achieved by considering amaterial with in-plane magnetic anisotropy of strength κ x [35]and a skyrmion characterized by an elliptical profile, as the result of defect engineering [36,37] . The Hamiltonianfor this new type of helicity qubit reads Hφ0¼κˆSz−hˆSzþVðˆφ0Þ, with the double-well potential given by Vðφ0Þ¼κxcos2ˆφ0−Ezcos ˆφ0þh⊥sin ˆφ0: ð4Þ The first two terms in Eq. (4)create a symmetric potential, and the third term describes a depth difference between the well created by an in-plane MFG of strength h⊥. The solutions of the eigenvalue problem 1Hφ0Ψnðφ0Þ¼EnΨnðφ0Þare2π-periodic functions calcu- lated numerically. The potential in the helicity representa- tion is schematically shown in Fig. 3(b) together with the first three levels. Close to the degeneracy point at ¯h¼1and forh⊥¼0, the two lowest energy functions Ψ0;1are symmetric and antisymmetric combinations of the two wave functions localized in each well located at φm¼tan−1ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16κ2x−E2zp =EzÞ. A finite h⊥acts as an energy bias creating a depth well difference, such that the ground and first-excited states are now localized indifferent wells. At¯h¼1, level anticrossing can be probed by applying either an electric field E z[see Fig. 3(c), upper panel] or a magnetic field gradient h⊥[see Fig. 3(c), lower panel]. The (a) (b) (c) (d) FIG. 2. The Sz-qubit properties. (a) Magnetic field ¯hand electric field Ezdependence of the transition frequency ωq, close to the degeneracy point ¯h¼0.5. The colored surface represents the values of ωqwhich satisfy the requirement of high anharmonicity. (b) Nonequidistant quantized energy levels and potential landscape. The qubit states are the ground state j0iand first excited state j1i with level spacing ℏω01¼ωqsmaller than transitions to higher states ℏω02,ℏω12. (c) Universal energy level anticrossing diagram close to the degeneracy point (dashed lines). The degeneracy is lifted by an electric field (upper panel) or increasing the magnetic field away from ¯h¼0.5(lower panel). At the degeneracy point, energy eigenstates are symmetric and antisymmetric superpositions of the skyrmion qubit states ðj0i/C6j1iÞ=ffiffiffi 2p . (d) A magnetic skyrmion with a circular profile stabilized in a magnetic nanodisk. TABLE II. Skyrmion qubit operation regime and lifetime. We use α¼10−5andT¼100mK. EF stands for electric field and MFG for magnetic field gradient. Qubit type Magnetic field External control ωq T1 T2 ω12 Tc Szqubit 8.9 mT EF ¼108mV=μm 25.6 GHz 0.27μs 0.49μs 310 GHz 2.50 K Helicity qubit 445 mT EF ¼296mV=μm 14.9 GHz 0.15μs 0.26μs 330 GHz 2.60 K Helicity qubit 445 mT MFG ¼1.73mT=nm 2.1 GHz 0.43μs 0.32μs 330 GHz 2.55 KPHYSICAL REVIEW LETTERS 127, 067201 (2021) 067201-3reduced qubit Hamiltonian under the two-level approxi- mation has the form of Eq. (3), where H0¼E1−E0and Xc¼geEzforh⊥¼0,o rXc¼gbh⊥forEz¼0. Constants H0,ge, and gbare found numerically. The helicity-qubit operation regime in physical units is given in Table II, using both Ezandh⊥as external control parameters. Qubit control. —A quantum coherent computation depends on the ability to control individual quantumdegrees of freedom. Here we propose microwave MFGsfor skyrmion qubit manipulation and gate operation. MFGsgive rise to additional Hamiltonian terms H extðtÞ¼ bfðtÞcosðωtþϕextÞcos ˆφ0, with fðtÞa dimensionless envelope function, or in terms of the qubit Hamiltonian, Hq ext¼bxðtÞˆσx, with bxðtÞ¼b0fðtÞcosðωtþϕextÞ. In the diagonal basis, the driven Hamiltonian is written as Hq¼ωq 2ˆσzþbxðtÞ½cosθˆσxþsinθˆσz/C138; ð5Þ with tan θ¼Xc=H0. To elucidate the role of the drive, we transform Hqinto the rotating frame, Hrot¼Δω 2ˆσzþΩ 2fðtÞ½cosϕextˆσxþsinϕextˆσy/C138;ð6Þ where Δω¼ωq−ωis the detuning frequency and Ω¼b0cosθ. Single-qubit operations correspond to rota- tions of the qubit state by a certain angle about a particularaxis. As an example, for ϕ ext¼0andΔω¼0, the unitary operator UxðtÞ¼e−ði=2ÞϑðtÞˆσxcorresponds to rotations around the xaxis by an angle ϑðtÞ¼−ΩRt 0fðt0Þdt0[38]. Rotations about the yaxis are achieved for ϕext¼π=2. Qubit coupling scheme. —A key component for realizing a scalable quantum computer is an interaction Hamiltonianbetween individual qubits. As a straightforward scheme forcoupling skyrmion qubits, we consider the interlayer exchange interaction in a magnetic bilayer mediated by a nonmagnetic spacer layer (see Fig. 1for a visualization). The interaction term is given by F int¼JintR rm1·m2[39], or in terms of the helicities, Hint¼−Jintcosðφ1−φ2Þ. The resulting Hamiltonian in the qubit basis contains both transverse and longitudinal couplings, Hint¼−Jx intˆσ1xˆσ2x−Jz intˆσ1zˆσ2z: ð7Þ Jintcan be tuned experimentally by changing the spacer thickness, while both Jx;z intallow for an independent control by tuning all three external fields h,Ez, and h⊥. This property is especially important in applications where bothlongitudinal and transverse couplings are desired, such as quantum annealing [38]. Noise and decoherence. —The interaction of the sky- rmion qubit with the environmental degrees of freedom is a source of noise that leads to decoherence. They result inOhmic damping terms for the collective coordinates φ 0and Sz[40], accompanied by random fluctuating forces ξithat enter the quantum Hamiltonian as ˆH→ ˆHþξφ0ˆφ0þ ξSzˆSz.ξiis fully characterized by the classical ensemble averages hξiðtÞi ¼ 0and hξiðtÞξjðt0Þi ¼ δijSiðt−t0Þ[34], and the correlator SiðtÞis defined via the fluctuation- dissipation theorem, SiðωÞ¼αiωcothðβω=2Þ, with αi constants proportional to the Gilbert damping α. In terms of the reduced qubit Hamiltonian one finds Hq¼ωq 2ˆσzþξxðtÞγxˆσxþξyðtÞγyˆσyþξzðtÞγzˆσz;ð8Þ where γiconstants which depend on the qubit type and ξx;y;zare linear combinations of ξφ0andξSz. (a) (b) (c) (d) FIG. 3. The helicity-qubit properties. (a) Electric field Ezand magnetic field gradient h⊥dependence of the transition frequency ωq, close to the degeneracy point ¯h¼1. The colored surface represents the values of ωqwhich satisfy the requirement of high anharmonicity. (b) Nonequidistant quantized energy levels and double-well potential landscape. The qubit states are the ground state j0i and first excited state j1iwith level spacing ℏω01¼ωqsmaller than transitions to higher states ℏω02,ℏω12. The potential barrier Vmis controlled by Ezand the well difference by h⊥. (c) Universal energy level anticrossing diagram close to the degeneracy point ¯h¼1. The degeneracy is lifted by an electric field (upper panel) or a magnetic field gradient (lower panel). (d) A magnetic skyrmion with anelliptical profile stabilized in a magnetic nanodisk. The elliptical profile is essential for realizing the double-well potential.PHYSICAL REVIEW LETTERS 127, 067201 (2021) 067201-4Within the Bloch-Redfield picture of two-level system dynamics, relaxation processes are characterized by the longitudinal relaxation rate Γ1¼T−1 1and the dephasing rateΓ2¼T−1 2. The latter is a combination of effects of the depolarization Γ1and of the pure dephasing Γφ, combined to a rate Γ2¼Γ1=2þΓφ, withΓ1¼γ2xSxðωqÞþγ2ySyðωqÞ andΓφ¼γ2zSzð0Þ[41]. The optimal regime for realizing both long coherence and high anharmonicity is close to the degeneracy point and for Xc≪H0. This translates to the requirement ¯h¼0.5andEz≪1for the Szqubit, and to ¯h¼1andEz,h⊥≪1for the helicity qubit. In Table IIwe present the expected qubit lifetimes for a modest choice of an ultralow Gilbert damping α¼10−5 andT¼100mK. A skyrmion qubit has a moderately high coherence time in the microsecond regime. This is com-parable to early measurements of the flux superconducting qubit and 2 orders of magnitude larger than the Cooper pair box[33]. The number of coherent Rabi frequency oscil- lations within the coherence time is ΩT 1∝105, inside the desired margins expected for superconducting qubits [34,42] . Several magnetic thin films exhibit ultralow Gilbert damping of the order of α∼10−4−10−5[43– 45]. In the sub-Kelvin qubit operational regime, Gilbert damping is expected to be even lower [46,47] . Coherence times can be further improved with the development ofcleaner magnetic samples and interfaces in engineered architectures, without trading off qubit anharmonicity and scalability. Readout techniques. —An essential part for implement- ing skyrmion-based quantum-computing architectures is areliable readout. Quantum sensing of coherent single-magnon techniques, based on quantum dot [48] or super- conducting qubit [49]sensors, is promising for the readout ofS z-qubit states, single magnetic excitations from the equilibrium configuration. On the other hand, helicity-qubit states represent two distinct skyrmion configurations with helicity values located at the two minima of the double-wellpotential of Eq. (4). Experimental observation of skyrmion helicity is possible using nitrogen-vacancy (NV) magne- tometry [50], allowing for a detector-single qubit coupling control by varying the NV sensor distance from the skyrmion. Resonant elastic x-ray scattering [51]techniques provide a direct observation of skyrmion helicity, and whencombined with ferromagnetic resonance measurements[52] can offer a promising single-qubit readout method. Finally, coupling a skyrmion to a magnetic force micros- copy resonator allows the detection of magnetic states,which appear as resonance frequency shift signals [53]. Conclusions. —We proposed a novel physical qubit plat- form based on magnetic nanoskyrmions in frustrated magnets. The skyrmion state, energy-level spectra, tran- sition frequency, and qubit lifetime are configurable andcan be engineered by adjusting external electric and magnetic fields, offering a rich operation regime with highanharmonicity. Microwave pulses were shown to generate single-qubit gates for quantum computing, and skyrmionmultiqubit schemes were considered for a scalable archi- tecture with tailored couplings. Whereas, nonvolatile read- out techniques can be employed for a reliable qubit state readout, using state-of-the-art magnetic sensing technol- ogy. We anticipate the considerable progress in the field ofskyrmionics will provide exciting new directions on the development of skyrmion qubits as promising candidates for quantum computing technology. We thank Martino Poggio, So Takei, Daniel Loss, Ivar Martin and Markus Garst for useful discussions. C. Psaroudaki has received funding from the European Union ’s Horizon 2020 research and innovation program under the Marie Sk łodowska-Curie Grant Agreement No. 839004. C. Panagopoulos acknowledges support fromthe Singapore National Research Foundation (NRF) NRF- Investigatorship (No. NRFNRFI2015-04) and Singapore MOE Academic Research Fund Tier 3 Grant No. MOE2018-T3-1-002. *cpsaroud@caltech.edu †christos@ntu.edu.sg [1] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O ’Brien, Nature (London) 464,4 5 (2010) . [2] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998) . [3] M. Grimm, A. Beckert, G. Aeppli, and M. Müller, PRX Quantum 2, 010312 (2021) . [4] J. Preskill, Quantum 2, 79 (2018) . [5] J. Clarke and F. K. Wilhelm, Nature (London) 453, 1031 (2008) . [6] Y. Alexeev et al. ,Phys. Rev. X Quantum 2, 017001 (2021) . [7] A. N. Bogdanov and C. Panagopoulos, Nat. Rev. Phys. 2, 492 (2020) . [8] T. Okubo, S. Chung, and H. Kawamura, Phys. Rev. Lett. 108, 017206 (2012) . [9] A. O. Leonov and M. Mostovoy, Nat. Commun. 6, 8275 (2015) . [10] S.-Z. Lin and S. Hayami, Phys. Rev. B 93, 064430 (2016) . [11] X. Zhang, J. Xia, Y. Zhou, X. 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PhysRevResearch.2.012045.pdf

PHYSICAL REVIEW RESEARCH 2, 012045(R) (2020) Rapid Communications Cluster multipole dynamics in noncollinear antiferromagnets Takuya Nomoto1,*and Ryotaro Arita1,2 1Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan (Received 11 March 2019; accepted 3 February 2020; published 25 February 2020) A systematic framework to investigate the spin dynamics in a noncollinear antiferromagnet is proposed. Taking Mn 3Sn as a representative example, we derive an effective low-energy model based on the multipole expansion of the magnetic structure, and investigate the uniform precession and the domain wall dynamics.We show that the solution for the effective model accurately reproduces the numerical calculation of theLandau-Lifshitz-Gilbert equations. Our results indicate that Mn 3Sn has preferable properties for applications to a racetrack memory and a spin torque oscillator, and thus is a promising candidate for spintronics devices byusing the multipole degrees of freedom. DOI: 10.1103/PhysRevResearch.2.012045 Introduction. In the field of spintronics, spin manipula- tion based on an antiferromagnet (AFM) has attracted muchattention because of its potential advantages over a ferro-magnet (FM) [ 1–7]. For example, due to the absence of net magnetization, AFM devices are relieved of the stray fieldproblem, which is one of the main obstacles to high-densityintegration. A maximum velocity of a domain wall inducedby a spin current, thermal gradient, and staggered field ismuch faster in collinear AFM than in FM [ 8–11], which is a favorable property for applications to racetrack memories. Atypical energy scale of AFM is also much higher than that ofFM, resulting in a fast switching of its magnetization [ 12,13] as well as a coherent precession with the THz frequency[14–17]. The ac signals generated by such steady motion can be extracted as the ac voltage through inverse spin-Hall effectsor as dipolar radiation in a special case [ 18,19]. Despite such fascinating properties, however, so far there have been few realizations of AFM devices. This is mainlybecause the Néel vector, the order parameter of collinearAFM, does not couple directly to the external field. Sincecollinear AFM usually possesses time-reversal symmetry, itdoes not show any directional signal associated with symme-try breaking such as the anomalous Hall effect and magneto-optical Kerr effect. For example, in a racetrack memory, itis necessary to detect each domain separated by the domainwalls, but it is impossible in conventional collinear AFM. Onepossibility to overcome the problem is to use a ferrimagnet[20–25]. Although it has features of both FM and AFM, a usual ferrimagnet shows a fast response only near its compen-sation point. In this Rapid Communication, we focus on another pos- sibility of AFM, namely, noncollinear AFM. Recently, it *nomoto@ap.t.u-tokyo.ac.jp 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.was shown that the Weyl AFM Mn 3Sn [26] has a tiny net magnetization about 2m μB/atom but shows large anomalous Hall and Nernst effects comparable to a conventional FM[27–32]. The spin texture in its Néel state is regarded as a ferroic order of a cluster octupole whose symmetry is thesame as a conventional dipole under hexagonal point groupsymmetry [ 33]. The related AFM Mn 3Ge also shows a large anomalous Hall effect and has a noncollinear spin texture[30,34]. Thus, one may expect that noncollinear AFM is a promising platform for magnetic devices since it is AFM andits spin dynamics is detectable by the same methods as FM. In contrast to FM and collinear AFM, theoretical stud- ies on the spin dynamics of noncollinear AFM are limited[35–38]. Especially, there is a lack of systematic methods to obtain its effective model so far. Here, we propose aframework to derive an effective model of noncollinear AFMbased on the cluster multipole theory [ 33,39,40]. In Mn 3Sn, the derived model is composed of two octupole degrees offreedom and reduced into the sine-Gordon model similar toFM and collinear AFM. We check the validity of the modelby comparing two phenomena to these in the original model:the domain wall dynamics and steady-state precession. Theagreement is very good at low energy, which means that thespin dynamics in Mn 3Sn is almost dominated by the octupole degrees of freedom. As expected, the domain wall shows ahigh maximum velocity without a Walker breakdown, and thecoherent precession shows a tunable frequency from sub-THzto THz. Our results indicate that Mn 3Sn is a good candidate with many desirable properties for applications, owing to itsoctupole degrees of freedom. Models. Here, we consider spin dynamics in the following Hamiltonian defined on the two-dimensional kagome lattice,which is known as a minimal model describing the Néel stateof Mn 3Sn [41–44], H=J/summationdisplay /angbracketleftia,jb/angbracketrightSia·Sjb+D/summationdisplay /angbracketleftia,jb/angbracketright/epsilon1abˆz·(Sia×Sjb) −K⊥ 2/summationdisplay ia(ˆKa·Sia)2, (1) 2643-1564/2020/2(1)/012045(6) 012045-1 Published by the American Physical SocietyTAKUYA NOMOTO AND RYOTARO ARITA PHYSICAL REVIEW RESEARCH 2, 012045(R) (2020) FIG. 1. (a) Spin configuration in the Néel state of Mn 3Sn, which is regarded as the ferroic order of the cluster octupole Ox. The nearly degenerate state corresponding to Oyis obtained by 90◦rotation of each spin. (b) Schematic picture of a domain wall. From x/prime=0t o L,Oxchanges sign from +1t o−1, and Oyappears when Ox/similarequal0, i.e., near the domain wall. The wall is profiled well with the two octupoles. where the suffixes i,jdenote a unit cell, a,b∈{A,B,C} denote a sublattice, and /epsilon1abis an antisymmetric tensor sat- isfying /epsilon1AB=/epsilon1BC=/epsilon1CA=1 [see Fig. 1(a)].Jand Drep- resent a nearest-neighbor exchange interaction ( J>0) and a Dzyaloshinskii-Moriya (DM) interaction, respectively. Theclassical ground state of Hdepends on the sign of D, and degenerate 120 ◦spin textures corresponding to the Néel states of Mn 3Sn are realized when Dis positive. The in- plane anisotropy K⊥>0 with ˆKa=(cosψa,sinψa,0) and (ψA,ψB,ψC)=(0,4π 3,2π 3) lifts the degeneracy, resulting in anOxoctupole as the ground state [ 45,48]. Here, the spin dynamics in Mn 3Sn is considered based on the Landau- Lifshitz-Gilbert (LLG) equations, which are formally writtenas ˙S ia=δH ¯hδSia×Sia−α SSia×˙Sia+Text ia, (2) where Text iarepresents the torque acting on the spin Sia, which comes from the external magnetic field or current in this RapidCommunication. αdenotes a Gilbert damping coefficient. In the numerical calculations, we follow previous studiesand set S=1,α=0.01,K ⊥=0.05J, and√ 3D=JorJ/3 [37,49,50]. Effective theory. Although Eq. ( 2) with Eq. ( 1) can be solved numerically, in order to grasp the physics and re-duce the computational cost for future applications, we thenderive an effective model describing the low-energy spindynamics in Mn 3Sn. When K⊥=0, each unit cell has D6h point group symmetry and the possible spin textures can be classified into its irreducible representations [ 33,52]. For example, the ground state shows a spin texture identified asthe cluster octupole O x/Oy, which belongs to E1girreducible representation, Oi=1√ 3/parenleftbig¯SiA+R4π 3¯SiB+R2π 3¯SiC/parenrightbig , (3) where ¯Sia=(Sx ia,Sy ia),Oi=(Oix,Oiy), and Rθis the two- dimensional rotation matrix. The other multipoles miμ(μ= 1,..., 7), corresponding to the other spin textures, are con- structed as linear combinations of spins in similar ways [ 53]. Using these transformations, we can derive the LLG equationsin the multipole representation from the original Eq. ( 2). An advantage in deriving such LLG comes from the fact thatTABLE I. Summary of the parameters appearing in the effective model ( 4). Domain wall width λdom, steady-state wall velocity vsteady, and relaxation time τrelax are respectively given by λ2 dom=κ/γ, ¯hvsteady=gμBBλdom/α,a n dτrelax=τ/α. Maximum wall velocity is dominated by Walker breakdown (¯ hvWB=λdomKz/2) in FM and spin wave ( ¯ hvSW=√¯hκ/τ)i nA F M /Mn 3Sn. Models ¯ hτ−1κ/a2 lat γ vmax FM Kz |J| K⊥ vWB AFM 8 |J|+Kz |J| K⊥ vSW Mn 3Sn 2√ 3D+6|J| (√ 3D+|J|)/2 K⊥ vSW the spin configurations corresponding to miμhave at least√ 3Dhigher energy than Ox/Oy. Thus, we can systematically extract an effective model only composed of Ox/Oyby inte- grating out the small miμdegrees of freedom. Then, the spin dynamics of the effective model can be understood in terms oftwo cluster octupoles. When parametrizing O i=|Oi|(cosϕi,sinϕi) and taking the continuum limit, we finally obtain the following equationof motion for ϕ(t,x /prime), τ¯h¨ϕ+α¯h˙ϕ−κ∂2ϕ+γ 2sin(2ϕ)=Text, (4) where the parameters are given by ¯ hτ−1=2√ 3D+6|J|,κ= a2 lat(√ 3D+|J|)/2, and γ=K⊥[54]. Here, alatis the distance between the nearest-neighbor spins and we have set S=1. We have also assumed that ϕis uniform along the a1direction [for the definitions of x/primeanda1, see Fig. 1(a)]. The force term Textgenerally depends on the external torque Text ia. Note that Eq. ( 4) is defined in the continuum space and can be scaled by renormalizing the stiffness parameter κ, and thus would be useful in micromagnetic simulations. To derive Eq. ( 4), we have assumed (1) the cluster multi- poles are slowly varying with respect to alat, (2) the cluster octupoles are energetically stable, and thus |Oix|,|Oiy|/greatermuch |miμ|, and (3) 1 /Sand K⊥/Jare also small. The minor multipole contributions are included within the lowest orderof these small parameters. (For details, see the SupplementalMaterial [ 53].) The above assumptions are plausible whenever the system possesses a robust low-energy collective motion.Thus, our approach would be applicable to a wide class ofmagnets unless they host a number of nearly degenerate spinconfigurations at low energy. Indeed, Eq. ( 4), the sine-Gordon form, is completely the same as in collinear FM and AFM.For example, let us consider the following Hamiltonian on thetwo-dimensional square lattice, H=−J/summationdisplay /angbracketlefti,j/angbracketrightSi·Sj+1 2/summationdisplay i/bracketleftbig Kz/parenleftbig Sz i/parenrightbig2−K⊥/parenleftbig Sx i/parenrightbig2/bracketrightbig ,(5) where Kz,K⊥>0, and J>0(J<0) for the collinear FM (AFM). Using this Hamiltonian with |J|>Kz/greatermuchK⊥,ϕ(t,x/prime) appearing in Eq. ( 4) respectively corresponds to the in-plane angle of the spin in FM and that of the Néel vector, defined asthe difference between the spins on two sublattices, in AFM.In the same manner, we can derive the effective model andidentify the parameters τ,κ, andγfor FM and AFM, which are summarized in Table I. The typical timescale of AFM and 012045-2CLUSTER MULTIPOLE DYNAMICS IN NONCOLLINEAR … PHYSICAL REVIEW RESEARCH 2, 012045(R) (2020) FIG. 2. Domain wall velocity ˙R(t). Staggered magnetic field, which is set to be gμBBstg=8×10−5J, is applied for t>1000¯ h/J. The open squares represent the results for collinear FM (blue) and AFM (red). The open circles represent those for Mn 3Sn with√ 3D=J(green) and√ 3D=J/3 (purple). The dashed black lines L1,L2,a n d L3indicate analytic solutions given in Eq. ( 6)f o rF M , Mn 3Sn(√ 3D=J/3), and AFM /Mn 3Sn(√ 3D=J), respectively. Mn 3Sn is given by O(¯hJ−1), which is usually much faster than that of FM of O(¯hK−1 z). As will be seen later, this results in a short time relaxation of the domain wall motion as well as aTHz coherent precession. Another notable point is that when J andDsatisfy√ 3D=J, all parameters in Eq. ( 4)a r et h es a m e in between collinear AFM and Mn 3Sn up to the first order of J/Kz. Thus, we can expect that the spin dynamics of collinear AFM and Mn 3Sn are essentially the same in this limit. Domain wall motion. In the following, we will see the validity of our effective model to calculate the domain walldynamics. It should be noted that, similar to collinear AFM,the torque coming from the uniform magnetic field cancelsout in each unit cell and does not drive the domain wall.Here, we simply apply the staggered magnetic field by addingH ext=−gμBBstg/summationtext ia(ˆKa·Sia)t oH, which results in an ef- fective torque as Text=−gμBBstgsinϕ[53,55]. To obtain a domain wall solution, we take the boundary condition suchthatϕ(t,0)=0 and ϕ(t,L)=π[see Fig. 1(b)]. Assum- ing an equilibrium solution with the profile cos ϕ(t,x /prime)= tanh [( x/prime−R)/λdom] and resubstituting it to the action by interpreting the constant of the integration Ras the time- dependent variable describing the domain wall center, weobtain ˙R(t)=v steady(1−e−t/τrelax), (6) which satisfies ˙R(0)=0. ¯hvsteady=gμBBstagλdom/αis the domain wall velocity in the steady state and τrelax=τ/α is the typical timescale to relax into it. Figure 2shows numerical results for the domain wall velocity obtained by solving Eq. ( 2)[53] and the analytic solutions given by Eq. ( 6). From the figure, we can see that the analytic solutions agree well with the numerical results exceptfor the small oscillating behavior in FM [ 56]. As expected, the relaxation time to reach v steady is much faster in AFM /Mn 3Sn than in FM, and the behavior of Mn 3Sn with√ 3D=Jis almost the same as AFM. Figure 2clearly shows that ourFIG. 3. Steady-state domain wall velocity ˙Ras a function of the staggered magnetic field. The open symbols are defined in the same way as in Fig. 2. The lines L1andL2show vsteady corresponding to FM/AFM/Mn 3Sn(√ 3D=J)a n dM n 3Sn(√ 3=J/3), respectively. L3,L4,a n d L5indicate the saturation values, i.e., vWBfor FM, vSWfor Mn 3Sn(√ 3D=J/3), and AFM /Mn 3Sn(√ 3D=J), respectively. effective model correctly represents the original model not only in FM /AFM but also in Mn 3Sn regardless of the value ofD. In Fig. 3, we show the field strength dependence of the steady-state velocity. At a low-field region, the domain wallvelocity is proportional to B stgand is almost on the lines vsteady=gμBBstgλdom/αin all cases. However, at a high-field region, the behavior in FM is different from the other cases,because of the presence (absence) of the Walker breakdown inFM (AFM /Mn 3Sn). The absence of the Walker breakdown in AFM can be understood as follows: The trigger of the Walkerbreakdown is the tilt of spins to the out-of-plane direction dueto the torque, which arranges the spins to the same direction.However, in contrast to FM, such a spin configuration lossesthe exchange energy of order O(J), and thus does not occur unless gμ BBstgexceeds J[9–11]. In Mn 3Sn, the situation is the same as AFM and the Walker breakdown does notoccur. Thus, the saturation velocity in AFM /Mn 3Sn is simply determined by the Lorentz boost of the equilibrium solutionand given by the spin-wave velocity ¯ hv SW=√κ/τwhile that in FM is given by the Walker breakdown ¯ hvWB=λdomKz/2, which are indicated in Fig. 3. Using the parameters 2 alat= 5.4Å , J=2.8m e V , D=0.64 meV , and S=3/2[37,57], we estimate vSW/similarequal2k m/si nM n 3Sn, which is slightly smaller than the collinear AFM such as 36 km /s of dielectric NiO [ 58] and 90 km /s of KFeS 2[59], but still faster than the highest record in FMs of 400 m /s[60]. Coherent precession of spins. Finally, we focus on the steady precession motion allowed in Mn 3Sn, which may be the source of a coherent THz signal. Here, we consider thesystem that contains a Mn 3Sn thin film sandwiched by two conventional FMs along the zdirection [ 38]. When the spin accumulation polarizing along ζexists at the interface, the torque expressed by the following form acts on the spin Sia, ¯hText ia=τFSia×ζ+τD SSia×(ζ×Sia), (7) 012045-3TAKUYA NOMOTO AND RYOTARO ARITA PHYSICAL REVIEW RESEARCH 2, 012045(R) (2020) FIG. 4. (a) Time evolution of space-averaged ˙ ϕ(t)w h e n τD= 0.02J. Red and blue lines respectively show the results by solving Eqs. ( 2)a n d( 4) numerically. (b) Time evolution of the polar angle θ(t) of each spin obtained by solving Eq. ( 2). (c) Time- and space- averaged /angbracketleft˙ϕ/angbracketrightand/angbracketleftθ/angbracketrightin the steady state. The slope of the red line is given by ¯ h/angbracketleft˙ϕ/angbracketright=τD/α. where the first term, called a fieldlike torque, represents the exchange interaction between the spins, while the secondterm, called a dampinglike torque, comes from the conserva-tion of the spin angular momentum through the dissipation[61,62]. Although both τ FandτDare proportional to the injected spin current [ 63], the first term does not drive the steady precession and we only take into account the secondterm in the following. Also, we set ζ=(0,0,1), resulting in the constant force T ext=τDin the effective model [ 53], and impose the periodic boundary condition on the system. Inthe effective model, we can simply neglect the x /primedependence ofϕ(t,x/prime), and then the model coincides with the second Josephson equation under a current bias [ 18,64]. Figure 4(a) shows the space-averaged ˙ ϕ(t) obtained by solving the original LLG ( 2) with the torque ( 7) and the effective model ( 4) with Text=τD, where τD=0.02J.W e can see that the coherent precession of octupoles is reallyrealized and it does not decay with time. The agreementbetween the original and the effective models is very good.The mechanism of such a steady precession can be understoodin the same way as in FM: The dissipation of the spin angularmomentum through the Gilbert damping exactly compensatesthe provided one through the dampinglike torque, namely, thedissipation of the accumulated spins. The velocity of theprecession in Mn 3Sn, however, is much higher than the FMbecause the dampinglike torque rather competes with the exchange Jand the DM interaction D[Fig. 4(b)] than the external field Bzor the anisotropy Kzin the case of FM. That implies that the precession frequency reaches O(J/¯h)i nt h e limit that all spins are along the zdirection. It is worth noting that the steady state ˙ ϕ(t) is not constant with time and oscillates as seen in Fig. 4(a). This comes from the out-of-plane anisotropy Kzin the case of collinear AFM [18], and the DM interaction plays a similar role in Mn 3Sn. In collinear AFM, only a small oscillation of ˙ ϕ(t) is detectable through the inverse spin-Hall effects while we can directlydetect the whole octupole precession motion such as throughthe magneto-optical Kerr effect [ 46] and an oscillation of the Hall voltage [ 53]. This is a clear advantage of Mn 3Sn over collinear AFM. Figure 4(c) shows space- and time-averaged /angbracketleft˙ϕ/angbracketrightand/angbracketleftθ/angbracketright (the polar angle of the spins) in the steady state. /angbracketleft˙ϕ/angbracketrightof the effective model is simply given by ¯ h/angbracketleft˙ϕ/angbracketright=τD/α, and again agrees well with the LLG calculations. The maxi-mum frequency f maxof the precession is achieved where all spins are along the zdirection and is estimated as fmax= (2√ 3D+6J)/h/similarequal7.2T H z[ 65], which is comparable to the magnon frequency of KFeS 2[59]. On the other hand, owing to the extremely small in-plane anisotropy of Mn 3Sn, the threshold frequency fthr∼O(K⊥/αh) is about 10 GHz [51,66]. Thus, the frequency in the range of three orders of magnitude may be available in Mn 3Sn. Conclusion. In this Rapid Communication, we develop a method to obtain a low-energy effective model of noncollinearAFM based on the cluster multipole theory and apply it to asimple model of Mn 3Sn. A comparison between the original and effective models shows good agreement both in the do-main wall dynamics and in the coherent steady precession ofspins. This means that the low-energy dynamics of Mn 3Sn is almost dominated by the octupole degrees of freedom andwe do not have to trace that of each spin, which enables usto reduce the computational cost. Our results show that theoctupole dynamics in Mn 3Sn is almost the same as that of the Néel vector in collinear AFM, which indicates that Mn 3Sn possesses advantages of AFM as well as of FM. Thus, Mn 3Sn would be a promising candidate for future applications inmultipole-based electronics. Acknowledgments. We are grateful to W. Koshibae, S. Miwa, Y . Otani, S. Nakatsuji, K. Yakushiji, and S. Yuasa formany valuable discussions. 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Here, we follow thenotation defined in our simplified model. [53] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevResearch.2.012045 for details of the deriva- tion of the effective model, numerical simulation and oscillationof the Hall voltage. 012045-5TAKUYA NOMOTO AND RYOTARO ARITA PHYSICAL REVIEW RESEARCH 2, 012045(R) (2020) [54] The mass term with γ∝K3 ⊥and sixfold symmetry in Ref. [ 37] is not predicted in our model. This comes from the differenceof the anisotropy terms in the original models (see Ref. [ 45]). [55] At low energy, the staggered field has almost the same effect as the dampinglike torque introduced by Eq. ( 7). [56] This oscillation originates from a long lifetime precession mode in FM, which is irrelevant to our arguments. [57] If we take into account the out-of-plane anisotropy K zin Mn 3Sn, we obtain a higher maximum velocity since ¯ hτ−1 changes from 2√ 3D+6Jto 2√ 3D+6J+Kz[53]. [58] M. T. Hutchings and E. J. 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PhysRevLett.104.147202.pdf

Domain-Wall Motion in Ferromagnetic Nanowires Driven by Arbitrary Time-Dependent Fields: An Exact Result Arseni Goussev, J. M. Robbins, and Valeriy Slastikov School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom (Received 15 February 2010; published 6 April 2010) We address the dynamics of magnetic domain walls in ferromagnetic nanowires under the influence of external time-dependent magnetic fields. We report a new exact spatiotemporal solution of the Landau-Lifshitz-Gilbert equation for the case of soft ferromagnetic wires and nanostructures with uniaxialanisotropy. The solution holds for applied fields with arbitrary strength and time dependence. We furtherextend this solution to applied fields slowly varying in space and to multiple domain walls. DOI: 10.1103/PhysRevLett.104.147202 PACS numbers: 75.78.Fg, 75.75. /C0c Introduction.— The motion of magnetic domain walls (DWs) in ferromagnetic nanowires has recently become asubject of intensive research in the condensed matter phys-ics community [ 1]. Manipulation of DWs by external magnetic fields, and, in particular, the question of howthe DW propagation velocity depends on the applied field,have drawn considerable attention [ 2–4]. In ferromagnetic nanowires, the dynamics of the orien- tation of the magnetization distribution, mðx; tÞ(normal- ized so that jmj¼ 1), is described by the Landau-Lifshitz- Gilbert (LLG) equation [ 5] @m @tþ/C11m/C2@m @t¼ð1þ/C112Þm/C2ðHðmÞþHaÞ;(1) where xis the coordinate along the nanowire, tis time, /C11is the Gilbert damping parameter, Hadenotes the applied magnetic field, and HðmÞ¼/C0 /C14E=/C14m, where EðmÞ¼A 2Z R/C12/C12/C12/C12/C12/C12/C12/C12@m @x/C12/C12/C12/C12/C12/C12/C12/C122 dxþK1 2Z Rð1/C0ðm/C1^xÞ2Þdx þK2 2Z Rðm/C1^yÞ2dx (2) is the reduced micromagnetic energy. Here, Ais the ex- change constant of the material, and K1,K2/C210are the anisotropy constants along the (easy) xand (hard) yaxes. The anisotropy constant along the zaxis is taken to be zero by convention. To date only one exact spatiotemporal [ 6] solution of the LLG equation has been reported in the literature, namely, the so-called Walker solution [ 7] [exact solutions in the absence of damping (i.e., for /C11¼0) are discussed in Ref. [ 8] ]. The analysis of Schryer and Walker [ 7] applies to the case where K2>0, i.e., where the anisotropy con- stants in the transverse plane are strictly unequal. This isappropriate for a thin film or thin strip geometry. Theapplied field is taken to be uniform in space, constant intime, and directed along the nanowire, i.e., H aðx; tÞ¼ Ha^x.F o r jHajless than a certain threshold HW, the so- called Walker breakdown field, a planar domain wall prop-agates rigidly along the nanostrip with velocity depending nonlinearly on Ha. In this Letter we present an exact spatiotemporal solu- tion of the LLG equation that, to our knowledge, has notbeen previously reported in the literature. We consider thecase of transverse isotropy, i.e., K 2¼0. This is appropriate for soft ferromagnetic nanowires whose cross-sectionaldimensions are comparable, as well as for uniaxial nano- wires whose easy axis lies along the wire. We take the applied field to lie along the nanowire, as in the case of theWalker solution, but allow for arbitrary time dependence,i.e.,H aðx; tÞ¼HaðtÞ^x. Exact solution of the LLG equation.— The boundary conditions appropriate for a domain wall with finite micro-magnetic energy EðmÞare given by mðx; tÞ!/C6 ^xasx! /C61.F o r K 2¼0the magnetization-dependent field His given by HðmÞ¼A@2m @x2þK1ðm/C1^xÞ^x: (3) We now take into account the fact that mhas its values on S2, and parametrize min terms of angles /C18ðx; tÞand/C30ðx; tÞ according to m¼ðcos/C18;sin/C18cos/C30;sin/C18sin/C30Þ. From Eqs. ( 1) and ( 3) we obtain the LLG equation in the equiva- lent form _/C18/C0/C11_/C30sin/C18þAð1þ/C112Þð/C3000sin/C18þ2/C180/C300cos/C18Þ¼0;(4a) /C11_/C18þ_/C30sin/C18þð1þ/C112Þð/C0A/C1800þAð/C300Þ2sin/C18 /C2cos/C18þK1cos/C18sin/C18þHaðtÞsin/C18Þ¼0;(4b) FIG. 1 (color online). Dynamics of domain walls. See text for discussion.PRL 104, 147202 (2010) PHYSICAL REVIEW LETTERSweek ending 9 APRIL 2010 0031-9007 =10=104(14) =147202(3) 147202-1 /C2112010 The American Physical Societywhere the overdot denotes @=@t and the prime denotes @=@x . We now look for a solution of Eq. ( 4) in the form /C18/C3ðx; tÞ¼/C180ðx/C0x/C3ðtÞÞ;/C30 /C3ðx; tÞ¼/C30/C3ðtÞ; (5) where /C180ðxÞ¼ 2 arctan exp ð/C0x=d 0Þ;d 0¼ffiffiffiffiffiffiffiffiffiffiffiffi A=K 1q :(6) /C180ðxÞdescribes the static domain wall in the absence of an applied field. The magnetization density determined by /C180ðxÞminimizes the micromagnetic energy EðmÞfor the specified boundary conditions. Substituting Eq. ( 6) into Eq. ( 4), and taking into account that /C180 0¼/C0 sin/C180=d 0 and/C1800 0¼sin2/C180=ð2d2 0Þ, we find that /C18/C3and/C30/C3satisfy the LLG equation ( 4) provided that x/C3ðtÞand/C30/C3ðtÞsatisfy _x/C3¼/C0/C11d 0HaðtÞ; _/C30/C3¼/C0HaðtÞ: (7) [In fact, ( 6) and ( 7) provide the only solution of the form (5).] Equations ( 5)–(7) constitute the main result of this Letter. They represent an exact solution of the LLG equa- tion, and describe a DW, with profile independent of theapplied field, propagating along the nanowire with velocity _x /C3while precessing about the nanowire with angular ve- locity _/C30/C3. No restrictions have been imposed on the strength of the applied magnetic field and no assumptions have been made about its time dependence. We now compare the precessing solution Eqs. ( 5)–(7) with the Walker solution [ 7]. The Walker solution is de- fined only for K2>0(the fully anisotropic case) and time- independent Haless than the breakdown field HW¼/C11K 2=2: (8) It is given by /C18Wðx; tÞ¼/C180/C18x/C0VWt /C13/C19 ;/C30 Wðx; tÞ¼/C30W;(9) where sin2/C30W¼Ha=HW (10) determines the (fixed) inclination of the DW plane and VW¼/C131þ/C112 /C11d0Ha;/C13 ¼/C18K1 K1þK2cos2/C30W/C191=2 (11) gives the DW velocity. There are several characteristic differences between the Walker solution and the precessing solution which shouldbe distinguishable experimentally. Foremost is the fact that the Walker solution exists only for constant applied fields whose strength does not exceed a certain threshold, so thatthe DW velocity is bounded. The precessing solution isdefined for time-dependent applied fields of arbitrary strength, so that the DW velocity, which for the precessingsolution is proportional to the field strength, can be arbi- trarily large. Next, while for the Walker solution the plane of the DW remains fixed, for the precessing solution itrotates about the nanowire at a rate proportional to H a. Finally, we observe that, for the Walker solution, the DWprofile contracts ( /C13> 1) or expands ( /C13> 1) in response to the applied field, whereas for the precessing solution theDW profile propagates without distortion. Spatially nonuniform applied fields and multiple do- main walls.— We now extend our results to applied fields that depend on both position along the nanowire andtime, i.e., H a¼Haðx; tÞ^x. For any (nonsingular) applied field, Eq. ( 4) is satisfied at xoutside the DW transition layer jx/C0x/C3ðtÞj /C29 d0(up to exponentially small terms). Assuming now that the field varies slowly across thetransition region, jH aðx; tÞ/C0Haðx/C3ðtÞ;tÞj/C28j Haðx/C3ðtÞ;tÞj forjx/C0x/C3ðtÞj&d0;(12) we obtain an approximate solution of the LLG equation: the magnetization density is given by Eqs. ( 5) and ( 6) with _x/C3¼/C0/C11d 0Haðx/C3ðtÞ;tÞ; _/C30/C3¼/C0Haðx/C3ðtÞ;tÞ:(13) The physical meaning of Eq. ( 13) is quite obvious: the DW is only sensitive to the applied field within the transitionlayer. This approximation can now be extended to the case of Nnonoverlapping DWs. Indeed, /C18 Nðx; tÞ¼XN n¼1/C180fð/C0 1Þnþ1½x/C0xnðtÞ/C138g; (14a) /C30Nðx; tÞ¼/C30/C22nðtÞ;n ¼/C22nminimizes jx/C0xnðtÞj; (14b) with xkþ1ðtÞ/C0xkðtÞ/C29d0fork¼1;...;N/C01, consti- tutes an approximate solution of the LLG equation giventhat _x n¼ð /C0 1Þn/C11d 0HaðxnðtÞ;tÞ; (15a) _/C30n¼/C0HaðxnðtÞ;tÞ; (15b) forn¼1;...;N. For the case of a spatially uniform ap- plied field Eqs. ( 14) and ( 15) describe the time evolution of NDWs such that any two adjacent DWs travel in opposite directions while rotating in the same direction (and withthe same angular velocity) around the nanowire. Figure 1 illustrates the dynamics of two ( N¼2) DWs. Conclusions.— In this Letter we have presented an exact spatiotemporal solution of the LLG equation that has not been previously reported in the literature. The validity ofPRL 104, 147202 (2010) PHYSICAL REVIEW LETTERSweek ending 9 APRIL 2010 147202-2the new solution requires no assumptions about the time dependence or strength of the applied field. We have then provided a natural extension of the solu- tion to physical situations in which the applied field varies slowly in space. An approximate solution of the LLGequation for the case of multiple domain walls has alsobeen obtained. A. G. acknowledges the support by EPSRC under Grant No. EP/E024629/1. [1] See, e.g., S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008); R. P. Cowburn, Nature (London) 448, 544 (2007). [2] Z. Z. Sun and J. Schliemann, Phys. Rev. Lett. 104, 037206 (2010).[3] X. R. Wang, P. Yan, and J. Lu, Europhys. Lett. 86, 67001 (2009); X. R. Wang, P. Yan, J. Lu, and C. He, Ann. Phys. (N.Y.) 324, 1815 (2009). [4] M. C. Hickey, Phys. Rev. B 78, 180412(R) (2008). [5] See, e.g., A. Hubert and R. Scha ¨fer,Magnetic Domains: The Analysis of Magnetic Microstructures (Springer, Berlin, 1998). [6] The only other exact solution of the LLG equation re- ported in the literature [Z. Z. Sun and X. R. Wang, Phys. Rev. Lett. 97, 077205 (2006)] appears in the problem of magnetization switching, where the magnetization density is considered to be uniform in space and is a function of time only, i.e., m¼mðtÞ. [7] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). [8] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys. Rep. 194, 117 (1990).PRL 104, 147202 (2010) PHYSICAL REVIEW LETTERSweek ending 9 APRIL 2010 147202-3

PhysRevA.85.032324.pdf

PHYSICAL REVIEW A 85, 032324 (2012) Superoperator analysis of entanglement in a four-qubit cluster state Yaakov S. Weinstein, Jay Feldman,*Jacob Robins,†Jason Zukus,‡and Gerald Gilbert Quantum Information Science Group, MITRE , 260 Industrial Way West, Eatontown, New Jersey 07224, USA (Received 24 November 2010; revised manuscript received 15 February 2012; published 19 March 2012) In this paper we utilize superoperator formalism to explore the entanglement evolution of four-qubit cluster states in a number of decohering environments. A four-qubit cluster state is a resource for the performanceof an arbitrary single-logical-qubit rotation via measurement-based cluster-state quantum computation. Weare specifically interested in the relationship between entanglement evolution and the fidelity with whichthe arbitrary single-logical-qubit rotation can be implemented in the presence of decoherence as this will haveimportant experimental ramifications. We also note the exhibition of entanglement sudden death (ESD) andask how severely its onset affects the utilization of the cluster state as a means of implementing an arbitrarysingle-logical-qubit rotation. DOI: 10.1103/PhysRevA.85.032324 PACS number(s): 03 .67.Mn, 03 .67.Bg, 03 .67.Pp I. INTRODUCTION Entanglement is a uniquely quantum mechanical phe- nomenon in which quantum systems exhibit correlations aboveand beyond what is classically possible. Entangled systems arethus an important resource for many quantum information pro-cessing protocols including quantum computation, quantummetrology, and quantum communication [ 1]. In the area of quantum computation, certain entangled states play a uniquerole as the basic resource for measurement-based quantumcomputation. The cluster state in particular allows for quantumcomputation to proceed via single-qubit measurements aftercreation of the cluster state [ 2]. An important area of research is to understand the possible degradation of entanglement due to decoherence. Decoher-ence, stemming from unwanted interactions between thesystem and environment, is a major challenge confronting ex-perimental implementations of quantum computation, metrol-ogy, and communication [ 3]. Decoherence may be especially detrimental to highly entangled states [ 4] and, indeed, much work has been done on studying the effects of decoherence oncluster states [ 5]. An extreme manifestation of the detrimental effects of decoherence on entangled states is “entanglement suddendeath” (ESD), in which entanglement within a system iscompletely lost in a finite time [ 6,7] despite the fact that the loss of system coherence is asymptotic. This aspectof entanglement has been well explored in the case ofbipartite systems and there are a number of studies lookingat ESD in multipartite systems [ 8–13] including the four-qubit cluster state [ 14]. In addition, there have been several initial experimental ESD studies [ 15]. In this paper we study the entanglement evolution of a four-qubit cluster state which can be used as the basic resourceto perform an arbitrary single (logical) qubit rotation viacluster-state quantum computation. We analyze the effects *Present address: Johns Hopkins University, Baltimore, MD 21201, USA. †Present address: University of Pennsylvania, Philadelphia, PA 19104, USA. ‡Present address: Princeton University, Princeton, NJ 08544, USA.of various decoherence models on the entanglement of the premeasurement state and compare the entanglement behaviorto the accuracy with which the decohered state can be usedto implement the desired arbitrary single-qubit rotation. Tocompletely characterize the effects of decoherence we makeuse of superoperator representations and aspects of quantumprocess tomography. Quantum process tomography is anexperimental protocol which is used to completely determine(open) system dynamics. The information gleaned fromquantum process tomography can, in turn, be used to determinea wealth of accuracy measures. One would expect that theproper working of cluster-state-based quantum computationwould be strongly dependent on the amount of entanglementpresent in the premeasurement cluster state. Thus, an explicitanalysis of the strength of this dependence, especially whenattempting to perform basic computational gates, is essentialfor experimental implementations of cluster-state quantumprotocols. A secondary aim of this paper is to analyze the effect of ESD on the implementation of the single-logical-qubit rotation. TheESD phenomenon is of interest on a fundamental level andimportant for the general study of entanglement. However, it isnot yet clear what the effect of ESD is on quantum informationprotocols. Are different quantum protocols helped, hurt, or leftintact by ESD? Previous results suggest a possible connectionbetween the loss of certain types of entanglement in the four-qubit cluster state and the fidelity with which measurementson the four-qubit state will lead to the desired state on theremaining, unmeasured qubits [ 14]. The current paper expands these results by exploring additional decoherence mechanismsand calculating state independent accuracy measures such asthe gate fidelity. Other explicit studies of the effect of ESDon quantum information protocols include the three-qubitphase flip code, a four-qubit decoherence free subspace anda three-qubit noiseless subsystem [ 13,16]. None of these studies find a correlation between the accuracy of the protocolimplementation and the advent of ESD. II. CLUSTER STATES The cluster state [ 17] is a specific type of entangled state that can be used as a resource for measurement-based quantumcomputation [ 2]. A cluster state can be constructed by rotating 032324-1 1050-2947/2012/85(3)/032324(9) ©2012 American Physical SocietyWEINSTEIN, FELDMAN, ROBINS, ZUKUS, AND GILBERT PHYSICAL REVIEW A 85, 032324 (2012) all qubits into the state |+/angbracketright =1√ 2(|0/angbracketright+| 1/angbracketright) and applying control phase ( CZ) gates, diag(1 ,1,1,−1), between desired pairs. In a graphical picture of a cluster state, qubits arerepresented by circles and pairs of qubits that have beenconnected via CZgates are connected by a line. A cluster state with qubits arranged in a two-dimensional lattice, suchthat each (nonedge) qubit has been connected via CZgates with its four nearest neighbors, suffices for universal QC. After constructing the cluster state, any quantum computa- tional algorithm can be implemented using only single-qubitmeasurements performed along axes in the x-yplane of the qubit, that is, the plane spanned by |+/angbracketright = 1√ 2(|0/angbracketright+| 1/angbracketright), |+i/angbracketright=1√ 2(|0/angbracketright+i|1/angbracketright). These processing measurements are performed by column, from left to right, until only the last column remains unmeasured. The last column contains theoutput state of the quantum algorithm which can be extractedby a final readout measurement. One can view each row of thecluster-state lattice as the evolution of a single-logical qubit intime. Two (logical) qubit gates are performed via connectionsbetween two rows of the cluster state. CZgates in particular are “built-in” to the cluster state and simple measurement on twoconnected qubits in different rows automatically implementsthe gate. Measurement of a physical qubit in the cluster state at an angle φfrom the xaxis in the x-yplane implements a rotation on the logical qubit given by X(πm)HZ(φ), where H= 1√ 2(11 1−1) is the Hadamard gate, and Z(α)[X(α)] is a z(x) rotation by an angle α[18]. The dependence of the logical operation on the outcome of the measurement is determinedby the value of m=0,1 for measurement outcome −1,+1, respectively. An arbitrary single-logical-qubit rotation can beimplemented via three such measurements yielding HZ/parenleftbig θ 1+πmθ1/parenrightbig X/parenleftbig θ2+πmθ2/parenrightbig Z/parenleftbig θ3+πmθ3/parenrightbig , where ( θ1,θ2,θ3) are the Euler angles of the rotation. As an example, by drawing the Euler angles according to the Haarmeasure, a random single-qubit rotation can be implemented. We explore an arbitrary single (logical) qubit cluster-based rotation performed on an arbitrary initial state in a decoheringenvironment. To construct the relevant cluster, a qubit isplaced in the desired initial state |ψ in(α,β)/angbracketright=cosα|0/angbracketright+ eiβsinα|1/angbracketright, where ρin(α,β)=|ψin(α,β)/angbracketright/angbracketleftψin(α,β)|. Three additional qubits (numbered 2–4) are rotated into the |+/angbracketrightstate and CZgates are then applied between the original qubit and 2, 2 and 3, and 3 and 4. The four-qubit initial (pure) stateis thus |ψ 4I(α,β)/angbracketright=CZ 34CZ 23CZ 12(|ψin(α,β)/angbracketright⊗| + /angbracketright⊗3)o r ρ4I(α,β)=|ψ4I(α,β)/angbracketright/angbracketleftψ4I(α,β)|. III. ENTANGLEMENT MEASURES To quantify and monitor entanglement in the above con- structed types of cluster states as they undergo decoherence,we use an entanglement measure known as the negativity N defined as the most negative eigenvalue of the partial transposeof the system density matrix [ 19]. There are a number of inequivalent forms of the negativity for any four-qubit system:the partial transpose may be taken with respect to any single-qubitN (j)or the partial transpose may be taken with respect to any two qubits N(j,k). The negativity thus defined does notdifferentiate different types of entanglement. Furthermore, due to the possible presence of bound entanglement, the disappear-ance of all measurable negativity does not guarantee that thestate is separable. However, the presence of negativity doesensure the presence of distillable entanglement in the system. A method of monitoring specifically four-qubit cluster type entanglement is via the expectation value of the statewith respect to an appropriate entanglement witness [ 20]. Entanglement witnesses are observables with positive or zeroexpectation value for all states not in a specified entanglementclass and a negative expectation value for at least one stateof the specified entanglement class. Entanglement witnessesmay allow for an efficient, though imperfect, means ofexperimentally determining whether entanglement is presentin a state (as opposed to inefficient state tomography). Thisis especially important for experiments with any more than afew qubits as it may be the only practical means of decidingwhether or not sufficient entanglement is present in thesystem. The entanglement witnesses used here are specificallydesigned to detect four-qubit cluster type entanglement of thekind exhibited by states of the form ρ 4I(α,β). In Ref. [ 21]a n entanglement witness is constructed for a cluster state in whichthe first qubit is |+/angbracketright. It is given by W +=1/2−ρ4I(π/4,0). For the current study we modify this witness by a phase rotationof angle βon the first qubit yielding witnesses of the form W β=1/2−e−iβσ1 z/2ρ4I(π/4,0)eiβσ1 z/2, (1) where σk zis the Pauli zspin operator on qubit kandβis the phase of the initial state |ψin(α,β)/angbracketright. This witness more accurately determines whether the cluster states of interest inthis work have any four-qubit cluster entanglement. IV . SUPEROPERATOR REPRESENTATION AND ACCURACY MEASURES We would like to completely describe the evolution of the single-logical qubit undergoing an arbitrary cluster-basedrotation in the presence of decoherence. To do so we needto account for both the decoherence and the measurementsof the (three) physical qubits. For this study we assume thatthere is no interaction between the qubits of the cluster state(beyond the initial conditional phase gates used to construct thecluster state). We further assume that all decoherence occursafter construction of the cluster state but before measurements.Measurement is done on each of the first three qubits in bases atangleθ i,i=1,2,3, from the positive xaxis. As noted above, the measurement bases are chosen so as to implement thedesired logical qubit rotation. After measurement the finalstate of the logical qubit resides on the fourth, unmeasured,physical qubit and is a function of the initial state of the logicalqubit|ψ in(α,β)/angbracketright, the decoherence strength p, and the three measurement angles θi:ρout=ρout(α,β,p,θ 1,θ2,θ3). To construct the dynamical superoperator of the one qubit logical gate we follow the method described in Refs. [ 3,22]. We construct the appropriate cluster, apply decohering evolution,and perform the desired measurements on a set of states|ψ 4I(α,β)/angbracketrightwhich span the one qubit Hilbert space (Hilbert 032324-2SUPEROPERATOR ANALYSIS OF ENTANGLEMENT IN A ... PHYSICAL REVIEW A 85, 032324 (2012) space dimension N=2). From this we can construct the N2×N2Liouvillian superoperator Swhere S(p,θ 1,θ2,θ3)ρin(α,β)=ρout(α,β,p,θ 1,θ2,θ3). (2) Note that in Liouvillian space, density matrices are column vectors of dimension N2×1. From the superoperator Swe can construct the corresponding N×NKraus operators following [23]. An analysis of the Kraus operator representation of the scenarios outlined below is done in Appendix B. There are two accuracy measures that we find useful for our analysis and that we use to compare the accuracy ofthe implemented gate to the evolution of the entanglement.These measures quantify how well the system performed thedesired operation and are thus vital in experimental work. Thefirst accuracy measure we utilize is the cluster-state fidelity of the four-qubit state ρ 4F(α,β,p ) before measurement but after decoherence as a function of p. This is given by Fc=Tr[ρ4I(α,β)ρ4F(α,β,p )†]. (3) This is a simple measure which tells how close the actual final state is to the desired one in the presence of decoherence. Thesecond accuracy measure is the gate fidelity of the attempted single (logical) qubit rotation U(θ 1,θ2,θ3). The gate fidelity quantifies the accuracy with which the attempted evolutionwas achieved independent of the initial state of the system.The superoperator allows us to calculate the gate fidelity via F g=Tr[S(0,θ1,θ2,θ3)S(p,θ 1,θ2,θ3)†], (4) where S(0,θ1,θ2,θ3)=U(θ1,θ2,θ3)⊗Conj(U(θ1,θ2,θ3)).(5) In the next three sections we look at decohering environ- ments of experimental interest: phase damping, amplitudedamping, and depolarization. In all three we explore theentanglement evolution as a function of decoherence strengthassuming that the decoherence occurs prior to measurement.Measurements are performed on the decohered state and thusthe evolution of the fidelity of the implemented operationcan be compared to the evolution of the entanglement. Ourgoal is to see what correlations exist between entanglementdegradation and the accuracy with which the cluster statecan be used to implement the desired single-logical-qubitrotation. We will also note occurrences of ESD and what effectthis phenomenon may have on the ability of the system toimplement the desired rotation. V . DECOHERING ENVIRONMENTS In this paper we discuss the effects of three different decohering environments on the four-qubit cluster state. Theyare independent qubit phase damping, amplitude damping,and depolarization. Each of these environments is completelydescribed by a Kraus operator representation. The Krausoperator representation for the phase damping environmentis given by K 1=/parenleftbigg10 0√1−p/parenrightbigg ,K 2=/parenleftbigg00 0√p/parenrightbigg , (6)for the amplitude damping environment K1=/parenleftbigg10 0√1−p/parenrightbigg ,K 2=/parenleftbigg0√p 00/parenrightbigg , (7) and for the depolarizing environment K1=⎛ ⎝/radicalBig 1−3p 40 0/radicalBig 1−3p 4⎞ ⎠,K /lscript=√p 2σ/lscript, (8) where the σ/lscriptare the Pauli spin operators /lscript=x,y,z . In each case we have defined a decoherence strength parameter p whose exact behavior as a function of time is left unspecifiedso as to accommodate various possible experimentally relevantbehaviors. As an example, one might have p=1−e −κτ, where τis time and κis the decay constant. In the case of independent qubit dephasing, this decoherence behavior woulddecay off-diagonal terms of the density matrix as a power ofe −κτand thus go to zero (i.e., p→1) only in the limit of infinite times. We also assume equal decoherence for all fourqubits. VI. RESULTS Starting with the general state ρ4I(α,β) we separately apply each of the decohering environments with arbitrarydecoherence strength pand determine the entanglement in the output state ρ 4F(α,β,p ). For the depolarizing environment the eigenvalues of the partially transposed states can be determinedanalytically. For the other environments the eigenvalues foreachαandpmust be determined numerically. The entan- glement evolution is shown in Figs. 1–3 and summarized in Table I. We also calculate cluster-state fidelity comparing the state before and after decoherence. The entanglement of the system before or after decoherence is completely independent of β.βarises as a rotation exp[−i β 2σz] on the state |ψin(α,0)/angbracketright. This rotation commutes with CZgates and, as single-qubit rotations do not effect the entanglement, the entanglement of the state |ψ4I(α,β)/angbracketright is independent of β.T h eσzrotation which gives rise to βalso commutes with the Kraus operators and of the phase dampingand amplitude damping environments. Thus, the entanglementof the cluster states under these types of decoherence is alsoindependent of β. The depolarizing environment, however, includes (two) Kraus operators proportional to σ xandσy which do not commute with the σzrotation. Nevertheless, the rotation amounts to simply reorienting σxandσyand thus, taken together, the σzrotation can be applied after the decoherence with no effect on the final state. We then apply measurements to the first three qubits along axes in the x-yplane at angles θ1,θ2,θ3from the xaxis. From this we determine the superoperator S(p,θ 1,θ2,θ3)o f the attempted arbitrary rotation. These superoperators areexplicitly shown in Appendix A.F r o m S(p,θ 1,θ2,θ3) we can compute the state independent gate fidelity as a function of theattempted rotation and the decoherence strength and determinethe output state for any input state. Explicit equations for thetwo fidelity measures are given in Table Iand are depicted in Figs. 4–6. There are a number of differences in the entangle- ment evolution between the three decohering environments. 032324-3WEINSTEIN, FELDMAN, ROBINS, ZUKUS, AND GILBERT PHYSICAL REVIEW A 85, 032324 (2012) TABLE I. Summary of entanglement and fidelity results for the three explored decohering environments. The columns show values of p at which ESD is exhibited, values of pwhere the entanglement witness cannot detect entanglement, gate fidelity, and the state fidelity, where q=p−1a n d ˜p=√1−p. ESD (N(j)=0) Tr[ ρWβ]=0 FgFC Dephasing N(1),N(1,2):p=2(√ 2−1) p/lessorsimilar0.51 16[10+6˜p+p(p−6˜p−7)1 32[16(1+˜p)+p(p−6˜p−14) +q(p+2˜p−2) cos 2 θ2 −p(p−2˜p−2) cos 4 α] N(1,3):p/similarequal0.938 +2q(p+2˜p−2) cos θ2 2cos 2θ3] Amplitude none p< 0.2 same as dephasing see figure damping Depolarizing all N(j):p/lessorequalslant0.45 p/lessorsimilar0.21 8((p−2){−4+p[7+p(p−6)]}1 32(p−2)2[3p(3p−5)+8−qpcos 4α] +q2p2cos 2θ2) Entanglement degrades most slowly in the dephasing envi- ronment before exhibiting ESD at high values of p.I nt h e amplitude damping environment the entanglement degradesmore quickly but never exhibits ESD, and in the depolarizingenvironment the entanglement degrades most quickly andESD is exhibited for low values of p. In addition, different states lose entanglement at different rates depending on thedecohering environment. For example, N (1,2)in all states degrades uniformly in the depolarizing environment but not 0 0.1 0.1 0.20.2 0.3 0.30.4 0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpa0 00 0.1 0.2 0.3 0.4 0.5 0.5 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpb 0 0.1 0.2 0.30.3 0.3 0.4 0.40.4 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpc 0.4 0.3 0.20.10 0.10.2 0.30.4 0.0 0.5 1.0 1.50.00.20.40.60.81.0 αpd FIG. 1. (Color online) Negativity as a function of dephasing strength pand initial state parametrized by α.( a )N(1), partial transpose taken with respect to the first qubit, (b) N(1,2), partial transpose taken with respect to the first two qubits, and (c) N(1,3), partial transpose taken with respect to qubits one and three. (d) Evolution of the expectation value of the entanglement witness as a function of initial state (the expectation value is not dependent on β) and decoherence strength. Notice that the dephasing strength at which the expectation value goes to zero is dependent on αand is well below the point where ESD is exhibited for N(1). The four-qubit cluster entanglement can only be observed at low levels of decoherence p/lessorsimilar0.5.in the other environments. This is most likely due to uniform effect of the depolarizing environment over the Bloch sphere. With respect to detecting entanglement via the entangle- ment witness, we find for all decohering environments that thedetection of four-qubit cluster entanglement goes to zero muchmore quickly than any of the entanglement measures (and goesto zero for the amplitude damping environment though no ESDis exhibited). This shows a quick demise specifically for thefour-qubit cluster entanglement (or indicates the inefficiency ofthe witnesses). The maximum decoherence for which detectionis still possible is about the same for the amplitude dampingand depolarizing and much higher for dephasing. 0 0.1 0.1 0.20.2 0.30.3 0.40.4 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpa 0.0 0.1 0.150.20.250.3 0.35 0.40.45 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpb 0.05 0.1 .15 0.2 0.250.30.30.30.35 0.350.35 0.40.40.40.450.45 0.45 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpc 0.2 0.100.10.2 0.30.4 0.0 0.5 1.0 1.50.00.20.40.60.81.0 αpd FIG. 2. (Color online) Evolution of various negativity measures as a function of initial state parametrized by αand amplitude damping strength p.( a )N(1), the entanglement goes to zero at α=π/2. (b)N(1,2)and (c) N(1,3), for these measures the entanglement goes to zero only in the limit of p→1. (d) Expectation value of four-qubit cluster state with respect to entanglement witness Wβ, withβ=0, as a function of initial state and decoherence strength. The four-qubit cluster entanglement goes undetected at very low decoherence strengths ( p< 0.2) despite the presence of some sort of entanglement for any nonzero p. 032324-4SUPEROPERATOR ANALYSIS OF ENTANGLEMENT IN A ... PHYSICAL REVIEW A 85, 032324 (2012) 0 0.1 0.1 0.20.2 0.30.30.40.4 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpa 0 0.1 0.2 0.3 0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpb 0 0.1 0.2 0.30.3 0.3 0.4 0.40.4 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpc 0.4 0.30.2 0.1 00.10.2 0.30.4 0.0 0.5 1.0 1.50.00.20.40.60.81.0 αpd FIG. 3. (Color online) Negativity as a function of depolarizing strength pand initial state parametrized by α:( a )N(1),( b )N(1,2),a n d (c)N(1,3). (d) Evolution of expectation value of entanglement witness as a function of initial state and decoherence strength. Notice thatthe evolution is similar to N (1)though the expectation value goes to zero well before ESD of N(1). The four-qubit cluster entanglement can only be observed at low levels of decoherence p/lessorsimilar0.2. When comparing the entanglement evolution to the evo- lution of the gate fidelity or cluster-state fidelity we findonly superficial correlations. In addition, we do not find anysignature of ESD in the fidelity functions. While clearly bothentanglement and fidelity decrease as the decoherence strengthincreases, these superficial correlations do not give rise to anyproblems regarding the viability of quantum computing. Wenote, however, that this does not prove that ESD is completelyirrelevant with respect to quantum computation in general, itsimply demonstrates that the effect of ESD on this specificprotocol is not manifest in the fidelity measure. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 θ2pa0.1 0.10.2 0.3 0.40.5 0.6 0.7 0.8 0.9 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpb FIG. 4. (Color online) (a) Gate fidelity of an arbitrary single-qubit rotation as a function of the rotation (parametrized by Euler angle θ2 withθ3set to zero; note that the gate fidelity is independent of θ1) and dephasing strength p. (b) Fidelity of premeasurement four-qubit cluster state as a function of initial state parametrized by α(this fidelity is independent of β)a n dp.0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 θ2pa0.1 0.20.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpb FIG. 5. (Color online) (a) Gate fidelity as a function of amplitude damping strength pand choice of rotation (parametrized by θ2 withθ3=0; note that the gate fidelity is independent of θ1). (b) Contour plot of four-qubit state fidelity as a function of amplitudedamping strength and initial state (parametrized by α; the fidelity is independent of β). The gate fidelity provides a state independent measure for the accuracy of the entire single-qubit rotation algorithm. Thisfidelity, and the explicit superoperators given in Appendix A, are vital information for those attempting to discern the possi-ble accuracy that can be achieved by invariably decoherentexperimental systems. We immediately note that the gatefidelity is the same in the dephasing and amplitude dampingenvironments F g A=Fg z. This is not surprising considering that the Kraus operators governing these decohering environmentsare very similar. Nevertheless, the cluster-state fidelity ofthe two environments are not at all similar as seen inFigs. 4and 5. This demonstrates the importance of utilizing multiple accuracy measures. In addition, the amplitude damp-ing environment does not cause ESD for any entanglementmeasure while dephasing does. This suggests that dephasingis more harmful to the entanglement types found in thefour-qubit cluster state than is amplitude damping. Rotatingthe physical system qubits, such that a dephasing environmentacts as an amplitude damping environment, could conserveentanglement though it would not increase the accuracy of theimplemented single-logical-qubit rotation. Both fidelity typesdecrease much more quickly in the depolarizing environmentthan in the other two environments. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 θ2pa 0.1 0.2 0.3 0.4 0.50.6 0.70.80.9 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 αpb FIG. 6. (Color online) (a) Gate fidelity for arbitrary single-qubit rotation in depolarization environment as a function of depolarization strength and the rotation (parametrized by θ2, in this case the gate fidelity does not depend either on θ1orθ3). (b) Fidelity of four-qubit cluster state as a function of initial state (parametrized by αasFC P does not depend on β) and depolarizing strength. 032324-5WEINSTEIN, FELDMAN, ROBINS, ZUKUS, AND GILBERT PHYSICAL REVIEW A 85, 032324 (2012) VII. CONCLUSION In conclusion, we have studied an attempted implemen- tation of an arbitrary single-qubit rotation via cluster-statequantum computation in a noisy environment. We specificallylooked at three different decohering environments: dephasing,amplitude damping, and depolarization. Such studies are vitalfor future experiments in cluster-state quantum computationas they will shed light on the types of errors that occur duringimplementation and can prescribe what types and strengths oferror are tolerable if attempting to achieve a certain accuracy ofimplementation. To this end we have provided both accuracymeasures of the implementation as a function of the attemptedrotation and the decoherence type and strength, as well asthe complete superoperator describing the entire process. Thesuperoperators specifically allow for the identification of thetype of error that will be manifest in the output state of anexperiment given the error models. The accuracy measuresand the superoperators are also vital for the determination ofcluster-based fault tolerance thresholds. In addition, we have studied the entanglement evolution of the four-qubit cluster state under the above mentioneddecohering environments. This behavior is important for anumber of reasons. First, multipartite entanglement evolutionunder decoherence is still very much an area of intense study.Here we explore a number of negativity measures (whichare bipartite measures) and the detection capabilities of theentanglement witness. The latter is a way to test for thepresence of cluster-type entanglement in experiments whichwill be necessary before proceeding with any cluster-basedalgorithm. Our results allow for comparison of entanglementevolution between the different decohering environments. Theentanglement evolution is compared to the fidelity measuresin an attempt to note any possible correspondence betweenthe two. The presence or lack of a correlation between fidelityand entanglement addresses the general question of the role of entanglement in quantum computation. Is entanglementintegral to any quantum computation, or is it simply abyproduct of large Hilbert spaces? One may surmise that therole of entanglement is especially vital in cluster-state quantumcomputation as the highly entangled cluster state is the basicresource for any algorithm. However, we showed that there areonly superficial correlations between fidelity and entanglementand specifically noted that the complete disappearance of en-tanglement upon sufficient decoherence, entanglement suddendeath, does not have any effect of the fidelity behavior. This isespecially manifest when comparing the evolution under thedephasing and amplitude damping environments. While thegate fidelity of the single-qubit arbitrary rotation is the samefor both of these environments, the entanglement evolution isvery different. All of this demonstrates that while entanglementis certainly necessary for universal quantum computation to beimplemented on a cluster state the amount of entanglement per seis not a good indicator as to how accurately the algorithm will be implemented. ACKNOWLEDGMENT We acknowledge support from the MITRE Technology Program under MIP Grant No. 20MSR053. APPENDIX A: ARBITRARY SINGLE-QUBIT ROTATION SUPEROPERATORS In this Appendix we provide the expressions for superop- erators describing the evolution of a single-logical qubit in acluster-based quantum computation attempting to implementan arbitrary rotation described by Euler angles ( θ 1,θ2,θ3)i n the different decohering environments. The superoperator forthe case of qubits in a dephasing environment is given by Sz=1 2⎛ ⎜⎜⎜⎝(1−qs2s3) −eiθ1q(c3−i˜pc2s3) −e−iθ1q(c3+i˜pc2s3) (1 +qs2s3) q(c2−i˜pc3s2)eiθ1q[qc2c3+i˜p(s2+s3)] −e−iθ1q[qc2c3+i˜p(s2−s3)]−q(c2−i˜pc3s2) q(c2+i˜pc3s2)−eiθ1q[qc2c3−i˜p(s2−s3)] e−iθ1q[qc2c3−i˜p(s2+s3)] −q(c2+i˜pc3s2) (1+qs2s3) eiθ1q(c3−i˜pc2s3) e−iθ1q(c3+i˜pc2s3) (1 −qs2s3)⎞ ⎟⎟⎟⎠, (A1) where q≡p−1,˜p≡√1−p, and we write sin θjand cos θjforj=1,2,3a ssjandcj. The superoperator for the amplitude damping environment is given by SA=1 2⎛ ⎜⎜⎜⎝(1+p)+q2s2s3 eiθ1q2(c3−i˜pc2s3) e−iθ1q2(c3+i˜pc2s3) (1 +p)−q2s2s3 q(c2−i˜pc3s2) eiθ1q[qc2c3+i˜p(s2+s3)] −e−iθ1q[qc2c3+i˜p(s2−s3)]−q(c2−i˜pc3s2) q(c2+i˜pc3s2)−eiθ1q[qc2c3−i˜p(s2−s3)] e−iθ1q[qc2c3−i˜p(s2+s3)] −q(c2+i˜pc3s2) −q(1+qs2s3) −eiθ1q2(c3−i˜pc2s3) −e−iθ1q2(c3+i˜pc2s3) −q(1−qs2s3)⎞ ⎟⎟⎟⎠.(A2) We note that the second and third rows of the amplitude damping superoperator are exactly the same as the second and third rows of the dephasing superoperator. The superoperator for the depolarizing environment is given by S P=1 2⎛ ⎜⎜⎜⎝1−q3s2s3 −e−iθ1q3(c3+iqc2s3) e−iθ1q3(−c3+iqc2s3) 1 +q3s2s3 −q2(c2+iqc3s2)eiθ1q3[qc2c3+i(s2+s3)] −e−iθ1q3[qc2c3+i(s2−s3)]q2(c2+iqc3s2) −q2(c2−iqc3s2)−eiθ1q3[qc2c3−i(s2−s3)] e−iθ1q3[qc2c3−i(s2+s3)] q2(c2−iqc3s2) 1+q3s2s3 e−iθ1q3(c3+iqc2s3) e−iθ1q3(c3−iqc2s3) 1 −q3s2s3⎞ ⎟⎟⎟⎠. (A3) 032324-6SUPEROPERATOR ANALYSIS OF ENTANGLEMENT IN A ... PHYSICAL REVIEW A 85, 032324 (2012) 0.2 0.4 0.6 0.8 1.0p0.20.40.60.81.0Aa 0.2 0.4 0.6 0.8 1.0p0.60.70.80.91.0 FIG. 7. (Color online) (Left) Kraus operator amplitudes as a function of dephasing strength. The highest Kraus operator amplitude decreases linearly until becoming equal to the amplitudes of the other three Kraus operators. The behavior of the highest and lowest amplitudes areindependent of all measurement angles while the middle two amplitudes depend slightly on θ 2andθ3. (Right) Fidelity (solid line) and correlation (dashed line) of first Kraus operator as a function of dephasing strength. The above superoperators promise to be useful for exper- imental realizations of this cluster-state protocol, includingquestions of fault tolerance, as they can be used to characterizea given environment. APPENDIX B: KRAUS OPERATOR REPRESENTATION In the main part of this paper we determined the super- operators for single-logical-qubit rotations in a cluster-basedquantum computer undergoing different types of decoherence.Recasting these superoperators in terms of Kraus operatorsgives additional insight into the evolution of the logicalinformation under the arbitrary qubit rotation as a functionof decoherence. To calculate the Kraus operators from thesuperoperator one first determines the Choi matrix. EachKraus operator K ais a Choi matrix eigenvector (unstacked so that its dimension is N×N), times the square root of the corresponding Choi matrix eigenvalue divided by N[23]. We define the amplitude of a given Kraus operator Aato be the square root of the Choi matrix eigenvalue divided by N, Aa=√λa/N. The higher the amplitude of a Kraus operator the more significant its effect on the overall system dynamics.This method of Kraus operator construction maximizes theamplitude of one (and hence the most significant) Krausoperator. Using Kraus operators, the complete evolution of the system is given by /summationdisplay aKa(p,θ 1,θ2,θ3)ρin(α,β)Ka(p,θ 1,θ2,θ3)†=ρout. (B1) Clearly, if there is only one Kraus operator it will be unitary with an amplitude of 1. In this way, unitarity of the evolutioncan be quantified by A 1, the amplitude of the first Kraus operator. In addition, the accuracy of the applied evolution canbe quantified by the fidelity or correlation of the first Krausoperator as compared to the desired unitary [ 22]. The fidelity is given by F 1=Tr[U†K1] Tr[U†U], (B2) and the correlation is given by C1=Tr[U†K1]/radicalBig Tr[U†U]Tr[K† 1K1]. (B3) The fidelity measure accounts for both decoherent losses, a change in purity, and coherent errors, what we might call achange in “direction” effected by the first Kraus operator. Thecorrelation is unaffected by a change in magnitude. 0.2 0.4 0.6 0.8 1.0p0.20.40.60.81.0Aa 0.2 0.4 0.6 0.8 1.0p0.50.60.70.80.91.0 FIG. 8. (Color online) (Left) Kraus operator amplitudes as a function of amplitude damping strength. The behavior of all amplitudes depend slightly on θ2andθ3. (Right) Fidelity (solid line) and correlation (dashed line) of first Kraus operator as a function of amplitude damping strength. Note that in this case the correlation does not remain constant implying a coherent affect on the dynamics of the system due to amplitude damping. 032324-7WEINSTEIN, FELDMAN, ROBINS, ZUKUS, AND GILBERT PHYSICAL REVIEW A 85, 032324 (2012) 0.2 0.4 0.6 0.8 1.0p0.20.40.60.81.0Aa 0.2 0.4 0.6 0.8 1.0p0.60.70.80.91.0 FIG. 9. (Color online) (Left) Kraus operator amplitudes as a function of depolarizing strength. The behavior of all amplitudes depend slightly on θ2. (Right) Fidelity (solid line) and correlation (dashed line) of first Kraus operator as a function of dephasing strength. We start with the Kraus operators of the dephasing envi- ronment. The amplitude of the Kraus operators as a functionof decoherence strength is shown in Fig. 7. The amplitude of the first Kraus operator decreases linearly and the amplitudeof the other three Kraus operators increase until, at p=1, the four Kraus operators have equal amplitudes. At that limit eachof the four Kraus operator matrices have an element equal to 1/√ 2 in one of the corners and all the other elements are zero. The fidelity of the first Kraus operator also decreaseslinearly with dephasing strength while the correlation remainsconstant at 1 (until very high p) implying purely decoherent evolution. We noted in the main part of the paper that the gate fidelities of a single-logical-qubit cluster-state-based arbitraryrotation in a dephasing environment and amplitude dampingenvironment are the same. Nevertheless, we find that theirKraus operator representations are very different. Two ofthe Kraus operator amplitudes in an amplitude dampingenvironment go to zero as p→1. The remaining two Kraus operator matrices have a one in the upper right or left cornerand zeros elsewhere. For values of p< 1, the amplitude of the first Kraus operator decreases faster in the amplitude dampingenvironment than in the dephasing environment. However, thisdescent slows as papproaches one. The second Kraus operatoralways plays a more significant role in the amplitude damping environment than in the dephasing environment. The fidelity of the first Kraus operator as function of amplitude damping strength pdecreases linearly (faster than the dephasing environment) before rounding off at high valuesofpwhile the correlation decreases, staying near one only at low values of p. This behavior, portrayed in Fig. 8, suggests that amplitude damping, despite being decoherent dynamics,has a coherent affect on the system dynamics. Rotating thesystem so that the amplitude damping acts as phase dampingmay increase the fidelity and correlation of the first Krausoperator but will not increase the gate fidelity of the attemptedlogical qubit rotation. Of all the decohering environments studied here, the first Kraus operator amplitude decreases fastest (and not linearly)in a depolarizing environment. The increase of the lowestamplitude Kraus operator is also not linear. However, inthe limit of p→1, the depolarizing environment is like the dephasing environment in that the amplitudes of all four Krausoperators converge to 0.5, as demonstrated in Fig. 9.T h e fidelity in a depolarzing environment also decreases fasterthan the other decoherent environments while the correlationremains constant at one, demonstrating that the evolution isentirely decoherent. [1] For a recent review see R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009). [2] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001). [ 3 ] M .N i e l s e na n dI .C h u a n g , Quantum Information and Computation (Cambridge University Press, Cambridge, 2000). [4] C. Simon and J. Kempe, Phys. Rev. A 65, 052327 (2002); W. Dur and H.-J. Briegel, Phys. Rev. Lett. 92, 180403 (2004); S. Bandyopadhyay and D. A. Lidar, P h y s .R e v .A 72, 042339 (2005). [5] M. Hein, W. Dur, and H.-J. Briegel, P h y s .R e v .A 71, 032350 (2005); O. Guhne, F. Bodoky, and M. Blaauboer, ibid. 78, 060301(R) (2008); D. Cavalcanti, R. Chaves, L. Aolita, L. Davidovich, and A. Acin, P h y s .R e v .L e t t . 103, 030502 (2009); L. Aolita, D. Cavalcanti, R. Chaves, C. Dhara, L. Davidovich, and A. Acin, P h y s .R e v .A 82, 032317 (2010).[ 6 ] L .D i o s i ,i n Irreversible Quantum Dynamics , edited by F. Benatti and R. Floreanini, Lecture Notes in Physics, V ol. 622 (Springer,Berlin, 2003), p. 157; P. J. Dodd and J. J. Halliwell, Phys. Rev. A69, 052105 (2004). [7] T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004); 97, 140403 (2006). [8] I. Sainz and G. Bjork, Phys. Rev. A 76, 042313 (2007). [9] L. Aolita, R. Chaves, D. Cavalcanti, A. Acin, and L. Davidovich, Phys. Rev. Lett. 100, 080501 (2008). [10] C. E. Lopez, G. Romero, F. Lastra, E. Solano, and J. C. Retamal, Phys. Rev. Lett. 101, 080503 (2008). [11] M. Yonac, T. Yu, and J. H. Eberly, J. Phys. B 39, 5621 (2006); 40, 545 (2007). [12] I. Sainz and G. Bjork, P h y s .R e v .A 77, 052307 (2008). [13] Y . S. Weinstein, P h y s .R e v .A 79, 012318 (2009); 82, 032326 (2010). 032324-8SUPEROPERATOR ANALYSIS OF ENTANGLEMENT IN A ... PHYSICAL REVIEW A 85, 032324 (2012) [14] Y . S. Weinstein, P h y s .R e v .A 79, 052325 (2009). [15] M. P. Almeida et al. ,Science 316, 579 (2007); J .L a u r a t ,K .S . Choi, H. Deng, C. W. Chou, and H. J. Kimble, P h y s .R e v .L e t t . 99, 180504 (2007); A .S a l l e s ,F .d eM e l o ,M .P .A l m e i d a ,M .H o r - Meyll, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich,Phys. Rev. A 78, 022322 (2008). [16] Y . S. Weinstein, P h y s .R e v .A 80, 022310 (2009). [17] H. J. Briegel and R. Raussendorf, P h y s .R e v .L e t t . 86, 910 (2001).[18] R. Raussendorf, D. E. Browne, and H. J. Briegel, P h y s .R e v .A 68, 022312 (2003). [19] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002). [20] B. M. Terhal, Phys. Lett. A 271, 319 (2000); M. 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PhysRevLett.103.077201.pdf

Combined Electron Resonance Driven by an All-Oscillating Potential of Patterned Magnets Nikolay I. Polushkin * Institute for Physics of Microstructures of Russian Academy of Sciences, 603950 GSP-105 Nizhniy Novgorod, Russian Federation (Received 6 January 2009; published 10 August 2009) A novel mechanism is proposed for the phenomenon of combined electron resonance. It is shown that the spatially localized microwave fields of an Fe stripe array mediate the intense electronic transitionsinvolving the changes in both spin and orbital quantum numbers when the electron moves along acyclotron orbit in a semiconductor (e.g., InGaAs-based) quantum well. This discovery bridges the fields ofspintronics and quantum computing, paving the way for conceptually new hybrid devices based onferromagnetic and semiconductor structured materials. DOI: 10.1103/PhysRevLett.103.077201 PACS numbers: 75.75.+a, 71.70.Di, 76.30. /C0v The recent progress in quantum computing devices based on semiconductor wells and dots [ 1] is associated with utilizing general quantum phenomena such as elec- tron spin resonance driven by either magnetic [ 2] or elec- tric [ 3–5] ac fields. The latter effect relates to another phenomenon, the so-called combined resonance (CR) [ 6– 8]. By general definition [ 6], the CR is a resonant transition involving the changes in both spin and Landau quantum numbers. A possible mechanism mediating the combined transitions relies on the spin-orbit (SO) interaction that leads to mixing of the orbital and spin motion in a periodic crystalline lattice [ 1–8]. This coupling breaks the usual selection rules, so the transitions can be excited from the statejl;#itojn;"i, where landnare the Landau quantum numbers ( n/C222l) and " ð#Þ denote the spin quantum num- bers, i.e., spin-up (-down). The ability of a quantum system to exhibit the CR depends on how the lattice period relates to the cyclotron radius Rc[6,7]. For better matching of these two length scales and for their manipulation as well, it would be desirable to artificially form the drive potential in the same medium or another one brought into close proximity to the medium where the electron moves. One fascinating discovery in this arena is associated with the commensurability effects in a two-dimensional electron gas subjected to a periodic electrostatic or magnetic potential [ 9,10]. Related effects were observed in other hybrid systems such as a superconductor- semiconductor [ 11] or a ferromagnet-superconductor [12]. However, despite broad interest in studying the inter- play of spin and charge dynamics, the basic question of the coupling between orbital (cyclotron) and spin motions in an external nonuniform potential remains open. In this Letter, I answer this question by predicting new strong effects of an external nonuniform potential on the electronic properties. Using a simple formalism, I demon- strate that an electron may change its Landau index via the interaction with an external potential hðr; tÞoscillating in both space and time. Such an all-oscillating potential hav- ing a spatial period /C3and angular frequency /C10can be considered as a perturbation of the wave function clof anelectron moving along an lth Landau orbit in a static magnetic field H. As the matrix elements of the perturba- tion operator exp½ið!l;#;n;"/C0/C10Þt/C138Rcn/C3ðrÞhðrÞclðrÞdrfor a transition l!nare nonvanishing at n/C222l, the coordinate-dependent function hðrÞallows for a nonzero rate of the transitions between the lth and nth Landau levels with spin rotation at the resonance that occurs when /C10¼!l;#;n;": (1) I find that the hðr; tÞcan be effectively generated by the dynamic magnetization in a stripe ferromagnetic structure (FMS) patterned in a thin Fe layer. I show that the pre-cession of the magnetization with a frequency equal to the eigenfrequency of the patterned layer /C10is able to induce intense combined electronic transitions outside the layer ata distance compared to its thickness. To demonstrate thefeasibility of the combined transitions, I envisage a real-istic situation where the electron is confined in a semicon-ductor quantum well (QW) from an engineered materialsuch as In xGa1/C0xAs, whose technology is well-established [13]. Importantly, an extrinsic engineered potential such as hðr; tÞcan be employed instead of the intrinsic SO inter- action effect to mediate the drastic changes in chargemotion, thereby providing a new route for spin-to-chargeconversion in quantum computing devices [ 5]. Another feature is that the stray fields of the patterned elementsquickly decay out of them in the lateral direction.Localizing the resonant transitions via strong and localizedac fields would be a necessary step for single-electron spin manipulation. I propose to generate such fields by amplify- ing and localizing a uniform microwave field applied to theFMS. The FMS-QW design and calculation geometry are shown in Figs. 1(a) and 1(b). An external microwave uniform field h ecos!toriented along the yaxis causes the resonant precession of the magnetization in the FMS at!¼/C10. The dynamic magnetization ~mðr; tÞ, in turn, gen- erates the stray field ~hðr; tÞwhich perturbs the electron cyclotron motion inside a QW placed at a distance z 0 from the FMS surface. The FMS-QW system is subjectedPRL 103, 077201 (2009) PHYSICAL REVIEW LETTERSweek ending 14 AUGUST 2009 0031-9007 =09=103(7) =077201(4) 077201-1 /C2112009 The American Physical Societyto the out-of-plane static magnetic field H. It is easier to satisfy the resonance condition ( 1) if the vertical field component Hzis taken to be smaller than the saturation magnetization 4/C25M s. With such a limitation, the contribu- tion of Hzto/C10can be neglected [ 14]. An FMS used in my calculation consists of periodically arranged ferromagneticstripes with 4/C25M s¼21 kG (bulk Fe) and grating filling w=/C3¼0:2, where wis the stripe width [Fig. 2(a)]. Values taken for the damping constant and ratio of the stripe thickness to the grating period are /C11¼0:0019 [15] and Lz=/C3¼0:01, respectively. Figure 2shows how (b) the resonant susceptibility /C31y;z¼4/C25my;z=heand (c) produced dynamic stray fields hy;zare distributed over /C3. An in-plane field Hx¼20 kOe was taken for plotting /C31y;zðyÞ, and hy;zðyÞwere calculatedat different distances z0from the FMS surface. The values of/C31were calculated numerically by reducing the Landau- Lifshitz-Gilbert equations to an infinite set of algebraic linear equations for Fourier components of /C31[16]. The dynamic stray field ~hðr; tÞwas found from the magneto- statics equations, taking into account the standard electro-magnetic boundary conditions at both interfacesz¼/C6L z=2. Then the field components in the region out- side the FMS ( z>L z=2) can be written as hyðzÞ¼2heX1 j¼1e/C0qjzsinhqjLz 2½/C31zjFyðzÞjðyÞþ/C31yjGyðzÞjðyÞ/C138; (2) where qj¼2/C25j=/C3,Fyj/C17Gzj¼sinqjy, and Gyj/C17 /C0Fzj¼/C0 cosqjy. One sees that outside the shaded re- gions in Fig. 2both the vertical hzand horizontal hyfield components quickly decay along the ydirection. To describe the electronic motion along a cyclotron orbit inside a QW, I additionally introduce a coordinate system r0/C17ðx0;y0;z0Þ, as shown in Fig. 1. In the presence of a tilted field ~Hz0and confining potential UðzÞ, the effective Hamiltonian of a conduction-band electron in a QW reads[17] ^H¼^H 0þ/C23½^p;~ez/C138^/C27; (3) where ^H0¼^p2=2m/C3/C0/C22^/C27z0Hz0þUðzÞ,m/C3is the ef- fective mass, ^p¼/C0i@rþj ej~AðrÞ=c,rot~A¼~Hz0,/C22/C17 /C22Bg/C3=2is a maximal value of the magnetic-moment pro- jection on the z0axis, ^/C27is the Pauli matrices, g/C3is the effective Lande ´gfactor, /C23is the SO interaction constant, and~ezis a unit vector along the zaxis. In the absence of the ac electric fields applied to the QW, the SO interaction term in Eq. ( 3) is time-independent, so that it cannot mediate any electronic transitions, though itcan affect the resonance condition ( 1) by shifting the energy levels [ 18]. The logic of the further formalism is as follows. I start with the treatments of the unperturbed Hamiltonian, ^H 0. Then I consider the perturbation of ^H0by the all-oscillating external potential ~hðr; tÞ. Finally, the effect of SO interaction is evaluated as a time-independent perturbation of ^H0. The stationary states for the unperturbed cyclotron mo- tion described by ^H0can be found from the equation ^H0cðr; sz0Þ¼Ecðr; sz0Þin which the envelope function can be represented as the product of the coordinate andspin parts, i.e., cðr; sz0Þ¼’ðrÞS"ð#Þðsz0Þ. By taking into account that ^/C27z0S"¼1and ^/C27z0S#¼/C01and choosing the gauge for the vector potential as ~A¼ð0;/C0zHxþxHz;0Þ, the coordinate part can be represented as ’ðrÞ¼ expðikyyÞfðx; zÞto yield the equation for fðx; zÞ[19]: /C0@2 2m/C3@2f @x2/C0/C20@2 2m/C3@2 @z2/C0UðzÞ/C21 f þ@2 2m/C3/C18~x R2z/C0z R2x/C192 f/C6/C22H z0f¼Ef; (4) FIG. 2 (color online). Distributions of the saturation magneti- zation (a), horizontal ( y) and vertical ( z) components of the dynamic susceptibility /C31(b), and the stray field (c) at Hx¼ 20 kOe . Resonant amplification of a uniform field is shown at different distances from the FMS surface. FIG. 1 (color online). (a) Calculation geometry. An electron is confined in a narrow QW placed a distance z0from the FMS surface where the precession magnetization is driven by anexternal microwave field h ecos!tin a tilted static magnetic field H. (b) Dynamic stray fields hy(brown contours) and hz (blue contours) from a ferromagnet (FM) perturb the electron cyclotron motion and mediate the combined transitions.PRL 103, 077201 (2009) PHYSICAL REVIEW LETTERSweek ending 14 AUGUST 2009 077201-2where RzðxÞ¼½c@=jej=HzðxÞ/C1381=2is the magnetic length in thex/C0y(z/C0y) plane, ~x¼x/C0x0, and x0¼Rz2kyis the xcoordinate of the orbital center. I assume that the QW is so narrow that aw/C28j~xjR2x=R2z. In this case, the depen- dence on zin (4) can be neglected. In the aw!0limit, the electron motion is purely two-dimensional in the x/C0y plane, and only Hzis responsible for the Landau quantiza- tion, while the Zeeman splitting is defined by the full fieldH. Then the two remaining variables can be separated to yield the equation for a harmonic oscillator with eigenval- uesE l¼@!cðlþ1=2Þþ@2/C252=2=a2w=m/C3/C6/C22H z0for a QW of infinite depth, where !c¼jejHz=m/C3=cis the cy- clotron frequency. Finally, the eigenfunctions are given by cð0Þ l¼ffiffiffiffiffiffiffiffiffiffiffi 2=awq eikxxsinð/C25z=a wÞ/C8lð~y=R zÞS"ð#Þðsz0Þ;(5) with /C8lð~/C24Þbeing the harmonic oscillator eigenfunctions where the variable xis switched with y. Let an electron initially be in its unperturbed state jl;#i that can be obtained by switching H, as shown in Fig. 3(a). To find how this state is perturbed by ~hðr; tÞ, I represent the perturbed state as a harmonic-oscillator-eigenfunction ex- pansion cðr; t; s z0Þ¼cð0Þ lðr; t; /C27 z0ÞþX k/C222lckðtÞcð0Þ kðr; t; /C27 z0Þ:(6)In the coordinate system used [Fig. 1(a)], the perturbation due to the Zeeman interaction with ~hðr; tÞreads [ 20] ^Vðr; t; /C27Þ¼/C0 /C22½^/C27yhyðyÞ/C0^/C27x0hzðyÞcos/C18/C138cos/C10 t;(7) where /C18is the angle between the z0andxaxes. Substituting (3), (6), and ( 7) into the wave equation i@@c=@t¼ð^H0þ ^VÞc, multiplying it by a complex conjugate wave function cð0Þ/C3 n, and integrating over the QW volume lead to the equation for an expansion coefficient ckðtÞ. Finally, one obtains for the transition probability from the lth to nth quantum level PnlðtÞ/C17jcnðtÞj2¼/C18t 2@/C192 ðAnl2þBnl2Þ; (8) where Anl¼ReVnly/C0ImVnlz,Bnl¼ReVnlzþImVlny, Vnlz¼/C22Inlzcos/C18,Vnly¼/C0/C22Inly, and InlzðyÞ¼R1 /C01hzðyÞð~/C24; z 0Þ/C8lð~/C24Þ/C8nð~/C24Þd~/C24. The combined transitions can be excited at /C10¼/C13½ðn/C0lÞHz=/C16/C6g/C3H=2Þ, where the signs ‘‘ þ’’ and ‘‘ /C0’’ denote the transitions l,"!n,# andl,#!n,", respectively, /C13¼17:6 GHz kOe/C01is the gyromagnetic ratio, and /C16is the ratio of the effective mass to the free-electron one. Figure 3(b) shows /C10(tilted blue line) and the frequency for a l,#!lþ1,"transition (solid nearly vertical lines) in an In0:53Ga0:47As-based QW ( /C16¼ 0:044,jg/C3j¼4:0)[13] as functions of Hxat a fixed H. The inset (b1) shows Hzversus Hxfor different Happlied to an In0:53Ga0:47AsQW. The combined transition is schemati- cally shown in Fig. 3(c). For the applications [ 1–5], one should provide a sufficient energy splitting of the spin states. The inset (b2) shows the energy difference /C1E between l,#andlþ1,"levels versus H. As seen, this quantity can be much larger than the thermal energy of theelectrons in the reservoir ( /C24100 mK ). Figure 3(d)shows a 3D plot of the CR intensity P 0;1for a j0;#i ! j 1;"itransition as a function of the stripe width w=2Rzand position of the orbital center y0=/C3at a spacing ofz0¼Lzfrom the FMS surface. The shown values of P0;1 are normalized to those for the intensity of resonant mag- netic transitions P0driven by an external microwave field in the absence of the FMS. The amplification of the CRintensity is due to resonant excitation of the magnetizationprecession in the FMS. As seen from Eq. ( 8), the CR in- tensity is defined by the matrix elements I l;lþ1. For in- stance, the matrix elements of the lowest transition j0;#i ! j1;"iprovide the maximal CR intensity at w¼2Rz. The electron motion over the FMS should be less sensitive to the nonuniformity of hðr; tÞwhen wis much smaller or, vice versa, much larger than 2Rz. The absence of the oscillations expected in P0;1as a function of wwith max- ima at 2Rz=w¼1þsð/C3=wÞ, where s¼0;1;2;3... (/C3=wis invariable), is explained by decaying the hðr; tÞ potential as expð/C0qjz0Þ, which is seen from Eq. ( 2). Note also that P0;1ðy0Þhas two maxima at y0¼/C60:1/C3over the structure period for wider stripes ( w=2Rz>1:5). The same behavior is shown in Fig. 3(e) by plotting P0;1ðy0Þat FIG. 3 (color online). (a) Intralevel electron transition with switching the field needed to prepare the initial electronic statefor the combined transition. (b) /C10(blue tilted line) and transition frequency ! l;#;lþ1;"in an InGaAs-based QW (nearly vertical solid lines) versus Hx. The transition occurs at /C10¼!l;#;lþ1;". The insets (b1) and (b2) show Hzversus Hxand the energy splitting versus H, respectively. (c) Schematic diagram of energy levels showing the combined transition (red arrows). (d) Intensity P0;1 of a 0, #!1,"combined transition versus y0=/C3andw=2Rz(3D plot). (e) P0;1versus y0=/C3atw=2Rz¼1:0and 3.0. (f) Intensity Pl;lþ1for various combined transitions l¼0;1;2;3;4versus w=2Rz.PRL 103, 077201 (2009) PHYSICAL REVIEW LETTERSweek ending 14 AUGUST 2009 077201-3w=2Rz¼1:0(solid line) and w=2Rz¼3:0(dashed line). I find, finally, that the maximum in the CR intensity shifts towards larger w=2Rzas/C25ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2lþ1p with raising a Landau index l¼0;1;2;3;4... [Fig. 3(f)]. This shift reflects the increase in cyclotron radius at higher l. Figure 4shows a hypothetical device for single-electron spin manipulation through the combined transitions. Thisdevice would consist of quantum dots (QD) formed with an electrostatic potential on top of patterned ferromagnetic elements [Fig. 4(a)]. By applying a bias voltage pulse to the gate electrode G, an electron taken from the reservoir and having, for instance, an initial state j0;#ienters the QD where it experiences a combined j0;#i ! j 1;"itransition under the local dynamic stray field produced by the FMS[Fig. 4(b)]. With the change in j cj2given by Eq. ( 5), the effective charge on the QD alters as /C1Q/ðe=r effÞRz, where reffis the effective distance from the QD to an electrometer (the electrode R). At reff/C24Rz,/C1Q/C24ecan be measured by means of a spin-to-charge conversion technique such as that proposed in Ref. [ 5]. In contrast to the other proposals [ 3–5], the readout frequency in our scheme is defined by not the electric component of the radiation but by the amplitude of ~hðr; tÞ. To experimentally separate the combined transitions governed by ~hðr; tÞ, one should eliminate the SO interac- tion effect on the CR intensity via the ac electric compo-nent of the radiation [ 6,7]. This can be done by placing the sample at a node of the ac electric component inside amicrowave cavity. It is also important that the ac electric component e acproduced by ~hðr; tÞis small compared to it, notably eac=h/C24/C3!=c= 2/C25/C2410/C04. In the absence of ac electric fields, the SO interaction term /C23½^p;~ez/C138in Eq. ( 3) gives a correction /C1!/C252jVSO l;lþ1j2=ð!c/C02/C22H z0=@Þ=@2to a frequency of an allowed l,#! lþ1,"transition, whereVSO l;lþ1¼ð2/C23sin/C18=R zÞR1 /C01/C24/C8lð/C24Þ/C8lþ1ð/C24Þd/C24 and /C24¼ ~x=R z. I find that /C1!/ð0:1–0:2Þ!l;#;lþ1;"under the condi- tions considered, notably l/C241andHz=Hx/C281, while a typical value of the SO constant is /C23/C245/C210/C010eV cm [21]. In summary, a novel mechanism is proposed for the phenomenon of combined electron resonance. It is shown that the interaction with a time-dependent nonuniform magnetic potential mediates resonant electronic transitions with changing both spin and orbital quantum numbers. Such an all-oscillating potential can be generated by the resonant magnetization precession in a patterned magnetic nanostructure. This finding illustrates how an external en- gineered potential can be used instead of intrinsic effectssuch as the SO interaction for spin manipulation in quan- tum wells and dots that serve as a basis of conceptually new devices in a united field of spintronics and quantum computing. This work was supported by the Russian Foundation for Basic Research (Grant No. 07-02-01305). The author thanks V. Ya. Aleshkin, V. I. Gavrilenko, and A. S. Mel’nikov for valuable discussions. *nip@ipm.sci-nnov.ru [1] R. Hanson and D. D. Awschalom, Nature (London) 453, 1043 (2008). [2] F. H. L. Koppens et al. , Nature (London) 442, 766 (2006). [3] K. C. Nowack et al. , Science 318, 1430 (2007). [4] M. Pioro-Ladrie ´reet al. , Nature Phys. 4, 776 (2008). [5] M. Friesen, Ch. Tahan, R. Joint, and M. A. Eriksson, Phys. Rev. Lett. 92, 037901 (2004). [6] E ´. I. Rashba, Sov. Phys. Usp. 7, 823 (1965). [7] B. D. McCombe, Phys. Rev. 181, 1206 (1969). [8] I. I. Lyapilin and A. E. Patrakov, Phys. Rev. B 75, 155320 (2007). [9] R. R. Gerhardts, D. Weiss, and K. v. Klitzing, Phys. Rev. Lett. 62, 1173 (1989). [10] P. D. Ye et al. , Phys. Rev. Lett. 74, 3013 (1995). [11] M. Berciu, T. G. Rappoport, and B. Janko, Nature (London) 435, 71 (2005). [12] J. E. Villegas et al. , Science 302, 1188 (2003). [13] H. Kosaka et al. , Electron. Lett. 37, 464 (2001). [14] T. S. Rahman and D. L. Mills, J. Appl. Phys. 53, 2084 (1982). [15] S. Scheck et al. , Phys. Rev. Lett. 98, 117601 (2007). [16] N. I. Polushkin, Phys. Rev. B 77, 180401(R) (2008). [17] Y. A. Bychkov and E ´. I. Rashba, J. Phys. C 17, 6039 (1984). [18] B. Das, S. Datta, and R. Reifenberger, Phys. Rev. B 41, 8278 (1990). [19] F. Stern and W. E. Howard, Phys. Rev. 163, 816 (1967). [20] Another part of the perturbation due to the interaction of the electron charge with ~hðr; tÞis not considered here. [21] T. Koga, J. Nitta, T. Akazaki, and H. Takayanagi, Phys. Rev. Lett. 89, 046801 (2002). FIG. 4 (color online). (a) Array of patterned FMs with gate- defined QD. (b) Hybrid FMS-QW device based on combinedtransitions; see the text for details.PRL 103, 077201 (2009) PHYSICAL REVIEW LETTERSweek ending 14 AUGUST 2009 077201-4

PhysRevLett.97.107204.pdf

Current-Driven Resonant Excitation of Magnetic Vortices Shinya Kasai,1Yoshinobu Nakatani,2Kensuke Kobayashi,1Hiroshi Kohno,3and Teruo Ono1 1Institute for Chemical Research, Kyoto University, Uji 611-0011, Japan 2University of Electro-communications, Chofu 182-8585, Japan 3Graduate School of Engineering Science, Osaka University, Toyonaka 560-8531, Japan (Received 5 April 2006; published 6 September 2006) A magnetic vortex core in a ferromagnetic circular nanodot has a resonance frequency originating from the confinement of the vortex core. By the micromagnetic simulation including the spin-transfer torque, we show that the vortex core can be resonantly excited by an ac (spin-polarized) current through the dotand that the resonance frequency can be tuned by the dot shape. The resistance measurement under the ac current successfully detects the resonance at the frequency consistent with the simulation. DOI: 10.1103/PhysRevLett.97.107204 PACS numbers: 85.75. /.0255d, 72.25.Ba, 72.25.Pn, 75.75.+a The manipulation of magnetization by spin currents is a key technology for future spintronics [ 1–14]. The under- lying physics is that spin currents can apply a torque on themagnetic moment when the spin direction of the conduc-tion electrons has a relative angle to the local magneticmoment. This leads us to the hypothesis that any type of spin structure with spatial variation can be excited by a spin-polarized current in a ferromagnet. The ideal exampleof such a noncollinear spin structure is a curling magneticstructure (‘‘magnetic vortex’’) realized in a ferromagneticcircular nanodot. Although this structure was theoreticallypredicted long ago [ 15], it was only recently confirmed by microscopic experiments that such a vortex exists with ananometer-scale core where the curling magnetization be- comes out-of-plane [ 16,17]. Subsequent intensive studies have clarified that, after switching off an in-plane magneticfield, a vortex core exhibits a spiral precession around thedot center during the relaxation process [ 18–20]. Thus, the nanodot functions as a resonator for vortex core motion.Thus far, magnetostatic interactions triggered by an exter-nal magnetic field have dominated the study of vortexdynamics; however, the abovementioned concept—vortex manipulation by a spin-polarized current—has just started [21]. Here we demonstrate that a magnetic vortex core can be resonantly excited by an ac current through a ferromag-netic circular dot when the current frequency is tuned to theeigenfrequency originating from the confinement of thevortex core in a dot. Our micromagnetic simulations withthe spin-transfer effect reveal in detail the motion during the excitation; an excited vortex core draws a spiral trajec- tory to settle in a steady orbital around the dot center. Wesucceeded in detecting the predicted resonance by resist-ance measurements. We observed efficient excitation by anelectric current due to the resonant nature and tunability ofthe resonance frequency based on the dot shape. By micro-magnetic simulations including the spin-transfer effect, weshow below that an ac spin-polarized current with the eigenfrequency of the resonator can resonantly excite a magnetic vortex core. Then we present the results of theexperimental detection of the resonance of a vortex core, as predicted by the simulation. Figure 1shows a scanning electron microscope image of the sample and the schematic configuration used for themeasurements. The samples were fabricated on thermallyoxidized Si substrates by the lift-off method in combina- tion with e-beam lithography. Each sample consists of a Permalloy ( Fe 19Ni81) dot and two 50 nm-thick Au wide electrodes. The thickness of the dot his 40 nm, and the radiusris varied to be r/.0136410, 530, and 700 nm. The existence of a vortex core in each dot was confirmed byconventional magnetic force microscopy. The current-induced dynamics of the vortex core was calculated bythe micromagnetic simulations based on the Landau- Lifshits-Gilbert (LLG) equation with a spin-transfer term [22,23]. The modified LLG equation is given by @m @t/.0136/.0255/.00130m/.0002Heff/.0135/.0011m/.0002@m @t/.0255/.0133us/.0001r/.0134m;(1) where mis a unit vector along the local magnetization, /.00130 the gyromagnetic ratio, Heffthe effective magnetic field Bias-T f=223 Hzf~100 MHz -1 GHz Lock-in amplifierV+ V-Iexc Imes 100 k ΩI=Imes+Iexc Internal oscillation1 kΩ 1 kΩSignal Generator 500 nm FIG. 1. Scanning electron microscope image of the sample along with a schematic configuration used for the measurements.The detection of the vortex excitation was performed by resist- ance measurements with a lock-in technique (223 Hz and current I mes/.013615/.0022A) under the application of an ac excitation current Iexc/.01363/.00021011A=m2.PRL 97,107204 (2006)PHYSICAL REVIEW LETTERSweek ending 8 SEPTEMBER 2006 0031-9007 =06=97(10)=107204(4) 107204-1 ©2006 The American Physical Societyincluding the exchange and the demagnetizing fields, and /.0011the Gilbert damping constant. The last term represents the spin-transfer torque, which describes the effect of spintransfer from conduction electrons to localized spins. This spin-transfer effect is a combined effect of the spatial nonuniformity of magnetization and the current flow[24]. The vector u/.0136/.0255 jPg/.0022 B=/.01332eMs/.0134, which has the dimension of velocity, is essentially the spin current asso- ciated with the electric current in a ferromagnet, where jis the current density, Pthe spin polarization of the current, g thegvalue of an electron, /.0022Bthe Bohr magneton, ethe electronic charge, and Msthe saturation magnetization. In the simulation, an electric current in a dot was assumedto be uniform, and an ac electric current in the form of j/.0136J 0sin2/.0025ft was applied, where J0is the current den- sity,fthe frequency of the ac current, and tthe time. The dot was divided into rectangular prisms of 4/.00024/.000240nm3; the magnetization in each of these was assumed to be constant. The typical material parameters for Permalloywere used: M s/.01361T, the exchange stiffness constant A/.0136 1:0/.000210/.025511J=m,P/.01360:7[25], and/.0011/.01360:01. First, we determined the eigenfrequency f0of the vortex core precession in the dot by calculating the free relaxa- tional motion of the vortex core from the off-centered position. The eigenfrequency depends on the aspect ratioh=r(the height hto the radius r) of the dot [ 18]. Then the simulations were performed by applying an ac current at a given frequency fin the absence of a magnetic field. Figure 2(a) shows the time evolution of the core position when an ac current ( f/.0136f0/.0136380 MHz andJ0/.0136 3/.00021011A=m2) is applied to a dot with r/.0136410 nm and h/.013640 nm . Once the ac current is applied, the vortex core first moves in the direction of the electron flow or spin current. This motion originates from the spin-transfer ef- fect. The off-centered core is then subjected to a restoring force toward the dot center. Furthermore, because of the gyroscopic nature of the vortex (the vortex moves perpen- dicular to the force), the core makes a circular precessional motion around the dot center [ 18]. The precession is am- plified by the current to reach a steady orbital motion where the spin transfer from the current is balanced with the damping, as depicted in Fig. 2(a). The direction of the precession depends on the direction of the core magneti- zation as in the motion induced by the magnetic field [18,21]. It should be noted that the radius of the steady orbital on resonance is larger by more than an order of magnitude as compared to the displacement of the vortex core induced by a dc current of the same amplitude [ 21]. Thus, the core is efficiently excited by the ac current due to resonance. We verified that the resonant excitation of the FIG. 2 (color). (a) Time evolution of the vortex under the ac current application. The magnetization direction m/.0136/.0133mx;my;mz/.0134 inside the dot on the xyplane was obtained by micromagnetic simulation. The 3D plots indicate mzwith themx-myvector plots superimposed. The plot on the left represents the initial state of the vortex core situated at the center of the dot with r/.0136410 nm . The 3D plots on the right show the vortex on the steady orbital at t/.013680:6, 81.5, and 82.3 ns after applying the ac current ( f0/.0136380 MHz andJ0/.01363/.00021011A=m2). These plots are close-ups of the square region around the dot center indicated by the black square in the plot on the left. The time evolution of the core orbital from t/.01360to 100 ns is superimposed only on the t/.013682:3n s plot. (b) Time evolutions of the vortex core displacement ( x) for three excitation frequencies f/.0136250, 340, and 380 MHz ( r/.0136410 nm andJ0/.0136 3/.00021011A=m2). (c) Radius of the steady orbital as a function of the frequency for the dots with r/.0136410, 530, and 700 nm.PRL 97,107204 (2006)PHYSICAL REVIEW LETTERSweek ending 8 SEPTEMBER 2006 107204-2vortex core presented above was not observed in the micro- magnetic simulations without the spin-transfer term even if we included a magnetic field generated by an ac current into the simulation, which indicates that the vortex core is excited not by the oersted field but by the spin-polarized current. Figure 2(b) shows the time evolutions of the xposition of the vortex core for three different excitation frequencies f/.0136250, 340, and 380 MHz. The steady state appears after around 30 ns on resonance ( f/.0136380 MHz ). Forf/.0136 340 MHz slightly off the resonance, the amplitude beats first, and then the steady state with smaller amplitude appears. The vortex core shows only a weak motion for f/.0136250 MHz , which is quite far from the resonance. The displacement amplitude for the nonresonant response is essentially the same as the case of the dc current applica- tion [ 21]. Figure 2(c)shows the radii of the steady orbitals as a function of the current frequency for the dots with r/.0136 410, 530, and 700 nm. Each dot exhibits the resonance at the eigenfrequency of the vortex motion. Our experimental detection method for the resonant excitation of a vortex core is based on the difference in resistance of the dot between the on- and off-resonance states as described below. In general, the resistance of ferromagnetic metals depends on the relative angle be- tween the magnetization and the measuring current— known as the anisotropic magnetoresistance (AMR) effect. Figure 3(a) shows the results of the magnetoresistance measurements at room temperature for the dot with r/.0136 700 nm . The resistance was measured by a lock-in tech- nique using a current of 15/.0022Aat 223 Hz. The magnetic field was applied perpendicular to the measuring current (H?I; the result is indicated by the blue curve) or parallel to the measuring current in the dot plane ( HkI; the result is indicated by the red curve). Here the results are plotted as a deviation in resistance ( /.0001Rkand/.0001R?) from the state in the zero magnetic field where the core exists at the center of the dot. The spin structure of the dot for each state is also indicated in the figure. Figure 3(a)clearly indicates that the resistance of the dot is highly correlated with the core position because of the AMR effect. The key feature is that the resistance change for H?I/.0133j/.0001R?j/.0134is larger than that forHkI/.0133j/.0001Rkj/.0134, as seen in the plot of j/.0001Rkj/.0255j/.0001R?j as a function of H[Fig. 3(b)]. This difference in the resistance change results from the symmetry breaking of the system because of the two electrodes attached to thedot. When the core is on resonance and the measurement time is considerably longer than the period of the core orbital motion, the experimentally measured resistance is the average value of the resistances for all the core posi- tions in the orbital shown in Fig. 2(a). This averaged resistance on resonance is expected to be smaller than that for the off-resonance state, in which the core remainsaround the dot center because /.0133j/.0001R kj/.0255j/.0001R?j/.0134<0,a s shown in Fig. 3(b). We detect the resonance in this manner.We measured the resistance of the dot while an ac excitation current was passed through it at room tempera-ture in the configuration shown in Fig. 1. The resistance measurements were performed by conventional lock-intechniques using a current of 15/.0022Awith a frequency of 223 Hz. The amplitude of the ac excitation current was 3/.0002 10 11A=m2. Figure 4(a)shows the resistances as a function of the frequency of the ac excitation current for the dotswith three different radii r/.0136410, 530, and 700 nm. A small but clear dip is observed for each dot; this signifiesthe resonance. Since the observed dip originates from theAMR effect averaged over the vortex orbital, the maximumsignal corresponding to the core motion along the dot edgeis expected to be about /.0133j/.0001R kj/.0255j/.0001R?j/.0134=2/.0025/.025510 m/.0010 on the basis of the result for r/.0136700 nm shown in Fig. 3(b). Thus, the observed signal amplitude ( 3m /.0010 ) approximately corresponds to the core orbital motionwhose radius is about 0:3r/.0025200 nm ; this amplitude is in the same range as the result of the simulation shown inFig. 2(c). The radius dependence of the resonance fre- ∆R//,∆R⊥ (mΩ) H (kOe)|∆R//|-|∆R⊥|( mΩ)H//I H⊥IH(a) (b)H H H -40-2002040 -200 -1.0 -0.5 0.0 0.5 1.0∆R// ∆R⊥ FIG. 3 (color). (a) Results of magnetoresistance measurements at room temperature for the dot with r/.0136700 nm . The red (blue) line is the result for HkI(H?I). The results are plotted as a deviation in resistance ( /.0001Rkand/.0001R?) from the state in the zero magnetic field where the core exists at the center of the dot. The spin structure in the dot for each state (at /.0006150 Oe denoted by the solid circles), which was determined by micromagnetic simulation, is also indicated. (b) j/.0001Rkj/.0255j/.0001R?jas a function of the magnetic field.PRL 97,107204 (2006)PHYSICAL REVIEW LETTERSweek ending 8 SEPTEMBER 2006 107204-3quency is well reproduced by the simulation, as shown in Fig. 4(b). In particular, for the dots with r/.0136700 nm , fair agreement is observed. The systematic deviation betweenthe experiments and simulation is possibly due to theinhomogeneous current distribution in the samples, whichis more pronounced for the smaller dots. Thus, we have demonstrated that a magnetic vortex core can be resonantly excited by an ac electric current. This phenomenon will facilitate detailed studies on the spin-transfer effect because of the simplicity of the system. Thevortex core experiences a well-defined potential that isdependent on the dot shape; this potential is not sensitive to edge roughness. State-of-the-art time-resolved imagingtechniques [ 19,20,26] can reveal the trajectory of the vor- tex core during excitation; this would lead to a better quantitative understanding. The present work was partly supported by MEXT Grants-in-Aid for Scientific Research in Priority Areasand JSPS Grants-in-Aid for Scientific Research. [1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [2] L. Berger, Phys. Rev. B 54, 9353 (1996). [3] M. Tsoi et al. , Phys. Rev. Lett. 80, 4281 (1998); 81, 493(E) (1998). [4] E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867 (1999). [5] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000). [6] M. Tsoi et al. , Nature (London) 406, 46 (2000). [7] L. Berger, J. Appl. Phys. 55, 1954 (1984). [8] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004). [9] M. Kla ¨uiet al. , Appl. Phys. Lett. 83, 105 (2003). [10] J. Grollier et al. , Appl. Phys. Lett. 83, 509 (2003). [11] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004). [12] N. Vernier et al. , Europhys. Lett. 65, 526 (2004). [13] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature (London) 428, 539 (2004). [14] E. Saitoh, H. Miyajima, T. Yamaoka, and G. Tatara, Nature (London) 432, 203 (2004). [15] A. Hubert and R. Scha ¨fer,Magnetic Domains (Springer, Berlin, 1998). [16] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science 289, 930 (2000). [17] A. Wachowiak et al. , Science 298, 577 (2002). [18] K. Yu. Guslienko et al. , J. Appl. Phys. 91, 8037 (2002). [19] J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A. Crowell, Phys. Rev. B 67, 020403(R) (2003). [20] S.-B. Choe et al. , Science 304, 420 (2004). [21] Motion of a magnetic vortex under a dc spin-polarized current was recently studied theoretically in J. Shibata, Y . Nakatani, G. Tatara, H. Kohno, and Y . Otani, Phys. Rev.B73, 020403(R) (2006). [22] A. Thiaville, Y . Nakatani, J. Miltat, and N. Vernier, J. Appl. Phys. 95, 7049 (2004). [23] A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki, Europhys. Lett. 69, 990 (2005); S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). [24] In Eq. ( 1), we have assumed the perfect adiabaticity in neglecting the nonadiabatic term. We have verified by numerical simulations that the effects of the nonadiabatic term are negligible for a vortex. [25] J. Bass and W. P. Pratt, Jr., J. Magn. Magn. Mater. 200, 274 (1999). [26] H. Stoll et al. , Appl. Phys. Lett. 84, 3328 (2004).400 300 200 100 0Frequency (MHz) 0.8 0.7 0.6 0.5 0.4 0.3 Radius ( µm) Simulation Experiment Average-20∆R (mΩ) 350 300 250 200 Frequency (MHz)-6-4-20 -4-20r = 700 nmr = 410 nm r = 530 nm(a) (b) FIG. 4 (color). (a) Experimental detection of the current- driven resonant excitation of a magnetic vortex core. The resis-tances are indicated as a function of the frequency of the ac excitation current for the dots with three different radii r/.0136410, 530, and 700 nm. (b) Radius dependence of the resonancefrequency. The blue rectangles and the red circles indicate thesimulation and the experimental results, respectively. The ex- perimental results for 8 samples are plotted. The red dashed line is the averaged value of the experimental data.PRL 97,107204 (2006)PHYSICAL REVIEW LETTERSweek ending 8 SEPTEMBER 2006 107204-4

PhysRevApplied.15.034085.pdf

PHYSICAL REVIEW APPLIED 15,034085 (2021) Selective and Tunable Excitation of Standing Spin Waves in a Magnetic Dielectric Film by Optical Guided Modes D.M. Krichevsky ,1,2,3, *D.O. Ignatyeva ,1,3,4V.A. Ozerov,1,2,3and V.I. Belotelov1,3,4 1V .I. Vernadsky Crimean Federal University, 295007 Simferopol, Crimea 2Moscow Institute of Physics and Technology (MIPT), 141700 Dolgoprudny, Russia 3Russian Quantum Center, 121353 Moscow, Russia 4Physics Department, Lomonosov Moscow State University, 119991 Moscow, Russia (Received 12 December 2020; revised 16 February 2021; accepted 22 February 2021; published 30 March 2021) We propose an approach for wavelength-selective excitation of perpendicular standing spin waves by optical guided modes in a magnetic dielectric film. TM-polarized guided waves with elliptical polarizationinduce an effective magnetic field due to the inverse transverse magneto-optical Kerr effect (ITMOKE). The ITMOKE is similar to the inverse Faraday effect, except for the induced magnetic field direction that is oriented in the film plane perpendicular to guided-mode propagation. Switching between different guidedmodes by tuning the wavelength or angle of the laser pulse significantly modifies the spatial distribution of the induced ITMOKE field and, therefore, selectively launches the standing spin-wave modes of different orders. Consequently, one may perform a tunable optical excitation of a certain spin-wave mode via a femtosecond laser pulse of the corresponding wavelength. The proposed approach broadens the spin-wave manipulation capabilities via optical means. DOI: 10.1103/PhysRevApplied.15.034085 I. INTRODUCTION Control and detection of spin waves in magnetic materi- als are the fundamental basis of spintronic devices, such as Boolean logic elements [ 1,2], memory cells [ 3,4], sensors [5,6], and elements of quantum computing [ 7]. Modern nanophotonics opens up opportunities to couple light and spin in nanoscale magnetic materials via excitation of var- ious types of optical modes [ 4,8–13]. This approach is a hot topic of current ultrafast all-optical magnetism, which allows nondissipative spin-wave manipulation. Among other methods, the inverse magneto-optical effects, like the inverse Faraday effect (IFE), are a unique technique for the nonthermal launch of spin precession by optical means [ 14]. The IFE manifests itself as an impact of circular light on spins [ 15,16]. The inverse Faraday effect is generally described in terms of the effective magnetic field induced by light that exists in a magnetic medium dur- ing light propagation through the medium. Previously, it was shown that the nonthermal laser-induced excitation of dielectric iron garnet films via the IFE resulted in the exci- tation of volume and bulk magnetostatic spin waves with controlled amplitude, wave numbers, and other parameters [17–19]. *krichevskii.dm@phystech.eduOn the other hand, the exchange spin modes, i.e., perpendicular standing spin waves (PSSW) in ferromag- netic films, have been a subject of extensive studies for decades [ 20–23]. High-order perpendicular standing spin- wave modes of up to 10 orders were experimentally observed in Ref. [ 24]. The PSSW frequency is much higher (about tens of GHz) than those for magnetostatic volumeor surface modes (basically about several GHz for iron gar- nets), which is crucial for various practical applications. At the same time, selective excitation of different PSSWs makes it possible to operate at several frequencies in the frame of a single device and opens up an opportunity for multichannel information processing. PSSWs have highly nonuniform alternating-sign mag- netization profiles, which makes their excitation rather complicated. Conventionally, PSSWs in thin magnetic films are excited by microwaves, which requires either nonsymmetric boundary conditions or a nonuniform mag- netic field [ 24]. However, since the spatial distribution of the external magnetic field is relatively smooth, in contrast to the PSSW profile, both approaches have low efficiency of high-order PSSW excitation and do not provide selec- tivity of the excited PSSW. In this respect, optical means allow the creation of highly nonuniform effective magnetic field distributions via inverse magneto-optical effects at the submicron scale, which can provide additional degrees of freedom in spin-wave control. Such a nonuniform distri- bution of the optical field inside the magnetic film can be 2331-7019/21/15(3)/034085(8) 034085-1 © 2021 American Physical SocietyKRICHEVSKY, IGNATYEVA, OZEROV, and BELOTELOV PHYS. REV. APPLIED 15,034085 (2021) achieved by excitation of different optical modes that gen- erally allow one to create a complicated effective magnetic field and increase its magnitude notably [ 19–25]. In accordance with the IFE, circularly polarized light induces an effective magnetic field in a magnetic medium [16,26]: Heff=−a 16πIm(E×E∗)=−a 8πE2n (1) where ais the magneto-optical constant responsible for gyrotropy, Eis the amplitude of the electric field of light, and nis the refractive vector of light ( n=k/k,kis the wave vector of light). It follows from Eq. (1)that, in a transparent medium, linearly polarized light does not gen- erate any effective magnetic field, as, in that case, the cross product (E×E∗)becomes zero. The situation becomes different if a magnetic medium is not transparent due to either optical losses or electron plasma. In this case, the refractive index of the medium acquires an imaginary part. As a result, the obliquely incident light of p-polarization becomes elliptically polarized inside the medium, i.e., a phase shift between its two orthogonal components, Exand Ez(see the coordinate system in Fig. 1), appears. It follows from Eq. (1)that, in this case, linearly polarized light can also induce the effective magnetic field; however, this field is orthogonal to the plane of light incidence, i.e., in the O-Y-axial direction [ 27]: Heff ITMOKE =−a 8πExEzsin/Delta1ϕ( n×N),( 2 ) where /Delta1ϕis the phase shift between Exand Ezfield com- ponents, Nis the normal vector to the surface of a magnetic sample (see Sec. 2 within the Supplemental Material [ 34]). For this reason, this effect can be referred to as the inverse transverse magneto-optical Kerr effect (ITMOKE). The ITMOKE is of prime importance to the optical control of spins, since it provides an optically induced in-plane mag- netic field, whereas, due to the IFE, the induce magnetic field is mainly out of plane. Optical modes in nanostructured media enhance the inverse magneto-optical effects and might modify them. However, there is a significant difference between the inverse magneto-optical effects observed at different opti- cal modes, cavities, and guided modes, as an example. The polarization of the optical cavity mode is the same for nor- mally incident light used to excite the mode. Consequently, a cavity mode excited by circularly polarized light also has circular polarization and induces the effective mag- netic field directed along the incident-light wave vector [25]. This is similar to the conventional IFE in the bulk. On the contrary, the guided modes’ polarization might be quite different with respect to excitation light [ 28]. In particular, the surface-plasmon polaritons have elliptical polarization and, therefore, induce the effective magnetic field directed FIG. 1. Schematic representation of the tunable and selective excitation of different PSSW modes via femtosecond laser pulses of different wavelengths. Femtosecond pump pulses and excited guided modes are shown with red, green, and blue colors, corre- sponding to different wavelengths. Black lines depict spin modes triggered by the corresponding optical guided mode, whereas cir-cles with arrows schematically show spin precession at different positions. orthogonally to the wave vector of the incident light and provide the ITMOKE [ 27]. While the IFE on the cavity modes is demonstrated in a one-dimensional magnetophotonic crystal with a Bi- substituted iron garnet (BiIG) cavity layer [ 25], the influ- ence of surface plasmons on spins is studied in a plas- monic structure based on gold grating deposited on a BiIG film [ 9]. Though the latter configuration belongs to the ITMOKE one, it was referred to in Ref. [ 9] as a modifi- cation of the IFE. Therefore, terminology in this area has not been established yet. In Ref. [ 10], it was shown theoret- ically that the frequency-comb technique might resonantly enhance the ITMOKE in a similar structure and launch magnetostatic spin waves. Surface-plasmon polaritons, due to their evanescent character, provide strong spatial localization (approxi- mately 100 nm) of the ITMOKE. However, this approach has several drawbacks. Due to high losses in metals, the utilization of plasmons decreases the quality factor of the resonances and leads to thermal heating [ 13,29]. At the same time, such systems are not tunable, since the parame- ters of the surface plasmons, such as field distribution and localization, cannot be varied notably. Here, we propose an all-dielectric magnetic structure to support optical guided waves for the excitation of PSSWs. It is shown analytically and numerically that the TM modes with elliptical polarization provide the ITMOKE in 034085-2SELECTIVE AND TUNABLE EXCITATION. . . PHYS. REV. APPLIED 15,034085 (2021) the magnetic structure characterized by strongly nonuni- form alternating-sign spatial distribution of the effective magnetic field. Moreover, the spatial distribution of the ITMOKE field can be significantly modified by the exci- tation of different optical modes. It allows for selective excitation of different PSSW modes and switching between them by variation of the optical pump parameters. II. RESULTS AND DISCUSSION For smooth media and without excitation of the optical modes, the ITMOKE requires lossy or metallic materials. However, if a laser pulse excites optical modes with ellip- tical polarization, then the ITMOKE might appear, even in transparent dielectric media. Let us consider TM-guided modes with nonzero Exand Ezcomponents of the electro- magnetic field that are phase shifted by π/2 and, therefore, the Evector circumscribes an ellipse in the propagation plane (Fig. 1). It follows from Eq. (2)that the TM modes can induce the effective magnetic field, Heff, directed per- pendicularly to the propagation plane of the guided mode that corresponds to the ITMOKE. One should notice that the TE-guided modes with only Eynonzero components do not produce an effective magnetic field. Apart from the ITMOKE, there is also another effect, the inverse Cotton-Mouton effect, which induces the effective magnetic field due to linearly polarized light. However,this effect appears only in the presence of the external mag- netic field. It strongly depends on the orientation of light, linear polarization, and crystallographic axes with respect to the external magnetic field. For the case of TM modes in a magnetic film of cubic crystal lattice considered here, the inverse Cotton-Mouton effect does not make any contri- bution to the spin dynamics in the optical waveguide (see Sec. 2 within the Supplemental Material [ 34]). The dispersion of the TM-guided modes of the planar waveguide with propagation constant βis determined by the following transcendental equation [ 30]: −p(2,m)d+tan−1/bracketleftbiggn2 2p(1,m) n2 1p(2,m)/bracketrightbigg +tan−1/bracketleftbiggn2 2p(3,m) n2 3p(2,m)/bracketrightbigg =− mπ, (3) where p(1,m)=(β2−n2 1k2 0)1/2,p(2,m)=(n2 2k2 0−β2)1/2, p(3,m)=(β2−n2 3k2 0)1/2,njare the refractive indices of the magnetic core (n2)and the surrounding dielectric claddings ( n1,n3),k0=2π/λ is the free-space wave num- ber,λis the wavelength, mis the integer that defines the order of the mode, and dis the core thickness. In the case of the TM modes, only the Hy,Ex,a n d Ezcomponents are present [ 30], so that the electromagnetic field of the eigenmode inside the core of the asymmetric waveguide is H(y,m)(x,z,t)=C/braceleftbigg −n2 1p(2,m) n2 2p(1,m)cos/bracketleftbigg p(2,m)/parenleftbigg z−d 2/parenrightbigg/bracketrightbigg +sin/bracketleftbigg p(2,m)/parenleftbigg z−d 2/parenrightbigg/bracketrightbigg/bracerightbigg ei(ωt−βx), E(z,m)(x,z,t)=Cβ ωn2 2/braceleftbigg −n2 1p(2,m) n2 2p(1,m)cos/bracketleftbigg p(2,m)/parenleftbigg z−d 2/parenrightbigg/bracketrightbigg +sin/bracketleftbigg p(2,m)/parenleftbigg z−d 2/parenrightbigg/bracketrightbigg/bracerightbigg ei(ωt−βx), E(x,m)(x,z,t)=− Cip(2,m) ωn2 2/braceleftbiggn2 1p(2,m) n2 2p(1,m)sin/bracketleftbigg p(2,m)/parenleftbigg z−d 2/parenrightbigg/bracketrightbigg +cos/bracketleftbigg p(2,m)/parenleftbigg z−d 2/parenrightbigg/bracketrightbigg/bracerightbigg ei(ωt−βx),(4) where Cis the magnitude of the guided optical wave. According to Eqs. (1)and (4), the TM modes induce a nonzero Heff y=−(a/16π)Im[ E(z,m)E∗ (x,m)−E(x,m)E∗ (z,m)] component of the effective magnetic field, Heff. For the symmetric claddings (n1=n3), it can be expressed as Heff (y,m)=(−1)m+1Heff (0,m)sin[2 p(2,m)(z−d)], (5) where Heff (0,m)=−a 16πC2iβp(2,m) ω2n4 2 is the magnitude of the effective magnetic field propor- tional to the optical field intensity. The Heff ycomponentis uniform along the O-Xand O-Ydirections and oscil- lates along the waveguide thickness. It is important that the guided modes of different orders, m, have a different spatial distribution of Heff (y,m)(z)inside the core. According to Eq. (3), the value of p(2,m)=[(m+μ1)π]/d, where the number 0 ≤μ1≤1 is determined by the phase: μ1=2 πtan−1/bracketleftbiggn2 2p(1,m) n2 1p(2,m)/bracketrightbigg . Consequently, the periodic part of Eq. (5)can be expressed as Heff (y,m)=(−1)m+1Heff (0,m)sin/bracketleftbigg2(m+μ1)π dz/bracketrightbigg .( 6 ) 034085-3KRICHEVSKY, IGNATYEVA, OZEROV, and BELOTELOV PHYS. REV. APPLIED 15,034085 (2021) Heff (y,m)(z)changes its sign along the film thickness, and the sign change takes place at a spatial scale of d/[2(m+μ1)], which is less than 100 nm for m>7. This is a unique feature of using the waveguide modes for effective mag- netic field generation. It is very difficult to establish such inhomogeneous alternating-sign magnetic fields by elec- tric currents or other means. Let us consider spin-wave dynamics driven by the guided modes and show that the oscillations of Heff (y,m)(z)provide selective and tunable excitation of different PSSW modes. Spin dynamics in a thin magnetic layer excited by ultra- short laser pulses via the inverse magneto-optical effects can be described by the Landau-Lifshitz-Gilbert equation [31]: dM dt=−γ(M×Heff)+α M/parenleftbigg M×dM dt/parenrightbigg ,( 7 ) where Mis the magnetization vector, γis the gyro- magnetic ratio of the magnetic medium, and αis the Gilbert damping constant. Writing Eq. (7)in the spher- ical coordinates ˜θandϕwith the polar axis along O-Z, one obtains a set of the linearized Landau-Lifshitz-Gilbertequations: ∂θ ∂t=α∂ϕ ∂t+γHϕ−γAexcM∂2ϕ ∂z2+γ/Delta1 tHeff (y,m)(z)δ( t), ∂ϕ ∂t=−α∂θ ∂t−γ/parenleftbigg H+4πM−2KU M/parenrightbigg θ+γAexcM∂2θ ∂z2, (8) where θ=(π/2)−˜θ,KUis the uniaxial anisotropy con- stant, Aexcis the exchange constant, and His the external magnetic field directed along O-X. In the absence of the external torque given by Heff (y,m)(z), the eigenmodes of Eq.(8)have the form of damped harmonic oscillations, with a damping parameter λnand precession frequency ωn, depending on the spin-wave wave number kn, according tothe following dispersion equations: λn=αγ/bracketleftbigg H+(2π+Aexck2 n)M−KU M/bracketrightbigg , ω2 n=γ2(H+AexcMk2 n) ×/parenleftbigg H+4πM−2KU M+AexcMk2 n/parenrightbigg −λ2 n.( 9 ) Let us assume no pinning of spins on the boundaries, i.e., free boundary conditions, which is generally the case for iron garnet and other magnetic dielectric films [ 32]. In this case, the spin eigenmodes θn(z,t)have the following form: θn(z,t)∝e−λntsinωnt/parenleftBigg cos knz,n=2, 4, 6, ... sin knz,n=1, 3, 5, .../parenrightBigg , (10) where kn=(πn/d),λnis the damping parameter, and ωnis the frequency of the PSSW of nth order defined by Eq. (9). Odd and even mode numbers correspond to the antisym- metric and symmetric solutions, respectively. Similar solu- tions can be found for ϕn(z,t)dependence; however, we focus our attention on θvariation, since the probe transient Faraday rotation proportional to θis usually measured in pump-probe experiments. The femtosecond pulse knocks the system out of the equilibrium θ=0 via the effective magnetic field Heff (y,m)(z). The magnetization dynamics after this action is described via the series of excited-spin eigenmodes: θ(z,t)=∞/summationdisplay n=1Anθn(z,t), (11) with the corresponding magnitudes, An, of PSSW modes of An=a0/integraldisplayd/2 −d/2Heff (y,m)(z)θn(z)dz, (12) where a0is the normalization constant. Taking into account Eqs. (6)and(12) Ancan be written as An∝/integraltextd/2 −d/2(−1)m+1sin/bracketleftbigg2(m+μ1)π dz/bracketrightbigg cosπn dzdz,n=2, 4, 6, ..., /integraltextd/2 −d/2(−1)m+1sin/bracketleftbigg2(m+μ1)π dz/bracketrightbigg sinπn dzdz,n=1, 3, 5, ....(13) It follows from Eq. (13) that the amplitude, An, of a certain PSSW mode is maximal when the spatial distribution of θn(z)is close to Heff (y,m)(z). Namely, if μ1=0.5, the opticalguided wave of mode number mexcites predominantly the PSSW of order n=2m+1, whereas other PSSWs are almost absent. Varying the wavelength of the incident 034085-4SELECTIVE AND TUNABLE EXCITATION. . . PHYS. REV. APPLIED 15,034085 (2021) laser pulse, one may excite optical guided modes of dif- ferent m[see Eq. (3)], which induce different profiles of effective magnetic field [see Eq. (6)] and, therefore, launch PSSWs of different orders n. We illustrate this process of tunable and selective PSSW excitation by numerical sim- ulations performed by solving Maxwell equations through the optical transfer-matrix approach [ 33]. The guided TM mode in a planar waveguide can be excited by a prism or a grating coupling, if phase- matching conditions are satisfied. Here, we assume the prism-coupling method. The investigated structure con- sists of a GaP prism, a 350-nm-thick SiO 2layer, and an optical waveguide core made of a 400-nm-thick bismuth iron garnet film on SiO 2substrate. The incidence angle for the optical pump pulse is fixed to 28°. The spectral posi- tions of the TM-guided modes correspond to high-quality resonant dips in the reflectance spectrum [Fig. 2(a)]. For instance, resonances at 1392, 799, and 558 nm correspond to TM1, TM2, and TM3 mode excitations inside the iron garnet film. The electromagnetic field distributions of these modes presented in Fig. 2(b) agree with the theoretical predictions obtained using Eq. (3).Hyand Heff yof the cor- responding TM1, TM2, and TM3 eigenmodes are depicted in Fig. S1 within the Supplemental Material [ 34]. The oscillating Ex,Ez,a n d Hypatterns in both O-Xand O- Zdirections result in the effective magnetic field Heff y(z) varying only along the O-Zdirection [Fig. 2(c)], with the period decreasing with the mode number m(which alsoagrees with the analytical description presented in Sec. 1 within the Supplemental Material [ 34]). Both Heff yand spin-wave profiles resemble a standing wave distributed between the SiO 2cladding. The shapes of the Heff yfields of TM1, TM2, and TM3 guided modes are very close to the PSSW eigenmodes of n=3, 5, and 7 orders, respectively [Figs. 3(a)–3(c)]. The rigorous numerical calculation of the excited PSSWs magnitudes An [Figs. 3(d)–3(f)] performed via Eq. (12) shows very high selectivity of the n=2m+1 PSSW excitation, with the magnitudes of the other modes being more than one order lower than that of the n=2m+1 mode. We also consider the case of pinned spins at the boundary of the magnetic waveguide and present a short summary in Sec. 3 within the Supplemental Material [ 34]. The frequency of the PSSWs depends on the exter- nal magnetic field applied to the structure, according to Eq.(9)[see Fig. 4(a)]. In the calculations, the following parameters of the BiIG film are used: Aexc=0.31×10−6erg cm,4πM=854 G,γ 2π=280THz G. KUis neglected due to its small contribution. The modes of lower orders are very close to each other (the spac- ing is several hundreds of MHz), whereas the higher-order modes are separated by several GHz. Considering the external magnetic field H=1000 Oe, one may estimate (a)(b) (c) TM1, = 1392 nmEEEy 2 0] [* -101 1000 nm TM2, = 799 nm 1000 nm GaP SiO2 BiIG SiO230 n m5 400 nmTM3, = 558 nm 1000 nmTM1, = 1392 nm 1000 nm TM2, = 799 nm 1000 nm TM3, = 558 nm 1000 nm –101Normalized amplitude zxyH/ Hy0 Normalized amplitude 500 750 1000 1250 1500020406080100 Reflectance( % ) Wavelength (nm)5587991392 GaP SiO2 BiIG SiO2GaP SiO2 BiIG SiO2 30 n m5 400 nm30 n m5 400 nm GaP SiO2GaP SiO2 BiIG SiO2GaP SiO2GaP SiO2 BiIG SiO2GaP SiO2GaP SiO2 BiIG SiO2GaP SiO2 30 n m5 400 nm30 n m5 400 nm30 n m5 400 nm FIG. 2. Calculated reflectance spectrum of the structure under consideration (a); Hyspatial distribution at t=0 for TM1, TM2, and TM3 inside the iron garnet layer of the waveguide (b); and corresponding time-independent Heff y(c). 034085-5KRICHEVSKY, IGNATYEVA, OZEROV, and BELOTELOV PHYS. REV. APPLIED 15,034085 (2021) Spin-wave mode order Spin-wave mode order Spin-wave mode order1.0 1.01.0 1.0 1.0 1.0 1.0 1.0(a) (b) (c) (d) (e) (f) FIG. 3. (a)–(c) Heff ycalculated from Eq. (4)and spin-wave profiles in the cross section of the waveguide. Blue lines represent Heff y; yellow lines are spin-wave profiles of n=3, 5, and 7 orders. Orange and purple lines are spin-wave profiles of n−2a n d n+2o r d e r s , respectively. (d)–(f) Bar plots of PSSW amplitudes An(normalized on the major spin-wave maxima) for the corresponding Heff yprofiles of (a)–(c). both frequencies, ωn,and the damping parameter, λn,using Eq.(8)and, therefore, obtain the spectrum of the magne- tization precession, S(ωspin), excited by light of frequency ωlightin the structure, see Fig. 4(b): S(ωspin,ωlight) =1 max nA2n(ωlight)/summationdisplay A2 n(ωlight)e−[(ωspin−ωn)2/λ2n]. (14)The spin-precession spectrum [Fig. 4(b)] shows that, as the Heff yfield is nonresonantly generated at all optical fre- quencies, the PSSW modes are excited by the incident TM-polarized optical radiation of any wavelength, too. This situation is quite the opposite of the smooth magnetic film without a prism coupler, where the linear polarization of light does not induce any IFE or ITMOKE at all. The spectra of the excited magnetization precession, therefore, can be tuned continuously through a change of the optical pump-light wavelength with certain resonance frequencies 5 1 01 52 0 Spin-wave frequency (GHz)500 750 1000 1250 1500Optical wavelength (nm) 0.00.20.40.60.81.0 0 200 400 600 800 1000 H (Oe)051015 f (GHz)n = 8 n = 7 n = 6 n = 5 n = 0(a)Normalized amplitude(b) n = 4 FIG. 4. Frequency of PSSW modes as a function of magnetic field (a) and the PSSW-mode normalized amplitude as a function of optical and spin-wave frequency (b) calculated for an external magnetic field of H=1000 Oe. 034085-6SELECTIVE AND TUNABLE EXCITATION. . . PHYS. REV. APPLIED 15,034085 (2021) of guided modes exciting a single PSSW resonance with the enhanced magnitude. The proposed iron garnet waveguide structure has sev- eral advantages. The first one is its selectivity. This means that the desired order of the spin-wave mode can be excited by the proper selection of an optical mode by simply changing the pump incident angle or wavelength. The last feature results in the second advantage, i.e., the spectral tunability of such a system without changing the structure’s properties (core thickness, materials of the claddings, etc.) Such spectral selectivity and tunability open up opportunities for multispectral spin-wave control. While in our recent study the consideration was focused on prism coupling, nanogratings can also be utilized for the excitation of optical modes in the waveguide. However, the efficiency of optical-mode excitation can be decreased due to scattering effects. It is noteworthy that the core thickness can be signifi- cantly decreased by utilizing highly optically dense mag- netic materials, such as optically active magnetic semi- conductors, for instance, (Ga,Mn )As, GaAs ( n∼3.5 at 700 nm), or some related compounds. In this case, the waveguiding structure can be encapsulated in the system of integrated electro-optical devices [ 35] with magnon-based logical elements [ 1]. As the frequency of the PSSWs can reach tens or even hundreds of GHz, the approach opens up opportunities for advanced high-frequency data processing beyond silicon technology. III. CONCLUSION We propose and theoretically demonstrate an efficient approach for a perpendicular standing spin-wave launch by optical TM-guided modes. The optical mode induces an effective magnetic field with a highly nonuniform alternating-sign distribution that results in the PSSW exci- tation of different orders. The system is both spectrally tunable and selective: at a given iron garnet film thick- ness, one may switch between several high-order PSSW modes via tuning the wavelength of the optical pump.Our approach opens up a route towards the excitation of high-frequency spin waves in iron garnets. ACKNOWLEDGMENTS This work is financially supported by the Russian Min- istry of Education and Science, Megagrant Grant No. N 075-15-2019-1934. The authors thank Dr. Elena Bazanova for help with English language editing. [1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hille- brands, Magnon spintronics, Nat. Phys. 11, 453 (2015). [2] K. Ganzhorn, S. Klingler, T. Wimmer, S. Geprägs, R. 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PhysRevB.87.224422.pdf

PHYSICAL REVIEW B 87, 224422 (2013) Angle-dependent spin-wave resonance spectroscopy of (Ga,Mn)As films L. Dreher,1,*C. Bihler,1E. Peiner,2A. Waag,2W. Schoch,3W. Limmer,3S. T. B. Goennenwein,4and M. S. Brandt1 1Walter Schottky Institut, Technische Universit ¨at M ¨unchen, Am Coulombwall 4, 85748 Garching, Germany 2Institut f ¨ur Halbleitertechnik, Technische Universit ¨at Braunschweig,Hans-Sommer-Straße 66, 38023 Braunschweig, Germany 3Institut f ¨ur Quantenmaterie, Universit ¨at Ulm, 89069 Ulm, Germany 4Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Walther-Meißner-Straße 8, 85748 Garching, Germany (Received 5 March 2013; published 25 June 2013) A modeling approach for standing spin-wave resonances based on a finite-difference formulation of the Landau- Lifshitz-Gilbert equation is presented. In contrast to a previous study [C. Bihler et al. ,Phys. Rev. B 79, 045205 (2009) ], this formalism accounts for elliptical magnetization precession and magnetic properties arbitrarily varying across the layer thickness, including the magnetic anisotropy parameters, the exchange stiffness, theGilbert damping, and the saturation magnetization. To demonstrate the usefulness of our modeling approach,we experimentally study a set of (Ga,Mn)As samples grown by low-temperature molecular-beam epitaxy bymeans of angle-dependent standing spin-wave resonance spectroscopy and electrochemical capacitance-voltagemeasurements. By applying our modeling approach, the angle dependence of the spin-wave resonance data canbe reproduced in a simulation with one set of simulation parameters for all external field orientations. We findthat the approximately linear gradient in the out-of-plane magnetic anisotropy is related to a linear gradient inthe hole concentrations of the samples. DOI: 10.1103/PhysRevB.87.224422 PACS number(s): 75 .50.Pp, 76 .50.+g, 75.70.−i, 75.30.Ds I. INTRODUCTION Due to their particular magnetic properties, including magnetic anisotropy,1–3anisotropic magnetoresistance4,5and magnetothermopower,6in past years ferromagnetic semicon- ductors have continued to be of great scientific interest in ex-ploring new physics and conceptual spintronic devices. 7–11The most prominent ferromagnetic semiconductor is (Ga,Mn)As,where a small percentage of Mn atoms on Ga sites introduceslocalized magnetic moments as well as itinerant holes whichmediate the ferromagnetic interaction of the Mn spins ( p-d exchange interaction). 12Both theoretical and experimental studies have shown that the magnetic anisotropy, i.e., thedependence of the free energy of the ferromagnet on themagnetization orientation, depends on the elastic strain andthe hole concentration in the (Ga,Mn)As layer, 12,13opening up several pathways to manipulate the magnetic anisotropy of(Ga,Mn)As. 14–16 A common spectroscopic method to probe the magnetic anisotropy of ferromagnets, in particular, (Ga,Mn)As, isangle-dependent ferromagnetic resonance (FMR), 17–23where FMR spectra are taken as a function of the orientationof the external magnetic field. If the magnetic propertiesof the ferromagnet are homogeneous, a zero wave vector(k=0) mode of collectively, uniformally precessing magnetic moments couples to the microwave magnetic field, e.g., in amicrowave cavity, allowing for detection of the magnetizationprecession. The resonance field of this mode, referred to as theuniform resonance magnetic field, depends on the employedmicrowave frequency and the magnetic anisotropy parameters.Thus, by recording FMR spectra at different orientationsof the external field with respect to the crystal axes, theanisotropy parameters can be deduced from the experiment.However, if the magnetic properties of a ferromagnetic layerare nonhomogeneous or the spins at the surface and interfaceof the layer are pinned, nonpropagating modes with k/negationslash=0, referred to as standing spin-wave resonances (SWRs), can beexcited by the cavity field and thus be detected in an FMR experiment. On one hand, this can hamper the derivation ofanisotropy parameters; on the other hand, a detailed analysisof these modes can elucidate the anisotropy profile of the layerand the nature of spin pinning conditions. Furthermore, theexcitation of spin waves is of topical interest in combinationwith spin pumping, 24–27i.e., the generation of pure spin currents by a precessing magnetization.28–30In this context, the exact knowledge of the magnetization precession amplitude asa function of the position coordinate within the ferromagnet isof particular importance. 24 Several publications report on SWR modes in (Ga,Mn)As with a mode spacing deviating from what is expected according to the Kittel model for magnetically homogeneous films withpinned spins at the surface. 31–36These results have been attributed to an out-of-plane anisotropy field linearly31,36or quadratically varying33–35as a function of the depth into the layer, as well as to specific spin pinning conditions at the surface and at the interface to the substrate.35While most of these studies have focused on the spacings of theresonance fields when modeling SWR measurements, inRef. 36a more sophisticated approach, based on a normal mode analysis, 37,38was employed to model resonance fields as well as relative mode intensities for the external field ori-ented along high-symmetry directions, assuming a circularlyprecessing magnetization. In this work, we present a more general modeling approach for SWR, based on a finite-difference formulation of theLandau-Lifshitz-Gilbert (LLG) equation. This approach holdsfor any orientation of the external magnetic field and accountsfor elliptical magnetization precession (Sec. II). It allows for a simulation of arbitrarily varying profiles of the magneticproperties across the thickness of the film, including vatiationsof the magnetic anisotropy parameters, the exchange stiffness,and the Gilbert damping parameter. As a result of thesimulation, we obtain the Polder susceptibility tensor as a 224422-1 1098-0121/2013/87(22)/224422(12) ©2013 American Physical SocietyL. DREHER et al. PHYSICAL REVIEW B 87, 224422 (2013) function of the depth within the ferromagnet. Based on this result, the absorbed power upon SWR and the magnetizationprecession amplitude as a function of the depth can becalculated for any orientation of the external magnetic field. We apply our modeling approach to a set of four (Ga,Mn)As samples epitaxially grown with different V/III flux ratios(Sec. III), motivated by the observation that V/III flux ratios of/lessorsimilar3 lead to a gradient in the hole concentration p, 39 which in turn is expected to cause nonhomogeneous mag- netic anisotropy parameters.31,36Electrochemical capacitance- voltage (ECV) measurements revealed a nearly linear gradientinpacross the thickness of the layers investigated. To show that our modeling approach is capable of simulatingSWR spectra for arbitrary magnetic field orientations, angle-dependent SWR data were taken and compared with themodel using one set of magnetic parameters for each sample,revealing gradients in the uniform resonance magnetic fields.We discuss the influence of the gradient in pon the observed uniform resonance field gradients as well as the possibleinfluences of strain and saturation magnetization gradients onthe observed out-of-plane anisotropy profile. It should be em-phasized, however, that the objective of this work is to show theusefulness of our modeling approach, while a detailed investi-gation of the origin of the gradient in the out-of-plane magneticanisotropy profile and therefore a detailed understanding of theparticular materials physics of (Ga,Mn)As is beyond the scopeof this study. Finally, we summarize our results and discussfurther potential applications of this work (Sec. IV). II. THEORETICAL CONSIDERATIONS In this section, we provide the theoretical framework necessary to describe the full angle dependence of the SWRspectra presented in Sec. III. Referring to the coordinate system depicted in Fig. 1, we start from the canonical expression for the free enthalpy density (normalized to the saturationmagnetization M) for a tetragonally distorted (Ga,Mn)As film: 13,20,40,41 G=const−μ0H·m+B001m2 z+B4⊥m4 z +B4/bardbl/parenleftbig m4 x+m4 y/parenrightbig +1 2B1¯10(mx−my)2. (1) φ0Θ0 123m2 m1m3≈1 m x||[100]y||[010]z||[001] SubstrateFerromagnet FIG. 1. (Color online) Relation between the two coordinate sys- tems employed. The ( x,y,z ) frame of reference is spanned by the cu- bic crystal axes, while the (1 ,2,3) coordinate system is determined by the equilibrium orientation of the magnetization (3 direction) and two transverse directions, the 2 direction being parallel to the film plane; the latter system is zandμ0Hdependent, as described in the text.Here, μ0His a static external magnetic field, B001is a uniaxial out-of-plane anisotropy parameter, reflecting shapeand second-order crystalline anisotropy, 13andB4⊥,B4/bardbl, and B1¯10are fourth-order crystalline and second-order uniaxial in-plane anisotropy parameters, respectively;1mx,my, and mzdenote the components of the normalized magnetization vectorm(z)=M(z)/M(z) along the cubic axes [100], [010], and [001], respectively. We assume the magnetic properties ofthe layer to be homogeneous laterally (within the xyplane) and inhomogeneous vertically (along z); the anisotropy parameters in Eq. (1)and the magnetization are consequently a function of the spatial variable z. To obtain the anisotropy parameters from Eq.(1)in units of energy density, it would therefore be required to know the zdependence and the absolute value of M. The minimum of Eq. (1) determines the equilibrium orientation of the magnetization, given by the angles θ 0=θ0(z) andφ0=φ0(z)( c f .F i g . 1). To describe the magnetization dynamics, we introduce a new frame of reference, (1 ,2,3), shown in Fig. 1, in which the equilibrium orientation of the magnetization m0coincides with axis 3. For small pertur- bations, the magnetization precesses around its equilibriumwith finite transverse components of the magnetization m i (i=1,2) as illustrated in the inset in Fig. 1. The transformation between the two coordinate systems is given in Appendix A by Eqs. (A1) and (A2) . We write, for the (normalized) magnetization, m=⎛ ⎜⎝0 01⎞ ⎟⎠ /bracehtipupleft /bracehtipdownright/bracehtipdownleft/bracehtipupright m0+⎛ ⎜⎝m1 m2 0⎞ ⎟⎠+O/parenleftbig m2 1,m22/parenrightbig . (2) The evolution of the magnetization under the influence of an effective magnetic field μ0Heffis described by the LLG equation42,43 ∂tm=−γm×μ0Heff+αm×∂tm, (3) where γis the gyromagnetic ratio and αa phenomenological damping parameter. The effective magnetic field is given by36 μ0Heff=− ∇ mG+Ds M∇2M+μ0h(t), (4) where ∇m=(∂m1,∂m2,∂m3) is the vector differential operator with respect to the components of m,Ds=2A/M is the exchange stiffness with the exchange constant A,∇2is the spatial differential operator ∇2=∂2 x+∂2 y+∂2 z, andh(t)= h0e−iωtis the externally applied microwave magnetic field with angular frequency ω;h(t) is oriented perpendicularly to μ0H. Since the magnetic properties are independent of xand y,E q . (3)simplifies to ∂tm=−γm×[−∇mG+Dsm/prime/prime+μ0h(t)]+αm×∂tm, (5) withm/prime/prime=∂2 zm, neglecting terms of the order of m2 i(fori= 1,2). By definition of the (1 ,2,3) coordinate system, the only nonvanishing component of ∇mGin the equilibrium is along the 3 direction. For small deviations of mfrom the equilibrium 224422-2ANGLE-DEPENDENT SPIN-W A VE RESONANCE ... PHYSICAL REVIEW B 87, 224422 (2013) we find44 ∇mG=⎛ ⎜⎝G11m1+G21m2 G12m1+G22m2 G3⎞ ⎟⎠, (6) where we have introduced the abbreviations Gi=∂miG|m=m0 andGij=∂mi∂mjG|m=m0; the explicit expressions for these derivatives are given in Appendix A. In the following, we calculate the transverse magnetization components assuming a harmonic time dependence mi= mi,0e−iωt. The linearized LLG equation, considering only the transverse components, reads /parenleftbiggH11H12 H21H22/parenrightbigg/parenleftbiggm1 m2/parenrightbigg −Ds/parenleftbiggm/prime/prime 1 m/prime/prime2/parenrightbigg =μ0/parenleftbiggh1 h2/parenrightbigg ,(7) where we have introduced the abbreviations H11=G11− G3−iαω/γ ,H12=H∗ 21=G12+iω/γ , and H22=G22− G3−iαω/γ . We have dropped all terms which are nonlinear inmiand products of miwith the driving field. Resonant uniform precession of the magnetization ( m/prime/prime i= 0) occurs at the so-called uniform resonance field μ0Huni(z), which is found by solving the homogeneous ( h=0) equation H11(z)H22(z)−H12(z)H21(z)=0 (8) ⇔(G11−G3)(G22−G3)−G2 12=/parenleftbiggω γ/parenrightbigg2 forμ0H, neglecting the Gilbert damping ( α=0). Equation (8)can be used to derive anisotropy parameters from angle- dependent FMR spectra. As extensively discussed by Baselgiaet al. , 44using Eq. (8)is equivalent to using the method of Smit and Beljers, which employs second derivatives of thefree enthalpy with respect to the spherical coordinates. 41,45,46 To illustrate the role of the uniform resonance field in the context of SWRs, we consider the special case wheremagnetization is aligned along the [001] crystal axis ( θ 0=0), before we deal with the general case of arbitrary field orien-tations. Neglecting the uniaxial in-plane anisotropy ( B 1¯10= 0) since this anisotropy is typically weaker than all otheranisotropies, 13,41we find G3=−μ0H+2B001+4B4⊥and G11=G22=G12=0, resulting in the uniform resonance field μ0H001 uni(z)=ω/γ+2B001(z)+4B4⊥(z). (9) To find the eigenmodes of the system, we consider the unperturbed and undamped case, i.e., α=0 and h=0i n Eq.(7). With m2=im1=˜mwe find the spin-wave equation Ds˜m/prime/prime+μ0H001 uni(z)˜m=μ0H˜m, (10) in agreement with Ref. 36. The relation of the anisotropy parameters defined in Ref. 36 to the ones used here is given by B001=K100 eff/M+B1¯10, B1¯10=−K011 u/M,2B4⊥=−K⊥ c1/M, and 2 B4/bardbl=−K/bardbl c1/M. Equation (10) is mathematically equivalent to the one- dimensional time-independent Schr ¨odinger equation, where the uniform resonance field corresponds to the potential, ˜mto the wave function, μ0Hto the energy, and Dsis proportional to the inverse mass. To calculate the actual precession amplitudeof the magnetization, the coupling of the eigenmodes ofEq.(10) to the driving field is relevant, which is proportional to the net magnetic moment of the mode. 36,38In analogy to a particle in a box, the geometry of the uniform resonance fieldas well as the boundary conditions determines the resonancefields and the spatial form of the precession amplitude. For theremainder of this work, we assume the spins to exhibit natural freedom at the boundaries of the film, i.e., ∂ z˜m=˜m/prime=0 at the interfaces,36,47since these boundary conditions have been shown to describe the out-of-plane SWR data of similarsamples well. 36To graphically illustrate the influence of the uniform resonance field on the SWR modes, we consider inFig. 2a ferromagnetic layer with a thickness of 50 nm with constant magnetic properties across the layer [Fig. 2(a)] and with a linearly varying uniform resonance field [Fig. 2(c)]; in both cases we assume D s=13 T nm−2, a value similar to that obtained in previous studies.36For these conditions, we (a) (b) (c) (d) m (arb. u.)~ m (arb. u.)~ FIG. 2. (Color online) Simulation to demonstrate the influence of the uniform resonance field μ0H001 union the SWR modes for m0||[001], assuming circular precession. In (a), μ0H001 uniis set to be constant across the layer, while in (c) it varies linearly [dashed(blue) lines], in analogy with a square potential and a triangular potential, respectively. Dotted (black) lines are the resonance fields, calculated assuming boundary conditions of natural freedom (seetext). Solid (red) lines show the eigenmodes of the system, i.e., the precession amplitude ˜mof the magnetization; for each mode the dotted line corresponds to ˜m=0. As shown in (a), for a constant uniform resonance field the first mode occurs at the uniform resonance field and exhibits a constant precession amplitude across the layer, i.e., an FMR mode. The second and third modes (higher order modes are not shown) exhibit a nonuniform magnetization profile. In order to couple to the driving field the modes need to have a finite netmagnetic moment. As shown in (a), the positive and negative areas of the second and third modes are equal, thus these modes are not visible in the SWR spectrum (b). This is in contrast to the case of the linearlyvarying uniform resonance field (c), where the mode profile is given by Airy functions, which have a nonzero net magnetic moment also for the second and third modes, resulting in a finite SWR intensity ofthese modes (d). Spectra in (b) and (d) were calculated by integrating over the eigenmodes ˜mand convoluting the square of the result with Lorentzians. 224422-3L. DREHER et al. PHYSICAL REVIEW B 87, 224422 (2013) numerically solve Eq. (10) by the finite-difference method described in Appendix B1, in order to obtain the resonance fields (eigenvalues) and the zdependence of the transverse magnetic moments (eigenfunctions). To which amount a modecouples to the driving field is determined by the net magneticmoment of the mode, which is found by integrating ˜m(z) over the thickness of the film. For the magnetically homogeneouslayer, the only mode that couples to the driving field is theuniform precession mode at μ 0H001 uni, since modes of higher order have a zero net magnetic moment [Fig. 2(a)], resulting in one resonance at the uniform resonance field [cf. Fig. 2(b)]. For the nonuniform layer, with μ0H001 uni(z) linearly varying across the film, the mode profile is given by Airy functions31,36,38and various nonuniform modes couple to the driving field, resultingin several SWRs, with their amplitude proportional to thesquare of the net magnetic moment 36,38of the corresponding mode [cf. Figs. 2(c) and2(d)]. We now turn to the general case of arbitrary field orienta- tions. Due to the magnetic anisotropy profile, the magnetiza-tion orientation is ap r i o r i unknown and a function of zand μ 0H. Furthermore, the assumption of a circularly precessing magnetization is not generally justified. To solve Eq. (7) for arbitrary field orientations, we employ a finite-differencemethod as outlined in Appendix B2. By solving Eq. (7),we obtain the z-dependent generalized Polder susceptibility tensor ¯ χ(μ 0H,z), which relates the transverse magnetization components Mi(z)=M(z)mi(z) with the components of the driving field by /parenleftbiggM1 M2/parenrightbigg =¯χ(μ0H,z)/parenleftbiggh1 h2/parenrightbigg . (11) In a microwave absorption measurement, the components Mi which are out of phase with the driving field are detected. The absorbed power density is related to the imaginary part of ¯χ(μ0H,z) and can be calculated by48 P=ωμ 0 2z0Im/braceleftbigg/integraldisplay0 −z0/bracketleftbigg (h∗ 1,h∗2)¯χ(μ0H,z)/parenleftbiggh1 h2/parenrightbigg/bracketrightbigg dz/bracerightbigg ,(12) where z0is the thickness of the ferromagnetic layer. Note that the position coordinate zis negative in the film (cf. Fig. 1). To obtain an impression of how gradients in different anisotropy parameters influence the SWR spectra, we plot inFig. 3simulated SWR spectra together with the magnetization precession cone as a function of the depth in the ferromagneticlayer. We assume a constant saturation magnetization (itsvalue is not relevant for the outcome of the simulation), aconstant exchange stiffness D s=35 T nm2unless otherwise specified, α=0.09, and B001=90 mT, B4||=− 50 mT, 90 03060 90 03060 SWR Intensity (arb. units)ψ (deg.) ψ (deg.) 200 600 400 μ0H (mT)ψμ0H [110][001](a ii) (a iii) (a iv) )iii b( )ii b( )i b( (b iv) )vi c( )iiic( )iic( )i c(Im(m1m2-m1m2) (arb. u.) **(a i)90 03060ψ (deg.) FIG. 3. (Color online) Atlas illustrating the influence of gradients in the anisotropy parameters on SWR spectra. In (a) all anisotropy parameters are kept constant with the values given in the text, except B001which is varied linearly. Correspondingly, in (b) and (c) B4⊥and B4||were varied linearly, respectively. Panels (i) show the first derivative of simulations using Eq. (12) with respect to μ0Hand panels (ii)-(iv) show the precession cone Im( m∗ 1m2−m1m∗ 2) in a color plot together with the uniform resonance field μ0Huni(z) (dashed blue lines) at three different external field orientations; the black dotted lines indicate the resonance field positions of the modes. Panel (a i) additionally shows the influence of a linear gradient in the exchange stiffness parameter on the spin-wave spectra, see text for further details and discussion. 224422-4ANGLE-DEPENDENT SPIN-W A VE RESONANCE ... PHYSICAL REVIEW B 87, 224422 (2013) B4⊥=15 mT. In Fig. 3(a), we assume B001to vary across the layer thickness according to B001(z)=B001−b001×zwith b001=− 0.8m T/nm. Figure 3(a i) shows the simulated SWR spectra calculated by taking the first derivative of Eq. (12) with respect to μ0Hfor different angles ψdefined in the inset in Fig. 3(c iv). We observe several SWR modes for μ0H||[001], with the number decreasing as μ0His tilted away from [001]. Atψ=40◦only one mode is visible, while for ψ=0◦we again observe multiple SWR modes. This observation can beunderstood by considering the uniform resonance fields as afunction of the depth for these orientations. In Figs. 3(a ii)– (a iv), we show the uniform resonance field [dashed (blue)line] for ψ=0 ◦,ψ=30◦, andψ=90◦, respectively, together with the magnetization precession cone Im( m∗ 1m2−m1m∗ 2)i n a contour plot as a function of depth and μ0H.A tψ=90◦, the uniform resonance field varies strongly across the film,which can be understood by considering Eq. (9). This results in several spin-wave modes, with their resonance fields indicatedby dotted lines. For other field orientations, the formula for the uniform resonance field can also be derived but results in a longer,more complex equation than Eq. (9). Important in this context is that positive values of B 001lead to an increase (decrease) in the resonance field for magnetization oriented perpendicular(parallel) to the film plane, accounting for the reversed sign ofthe slopes of μ 0Huniin Figs. 3(a ii) and 3(a iv). Consequently, in between these two extreme cases μ0Hunimust be constant across the layer for some field orientation, in our case for ψ= 30◦, resulting in a single SWR mode [cf. Figs. 3(a i) and 3(a iii). In addition to the SWR simulations with constant Ds,w ep l o t in Fig. 3(a i) simulated SWR spectra with Dsvarying linearly across the film, with Ds=35–65 T nm2[dotted (blue) lines] andDs=35–5 Tnm2[dashed (green) lines]. A decreasing Ds leads to a decreasing spacing in the modes, and vice versa, for an increasing Ds, as can be seen, e.g., for μ0H||[001]. In Fig. 3(b), we consider the case where all magnetic parameters are constant with the values given above, exceptB 4⊥(z)=B4⊥−b4⊥×zwithb4⊥=− 0.4m T/nm. As is evident from Eq. (9), this results in the same slope of μ0Huni forψ=90◦as in the case above where we varied B001 only [cf. Fig. 3(a iv) and 3(b iv)]. In contrast to the case depicted in Fig. 3(a), however, here for ψ=0◦the uniform resonance field is constant. This can be understood whenevaluating the parameters that enter into the calculation ofthe uniform resonance field [Eq. (8)]. Ifmis in the film plane, none of the parameters in Eqs. (A4) –(A6) depends onB 4⊥, resulting in a constant uniform resonance field for ψ=0◦.A smis tilted away from the film plane, B4⊥enters into some of the terms in Eqs. (A4) –(A6) . As a consequence, μ0Hunivaries, first such that it increases [cf. Fig. 3(b iii)] and, finally, such that it decreases as a function of the depth[cf. Fig. 3(b iv)]. Finally, we discuss the case where all parameters are con- stant except B 4||(z)=B4||−b4||×zwithb4||=− 0.4m T/nm [Fig. 3(c)]. Here, μ0Huniis constant for ψ=90◦as predicted by Eq. (9).A smis tilted away from [001] a varying B4||leads to a varying uniform resonance field as shown in Figs. 3(c ii) and 3(c iii). Here, a sign reversal of the slope as is the case inFigs. 3(a) and3(b) does not take place and multiple resonances occur, starting at ψ=60 ◦[Fig. 3(c i)].III. EXPERIMENTAL RESULTS AND DISCUSSION (Ga,Mn)As samples with a nominal Mn concentration of ≈4% were grown on (001)-oriented GaAs substrates by low- temperature molecular-beam epitaxy at a substrate temperatureof 220 ◦C using V/III flux ratios of 1.1, 1.3, 1.5, and 3.5, referred to as samples A, B, C, and D, respectively. The layer thicknesswas 210–280 nm as determined from the ECV measurements(cf. Fig. 4). For samples with V/III flux ratios of /lessorsimilar3 a gradient in the hole concentration has been reported, 39hence this set of samples was chosen to study the influence of a gradient inpon the out-of-plane magnetic anisotropy. Further details on the sample growth can be found in Refs. 39and41. The hole concentration profiles of the as-grown (Ga,Mn)As layers were determined by ECV profiling using a BioRadPN4400 profiler with a 250-ml aqueous solution of 2.0 gNaOH +9.3 g EDTA as the electrolyte. For further details on the ECV analysis see Ref. 39. The results of the ECV mea- surements for the layers investigated are shown in Fig. 4(a). Except for the sample with V /III=3.5, they reveal a nearly linearly varying hole concentration across the layer thicknesswith different slopes and with the absolute value of the holeconcentration at the surface of the layer varying by about20%. The profiles are reproducible within an uncertainty ofabout 15%. Secondary ion mass spectroscopy measurementsof similar samples showed that the Mn content can vary by upto 40% across the sample depth. 39 To investigate the magnetic anisotropy profiles of the samples, we performed cavity-based FMR measurements,using a Bruker ESP300 spectrometer operating at a microwavefrequency of 9.265 GHz ( Xband) with a microwave power of 2m Wa t T=5 K; we used magnetic field modulation at a frequency of 100 kHz and an amplitude of 3.2 mT. Since we (a) (b) FIG. 4. (Color online) (a) The hole concentration in the different (Ga,Mn)As samples is shown as a function of the depth within thelayers as determined by ECV profiling. (b) Uniform resonance fields μ 0H001 uni(z) for the four samples obtained from simulations for the out-of-plane orientation of the external field ( ψ=90◦) as a function of the depth. 224422-5L. DREHER et al. PHYSICAL REVIEW B 87, 224422 (2013) ψ[001] [110]μ0Hext SimulationExperiment(a) (b) (d) (c) V/III=1.5 V/III=3.5V/III=1.3 V/III=1.1 FIG. 5. (Color online) The spin-wave resonance data [dotted (blue) lines] are shown together with simulations [solid (red) lines] using the numerical procedure described in the text and in Appendix B2. Data were obtained as a function of the external magnetic field orientation and magnitude for samples with a V/III flux ratio of (a) 1.1, (b) 1.3, (c) 1.5, and (d) 3.5. The rotation angle ψis defined in the inset and the parameters used for the simulations are summarized in Table I. are mainly interested in the out-of-plane magnetic anisotropy, we recorded spectra for external magnetic field orientationswithin the crystal plane spanned by the [110] and [001] crystalaxes in 5 ◦steps (cf. the inset in Fig. 5). For each orientation, the field was ramped to 1 T in order to saturate the magnetizationand then swept from 650 to 250 mT; the spectra for the samplesinvestigated are shown in Fig. 5. We start by discussing qualitative differences in the spectra. Samples A and B exhibit several pronounced resonances forthe external field oriented along [001], which we attribute tostanding SWRs [Figs. 5(a) and 5(b)]. For these samples, the [001] direction is the magnetically hardest axis since at thisorientation the resonance field of the fundamental spin-wavemode is larger than at all other orientations. As the externalfield is rotated into the film plane, the resonance position of thismode gradually shifts to lower field values as expected for apronounced out-of-plane hard axis. In contrast, samples C andD exhibit the largest resonance fields for a field orientationof 50 ◦–60◦[Figs. 5(c) and 5(d)] pointing to an interplay of second- and fourth-order out-of-plane anisotropy withdifferent signs of the corresponding anisotropy parameters.These samples exhibit SWRs as well, however, they are lesspronounced than for samples A and B.To quantitatively model the spin-wave spectra we numeri- cally solve for each magnetic field orientation the spin-waveequation, (7), by the finite-difference method as outlined in Appendix B2. Although this method allows for the modeling of the SWR for arbitrary profiles of the anisotropy parameters,the exchange stiffness, the Gilbert damping parameter, andthe saturation magnetization, we assume the parameters tovary linearly as a function of z. This approach is motivated by the linear gradient in the hole concentration, which infirst approximation is assumed to cause a linear gradient inthe anisotropy parameters, resulting in the SWRs observedin the samples. 31,36In Table I, we have summarized the parameters used in the simulation for the different samples.Parameters in capital letters denote the value at the surfaceof the sample, while those in lowercase letters denote theslope of this parameter; e.g., the zdependence of the second- order, uniaxial out-of-plane anisotropy parameter is given byB 001(z)=B001−b001×z. We estimate an error margin of about ±20% and ±5 mT for the slopes and absolute values of the anisotropy parameters, respectively. The reason for thisuncertainty is that both, a gradient in D sand a gradient in the anisotropy parameters can affect the SWR mode spacing, aswe discuss below. The layer thickness used for the simulation 224422-6ANGLE-DEPENDENT SPIN-W A VE RESONANCE ... PHYSICAL REVIEW B 87, 224422 (2013) TABLE I. Simulation parameters and their zdependence of the samples under study as obtained by fitting the simulations to the SWR measurements. For anisotropy parameters capital letters denote the value at the surface of the film and lowercase letters the slope as described in the text. For sample A, the first value of b001was used for the first 100 nm and the second one for the remaining layer. In addition to the anisotropy parameters, the saturation magnetization is also assumed to vary linearly across the layer, while its absolute value is unknown andnot important for the SWR simulations. B001 b001 B4/bardbl b4/bardbl B4⊥ b4⊥ Ds∂M(z) ∂zM (0) Sample V/III (mT) (mT nm)( m T ) (mT nm)( m T ) (mT nm)( T n m2) α (1 μm) A 1.1 90 −0.1,−0.3 −50 0.05 25 −0.3 35 0.09 −3 B 1.3 130 −0.5 −50 0 0 0 20 0.06 −4 C 1.5 75 −0.4 −55 −0.04 −15 0 40 0.11 −4 D 3.5 91 −0.3 −55 −0.04 −15 0 20 0.09 −3 can be inferred from Fig. 4(a) and was determined from the ECV data under the assumption that at the position where thehole concentration rapidly decreases, the magnetic propertiesof the layer abruptly undergo a transition from ferromagneticto paramagnetic. For simulations, we divided each film inton=100 layers, with constant magnetic properties within each layer. For the gyromagnetic ratio we used γ=gμ B/¯h, with g=2.21 As a result of the simulation we obtain the Polder sus- ceptibility tensor ¯ χ(μ0H,z) and the transverse magnetization components as a function of zandμ0H. Additionally, we obtain the zdependence of the uniform resonance field by solving Eq. (8)for each field orientation. In an SWR absorption experiment with magnetic field modulation, theobtained signal is proportional to the first derivative of theabsorbed microwave power with respect to the magnetic field.Thus, we calculate the absorbed power using the simulatedsusceptibility and Eq. (12) and numerically differentiate the result in order to compare the simulated SWR spectra withthe experiment. Additionally, we use a global scaling factor,accounting, e.g., for the modulation amplitude, which is thesame for all field orientations, and we multiply all the simulateddata with this factor. In Fig. 5, we plot the experimental data together with the simulations using the parameters given inTable I, demonstrating that a reasonable agreement between theory and experiment can be found with one set of simulationparameters for all magnetic field orientations for each sample. We now discuss the angle dependence of the SWR spectrum of sample A shown in Fig. 5(a) based on the uniform resonance field and the resulting magnetization mode profile obtainedfrom the simulation. To this end, we plot in Figs. 6(a)–5(c) the magnetization precession amplitude Im( m ∗ 1m2−m1m∗ 2) for selected external field orientations as a function ofthe depth and external magnetic field in a contour plot, 0 )c( )b( (a)Im(m1m2-m1m2) (10-5) ** 0 1.2Im(m1m2-m1m2) (10-5) ** 0 0.3Im(m1m2-m1m2) (10-5) ** 0 0.53 )f( )e( )d(001arb.(a) FIG. 6. (Color online) Simulated magnetization mode profile and uniform resonance field of sample A. The contour plots show the magnetization precession amplitude Im( m∗ 1m2−m1m∗ 2) as a function of the position within the film and the external magnetic field for the external field aligned (a) along [001], (b) at an angle of 50◦with respect to [110] (cf. the inset in Fig. 5), and (c) along [110]. Dashed (blue) lines in (a–c) show the uniform resonance field, obtained by numerically solving Eq. (8)for each given field orientation. Dotted (black) lines in (a) indicate the resonance magnetic fields. (d–f) A magnification of the data [dotted (blue) lines] and simulation [solid (red) lines] from Fig. 5(a) shown using the same scale for all orientations. In (e), a simulation with a different set of parameters is shown for comparison (black, dashed line), see text. 224422-7L. DREHER et al. PHYSICAL REVIEW B 87, 224422 (2013) together with the corresponding uniform resonance field. In Figs. 6(d)–5(f), we show for each external field orientation a magnification of the corresponding SWR spectrum togetherwith the simulation. Note that in contrast to the normal-modeapproach (Appendix B1) used to calculate the modes in Fig. 2, where the coupling of each mode to the cavity field has to befound by integration, the approach elaborated in Appendix B2, directly yields the transverse magnetization components,already accounting for the coupling efficiency and the linewidth. Further, the approach presented in Appendix B2,i s also valid when the difference in the resonance fields of twomodes is comparable to or smaller than their line width, incontrast to the normal-mode approach. 38 If the external field is parallel to the surface normal ( ψ= 90◦), the uniform resonance field varies by about 350 mT across the film thickness [cf. the dashed line in Fig. 6(a)], resulting in several well-resolved standing spin-wave modes.The SWR fields are plotted as dotted lines in Fig. 6(a); since the spacing of the resonance fields is larger than the SWR linewidth, the modes are clearly resolved [cf. Figs. 6(a) and5(d)]. In the simulation two regions with different b 001values were used in order to reproduce the spacing of the higher orderspin-wave modes found in the experiment. Using the sameslope as in the first 100 nm for the entire layer would leadto a smaller spacing between the third-order and the higherorder modes. Instead of defining two regions with differentslopes b 001, a gradient in the exchange stiffness with a positive slope could also be used to model the experimentally foundmode spacing as discussed in the context of Fig. 3. Since the exchange interaction in (Ga,Mn)As is mediated by holes 12 andpdecreases across the layer, we refrain from modeling our results with a positive gradient in Ds. Further, the results in Ref. 36rather point to a negative gradient in Dsin a similar sample. However, a decreasing Mn concentration as a functionof the depth could lead to an increase in D s.34 Finally, we note that, since B1¯10=0 in the simulations, the magnetization precesses circularly for ψ=90◦and thus Im(m∗ 1m2−m1m∗ 2)=2s i n2τ,49with the precession cone angle τ. For all other orientations, mprecesses elliptically, which is accounted for in our simulations. In the simulationsof the precession amplitudes, we have assumed an externallyapplied microwave magnetic field with μ 0h=0.1m T . At an external field orientation of ψ=50◦the uniform resonance field is nearly constant across the layer, andconsequently only one SWR mode is observed with analmost-uniform magnetization precession across the layer[cf. Fig. 6(b)]. The precession amplitude is a measure for the SWR intensity. While the fundamental mode at ψ=90 ◦ exhibits a larger precession cone at the interface, it rapidly decays as a function of the depth, in contrast to the nearlyuniform precession amplitude for ψ=50 ◦. Since the entire layer contributes to the power absorption, the SWR modeatψ=50 ◦is more intense than the fundamental mode forψ=90◦, which is indeed observed in the experiment [cf. Figs. 6(d) and5(e)]. For the magnetic field within the film plane [ ψ=0◦; cf. Fig. 6(c)], the uniform resonance field again varies linearly across the film, however, in a less pronounced way thanfor the out-of-plane field orientation and with an oppositesign of the slope. The sign reversal of the slope can beunderstood in terms of the uniaxial out-of-plane anisotropy parameter B 001: positive values of these parameters lead to an increase (decrease) in the resonance field for the magnetizationoriented perpendicular (parallel) to the film plane, accountingfor the slopes of the uniform resonance fields in Fig. 6. Since the gradient in the uniform resonance field is lesspronounced for ψ=0 ◦than for ψ=90◦,t h es p i n - w a v e modes are not resolved for ψ=0◦, since their spacing is smaller than the SWR line width, leading to one rather broadline [cf. Figs. 6(c) and 5(f)]. A steeper gradient in B 4||,i n combination with a different Gilbert damping (or with anadditional inhomogeneous damping parameter) and amplitudescaling factor, could improve the agreement of simulation andexperiment in the in-plane configuration, as discussed later. Adetailed study of the in-plane anisotropy profile is, however,beyond the scope of this work. Given that the presentedsimulations were obtained with one set of parameters, theagreement of theory and experiment is reasonably good alsofor the in-plane configuration, since salient features of theSWR line shape are reproduced in the simulation. Having discussed the angle dependence of the SWR spectra, we turn to the zdependence of the out-of-plane anisotropy of sample A. Our simulations reveal that it is governed by thezdependence of both B 001(z) andB4⊥(z). Assuming only a gradient in B001results in a reasonable agreement of theory and experiment for the external field oriented along [001] and [110]but fails to reproduce the spectra observed for the intermediatefield orientations, e.g., ψ=50 ◦. This is illustrated by the dashed (black) line in Fig. 6(e), which represents simulations with a constant B4⊥(z)f o rψ=50◦. As can be seen, this simulation produces several SWRs, whereas in the experimentonly one resonance is present, which is better reproduced bythe simulation with both B 001(z) andB4⊥(z) varying across the layer. We now discuss the anisotropy parameters of all samples. In contrast to sample A, the out-of-plane anisotropy profileof all other samples appears to be governed by a gradient inB 001(z). As already discussed qualitatively, the hard axis of the samples is determined by an interplay of B001andB4⊥. For samples A and B B4⊥is positive and 0, leading to an out-of-plane hard axis. In contrast, samples C and D exhibitan e g a t i v e B 4⊥, leading to a hard axis between out-of-plane and in-plane. The B4||parameter is negative and of similar magnitude for all samples. Since the out-of-plane anisotropy profile of sample A is governed by B001(z) and B4⊥(z), a comparison of the out-of-plane anisotropy profile among all samples based on anisotropy parameters is difficult. We therefore compare the uniform resonance fields, where both anisotropy parametersenter. As is evident from Fig. 6, the strongest influence of the magnetic inhomogeneity of the layers on the uniformresonance fields is observed for the external field along [001].To compare the hole concentration profile in Fig. 4(a) with the anisotropy profile, we therefore plot in Fig. 4(b) thez dependence of the uniform resonance field μ 0H001 unifor this field orientation. The figure demonstrates that the gradient inμ 0H001 uniis correlated with the gradient in p. For the sample with the strongest gradient in pthe gradient in μ0H001 uniis also most distinct, while the samples with a weaker gradientinpexhibit a less pronounced gradient in μ 0H001 uni. However, 224422-8ANGLE-DEPENDENT SPIN-W A VE RESONANCE ... PHYSICAL REVIEW B 87, 224422 (2013) for sample D, exhibiting a nearly constant p, we still observe standing SWRs for μ0H||[001] [Fig. 5(d)], reflected in a slight gradient of μ0H001 uni. This observation suggests that addition- ally other mechanisms lead to a variation of the anisotropyprofile. One possibility would be a gradient in the elastic strainof the layer, due to a nonhomogeneous incorporation of Mnatoms in the lattice. However, x-ray diffraction measurementsof this sample, in combination with a numerical simulationbased on dynamic scattering theory, reveal a variation of thevertical strain /Delta1ε zzas small as 3 ×10−5across the layer. According to the measurements in Ref. 13, such a variation in strain would lead to a variation of the B001parameter by a few milliteslas only, insufficient to account for the variationofμ 0Huniby almost 100 mT across the layer. A more likely explanation seems to be a variation of the saturationmagnetization, which should also influence the anisotropy parameters. In the simulation, a nonhomogeneous saturation was assumed, potentially explaining also the observed gradientin the anisotropy parameters and therefore in the uniformresonance field. In contrast to the out-of-plane anisotropy parameters, B 4|| was found to depend only weakly on z, for all samples except sample B, where it was constant. Additionally, B1¯10, typically of the order of a few milliteslas,13might have an influence and interplay with B4||in determining the in-plane anisotropy. Here, however, we focus on the out-of-plane anisotropy andtherefore neglect B 1¯10in our simulations. An in-plane rotation of the external field would be required for a more accurate mea-surement of B 4||andB1¯10but is outside the scope of this work. According to the valence-band model in Ref. 12,a n oscillatory behavior of the magnetic anisotropy parametersis expected as a function of p. Therefore, depending on the absolute value of p, different values for, e.g., ∂B 001/∂p are expected. In particular, there are regions where a anisotropyparameter might be nearly independent of pand other regions with a very steep pdependence. Since the absolute value of pis unknown, a quantitative discussion of the pdependence of the obtained anisotropy parameters based on the model inRef. 12is not possible. In addition to p,t h ep-dexchange integral, 12which may also vary as a function of the depth in a nonhomogeneous film, also influences the anisotropy parameters,12further complicating a quantitative analysis. For all samples, we used a constant exchange stiffness Dsin our modeling. As alluded to above, there is some ambiguity in this assumption, since the exchange stiffnessand the gradient in the anisotropy both influence the modespacing. For simplicity, however, we intended to keep asmany simulation parameters as possible constant. The absolutevalues obtained for the exchange stiffness agree within a factorof 2 with the ones obtained in previous experiments 36,50but are a factor of 2–4 larger than theoretically predicted.51For the reasons discussed above, there is a large uncertainty alsoin the derivation of the absolute value of D sfrom standing spin-wave modes in layers with a gradient in the magnetic anisotropy constants. In order to use one parameter set for all field orientations, the Gilbert damping parameter was assumed to be isotropicin the simulations. The modeling of the SWR data couldbe further improved by assuming a nonisotropic damping,its value being larger for μ 0H||[110] than for μ0H||[001](cf. Fig. 5). This, however, only improves the result when assuming a field-orientation-dependent scaling factor for theamplitude, which could be motivated, e.g., by the assumptionthat the microwave magnetic field present at the sampleposition depends on the sample orientation within the cavity.The absolute values of αdetermined here are comparable with the ones obtained in ultrafast optical experiments 52but are larger than the typical α=0.01...0.03 values found by frequency-dependent FMR studies.53,54As already alluded to, inhomogeneous line-broadening mechanisms may play a dom-inant role, 54in particular, for as-grown samples.55We therefore assume that the values for αobtained in this study overestimate the actual intrinsic Gilbert damping. A frequency-dependentSWR study would be required to determine the intrinsic α. Such a study could possibly also reveal a p-dependent αas theoretically predicted. 55In our study, assuming a z-dependent αdid not improve the agreement between simulation and experiment, corroborating the conjecture that inhomogeneousbroadening mechanisms dominate the line width and thereforeobscure a possible zdependence of α. IV . SUMMARY We have presented a finite-difference-type modeling ap- proach for standing SWRs based on a numerical solution of theLLG equation. With this generic formalism, SWR spectra canbe simulated accounting for elliptical magnetization preces-sion, for arbitrary orientations of the external magnetic field,and for arbitrary profiles of all magnetic properties, includinganisotropy parameters, exchange stiffness, Gilbert damping,and saturation magnetization. The approach is applicable notonly to (Ga,Mn)As but to all ferromagnets. Four (Ga,Mn)As samples, epitaxially grown with V/III flux ratios of 1.1, 1.3, 1.5, and 3.5, were investigated by ECVand SWR spectroscopy, revealing a correlation of a lineargradient in the hole concentration with the occurrence ofstanding SWRs, in particular, for the external field orientedout of plane. Using the presented modeling approach, theSWR spectra could be reproduced in a simulation with oneparameter set for all external field orientations. The simulationresults demonstrate that the profile of the out-of-plane uniformresonance field is correlated with the hole concentrationprofile. However, our measurements and simulations show thata nonuniform hole concentration profile is not the only causethat leads to the observed nonuniform magnetic anisotropy;possibly, a variation in the saturation magnetization alsoinfluences the anisotropy parameters. To gain a quantitativeunderstanding of this issue, more samples with known holeconcentrations would be required, where both the absolutevalues and the profiles of pare varied. Such a study was, however, outside the scope of this work. Besides the modeling of SWR intensities and line widths, the presented formalism yields the magnetization precessionamplitude as a function of the position within the ferromagnet.It can therefore be used to investigate spin-pumping inten-sities in (Ga,Mn)As/Pt bilayers. 27The spin-pumping signal, detected as a voltage across the Pt layer, should be proportionalto the magnetization precession cone in the vicinity of the(Ga,Mn)As/Pt interface. By measuring the spin-pumpingsignal as well as the SWR intensities of (Ga,Mn)As/Pt and 224422-9L. DREHER et al. PHYSICAL REVIEW B 87, 224422 (2013) by using our modeling approach, it should be possible to investigate to what extent a magnetization mode which islocalized at a certain position within the (Ga,Mn)As layercontributes to the spin-pumping signal. ACKNOWLEDGMENTS This work was supported by the Deutsche Forschungs- Gemeinschaft via Grant No. SFB 631 C3 (Walter SchottkyInstitut) and Grant No. Li 988/4 (Universit ¨at Ulm). APPENDIX A: COORDINATE TRANSFORMATION AND FREE ENTHALPY DERIV ATIVES The transformation between the crystallographic coordi- nate system ( x,y,z ) and the equilibrium system (1,2,3) is given by ⎛ ⎜⎝mx my mz⎞ ⎟⎠=T⎛ ⎜⎝m1 m2 m3⎞ ⎟⎠, (A1) with T=⎛ ⎜⎝cosθ0cosφ0−sinφ0sinθ0cosφ0 cosθ0sinφ0 cosφ0 sinθ0sinφ0 −sinθ0 0 cos θ0⎞ ⎟⎠. (A2) The derivatives of the free enthalpy density, Eq. (6), with respect to the magnetization components are G3=∂m3G|m=m0=−μ0H3+2B001cos2θ0 +B1¯10(sinθ0cosφ0−sinθ0sinφ0)2 +4B4⊥cos4θ0+4B4/bardblsin4θ0(cos4φ0+sin4φ0), (A3) G21=G12=∂m1∂m2G|m=m0 =cosθ0(1−2 cos2φ0)[B1¯10 +12B4/bardblsin2θ0cosφ0sinφ0], (A4) G11=∂m1∂m1G|m=m0=2B001sin2θ0 +12 cos2θ0sin2θ0[B4⊥+B4/bardbl(cos4φ0+sin4φ0)] +B1¯10cos2θ0(cosφ0−sinφ0)2, (A5) G22=∂m2∂m2G|m=m0=2B1¯10(sinφ0+cosφ0)2 +24B4/bardblsin2θ0cos2φ0sin2φ0. (A6) APPENDIX B: FINITE-DIFFERENCE METHOD In this Appendix, we describe how the spin-wave equation can be numerically solved by the finite-difference method.We start with the simple case of a circularly precessingmagnetization, neglecting Gilbert damping and the drivingfield (Sec. B1). Then we turn to the general case, where the magnetization precesses elliptically and the Gilbert dampingas well as the driving field is included (Sec. B2).1. The one-dimensional, homogeneous, undamped case Here, we describe how the resonance fields and the spin- wave modes can be found, assuming a circularly precessingmagnetization m 2=im1=˜m, a constant exchange stiffness, and az-independent equilibrium magnetization. This case has been considered in Ref. 36using a semianalytical approach to solve the spin-wave equation, Eq. (10). The approach considered here is slightly more general, as it is straightforwardto determine resonance fields and eigenmodes of the systemfor an arbitrary zdependence of the uniform resonance field. To solve Eq. (10), we divide the ferromagnetic film into a finite number nof layers with equal thickness land constant magnetic properties within each of these layers. Thezdependence of ˜mandμ 0H001 uniis thus given by an index j=1...n . Within each of these layers the uniform resonance field and ˜m(z) are thus constant and given by the values μ0H001,j uni=:Kjand ˜mj, respectively. The second derivative of˜mis approximated by ˜m/prime/prime(z=j·l)≈˜mj−1−2˜mj+˜mj+1 l2. (B1) Consequently, Eq. (10) is converted to the homogeneous equation system ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝......... ... K j−1+2d −d 0 ... ... −dKj+2d −d. . . ... 0 −dKj+1+2d. . . .........⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎠ ×⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝... ˜m j−1 ˜mj ˜mj+1 ...⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=μ 0H⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝... ˜m j−1 ˜mj ˜mj+1 ...⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎠, (B2) with the abbreviation d=−D s/l2. The boundary condition of natural freedom36(von Neumann boundary condition) reads ˜m0=˜m1and ˜mn−1=˜mnand can be incorporated into Eq.(B2) . Since the matrix on the left-hand side of Eq. (B2) is sparse, it can be efficiently diagonalized numerically, yieldingthe resonance fields (eigenvalues) and the correspondingmodes (eigenvectors). After diagonalizing the matrix, therelevant resonance fields are found by sorting the eigenvaluesand considering only the modes with positive resonance fields,corresponding to the bound states in the particle-in-a-boxanalogon. The SWR amplitude of each mode is proportional toits net magnetic moment; thus, the amplitudes can be found byintegrating the (normalized) eigenmodes. The mode profile,the resonance fields, and the SWR intensities are illustrated inFig. 2for a constant and a linearly varying uniform resonance field. The finite line width of the SWR modes can be accountedfor by assuming a Lorentzian line shape for each mode with acertain line width and with the resonance fields and intensitiescalculated as described above. 36Note that this approach to 224422-10ANGLE-DEPENDENT SPIN-W A VE RESONANCE ... PHYSICAL REVIEW B 87, 224422 (2013) derive resonance fields and intensities is only valid if the mode separation is large compared with the line width of themodes; this restriction does not apply to the model presentedin Appendix B2. 2. The general case To solve Eq. (7)for arbitrary μ0Hand arbitrarily varying magnetic properties, we again divide the ferromagnetic filminto a finite number nof layers with equal thickness land constant magnetic properties within each of these layers. Incontrast to the case in Appendix B1, where only the uniform resonance field was varied across the layer, here potentially allmagnetic properties entering Eq. (7)can be assumed to be z dependent. Additionally, the components of the driving fieldμ 0hi(i=1,2) can also vary as a function of z, since the (1 ,2,3) frame of reference is zdependent and thus the projections of the driving field have to be calculated for each layer. The z dependence of the components mi(i=1,2), of the parameters H11,H12,H21, andH22(defined in Sec. II) and the exchange stiffness is thus given by the index j=0...n ; the second derivative of each of the components miis approximated as in Eq.(B1) . The linearized LLG equation, Eq. (7), is thus converted into the inhomogeneous equation system ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝.................. ...H j−1 11−2dj−1Hj−1 12 dj−100 0 ... ... Hj−1 21 Hj−1 22−2dj−10 dj−100 ... ... dj0 Hj 11−2djHj 12 dj0 ... ... 0 djHj 21 Hj 22−2dj0 dj... ... 00 dj+10 Hj+1 11−2dj+1Hj+1 12 ... ... 00 0 dj+1Hj+1 21 Hj+1 22−2dj+1... ..................⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝... m j−1 1 mj−1 2 mj 1 mj2 mj+1 1 mj+1 2 ...⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ =μ 0⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝... h j−1 1 hj−1 2 hj1 hj2 hj+1 1 hj+1 2 ...⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (B3) with the abbreviation dj=−Dj s/l2. At the boundaries of the magnetic film we again assume the spins to exhibit naturalfreedom, m 0 i=m1 iandmn i=mn+1 i. To simulate a spin-wave spectrum for a given orientation of the external field and a given profile of the magneticproperties, we numerically sweep the magnetic field andcalculate the equilibrium magnetization orientation for all indices j=0...n at a given external field. The inverse of the matrix in Eq. 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PhysRevB.64.012411.pdf

Precessional effects in the linear dynamic susceptibility of uniaxial superparamagnets: Dependence of the ac response on the dissipation parameter W. T. Coffey * Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland D. S. F. Crothers Departments of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 1NN, United Kingdom Yu. P. Kalmykov Centre d’Etudes Fondamentales, Universite ´de Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan Cedex, France S. V. Titov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryazino, Moscow Region 141120, Russian Federation ~Received 9 April 2001; published 13 June 2001 ! It is shown that the low-frequency relaxation spectrum of the linear dynamic susceptibility of uniaxial single domain particles with a uniform magnetic field applied at an oblique angle to the easy axis can be used todeduce the value of the damping constant. DOI: 10.1103/PhysRevB.64.012411 PACS number ~s!: 75.50.Tt, 05.40.Ca, 76.20. 1q, 76.50. 1g A single domain ferromagnetic particle is characterized by an internal potential, having several local states of equi-librium with potential barriers between them. If the particlesare small ~;10 nm !so that the potential barriers are rela- tively low, the magnetization vector Mmay cross over the barriers due to thermal agitation. The ensuing thermal insta-bility of the magnetization results in the phenomenon ofsuperparamagnetism. 1This problem is important in informa- tion storage, rock magnetism, and the magnetization reversalobserved in isolated ferromagnetic nanoparticles. 2The dy- namics of the magnetization Mof a superparamagnetic par- ticle is usually described by the Landau-Lifshitz or Gilbert~LLG!equation 3,4 2tNd dtM5b~a21Ms@M3H#1@@M3H#3M#!,~1! where tN5b~11a2!Ms 2ga~2! is the free Brownian motion diffusion time of the magnetic moment, ais the dimensionless damping ~dissipation !con- stant,Msis the saturation magnetization, gis the gyromag- netic ratio, b5v/(kT),vis the volume of the particle, and the magnetic field Hconsists of applied fields ~Zeeman term!, the anisotropy field Ha, and a random white-noise field accounting for the thermal fluctuations of the magneti-zation of an individual particle. Here the internal magnetiza-tion of a particle is assumed homogeneous. Surface and‘‘memory’’ effects are also omitted in Eq. ~1!. These as- sumptions are discussed elsewhere ~e.g., Refs. 5–7 !. Further- more, the description of the relaxation processes in the con-text of Eq. ~1!does not take into account effects such as macroscopic quantum tunneling ~a mechanism of magnetiza- tion reversal suggested in Ref. 1 !. These effects are impor-tant at very low temperatures 8,9and necessitate an appropri- ate quantum-mechanical treatment, e.g., Refs. 10–12. The various regimes of relaxation of Min superparamag- netic particles are governed by a. In general, ais difficult to estimate theoretically, although a few experimental methodsof measuring a@such as ferromagnetic resonance ~FMR !and the angular variation of the switching field, e.g., Refs. 7 and8#have been proposed. Yet another complementary and po- tentially promising technique, viz., the nonlinear response of single domain particles to alternating ~ac!stimuli, has recently 13been suggested in order to evaluate a. In particu- lar, it has been shown in Ref. 13 that for uniaxial particleshaving a strong ac field applied at an angle cto the easy ~Z! axis, the nonlinear response truncated at terms cubic in the acfield is particularly sensitive to the value of a. On the other hand, the linearresponse to the ac field does not exhibit such behavior. The explanation of this is reasonably straightfor-ward: the linear ac response may simply be calculated fromthe after effect solution following the removal of a weakuniform field applied at an angle cto the easy axis. Thus the superparamagnetic ~greatest !relaxation time tis that of a particle with simple uniaxial anisotropy, which is given byBrown’s 4expression t;tNAp 2s3/2es,s@1, ~3! where s5bKis the barrier height parameter and Kis the anisotropy constant. Equation ~3!yields the approximate po- sition of the peak in the imaginary part x9~v!of the complex susceptibility x(v)5x8(v)2ix9(v) in linear response. The most striking feature of Eq. ~3!is that twhen normalized by tNis independent of a. The physical reason for this is the lack of coupling between the transverse and longitudinalmodes in the linear response when a weak ac field alone isapplied at an angle cto theZaxis. If one proceeds, however,PHYSICAL REVIEW B, VOLUME 64, 012411 0163-1829/2001/64 ~1!/012411 ~4!/$20.00 ©2001 The American Physical Society 64012411-1to the next term in the response which yields the so-called third-order susceptibility x(3)(v), then a strong dependence of the imaginary part of x(3)(v)o naappears.13The expla- nation of this lies in the coupling between the longitudinaland transverse ~or processional !modes in the nonlinear re- sponse. Thus one can evaluate afrom measurements of the nonlinear ac response. Here unlike Ref. 13, we consider a uniaxial particle in a strong uniform field H0applied at an angle cto the anisot- ropy axis of the particle. Hence the system in the absence ofthe ac perturbation unlike that of 13is nonaxially symmetric, thus we expect precessional effects due to coupling of thetransverse and longitudinal modes to appear even in the lin- ear response to a small ac field H(t) superimposed on H 0. Indeed, the limiting values of a, viz., a!‘anda!0, cor- respond to the high-damping and the low-damping limits inthe Kramers escape rate theory. 14The coupling effect is made manifest in the formulas for the Kramers escape rate orinverse of the greatest relaxation time tof the magnetization for both intermediate to high damping ~IHD!and very low damping ~VLD!which apply to nonaxially symmetric poten- tials of the magnetocrystalline anisotropy5,6~see below !. Now we recall that the Fokker-Planck equation ~FPE!for the probability density distribution WofM~Ref. 4 !corre- sponding to Eq. ~1!is4,15 2tN] ]tW5DW1b$a21u@„V3„W#1„~W„V!%, ~4! where „andDare the gradient and the Laplacian on the surface of the unit sphere, respectively, uis the unit vector directed along M, andV(M) is the free-energy density. Here, in the absence of the ac field, Vis given by bV52s@cos2q12h~sinccoswsinq1cosccosq!#, ~5! where qandware the polar and azimuthal angles, respec- tively, and h5MsH0/(2K) is the dimensionless external field parameter. The free energy in Eq. ~5!has a bistable structure with minima at n1andn2separated by a potential barrier containing a saddle point at n0.15If (a1(i),a2(i),a3(i)) denote the direction cosines of MandMis close to a sta- tionary point niof the free energy, then V(M) can be ap- proximated to second order in a(i)as4 V5Vi11 2@c1~i!~a1~i!!21c2~i!~a2~i!!2#. ~6! Substituting Eq. ~6!into Eq. ~5!, the FPE may be solved near the saddle point yielding4,15 t5tIHD;HV0 2pv0@v1eb~V12V0!1v2eb~V22V0!#J21 ,~7! where vi25g2Ms22c1(i)c2(i)(i51,2) and v025 2g2Ms22c1(0)c2(0)are the squares of the well and saddle an- gular frequencies, respectively, andV05b 4tN@2c1~0!2c2~0!1A~c2~0!2c1~0!!224a22c1~0!c2~0!#. Equations for ci(j)andViare given elsewhere.15Equation ~7! is similar to the IHD formula derived by Kramers14and ap- plies when the energy loss per cycle at the saddle point en- ergy of the motion of the magnetic moment DE@kT.I f DE!kT~VLD!, we have for the escape from a single well5,16 t5tLD;pkT v1DEeb~V02V1!. ~8! @Here instead of numerical evaluation of DE, we have used an approximation DE’avuV0u~Ref. 5 !#. The IHD and VLD limits correspond to a>1 and a<0.01, respectively. How- ever, for crossover values of a~about a’0.1!neither the IHD formula ~7!nor the VLD, Eq. ~8!, can yield reliable quantitative estimates. Thus a more detailed analysis isnecessary. 17 Equations ~7!and~8!applied to the potential given by Eq. ~5!yield the greatest relaxation time tin the appropriate limits ~IHD, VLD !for a strong uniform field H0applied at an angle cto theZaxis8,18;tis effectively identical to the integral relaxation time ~in linear response, the correlation time!19if the strength of H0is smaller than the reduced criti- cal fieldhcat which depletion of the shallower of the two potential wells of the bistable potential occurs ~for example, hc’0.17 in the axially symmetrical case19!. As shown in Refs. 8 and17, the asymptotes ~7!and~8!are in excellent agreement with the exact numerical results from the FPE ~4!. Equations ~7!and~8!can also successfully reproduce the experimental angular variation of the switching field for in-dividual Co and BaFeCoTiO particles and thus allows one toevaluate a.8 Equations ~7!and~8!fort, which now exhibit strong a dependence, suggest that the frictional dependence of the relaxation process may also be observed and used for theevaluation of ain the linear ac response of the system. In order to verify our conjectures concerning the adependence of the linear response to a small ac field H(t)~i.e., assuming bMsH!1!, we have calculated using linear-response theory the complex magnetic susceptibility x~v!of the system. The susceptibility was calculated by using a matrix continued-fraction solution 20,21of the system of moments @the expecta- tion values of the spherical harmonics ^Yl,m&(t)#governing the kinetics of the magnetization M~the moment system can be obtained either from the FPE or from the LLGequation 22!. The details of the calculation can be found elsewhere20,21; it is assumed that H(t) is directed along H0. The plots of Re $x(v)%and log 10@2Im$x(v)%#vs log10(vtN) are shown in Figs. 1–3 for a wide range of frequency, biasfield strength, and damping ~the calculations were carried out for vbMs2N051;N0is the number of particles per unit vol- ume!. The results indicate that a marked dependence of x~v! onaexists and that three distinct dispersion bands appear in the spectrum. Furthermore, the characteristic frequency andthe half-width of the low-frequency relaxation band ~LRB! are determined by the characteristic frequency vob;t21ofBRIEF REPORTS PHYSICAL REVIEW B 64012411 012411-2the overbarrier relaxation mode. As adecreases, this peak shifts to higher frequencies and reaches its limiting value tLD21. In addition, a far weaker second relaxation peak ap- pears at high frequencies ~HF!. This HF relaxation band ~HRB!is due to the intrawell modes @forc50 and s@1, the characteristic frequency of this relaxation peak is vwell ’2s(11h)/tN~Ref. 19 !#. The third FMR peak due to the excitation of transverse modes having frequencies close to the precession frequency vprof the magnetization appears only at low damping and strongly manifests itself at HF. As adecreases, the FMR peak shifts to higher frequencies since vpr;a21. Moreover, at c50o rc5p, the FMR peak dis- appears because the transverse modes no longer take part inthe relaxation process. The dependence of the linear responseon the bias-field strength is demonstrated in Fig. 3. Here, theeffect of the depletion 19,23of the shallower of the two poten- tial wells of a bistable potential ~5!by a bias field is appar- ent: at fields above the critical field hcat which the depletion occurs, it is possible to make the LF peak disappear ~curves 3 and 3 8!. Such behavior of x~v!implies that if one is inter- ested solely in the low-frequency ( vt<1) part of x~v!,where the effect of the HF modes may be completely ignored ~so that the relaxation of the magnetization at long times may be approximated by a single exponential with the character-istic time t!, then the Debye-like relaxation formula, viz., x~v!5xst2Dxhf 11ivt1Dxhf, ~9! yields an accurate description of the LF spectra ~see Figs. 1–3!. Here tis given by Eqs. ~7!and~8!in the IHD and VLD limits, respectively, xst5x(0) is the static susceptibil- ity, and Dxhfis the contribution of the HF transverse and longitudinal modes. The values of xstandDxhfdepend on j, c, and sand can be measured experimentally, calculated numerically, and/or estimated theoretically ~an example of such theoretical estimations of xstandDxhfforc50 has been given by Garanin19!. Our calculations indicate that Eqs.~7!–~9!yield an adequate description of the LF spectra fors>3. We have demonstrated that it is unnecessary to resort to the nonlinear response in order to observe large precessionaleffects in the relaxation processes of uniaxial superparamag-nets. All that is required is to superimpose a strong bias field H 0at an angle cto the easy axis of the uniaxial particle, thus ensuring that the system is nonaxially symmetric, and then to calculate the linear response to a perturbing ac field H(t). It follows that the nonaxial symmetry causes the various damp-ing regimes ~IHD and VLD !of the Kramers problem to ap- pear unlike in an axially symmetric potential, where the for-mula for t@for example, Eq. ~2!#is valid for all abecause t/tNis independent of a. We remark that the intrinsic a dependence of x~v!for the oblique field configuration serves as a signature of the coupling between the longitudinal andprecessional modes of the magnetization. Hence, it should bepossible to determine the evasive damping coefficient frommeasurements of the linearresponse, e.g., by fitting the theory to the experimental LF dependence of x~v!on the angle cand the bias strength H0, so that the sole fitting FIG. 1. Re $x(v)%vs log10(vtN) from the IHD ( a51) to the VLD ( a50.001) limits for s510,h50.1, and c5p/4. Curves 1–4: exact numerical calculations of x~v!based on the results of Refs. 20 and 21. Stars and filled circles: Eq. ~9!with Dxhf’0.023 andtfrom Eqs. ~7!and~8!, respectively. FIG. 2. The same as in Fig. 1 but for log10@2Im$x(v)%#. FIG. 3. log10@2Im$x(v)%#vs log10(vtN) for s510,c5p/4, a51.0~IHD: solid lines 1, 2, and 3 !, and a50.01~low damping: dashed-dotted lines 1 8,28, and 3 8!. Lines 1, 1 8(h50.01), 2, 2 8(h 50.17), and 3, 3 8(h50.4) are exact numerical calculations. Stars and filled circles: Eqs. ~9!withtfrom Eqs. ~7!and~8!, respectively.BRIEF REPORTS PHYSICAL REVIEW B 64012411 012411-3parameter is a. Just as in the nonlinear response,13acan be determined at different T, yielding its temperature depen- dence. This is of importance because of its implications inthe search for other mechanisms of magnetization reversal ofM~e.g., macroscopic quantum tunneling 9,27!, as a knowl- edge of aand itsTdependence allows the separation of the various relaxation mechanisms. Moreover, such experimentsare much more easily accomplished than those for the non-linear response of Ref. 13. The results we have obtainedsuggest that the experimental measurements of linear andnonlinear susceptibility of fine particles ~e.g., Refs. 24–26 ! should be repeated for a strong bias-field configuration. The results we have presented pertain to noninteracting superparamagnetic particles with easy axes oriented alongtheZaxis of the laboratory system of coordinates. If the easyaxes are randomly distributed in space, further averaging must be carried out in order to calculate x~v!. In the calcu- lations, we have also assumed that all the particles are iden-tical; in order to account for polydispersity, one must aver-age x~v!over the appropriate distribution function ~e.g., over the particle volumes; see for details Refs. 25, 26, and 28 !. Furthermore, the neglect of interparticle interactions in thepresent model suggests that the results we have obtained are applicable for systems where the effects of the dipole-dipoleand exchange interactions may be ignored, such as individualnanoparticles ~e.g., Refs. 2 and 8 !and diluted solid suspen- sions of nanoparticles ~e.g., Ref. 26 !. The support of the Enterprise Ireland Research Collabo- ration Fund 2000 is gratefully acknowledged. *Corresponding author. 1C. P. Bean and J. D. Livingston, J. Appl. Phys. 30, 120S ~1959!. 2W. Wernsdorfer, B. Doudin, D. Mailly, K. Hasselbach, A. Benoit, J. Meier, J.-Ph. Ansermet, and B. Barbara, Phys. Rev. Lett. 77, 1873 ~1996!. 3T. L. Gilbert, Phys. Rev. 100, 1243 ~1956!. 4W. F. Brown, Jr, IEEE Trans. Magn. 15, 1196 ~1979!. 5I. Klik and L. Gunther, J. Stat. Phys. 60, 473 ~1990!. 6I. Klik and L. Gunther, J. Appl. Phys. 67, 4505 ~1990!. 7Yu. L. Raikher and M. I. Shliomis, Adv. Chem. Phys. 87, 595 ~1994!. 8W. T. Coffey, D. S. F. Crothers, J. L. Dormann, Yu. P. Kalmykov, E. C. Kennedy, and W. Wernsdorfer, Phys. Rev.Lett.80, 5655 ~1998!. 9B. Barbara, L. C. Sampaio, J. E. Wegrowe, B. A. Ratnam, A. Marchand, C. Paulsen, M. A. Novak, J. L. Tholence, M. Uehara,and D. Fruchart, J. Appl. Phys. 73, 6703 ~1993!. 10D. A. Garanin, Physica A 172, 470 ~1991!. 11D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B 56, 11 102 ~1997!. 12V. L. Safonov, J. Appl. Phys. 85, 4370 ~1999!; J. Magn. Magn. Mater.195, 526 ~1999!. 13J. L. Garcia-Palacios and P. Svendlindh, Phys. Rev. Lett. 85, 3724 ~2000!. 14H. A. Kramers, Physica ~Utrecht !7, 284 ~1940!.15L. J. Geoghegan, W. T. Coffey, and B. Mulligan, Adv. Chem. Phys.100, 475 ~1997!. 16W. T. Coffey, Adv. Chem. Phys. 103, 259 ~1998!. 17W. T. Coffey, D. A. Garanin, and D. J. McCarthy, Adv. Chem. Phys.117, 483 ~2001!. 18W. T. Coffey, D. S. F. Crothers, J. L. Dormann, L. J. Geoghegan, and E. C. Kennedy, J. Phys.: Condens. Matter 10, 3249 ~1998!. 19D. A. Garanin, Phys. Rev. E 54, 3250 ~1996!. 20Yu. P. Kalmykov and S. V. Titov, Fiz. Tverd. Tela ~St. Peters- burg!40, 1642 ~1998!@Phys. Solid State 40, 1492 ~1998!#. 21Yu. P. Kalmykov and S. V. Titov, Fiz. Tverd. Tela ~St. Peters- burg!42, 893 ~2000!@Phys. Solid State 42, 918 ~2000!#. 22Yu. P. Kalmykov and S. V. Titov, Phys. Rev. Lett. 82, 2967 ~1999!. 23W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and J. T. Waldron, Phys. Rev. B 51, 15 947 ~1995!. 24J. I. Gittleman, B. Abeles, and S. Bozowski, Phys. Rev. B 9, 3891 ~1974!. 25T. Bitoh, K. Ohba, M. Takamatsu, T. Shirane, and S. Chikazawa, J. Phys. Soc. Jpn. 64, 1311 ~1995!. 26P. Jo¨nsson, T, Jonsson, J. L. Garcia-Palacios, and P. Svedlindh, J. Magn. Magn. Mater. 222, 219 ~2000!. 27W. Wernsdorfer, E. B. Orozco, B. Barbara, A. Benoit, and D. Mailly, Physica B 280, 264 ~2000!. 28Yu. L. Raikher and V. I. Stepanov, Phys. Rev. B 55, 15 005 ~1997!.BRIEF REPORTS PHYSICAL REVIEW B 64012411 012411-4

PhysRevB.91.214434.pdf

PHYSICAL REVIEW B 91, 214434 (2015) Current-driven asymmetric magnetization switching in perpendicularly magnetized CoFeB/MgO heterostructures Jacob Torrejon,1,2Felipe Garcia-Sanchez,3Tomohiro Taniguchi,4Jaivardhan Sinha,1Seiji Mitani,1 Joo-V on Kim,3and Masamitsu Hayashi1,* 1National Institute for Materials Science, Tsukuba 305-0047, Japan 2Unit ´e Mixte de Physique CNRS/Thales, 1 Avenue Augustin Fresnel, 91767 Palaiseau, France 3Institut d’Electronique Fondamentale, UMR CNRS 8622, Universit ´e Paris-Sud, 91405 Orsay, France 4National Institute of Advanced Industrial Science and Technology (AIST), Spintronics Research Center, Tsukuba, Ibaraki 305–8568, Japan (Received 27 October 2014; revised manuscript received 21 April 2015; published 29 June 2015) The flow of in-plane current through ultrathin magnetic heterostructures can cause magnetization switching or domain-wall nucleation owing to bulk and interfacial effects. Within the magnetic layer, the current cancreate magnetic instabilities via spin transfer torques (STT). At interface(s), spin current generated from thespin Hall effect in a neighboring layer can exert torques, referred to as the spin Hall torques, on the magneticmoments. Here, we study current-induced magnetization switching in perpendicularly magnetized CoFeB/MgOheterostructures with a heavy metal (HM) underlayer. Depending on the thickness of the HM underlayer, wefind distinct differences in the in-plane field dependence of the threshold switching current. The STT is likelyresponsible for the magnetization reversal for the thinner underlayer films whereas the spin Hall torques cause theswitching for thicker underlayer films. For the latter, we find differences in the switching current for positive andnegative currents and initial magnetization directions. We find that the growth process during the film depositionintroduces an anisotropy that breaks the symmetry of the system and causes the asymmetric switching. Thepresence of such symmetry-breaking anisotropy enables deterministic magnetization switching at zero externalfields. DOI: 10.1103/PhysRevB.91.214434 PACS number(s): 85 .75.−d,75.70.Tj,72.25.Pn I. INTRODUCTION Spin transfer torques (STT), which represent the transfer of spin angular momentum from a spin-polarized current tolocal magnetization, are now well established for their useto control magnetization [ 1,2]. STT has been exploited in magnetic tunnel junctions (MTJs) for developing advancednonvolatile memory (MRAM). One of the main challengesto achieve reliable operation of MRAM is to increase themargin of reading and writing current, which requires highmagnetoresistance ratio and low writing current. Alternatively, a three-terminal device can be used to overcome this problem by separating the circuit for readingand writing [ 3–7]. For such device, one can make use of the recently discovered spin-orbit effects to trigger magnetizationswitching [ 3,8]. In particular, the spin Hall effect (SHE) in heavy metal (HM) layers [ 9] can generate sufficiently large spin current to manipulate magnetic moments of a magneticlayer adjacent to the HM layer. The torque on the magneticmoments exerted by the spin current is referred to as thespin Hall torque. Intuitively, the action of STT and spin Halltorques on magnetization is governed by the same physics,however, the underlying processes related to the latter and thedifference between the two torques are not clear and requirefurther thorough study [ 10–14]. For STT driven magnetization switching, it is beneficial to use MTJs with perpendicularly magnetized “free” layer toachieve fast and low-current magnetization switching [ 15–17]. With regard to magnetization switching of a perpendicularly *hayashi.masamitsu@nims.go.jpmagnetized layer with in-plane current via the spin-orbiteffects, one needs to apply an in-plane field directed alongthe current in order to reverse the magnetization direction[3,8,18]. The need to apply such in-plane field may require additional costly processing for developing devices and thuswould preferably be avoided. On this front, it has been recentlydemonstrated that magnetization switching can be triggeredvia the spin Hall torque in the absence of any magnetic fieldby using sophisticated device structuring [ 19,20]. In order to fully utilize spin Hall torque driven magnetization switching for technological device applications, the underlying physicsof the switching process needs to be further clarified. Here, we report magnetization switching in wires patterned from CoFeB/MgO heterostructures with heavy metal (HM)underlayers. We study the threshold current needed to reversemagnetization as a function of pulse amplitude, pulse length,and in-plane magnetic field. Distinct differences are found inthe in-plane field dependence of the switching current betweenSTT and spin Hall torque driven processes. Direct currentflowing through the magnetic layer can cause instability ofthe magnetic moments via STT and consequently can resultin magnetization switching, however, with no difference inthe switching probability against the current flow directionor initial magnetization direction. In contrast, for spin Halltorque driven magnetization switching, the switching currentis different for positive and negative currents and initialmagnetization directions. We find that a tilt in the uniaxialanisotropy axis, first reported by You et al. [20] to show that such effect enables spin Hall torque switching at zero field,develops during the film deposition process, and is found to beresponsible for the asymmetric magnetization switching withcurrent. 1098-0121/2015/91(21)/214434(10) 214434-1 ©2015 American Physical SocietyJACOB TORREJON et al. PHYSICAL REVIEW B 91, 214434 (2015) II. EXPERIMENTAL RESULTS A. Experimental setup The heterostructures studied here are the same with those reported in [Ref. 21]. The film stack Sub. |dHM|1 CoFeB |2 MgO|1 Ta (figures indicate film thicknesses in nanometers) is sputtered onto thermally oxidized Si substrates (SiO 2is 100 nm thick). We have studied a number of materials forthe HM underlayer (TaN, Hf, W) and found similar results.Representative results from the TaN underlayer films aremostly reported here. TaN is formed by reactively sputteringTa in a mixed gas atmosphere of Ar and N 2[22]: the atomic concentration is Ta 48±5N52±5for the results shown here. The underlayer thickness dis varied within the substrate using a linear shutter during the sputtering. Wires are patternedusing optical lithography and Ar ion etching and a subsequentliftoff process is employed to form electrical contacts made by10 Ta|100 Au (units in nanometers). The width and the length of the patterned wires are 5 µm and 20–30µm, respectively. Figure 1(a) shows a typical optical microscopy image of the patterned wires and the definition of the coordinate axes.A pulse generator is connected to one of the contacts to apply -40 -20 20 40-1.0-0.50.00.51.0Probability Pulse amplitude (V)M+z M -z-0.6 -0.4 0.4 0.6 -50 -25 25 50-1.0-0.50.00.51.0 M+z M -zProbability Pulse amplitude (V)-0.8 -0.4 0.4 0.8-400 -200 0 200 400-12012 TaN (nm) 0.5 3.6Kerr intensity (a.u.)HZ (Oe) JN(A/cm2) x108JN(A/cm2) x108NaT mn 6.3 )d( NaT mn 5.0 )c((b) (a) y x z30m +I FIG. 1. (Color online) (a) Optical microscopy image of the wire used to study current-induced magnetization switching. The dark regions indicate the magnetic film, the bright regions correspond tothe substrate surface and the yellow regions represent the Ta |Au electrodes. A pulse generator is connected to the left electrode. (b) Out-of-plane hysteresis loops measured using Kerr microscopyfor Sub. |dTaN|1C o F e B |2M g O |1T a : d=3.6 nm (red circles) andd=0.5 nm (black squares). (c), (d) Magnetization switching probability as a function of pulse amplitude for initial magnetizationconfigurations pointing along +z(black squares) and −z(red circles) for the two devices shown in (b). Positive and negative probability corresponds to initial magnetization direction pointing along +zand −z, respectively. A pulse train consisting of five 100-ns-long pulses is applied. Representative Kerr images captured after the application of±50 V (c) and ±35 V (d) are included at the corresponding corners of each panel. Results are from substrates placed in the “left” position defined in Fig. 5(a). No external magnetic field is applied during the current pulse application for the results shown in (c)and (d).constant amplitude voltage pulses (0.5–100 ns long) to the wire. Positive current corresponds to current flow along the+xdirection. We use Kerr microscopy to study magnetization reversal driven by magnetic field and/or current. The magnetic easy axis of the films points along the film normal owing to the perpendicular magnetic anisotropy (PMA)developed at the CoFeB/MgO interface [ 15,22]. Figure 1(b) shows magnetization hysteresis loops of two TaN underlayerfilms measured using Kerr microscopy. The difference in theKerr intensity for magnetization pointing along + Zand−Zis opposite for the two samples shown in Fig. 1(b). This is due to an optical interference effect that depends on the film thicknessas well as the thickness of the thermally oxidized Si (see[Ref. 21] for details). H SW, the average (absolute) out-of-plane field (HZ) needed to switch the magnetization from +zto−z and vice versa, is ∼100 Oe for the two films shown in Fig. 1(b): typical values of HSWrange between ∼50–500 Oe for all films studied. Note that HSWrepresents the field needed to nucleate reversed domains; once a reversed domain forms, domain-wallpropagation takes place to magnetize the entire wire (the wallpropagation field is ∼5t o∼30 Oe). The variation of H SW is mostly related to the strength of PMA for each film: the magnetic and electrical properties of all films studied here canbe found in [Ref. 21]. Current-induced magnetization switching is studied using the captured Kerr images. To determine the threshold cur- rent for magnetization switching, the following sequence is performed. (1) A large out-of-plane field ( H Z) is applied to uniformly magnetize the wire along the zdirection. (2) The out-of-plane field is reduced, typically to zero unless notedotherwise, and an in-plane field directed along x(H X)o ry (HY) is applied. Then, a Kerr image of the uniform state is captured to obtain a reference image. (3) Current is injected tothe wire by applying voltage pulse(s) from the pulse generator.The pulse is either a single pulse or a pulse train with eachpulse separated in time by ∼10 ms. The pulse length is fixed to 100 ns unless noted otherwise. After the application of thevoltage pulse(s), a second Kerr image is captured. The firstimage captured in (2) is subtracted from this second image toacquire the “subtracted image,” which we use to calculate thearea where the magnetization direction reversed upon the pulseapplication. The switching probability ( P SW) is calculated by dividing the area where the magnetization switched with thearea of the wire. This process [(1)–(3)] is repeated five timesto acquire statistics: the switching probability shown hereaftercorresponds to the mean of P SWof the five measurements. B. Current-induced magnetization reversal Figures 1(c) and 1(d) display the probability of magne- tization switching as a function of pulse amplitude for thetwo devices shown in Fig. 1(b). For illustration purposes, we multiply the probability by −1 when the initial magnetization direction points along the −zdirection (red circles). At the corners of each graph, representative Kerr images correspond-ing to the magnetic state after the pulse application for largepulse amplitudes are shown for both current directions andinitial magnetization configurations. From these images, it canbe seen that the switching characteristics depend on the filmstructure. Wires with thin TaN underlayers [Fig. 1(c)]s h o wa 214434-2CURRENT-DRIVEN ASYMMETRIC MAGNETIZATION . . . PHYSICAL REVIEW B 91, 214434 (2015) 02460.20.40.60.8 +I IJNC (A/cm2) x108 TaN thickness (nm)0246 TaN thickness (nm) +I I(a)M||z (b)M|| +z FIG. 2. (Color online) Threshold current density ( JNC) as a func- tion of TaN underlayer thickness. The initial magnetization direction points along −z(a) and +z(b). Solid and open symbols represent positive and negative JNC, respectively. A pulse train consisting of five 100-ns-long pulses is applied. Results are from substrates placed in the “left” position defined in Fig. 5(a). No external magnetic field is applied during the current pulse application. symmetric nucleation process with respect to the current flow direction and the initial magnetization configuration: abovethe threshold voltage, the switching probability increases andsaturates to ∼0.5. For thicker TaN underlayer films [Fig. 1(d)], the probability is asymmetric with respect to the currentdirection and the initial magnetization configuration. For initial magnetic states pointing along +z(−z), the switching probability is lower for negative (positive) current. These results indicate that different mechanisms are in- volved in the magnetization reversal process depending on thethickness of the underlayer. Figure 2shows the TaN underlayer thickness dependence of the threshold current density ( J NC) that flows through the underlayer. We define JNCas the min- imum current density needed to achieve switching probabilityexceeding 25%. J NCis calculated using the threshold pulse amplitude, the resistance of the wire, the thickness and theresistivity ( ρ) of the CoFeB layer ( ρ∼160μ/Omega1cm) and the HM underlayer ( ρ∼375μ/Omega1cm for Ta 48N52)[21]. The solid and open symbols in Fig. 2represent positive and negative JNC, respectively; here we show −JNCfor negative current to compare the absolute value with that of positive current.The dependence of J NCon the initial magnetization states is shown in Figs. 2(a) and2(b). The asymmetry in the threshold current density with respect to the current flow direction andthe initial magnetization direction reduces to near zero whenthe TaN underlayer thickness is below ∼1n m .T h ed e g r e eo f asymmetry is nearly constant when the underlayer thicknessis larger than ∼2 nm. This trend qualitatively agrees with the underlayer thickness dependence of the “effective field” dueto the spin Hall torque [ 21,23] [see Figs. 4(a) and4(b)]. When the thickness of the TaN underlayer is thinner than its spindiffusion length, the effective field is nearly zero. In contrast,if the underlayer thickness is larger than its spin diffusionlength, ∼2.5 nm for TaN [ 13,21], the effective field saturates and becomes constant against the thickness. We thus infer thatthe magnetization switching for the thicker underlayer films isdue to the spin Hall torque at the HM |CoFeB interface, whereas the switching for the thin underlayer films is dominated by spintransfer torque within the CoFeB layer [ 24–26]. Note that the thickness at which the effective field saturates is larger than theFIG. 3. (Color online) In-plane field dependence of the threshold current density ( JNC). The field direction is along (a), (c) and transverse to (b), (d) the current flow. The underlayer is TaN: its thickness is 0.5 nm (a), (b) and 6.6 nm (c), (d). Black squares andred circles represent initial magnetization direction along +zand −z, respectively. A pulse train consisting of five 100-ns-long pulses is applied. Results are from substrates placed in the “left” positiondefined in Fig. 5(a). thickness at which the degree of asymmetry of the threshold current becomes constant; the origin of this difference requiresfurther investigation. C. In-plane field dependence of the threshold current To gain insight into the respective roles of the spin transfer torques and the spin Hall torques for driving magnetizationreversal, we have studied the threshold current as a function ofin-plane external fields. Figure 3shows J NCas a function of in-plane field along x(HX) andy(HY) for films with thin and thick TaN underlayer films. The squares and circles representinitial magnetization pointing along +zand−z, respectively. For the thin underlayer films [Figs. 3(a) and 3(b)],J NC is symmetric with respect to the in-plane field. Figure 3(a) shows that magnetization switching is assisted by +HXfor positive current when the initial magnetization direction pointsalong −z.J NCtends to saturate as the magnitude of HX is increased. In contrast, Fig. 3(b) shows that the threshold current is strongly influenced by HYwithin the same field range: the difference in JNCfor initial magnetization pointing along +zand−zincreases with increasing |HY|. For these films, the current-induced effective field due to the spin Halleffect is small and we can therefore assume that the STT(current through the magnetic layer) plays the dominant rolein the magnetization reversal process. Theoretically, it has beenreported that STT can amplify spin waves in uniform magneticstate that can result in domain-wall nucleation, or partialmagnetization reversal, when large enough current is applied 214434-3JACOB TORREJON et al. PHYSICAL REVIEW B 91, 214434 (2015) -6000600HY (Oe) M || +z M || -z -6000600HX (Oe)(a) (b) (c) (d)0-1-2bJ / aJ 024601530 I+ I-HX*(Oe) TaN thickness (nm) FIG. 4. (Color online) The fieldlike ( /Delta1HY) (a) and the damping- like (/Delta1HX) (b) components of the current-induced effective field plotted against the TaN underlayer thickness (source: [Ref. 21]). Black squares and red circles correspond to magnetization directedalong +zand−z, respectively. The effective field is normalized by the current density J Nthat flows through the TaN layer. (c) Ratio of the fieldlike component to the dampinglike component bJ/aJ= −mZ/Delta1HY//Delta1H Xplotted against the TaN underlayer thickness. (d) TaN thickness dependence of the offset field HX∗. Solid and open symbols correspond to HX∗estimated using positive and negative currents. [25,26]. In such cases, the threshold current needed to cause magnetization switching does not, as a first approximation,depend on small (compared to the anisotropy field) in-planeapplied field [ 27]. Further study is required to identify the origin of the in-plane field dependence. For the thicker underlayer films, the threshold current density exhibits a different in-plane field dependence. Asdescribed above, J NCis different for initial magnetization pointing along +zand−zin the absence of external field. This difference in JNC, for a given current direction, reverses when a small in-plane field directed along the +xdirection is applied [Fig. 3(c)]. The field needed to match JNCfor positive and negative currents, termed the offset field ( HX∗) hereafter, is ∼20–25 Oe for the sample shown in Fig. 3(c). The offset field HX∗is plotted as a function of the TaN underlayer thickness in Fig. 4(d). We find that HX∗increases with the TaN underlayer thickness: the reason behind this will be discussed in Sec. III in connection with the ratio of the fieldlike [ /Delta1HY,F i g . 4(a)] and the dampinglike [ /Delta1HX,F i g . 4(b)] components of the spin Hall effective field, shown in Fig. 4(c).Previously, it has been reported that a nonzero HXis needed to switch the magnetization directed along the film normal within-plane current [ 3,8,18]. Here, owing to the nonzero H X∗, magnetization switching can be triggered at zero magneticfield. Note that the threshold current dependence on H Xis consistent with the negative spin Hall angle of the underlayer[13,21]: the threshold current is smaller when the direction of H X−HX∗matches that of the dampinglike component of the spin Hall effective field compared to the opposite case. Thedampinglike component of the spin Hall effective field pointsalong the −xdirection for positive current and magnetization pointing along +z: it points in the opposite direction if the current or the magnetization direction is reversed; see Fig. 4(b). For in-plane field ( H Y) applied perpendicularly to the current flow, JNCis found to vary more or less linearly with HY[Fig. 3(d)]. The dependence of JNConHYis compared to model calculations in Sec. IIIto discuss its relationship with the sign of the fieldlike spin Hall torque. D. Dependence on the film deposition conditions The zero-field switching found here indicates that the symmetry of the system is broken for the thick underlayerfilms. We find that the symmetry-breaking factor arises duringthe film deposition (sputtering) process. Figure 5(a) shows schematic of the inside of sputtering chamber with focus on the relation between the substrate position and the sputteringtarget. The same coordinate axes shown in Fig. 1(a) are illustrated in Fig. 5(a)for reference. Three substrates are placed for film deposition and we find that the asymmetry in theswitching with current changes depending on the positionof the substrate. Figures 5(b) and 5(c) show Kerr images after voltage pulses are applied to the wire when the initialmagnetization direction is set along −z(the films have 3.6–nm- thick TaN underlayer). When the substrate is positioned alongthe+ydirection, denoted as “left” in Fig. 5(a), the switching probability (i.e., the area with brighter contrast) is larger fornegative current [Fig. 5(b)]. This asymmetry is the same with that shown in Figs. 1–3. In contrast, when the substrate is placed along the −ydirection [referred to as the “right” position in Fig. 5(a)], the asymmetry reverses: the switching probability is now larger for the positive current. The pulseamplitude dependence of the switching probability is shown inFig.5(d), which clearly shows the difference in the asymmetry. We have also studied current-induced magnetization switchingfor wires whose long axis is directed along the yaxis (Fig. 6). In such case, we find little difference in the switching currentfor positive/negative currents and the initial magnetizationalong±z. The asymmetric magnetization switching is also found in other heavy metal underlayer films (Hf and W). As shownin Fig. 7, the asymmetry of the switching with respect to the current flow direction and the initial magnetization directionis the same for all underlayer films as long as the positionof the substrate is kept same. Note that the sign of the spinHall angle for the heavy metals used here is the same, whereasthe Dzyaloshinskii-Moriya interaction (DMI) [ 28,29]a tt h e underlayer|CoFeB layer interface changes its sign between Hfand W [ 21]. 214434-4CURRENT-DRIVEN ASYMMETRIC MAGNETIZATION . . . PHYSICAL REVIEW B 91, 214434 (2015) FIG. 5. (Color online) (a) Schematic illustration of inside the sputtering chamber where the relative position of the substrates and the target is shown. Three ∼1×1c m2square substrates, separated by∼0.15 cm along the ydirection, are placed ∼10 cm away from the target. (b), (c) Kerr images after application of ±32 V voltage pulses for devices made of Sub. |3.6 nm TaN |1C o F e B |2M g O |1T a substrates placed at different positions: (b) “Left” position and (c) “right” position defined in (a). The top and bottom images correspond to images when positive and negative voltage pulses are applied,respectively. (d) Magnetization switching probability as a function of pulse amplitude for the two devices shown in (b) and (c). The initial magnetization direction points along −z. A pulse train consisting of five 100-ns-long pulses is applied for (b)–(d). No external magnetic field is applied during the current pulse application. FIG. 6. (Color online) Pulse amplitude dependence of magneti- zation switching probability for Sub. |2.9 TaN |1C o F e B |2M g O |1T a (units in nm). The patterned wires’ long axis is directed along x(a) and y(b). Results are from substrates placed in the “left” position defined in Fig. 5(a). Positive and negative probability corresponds to initial magnetization direction pointing along +zand−z, respectively. No external magnetic field is applied during the current pulse application.-24 -16 16 24-1.0-0.50.00.51.0 M+z M -z Pulse amplitude (V)-16 -8 8 16-1.0-0.50.00.51.0 Pulse amplitude (V)M+z M -zProbability3.1 nm W 5.9 nm Hf ) b( )a( FIG. 7. (Color online) Magnetization switching probability as a function of pulse amplitude for initial magnetization configurations pointing along +z(black squares) and −z(red circles) for devices with different heavy metal underlayers. The films are Sub. |dX|1 CoFeB |2M g O |1 (units in nanometers), with X =5.9 nm Hf (a) and 3.1 nm W (b). A pulse train consisting of five 100-ns-long pulses is applied. Positive and negative probability corresponds to initialmagnetization direction pointing along +zand−z, respectively. Results are from substrates placed in the “left” position defined in Fig. 5(a). No external magnetic field is applied during the current pulse application. E. Effect of the out-of-plane field A nonzero out-of-plane magnetic field can introduce dif- ference in the switching probability for initial magnetizationpointing along +zand−z. Figure 8shows the pulse amplitude dependence of the switching probability when the out-of-planefield (H Z) is varied. As evident, the switching probability is larger for both current flow directions when HZassists the switching process, i.e., when HZis pointing opposite to the initial magnetization direction. However, these results showthatH Zby itself cannot induce difference in the switching for positive and negative currents. The maximum residual fieldfrom the electromagnet at the sample position is ∼1O e . F. Pulse-length dependence and repeated switching measurements The magnetization switching observed here may be influ- enced by subsequent motion of nucleated domain walls drivenby current [ 30,31]. To study whether the asymmetry of J NC with the current and initial magnetization directions is due to the motion of domain walls, we have studied the pulse-lengthdependence of J NC. If any subsequent domain-wall motion is causing the asymmetry, such effect should diminish whenthe pulse length is reduced since the distance the domainwall travels will also decrease. Figure 9(a) shows J NCas a function of pulse length ( tP) for the device shown in Fig. 1(d), in which we consider spin Hall torque is responsible forthe switching. A pulse train consisting of five t Pns-long pulses, each separated by 10 ms, is applied. The differenceinJ NCfor positive and negative currents as well as that for initial magnetization pointing along +zand−zremains the same even for pulse length of 10 ns. We have observed suchasymmetry in other devices for pulse length as small as 5 ns.Thus, these results show that the asymmetry is predominantlycaused by the nucleation process and not the subsequentdomain-wall motion. In Fig. 9(b), we show that the switching process can be deterministic even in the absence of magnetic field. A 214434-5JACOB TORREJON et al. PHYSICAL REVIEW B 91, 214434 (2015) -40 -30 30 40M +z M-z Pulse amplitude (V)-40 -30 30 40M +z M-z Pulse amplitude (V)-40 -30 30 40-1.0-0.50.00.51.0M +z M-zProbability Pulse amplitude (V)eO 5 )c( eO 0 )b( eO 5- )a( FIG. 8. (Color online) Magnetization switching probability as a function of pulse amplitude for Sub. |2.9 TaN |1C o F e B |2M g O |1 Ta (units in nm). The out-of-plane field HZis varied: HZ∼− 5( a ) ,∼0( b ) ,a n d ∼5 Oe (c). Positive and negative probability corresponds to initial magnetization direction pointing along +zand−z, respectively. Results are from substrates placed in the “left” position defined in Fig. 5(a). pulse train consisting of five 100-ns-long pulse is used for each “pulse” shown in the top panel. The sign of the pulsetrain is altered each time. We have chosen the same deviceshown in Fig. 1(d) in which the asymmetry is large so that -0.50.00.5JN (A/cm2) (x108) -250 0123456789 1 0025I Iteration-101mZ -1010.40.60.8 11 0 1 0 0-0.8-0.6-0.4M +z M-zJNC (A/cm2) x108 Pulse length (ns) (a) (b) FIG. 9. (Color online) (a) Threshold current density ( JNC)v s pulse length ( t) at zero external field for Sub. |3.6 nm TaN |1C o F e B |2 MgO|1 Ta. A pulse train consisting of five tns-long pulses, each separated by 10 ms, is applied. Black squares and red circles show JNCwhen the initial magnetization direction is along +zand−z, respectively. (b) Sequences of voltage pulses applied to the wire (top panel) and the resulting Kerr contrast ( /Delta1I) calculated from the Kerr images. The corresponding magnetic state (1: along +z,−1: along −z) is shown in the right axis. A pulse train consisting of five 100-ns-long pulses, each separated by 10 ms, is applied at each pulse shown in the top panel. Middle and bottom panels of (b) showchanges in the Kerr contrast for initial magnetization pointing along +zand−zat the beginning of the sequence, respectively. No external magnetic field is applied during the current pulse application. Resultsare from substrates placed in the “left” position defined in Fig. 5(a).full switching of magnetization takes place upon the pulse application (if the asymmetry is small, it is difficult to reversethe entire area of the wire just with the current pulse). Themiddle and bottom panels of Fig. 9(b) show the variation of the magnetic state, inferred from the Kerr images, withsuccessive pulse application. The state at the beginning (i.e.,“iteration 0”) has different orientation for the middle andbottom panels. When the magnetization is pointing along +z (−z), positive (negative) current can trigger magnetization reversal. Full switching of the wire magnetization is observedwhen appropriate pulse is applied. When a “wrong” pulse isapplied, as shown at “iteration 1” in the bottom panel, wedo not find random nucleation due to thermal activation, thusshowing the robustness of this switching scheme. III. MODEL CALCULATIONS A. Macrospin model To gain insight of the asymmetric magnetization switching with current and the in-plane field dependence of JNC,w e show results from model calculations using the Landau-Lifshitz-Gilbert (LLG) equation. We find that if we assumea uniaxial magnetic anisotropy that is tilted away from thenormal of the film plane, a mechanism first suggested in[Ref. 20], many of our experimental results can be explained. Similar results can be obtained if a unidirectional anisotropypointing along the wire’s long axis is assumed. However,with this assumption, H X∗will simply be defined by the unidirectional anisotropy field and it is difficult to explain someof the experimental results, for example, the TaN underlayerthickness dependence of H X∗[Fig. 4(d)]. The LLG equation that includes the spin Hall torques reads as ∂ˆm ∂t=−γˆm×(/vectorHK+/vectorHEXT+aJ(ˆm׈p)+bJˆp) +αˆm×∂ˆm ∂t, (1) where ˆmis a unit vector representing the magnetization direction, tis time, γis the gyromagnetic ratio, and αis the Gilbert damping parameter. /vectorHKand/vectorHEXTrepresent the uniaxial anisotropy field and the external magnetic field,respectively. We set the axis of the uniaxial anisotropy field to be oriented along a unit vector ˆk, i.e., /vectorH K=HK(ˆm·ˆk)ˆk.T h e coordinate system employed in the calculations is the same asthat shown in Fig. 1(a). 214434-6CURRENT-DRIVEN ASYMMETRIC MAGNETIZATION . . . PHYSICAL REVIEW B 91, 214434 (2015) The effect of current is coded in the parameters aJand bJ.aJis the dampinglike component [ 1,2]o ft h es p i n Hall effective field, whereas bJcorresponds to the fieldlike component [ 32]. We assume that aJandbJare proportional to current that flows through the wire. Unit vector ˆprepresents the spin direction of the electrons that impinge upon themagnetic layer (FM) generated within the heavy metal layer(HM) via the spin Hall effect. Positive current correspondsto current flow along the +xdirection. For positive current, we set ˆp=(0,1,0) as this represents the spin direction of the electrons entering the CoFeB layer via the spin Hall effectin heavy metal layers with negative spin Hall angle suchas Ta and W. We vary a JandbJto study the effect of current. Current and field are applied to the system and theresulting equilibrium magnetization direction is calculated fora 100 ns-long current pulse. In order to cause magnetizationswitching within reasonable values of a J, we use a reduced uniaxial anisotropy field [ 18], i.e., HK∼530 Oe. Figure 10shows results of model calculations when the uniaxial anisotropy axis is tilted in the yzplane, i.e., ˆk=(0,sinβ,cosβ). Here, we set the tilt angle βto be 2°away from thezaxis. Figures 10(a) and10(b) show the zcomponent of magnetization as a function of aJ. The sign of bJis opposite for Figs. 10(a) and 10(b) . As evident, the zcomponent of the magnetization ( mZ) rotates toward the film plane as aJis increased. In many cases, we find an abrupt transition of themagnetic state from the film normal to the film plane. Oncethe magnetization points along the film plane, it can moveback to its original direction or it can move to the oppositeside of the zaxis, resulting in magnetization switching, after the current is turned off due to thermal activation. We thusdefine the threshold a J(aJC) as the minimum aJneeded to cause the absolute value of mZto be less than 0.15: this value is justified by micromagnetic simulations shown in the nextsection. Note that in some cases [e.g., Fig. 10(b) ], we find the equi- librium m Zduring the current application jumps to the equil- ibrium position of the other branch (i.e., opposite to theinitial direction). This indicates deterministic switching of themagnetization, not the probabilistic switching as described -800 0 800-1.0-0.50.00.51.0 aJ (Oe)-800 0 800-1.0-0.50.00.51.0mZ aJ (Oe)M || z M || -z -100 0 100-400-2000200400aJC (Oe) HY (Oe)-100 0 100 HY (Oe)-100 0 100 HY (Oe)-100 0 100-400-2000200400M|| z M || -zaJC (Oe) HX (Oe)-100 0 100 HX (Oe)-100 0 100 HX(Oe)bJ=aJ bJ=0 bJ=aJ)b( )a( )d( )c( (h) )g( )f((e) HX*HX* HX*bJ=aJ bJ=aJ FIG. 10. (Color online) (a), (b) zcomponent of the equilibrium magnetization when current and in-plane magnetic field are turned on plotted as a function of aJ, the dampinglike component of the spin Hall torque. The fieldlike component of the spin Hall torque bJis set to −aJ (a) and aJ(b). The horizontal blue dashed lines indicate |mZ|=0.15, which is used to define aJC. (c)–(h) aJCas a function of HX(c)–(e) and HY(f)–(h). The fieldlike component bJis varied: bJ=−aJ(c), (f), bJ=0 (d), (g), and bJ=aJ(e), (h). For all plots, black squares and red circles represent calculation results when the initial magnetization direction points along +zand−z, respectively. HK=528 Oe, α=0.05, the uniaxial anisotropy axis (direction defined by a unit vector ˆk) is tilted 2°toward the yaxis, i.e., ˆk=(0,sinβ,cosβ) with β=2d e g . 214434-7JACOB TORREJON et al. PHYSICAL REVIEW B 91, 214434 (2015) above: the direction of switching is predefined during the current application. Interestingly, such deterministic switchingwill diminish as a Jis further increased since the equilib- riummZduring the current application favors the direction along the film plane, resulting in the probabilistic switch-ing. Such drop in the switching probability with increasingcurrent density is also found in experiments [see, e.g.,Fig. 5(d)]. Figures 10(c) –10(e) show the in-plane field ( H X) depen- dence of aJCwhen the fieldlike component ( bJ) is varied. The asymmetric magnetization switching with nonzero HX∗ is reproduced here with the tilt angle βset to 2°. The sign of HX∗is independent of the size and sign of bJ. The negative HX∗shown in Figs. 10(c) –10(e) is found experimentally in samples deposited in the “right” position defined in Fig. 5(a). Due to the nonzero tilting of the anisotropy axis that breaksthe symmetry of the system, a JCis different for positive and negative currents for a given initial magnetization direction atzero magnetic field. Interestingly, H X∗not only depends on the tilting angle ( β), but also on the relative size of the fieldlike and dampinglikecomponents of the spin Hall torque, that is, the size of b J/aJ. The model shows that HX∗exhibits a complex dependence on bJ/aJ:HX∗takes a maximum when bJ=−aJand it drops as|bJ|further increases. Experimentally, we have previously studied the underlayer thickness dependence of the spin Hall torque using the harmonic Hall measurements [ 21,23]: Figs. 4(a) and 4(b) show the fieldlike ( /Delta1HY=bJˆp·ˆy) and the dampinglike [ /Delta1HX=aJ(ˆm׈p)·ˆx] components of the spin Hall effective field, respectively. The ratio of the twocomponents −m Z/Delta1HY//Delta1H Xis equal to bJ/aJand is plotted in Fig. 4(c). Although the number of data is limited, the thickness dependence of HX∗, plotted in Fig. 4(d), shows that it more or less scales with bJ/aJ. These results show that HX∗is not only a function of the sample position during the sputtering,but also dependent on the characteristics of the spin Halltorque. The detailed difference between the model calculationsand the experimental results requires further thorough studyofH X∗. We have also studied the in-plane field dependence of aJC when the direction of the uniaxial anisotropy axis ( ˆk) is varied. When the tilt direction is inverted in the yzplane, i.e., ˆk= (0,−sinβ,cosβ), the sign of HX∗reverses. Experimentally, HX∗changes its sign when the position of the substrate during the sputtering is changed, as shown in Fig. 5. These results indicate that the tilt angle depends on the substrate position.H X∗is zero and the asymmetric magnetization switching disappears when the tilt direction is set along the xzplane, i.e., ˆk=(sinβ,0,cosβ). This is in agreement with the results shown in Fig. 6, where the asymmetry diminishes when the wire’s long axis is oriented along the tilt direction (i.e., alongtheyaxis). It is somewhat counterintuitive to understand why an offset field in the xdirection ( H X∗) emerges [e.g., Fig. 3(c)] when the uniaxial anisotropy field is tilted along the yzplane with a tilt angle β. One way to understand this is to view the incoming spins diffusing from the HM layer into themagnetic layer in the frame along the tilted anisotropy axis.The polarization ˆpdirected along the +ydirection in the lab frame has to be changed to ˆp /prime=(0,cosβ,sinβ) in a rotated FIG. 11. (Color online) Micromagnetic simulations of spin Hall torque driven magnetization switching. (a), (b) aJ(the damping- like component of the spin Hall torque) dependence of the z component of magnetization ( mZ) at the end of 1-ns pulse (a) and the switching probability calculated from the magnetic state 20 ns after the pulse is turned off (b). Black squares and red circles represent initial magnetization along +zand−z, respectively. Parameters used in the calculations are: saturation magnetization MS=1250 emu /cm3, exchange constant A=3.1×10−6erg/cm, uniaxial magnetic anisotropy energy K=10.15×106erg/cm3, Gilbert damping α=0.05, and the fieldlike component of the spin Hall torque bJ=aJ. The dimension of the simulated element is 2000×500×1n m3with a discretization cell of ∼2×2×1n m3. The anisotropy axis is tilted along the yzplane by 1°. Inset to (a): simulated magnetization image 20 ns after a pulse of aJ=368.6O e is turned off: the initial magnetization is along −z. frame defined by the tilted anisotropy axis. The polarization possesses a nonzero component (i.e., sin β) along the easy axis that can cause the difference in the switching currentfor opposite initial magnetization directions and currentflow directions, similar to conventional spin transfer torqueswitching of parallel/antiparallel magnetization. The tiltedanisotropy axis thus breaks the symmetry along the zdirection, which in turn manifests itself as an offset field in the x direction. The bottom panels of Fig. 10show the H Ydependence of aJCfor different values of bJ. When the sign of bJis opposite to that of aJ[Fig. 10(f) ],aJCmonotonically varies with HY. This is in agreement with the experimental results shown inFig.3(d). The slope of a JCversus HYaround zero field changes as the size and sign of bJis varied [Figs. 10(f) –10(h) ]. These results show that the slope of JNCversus HYaround zero field roughly gives the sign of the fieldlike torque ( bJ). B. Micromagnetic simulations We have performed micromagnetic simulations to validate the macrospin model used to describe the experimental results.The micromagnetic code “ MUMAX ”[33]i su s e df o rt h e simulations. The magnetic parameters used in the simulationsare described in the caption of Fig. 11: the parameters are chosen so that the magnetic anisotropy is the same with thatused in the macrospin calculations (Fig. 10). The definition of the coordinate axis is drawn in the inset to Fig. 11(a) .T h e anisotropy axis is tilted along the yzplane by 1°. Here, we use b J=aJsince this condition gives the largest difference in the switching current for opposite initial magnetization directionsat zero field in the macrospin model. 214434-8CURRENT-DRIVEN ASYMMETRIC MAGNETIZATION . . . PHYSICAL REVIEW B 91, 214434 (2015) The procedure of simulation is the following: a temperature pulse of 700 K and duration of 0.2 ns is first applied to auniform magnetic state to mimic the experimental condition,i.e., thermal agitation of the magnetization. A pulse current of1 ns is applied to study the magnetic state during the currentapplication. The current flows along the +xdirection. We have checked the effect of the current pulse length and find that 1 nsis long enough to study the switching process in most cases.The current is then turned off and the system is relaxed tostudy the final state of the magnetization. The equilibriummagnetic state during the current application for positivecurrent is plotted in Fig. 11(a) for initial magnetization states along∼+z(black squares) and ∼−z(red circles). The results are equivalent to those of macrospin calculations [Fig. 10(b) ]. When the initial magnetization points along ∼−z, there is a critical a Jabove which magnetization switches its direction during the current application. This is equivalent to thedeterministic switching found in the macrospin calculations.When the current is further increased, the magnetization fallscloser to the film plane. The switching probability after the current is turned off and the system is relaxed is shown in Fig. 11(b) as a function of a J for both initial magnetic states. The switching probability is obtained from the area of the element that switched divided bythe whole area, similar to the method used in the experiments.For initial magnetization pointing along −z, we find full (i.e., deterministic) switching of the magnetization above a J∼400 Oe. For the opposite initial magnetic state (along +z), the switching probability saturates at ∼0.5 for large aJ. Note that probability ∼0.5 corresponds to a multidomain state a ss h o w ni nt h ei n s e tt oF i g . 11(a) . We find that if |mZ|during the current application is less than ∼0.13, denoted by the blue dashed line in Fig. 11(a) , domain walls can nucleate during the relaxation process and the final state is a multidomainstate. In other words, if |m Z|is larger ∼0.13, the final state possesses the same magnetization configuration with the initialmagnetic state unless the deterministic switching occurs. Thisjustifies our assumption on using m Z=0.15 for calculating the threshold aJfor magnetization switching in the macrospin model. The features found in the simulations are in agreementwith experiments, where full switching of magnetization isobserved only in one of the starting conditions for a givencurrent direction, while the other only produces a multidomainstate, i.e., partial magnetization switching. IV . DISCUSSION Aside from the tilted uniaxial anisotropy which we consider breaks the symmetry in our system, other factors can alsocause the asymmetry in magnetization reversal with current.Recently, it has been reported that a gradient in the magneticanisotropy across the wafer can break the symmetry and enablezero-field switching. Here, as the underlayer thickness is variedalong the xdirection, it creates a gradient in the magnetic anisotropy and the saturation magnetization across the wafer.This is in contrast to [Ref. 19] in which the gradient is created along the yaxis in our definition [see Fig. 5(a)]. We thus consider that the effect of the out-of-plane fieldlike torqueproposed in [Ref. 19] may be minor here.The asymmetric shape of the patterned wire [Fig. 1(a)], where the right side of the wire is connected to a region withlow magnetic anisotropy due to prior etching of half the MgOlayer and the Ta capping layer before the Ta |Au pad formation, can result in preferential current-induced injection of domainwalls from the right side of the wire [ 34]. We have thus tested symmetric structures with large pads attached to both sides ofthe wire and have found that the asymmetry is not altered. The DMI can play a role in the nucleation process [31,35,36]. As reported in [Ref. 36], for a uniform initial magnetization state, the DMI is relevant near the edge ofthe wire where the magnetization is tilted. We find littleevidence of nucleation events taking place preferentially fromthe edges of the wire for many of the films studied here. Oneexception is the W underlayer films, where we find preferentialnucleation from the edges when a relatively large (a fewhundred Oersteds) in-plane field along the wire’s long axis(H X) is applied. However, the nucleated region is limited to the edge of the wire (near the Ta |Au electrodes) and cannot explain the full reversal that occurs within the wire. As shownin Fig. 7, the asymmetric magnetization switching with current occurs in a similar fashion for the Hf and W underlayer films,which possess opposite sign of the interface DMI [ 21]. We thus infer that the DMI is not the main source of the asymmetricswitching. V . CONCLUSION In summary, we have studied current-driven magnetiza- tion switching in perpendicularly magnetized CoFeB/MgOheterostructures with heavy metal underlayers (TaN). Thethreshold current needed to reverse the magnetization directionis studied as a function of film structure, pulse amplitude, pulselength, and in-plane magnetic field. From the in-plane mag-netic field dependence we find that magnetization switchingtakes place via spin transfer torque within the CoFeB layerwhen the underlayer thickness is small, whereas the switchingoccurs due to spin Hall torque for thicker underlayer films. Forspin Hall torque driven magnetization reversal, the thresholdcurrent is different for positive and negative currents as wellas the initial magnetization directions (pointing along +zor −z). We attribute such asymmetry of the switching current to a tilting of the uniaxial anisotropy axis, away from the normalof the film plane, which develops during the film depositionprocess (sputtering). The asymmetry depends on the relativeposition of the substrate and the center of the sputtering target,suggesting an extrinsic origin. Just a few degrees of the tiltingcan break the symmetry to enable zero field switching ofperpendicularly magnetized thin films using in-plane current. ACKNOWLEDGMENTS We thank G. Tatara for helpful comments on the experimen- tal results and J. Kim and T. Devolder for technical support.This work was partly supported by the Japanese Ministry ofEducation, Culture, Sports, Science and Technology (MEXT)R & D Next-Generation Information Technology and theGrant-in-Aid for Young Scientists (A), the Agence Nationalede la Recherche under Contract No. ANR-11-BS10-003(NanoSWITI). 214434-9JACOB TORREJON et al. PHYSICAL REVIEW B 91, 214434 (2015) [1] J. C. Slonczewski, J. Magn. Magn. Mater. 159,L1(1996 ). [2] L. Berger, Phys. Rev. B 54,9353 (1996 ). [3] L. Liu, C.-F. Pai, Y . Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336,555(2012 ). [4] C. F. Pai, L. 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PhysRevMaterials.1.061401.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW MATERIALS 1, 061401(R) (2017) Mode-dependent damping in metallic antiferromagnets due to intersublattice spin pumping Qian Liu,1H. Y . Yuan,2Ke Xia,1,3and Zhe Yuan1,* 1The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, 100875 Beijing, China 2Department of Physics, South University of Science and Technology of China, Shenzhen, Guangdong 518055, China 3Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China (Received 26 April 2017; published 13 November 2017) Damping in magnetization dynamics characterizes the dissipation of magnetic energy and is essential for improving the performance of spintronics-based devices. While the damping of ferromagnets has been wellstudied and can be artificially controlled in practice, the damping parameters of antiferromagnetic materials arenevertheless little known for their physical mechanisms or numerical values. Here we calculate the dampingparameters in antiferromagnetic dynamics using the generalized scattering theory of magnetization dissipationcombined with the first-principles transport computation. For the PtMn, IrMn, PdMn, and FeMn metallicantiferromagnets, the damping coefficient associated with the motion of magnetization ( α m) is 1–3 orders of magnitude larger than the other damping coefficient associated with the variation of the Néel order ( αn), in sharp contrast to the assumptions made in the literature. DOI: 10.1103/PhysRevMaterials.1.061401 Damping describes the process of energy dissipation in dynamics and determines the time scale for a nonequi-librium system relaxing back to its equilibrium state. Formagnetization dynamics of ferromagnets (FMs), the dampingis characterized by a phenomenological dissipative torqueexerted on the precessing magnetization [ 1]. The magnitude of this torque, which depends on material, temperature, andmagnetic configurations, has been well studied in experiment[2–10] and theory [ 11–16]. Recently, the magnetization dynamics of antiferromagnets (AFMs) [ 17–20], especially those controlled by an electric or spin current [ 21–32], has attracted lots of attention in the process of searching for high-performance spintronic devices.However, the understanding of AFM dynamics, in particularthe damping mechanism and magnitude in real materials, isquite limited. The magnetization dynamics of a collinear AFMcan be described by two coupled Landau-Lifshitz-Gilbert(LLG) equations corresponding to the precessional motionof the two sublattices, respectively [ 33], i.e. ( i=1,2), ˙m i=−γmi×hi+αimi×˙mi, (1) where γis the gyromagnetic ratio, miis the magnetization direction on the ith sublattice, and ˙mi=∂tmi.hiis the effective magnetic field on mi, which contains the anisotropy field, the external field, and the exchange field arising fromthe magnetization on both sublattices. The last contribution toh imakes the dynamic equation of one sublattice coupled to the equation of the other one. Specifically, if the free energyof the AFM is given by the following form, F[m 1,m2]≡ μ0MsVE[m1,m2], with the permeability of vacuum μ0,t h e magnetization on each sublattice Ms, and the volume of the AFM V, one has hi=−δE/δmi.αiin Eq. ( 1)i st h e damping parameter representing the dissipation rate of themagnetization m i. Due to the sublattice permutation symme- try, the damping magnitudes of the two sublattices should beequal. This approach has been used to investigate the AFM *Corresponding author: zyuan@bnu.edu.cnresonance [ 33,34], temperature-gradient-induced domain wall (DW) motion [ 35], and spin-transfer torques in an AFM |FM bilayer [ 36]. An alternative way to deal with the AFM dynamics is introducing the net magnetization m≡m1+m2and the Néel order n≡m1−m2so that the precessional motion of m andncan be derived from the Lagrangian equation [ 26]. The damping effect is then included artificially with twoparameters α mandαnthat characterize the dissipation rate ofmandn, respectively. This approach is widely used to investigate the spin superfluid in an AFM insulator [ 37,38], an AFM nano-oscillator [ 39], and DW motion induced by an electrical current [ 26,40], spin waves [ 41], and spin-orbit torques [ 42,43]. Using the above definitions of mandn, one can reformulate Eq. ( 1) and derive the following dynamic equations: ˙n=(γhm−αm˙m)×n+(γhn−αn˙n)×m, (2) ˙m=(γhm−αm˙m)×m+(γhn−αn˙n)×n, (3) where hnandhmare the effective magnetic fields exerted on n andm, respectively. They can also be written as the functional derivative of the free energy [ 26,41], i.e., hn=−δE/δnand hm=−δE/δm. The damping parameters in Eqs. ( 1)–(3) have the relation αn=αm=α1/2=α2/2[36]. Indeed, the assumption αm=αnis commonly adopted in the theoretical study of AFM dynamics with only a few exceptions, whereα mis ignored in the current-induced skyrmion motion in AFM materials [ 44] and the magnon-driven DW motion [ 45]. However, the underlying damping mechanism of an AFM andthe relation between α mandαnhave not been fully justified yet [46,47]. In this paper, we generalize the scattering theory of mag- netization dissipation in FMs [ 48] to AFMs and calculate the damping parameters from first principles for metallic AFMsPtMn, IrMn, PdMn, and FeMn. The damping coefficients inan AFM are found to be strongly mode dependent, with α m up to 3 orders of magnitude larger than αn. By analyzing the dependence of damping on the disorder and spin-orbit coupling 2475-9953/2017/1(6)/061401(6) 061401-1 ©2017 American Physical SocietyRAPID COMMUNICATIONS QIAN LIU, H. Y . YUAN, KE XIA, AND ZHE YUAN PHYSICAL REVIEW MATERIALS 1, 061401(R) (2017) (SOC), we demonstrate that αnarises from SOC in analog to the Gilbert damping in FMs, while αmis dominated by the spin pumping effect between sublattices. Theory. In analog to the scattering theory of magnetization dissipation in FMs [ 48], the damping parameters in AFMs, αnandαm, can be expressed in terms of the scattering matrix. Following the previous definition of the free energy, the energydissipation rate of an AFM reads ˙E=−μ 0MsV˙E=μ0MsV/parenleftbigg −δE δm·˙m−δE δn·˙n/parenrightbigg =μ0MsV(hm·˙m+hn·˙n). (4) By replacing the effective fields hmandhnby the time derivative of magnetization order and Néel order using Eqs. ( 2) and ( 3), one arrives at [ 49] ˙E=μ0MsV γ(αn˙n2+αm˙m2). (5) If we place an AFM between two semi-infinite nonmag- netic metals, the propagating electronic states coming fromthe metallic leads are partly reflected and transmitted. Theprobability amplitudes of the reflection and transmission formthe so-called scattering matrix S[50]. For such a scattering structure with only the order parameter nof the AFM varying in time (see the insets of Fig. 1), the energy loss that is pumped into the reservoir is given by ˙E=¯h 4πTr(˙S˙S†)=¯h 4πTr/parenleftbigg∂S ∂n∂S† ∂n/parenrightbigg ˙n2≡Dn˙n2.(6) Here we define Dn≡(¯h/4π)Tr[(∂S/∂n)(∂S†/∂n)]. Compar- ing Eqs. ( 5) and ( 6), we obtain Dn=μ0MsA γαnL, (7) where we replace the volume Vby the product of the cross- sectional area Aand the length Lof the AFM. We can express αmin the same manner, Dm=μ0MsA γαmL, (8) withDm≡(¯h/4π)Tr[(∂S/∂m)(∂S†/∂m)]. Using Eqs. ( 7) and (8), we calculate the energy dissipation as a function of the length Land extract the damping parameters αn(m)via a linear-least-squares fitting. Note that the above formalismcan be generalized to include noncollinear AFM, such asDWs in AFMs, by introducing the position-dependent orderparameters n(r) and m(r). It can also be extended for the AFMs containing more than two sublattices, which may notbe collinear with one another [ 51]. For the latter case, one has to redefine the proper order parameters instead of nandm [52]. First-principles calculations. The above formalism is im- plemented using the first-principles scattering calculation andis applied here in studying the damping of metallic AFMsincluding PtMn, IrMn, PdMn, and FeMn. The lattice constantsand magnetic configurations are the same as in the reportedfirst-principles calculations [ 53]. Here we take tetragonal PtMn as an example to illustrate the computational details. A finitethickness ( L) of PtMn is connected to two semi-infinite Au 0 5 10 15 20 25 30 L (nm)0102030Dm/(0Ms A) (nm) 0 0.5 1 SOC Factor00.40.8 m 103 n0.020.030.04Dn/(0Ms A) (nm)Pt Mn1Mn2m1 m2 m1 m2 mm ξSO=0 ξSO=0(a) (b) ξSO≠0 ξSO≠0abc FIG. 1. Calculated energy dissipation rate as a function of the length of PtMn due to variation of the order parameters n(a) and m (b).Ais the cross-sectional area of the lateral supercell. Arrows in each panel illustrate the dynamical modes of the order parameters.The empty symbols are calculated without spin-orbit interaction. The inset of panel (a) shows the atomic structure of PtMn with collinear AFM order. The inset in (b) shows calculated α nandαmas a function of the scaled SOC strength. The factor 1 corresponds to the real SOC strength that is determined by the derivative of the self-consistent potentials. leads along (001) direction. The lattice constant of Au is made to match that of the aaxis of PtMn. The electronic structures are obtained self-consistently within the density functionaltheory implemented with a minimal basis of the tight-bindinglinear muffin-tin orbitals (TB LMTOs) [ 54]. The magnetic moment of every Mn atom is 3 .65μ Band Pt atoms are not magnetized. To evaluate αnandαm, we first construct a lateral 10 ×10 supercell including 100 atoms per atomic layer in the scatteringregion, where the atoms are randomly displaced from theirequilibrium lattice sites using a Gaussian distribution withthe rms displacement /Delta1[15,55]. The value of /Delta1is chosen to reproduce typical experimental resistivity of the correspondingbulk AFM. The scattering matrix Sis obtained using a first-principles “wave-function matching” scheme that is alsoimplemented with TB LMTOs [ 56], and its derivative is obtained by the finite-difference method [ 49]. Figure 1(a) shows the calculated energy-pumping rate D nof PtMn as a function of Lfornalong the caxis with/Delta1/a=0.049. The total pumping rate (solid symbols) 061401-2RAPID COMMUNICATIONS MODE-DEPENDENT DAMPING IN METALLIC . . . PHYSICAL REVIEW MATERIALS 1, 061401(R) (2017) increases linearly with increasing the volume of the AFM. A linear-least-squares fitting yields αn=(0.67±0.02)×10−3, as plotted by the solid line. The finite intercept of the solidline corresponds to the interface-enhanced energy dissipation,which is essentially the spin pumping effect at the AFM |Au interface [ 57,58]. The Néel-order-induced damping α ncom- pletely results from SOC. If we artificially turn SOC off,the calculated pumping rate is independent of the volumeof the AFM, indicating α n=0. This is because the spin space is decoupled from the real space without SOC and theenergy is then invariant with respect to the direction of n.T h e spin-pumping effect is nearly unchanged by the SOC. The energy-pumping rate D mof PtMn with nalong the caxis is plotted in Fig. 1(b), where we find three important features: (1) The extracted value of αm=0.59±0.02, which is nearly 1000 times larger than αn. (2) Turning SOC off only slightly increases the calculated αm, indicating that SOC is not the main dissipative mechanism of αm. The difference between the solid and empty circles in Fig. 1(b) can be attributed to the SOC-induced variation of electronic structure near the Fermilevel. To see more clearly the different influence of SOC on α m andαn, we plot in the inset of Fig. 1(b)the calculated damping parameters as a function of SOC strength. Indeed, as the SOCstrength ξ SOis artificially tuned from its real value to zero, αn decreases dramatically and tends to vanish at ξSO=0, while αmis less sensitive to ξSOthanαn. (3) The intercepts of the solid and dashed lines are both vanishingly small, indicating that this specific mode does not pump spin current into thenonmagnetic leads. The pumped spin current from an AFMgenerally reads I pump s∝n×˙n+m×˙m[58]. For the mode depicted in Fig. 1(b), one has ˙n=0 and ˙m/bardblmsuch that Ipump s=0. To explore the disorder dependence of the damping param- etersαnandαm, we further perform the calculation by varying the rms of atomic displacements /Delta1. Figure 2(a)shows that the calculated resistivity increases monotonically with increasing/Delta1. The resistivity ρ cwith nalong the caxis is lower than ρa with nalong the aaxis. The anisotropic magnetoresistance (AMR) defined by ( ρa−ρc)/ρcis about 10%, which slightly decreases with increasing /Delta1, as plotted in the inset of Fig. 2(a). The large AMR in PtMn is useful for experimental detectionof the Néel order. The calculated AMR seems to be an order ofmagnitude larger than the reported values in literature [ 59–61]. We may attribute the difference to the surface scattering inthin-film samples and other types of disorder that have beenfound to decrease the AMR of ferromagnetic metals and alloys[62]. α nof PtMn plotted in Fig. 2(b)is of the order of 10−3, which is comparable with the magnitude of the Gilbert damping offerromagnetic transition metals [ 2–4,15]. For nalong the a axis,α nshows a weak nonmonotonic dependence on disorder, while αnfornalong the caxis increases monotonically. With the relativistic SOC, the electronic structure of an AFMdepends on the orientation of n. When nvaries in time, the occupied energy bands may be lifted above the Fermi level.Then a longer relaxation time (weaker disorder) gives rise to alarger energy dissipation, corresponding to the increase in α n with decreasing /Delta1at small /Delta1. It is analogous to the intraband transitions accounting for the conductivitylike behavior ofGilbert damping at low temperature in the torque-correlation0.51.01.52.0 n (10-3)(a) 80160240 ( cm) n//a(b) n//c4.6 5.4 6.2 /a (10-2)01020 AMR (%) 4.6 5.0 5.4 5.8 6.2 /a (10-2)0.20.40.60.8 m4 6 8 10 12 (105-1 m-1)0.20.50.8 m(b) (c) FIG. 2. Calculated resistivity (a) and damping parameters αn (b) and αm(c) of PtMn as a function of the rms of atomic displacements. The red squares and black circles are calculated with nalong the aandcaxis, respectively. The inset of (a) shows the calculated AMR. αmis replotted as a function of conductivity in the inset of (c). The blue dashed line illustrates the linear dependence. model [ 11,12]. Sufficiently strong disorder renders the system isotropic, and the variation of ndoes not lead to electronic excitation, but scattering of conduction electrons by disorderstill dissipates energy into the lattice through SOC. Thehigher the scattering rate, the larger the energy dissipation ratecorresponding to the contribution of the interband transitions[11,12]. Therefore, α nshares the same physical origin as the Gilbert damping of metallic FMs. The value of αmis about 3 orders of magnitude larger thanαn, and it decreases monotonically with increasing the structural disorder, as shown in Fig. 2(c). This remarkable difference can be attributed to the energy involved in thedynamical motion of mandn. While the precession of n only changes the magnetic anisotropy energy in an AFM,the variation of mchanges the exchange energy that is in magnitude much larger than the magnetic anisotropy energy. Physically, α mcan be understood in terms of spin pumping [63,64] between the two sublattices of an AFM. The sublattice m2pumps a spin current that can be absorbed by m1, resulting in a damping torque exerted on m1asα/primem1×[m1×(m2× ˙m2)]. Here α/primeis a dimensionless parameter to describe the strength of the spin pumping. This torque can be simplifiedto beα /primem1×˙m2by neglecting the higher-order terms of the total magnetization m. In addition, the spin pumping by m1 061401-3RAPID COMMUNICATIONS QIAN LIU, H. Y . YUAN, KE XIA, AND ZHE YUAN PHYSICAL REVIEW MATERIALS 1, 061401(R) (2017) TABLE I. Calculated resistivity and damping parameters for the Néel order nalong the aandcaxis. AFM n ρ(μ/Omega1cm) αn(10−3) αm PtMn aaxis 119 ±5 1.60 ±0.02 0.49 ±0.02 caxis 108 ±4 0.67 ±0.02 0.59 ±0.02 IrMn aaxis 116 ±2 10.5 ±0.2 0.10 ±0.01 caxis 116 ±2 10.2 ±0.3 0.10 ±0.01 PdMn aaxis 120 ±8 0.16 ±0.02 1.1 ±0.10 caxis 121 ±8 1.30 ±0.10 1.30 ±0.10 FeMn aaxis 90 ±1 0.76 ±0.04 0.38 ±0.01 caxis 91 ±1 0.82 ±0.03 0.38 ±0.01 also contributes to the damping of the sublattice m1that is equivalent to a torque α/primem1×˙m1exerted on m1. Taking the intersublattice spin pumping into account, we are able to deriveEqs. ( 2) and ( 3) and obtain the damping parameters α n= α0/2 and αm=(α0+2α/prime)/2[49]. Here α0is the intrinsic damping due to SOC for each sublattice. It is worth notingthat the spin pumping strength within a metal is proportionalto its conductivity [ 65–67]. We replot α mas a function of conductivity in the inset of Fig. 2(c), where a general linear dependence is seen for nalong both the aaxis and caxis. We list in Table Ithe calculated ρ,αn, andαmfor typical metallic AFMs including PtMn, IrMn, PdMn, and FeMn. ForIrMn, α mis only 10 times larger than αn, while αmof the other three materials are about 3 orders of magnitude larger thantheirα n. Antiferromagnetic resonance. Keffer and Kittel formulated antiferromagnetic resonance (AFMR) without damping [ 33] and determined the resonant frequencies that depend on theexternal field H ext, exchange field HE, and anisotropy field HA,ωres=γ[Hext±√HA(2HE+HA)]. Here we follow their approach, in which Hextis applied along the easy axis and the transverse components of m1andm2are supposed to be small. Taking both the intrinsic damping due to SOC and spinpumping between the two sublattices into account, we solvethe dynamical equations of AFMR and find the frequency-dependent susceptibility χ(ω) that is defined by n ⊥(ω)= χ(ω)·h⊥(ω). Here n⊥andh⊥are the transverse components of the Néel order and microwave field, respectively. Theimaginary part of the diagonal element of χ(ω) withH ext=20 kOe is plotted in the inset of Fig. 3, where two resonance modes can be identified. The precessional modes for the positive ( ωR) and negative frequency ( ωL) are schematically depicted in Fig. 3. The linewidth of the AFMR /Delta1ωcan be determined from the imaginary part of the (complex) eigenfrequency[68] by solving det |χ −1(ω)|=0 and is plotted in Fig. 3as af u n c t i o no f Hext. Without Hext, the two modes have the same linewidth. A finite external field increases the linewidth ofω Rand decreases that of ωL, both linearly. By including the spin pumping between two sublattices, both the linewidth atH ext=0 and the slope of /Delta1ωas a function of Hextincrease by0 1 02 03 04 0 Hext (kOe)00.040.080.12 (THz)2.0 2.4 (THz)-1.6 -1.2-Im (arb.units)Hext=20 kOe m=nm=103 nm1 m2 m1 m2 1 1 Hext Hext ωL // // ωR ωL ωR FIG. 3. Linewidth of AFMR as a function of the external magnetic field. The black dashed lines and red solid lines are calculated with αm=αnandαm=103αn, respectively. Inset: The imaginary part of susceptibility as a function of the frequency for the external magnetic field Hext=20 kOe and αm=103αn. The cartoons illustrate the corresponding dynamical modes. Here we use HE=103 kOe,HA=5 kOe, and αn=0.001. a factor of about 3.5, which indicates that the spin-pumping effect between the two sublattices plays an important role inthe magnetization dynamics of metallic AFMs. Conclusions. We have generalized the scattering theory of magnetization dissipation in FMs to be applicable for AFMs.Using first-principles scattering calculation, we find the damp-ing parameter accompanying the motion of magnetization ( α m) is generally much larger than that associated with the motion ofthe Néel order ( α n) in the metallic AFMs PtMn, IrMn, PdMn, and FeMn. While αnarises from the spin-orbit interaction, αmis mainly contributed by the spin pumping between the two sublattices in an AFM via exchange interaction. TakingAFMR as an example, we demonstrate that the linewidth can besignificantly enhanced by the giant value of α m. Our findings suggest that the magnetization dynamics of AFMs shall berevisited with the damping effect properly included. Note added in proof . Recently, we became aware of a preprint [ 69], in which the intersublattice spin pumping is also found to play an important role in the spin transport across anAFM|NM or ferrimagnet|NM interface. Acknowledgments. We would like to thank the helpful discussions with X. R. Wang. This work was financiallysupported by the National Key Research and DevelopmentProgram of China (Contract No. 2017YFA0303300) and theNational Natural Science Foundation of China (Grants No.61774018, No. 61704071, No. 61774017, No. 11734004, andNo. 21421003). Z.Y . acknowledges the Recruitment Programof Global Youth Experts. Q.L. and H.Y .Y . contributed equally to this work. [1] T. L. Gilbert, A phenomenological theory of damping in ferromagnetic materials, IEEE Trans. Magn. 40,3443 (2004 ).[2] B. Heinrich and Z. Frait, Temperature dependence of the FMR linewidth of iron single-crystal platelets, Phys. Status Solidi B 16,K11 (1966 ). 061401-4RAPID COMMUNICATIONS MODE-DEPENDENT DAMPING IN METALLIC . . . PHYSICAL REVIEW MATERIALS 1, 061401(R) (2017) [3] S. M. Bhagat and P. Lubitz, Temperature variation of ferromag- netic relaxation in the 3 dtransition metals, Phys. Rev. B 10,179 (1974 ). [4] B. Heinrich, D. J. 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PhysRevLett.119.207202.pdf

Double-Exchange Interaction in Optically Induced Nonequilibrium State: A Conversion from Ferromagnetic to Antiferromagnetic Structure Atsushi Ono and Sumio Ishihara Department of Physics, Tohoku University, Sendai 980-8578, Japan (Received 29 April 2017; revised manuscript received 19 September 2017; published 16 November 2017) The double-exchange (DE) interaction, that is, a ferromagnetic (FM) interaction due to a combination of electron motion and the Hund coupling, is a well-known source of a wide class of FM orders. Here, weshow that the DE interaction in highly photoexcited states is antiferromagnetic (AFM). Transient dynamics of quantum electrons coupled with classical spins are analyzed. An ac field applied to a metallic FM state results in an almost perfect N´ eel state. A time characterizing the FM-to-AFM conversion is scaled by light amplitude and frequency. This hidden AFM interaction is attributable to the electron-spin coupling under nonequilibrium electron distribution. DOI: 10.1103/PhysRevLett.119.207202 Ultrafast optical manipulation of magnetism is widely accepted as a fascinating research topic in modern con- densed matter physics [1–3]from the viewpoints of fundamental physics and technological applications owingto the recent significant progress in optical laser techniques.Beyond the ultrafast demagnetization due to a rapid spin- temperature increase [4], various controls of magnetism, often utilizing photoinduced magnetic phase transition,have been demonstrated as promising strategies in subpico- second time scales [1,5–7]. The most efficient and direct method is by adjusting the magnetic exchange interactionsacting on electron spins by light [8,9]. This subject in highly nonequilibrium states essentially concerns the microscopic nature of electrons, e.g., the band structure, electron correlation, and relaxation processes. Among a number of exchange couplings, the double- exchange (DE) interaction is widely recognized as arepresentative microscopic source of the ferromagnetic(FM) phenomena. The DE interaction was originally proposed by Zener and Anderson-Hasegawa for FM oxides [10–12]. Elemental constituents of the DE interaction are mobile electrons and electron spins localized at lattice sites. The intra-atomic FM interaction, that is, the Hund coupling (J H), connects these two constituents. When the Hund coupling is sufficiently larger than the electron hopping ( t) for the mobile electrons, the spins align ferromagnetically [see Fig. 1(a)], and thus electronic transport strongly correlates with magnetism. This correlation in the DEinteraction has been observed ubiquitously in a wide variety of magnets and magnetic phenomena, such as colossal magnetoresistance [13],f-electron ferromagnet- ism[14], molecular magnets [15], anomalous Hall effect [16], Skyrmion physics [17], and spintronics devices [18]. This electron-spin coupling also provides a promising route to the ultrafast optical manipulation of magnetism owing to the direct connection between the electrons and light. A number of the photoinduced magnetizationchanges have been confirmed experimentally [19–25] and theoretically [26–31]in magnets, in which the DE interaction works in equilibrium states. In most cases, the laser light is applied into a narrow-band insulating phaseassociated with the antiferromagnetic (AFM) order, which is realized through the interactions additional to the original DE system. The experimentally observed formations of ametallic FM state are explained well within a naive extension of the DE interaction to the photoexcited states [28,29] ; kinetic motions of photogenerated carriers align spins ferromagnetically associated with an increase of theelectronic bandwidth. In this Letter, in contrast to a naive extension of the DE interaction picture, we show that the DE interaction in highly optically excited states is AFM [see Fig. 1(b)]. We analyze the minimal model for the DE interaction, con-sisting of classical spins and quantum electrons, in which no explicit AFM interactions are included. Coupled time- dependent equations are solved numerically in finite-size (a) (b) FIG. 1. Illustrations of the DE interaction, calculated spin configurations, and calculated intensity maps of the spin structurefactors in the momentum space in (a) the equilibrium FM state,and (b) the transient photoexcited AFM state. Long and shortbold arrows at left represent localized spins and mobile electrons,respectively. Two-dimensional square lattice is adopted in thecalculations.PRL 119, 207202 (2017) PHYSICAL REVIEW LETTERSweek ending 17 NOVEMBER 2017 0031-9007 =17=119(20) =207202(5) 207202-1 © 2017 American Physical Societyclusters. We introduce the continuous wave (cw) field, in which the frequency is chosen to induce the intraband electronic excitations. It is found that an initial metallic FMstate is converted to an AFM state. A time scale character- izing the FM-to-AFM conversion is controlled by light amplitude and frequency, as well as spin damping. Severaltypes of effective and realistic photoexcitations are pro-posed. The photoinduced AFM state is well demonstrated using a tight-binding model with a nonequilibrium electron distribution. Possible observation methods are proposed. The DE model we analyze describes the itinerant electrons coupled with the localized spins. This is defined as H¼−X hijistijc† iscjs−JHX iss0Si·c† isσss0cis0; ð1Þ where c† is(cis) is the creation (annihilation) operator for an electron at site iwith spin sð¼↑;↓Þ,σare the Pauli matrices, andSiis a localized spin operator with magnitude S. The first term ðHtÞrepresents the electron hopping between the nearest-neighbor sites with the hopping integral tij, and the second term ðHHÞrepresents the Hund coupling with JHð>0Þ. The total numbers of sites and electrons, and the electron density are represented by NL,Ne,a n dn≡Ne=NL, respectively. The time-dependent vector potential AðτÞis introduced as the Peierls phase as tij→te−iAðτÞ·ðri−rjÞwith the positionvector riof site i. The lattice constant, elementary charge, and Planck constant are set to 1, and the Coulomb gauge is adopted. The Hamiltonian in Eq. (1)without AðτÞin equilibrium has been studied well so far [32], and the FM metallic state is realized in a wide parameter range aroundn¼0.5and large J H=tð≳2Þ. No AFM interactions are included explicitly [33]. The ground and transient states are examined numerically in finite-size clusters [29,34] ,i nw h i c h Siare treated as classical spins, justified in the limit of large S. The electron operators ψνðτÞand energies ενðτÞare obtained by diagonal- izing the Hamiltonian, and the electronic wave function is calculated as jΨðτÞi ¼QNe ν¼1ψ† νðτÞj0iwith the vacuum j0i. The field operators at τþδτwith small time interval δτis generated as ψ† νðτþδτÞ¼eiHðτÞδτψ† νðτÞe−iHðτÞδτ. Dynamics of the classical spins are calculated using the Landau- Lifshitz-Gilbert equation, _Si¼heff i×SiþαSi×_Si, where heff iðτÞ¼−hΨðτÞj∂H=∂SijΨðτÞiandαare an effective field and a damping constant, respectively. The two-dimensional square lattice of NL¼L2sites ( L≤16) with the periodic (antiperiodic) boundary condition along the x(y) direction are adopted. The cluster sizes are sufficient to obtain theresults with high reliability as shown in the Supplemental Material [34]. A small randomness is introduced in S iat each site in the initial state, in which the maximum deviation in thepolar angle is δθ¼0.1corresponding to thermal fluctuation at temperature of approximately 0.001t[29,34] . For most of the numerical calculations, we utilize L¼8,n¼0.5,SJ H=t¼4, andSα¼1. We confirmed that the characteristic results shown below are observed in a wide parameter range. For a typical value of t¼0.5eV in the manganese oxides, a time unit of τ¼1=tis approximately 8 fs. First, we introduce the transient dynamics induced by the cw light represented by AðτÞ¼ð A0=ωÞθðτÞsinðωτÞwith frequency ωand amplitude A0[35]. We chose ω=t¼1and A0¼A0ðˆxþˆyÞ, where ˆx(ˆy) is a unit vector along x(y). The detailed A0=ωdependence is shown later. The time profiles of the energies, electronic bands, and spin structure factors SðqÞ¼N−2 LP i;jeiq·ðri−rjÞSi·Sjare presented in Figs. 2(a),2(b), and 2(c), respectively. Figure 2(c)displays the main result; the dominant spin structure is interchanged from FM to AFM states, in which Sðπ;πÞis approximately 90% of its maximum value. Intensity maps of SðqÞat τt¼0, 50, 70, and 300 are shown in Figs. 1(a),2(d),2(e), and1(b), respectively. An animation of the real-space spin dynamics is presented in the Supplemental Material [34]. This FM-to-AFM conversion is clearly in contrast to the photodoping effect in the DE model, in which the enhance-ment of the FM interaction is expected [19,27,29] . The photoinduced dynamics shown in Figs. 2(a)–2(c)is summarized as follows. (i) ( τ<0): Before photoirradia- tion, the metallic FM state is realized because of the DE interaction [see Fig. 1(a)]. The lower and upper bands are identified as the major- and minor-spin bands, respectively.The separations between the band centers and each band width ( W) are 2SJ Hand8t, respectively. The Fermi level is located at the middle of the lower band, indicating a half-metallic ferromagnet [36]. (ii) ( 0≲τt≲30): After turning (a) (b) (c)(d) (e) (f) FIG. 2. Time profiles of the electronic and spin structures induced by the cw light, where A0is parallel to ˆxþˆy. (a)AðτÞ, hHi,hHti, and hHHi, (b) energy levels ( εν), and electron population ( hnνi), and (c) Sð0;0ÞandSðπ;πÞ. (d)–(f) Intensity maps of SðqÞ. We chose τt¼50andA0∥ˆxþˆyin (d), τt¼70and A0∥ˆxþˆyin (e), and τt¼50andA0∥ˆxin (f). Other parameter values are A0=t¼2andω=t¼1.PRL 119, 207202 (2017) PHYSICAL REVIEW LETTERSweek ending 17 NOVEMBER 2017 207202-2on the cw field, hHtistarts oscillating with a frequency of 2ω. The electrons are excited inside the lower band, and the occupied ( hnνi∼1) and unoccupied ( hnνi∼0) levels are intermingled inside the lower band. Changes in the elec-tronic state at an early stage are explained through thedynamical localization (DL) phenomenon, as shown later. (iii) ( 30≲τt≲60): Abrupt reductions of WandSð0;0Þ occur cooperatively, which promote the changes in theelectron distribution inside the lower band further. Theelectrons distribute almost uniformly in the lower band with hn νi∼0.5. The upper band is almost empty, implying that the injected energy is much lower than the upper boundof the energy spectrum. The time when Sð0;0Þsteeply decreases is termed τ F. The transient spin structure depends on the polarization of light [see Fig. 2(f)forA0¼ffiffiffi 2p A0ˆx]. (iv) ( 60≲τt≲150):Sðπ;πÞappears and increases; The time when Sðπ;πÞsteeply increases is termed τAF. A time lag between τFand τAFis explained further later. (v) ( 150≲τt): An AFM steady state is realized, and the gap between the two bands is approximately 2SJH. The spin structure and the intensity map of SðqÞare shown in Fig.1(b). Next, we show the key factors that control the times characterizing the FM-to-AFM conversion. As shown in the detailed αdependence presented in the Supplemental Material, the time scales for the FM-to-AFM conversionincrease with decreasing α, as expected. Here, we show that A 0andωare the additional key parameters controlling the conversion times. The time profiles of W, electron number in the upper band ( Nupper e),Sð0;0Þ, and Sðπ;πÞare presented for several values of A0in Figs. 3(a)–3(d) atfixed ω. The decrease in Sð0;0Þis promoted with increas- ingA0. A steplike feature appears in the time profiles in W atW∼3. The time when Wdecreases steeply and that around the edge of the steplike feature correspond to τFand τAF, respectively [see bold arrows in Figs. 3(a),3(c), and3(d) forA0=t¼1.55]. At around τF, electrons are excited from the lower to upper bands by the excess energydue to the FM order destruction, as indicated in Fig. 3(b). Then, the electrons relax to the lower band associated withdevelopment of Sðπ;πÞat around τ AF. The electron excitation and relaxation between the lower and upperbands are attributed to the Hund coupling. Because of theseintricate interband excitation and relaxation processes, τ AF does not show monotonic dependence on A0. On the other hand, τFis well scaled by A0=ω, as shown in Fig. 3(e); data sets can be fitted by function ðA0=ω−cÞγwith numerical constants cð∼1.1–1.3) and γð∼−1Þ. A finite cimplies that the threshold values of A0=ωexist for the FM-to-AFM conversion. Here, we briefly point out that the transient dynamics just after turning on the cw light are understood in the generalizedDL phenomenon, which was originally proposed in thenoninteracting system under the cw field [37–39].T h e averaged kinetic energy in the early part of the time domain(ii) is plotted as functions of A 0=ωin Fig. 3(f) [40] . We define K≡ðΔTÞ−1R ΔTdτhHtiwith the time interval ΔTand the kinetic energy before irradiation K0. The calculated data sets are scaled by a universal curve, and can be fitted by thezeroth-order Bessel function J 0ðA0=ωÞpredicted by the DL theory. Deviation of the numerical data from J0ðA0=ωÞis seen in A0=ω≳1.25. This is attributable to the spin structure change which is beyond the DL scope. After the early part ofthe time domain (ii), corresponding to τ≳10=tin Fig. 2(b), fitting of the numerical data by J 0ðA0=ωÞdoes not work, because the spin structure starts changing. The photoinduced FM-to-AFM conversion occurs not only by the cw light, but also by various realistic methodsof light irradiation. Instead of the cw field, we introduce asudden quench of the vector potential simply modeled asAðτÞ¼A 1θðτÞ, which is equivalent to the electric field pulse EðτÞ¼−A1δðτÞ. This asymmetric pulse causes a non- adiabatic momentum shift of electrons by δk¼RdτEðτÞ, which induces the population inversion [41]. The popula- tion inversions induced by light have been studied in avariety of interacting electron systems [42–44]. The time profiles of the electronic energy bands, electron population, andSðqÞare presented in Figs. 4(a)and4(c), in which we chose δk¼ðπ;πÞ[35]. Immediately after pulse irradiation, the population inversion is realized inside the lower band asexpected, and WandSð0;0Þare reduced. Then, the electrons distribute almost uniformly in the narrow lowerband, and Sðπ;πÞemerges at τt∼50. Finally, the metallic FM state is recovered, and the electrons thermalize.Another type of effective light irradiation is a combina-tion of a pulse field and a delayed cw field modeled as(a) (e) (f)(b) (c) (d) FIG. 3. (a) –(d) Time profiles of the bandwidth, electron number density in the upper band, Sð0;0Þ, and Sðπ;πÞinduced by cw lights for several values of A0. We chose ω=t¼1. (e) τFplotted as functions of A0=ωfor several sets of ðSα;δθÞ. The bold lines represent the function ðA0=ω−cÞγ. (f) The normalized kinetic energy ( K=K 0) averaged between τt¼400–500(see text) plotted as functions of A0=ω. The bold line represents the zeroth-order Bessel function J0ðA0=ωÞ.PRL 119, 207202 (2017) PHYSICAL REVIEW LETTERSweek ending 17 NOVEMBER 2017 207202-3EðτÞ¼−∂τAðτÞ¼−A1δðτÞ−A0cos½ωðτ−τ0Þ/C138θðτ−τ0Þ with delay time τ0. As shown in Figs. 4(b) and4(d), the pulse field generates population inversion inside the lower band, and the subsequent cw field maintains the AFM state. In contrast, in the case without the subsequent cw field (A0¼0),Sðπ;πÞdisappears gradually [a dotted line in Fig. 4(d)]. An advantage in this pulse-cw combination is that a 1 order weaker A0is required to maintain the AFM state than the A0value in the case where the cw field is only introduced (see Fig. 2). The spin conversion by use of the pulse field might be more realistic rather than the cw light. Now, we focus on the photoinduced AFM steady state. Instead of a rigorous analysis of this nonequilibrium state inthe open many-body system, which is beyond the scope of the present work, we evaluate the energies in the idealized FM and AFM states under a hypothetic electron distribu- tion. The transient electronic density of states (DOS) and the electron population in the FM state ( τ¼0) and photoinduced AFM state ( τ¼300=t) are shown in Figs. 5(a)and5(b), respectively, in which the cw field is applied. In contrast to the equilibrium FM state, where the electrons occupy from the bottom to the Fermi level, theelectrons in the AFM state distribute almost uniformly, as suggested previously. Thus, we introduce the idealized FM and AFM orders in Eq. (1), and the uniform electron distribution in the lower band, that is, hn νi¼n(hnνi¼0) for level νbelonging to the lower (upper) band. The total energies in the FM ( EF) and AFM ( EAF) evaluated in the thermodynamic limit of a one-dimensional chain, two-dimensional square lattice, and three-dimensional cubic lattice are shown in Figs. 5(c) and5(d). The AFM state gives low energy throughout the parameter region of J H andnin the three lattice types, implying that the non- equilibrium electron distribution plays a major role on the transient AFM state. This is attributable to the fact that both the difference between the band centers in the FM state and the energy gap in the AFM state are approximately 2SJH [see dotted lines in Figs. 5(a)and5(b)].Experimental confirmations are indispensable for estab- lishing the present proposal. Perovskite manganites La1−xSrxMnO 3(x∼0.3) and layered manganites are the possible target materials for the metallic ferromagnets because of the DE interaction. Rather than the cw light, the use of pulse field might be realistic for the spin conversion in the present laser performance [35]. A uni- form electron distribution is not required inside the wide electronic band in the initial FM state, because a dynamical cooperation between the band narrowing and FM-to-AFM conversion promotes uniform electron distribution. The observation of the AFM Bragg peak through the magnetic x-ray diffraction is a direct method for observing the transient AFM state. The disappearance of the magneto- optical Kerr signal and appearance of the two-magnon Raman scattering confirm the vanishing of the FM order and the emergence of the AFM order, respectively. The angle-resolved photoemission spectroscopy technique will be able to succeed in acquiring the expected band narrow- ing, electron population change, and band folding due to emergence of the AFM state. The authors would like to thank S. Iwai, M. Naka, H. Nakao, T. Arima, and A. Fujimori for fruitful discussions. This work was supported by MEXT KAKENHI, Grants No. 26287070, No. 15H02100, and No. 17H02916. Some of the numerical calculations were performed using the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo. (a) (c)(b) (d) FIG. 4. Time profiles of the energy levels ( εν), electron distributions ( hnνi),Sð0;0Þ, and Sðπ;πÞwith (a),(c) the pulse electric field and (b),(d) the combination of pulse and cw fields(see text). A dotted line in (d) represents Sðπ;πÞwithout A 1. Here, A1¼πðˆxþˆyÞandSα¼1in (a),(c); A1¼πðˆxþˆyÞ, A0=t¼0.3ðˆxþˆyÞ,τ0t¼50, and Sα¼0.1in (b),(d); and ω=t¼1in (a) –(d).(a) (b) (c) (d) FIG. 5. (a), (b) DOS at τt¼0(FM state), and at τt¼300 (AFM state) when the cw field is introduced. Shadedareas represent the electron distribution. Other parametervalues are A 0=t¼2andω=t¼1. Dotted lines represent DOS calculated from Eq. (1)where the idealized FM or AFM structures are introduced. (c),(d) Energy differences betweenthe FM and AFM structures. We chose n¼0.5in (c), and SJ H=t¼8in (d). Broken, dashed, and bold lines represent the one-dimensional chain, square lattice, and cubic lattice,respectively.PRL 119, 207202 (2017) PHYSICAL REVIEW LETTERSweek ending 17 NOVEMBER 2017 207202-4[1] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010) . [2] Y. Tokura, J. Phys. Soc. Jpn. 75, 011001 (2006) . [3] H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and P. Werner, Rev. Mod. 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PhysRevB.92.165113.pdf

PHYSICAL REVIEW B 92, 165113 (2015) Probing excitations in insulators via injection of spin currents Shubhayu Chatterjee1and Subir Sachdev1,2 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Received 2 July 2015; published 12 October 2015) We propose a spin transport experiment to measure the low-energy excitations in insulators with spin degrees of freedom, with a focus on detecting ground states that lack magnetic order. A general formalism to computethe spin current from a metal with a nonequilibrium distribution of spins to an insulator is developed. It is appliedto insulating states with and without long range magnetic order, and salient features in the spin conductance arenoted. DOI: 10.1103/PhysRevB.92.165113 PACS number(s): 75 .10.Kt,72.25.Mk,72.25.Pn I. INTRODUCTION Observation of fractionalized excitations in insulating spin systems has been a long-sought goal in physics. Such quantumspin liquid states, if realized in nature, would be a new quantumphase of matter with exotic properties. Certain candidatematerials have strong experimental evidence for exhibitingspin liquid ground states. For example, thermal conductivityexperiments on insulating frustrated triangular lattice organicsalts by M. Yamashita et al. [1] indicate the presence of mobile gapless excitations. Inelastic neutron scattering experimentson single crystals of Herbertsmithite, a kagome lattice spin-half Heisenberg antiferromagnet by Han et al. [2] provide evi- dence for the presence of a continuum of fractionalized spinonexcitations. Numerical studies on the triangular [ 3,4] and kagome [ 5] lattice Heisenberg models also indicate the possi- bility of spin liquid ground states in certain parameter regimes. In spite of promising evidence for observation of spin liquids from several experiments [ 1,2,6,7], the exact nature of experimentally realized ground states, and in particular,the presence of a spin gap is still unclear. In this paper,we propose a transport experiment which can probe themobile spin-carrying excitations of the system at low energies;these experiments are similar in spirit to those discussedrecently by Takei et al. [8–10] and collaborators [ 11–13]f o r materials with magnetic order. Recent advances in spintronics[14,15] have made it possible to create a spin accumulation at boundaries of metals via the spin Hall effect. We proposeto use this nonequilibrium accumulation of spins to injecta spin current into an insulating state with spin degrees offreedom. We assume that the spin-orbit interaction within theantiferromagnet itself is small, and so the injected spin currentwill be equal to the spin current emerging from the otheredge of the antiferromagnet. The spin current is a functionof the spin-accumulation voltage in the metal. Therefore, bymeasuring the spin current as a function of this voltage, andlooking at thresholds and exponents, we can comment on thepresence of spin gaps and the low-energy dispersion of thefractionalized spin-half excitations. The rest of the paper is organized as follows. In Sec. II,w e describe the geometry of our setup, and develop a formalismto evaluate the spin current injected into a magnetic insulatorfrom a metal. In Sec. III, we apply the formalism to evaluate the spin current into an antiferromagnet with collinear N ´eel order. In Sec. IV, we first analytically calculate for the spincurrent into insulating states with no long-range magnetic order, including both valence bond solid states and spin liquidstates. Then we go beyond the analytical approximations, andnumerically identify some broad features in the spin conduc-tance for a spin liquid ground state [ 16] on the kagome lattice, which is a candidate state for Herbertsmithite [ 2,17]. Details of relevant calculations are contained in the appendices. II. FORMALISM TO EV ALUATE SPIN CURRENT A. Generation and detection of spin current We begin with a brief discussion of the spin Hall effects, which we shall use to generate and detect spin currents, andthen describe the exact geometry of spin injector and detectorwe use. A charge current passed through a paramagneticmaterial can drive a transverse spin current in the presence ofstrong intrinsic spin-orbit coupling or skew scattering by spin-orbit coupled disorder [ 18–21]. The spin current impinging on the boundary is given by J S=/planckover2pi1 2eθSHJC, where JCis the charge current density and θSHis the spin Hall angle, and sets up a spin accumulation at the boundary, that has been measuredin experiments for both metals [ 22,23] and semiconductors [24–26]. The reciprocal process, where injecting a spin current into a spin-orbit coupled paramagnetic material sets up acharge current (or voltage) transverse to the spin current—theinverse spin Hall effect—has also been observed [ 22,26,27]. Furthermore, both processes have been used simultaneouslyto transmit electrical signals across a magnetic insulator[23]. Theoretical predictions for the spin superfluid transport through a ferromagnetic [ 8] and antiferromagnetic [ 9] insulator sandwiched between two metallic reservoirs have been workedout in the linear response regime. Taking phenomenologicalGilbert damping into account, the spin current density J r S pumped into the right reservoir as a function of the spin accumulation voltage Vis given by [ 8,9] Jr S=V 4πg↑↓ lg↑↓ r g↑↓ l+g↑↓ r+gα, (1) where g↑↓ l(r)is the spin flip conductance at the left (right) interface, and gαquantifies the loss in spin current due to Gilbert damping. Let us consider an analogous geometry, where an insulating block with spin degrees of freedom is placed in between twometallic reservoirs, as shown in Fig. 1. A charge current in 1098-0121/2015/92(16)/165113(12) 165113-1 ©2015 American Physical SocietySHUBHAYU CHATTERJEE AND SUBIR SACHDEV PHYSICAL REVIEW B 92, 165113 (2015) FIG. 1. (Color online) Geometry for generation and detection of spin current. (a) Spin accumulation via the spin Hall effect, and injection at the left interface. (b) Spin current detection via the inverse spin Hall effect in the right metallic reservoir. the left metallic reservoir, in the presence of strong spin-orbit coupling, will create a nonequilibrium accumulation of spinat the metal-insulator boundary. We assume that there are nothermal gradients, and that the spin accumulation can be wellmodeled by different chemical potentials μ ↑andμ↓in the Fermi Dirac distribution at temperature Tfor the spin-up and spin-down electrons. The left metal reservoir will subsequentlyrelax by sending a spin current into the spin insulator. Weassume negligible loss of spin current inside the insulator,so that the spin current sets up a spin accumulation at theinsulator-metal boundary on the right. If the metallic reservoiron the right was initially in thermal equilibrium at T,t h e accumulated spin density at the boundary will drive a chargecurrent via the inverse spin Hall effect. This charge current, orthe associated voltage can be detected, and therefore we canfind the spin current by measuring charge currents (or voltages)in both metallic reservoirs. B. General expression for spin current Let us choose xas the longitudinal direction which is normal to the interfaces, and zas the spin-quantization axis. We shall evaluate the spin current (corresponding toangular momentum in the zdirection) crossing the left metal-insulator interface when V=μ ↑−μ↓>0. To make analytical progress, we assume a clean interface between themetal and the insulator, with translational invariance in theplane of the interface. The metallic reservoir is assumed tobe a Fermi liquid with quadratic dispersion and Fermi energy /epsilon1 F, so that nσ(/epsilon1)=(eβ(/epsilon1/vectork−μσ)+1)−1with/epsilon1/vectork=/vectork2 2m(setting /planckover2pi1=1). We shall always work in the regime where T,V/lessmuch/epsilon1F, and henceforth set μ↑=μ, so that μ↓=μ−V, to simplify notations. We assume that the electron spin /vectorSein the metal interacts with the boundary spins of the insulator, located at interface lattice sites /vectorXj, via a local spin-rotation symmetric local Hamiltonian Hint=J/summationdisplay j/vectorSe·/vectorSjδ(/vectorxe−/vectorXj). (2) Let the insulator have exact eigenstates {|n/angbracketright}; then its initial state is described by the equilibrium density matrix/summationtext ne−βEn Z|n/angbracketright/angbracketleftn|. For the metal, periodic boundary conditions in a large box of volume V=LxA⊥is assumed, where A⊥is the interface area. We now use Fermi’s golden rule to calculatethe rate of scattering of a right-moving electron state |/vectork1,↑/angbracketright to a left-moving electron state |/vectork2,↓/angbracketright. The matrix element for scattering to a final state |m/angbracketrightof the insulator is given by /angbracketleft/vectork2,↓;m|Hint|/vectork1,↑;n/angbracketright =J 2V/summationdisplay jei/vectorq·/vectorXj/angbracketleftm|S+ j|n/angbracketright,defining /vectorq=/vectork1−/vectork2.(3) Defining ω(/vectork1,/vectork2)=/epsilon1/vectork1,↑−/epsilon1/vectork2,↓as the energy transfer, the rate of scattering Ris R=2π/summationdisplay m,n1 Ze−βEn|/angbracketleft/vectork1,↑;n|Hint|/vectork2,↓;m/angbracketright|2 ×δ/parenleftbig En+/epsilon1/vectork1,↑−Em−/epsilon1/vectork2,↓/parenrightbig =πJ2 2L2xA⊥S−+/parenleftBigg /vectorq⊥,ω=2/vectork1·/vectorq−/vectorq2 2m/parenrightBigg , (4) where S−+(/vectorq⊥,ω) is the dynamic spin structure factor of the insulator at the interface, defined as S−+(/vectorq⊥,ω)=1 A⊥/summationdisplay l,je−i/vectorq⊥·(/vectorXl−/vectorXj) ×/integraldisplay∞ −∞dt eiωt/angbracketleftS− l(t)S+ j(0)/angbracketrightthermal.(5) The spin current crossing the boundary for this scattering event isqx m.I fw eh a v e Rsuch events per unit time, then the net spin current crossing the boundary is justqxR m. Summing over all initial electron and final states consistent with phase spaceconstraints, the current I spin,↑due to up-spin electrons getting reflected to down-spin ones is Ispin,↑=πJ2A⊥ 2m/integraldisplay k1x>0ddk1 (2π)d/integraldisplay qx>k1xddq (2π)dn↑/parenleftbig /epsilon1/vectork1/parenrightbig ×/bracketleftbig 1−n↓/parenleftbig /epsilon1/vectork1−/vectorq/parenrightbig/bracketrightbig qxS−+/parenleftBigg /vectorq⊥,ω=2/vectork·/vectorq−/vectorq2 2m/parenrightBigg .w (6) At nonzero T, the reverse process where spin-down electrons get reflected to spin-up ones contribute analogously a spin 165113-2PROBING EXCITATIONS IN INSULATORS VIA . . . PHYSICAL REVIEW B 92, 165113 (2015) FIG. 2. (Color online) Allowed phase space for scattering of an electron with given initial momentum. current Ispin,↓given by Ispin,↓=πJ2A⊥ 2m/integraldisplay k1x>0ddk1 (2π)d/integraldisplay qx>k1xddq (2π)dn↓/parenleftbig /epsilon1/vectork1/parenrightbig ×/bracketleftbig 1−n↑/parenleftbig /epsilon1/vectork1−/vectorq/parenrightbig/bracketrightbig qxS+−/parenleftBigg /vectorq⊥,ω=2/vectork·/vectorq−/vectorq2 2m/parenrightBigg . (7) The net spin current is therefore given by the difference of the two contributions listed above: Ispin=Ispin,↑−Ispin,↓. (8) C. Simplifications for certain physically relevant structure factors The expression for the spin current can be considerably simplified once we note that at T→0, scattering is essentially restricted within an energy window of V.F o r ω/lessorsimilarV,w e assume that the dynamic structure factor S+−(/vectorq⊥,ω) assumes large values only for small |/vectorq⊥|. This is physically relevant for several systems where excitations at large momenta typicallyhave large energy cost. As Fig. 2shows, if the system does not have excitations at ω/lessorsimilarVfor|/vectorq ⊥|/greaterorsimilar/Lambda1, then scattering is restricted within a patch of dimensionsV vF×/Lambda1d−1,vFbeing the Fermi velocity. To exploit this, we approximate the initial momentum /vectork1≈kFˆn, and linearize the energy transfer ωabout the point of elastic scattering as follows: /vectorq=2kF(ˆn·ˆx)−δqxˆx−/vectorq⊥, (9) ω(/vectork1,/vectorq)=vF[(ˆn·ˆx)δqx−ˆn·/vectorq⊥]+O/parenleftbig δq2 x,q2 ⊥/parenrightbig . We also assume that the electronic density of states ν(/epsilon1F)i s approximately a constant near the Fermi surface for δqx,q⊥/lessmuch kF. Leaving the details of calculation to Appendix A, these simplifications lead to the following form of the spin currentfor spin-up electrons flipping to spin-down ones. Ispin,↑=πJ2A⊥ν(/epsilon1F) 2/integraldisplaydω 2πdd−1q⊥ (2π)d−1 ×(V−ω) 1−e−β(V−ω)S−+(/vectorq⊥,ω). (10) Analogous manipulations for the reverse process lead to Ispin,↓=πJ2A⊥ν(/epsilon1F) 2/integraldisplaydω 2πdd−1q⊥ (2π)d−1 ×(V+ω) eβ(V+ω)−1S+−(/vectorq⊥,ω). (11) These expressions make it transparent that as T→0, only up- spin electrons flipping to down-spin ones contribute the energywindow (0 ,V). The reverse process is always exponentially suppressed as there must be an energy gain of at least Vfor a down-spin electron to flip to an up-spin one due to phase spaceconstraints. The net spin current is, as described in Eq. ( 8), the difference of the above two currents. This formalism can be extended to cases where the quasiparticle excitation energy has minima at large transverse momenta {/vectorQ ⊥}(with magnitude of a−1where ais the micro- scopic lattice length scale), provided the different /vectorQ⊥are well separated from each other. This is typically true for systemswith quasiparticle bands, as the momenta difference betweenthe band minima are of the order of a −1. For example, cubic lattice antiferromagnets with a two-dimensional boundaryhave spin-wave excitations about the ordering wave vector /vectorQ AF ⊥=π a(0,1,1). Referring the reader to Appendix Aagain for the details, here we just state the main result. The effect of inelastic scattering about large transverse momenta /vectorQ⊥is to scale the spin current by an overall O(1) angular factor fang(kF/Q⊥), so that Eq. ( 10)f o rIspin,↑is now modified to Ispin,↑=πJ2A⊥ν(/epsilon1F) 2/summationdisplay /vectorQ⊥fang(kF/Q⊥) ×/integraldisplaydω 2πdd−1q⊥ (2π)d−1(V−ω) 1−e−β(V−ω)S−+(/vectorq⊥,ω),(12) where the angular factor, coming from kinematical constraints, is given by fang(kF/Q⊥) =/integraldisplay ˆn·ˆx/greaterorequalslant0 k2 F(ˆn·ˆx)2+2kF(/vectorQ⊥·ˆn)/greaterorequalslantQ2 ⊥d/Omega1 Sd−1 ×/parenleftBigg 1+kF(ˆn·ˆx) /parenleftbig k2 F(ˆn·ˆx)2+2kF(/vectorQ⊥·ˆn)−Q2 ⊥/parenrightbig1/2/parenrightBigg .(13) In Eq. ( 13),Sd−1is the sphere in Rd, and one can check that for Q⊥=0 the angular factor reduces to unity, as desired. One can also check the limit Q⊥/greatermuchkF, in which case scattering of the electron by /vectorq⊥≈/vectorQ⊥is excluded by phase space constraints andfang(kF/Q⊥)→0. Equation ( 11) also undergoes similar modifications, and putting these together we obtain our main 165113-3SHUBHAYU CHATTERJEE AND SUBIR SACHDEV PHYSICAL REVIEW B 92, 165113 (2015) result of this section: Ispin=πJ2A⊥ν(/epsilon1F) 2/summationdisplay /vectorQ⊥fang(kF/Q⊥) ×/integraldisplaydω 2πdd−1q⊥ (2π)d−1/bracketleftbigg(V−ω) 1−e−β(V−ω)S−+(/vectorq⊥,ω) −(V+ω) eβ(V+ω)−1S+−(/vectorq⊥,ω)/bracketrightbigg . (14) We once again carefully note that this formalism for extension of the spin current calculation to a set of different{/vectorQ ⊥}works only when the different points are well isolated in the Brillouin zone of spin-carrying excitations of the insulator.Physically, this implies that the different momentum patches(to which the electron is scattered) do not overlap with eachother. If they start to overlap, then we would count the samefinal electron state multiple times and overestimate the spincurrent. III. SPIN CURRENT FOR ORDERED ANTIFERROMAGNETS In this section, we apply the formalism developed in Sec. IIto calculate the spin current from the metallic reservoir to an ordered collinear antiferromagnet, deep in the N ´eel phase. We assume d=3, so that a symmetry-broken state can occur at T> 0. The results can also be generalized tod=2a tT=0. In the following sections, we illustrate evaluation of the current with the simplest scenario—a cubiclattice antiferromagnet with ordering wave vector /vectorQ AF= π a(1,1,1), so that /vectorQAF ⊥=π a(0,1,1). We split our analysis into two sections, corresponding to the N ´eel order pointing perpendicular and parallel to the spin-quantization axis in themetal, and add up the contributions due to elastic reflectionfrom the static magnetic moments, and the inelastic reflectiondue to spin-wave excitations, to find the net spin current. A. N ´eel order perpendicular to spin quantization axis in the metal 1. Elastic contribution In order to contribute the elastic spin-flip scattering from the metal-antiferromagnet interface, we replace the fluctuatingspin operators at the boundary by static moments, resemblingthe classical ground state. For N ´eel order along ˆy, which is normal to the spin-quantization axis ˆzin the metal reservoir, we can write the Hamiltonian as H int=J/summationdisplay j/vectorSe·/vectorSjδ(/vectorx−/vectorXj) →JS/summationdisplay jSye−i/vectorQ⊥·/vectorXjδ(/vectorx−/vectorXj). (15) We use Fermi’s golden rule again to find the rate of scattering of spin-flip scattering of electrons at the interface: R=2π|/angbracketleft/vectork2,↓|Hint|/vectork1,↑/angbracketright|2δ/parenleftbig /epsilon1/vectork1−/epsilon1/vectork2/parenrightbig =πJ2 4L2xδ/vectorq⊥,/vectorQ⊥δ/parenleftbig /epsilon1/vectork1−/epsilon1/vectork1−/vectorq/parenrightbig . (16)Following an analogous procedure of finding the spin current due to this scattering event, and summing over all initial andfinal states consistent with phase space restrictions, we arriveat the following expression for the elastic contribution I spinin terms of fang(kF/QAF ⊥): Iel spin,↑=fang(kF/QAF ⊥)πJ2A⊥ 4ν(/epsilon1F)V 1−e−βV, (17) Iel spin,↓=fang(kF/QAF ⊥)πJ2A⊥ 4ν(/epsilon1F)V eβV−1, (18) Iel spin=Iel spin,↑−Iel spin,↓=fang(kF/QAF ⊥)πJ2A⊥ν(/epsilon1F) 4V. (19) Note that the elastic contribution to the current is proportional to the number of propagating modes at the Fermi surface, givenbyν(/epsilon1 F)V. So this contribution is similar to what one would obtain by using the Landauer formalism, as had been done foran analogous geometry by Takei et al. [9]. 2. Inelastic contribution The inelastic contribution can be directly evaluated by application of Eq. ( 14), as the ordered antiferromagnet deep in the N ´eel phase has spin-wave excitations that have minimum energy about /vectorQ⊥=0 and /vectorQ⊥=/vectorQAF ⊥, which are well sepa- rated in the insulator Brillouin zone. We work in the T→0 limit, which implies that the insulator is initially in its groundstate. Therefore ω/greaterorequalslant0 in the dynamic structure factors, and we can drop the contribution from I inel spin,↓to the spin current. We use the Holstein-Primakoff transformation to diago- nalize the Hamiltonian and evaluate S−+(/vectorq⊥,ω). Leaving the details to Appendix B, the dynamic structure factor in the small |/vectorq⊥|andT→0 limit is given by (for ω> 0, setting a=1) S−+(/vectorq⊥,ω)=πq⊥ 8√ 2δ(ω−vsq⊥), (20) where vsis the speed of spin waves in the antiferromagnet. We can plug this back into Eq. ( 14), and we obtain the inelastic contribution to be Iinel spinT→0=πJ2A⊥ν(/epsilon1F) 2[1+fang(kF/QAF ⊥)]V4 384√ 2πv3s. (21) We now add up the contributions from Eqs. ( 19) and ( 21)t o find the net spin current when the N ´eel order is perpendicular to the spin-quantization axis in the metal. IspinT→0=πJ2A⊥ν(/epsilon1F) 4/bracketleftbigg fang(kF/QAF ⊥)V +[1+fang(kF/QAF ⊥)]V4 192√ 2πv3s/bracketrightbigg .(22) B. N ´eel order parallel to spin quantization axis in the metal 1. Elastic contribution For N ´eel order along ˆz, which is normal to the spin- quantization axis ˆzin the metal reservoir, we can write the 165113-4PROBING EXCITATIONS IN INSULATORS VIA . . . PHYSICAL REVIEW B 92, 165113 (2015) Hamiltonian as Hint=J/summationdisplay j/vectorSe·/vectorSjδ(/vectorx−/vectorXj) →JS/summationdisplay jSze−i/vectorQ⊥·/vectorXjδ(/vectorx−/vectorXj). (23) In this case, the Hamiltonian Hintcommutes with the z component of the electron spin, and therefore cannot flip it.Therefore there is no elastic contribution to the spin current. 2. Inelastic contribution For the inelastic contribution, we again use the T→0 limit of Eq. ( 14). The dynamic structure factor is evaluated in an analogous manner to the previous Sec. III A 2 , and is essentially identical to Eq. ( 20) barring a constant extra prefactor. We find that the net spin current when the N ´eel vector is along the spin-quantization axis is given by IspinT→0=Iinel spinT→0=πJ2A⊥ν(/epsilon1F) 2 ×[1+fang(kF/QAF ⊥)]V4 96√ 2πv3s. (24) IV . SPIN CURRENT FOR SYSTEMS WITH NO MAGNETIC ORDER In this section, we shall apply the formalism from Sec. II to evaluate the spin current into states with no long-range magnetic order. Some candidate phases for Mott insulators with unbroken spin-rotation symmetry are described by spin-half quasiparticles or spinons, coupled to an emergent gaugefield. In the deconfined phase of the gauge field, the latticesymmetry is unbroken and the ground state is a spin liquid[16]. The spinons can propagate as independent quasiparticles and carry a spin current. In the confined phase, the groundstate might spontaneously break translation symmetry of thelattice, resulting in a valence bond solid (VBS) state [ 28] with short-range order. In this case, the low-lying excitations withnonzero spin are spin triplets or triplons, which are gappedexcitations that carry the spin current.A. VBS states with triplon excitations At low energies, the structure factor will be dominated by single triplon excitations. Let us assume that the triplon has agap/Delta1 Tand a quadratic dispersion, so the dynamic structure factor can be approximated by S−+(/vectorq⊥,ω)≈Cδ(ω−/Delta1T−γ/vectorq2 ⊥), (25) Here we also assume that the prefactor Cis independent of ωand/vectorq⊥. Now we again use the T→0 limit of Eq. ( 14)t o compute the spin current. For a d-dimensional system with a (d−1)-dimensional boundary, we find that the spin current is given by IspinT→0=πJ2A⊥CSd−1γ1−d/2ν(/epsilon1F) (2π)dd(d+1) ×(V−/Delta1T)d/2+1/Theta1(V−/Delta1T). (26) As expected, there is a threshold at V=/Delta1T, as energy conservation implies that no triplons can be excited when V is less than the triplon gap. Above the cutoff, the spin currenthas a power law behavior with voltage with an exponent thatdepends on the dimensionality dof the system. For instance, ind=3, the exponent is 5 2. B. Spin liquids with spinon excitations We first approach the problem analytically by using a low- energy effective theory to calculate the two-spinon structurefactor. We use a mean-field approach where the spinons are freequasiparticles in the system, and have negligible coupling toother excitations which do not carry spin (such as visons, whichare vortices of the emergent gauge field). For a given spinondispersion /epsilon1 /vectork, the free-spinon Green’s function in imaginary time is given by Gs(/vectork,iω n)=1 iωn−/epsilon1/vectork, (27) where ωnis a Matsubara frequency which is determined by bosonic or fermionic statistics of the spinons. We can calculatethe structure factor from the dynamic susceptibility χ −+,g i v e n by χ−+(/vectorq⊥,iωn)=−1 βV/summationdisplay /vectork,i/Omega1 nGs(−/vectork,−i/Omega1n)Gs(/vectork+/vectorq⊥,i/Omega1n+iωn) =/integraldisplayd2k (2π)2/parenleftBigg 1+nB(/epsilon1/vectork)+nB(/epsilon1/vectork+/vectorq⊥) −iωn+/epsilon1/vectork+/epsilon1/vectork+/vectorq⊥/parenrightBigg (for bosonic spinons) T→0→/integraldisplayd2k (2π)21 (−iωn+/epsilon1/vectork+/epsilon1/vectork+/vectorq⊥), (28) which, in turn, leads to the following result for the zero-temperature limit of the dynamic structure factor: S−+(/vectorq⊥,ω)=1 1−e−βωIm[χ−+(/vectorq⊥,iωn→ω+iη)] T=0,ω>0→ lim T→0Im[χ−+(/vectorq⊥,iωn→ω+iη)]=π/integraldisplayd2k (2π)2δ(ω−/epsilon1/vectork−/epsilon1/vectork+/vectorq⊥). (29) 165113-5SHUBHAYU CHATTERJEE AND SUBIR SACHDEV PHYSICAL REVIEW B 92, 165113 (2015) Intuitively, this follows from the fact that spinons are always excited in pairs and they share the momentum transferredfrom the electron at the interface. At T=0, the spin liquid is initially in its ground state, so we only have contributionsfrom two spin-up spinons that have center-of-mass momentum /vectorq ⊥. Equation ( 29) is the main result of this section, which we shall use to find the forms of the spin current for certainspin liquids with free-spinon bands in the mean-field picture,and then figure out how the spin current scales with thespin-accumulation voltage Vfor arbitrary spinon dispersions and dimensionality of the system. 1. Gapped spinons with quadratic bands Let us consider the case of gapped spin liquids in two dimensions with a spinon gap /Delta1s, where the lowest spinon band has a quadratic dispersion about a minima at /vectork=/vectorQ⊥with an effective mass of m∗, so that the spinon Green’s function is given by Gs(/vectork,iω n)=1 iωn−/Delta1s−(/vectork−/vectorQ⊥)2 2m∗. (30) This is true for several ansatz spin liquid ground states [ 29,30], including, for instance, the Q1=Q2state of the Z2spin liquid state on the kagome lattice [ 16], where the gap and the effective mass are given in terms of the mean-field parameters λandQ, and the antiferromagnetic coupling between nearest neighborsJ AFby /Delta1s=/radicalBig λ2−3J2 AFQ2,and1 m∗=3J2 AFQ2 2/Delta1s.(31) Equation ( 29) now leads to the following expression for the structure factor: S−+(/vectorq⊥,ω)=/integraldisplayd2k (2π)2δ/parenleftBigg ω−2/Delta1s−(/vectork−/vectorQ⊥)2 2m∗ −(/vectork+/vectorq⊥−/vectorQ⊥)2 2m∗/parenrightBigg =m∗ 4/Theta1/parenleftbigg ω−2/Delta1s−/vectorq2 ⊥ 4m∗/parenrightbigg . (32) In general, we may have several spinon bands with minima at different /vectorQ⊥with the same gap /Delta1s, so we sum over all of them to find the net spin current via Eq. ( 14)i nt h e T→0 limit. Ispin=πJ2A⊥ν(/epsilon1F) 2/summationdisplay /vectorQ⊥fang(kF/Q⊥) ×/integraldisplaydω 2πdd−1q⊥ (2π)d−1(V−ω)S−+(/vectorq⊥,ω) =ηJ2A⊥ν(/epsilon1F)(m∗)2 48π2⎛ ⎝/summationdisplay /vectorQ⊥fang(kF/Q⊥)⎞ ⎠ ×(V−2/Delta1s)3/Theta1(V−2/Delta1s) =η2(V−2/Delta1s)3/Theta1(V−2/Delta1s), (33) where we have absorbed all constant prefactors in η2to explicitly show the dependence on V. As expected, thereis a cutoff at twice the spinon gap, i.e., no spin current for V/lessorequalslant2/Delta1s, and a power law behavior above the threshold. Note that in the calculation above, we assume that both spinons come from bands that have minima at identical /vectorQ⊥. However, even if they come from different bands, say one with minima at /vectorQ⊥,1and the other with /vectorQ⊥,2, they will just contribute to add extra prefactors of fang[kF/(|/vectorQ⊥,1+/vectorQ⊥,2|)] in the expression for the spin current, but would not changeeither the threshold or the power law behavior. However, ifthe bands have different spinon gaps, say /Delta1 s,1and/Delta1s,2, then we expect the spin current to show a second threshold whenthe spin-accumulation voltage Vcrosses /Delta1 s,1+/Delta1s,2,a st h e scattering process then now excites spinons from both bands. 2. Gapless spinons at Dirac points Let us consider spin liquids described by gapless fermionic spinons at discrete Dirac points {/vectorQ⊥}in the Brillouin zone. The spinon dispersion is then given in terms of the spinon velocity vat a Dirac point at /vectorQ⊥by Gs(/vectork,iω n)=1 iωn−v|/vectork−/vectorQ⊥|. (34) This is again conjectured to be true for certain spin-liquid ansatz, for example, the π-flux state [ 31] of the Heisenberg antiferromagnetic Hamiltonian on a 2D square lattice, whichhas been argued to be stable against U(1) gauge fluctuations [32]. We again use Eq. (29) to evaluate the structure factor. S −+(/vectorq⊥,ω)=/integraldisplayd2k (2π)2δ/parenleftBig ω−v|/vectork|−v|/vectork+/vectorq⊥|/parenrightBig =1 8πv2ω2−v2q2 ⊥/2/radicalBig ω2−v2q2 ⊥/Theta1(ω−v|/vectorq⊥|).(35) We now use Eq. ( 14) to find the net spin current for T→0. Ispin=J2A⊥ν(/epsilon1F) 480π2v2V5=η1V5, (36) where we have again absorbed all prefactors in η1to make the Vdependence explicit. The current takes a nonzero value for anyV> 0, as there is no gap to a two-spinon excitation. We have evaluated the current for a single Dirac point, althoughextensions to multiple Dirac points with different velocitiescan be done in an exact analogy with the previous section, andwill not affect the threshold or the exponent in the power law. 3. Generic spinon dispersions and spatial dimensions In this section, we are going to generalize the above results for given spinon dispersion in d=2 to generic dispersions and arbitrary space dimensions d−1 of the metal-insulator boundary using scaling arguments. Although this approachdoes not give us the exact prefactors, it is sufficient to find outthe characteristic dependence I spinonV. We would require that the lowest spinon band has minima at discrete points in theBrillouin zone, which are well separated from each other. Westart off with gapless spin liquids with power law dispersions,and find that the exponent of Vis directly related to the power law in the dispersion and the dimensionality of the system.Our results easily generalize to gapped spin liquids. 165113-6PROBING EXCITATIONS IN INSULATORS VIA . . . PHYSICAL REVIEW B 92, 165113 (2015) Let the spinons have a dispersion given by /epsilon1(/vectork)=vα|/vectork|α. (37) The two-spinon structure factor is proportional to an integral over the allowed phase space consistent with energy conser-vation. S −+(/vectorq⊥,ω)∼/integraldisplay kd−2dk d/Omega1 d−2δ(ω−vα|/vectork|α −vα|/vectork+/vectorq⊥|α). (38) The solutions for k(when the δfunction is nonzero) can be written in terms of a dimensionless scaling function/Phi1(v αqα ⊥/ω)a s k=q⊥/Phi1(vαqα ⊥/ω). (39) Theδfunction in ωcan be rewritten as a δfunction in kas follows (in terms of another dimensionless function /Phi11which comes from the Jacobian): δ(ω−vα|/vectork|α−vα|/vectork+/vectorq⊥|α) =δ[k−q⊥/Phi1(vαqα ⊥/ω)]/[vαqα−1 ⊥/Phi11(vαqα ⊥/ω)].(40) Now we can see how the dynamic structure factor scales without explicitly evaluating the integral. S−+(/vectorq⊥,ω)∼qd−1−α ⊥/Psi1(vαqα ⊥/ω). (41) The dimensionless scaling function /Psi1must involve a /Theta1 function of the form /Theta1(ω−ζvαqα ⊥), where ζis some arbitrary numerical constant that depends upon the exact dispersion.This follows from the fact that a large center-of-mass mo-mentum will inevitably result in a large energy for the spinonpair which is precluded by energy conservation. Here, we areassuming that ωis small enough so that both the spinons come from the bottom of the band(s). Finally, we turn to the T→0 limit of Eq. ( 14) again to find the spin current. I spin∼/integraldisplayV 0(V−ω)dω/integraldisplay dq⊥qd−2 ⊥d/Omega1d−2S−+(/vectorq⊥,ω).(42) Because of the /Theta1function in S−+(/vectorq⊥,ω), the momentum integral is restricted to q/lessorequalslant(ω/vα)1/α, so dimensional analysis tells us that/integraldisplay dq⊥qd−2 ⊥d/Omega1d−2S−+(/vectorq⊥,ω)∼(ω/vα)(2d−2−α)/α.(43) The integral over ωscales as V2, so the final result after putting all this information together is Ispin∼V2×V(2d−2−α)/α=V1+2(d−1)/α. (44) As a check, let us see if the scaling matches the previous two exact calculations. In both cases, we have d−1=2. For the gapped Z2spin liquid in the limit of the gap /Delta1s→0, we have α=2 and hence, Ispin∼V1+2(3−1)/2=V3. For the gapless U(1) spin liquid with α=1, we have Ispin∼V1+2(3−1)/1= V5. For generalizing to gapped spin liquids with a spin gap of /Delta1s, all we need to do is make the following replacement in all the previous calculations: ω→ω−2/Delta1s. (45)This in turn tells us that the spin current is given by Ispin∼(V−2/Delta1s)1+2(d−1)/α/Theta1(V−2/Delta1s). (46) Equation ( 46) is the main result of this section. It shows that by measuring the spin current as a function of voltage, it ispossible to deduce both the nature of the spin gap as wellas the effective dispersion of the low-energy excitations. Notethat at the level of low-energy effective field theory, the currentdoes not depend on the detailed structure of the lattice, but onlyon the effective continuum dispersion, as expected. C. Numerical results for a model Z2spin liquid state on the kagome lattice In this section, we extend the previous results for a gapped Z2spin-liquid state via numerical calculations. As a model state, we choose the Q1=Q2ground state on the kagome lattice, described by Sachdev [ 16]. The reason for choosing this state for further investigation is that the dynamicalstructure factor measured in neutron scattering experimentson Herbertsmithite single crystals [ 2] is in good qualitative agreement with the calculations in the Q 1=Q2ground state by Punk et al.[17]. Following Sachdev [ 16], we use a large Nexpansion technique based on the symplectic group Sp( N). To generalize S− iS+ jto Sp(N), we just extract the part of the Sp( N) invariant scalar product /vectorSi·/vectorSj[16] that corresponds to1 2S− iS+ j.I nt e r m s of the flavor indices mof the Schwinger bosons that make up the spins, it can be written as S− iS+ j=1 2N2/summationdisplay m1,m2/parenleftbig b† im1↓bim2↑b† jm2↑bjm1↓ +b† im1↓bim2↑b† jm1↑bjm2↓/parenrightbig . (47) Note that this reduces exactly to S− iS+ jofSU(2) when we have a single flavor. To simplify the expression, we note that the N flavors are decoupled in the N=∞ mean-field theory, and each of the Nflavors has an identical Hamiltonian. Therefore, each flavor gives the same contribution, which just cancels offthe extra factor of N 2, and we just need to calculate each term for a single flavor. The spinon operators that diagonalize themean-field Hamiltonian are linear in the bandb †operators, hence the correlation function factorizes as follows: /angbracketleftS− iS+ j/angbracketright=1 2(/angbracketleftb† i↓bj↓/angbracketright/angbracketleftbi↑b† j↑/angbracketright+/angbracketleftb† i↓b† j↑/angbracketright/angbracketleftbi↑bj↓/angbracketright). (48) Moving to Fourier space and keeping only terms that give contributions to ω> 0 after analytic continuation, we find that the dynamic susceptibility is given by χ−+(/vectorq⊥,iωn)=1 2Ns/summationdisplay /vectork,i/Omega1 n[Ujl(−/vectork)Vjm(/vectork+/vectorq⊥) +Vjl(−/vectork)Ujm(/vectork+/vectorq⊥)]U∗ il(−/vectork)V∗ im(/vectork+/vectorq⊥) ×Gl(−/vectork,−i/Omega1n)Gm(/vectork+/vectorq⊥,iωn+i/Omega1n), (49) where /vectorq⊥belongs to the extended Brillouin zone, Nsis the total number of sites, U,V are the Bogoliubov matrices that diagonalize the mean-field Hamiltonian, and we have 165113-7SHUBHAYU CHATTERJEE AND SUBIR SACHDEV PHYSICAL REVIEW B 92, 165113 (2015) /Slash/Minus/Plus/LParen /RParen/LParen /RParen FIG. 3. (Color online) Momentum integrated structure factor for theQ1=Q2ground state of the Z2spin liquid on the kagome lattice. implicitly summed over all sublattice indices {i,j,l,m }.W e are going to use Eq. ( 49) to numerically evaluate the exact mean-field structure factor. As a side note, we mention that inthe low-energy limit, where /vectorkis close to the bottom of a spinon band/vectorQ ⊥, and/vectorq⊥is also small, so that the sum of the two spinon energies satisfies the energy constraint, we can approximate the elements of the UandVby their values at /vectorQ⊥, and then we recover the dynamic structure factor evaluated in Eq. ( 32). We first plot the momentum-integrated structure factor S−+(ω)=1 Ns/summationtext /vectorqS−+(/vectorq,ω) as a function of energy ωin Fig.3. We assume mean-field parameters λ=0.695 and Q1= Q2=0.4 in the units of JAF, which are notself-consistently determined, and lead to a spinon gap of /Delta1s≈0.5. We note two specific features, the jump at ω≈0.75 and the peak at ω≈1.3. Both these features can be understood using the band structure of the spinons for this ground state. Thespinon spectra has a flat band with /epsilon1 /vectork=λ, and once we have ω/greaterorequalslantλ+/Delta1s, we can excite two spinons, one of them being at any momentum on the flat band. The second peak presumablycomes from both spinons coming from the flat band, butis slightly smeared out by the Bogoliubov matrices and thefinite width Lorentzian approximation for the δfunction in the numerics. If we go up to energy scales of V≈J AF/lessmuch/epsilon1F (this is reasonable as JAF≈200 K for Herbertsmithite [ 33], but typical /epsilon1F≈104K), we now can have contributions to the current at large values of δqxandq⊥. In order to investigate the contributions properly, we need to numerically evaluatethe spin current starting with the T→0 limit equation ( 6). We next plot the spin current, evaluated numerically, in Fig. 4as a function of the spin accumulation voltage V.T h e Fermi-liquid parameters chosen for the plot below are k F=2 (units of inverse lattice spacing), and /epsilon1F/JAF=100. As expected, we observe the effects of the two features in the dynamic structure factor on the spin current, which isroughly an integral over the structure factor. The steplike jumpin the structure factor leads to a change in slope in the currentaround V≈0.7, and the spike leads to a steplike jump around V≈1.3, after which the current saturates. The observation of these two distinct features in the spin current would be strongevidence in favor of the Q 1=Q2Z2spin liquid on the kagome lattice. We note that the Q1=−Q2ground state [ 16] does not/( ) FIG. 4. (Color online) Spin current as a function of spin accumu- lation voltage for the Q1=Q2ground state of the Z2spin liquid on the kagome lattice. have any flat spinon band, and is hence not expected to show any such feature in the spin current. V . CONCLUSION AND OUTLOOK In summary, we proposed the use of spin currents as a gateway to probe the nature of excitations in magneticinsulators. Measurement of the spin current as a function of thespin-accumulation voltage can throw light on the dispersionof the low-lying excitations and gap above the ground state. Inparticular, we showed at that in the zero-temperature limit, thethreshold and scaling of spin current with voltages may be usedeffectively to search for spin-liquid ground states in magneticinsulators. Finally, we focused on a particular spin-liquidground state, which is a candidate state for Herbertsmithite[17], and identified some broad features in the spin current which can help to identify that state. The spin current is a valuable probe, because once injected into the insulator, the total spin is conserved in absence ofspin-orbit coupling and random field impurities. We anticipatethat it may be interesting to study how the presence of disorderin the interface, or the presence of non-spin-carrying low-lyingexcitations in the insulator, which couple to the mobile spin-carrying modes (for example, visons coupling to spinons [ 17] in spin liquids) affect the spin current. Note added in proof . Recently, we came across another proposal [ 34] for a similar experiment. The authors suggest measuring spin currents to detect gapless spin liquids withFermi surfaces or Dirac cones. Although we use a differenttool for our calculations, our results agree with theirs at T=0 for Dirac spin liquids. For collinear antiferromagnets, wefind that they have missed the elastic contribution which willdominate the spin current at low temperatures, and our inelasticcontributions are the same. We present additional exactcalculations and scaling arguments for gapped and gaplessspin liquids for different spinon dispersions and differentdimensions, as well as VBS states. We also numericallyinvestigate an attractive experimental candidate and go beyondlow-energy scalings to identify specific features in the spincurrent at higher energies. 165113-8PROBING EXCITATIONS IN INSULATORS VIA . . . PHYSICAL REVIEW B 92, 165113 (2015) ACKNOWLEDGMENTS Discussions with So Takei, Yaroslav Tserkovnyak, and Amir Yacoby helped motivate this research. We thank De-banjan Chowdhury, Soonwon Choi, and Bertrand Halperin forvaluable discussions, and especially Matthias Punk for helpwith numerics. This research was supported by the NSF underGrant No. DMR-1360789, the Templeton Foundation, andMURI Grant No. W911NF-14-1-0003 from ARO. Research atPerimeter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontariothrough the Ministry of Research and Innovation. APPENDIX A: DETAILS OF SPIN CURRENT CALCULATIONS (FROM SEC. II C) We begin with the linearized energy transfer ω(ˆn,/vectorq⊥,δqx) in Eqs. ( 9), and write the spin current from Eq. ( 6)a s Ispin,↑=πJ2A⊥ 2m/integraldisplay k1x>0ddk1 (2π)d/integraldisplay qx>k1xddq (2π)dnF/parenleftbig /epsilon1/vectork1/parenrightbig ×/parenleftbig 1−nF/bracketleftbig /epsilon1/vectork1+V−ω(ˆn,/vectorq⊥,δqx)/bracketrightbig/parenrightbig ×qxS−+[/vectorq⊥,ω(ˆn,/vectorq⊥,δqx)]. (A1) The integral over |/vectork1|can now be evaluated, as everything else depends only on the direction ˆnof the initial momentum, and the momentum transfer /vectorq. Assuming that the density of states ν(/epsilon1F) is approximately a constant close to the Fermi surface, we have /integraldisplay k1x>0ddk1 (2π)dnF/parenleftbig /epsilon1/vectork1/parenrightbig/parenleftbig 1−nF/bracketleftbig /epsilon1/vectork1+V−ω/parenleftbigˆn,/vectorq⊥,δqx/parenrightbig/bracketrightbig/parenrightbig ≈ν(/epsilon1F)/integraldisplay ˆn·ˆx>0d/Omega1 Sd−1V−ω(ˆn,/vectorq⊥,δqx) 1−e−β[V−ω(ˆn,/vectorq⊥,δqx)]. (A2) We can further simplify equation ( A1) by getting rid of qx in favor of ω. For given /vectorq⊥andˆn,dω=vF(ˆn·ˆx)d(δqx) and qx≈2kF(ˆn·ˆx), implying dqxqx m≈−d(δqx)2kF(ˆn·ˆx) m=−2dω. (A3) This is independent of the direction of initial momentum ˆn. Further, note that the constraint qx>kF(ˆn·ˆx) is guaranteed to be satisfied by energy conservation, which requires smallδqx. By our assumption that S−+(/vectorq⊥,ω) is insignificant for large|/vectorq⊥|, a change of energy due to large δqxcannot be offset by another due to large |/vectorq⊥|. The only problem arises when ˆn·ˆxis very small, but those are insignificant portions of the phase space that we can neglect. Therefore, all dependenciesof the /vectorqintegral on ˆnare removed, and this enables us to do the angular integral. Using/integraltext ˆn·ˆx>0d/Omega1 Sd−1=1 2, we recover the simplified expression stated in Eq. ( 10): Ispin,↑=πJ2A⊥ν(/epsilon1F) 2/integraldisplaydω 2πdd−1q⊥ (2π)d−1 ×(V−ω) 1−e−β(V−ω)S−+(/vectorq⊥,ω). (A4) The calculation for Ispin,↓[Eq. ( 11)] is analogous, with the only change coming from the different occupancies of the initial andfinal states. We now discuss the case when the dynamic structure factor has a minima at large transverse momentum /vectorQ ⊥. The trick is to note that although the momentum transfer can be large,the energy transfer at low temperatures is always small, i.e.,ω/lessorsimilarV/lessmuch/epsilon1 F. Therefore, we can expand in small parameters about the point of elastic scattering. To do so, we first solve fora longitudinal momentum transfer q x0which satisfies /epsilon1/vectork1= /epsilon1/vectork1−/vectorQ, where /vectorQ=qx0ˆx+/vectorQ⊥. /epsilon1/vectork1−/epsilon1/vectork1−/vectorQ=2kF[/vectorQ⊥·ˆn+qx0(ˆn·ˆx)]−/parenleftbig Q2 ⊥+q2 x0/parenrightbig =0 ⇒qx0=kF(ˆn·ˆx)+/bracketleftbig k2 F(ˆn·ˆx)2 +2kF(/vectorQ⊥·ˆn)−Q2 ⊥/bracketrightbig1/2. (A5) The contraint qx0/greaterorequalslantkF(ˆn·ˆx), required for reflection, implies that only the positive square root can contribute. As qxis real, only some values of ˆnare relevant. Specifically, we require k2 F(ˆn·ˆx)2+2kF(/vectorQ⊥·ˆn)/greaterorequalslantQ2 ⊥. (A6) We need to evaluate the angular integral over angular regions of the Fermi surface consistent with the above constraint. IfQ ⊥, the magnitude of the transverse scattering wave vector, is too large compared to the Fermi momentum kF, then there is no scattering consistent with energy conservation, and hencethere is no spin current due to this process. Now, we can expand about the solution for elastic scattering for small ω, and keep only linear terms in δq xand/vectorq⊥. /vectorq=(qx0−δqx)ˆx+/vectorQ⊥−/vectorq⊥, ω=1 m{[qx0−kF(ˆn·ˆx)]δqx+(/vectorQ⊥−kFˆn)·/vectorq⊥}+O/parenleftbig δq2 x,q2 ⊥/parenrightbig , w h e r ew eu s e d2 kF[/vectorQ⊥·ˆn+qx0(ˆn·ˆx)]−/parenleftbig Q2 ⊥+q2 x0/parenrightbig =0. (A7) We revert to our previous formalism, and replace the integral over qxby an integral over ω, with the only change being in the prefactor appearing in the angular integral. For fixed /vectorq⊥andˆn,dω=1 m[qx0−kF(ˆn·ˆx)]δqx, andqx≈qx0, implying that dqxqx me≈−d(δqx)qx0 me=dω/parenleftbiggqx0 kF(ˆn·ˆx)−qx0/parenrightbigg =−/parenleftBigg 1+kF(ˆn·ˆx) /bracketleftbig k2 F(ˆn·ˆx)2+2kF(/vectorQ⊥·ˆn)−Q2 ⊥/bracketrightbig1/2/parenrightBigg dω. Note that for |/vectorQ⊥|/lessmuchkF, we get back our previous result, which corresponds to scattering at /vectorQ⊥=0. This acts as a check on the above calculation, and also shows that the calculation can be generalized as long as we have low-energy excitations in thespin system about a set of isolated points in momentum space which are well separated in the Brillouin zone. 165113-9SHUBHAYU CHATTERJEE AND SUBIR SACHDEV PHYSICAL REVIEW B 92, 165113 (2015) The factor multiplying dωwill change the result of the angular integral over initial momenta, but the remaining calculation remains unchanged, and we have I/vectorQ⊥ spin,↑=πJ2A⊥ν(/epsilon1F) 2/integraldisplay k2 F(ˆn·ˆx)2+2kF(/vectorQ⊥·ˆn)/greaterorequalslantQ2 ⊥d/Omega1 Sd−1/parenleftBigg 1+kF(ˆn·ˆx) /bracketleftbig k2 F(ˆn·ˆx)2+2kF(/vectorQ⊥·ˆn)−Q2 ⊥/bracketrightbig1/2/parenrightBigg ×/integraldisplaydω 2πdd−1q⊥ (2π)d−1(V−ω) 1−e−β(V−ω)S−+(/vectorq⊥,ω) =πJ2A⊥ν(/epsilon1F) 2fang(kF/Q⊥)/integraldisplaydω 2πdd−1q⊥ (2π)d−1(V−ω) 1−e−β(V−ω)S−+(/vectorq⊥,ω), (A8) where fang(kF/Q⊥) is the angular integral referred to in Eq. ( 13). Typically, kFandQ⊥have the same order of magnitude, and then the angular integral is an overall factor of O(1) (the exact value is determined by the constraints set by the ordering wave vector /vectorQ⊥). Taking into account that there can be multiple such minima in the dynamic structure factor at large finite momenta {/vectorQ⊥}, and scattering to momenta patches around these minima are independent as long as the minima are well separated, we arrive atEq. ( 12), stated below for the sake of completeness. I spin,↑=πJ2A⊥ν(/epsilon1F) 2/summationdisplay /vectorQ⊥fang(kF/Q⊥)/integraldisplaydω 2πdd−1q⊥ (2π)d−1(V−ω) 1−e−β(V−ω)S−+(/vectorq⊥,ω). (A9) The expression for Ispin,↓follows in exact analogy to the above calculation. APPENDIX B: S−+/parenleftbig/vectorq⊥,ω/parenrightbigFOR AN ANTIFERROMAGNETIC INTERFACE (FROM SEC. III A 2 ) We evaluate the dynamic structure factor for a N ´eel-ordered state on a d-dimensional cubic lattice using the Holstein- Primakoff transformation. First, let us consider the case when the N ´eel vector points parallel to the spin-quantization axis in the metal (chosen to be ˆz). We have up-spins on sublattice Aand down-spins on sublattice B, with the total number of spins being 2N, and the coordination number of each spin is z=2d. Therefore we define i∈A, S− i=a† i(2S−a† iai)1/2;S+ i=(2S−a† iai)1/2ai,andSz i=S−a† iai, (B1) i∈B, S+ i=b† i(2S−b† ibi)1/2;S+ i=(2S−b† ibi)1/2bi,andSz i=−S+b† ibi and do an expansion in 1 /S. The Heisenberg Hamiltonian HAF=JAF/summationtext /angbracketleftij/angbracketright/vectorSi·/vectorSjcan be written in terms of the Holstein Primakoff bosons as HAF=−JAFNS2z+JAFSz/summationdisplay /vectork[a† /vectorka/vectork+b† /vectorkb/vectork+γ/vectork(a/vectorkb−/vectork+a† /vectorkb† −/vectork)]+O(S0),where γ/vectork=1 z/summationdisplay δ∈n.nei/vectork·/vectorδ. (B2) This can be diagonalized by a Bogoliubov transformation, using a/vectork=u/vectorkα/vectork+v/vectorkβ† −/vectork,b /vectork=u/vectorkβ/vectork+v/vectorkα† −/vectork withu/vectork=u−/vectork=cosh(θ/vectork),v /vectork=v−/vectork=sinh(θ/vectork),and tanh(2 θ/vectork)=−γ/vectork. (B3) The Hamiltonian is diagonal in terms of the Bogoliubov quasiparticles, HAF=−JAFNS2z−JAFNSz+/summationdisplay /vectorkE/vectork(α† /vectorkα/vectork+β† /vectorkβ/vectork+1),E /vectork=JAFSz/radicalBig 1−γ2 /vectork. S−+(/vectorq⊥,ω) may now be calculated from definition using the expression for the spin operators in terms of the quasiparticle creation and annihilation operators. After some algebra, we find S−+(/vectorq⊥,ω)=2πS(u/vectorq⊥+v/vectorq⊥)2{δ(ω−E/vectorq⊥)[1+n(β/vectorq⊥)]+δ(ω+E/vectorq⊥)n(α−/vectorq⊥)}. (B4) AtT=0, only the δfunction with positive ωcontributes to the spin current as there are only quasiparticles initially in the system. For low momenta, we have E/vectorq⊥≈vs|/vectorq⊥|,and (u/vectorq⊥+v/vectorq⊥)2=cosh(2 θ/vectorq⊥)+sinh(2 θ/vectorq⊥)=/radicalBigg 1−γ/vectorq⊥ 1+γ/vectorq⊥=q⊥ 2√ d, (B5) 165113-10PROBING EXCITATIONS IN INSULATORS VIA . . . PHYSICAL REVIEW B 92, 165113 (2015) which leads to the following expression for the dynamic structure factor for the T=0 antiferromagnet: S−+(/vectorq⊥,ω)=πSq ⊥√ dδ(ω−vsq⊥). (B6) For the case when the N ´eel order is perpendicular to the spin-quantization axis in the metal, we assume that spins on sublattice Aare pointing in the ˆydirection and the spins on sublattice Bare pointing in the −ˆydirection. In this case, we can still use the Holstein-Primakoff representation of spins after doing a π/2 rotation of our coordinate system with respect to the xaxis. In the rotated coordinate system XYZ we have X=x, Y=−z, andZ=y. Remembering that our original definitions of S±were with respect to the old axes, let us denote our spin operators by /Sigma1in the new set of axes. 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PhysRevB.86.180506.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 86, 180506(R) (2012) Vortex dynamics in ferromagnetic superconductors: Vortex clusters, domain walls, and enhanced viscosity Shi-Zeng Lin, Lev N. Bulaevskii, and Cristian D. Batista Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 19 January 2012; revised manuscript received 16 November 2012; published 28 November 2012) We demonstrate that there is a long-range vortex-vortex attraction in ferromagnetic superconductors due to polarization of the magnetic moments. V ortex clusters are then stabilized in the ground state for low vortexdensities. The motion of vortex clusters driven by the Lorentz force excites magnons. This regime becomesunstable at a threshold velocity above which domain walls are generated for slow relaxation of the magneticmoments and the vortex configuration becomes modulated. This dynamics of vortices and magnetic momentscan be probed by transport measurements. DOI: 10.1103/PhysRevB.86.180506 PACS number(s): 74 .25.Uv, 74 .25.F−,7 4.25.Ha, 74 .25.N− Introduction. Superconductivity (SC) and magnetism are at the heart of modern condensed matter physics. While theyseem to be antagonist according to the standard BCS theory,a large family of magnetic superconductors was discovered inthe past decades. Examples include the coexistence of antifer-romagnetism or helical ferromagnetic (FM) order in ternarysuperconducting compounds, 1uniform ferromagnetism in triplet superconductors,2–4and antiferromagnetism in the ReNi2B2C borocarbides5(Rerepresents a rare-earth element) and in the recently discovered iron-based superconductors.6 The interplay between SC and magnetism allows to control thesuperconducting properties through the magnetic subsystem,and vice versa. These phenomena open new possibilitiesfor applications to superconducting electronics and magneticstorage devices. 7,8 The Abrikosov vortices of superconductors are a natural link between the superconducting condensate and the magneticmoments (MMs). V ortices are induced either by externalmagnetic fields or by the MMs. 9On the other hand, the magnetic subsystem supports collective spin waves and topo-logical excitations that are domain walls. Because vorticesare magnetic objects, they are expected to interact stronglywith MMs via Zeeman coupling. Indeed, as we discuss below,vortex motion can drive magnetic domain walls. The MMs provide a novel handle to control the vortex be- havior in the static and dynamic regimes. It was demonstratedthat magnetic domains induce a vortex pinning that is 100times stronger than the one induced by columnar defects. 10 In the flux flow regime, vortex motion radiates magnons bytransferring energy into the magnetic system. This effect hasbeen recently proposed by Shekhter et al. for antiferromagnetic superconductors. 11By assuming a rigid vortex lattice and fast relaxation of the MMs, it is demonstrated that Cherenkov radiation of magnons occurs when the vortex lattice velocity v satisfies G·v=/Omega1(G), where Gis the vortex lattice wave vector and /Omega1(G) is the magnon dispersion. This emission gives an additional contribution to the vortex viscosity thatmanifests as a voltage drop in the I-Vcharacteristics. Thus, the overall dissipation is reduced for a given current. V ortexmotion can also be used to probe the spectrum of excitationsin the magnetic subsystem. 12 Several questions remain to be addressed. It is known that intrinsic nonlinear effects of the magnetic subsystem becomeimportant for high energy magnon excitations. However, it is unclear if magnon excitations remain stable in this nonlinearregime. On the other hand, the interaction between themagnetic subsystem and vortices may become comparableor even stronger than the intervortex repulsion. Therefore,the vortex lattice may be modified by this effect. Finally,the dominant dissipation mechanism of vortices when domainwalls are excited by the vortex motion is unknown. Here we study the vortex dynamics in FM superconductors. The Zeeman coupling between vortices and MMs inducesan additional vortex-vortex attraction that is comparableto the intervortex repulsion for a large enough magneticsusceptibility. This attraction leads to the formation of vortexclusters at low vortex densities. We also show that magneticdomain walls are created when vortex clusters driven bythe Lorenz force reach a threshold velocity. The interactionbetween domain walls and vortices greatly enhances the vortexviscosity and causes hysteresis in the dynamics of the wholesystem. The vortex configuration is modulated by the domainwalls. Model. Uniform FM order and SC suppress each other because of the exchange and electromagnetic coupling be-tween the MMs and Cooper pairs. 1However, they could coexist in triplet FM superconductors,2–4such as UGe 2, layered magnetic superconductors consisting of FM and SClayers, 13,14such as Sm 1.85Ce0.15CuO 4, or artificial bilayer systems.8,15Here we study the vortex dynamics in these FM superconductors. An applied dc magnetic field perpendicularto the ferromagnetic easy axis creates a vortex lattice that isdriven by a dc in-plane current. We use the approximationof straight vortex lines and the description of vortices is twodimensional. The total Gibbs free energy functional of the system, in terms of the vector potential A, magnetization M, and vortex position R i=(xi,yi), is G(A,M,Ri)=d/integraldisplay dr2(gsc+gM+gint)+1 8π/integraldisplay outdr3B2, (1) where dis the thickness of the system and the last term is the magnetic energy outside the superconductor. The energyfunctional density for the SC subsystem in the London 180506-1 1098-0121/2012/86(18)/180506(5) ©2012 American Physical SocietyRAPID COMMUNICATIONS LIN, BULAEVSKII, AND BATISTA PHYSICAL REVIEW B 86, 180506(R) (2012) approximation is gsc(A)=B2 8π−B·Hext 4π+1 8πλ2 L/parenleftbigg/Phi10 2π∇φ−A/parenrightbigg2 , (2) with B=∇×A.φis the superconducting phase, Hextis the applied magnetic field, λLis the London penetration depth, and/Phi10=hc/2eis the flux quantum. The energy functional density of the magnetic subsystem is gM=J 2(∇M)2−JA 2M2 x, (3) where JandJAare the exchange and anisotropy parameters. The easy axis is taken along the xdirection. We assume that the magnitude of the magnetic moment is conserved,|M|=M s, where Msis the saturated magnetization value. Because of the anisotropy, the magnetic Hamiltonian has twodegenerate minima and supports stable domain walls. TheZeeman interaction between MMs and SC is g int=−B·M. (4) The vortex axis is taken along the zdirection. The straight vortex lines approximation is valid when d/lessmuchλLord/greatermuchλL. The spreading of magnetic field associated with vortices nearthe surface of superconductors has to be taken into accountford∼λ L.16By minimizing gsc+gintwith respect to A,w e obtain the magnetic field associated with vortices λ2 L∇×∇×(B−4πM)+B=/Phi10/summationdisplay iδ(r−Ri)ˆz. (5) Mz(k)=Bz(k)˜χzz(k) in the linear response region when Mz/Ms/lessmuch1. AsλLis much larger than the magnetic correla- tion length ξm∼√J/JA, we can use a local approximation for ˜χzz(k)/similarequal1/JA=χ0/(1+4πχ 0). The uniform susceptibility χ0∝/angbracketleftMz(k=0)Mz(k=0)/angbracketrightdiverges at JA=4π, which signals an instability of the magnetic subsystem. The FMordering along the xdirection coexists with superconductivity only when J A>4π.17 According to Eq. (5), the magnetic field of a vortex at Riis Bz(k,Ri)=/Phi10 1+λ2ek2exp(ik·Ri), (6) with a renormalized penetration depth λe≡λL/√1+4πχ 0. Attraction between vortices via MMs. We calculate now the interaction between two vortices at RiandRj. V ortices interact with each other through the exchange of massive photonsdescribed by g sc, which leads to a short-range repulsion. As was first discussed by Pearl, vortices also interact through theexchange of massless photons outside the SC, as described bythe last term in Eq. (1). This contribution leads to a long-range repulsion. 18,19The total repulsion energy is Ur(R)=/Phi12 0d 8π2λ2eK0/parenleftbiggR λe/parenrightbigg +/Phi12 0 8π/Lambda1/bracketleftbigg H0/parenleftbiggR /Lambda1/parenrightbigg −Y0/parenleftbiggR /Lambda1/parenrightbigg/bracketrightbigg , (7) with R≡Ri−Rjand/Lambda1=2λecoth[d/λe] is the modified Pearl length. Kiis the modified Bessel function, H0is the Struve function, and Y0is the Weber function.FIG. 1. (Color online) V ortex-vortex interaction potential for different values of χ0according to Eqs. (7)and (8). Attraction is induced due to the Zeeman coupling between vortices and MMs,and the long-range repulsion arises from the electromagnetic fields outside the SC. Av o r t e xa t Ripolarizes the surrounding MMs. This effect leads to an effective attraction to a vortex at Rj. The magnetic energy due to the presence of vortices is d/integraltext dr2(gM+gint), withBv z=Bz(Ri)+Bz(Rj) and Mv z=χ0Bv z/(1+4πχ 0). The contribution from the gradient term in Eq. (3)is much smaller than the anisotropic contribution because kξm/lessmuch1 withk∼1/λe.B yu s i n g M2 x+M2 z=M2 s, we obtain the attractive interaction Ua(R)=−d 2/integraldisplay dr2Bv zMv z=−dχ0/Phi12 0R 4π(1+4πχ 0)λ3eK1/parenleftbiggR λe/parenrightbigg . (8) In the presence of attraction, the repulsion through the electromagnetic fields outside the SC in Eq. (7)cannot be neglected because it prevents the formation of a single cluster.The physics here is similar to the laminar phase in conventionaltype I superconductors. 20 The effect of finite velocity von the vortex-vortex inter- action is negligible because ˜ χzzdepends weakly on vfor ξm/λe/lessmuch1. The attractive component is comparable to the repulsion for χ0∼1 and the energy minimum takes place at Rm∼λe. Figure 1shows the energy of two vortices separated by a distance R.F o rχ0∼1, the net interaction is attractive for large separations λe<R</Lambda1 and repulsive at short distances R<λ e. There is also a long-range repulsion for R>/Lambda1 due to the surface effect. Since the susceptibility χ0decreases with JA, the attractive component drops as anisotropy increases. The intervortex interaction becomes purely repulsive forχ 0/lessmuch1. Excitation of domain walls. We introduce the equation of motion for MMs and vortices that is used in the numericalsimulation. The FM subsystem is described by the Landau-Lifshitz-Gilbert equation 21 ∂tm=−γm×Beff+αm×∂tm, (9) where γis the gyromagnetic ratio, m=M/Msis the nor- malized MM, αis the damping coefficient, and the effective magnetic field is Beff=−δ[gM+gint]/δM. The vortex sub- system is described by the time-dependent Ginzburg-Landau 180506-2RAPID COMMUNICATIONS VORTEX DYNAMICS IN FERROMAGNETIC ... PHYSICAL REVIEW B 86, 180506(R) (2012) equations ¯h2 2mD∂t/Psi1=−/bracketleftbigg αs/Psi1+β|/Psi1|2/Psi1+¯h2 2m/parenleftbigg i∇+2π /Phi10A/parenrightbigg2 /Psi1/bracketrightbigg , (10) σ c∂tA=Js+Jext−c 4π∇×(∇×A−4πM), (11) with the supercurrent Js=e¯h im(/Psi1∗∇/Psi1−/Psi1∇/Psi1∗)−4e2 mc|/Psi1|2A, (12) Dis the diffusion coefficient, σis the conductivity in the normal state, Jextis the external current, and the other parameters are defined according to the usual convention. TheMMs stop responding to the vortex motion when the averagemagnetic field, ¯B z≈nv/Phi10withnvbeing the vortex density, is larger than the saturation value, Bs=MsJA, and the two subsystems become decoupled. Therefore, we shall considerthe interesting region ¯B z<Bs. In the long wavelength and weak damping α/lessmuch1 limits, the magnon dispersion for the FM system of Eq. (9)is /Omega12=ω2 0+v2 sk2,v s=γMs/radicalBig/parenleftbig 2−m2 z0/parenrightbig JAJ, (13) ω2 0=J2 Aγ2M2 s/parenleftbig 1−m2 z0/parenrightbig/bracketleftbig 1+iα/parenleftbig 2−m2 z0/parenrightbig/parenleftbig 1−m2 z0/parenrightbig−1/2/bracketrightbig , (14) where mz0is thezcomponent of the MMs in the ground state andvsis the magnon velocity. Re( ω0) is the energy gap and Im(ω0) is the magnon relaxation rate. Re( ω0)=100 GHz and vs=50 m/s for typical ferromagnets.22 We then establish general relations of the energy transfer between MMs and vortices. The vortex velocity acquires an acpart, ˜v i, because of the interaction between vortices and MMs, vi=¯v+˜vi. The energy balance for the whole system reads η¯v2+η/angbracketleft˜vi2/angbracketrighti,t+1 nvα Msγ/angbracketleftbigg/integraldisplay dr2(∂tM)2/angbracketrightbigg x,t=FL·¯v,(15) where /angbracketleft ···/angbracketright i,tdenotes the average over vortices and time, and /angbracketleft ···/angbracketright x,tdenotes the average over space and time. The first and second terms on the left-hand side (lhs) correspond to Bardeen-Stephen (BS) damping with coefficient η=/Phi1 2 0σ/(2πc2ξ2),20 where ξ=/radicalbig ¯h2/(2m|αs|) is the coherence length. The third term on the lhs accounts for the dissipation due to precessionof MMs. The term on the right-hand side is the work done bythe Lorentz force F L. The effective viscosity ηeff=FL/¯vis enhanced due to the interaction between vortices and MMs, ηeff=η+η ¯v2/angbracketleft˜vi2/angbracketrighti,t+1 nv¯v2α Msγ/angbracketleftbigg/integraldisplay dr2(∂tM)2/angbracketrightbigg x,t.(16) Off resonance, the contribution of the magnetic damping is small, thus ¯v≈FL/η. Since FL=Jext/Phi10/candE= ¯vnv/Phi10/cwith an external current Jextand electric field E,t h e underlying dynamics can be probed by an I-Vmeasurement. The effect of magnons on the vortex dynamics depends on the vortex density. When the average intervortex distance issmaller than the value corresponding to the potential minimum,n v<1/R2 m, the attraction between vortices dominates. V or- tices form circular clusters with the internal triangular structurein the ground state, as shown in Fig. 2(a) obtained from our simulations.23The distance between neighboring vortices inside the cluster is of order λe, and the separation between neighboring clusters is of order/radicalbig πR2c/(nvλ2e), with a cluster radius given by Rc≈/Lambda1[−ua/(3ur)]1/3.24The attractive, ua< 0, and repulsive, ur>0, energies are defined in Fig. 1. The vortex clusters start to merge and more complex vortexconfigurations, such as stripes, are possible for larger valuesofn v.T h e H=Hc1transition from the uniform Meissner state to the state with vortex clusters is of first order,25–27 in contrast to the second order phase transition expected for conventional type II superconductors.20V ortex clusters have been observed experimentally in conventional superconduc-tors with intervortex attraction, such as Nb (see Ref. 28for a review). For finite transport current, each cluster moves as a whole driven by the Lorentz force and polarizes the MMs along itsway. The MMs relax to their positions of equilibrium afterthe vortex cluster leaves that region. The polarization andexcitation of magnons, and subsequent relaxation of MMs thuscauses vortex dissipation through the magnetic subsystem. 29 The static structure of the vortex clusters remains the same foras m a l l vbecause the change of the vortex-vortex interaction is negligible for ξ m/λe/lessmuch1. Here we derive a resonant condition between the motion of vortex clusters and magnon emission. The magnetic fielddistribution produced by the vortex motion has a dominantwave vector G x=2π/R m, with Rm≈λeas shown in Fig. 1. The unperturbed ordered state has Mz0=0. The resonant condition Gxv=/Omega1(Gx) gives a resonant velocity for vortices moving along the xdirection, vt=γMs/radicalBigg 2JAJ+R2mJ2 A 4π2. (17) This linear analysis is correct as long as the canted MMs satisfy the condition that Mzc≈/Phi10/(JAR2 m)/lessmuchMs[orJA/greatermuch /Phi10/(R2 mMs)]. The oscillation amplitude of MMs and the ac part of the vortex velocity are greatly enhanced in resonance and ηeff increases according to Eq. (16). Two competing processes are involved in the magnetic subsystem: the energy inputfrom vortex motion and the magnetic relaxation. For largedissipation ( α/greatermuch1), the excited magnon is quickly dissipated and the vortex cluster with canted MMs remains stable. Onthe contrary, the incoming energy accumulates for weakmagnetic dissipation, α/lessmuch1, and increases with time. This effect leads to an instability of the magnon excitations thathas been discussed decades ago both experimentally 30and theoretically.31–33For a large enough oscillation amplitude, the MMs are no longer restricted to one of the symmetry-breaking states (there are two degenerate ground states with M x0=±Ms√ 1−m2 z0) and they can flip to the other ground state (with opposite Mx0). Domain walls are then created as shown in Figs. 2(b)–2(d).Mzbecomes large inside the domains walls and this effect increases the coupling betweenthe magnetic subsystem and vortices. For v/greatermuchv t, the cluster structure evolves into vortex stripes along the driving direction[Figs. 2(b)–2(d)]. The domain walls are oriented along the vortex stripes due to the strong attraction between vortices 180506-3RAPID COMMUNICATIONS LIN, BULAEVSKII, AND BATISTA PHYSICAL REVIEW B 86, 180506(R) (2012) FIG. 2. (Color online) (a), (b) Development of the amplitude of the superconducting order parameter |/Psi1|, magnetic structure ( xcomponent of the magnetic moment: Mx), and Bzas the current increases. The vortex positions correspond to the regions with suppressed |/Psi1|. (a) Static configuration with Jext,y=0. In (b)–(d), domain walls are created and the vortex configuration is modulated. |/Psi1|is suppressed (top row) and Bzis maximal (bottom row) in the normal core of vortices. MMs are canted by vortices so Mxis reduced (middle row). and domain walls. V ortex stripes for large driving forces and random pinning potentials have also been observed innumerical simulations without MMs. 34As vortex clusters drive domain walls, the dissipation increases and the vortex velocity(voltage) drops, as shown in Fig. 3. The threshold velocityΔ FIG. 3. (Color online) Difference between the electric fields induced with and without magnetic moments as a function of currentJ ext,y,/Delta1E=EM−EB,w h e r e EMis the electric field for the system with magnetic moments and EBis the electric field for the system without magnetic moments. The vortex viscosity increases whendomain walls are created, resulting in a drop of the electric field (vortex velocity).obtained from simulations where the domain walls are created is compatible with that estimated from Eq. (17). Discussions. The magnetic susceptibility is small, χ0/lessmuch1, in bulk FM superconductors such as UGe 2.35Thus, the attraction between vortices is negligible and the ground stateis a triangular vortex lattice. In the flux flow regime, thevortex lattice is resonant with the oscillations of the MMswhen G·v=/Omega1(G). We predict an enhancement of the vortex viscosity at resonance, which can be probed by the I-V measurement. A large susceptibility, χ 0∼1, is needed to realize the vortex cluster configuration. This requirement canhe fulfilled by some cuprate superconductors with rare-earthelements ( Re), such as ReBa 2Cu3Ox, where Reions order antiferromagnetically below TN∼1 K. Spins are free from the molecular field above the N ´eel temperature TN∼1K and can be easily polarized36,37to mediate the attraction between vortices in the low magnetic field region. The vortexcluster phase can also be achieved in heterostructures ofsuperconductors and ferromagnets with large susceptibility. 38 On the other hand, random pinning centers may preventthe formation of vortex clusters because pinning is strongfor a small vortex densities. However, vortex motion inthe flux flow regime quickly averages out the effect of 180506-4RAPID COMMUNICATIONS VORTEX DYNAMICS IN FERROMAGNETIC ... PHYSICAL REVIEW B 86, 180506(R) (2012) random pinning centers39,40and the cluster structure may be recovered. Acknowledgment. We are indebted to V . Kogan, B. Maiorov, M. Weigand, C. J. Olson Reichhardt,and C. Reichhardt for helpful discussions. The present work is supported by the Los Alamos Laboratory di-rected research and development program with ProjectNo. 20110138ER. 1L. N. Bulaevskii, A. I. Buzdin, M. L. Kulic, and S. V . Panjukov, Adv. Phys. 34, 175 (1985). 2S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R. Julian,P. Monthoux, G. G. Lonzarich, A. Huxley, I. Sheikin, D. Braith-waite, and J. Flouquet, Nature (London) 406, 587 (2000). 3D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flouquet, J. P. Brison, E. Lhotel, and C. Paulsen, Nature (London) 413, 613 (2001). 4C. Pfleiderer, M. Uhlarz, S. M. Hayden, R. V ollmer, H. v.Lohneysen, N. R. Bernhoeft, and G. G. Lonzarich, Nature (London) 412, 58 (2001). 5P. C. Canfield, P. L. Gammel, and D. J. Bishop, Phys. Today 51(10), 40 (1998). 6J. H. Chu, J. G. Analytis, C. Kucharczyk, and I. 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PhysRevB.100.104412.pdf

PHYSICAL REVIEW B 100, 104412 (2019) Enhancement of ultrafast demagnetization rate and Gilbert damping driven by femtosecond laser-induced spin currents in Fe 81Ga19/Ir20Mn 80bilayers Wei Zhang ,1,2Qian Liu,3Zhe Yuan,3Ke Xia,3Wei He,1Qing-feng Zhan,4Xiang-qun Zhang,1and Zhao-hua Cheng1,2,5,* 1State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing 100875, China 4State Key Laboratory of Precision Spectroscopy, School of Physics and Materials Science, East China Normal University, Shanghai 200241, China 5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China (Received 24 October 2018; revised manuscript received 19 July 2019; published 9 September 2019) In spintronics applications, ultrafast spin dynamics have to be controlled at femtosecond timescales via femtosecond laser radiation. At such ultrafast timescales, the effect of the Gilbert damping factor αon ultrafast demagnetization time τMshould be considered. In previous explorations for the relationship between these two parameters, it was found that the theoretical calculations based on the local spin-flip scattering model donot agree with the experimental results. Here, we find that in Fe 81Ga19(FeGa)/Ir 20Mn 80(IrMn) bilayers, the unconventional IrMn thickness dependence of αresults from the competition between spin currents pumped from the ferromagnetic (FM) FeGa layer to the antiferromagnetic (AFM) IrMn layer and those pumped fromthe AFM layer to the FM layer. More importantly, we establish a proportional relationship between the changeof the ultrafast demagnetization rate and the enhancement of Gilbert damping induced by the spin currentsvia interfacial spin chemical potential μ s. Our work builds a bridge to connect the ultrafast demagnetization time and Gilbert damping in ultrafast photoinduced spin-current-dominated systems, which not only explainsthe disagreement between experimental and theoretical results in the relation of τ Mwithαbut provides further insight into ultrafast spin dynamics as well. DOI: 10.1103/PhysRevB.100.104412 I. INTRODUCTION The understanding of spin dynamics from nanosecond down to femtosecond timescales is an essential task towardsthe realization of ultrafast spintronic devices in the frequencyrange from gigahertz to terahertz [ 1,2]. The study of ultra- fast demagnetization time τ Mis one of the most challeng- ing problems in laser-induced ultrafast spin dynamics. TheGilbert damping factor αis of the utmost importance for high-frequency switching of spintronic devices. Since bothτ Mandαrequire a transfer of angular momentum from the electronic system to the lattice, the unification of these twoseemingly unrelated parameters can facilitate the explorationof the microscopic mechanism of laser-induced ultrafast spindynamics. An inversely proportional relationship between τ M andαwas predicted by theoretical calculations based on the local phonon-mediated Elliott-Yafet scattering mechanism[3–5] as well as the stochastic Landau-Lifshitz-Bloch (LLB) model [ 6]. However, the relationship between τ Mandαhas been debated for over one decade [ 7]. Until now, all experi- mental results have shown that τMincreases with α[8–12]. Apart from the local spin-flip scattering mechanism [ 13], we proposed that the nonlocal spin currents should be taken *To whom all correspondence should be addressed: zhcheng@iphy.ac.cninto account to coordinate the contradiction in the relationship between τMandα. Previous work suggested that the superdif- fusive spin current contributed to ultrafast demagnetization[14], while the Gilbert damping could also be enhanced via nonlocal spin currents in ferromagnetic (FM)/nonmagnetic(NM) [ 15] and FM/antiferromagnetic (AFM) heterostruc- tures [ 16]. Femtosecond laser irradiation of ferromagnetic thin films is a fascinating novel approach to create largespin currents [ 17,18]. Figure 1(a) shows that in the case of time-resolved magneto-optical Kerr effect (TRMOKE) exper-iments, hot electrons excited by femtosecond laser pulses cantravel at high velocities and over tens of nanometers throughthe films. The difference of mean free path between spin-majority and spin-minority hot electrons in ferromagnetic thinfilms generates superdiffusive spin currents on femtosecondtimescales. Such spin currents dissipated at the interface of theheterostructure result in the out-of-equilibrium spin accumu-lation represented by spin chemical potential μ s. Moreover, Fig. 1(b) shows that the damped magnetization precession around the effective field could be influenced via spin current.Tveten et al. [19] predicted that the ultrafast demagnetization timeτ Mcould be described in the language of spin-current- induced damping αspin magnetic heterostructures based on the electron-magnon scattering theory. However, the experi-mental evidence on the connection of ultrafast demagnetiza-tion time with damping driven by femtosecond laser-inducedspin currents is not yet understood. 2469-9950/2019/100(10)/104412(11) 104412-1 ©2019 American Physical SocietyWEI ZHANG et al. PHYSICAL REVIEW B 100, 104412 (2019) FIG. 1. Basic concept of both ultrafast demagnetization and spin precession induced by spin currents. (a) The excitation of fem- tosecond laser pulse transforms slow majority-spin delectrons (red) into fast spelectrons, thereby launching a spin current towards the AFM layer. The spin current crossing the interface results in the spin accumulation at the interface represented by spin chemicalpotential μ s. (b) The typical time evolution of magnetization after femtosecond laser irradiation measured by TRMOKE experiment. II. RESULTS A. Sample properties Ir20Mn 80(tIrMn)/Fe 81Ga19(10-nm) bilayers [ 20] were de- posited on optically transparent single-crystalline MgO (001)substrates in a magnetron sputtering system with a basepressure below 3 ×10 −7Torr. The substrates were annealed at 700 °C for 1 h in a vacuum chamber and then held at 250 °Cduring deposition. FeGa layers were obliquely deposited at anincidence angle of 45°. The IrMn layers were deposited whilecontinuously rotating the substrates. In order to induce anexchange bias along the FeGa [010] direction, a magnetic fieldof 500 Oe provided by a permanent magnet was applied alongthe MgO [110] axis during growth. After deposition, a 3-nmprotective Ta layer was deposited on the samples to avoidoxidation. The static longitudinal Kerr loops of Fe 81Ga19 (10 nm) /Ir20Mn 80(tIrMn) along FeGa [010] direction with var- ious AFM IrMn thicknesses ( tIrMn) at room temperature were acquired using a laser diode with a wavelength of 650 nm. Figure 2(a) shows the longitudinal Kerr loops of Fe 81Ga19 (10 nm) /Ir20Mn 80(tIrMnnm) along FeGa [010] direction with various AFM IrMn thicknesses ( tIrMn) at room temperature, whereas the thickness of the FM FeGa layer was fixed at10 nm. For t IrMn/lessorequalslant2 nm, the width of the hysteresis loops is FIG. 2. Static magnetic properties of MgO/Fe 81Ga19(10 nm) / Ir20Mn 80(tnm) bilayers. (a) Longitudinal-MOKE loops with various thicknesses of IrMn layer tIrMn. (b) Coercivity Hcand exchange bias field Hebas a function of IrMn layer thickness tIrMn. enlarged with no obvious shift along the xaxis, implying that the thickness of the IrMn layer is too thin to form an anti-ferromagnetic order for pinning the magnetization reversal ofFeGa [ 21] [insert in Fig. 2(b) (left)]. For t IrMn>2 nm, the an- tiferromagnetic orders are well established, and consequently,the antiferromagnetic moments pin FM moments in reverse toinduce a unidirectional anisotropy [insert in Fig. 2(b) (right)]. The loops therefore evidently exhibit exchange bias behavior.The exchange bias field achieves a value of about 60 Oe when t IrMn>2 nm, while the largest value of coercivity ( ∼72 Oe) occurs at tIrMn=2n m . B. TRMOKE measurements for ultrafast demagnetization and Gilbert damping We performed the polar TRMOKE experiment to measure ultrafast demagnetization time under a saturated applied fieldof 20 kOe in the normal direction of the samples [ 22]. The details of the TRMOKE experiment are described inAppendix A. Figure 3(a) shows the demagnetization curves for various IrMn thicknesses with a maximum magnetization 104412-2ENHANCEMENT OF ULTRAFAST DEMAGNETIZATION … PHYSICAL REVIEW B 100, 104412 (2019) FIG. 3. Ultrafast demagnetization. (a) Ultrafast demagnetization curves with various IrMn layer thicknesses. The solid lines rep- resent the fitting results by Eq. ( 1) in the text. The insert shows the configuration of the measurement for ultrafast demagnetization. (b) Ultrafast demagnetization time as a function of IrMn layer thickness. quenching of ∼10% [ 23,24]. The temporal changes of the Kerr signals /Delta1θk(t) were normalized by the saturation value θkjust before the pump laser excitation. The time evolution of magnetization on subpicosecond timescales can be fittedaccording to Eq. ( 1) in terms of the three-temperature model (3TM) [ 17]: −/Delta1M(t) M =/braceleftbigg/bracketleftbigg/parenleftbiggA1 (t/τ0+1)0.5−A2τE−A1τM τE−τMe−t τM −τE(A1−A2) τE−τMe−t τE/parenrightbigg /Theta1(t)/bracketrightbigg ∗G(t,τG)/bracerightbigg ∗G(t,τG),(1) where∗G(t,τG) represents the convolution product with the Gaussian laser pulse profile, τGis the FWHM of the laser pulses, τMis a step function, and τMis the Dirac δfunction. A1represents the value of/Delta1M(t) Mafter equilibrium between electrons, spins, and lattices. A2is proportional to the initial electron temperature rise. Here, we used the 780-nm laseras the pump pulse to excite the magnetic system out ofequilibrium, while the 390-nm laser pulse was used as aTABLE I. Values of the main fit parameters of ultrafast demag- netizations curves for various thicknesses of the samples. tIrMn(nm) τM(fs) τE(fs) τ0(ps) τG(fs) A1 A2 0 220 ±10 500 5 350 0.8 2 1 160 ±10 500 6 350 0.8 2 2 120 ±10 500 7 350 0.8 2 3 145 ±10 500 4 350 0.8 2 5 200 ±10 500 5 350 0.8 2 probe beam. Therefore, in Eq. ( 1), the state filling effects during pump-probe experiment are neglected due to the dif-ferent wavelengths of pump and probe beams used in thisstudy. The cooling time by heat diffusion is described byτ 0, which should be about 1 order of magnitude larger than τErepresenting the timescale of electron-phonon interactions. The best-fitted value of τE=500 fs for all samples is in good agreement with that of previous reports [ 18]. The fitting parameters in Eq. ( 1) are shown in Table I, from which one notes the pulse width is 350 fs for all the samples. In ourexperimental setup, the time resolution is about 80 fs. In orderto obtain a high time resolution, we measured the ultrafastdemagnetization with a very fine step of time delay (15 fs).The values of ultrafast demagnetization time (120–220 fs)obtained from Eq. ( 1) are defined as the time needed for the magnetization to reach a level of e −1of its maximum demagnetization. The time needed for magnetization to reachits maximum demagnetization ( >500 fs) should be longer than the time extracted from Eq. ( 1). A similar result was reported by V odungbo et al. [25]. The very large temporal stretching of the laser pulse up to 430 fs was attributed to theconversion of the incident laser pulse into a cascade of hotelectrons. This could be one of the possible reasons resultingin the spread of laser pulse on the samples in this study. Viachanging the single parameter τ Mwe can accurately reproduce the experimental results for various samples. The ultrafastdemagnetization time τ Mwas observed to decrease from 220±10 fs for tIrMn=0 nm to 120 ±10 fs for tIrMn=2n m , and then increase with further increasing tIrMn[Fig. 3(b)]. The precessional frequency and damping factor can be derived by means of the TRMOKE signals as well [ 26,27]. Figure 4(a) shows the typical time evolution of the polar component of magnetization after pump laser excitation atdifferent fields applied along the [110] direction of FeGa for t IrMn=2 nm. It is observed clearly that the spin precession process can obviously be influenced by applied fields. Theexact values for fwith various applied fields can be obtained using the damped harmonic function added to an exponential-decaying background: /Delta1M(t)=A+Bexp(−vt)+Cexp/parenleftbigg −t τ/parenrightbigg sin(2πft+ϕ), (2) where Aand Bare the background magnitudes, and vis the background recovery rate. C,τ,f,andϕare the mag- netization precession amplitude, relaxation time, frequency,and phase, respectively. The field dependence of frequency f extracted from the fitting procedure is shown in Fig. 4(b).W e note that the experimental f-Hrelation can be reproduced very 104412-3WEI ZHANG et al. PHYSICAL REVIEW B 100, 104412 (2019) FIG. 4. Spin precession. (a) TRMOKE signals of FeGa/IrMn bilayers with tIrMn=2 nm in various applied fields. (b) Precessional frequency as a function of applied fields. (c) Effective Gilbert damping constant as a function of applied fields. well by the Kittel equation ( 3)[27]: /parenleftbigg2πf γ/parenrightbigg2 =1 M2sH1H2, (3) with H1=− 2Kout+4πM2 s+2Kucos2ϕM+2K1−K1sin2 2ϕM+HM scos(ϕM−ϕH)+KebcosϕMand H2=2K1cos4ϕM+2Kucos2ϕM+MsHcos(ϕM−ϕH) +KebcosϕM. Andγ=γeg/2 is the gyromagnetic ratio. ϕMandϕHare the angles of in-plane equilibrium Mand Hwith respect to the FeGa [010] easy axis. K1,Ku,Keb,andKOutare the in-plane magnetocrystalline, uniaxial, unidirectional, and out-of-plane magnetic anisotropy constants of FeGa films, respec-tively. The value of the magnetocrystalline anisotropy con-stant is K 1=4.5×105erg/cm3for the samples with various AFM layer thicknesses during the fitting procedure and theuniaxial magnetic anisotropy constant K u=(1.5±0.3)× 105erg/cm3.F o r tIrMn=3 and 5 nm, the unidirectional mag- netic anisotropy constant of Keb=3×104erg/cm3has to be included for more accurate fitting, although it is 1 orderof magnitude smaller than those of magnetocrystalline anduniaxial anisotropy. The effective Gilbert damping factor α effshown in Fig. 4(c) is determined from the relaxation time τby Eq. ( 4)[28]: αeff=2/τγ(H1+H2). (4) Since the overall effective damping factor αeffconsists of intrinsic damping and extrinsic damping, whereby the sec-ond one arises from both the two-magnon-scattering andthe dephasing effect in the samples, the overall effectiveGilbert damping factor decreases monotonously to a constantvalue with increasing the applied field [Fig. 4(c)]. As one of the mainly extrinsic contributions, the two-magnon-scatteringinduced damping has been extensively studied in exchange-biased heterostructures [ 29–34]. The mature theory was de- veloped to explain the two-magnon scattering process dueto spatial fluctuations of anisotropy and exchange bias field[30,35]. The two-magnon scattering process comes from the scatterings of the uniform ( k=0) precession mode into nonuniform modes (k /negationslash=0 magnons) that are degenerate in frequency. This process is described by the Hamiltonian, inwhich the spatial fluctuation in the exchange coupling causedby interface roughness determines the scattering strength. Theroughness gives rise to a large fluctuating field because theFM magnetization interacts alternatively with one or the otherAF sublattice via the atomic exchange coupling. It is a well-known relaxation mechanism effective in exchange-biasedheterostructures due to the interface roughness occurring onthe short length scales. When a low external field comparablewith the exchange bias field was applied, the two-magnonscattering effect resulted in the increase of Gilbert dampingwith the exchange bias field according to previous reports[33,34]. However, as shown in Ref. [ 36], a strong enough applied field can be used to exclude the contributions from thetwo-magnon scattering, where the value of Gilbert dampingfactor keeps as a constant with various two-magnon-scatteringstrength. Based on this result, a similar method using strongenough external fields was applied in this study to excludethe two-magnon-scattering effect. Moreover, previous worksshow that the two-magnon-scattering induced damping in-creases with precession frequency because of the increaseddegeneracy of spin waves [ 37,38]. Our work demonstrated that the damping factor keeps almost a constant value at highenough applied fields, indicating the minor contributions from 104412-4ENHANCEMENT OF ULTRAFAST DEMAGNETIZATION … PHYSICAL REVIEW B 100, 104412 (2019) the two-magnon-scattering to Gilbert damping. Besides, it has been demonstrated previously that the two-magnon-scatteringcontributions decrease monotonously with increasing the filmthickness [ 33,34]. This again disagrees with the tendency of thickness dependence of damping at high applied fieldshown in Fig. 5(c). Therefore, in this study, the two-magnon- scattering strength was suppressed effectively by applyinga high enough external field. On the other hand, inhomo-geneities in FeGa thin film may cause variations in the localmagnetic anisotropy field, which leads to the variations ofspin orientations when the external field is not large enoughand gives rise to the enhanced damping arising from the spindephasing effect [ 28]. However, an applied field ( ∼kOe) much larger than the anisotropy field makes the spin orientationuniform; as a result, the dephasing effect is suppressed largely.Based on the above analysis, the intrinsic part of dampingis independent of the external field or precession frequency,while the extrinsic part including both the dephasing effectand the two-magnon-scattering effect are field dependent. Inorder to avoid the effect of the extrinsic damping factor, theintrinsic damping factors were obtained by fitting the overalldamping factor as a function of applied fields with Eq. ( 5) [39,40], shown as the red line in Fig. 4(c): α eff=α+α1e−H/H0, (5) where αandα1e−H/H0are the intrinsic and extrinsic parts of the damping factor, respectively. For the derivation of spin precessional frequency as well as the Gilbert damping, the similar producers as shown abovewere adapted to various samples. Figure 5(a) shows the precessional frequency from oscillation curves with variousIrMn thicknesses. Since the exchange bias field and coercivityare much weaker than the applied fields, the f-Hcurves of FeGa films are therefore slightly different with various AFMlayer thicknesses, which is in contrast to the observation thatthe enhanced uniaxial anisotropy of Fe/CoO bilayers [ 28] greatly increases the precessional frequency. More impor-tantly, we find the effective damping factor α effdecreases with applied fields [Fig. 5(b)]. The solid lines represent the fitting expression shown as Eq. ( 5). Interestingly, the effective Gilbert damping factors drop to as nearly a constant value asthe intrinsic damping factor when the applied fields increaseenough to suppress the extrinsic contributions as stated above. The values of the intrinsic damping factor as a function of the thickness of the IrMn layer are illustrated in Fig. 5(c).I t increases first and reaches the maximum value with the thick-ness of the IrMn layer at t IrMn=2 nm and finally decreases with further increasing the thickness of the IrMn AFM layer.A drastic change of 2.5 times for damping occurs at t IrMn= 2 nm. Similarly, Azzawi et al. showed around 2 times en- hancement of damping in NiFe/Pt bilayers when a continuousPt capping layer is just forming at 0.6 nm by TRMOKE mea-surements [ 41]. Moreover, once a continuous IrMn layer is forming at 2 nm, the accompanied strong intrinsic anisotropyof AFM would contribute partly to the damping enhancementsuperimposed to the spin pumping effect. This has beendemonstrated previously by Zhang et al. , where the damp- ing of Py/IrMn bilayers is 3 or 4 times larger than that inthe Py/Cu/IrMn samples [ 42]. Based on the discussions in Fig. 4, we can exclude the extrinsic mechanisms such as the FIG. 5. Frequency and damping of spin precession. (a) Fre- quency of spin precession as a function of applied fields with various IrMn thicknesses. The solid lines represent the fitting results by Kittle equations. (b) Effective Gilbert damping constants as a function ofapplied fields with various IrMn thicknesses. (c) Intrinsic Gilbert damping as a function of IrMn thickness. two-magnon-scattering and the dephasing effect as the dom- inant contributions to the damping process when the exter-nal fields are high enough [ 43]. Besides, FeGa alloys are 104412-5WEI ZHANG et al. PHYSICAL REVIEW B 100, 104412 (2019) particularly interesting because of their magnetoelastic prop- erties [ 44]. The acoustic waves possibly are triggered by ultra- short laser, and as a result, spin precession would be excitednonthermally via a magnetoelastic effect [ 45]. However, this effect can be excluded based on the following reasons: first,the external field has to be applied along with the hard axisof FeGa; otherwise, the magnetization precession cannot beinduced. It agrees with the fact that the canted magnetizationfrom the easy axis is necessary when the spin precessionarising from instantaneous anisotropy change accompaniedby ultrafast demagnetization occurs [ 26]. In contrast, the occurrence of spin precession from the magnetoelastic effectis independent of initial magnetization orientation. Second,in order to check the contribution of the resonance modefrom the magnetoelastic effect, we performed a fast Fouriertransform in Appendix B. Only the uniform field-dependent precession mode was excited at the present study. This is notthe expected behavior for the acoustically induced modulationof the magneto-optical effects. Therefore, the magnetoelasticeffect of FeGa was largely suppressed in this study. Thisis probably because the laser fluence of around 1 mJ /cm 2 is not high enough to induce a large amplitude of strain pulse. According to Ref. [ 45], the oscillation amplitude of the acoustic mode increases linearly with the laser energy densitywithin the probed range. Moreover, the FeGa material with athickness as thick as 60 nm is preferred to induce an obviousmagnetoelastic behavior [ 46], while 10 nm at the present ex- periment is probably too thin. As a result, the intrinsic damp-ing can be influenced by the following parameters: ( 1)t h e magnetocrystalline anisotropy of FM [ 47], (2) the exchange bias field [ 30,31,36], and ( 3) the spin pumping effect at the interface between FM and AFM [ 15,16,42,48]. In the case of FeGa/IrMn bilayers, the magnetocrystalline anisotropy con-stant of FeGa K 1=4.5×105erg/cm3obtained from Figs. 4 and 5is invariant with the AFM layer thickness. Moreover, referring to Fig. 2(b), it seems that there is no direct rela- tionship between the intrinsic damping factor and the ex-change bias field H eb. When the applied field is far higher than the exchange bias field, both the precessional frequencyand the damping factor show independence of exchange biasfield [ 36]. Therefore, the IrMn thickness dependence of the intrinsic damping is not attributed to the magnetocrystallineanisotropy and the exchange bias field. Due to the strongspin-orbit coupling of the heavy metal (HM) Ir in the IrMnalloy, the contribution of spin pumping to the damping factormust be taken into account. It is noteworthy that the IrMnthickness dependence of damping in FeGa/IrMn is differentfrom that in other normal FM/HM bilayers, where the damp-ing factor increases monotonically with the thickness of theHM layer and approaches a saturation value [ 49]. However, the damping of the FeGa ferromagnetic layer decreases againafter reaching a peak value at t IrMn=2 nm. The change of the damping factor is always accompanied by the spin currenttransfer between FM and AFM layers. More spin currentsabsorbed by the neighboring layer result in larger dampingin the FM layer. An unconventional decrease of the dampingfactor implies that not only does the effect of heavy metal Irin IrMn alloy have to be taken into account but also the anti-ferromagnetic magnetization. The heavy metal Ir serves as aperfect spin sink to absorb the spin currents and consequentlyincreases the damping in FeGa, while the antiferromagnetic magnetization in IrMn serves as a new source to compensatefor the dissipation of magnetization precession and decreasesthe damping of FeGa. C. First-principles calculations for IrMn layer thickness dependence of Gilbert damping To understand the behavior of the IrMn thickness- dependent damping factor, we calculated the damping fac-tor using the scattering theory of magnetization dissipationcombined with the first-principles electronic structure [ 50]. The calculated FM/AFM bilayer structure shown in Fig. 6(a) is the same as that in the experiment. Here, the magneticmoments of AFM sublattices serve as not only a spin sinkto absorb the spin current pumped from the adjacent FMlayer but also a spin current emitter to partly cancel the spinpumping effect of the FM. The interfacial exchange couplingforces the magnetic moments of the IrMn sublattices in a fewlayers near the interface to precess following the adjacent FM,generating spin currents back into the FM layer [Fig. 6(b)]. Based on this model, the enhancement of damping due tothe spin current α sp=/Delta1α=αtIrMn−αtIrMn=0n m as a function of IrMn thickness was calculated and shown as the solidcircle in Fig. 6(c). It increases first to a peak value at t IrMn= 2 nm and then drops with further increasing the IrMn layerthickness. When t IrMn/lessorequalslant2 nm, the thickness of the IrMn layer is too thin to establish the antiferromagnetic order, whichcan be supported by the negligible exchange bias as shownin Fig. 2(b). In this case, the pumped spin current from the AFM back into the FM to partially cancel the spin pumpingeffect by the FM is largely reduced because of the disorder ofthe antiferromagnetic moments, as illustrated on the left sidein Fig. 6(b). In this region, therefore, the magnetic moments in the AFM serve as a perfect spin sink to absorb the spincurrent pumped from the adjacent FM, resulting in a signif-icant enhancement in the damping factor. For the sampleswith the thickness of IrMn t IrMn>2 nm, however, the anti- ferromagnetic order is well established and the accompaniedexchange bias is remarkably large [see Fig. 2(b) and its insert]. Because of the exchange coupling between FM and AFM atthe interface, the magnetic moments of the AFM sublattices ina few layers near the interface are forced to precess followingthe magnetic moment of the FM, while those far away fromthe interface would stay static. Such an exchange spring effectat the interface caused spin precession in the AFM layer,and consequently, spin currents would be transferred fromAFM to the FM layer. Moreover, these spin currents fromthe AFM would be enhanced due to the coherent precessionof magnetization in different sublattices, as illustrated in theright side of Fig. 6(b). The exchange spring-effect-induced precession of the AFM has two effects: (1) the AFM hasintrinsic damping that increases the overall damping of theFM/AFM bilayer, and (2) the precessional motion of magneticmoments in AFM sublattices pumps spin currents into the FM,which partly cancels the spin pumping by the FM. As a result,the overall damping of the bilayers is reduced. From the solidcircles in Fig. 6(c), one can see that the damping decreases with increasing t IrMn when tIrMn>2 nm, indicating that the latter effect of the pumped spin currents is dominant over the 104412-6ENHANCEMENT OF ULTRAFAST DEMAGNETIZATION … PHYSICAL REVIEW B 100, 104412 (2019) FIG. 6. Results of first-principles calculations. (a) Illustration of the ferromagnet (FM)/antiferromagnet (AFM) structure employed to investigate the spin transport. (b) The configuration of the IrMn magnetic moments located at the first layer near the interface. (c) The calculated damping enhancement as a function of the thickness of the antiferromagnetic IrMn. The solid circles show the calculated damping enhancement with the precession of AFM magnetic moments. The solid diamonds show the calculated damping enhancement with perfectly static AFMordered IrMn without precession, while the solid triangles correspond to the calculated values using a static paramagnetic IrMn layer with vanishing Néel order. (d) The experimental damping enhancement as a function of the thickness of antiferromagnetic IrMn. intrinsic damping. Besides, by comparing the calculated and experimental values [Figs. 6(c) and 6(d)], one can find that the calculated Gilbert damping is larger than the experimentalone for t IrMn=1 nm. The reason for the deviation is the assumption of a perfectly flat FeGa/IrMn interface in thecalculation, which leads to a larger spin current pumped fromthe FM. Unfortunately, it is almost impossible to fabricate theperfectly flat film when the thickness is less than 1 nm. In order to separate the contribution of the precession of the magnetic moment of the AFM sublattice to damping, wealso calculated the damping by assuming perfectly static AFMordered IrMn without precession [solid diamonds in Fig. 6(c)] and a paramagnetic IrMn layer with vanishing Néel order[solid triangles in Fig. 6(c)]. The calculated results demon- strate that if the magnetic moments of the AFM sublatticeeither do not precess or align randomly, the IrMn layers serveonly as a perfect spin sink to absorb the spin currents pumpedfrom the adjacent FM, resulting in a significant enhancementof damping. The damping increases monotonically to a satu-ration value with IrMn thickness, which is similar to that ofheavy metals [ 49]. D. Relationship between ultrafast demagnetization rate and Gilbert damping induced by nonlocal spin currents The central strategy of our study is to establish a direct correlation between τMandα.According to Figs. 3(b) and 5(c), we find that the femtosecond laser-induced ultrafast demagnetization time τMand the Gilbert damping αshow an opposite IrMn thickness dependence in FeGa/IrMn bilayers.By plotting τ Mversus αas shown in Fig. 7(a), one can clearly observe that the value of τMdecreases with α, suggesting that spin transport acts as an additional dissipation channelfor accelerating the ultrafast demagnetization and enhancingthe damping. The damping factor α tIrMnfortIrMn>0n m is ascribed to the spin pumping effect induced by variousAFM thicknesses α spand the contribution from the FM it- self,αtIrMn=0n m.To give further insight into the relationship, we replotted Fig. 7(a) by using the change of the ultra- fast demagnetization rate /Delta11 τM=1 τM|tIrMn−1 τM|tIrMn=0n m ver- sus the enhancement of Gilbert damping αsp=/Delta1α=αtIrMn− αtIrMn=0n m induced by the spin current. An approximately linear relationship is confirmed and shown in Fig. 7(b), which can be fitted using Eq. ( 6): /Delta11 τM=μs ¯h/Delta1α, (6) where /Delta11 τM,/Delta1αrepresents the enhancement of ultrafast de- magnetization rate and Gilbert damping induced by the spincurrent, respectively, μ sis the spin chemical potential, and ¯his the Planck constant. (For the derivation of Eq. ( 6), please see Appendix Dfor details). A reasonable value of μs≈1e V , which is similar to that of spin splitting in 3 dtransition metals, was obtained by the linear fitting using Eq. ( 6). The spin chemical potential μsis proportional to spin accumulations at the interface between different layers. It con-tributes largely to ultrafast demagnetization according to themodel of laser-induced ultrafast superdiffusive spin transportin layered heterostructures [ 14,51]. There is a large difference in velocities or lifetimes for spin-dependent hot electrons[52]. As a result, the transport properties of hot electrons are spin dependent. For instance, the minority-spin electrons ex-cited by an ultrashort laser survive for only a very shorttime, and they decay to nonmobile bands approximately at theposition they were excited. Instead, majority-spin electronshave longer lifetimes and higher velocities, so they leavefast from the excitation region after being created, in part aresult of the demagnetization process. Because the directions 104412-7WEI ZHANG et al. PHYSICAL REVIEW B 100, 104412 (2019) FIG. 7. (a) Ultrafast demagnetization time as a function of Gilbert damping. (b) The variation of ultrafast demagnetization rateas a function of Gilbert damping enhancement. The red line indicates the fitting via Eq. ( 6) in the text. of motion for all the electrons are random, they can obtain a velocity directed back towards the ferromagnetic film. Asecond part of the demagnetization is ascribed to the backflowof spin-minority electrons from the substrate or the neigh-boring layer. Spin-majority electrons entering the ferromag-netic layer will find good transport properties and continuediffusing without severely decaying. However, spin-minorityelectrons experience a considerable worsening of the transportproperties as soon as they enter the ferromagnetic layer. Theconsequence is that they are trapped at the entrance of theferromagnetic layer, giving rise to the spin accumulations atthe interface. Nevertheless, the quantitative description forspin accumulations during ultrashort laser-induced demagne-tization in heterostructures is still lacking. This work aimsat filling this gap by relating ultrafast demagnetization timeand Gilbert damping. A detailed calculation for the value of 1eV for spin chemical potential obtained in this experiment ishighly desirable. The nonlocal spin currents dissipated at the interface of FeGa/IrMn open an additional channel to accelerate the ul-trafast demagnetization and enhance the Gilbert damping.However, in the case of the sample with t IrMn=0 nm with- out the assistant AFM layer, both the local spin-flip andnonlocal spin transport mechanisms probably contribute tothe ultrafast demagnetization in the ferromagnetic layer. Forinstance, based on the breathing Fermi-surface model of the Gilbert damping and the Elliott-Yafet relation for the spin-relaxation time, a relation shown as Eq. ( 7) is established between the conductivitylike Gilbert damping αand ultrafast demagnetization time τ M[10]: τM=M γFelpb2α. (7) Taking the values of τM|tIrMn=0n m andα|tIrMn=0n m as 220 fs and 0.004, respectively, a value of α/τ M=1.8×1010s−1 is derived. This value is reasonable and agrees well with that of 3 dtransition metal Ni calculated by the breathing Fermi-surface model [ 53], indicating that the ultrafast de- magnetization of ferromagnetic FeGa film itself is mainlygoverned by the local spin-flip scattering events. Nonetheless,we note that ultrafast demagnetization in the ferromagneticlayer was accelerated and the Gilbert damping was enhancedvia the interfacial spin accumulations once the IrMn layer wasattached. III. CONCLUSIONS The unconventional IrMn thickness dependence of αis attributed to the cancellation of the spin currents pumped fromthe AFM IrMn layer to the FM FeGa layer. We establisha proportional relationship between the change of ultrafastdemagnetization rate and the enhancement of Gilbert dampinginduced by the spin currents via the interfacial spin chemicalpotential. This result can facilitate the utilization of ultrafastspintronic devices in the terahertz region. ACKNOWLEDGMENTS This work is supported by the National Key Re- search Program of China (Grants No. 2015CB921403, No.2016YFA0300701, and No. 2017YFB0702702), the NationalNatural Sciences Foundation of China (Grants No. 91622126,No. 51427801, and No. 51671212), and the Key ResearchProgram of Frontier Sciences, CAS (Grants No. QYZDJ-SSW-JSC023, No. KJZD-SW-M01, and No. ZDYZ2012-2).The work at Beijing Normal University is partly supported bythe National Natural Sciences Foundation of China (GrantsNo. 61774017, No. 61704018, and No. 11734004), the Re-cruitment Program of Global Youth Experts, and the Funda-mental Research Funds for the Central Universities (Grant No.2018EYT03). The work at East China Normal University ispartly supported by the National Natural Sciences Foundationof China (Grant No. 11874150). APPENDIX A: TIME-RESOLVED MAGNETO-OPTICAL KERR EFFECT MEASUREMENTS In this study, the dynamical process of fast and ultra- fast spin dynamics was measured by time-resolved magneto-optical Kerr effect (TRMOKE) measurements. The experi-ments were carried out using an all-optical pump-probe tech-nique. A train of optical pulses with a wavelength of 780 nm,55-fs duration, and 100 nJ /pulse is generated at 5.2 MHz repetition rate by a Ti:sapphire oscillator (Femtolaser, XL-100). A 200- μm thickness beta barium borate (BBO) crystal 104412-8ENHANCEMENT OF ULTRAFAST DEMAGNETIZATION … PHYSICAL REVIEW B 100, 104412 (2019) FIG. 8. Scheme of TRMOKE experiment for spin precession dynamics. was used to double the frequency of the femtosecond laser. The laser beam from the source is split into both 780- and390-nm beams. We use the 780-nm laser as the pump pulseto excite the magnetic system out of equilibrium, while the390-nm laser pulse was used as a probe beam to measure thesubsequent magnetization dynamics with the timescale fromsubpicosecond to nanosecond. The pump laser beam is muchstronger than the probe, with an intensity ratio of about 100for all the measurements. Both the pump and probe beamsare incident along the normal axis ( zaxis) of the samples. The detection geometry is only sensitive to the out-of-planecomponent of the magnetization M z. For fast spin dynamics, we applied various external fields along the Fe 81Ga19[110] direction to trigger the spin precession, while a large enoughfield of about 20 kOe was applied along the Fe 81Ga19[001] direction to obtain the ultrafast demagnetization curves. Weadjusted the pump laser fluence from 1 to 1 .25 mJ/cm 2to ob- tain the same maximum quenching for various samples. Thepump and probe beams are focused onto the samples with spotdiameters of ∼10 and ∼5μm, respectively, via an objective lens. For the spin precession measurements, the scheme ofthe TRMOKE experiment is illustrated in Fig. 8. The signals are sensitive to the polar component of magnetization afterpump laser excitation at different fields applied along the[110] direction of FeGa. APPENDIX B: FAST FOURIER TRANSFORM ANALYSIS The ferromagnetic FeGa is a famous material for its mag- netoelastic properties. After femtosecond laser irradiation, anexternal field-independent resonance mode is triggered due tothe excitation of coherent acoustic phonons. However, onlyone field-dependent resonance mode was excited in this studyaccording to fast Fourier transform analysis in Fig. 9. APPENDIX C: FIRST-PRINCIPLES CALCULATIONS The electronic structure of the FeGa/IrMn bilayer is cal- culated self-consistently using the local density approxima-tion of the density functional theory. The spin-dependentpotentials, charge and spin densities are obtained with theminimal basis of tight-binding linear muffin-tin orbitals. Inthe calculation of the total damping, the scattering region,consisting of the repeated FeGa/IrMn bilayers, is connectedto two semi-infinite Cu leads. We have introduced the thermallattice disorder into a 4 ×4 supercell and displaced the atoms FIG. 9. Fourier transform spectra measured between 0.85 and 3.0 kOe for tIrMn=2n m . in the scattering region randomly away from their equilibrium positions with a Gaussian distribution. The root-mean-squareatomic displacements of the Gaussian distribution are deter-mined using a simple Debye model with a Debye temperatureof 470 K. The two-dimensional Brillouin zone of the supercellis sampled by a 24 ×24kmesh corresponding to the 96 × 96 mesh for the Brillouin zone for the 1 ×1 unit cell. The effect of magnons in the FM FeGa is neglected in our calcu-lation. This is because the magnetic damping is dominated byelectrons at the Fermi level in metals, which can efficientlytransfer spin angular momentum into the orbital motion viaspin-orbit interaction. In metals and alloys, the influence ofmagnon-phonon coupling is negligible, except for near theCurie temperature [ 54]. If magnetization precession occurs only in the FM FeGa layer, the calculated damping enhancement does not sensi-tively depend on the specific order of the AFM IrMn. Herewe take two limits: the perfectly antiferromagnetic orderedIrMn and the paramagnetic IrMn. (The magnetic moments ofMn are randomly distributed such that both the Néel orderand total magnetization vanish). The damping enhancementscalculated for the two cases are nearly identical, where thedamping factor is enhanced and saturates at a thickness of2 nm. It indicates that the pumped spin current by the pre-cessional FeGa is immediately absorbed by the IrMn layer.The large moment on the Mn atom can absorb the pumpedtransverse spin current efficiently. On the other hand, the AFMIrMn is forced to precess due to the interfacial exchangecoupling; however, the efficiency of the spin current genera-tion by AFM depends on its specific order. It is suppressedlargely in the case of paramagnetic IrMn because of the 104412-9WEI ZHANG et al. PHYSICAL REVIEW B 100, 104412 (2019) cancellation via magnetic moments with various orientations shown on the left side of Fig. 6(b) in the main text. In contrast, the efficiency of the spin current generation by theAFM is enhanced remarkably by the coherent precession ofthe ordered magnetic moments shown in the right side ofFig.6(b) in the main text. The cone angle of precessional IrMn is modeled to exponentially decay from the interface witha typical decay length of 2 nm. The precessional AFM hasmainly two contributions to the damping enhancement of thebilayer. First, the AFM has intrinsic damping that increasesthe total energy loss during the magnetization dynamics. Thesecond effect is that the precessional AFM pumps spin currentinto the FM that cancels partly the spin pumping by the FMand decreases the damping enhancement. APPENDIX D: DERIVATION OF EQ. ( 6)I N THE MAIN TEXT It is well known that the magnetic moment /vectorMsis propor- tional to the spin angular momentum /vectorSvia gyromagnetic ratio γ=gμB ¯h, /vectorMs=γ/vectorS, (D1) where gis the Landé factor andμBis the Bohr magneton. Normally, we take /vectorM=V/vectorMsas the total magnetic moments, where Vis the volume of the atom. τMis the ultrafast demagnetization time. Therefore, the value of1 τMis taken as the demagnetization rate. The de- magnetization is always accompanied by dissipation of thespin angular momentum, and hence the rate of spin angularmomentum dissipation is /vectorm γ·1 τM. (D2)On the other hand, the spin current−→jsper unit area generated by spin pumping effect reads −→js=1 4πgeff−→μs, (D3) where geffis the effective interfacial spin-mixing conductance including the influence of the backflow spin current fromthe AFM IrMn to FeGa, and /vectorμ sis the spin-accumulation- driven chemical potential. The pumped spin current across the interface is−→Is=−→jsA, where Ais the area of the interface: geff=4πMsd/Delta1α gμB, (D4) where dis the thickness of the ferromagnetic layer, /Delta1α= αtIrMn−αtIrMn=0 nm is the enhancement of Gilbert damping induced by the absorption and generation of spin current viavarious IrMn thicknesses. 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PhysRevB.82.155324.pdf

Random walk approach to spin dynamics in a two-dimensional electron gas with spin-orbit coupling Luyi Yang, J. Orenstein, and Dung-Hai Lee Department of Physics, University of California, Berkeley, California 94720, USA and Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA /H20849Received 30 July 2010; revised manuscript received 27 September 2010; published 26 October 2010 /H20850 We introduce and solve a semiclassical random walk /H20849RW /H20850model that describes the dynamics of spin polarization waves in zinc-blende semiconductor quantum wells. We derive the dispersion relations for thesewaves, including the Rashba, linear and cubic Dresselhaus spin-orbit interactions, as well as the effects of anelectric field applied parallel to the spin polarization wave vector. In agreement with calculations based onquantum kinetic theory /H20851P. Kleinert and V. V. Bryksin, Phys. Rev. B 76, 205326 /H208492007 /H20850/H20852, the RW approach predicts that spin waves acquire a phase velocity in the presence of the field that crosses zero at a nonzero wavevector, q 0. In addition, we show that the spin-wave decay rate is independent of field at q0but increases as /H20849q−q0/H208502forq/HS11005q0. These predictions can be tested experimentally by suitable transient spin grating experiments. DOI: 10.1103/PhysRevB.82.155324 PACS number /H20849s/H20850: 72.25. /H11002b, 72.10. /H11002d I. INTRODUCTION Spin-orbit /H20849SO/H20850coupled two-dimensional electron sys- tems are of great interest, both as model systems and as theactive component of devices that control electron spin withelectric fields. 1Unfortunately, the potential of the SO inter- action to control electron spin comes with a price—the SOterms in the Hamiltonian break SU /H208492/H20850spin symmetry. The violation of SU /H208492/H20850means that electron-spin polarization is not conserved, decaying instead with a characteristic spinmemory time /H9270s. The mechanism by which SO coupling leads to spin memory loss has been intensively investigatedin two-dimensional electron gases /H208492DEGs /H20850in semiconduc- tor quantum wells /H20849QWs /H20850, as described in recent reviews. 2,3 In GaAs QWs and related systems, breaking of inversion symmetry allows SO coupling that is linear in the electronwave vector k. 4–6The SO terms in the Hamiltonian can be viewed as effective magnetic fields that act only on the elec-tron spin, with magnitude and direction that vary with k. The loss of spin memory in the effective magnetic field, b/H20849k/H20850, takes place through the D’yakonov-Perel’ /H20849DP/H20850 mechanism. 7–10In this process the electron spin precesses during its ballistic motion between collisions; each time it isscattered b/H20849k/H20850and consequently the precession vector, /H9024/H20849k/H20850, change. The net result is exponential decay of spin polariza-tion at a rate approximately equal to /H9024 2/H9270, where /H9270is the mean time between collisions. There exist two distinct contributions to b/H20849k/H20850, the Rashba term5,6arising from asymmetry of the confining potential and the Dresselhaus term11originating in the intrinsic inversion asymmetry of the GaAs crystal structure. A prescription forlengthening spin lifetime in QWs of III-V semiconductors bytuning the Rashba coupling strength /H20849 /H9251/H20850to equal the linear Dresselhaus coupling /H20849/H92521/H20850was proposed by Schliemann et al.12Recently it was recognized that this mechanism amounts to a restoration of SU /H208492/H20850symmetry even in the pres- ence of anisotropic SO interactions.13The main purpose of this paper is to assess theoretically to what extent tuning SOinteractions can be expected to increase the distance overwhich electron-spin polarization can propagate without decay. The potential to extend the spin propagation length de- spite DP spin memory decay is based on the strong correla-tion between the electron’s displacement in space and therotation of its spin on the Bloch sphere. An important steptoward a quantitative theory of such correlations was madeby Burkov et al. 14and Mishchenko et al. ,15who derived equations of motion that describe the coupling of spin andcharge current degrees of freedom in /H20849001 /H20850GaAs QWs. Ini- tially only the linear Rashba SO coupling was examined,subsequently Bernevig et al. 13and Stanescu and Galitski16 extended the theory to include the linear and cubic Dressel- haus terms, respectively. The equations of motion can be solved to obtain the nor- mal modes of the coupled system, which are waves of mixedelectrical current and spin polarization. There exist four suchmodes, reflecting three spin degrees of freedom /H20849S x,Sy, and Sz/H20850and the charge density, n. For wave vectors, q, parallel to the directions /H20851110/H20852and /H2085111¯0/H20852, the four modes decouple into two pairs; in one the spin precesses in a plane containing q and the normal direction zˆ, in the other the current is coupled to the component of in-plane spin polarization perpendiculartoq. The spin precession mode is the one relevant to spin po- larization memory. For example, the decay rate of this modeatq=0 is precisely the DP decay rate, 1 / /H9270s. In the absence of spin-space correlation, the decay rate, /H9253q, of a spin polariza- tion wave would increase monotonically with q, i.e., /H9253q=1 //H9270s+Dsq2, where Dsis the spin diffusion coefficient. Instead, it was predicted14that for Rashba SO coupling the minimum decay rate occurs at nonzero wave vector, at whichpoint /H9253qis approximately half the DP rate. Bernevig et al.13 showed theoretically that the minimum /H9253qis further reduced when both Rashba and linear Dresselhaus interactions arenonzero and vanishes when the strength of the two couplingsis equal. The resulting “persistent spin helix” /H20849PSH /H20850was shown to be a conserved quantity of a newly found SU /H208492/H20850 symmetry that arises when /H9251=/H92521and the cubic DresselhausPHYSICAL REVIEW B 82, 155324 /H208492010 /H20850 1098-0121/2010/82 /H2084915/H20850/155324 /H208497/H20850 ©2010 The American Physical Society 155324-1term /H20849/H92523/H20850is zero.13However, Stanescu and Galitski16 showed that perfect SU /H208492/H20850is broken when /H92523/HS110050, leading to large, but not infinite, PSH lifetime. Recently, using the tran-sient spin grating technique, Koralek et al. 17observed the PSH mode experimentally by independently tuning theRashba and linear Dresselhaus couplings. The question that arises is whether the PSH effect can be exploited to lengthen the distance that a packet of spin po-larization can propagate in an applied electric field. In thispaper we address this question by analyzing the effects of anin-plane electric /H20849E/H20850field on the spin-precession modes. We focus on E /H20648q, which is the orientation relevant to the drift of spin polarization. To predict the spin memory length it isnecessary to determine how the applied field modifies boththe real /H20849R/H20850and imaginary /H20849I/H20850parts of the normal-mode frequency, /H9275/H20849q/H20850of spin-polarization modes. The real part is related to the drift velocity whereas the imaginary part isrelated to the lifetime. The modification of R/H20853 /H9275/H20849q/H20850/H20854is linear inE/H20849to lowest order /H20850, whereas the affect of EonI/H20853/H9275/H20849q/H20850/H20854is quadratic. Kleinert and Bryksin18,19recently have treated this to problem to linear order in E, using quantum kinetic theory, and obtained results for R/H20853/H9275/H20849q/H20850/H20854. In this work, we derive and solve equations of motion to quadratic order in Eusing a random walk /H20849RW /H20850approach that is different from previous treatments of this problem.The advantages of our approach are physical transparencyand mathematical simplicity. We construct a semiclassicalrandom walk model that tracks the electron’s motion in realspace and the propagation of its spin on the Bloch sphere. InSec. II, we introduce the random walk model, derive the equations of motion in the absence of an Efield, and solve for the spin-wave dispersion relations. We compare the re-sults thus obtained with the earlier quantum kinetic theoryapproaches. 13,16In Sec. III, we include an in-plane Efield, obtaining the equations of motion and the dispersion rela-tions to quadratic order. We use the dispersion relations toanalyze the motion of a spin-polarization packet in the pres-ence of the in-plane field, for different regimes of fieldstrength. We illustrate the results by focusing on representa-tive SO couplings: linear Dresselhaus coupling only, theSU/H208492/H20850case where Rashba and Dresselhaus terms are equal, and the case of SU /H208492/H20850broken by a small cubic Dresselhaus term. A brief summary is given in Sec. IV. II. RANDOM WALK MODEL As mentioned above, as an electron propagates between scattering events, SO coupling causes its spin to precess.Thus, as the electron performs an RW in real space, its spinperforms an RW on the Bloch sphere. We consider a 2Delectron gas with both structure and bulk inversion asymme-try. The SO Hamiltonian for conduction band electrons in aIII-V semiconductor QW grown in the /H20851001 /H20852direction /H20849taken aszˆdirection /H20850is given by H SO=/H9024·s, /H208491/H20850 where/H9024=2kF/H20877xˆ/H20875/H9251−/H92521−2/H92523/H20849vx2−vy2/H20850 vF2/H20876vy −yˆ/H20875/H9251+/H92521−2/H92523/H20849vx2−vy2/H20850 vF2/H20876vx/H20878, /H208492/H20850 s=/H6036/H9268/2 is the electron spin, vxandvyare the components of velocity in the /H20851110/H20852and /H2085111¯0/H20852directions, /H9251,/H92521, and/H92523are dimensionless quantities describing the strength of theRashba, linear, and cubic Dresselhaus spin-orbit couplings,respectively, and k Fis the Fermi wave vector. Spins precess about the effective SO field according to ds dt=/H9024/H11003s. /H208493/H20850 We assume that the impurity potential is short range so that there is no correlation between the scattering events. Inthe absence of the Efield, electrons perform an isotropic 2D random walk with vn/H20849velocity between the nth and /H20849n+1/H20850th scattering events /H20850given by vFtˆn, where tˆn=/H20849cos/H9258,sin/H9258/H20850is a random two-dimensional unit vector with a uniform probability density pn/H20849/H9258/H20850=1 /2/H9266. The dis- placement from nth to /H20849n+1/H20850th step is given by rn+1−rn=vn/H9270, /H208494/H20850 where /H9270is the electron scattering time. In the following we consider /H9024/H9270, the change in angle of the electron’s spin be- tween scattering events, as a small parameter. In this case wecan obtain from Eq. /H208493/H20850the change in the spin direction during the mean-free time as a series expansion in /H9024 /H9270, /H9004sn/H11013sn+1−sn=/H9024n/H9270/H11003sn+1 2/H9024n/H9270/H11003/H20849/H9024n/H9270/H11003sn/H20850,/H208495/H20850 where we retain terms to second order. LetPn/H20849r/H20850be the probability that after nsteps of random walk the electron arrives at position rand Dn/H20849r;s/H20850be the conditional probability that given the electron is at r, its spin iss. The joint probability Pn/H20849r/H20850Dn/H20849r;s/H20850satisfies the following recursion relation: Pn+1/H20849r/H20850Dn+1/H20849r;s/H20850=/H20855Pn/H20849r−vn/H9270/H20850Dn/H20849r−vn/H9270;s−/H9004sn/H20850/H20856, /H208496/H20850 where /H20855/H20856 denotes average over tˆn, i.e., /H20855An/H20856=/H2084802/H9266An/H20849/H9258/H20850pn/H20849/H9258/H20850d/H9258. Once Pn/H20849r/H20850Dn/H20849r;s/H20850is determined, the magnetization can be obtained from the following inte-gral on the Bloch sphere: m n/H20849r/H20850=/H20885 S2sPn/H20849r/H20850Dn/H20849r;s/H20850d/H9018. /H208497/H20850 By substituting Eq. /H208496/H20850into Eq. /H208497/H20850, we obtain, mn+1/H20849r/H20850=/H20883/H20885 S2sPn/H20849r−vn/H9270/H20850Dn/H20849r−vn/H9270;s−/H9004sn/H20850d/H9018/H20884. /H208498/H20850 Taylor series expansion on the right hand side of Eq. /H208498/H20850 yieldsYANG, ORENSTEIN, AND LEE PHYSICAL REVIEW B 82, 155324 /H208492010 /H20850 155324-2mn+1/H20849r/H20850=/H20883/H20885 S2/H20875s+/H9024n/H9270/H11003s+1 2/H9024n/H9270/H11003/H20849/H9024n/H9270/H11003s/H20850/H20876 /H11003/H20877Pn/H20849r/H20850Dn/H20849r;s/H20850−vn/H9270·/H11612/H20851Pn/H20849r/H20850Dn/H20849r;s/H20850/H20852 +1 2vn/H9270·/H11612/H11612/H20851Pn/H20849r/H20850Dn/H20849r;s/H20850/H20852·vn/H9270/H20878d/H9018/H20884. /H208499/H20850 Again retaining terms to second order, we can write mn+1=I1+I2+I3, /H2084910/H20850 where I1=/H20883/H20885 S2s/H20877Pn/H20849r/H20850Dn/H20849r;s/H20850−vn/H9270·/H11612/H20851Pn/H20849r/H20850Dn/H20849r;s/H20850/H20852 +1 2vn/H9270·/H11612/H11612/H20851Pn/H20849r/H20850Dn/H20849r;s/H20850/H20852·vn/H9270/H20878d/H9018/H20884, /H2084911/H20850 I2=/H20883/H20885 S2/H20851/H9024n/H9270/H11003s/H20852/H20853Pn/H20849r/H20850Dn/H20849r;s/H20850 −vn/H9270·/H11612/H20851Pn/H20849r/H20850Dn/H20849r;s/H20850/H20852/H20854d/H9018/H20884, /H2084912/H20850 and I3/H20849r/H20850=/H20883/H20885 S2/H208751 2/H9024n/H9270/H11003/H20849/H9024n/H9270/H11003s/H20850/H20876/H20853Pn/H20849r/H20850Dn/H20849r;s/H20850/H20854d/H9018/H20884. /H2084913/H20850 Upon performing the average over tˆn, all terms that linear in vnor/H9024nvanish by symmetry, leading to, I1=mn+/H9016op/H92702mn, /H2084914/H20850 I2=−xˆ/H20855/H9024nyvnx/H20856/H92702/H11509mnz /H11509x+yˆ/H20855/H9024nxvny/H20856/H92702/H11509mnz /H11509y +zˆ/H20873/H20855/H9024nyvnx/H20856/H11509mnx /H11509x−/H20855/H9024nxvny/H20856/H11509mny /H11509y/H20874/H92702, /H2084915/H20850 I3=−/H92702 2/H20849xˆ/H20855/H9024yn2/H20856mnx+yˆ/H20855/H9024xn2/H20856mny+zˆ/H20855/H9024n2/H20856mnz/H20850, /H2084916/H20850 where /H9016op/H110131 2/H20873/H20855vx2/H20856/H115092 /H11509x2+/H20855vy2/H20856/H115092 /H11509y2/H20874. /H2084917/H20850 Taking the continuum limit mn→m/H20849t/H20850,/H20849mn+1−mn/H20850//H9270 →dm/dt, and substituting into Eq. /H2084910/H20850, we obtain the equa- tion of motion for the magnetization vector. Resolving thevector equation into components yields three scalar equa-tions, 1 /H9270/H11509mx /H11509t=/H9016opmx−1 2/H20855/H9024y2/H20856mx−/H20855/H9024yvx/H20856/H11509mz /H11509x, /H2084918/H208501 /H9270/H11509my /H11509t=/H9016opmy−1 2/H20855/H9024x2/H20856my+/H20855/H9024xvy/H20856/H11509mz /H11509y, /H2084919/H20850 1 /H9270/H11509mz /H11509t=/H9016opmz−1 2/H20855/H90242/H20856mz+/H20855/H9024yvx/H20856/H11509mx /H11509x−/H20855/H9024xvy/H20856/H11509my /H11509y. /H2084920/H20850 Solving the equations of motion for eigenmodes with wave vector parallel to xˆyields the dispersion relation, i/H9275/H11006/H20849q/H20850 /H9270=1 4/H208492/H20855/H90242/H20856−/H20855/H9024x2/H20856/H20850 +1 2/H20855vx2/H20856q2/H11006/H20881/H20855/H9024x2/H208562 16+q2/H20855/H9024yvx/H208562. /H2084921/H20850 This dispersion relation corresponds to modes in which the spin polarization spirals in the x-zplane. Note that /H9275/H20849q/H20850is purely imaginary so that for all wave vectors the spin-polarization wave decays exponentially with time. However,the dispersion relation differs from ordinary diffusion, wherei /H9275/H110081//H9270+Dq2. The difference can be traced to the terms in Eq. /H2084915/H20850that are proportional to the first derivative of spin density with respect to position—these terms are absent inthe usual diffusion equation. The coefficients of these addi-tional terms are the cross-correlation functions, /H20855/H9024 xvy/H20856and /H20855/H9024yvx/H20856, which shows explicitly that the anomalous diffusion is a consequence of the correlation between the electron’smotion in real space and the propagation of its spin on theBloch sphere. In the SU /H208492/H20850case /H20849 /H9251=/H92521and/H92523=0/H20850, Eq. /H2084921/H20850simplifies to i/H9275/H11006/H20849q/H20850=1 4vF2/H9270/H20849q/H11006q0/H208502/H11013D/H20849q/H11006q0/H208502, /H2084922/H20850 where D/H11013vF2/H9270/4 and q0/H110134kF/H92521. The vanishing decay rate of the /H9275−mode at q=q0indicates the appearance of a con- served quantity—a helical spin-polarization wave or persis-tent spin helix. 13 The dispersion relations obtained above for the spiral po- larization waves are the same as those obtained previously,including the cubic Dresselhaus term. 13,16We note, however, that while the RW approach accurately describes the spiralcoupling of x-zcomponents of spin, it does not capture the coupling between charge current and the ycomponent of spin that appears in the quantum kinetic formulation. This isbecause the RW approach does not include relaxation to theequilibrium state. In other words, between consecutive scat-tering events the electron’s spin precesses about b/H20849k/H20850but has no tendency to spiral in toward it. Thus the well-knowncurrent-induced spin-polarization /H20849CISP /H20850effect 15is not pre- dicted. To recover CISP requires adding to Eq. /H208493/H20850a phenom- enological Gilbert damping term, ds dt=/H9261Gs/H11003/H20849/H9024/H11003s/H20850, /H2084923/H20850 where /H9261Gis the damping parameter.RANDOM WALK APPROACH TO SPIN DYNAMICS IN A … PHYSICAL REVIEW B 82, 155324 /H208492010 /H20850 155324-3III. SPIN HELIX DYNAMICS IN THE PRESENCE OF AN ELECTRIC FIELD In this section, we explore how the spin dynamics change in the presence of an Efield parallel to the wave vector of the spin spiral. To include the effect of Ewe add a drift term to the velocity at each random walk step, vn=vFtˆn+vdxˆ, /H2084924/H20850 where vdis the drift velocity assumed to be a linear function ofE. We assume further that the electric field does not change the shape of the impurity potential and therefore thescattering probability density is still uniform. The drift velocity modifies the precession vector, adding a fixed precession /H9024 d/H11013−2yˆkF/H20875/H9251+/H92521−2/H92523/H20849vx2−vy2/H20850 vF2/H20876vd, /H2084925/H20850 to/H9024nat each step of the random walk. Substituting and following the same strategy as before, we obtain I1/H20849E/H20850=I1−vd/H9270/H11509m /H11509x, /H2084926/H20850 I2/H20849E/H20850=I2+/H9024d/H9270/H11003m, /H2084927/H20850 I3/H20849E/H20850=I3, /H2084928/H20850 where the I1,2,3/H20849E/H20850are the quantities I1,2,3evaluated in the presence of the electric field. The field alters the equations ofmotion in two ways. First, new terms appear that are linear inE. The new term added to I 1converts the time derivative of mto the convective derivative that is the time derivative in a frame moving with the drifting electrons. The term added toI 2indicates that the Efield introduces uniform precession about the yˆaxis, when viewed in the frame comoving with vd. The second type of modification is quadratic in E; the field increases /H20855/H9024y2/H20856by the additive factor /H9024d2and the mean- square velocity /H20855vx2/H20856by the factor /H20855vd2/H20856. Solving for normal modes with wave vector parallel to xˆ, we obtain i/H9275/H11006/H20849q/H20850=1 4/H208492/H20855/H90242/H20856−/H20855/H9024x2/H20856/H20850/H9270+1 2/H20855vx2/H20856/H9270q2 +ivdq/H11006/H20881/H20855/H9024x2/H208562/H92702 16+/H20849q/H20855/H9024yvx/H20856/H9270+i/H9024d/H208502. /H2084929/H20850 To linear order in E, this dispersion relation is the same as that obtained by Kleinert and Bryksin.18,19In the presence of the electric field /H9275/H20849q/H20850acquires a real part, which describes the propagation of spin polarization. Equation /H2084929/H20850also de- scribes the modifications of the spin-polarization lifetimethat appear at second order in E. In the following we discuss the spin dynamics that emerge from this dispersion relationfor representative SO Hamiltonians.A. SU(2) case For the case of /H9251=/H92521,/H92523=0, the dispersion relation sim- plifies to i/H9275/H11006/H20849q/H20850=D/H208491+2/H92612/H20850/H20849q/H11006q0/H208502+ivd/H20849q/H11006q0/H20850, /H2084930/H20850 where /H9261/H11013vd/vF. To distinguish the lifetime and propagation effects we write the dispersion relation in the form i/H9275/H20849q/H20850=/H9253/H20849q/H20850+i/H9278˙/H20849q/H20850, /H2084931/H20850 where /H9253/H20849q/H20850is the decay rate and /H9278˙/H20849q/H20850is the rate of phase advance. The real and imaginary parts of i/H9275−/H20849q/H20850, correspond- ing to the longer lived of the two modes, are plotted in Fig.1. As is apparent from Fig. 1/H20849a/H20850, the spin-polarization life- time, 1 / /H9253−/H20849q/H20850remains infinite at the PSH wave vector, de- spite the presence of the electric field. This result is consis-tent with the theoretical prediction that at the SU /H208492/H20850point the spin helix generation operators commute with all perturba-tion terms that are not explicitly spin dependent. 13However, the field increases the effective diffusion constant by the fac-tor/H9261 2so that the decay rate for q/HS11005q0increases rapidly when the drift velocity approaches the thermal velocity of the elec-trons. The spin helix generation operators will not commutewith the Hamiltonion if there exists a spatial disorder of SOinteractions. 20,21 The rate of phase advance /H20851plotted in Fig. 1/H20849b/H20850/H20852vanishes atq=q0, i.e., the PSH is stationary, despite the fact that the Fermi sea of electrons is moving by with average velocity vd. Moreover, spin spirals with q/H11021q0will appear to move backward, that is, opposite to the direction of electron flow.Although unusual, this property can be understood by con-sidering the spin dynamics in a frame moving with velocity vd. In this frame Eparallel to xˆis perceived as a precession vector /H9024d=−4/H92521vdyˆ=−vdq0yˆ. Therefore in the moving frame /H9278/H11006/H20849x/H11032,t/H11032/H20850=/H11006qx/H11032−vdq0t. Transforming back to the laboratory frame then yields /H9278˙/H11006/H20849q/H20850=vd/H20849q/H11006q0/H20850. The nature of spin propagation at the SU /H208492/H20850symmetry point can be made more clear if we Fourier transform fromthe wave vector to spatial domain. If we inject a /H9254-function stripe of zpolarized spins at x=0, the space-time evolution ofSzis proportional to the propagator, Gz/H20849x,t/H20850, where3.0 2.5 2.0 1.5 1.0 0.5 0.0γ−/D q02 2.0 1.5 1.0 0.5 0.0 q/q0λ 0 0.5 1 1.5 2 -0.8-0.40.00.40.8φ− 2.0 1.5 1.0 0.5 0.0 q/q0q0vd 0.2 0.4 0.6 0.8 . (a) (b) FIG. 1. /H20849Color online /H20850The dispersion relations for /H20849a/H20850the decay rate and /H20849b/H20850the rate of phase change of the SO enhanced mode in the SU /H208492/H20850case. /H20849a/H20850The decay rate /H9253−/H20849q/H20850increases with the drift velocity /H20849/H9261/H11013vd/vF/H20850but always vanishes at the resonant wave vec- torq0./H20849b/H20850The rate of phase change /H9278˙−/H20849q/H20850is proportional to the drift velocity vdand it crosses zero at the resonant wave vector q0.YANG, ORENSTEIN, AND LEE PHYSICAL REVIEW B 82, 155324 /H208492010 /H20850 155324-4Gz/H20849x,t/H20850/H11008/H20885dqeiqx/H20849A+e−i/H9275+t+A−e−i/H9275−t/H20850, /H2084932/H20850 where A+andA−are the weighting factors for the passive and active modes, respectively, and A+=A−=1 /2 in the SU/H208492/H20850case. Upon substituting the dispersion relations /H9275/H11006/H20849q/H20850, we obtain Gz/H20849x,t/H20850/H110081 /H20881Dtcos/H20849q0x/H20850exp/H20875−/H20849x−vdt/H208502 4Dt/H20876. /H2084933/H20850 The spin propagator is the product of a Gaussian envelope function and a static spin wave with wave vector q0. The envelope function is the one-dimensional diffusion propaga- tor with width proportional to /H20881Dtand drift velocity vd.A n illustration of the space-time evolution described by thispropagator is provided Fig. 2, for a drift velocity vd=2Dq0. Note that the phase of the spin wave modulated by theGaussian envelope remains stationary as the packet drifts anddiffuses. This contrasts with the more familiar wave packet,where the modulated wave and envelope functions bothpropagate, albeit with velocities that may differ. B. SU(2) broken by cubic Dresselhaus term When SU /H208492/H20850is exact, the integral of the Gaussian enve- lope function is conserved, even in the presence of an Efield. However, Stanescu and Galitski16have shown theoretically that/H92523, which is nonzero in real systems, breaks SU /H208492/H20850. Ko- ralek et al.17verified experimentally that /H92523is indeed the factor that limits PSH lifetime in experiments on /H20849001 /H20850GaAs quantum wells. In this section we calculate the dispersionrelation and spin packet time evolution in the presence of asmall cubic Dresselhaus term. It was shown previously that when /H92523is small, the maximum lifetime occurs when the Rashba interaction /H9251=/H92521−/H92523/H20849Ref. 16/H20850. We consider a QW with Rashba cou- pling tuned to this value and assume that /H92523/H11270/H92521. This con- dition is met in QWs in the 2D limit, where kFd/H112701/H20849dis the well width /H20850. In this case the dispersion relation in the pres- ence of the electric field can be written asi/H9275/H11006/H20849q/H20850/H110616DkF2/H925232+D/H20849q/H11006q0/H208502+ivd/H20849q/H11006q0/H20850/H11007ivd/H9004q, /H2084934/H20850 where q0/H110134kF/H20849/H92521−/H92523/H20850and/H9004q=2kF/H92523. Performing the Fou- rier transform to obtain the space-time evolution of a spinpacket, we obtain G z/H20849x,t/H20850/H110081 /H20881Dte−6DkF2/H925232tcos/H20849q0x−vd/H9004qt/H20850exp/H20875−/H20849x−vdt/H208502 4Dt/H20876. /H2084935/H20850 In the presence of the cubic Dresselhaus interaction the inte- gral of the Gaussian envelope is no longer conserved. Thedecay rate can be written in the form /H9253=3 8Dq02/H20873/H92523 /H92521/H208742 , /H2084936/H20850 illustrating that although the decay rate is nonzero, it is re- duced relative to the DP relaxation rate by a factor/H11015/H20849 /H92523//H92521/H208502. This ratio is expected theoretically,22and has been verified experimentally,17to be determined by the rela- tion, /H92523 /H92521=kF2d2 4/H92662. /H2084937/H20850 For quite reasonable QW parameters a /H92523to/H92521ratio of 1:100 can be achieved, equivalent to a lifetime enhancementrelative to the DP spin memory time on the order of 10 4. C. Linear Dresselhaus coupling Finally, we consider a fully symmetric well in which only the linear Dresselhaus coupling exists. To make comparisonwith the SU /H208492/H20850situation, we set the strength of the linear Dresselhaus coupling be 2 /H92521so that the resonant wave vec- tor is at q/H11229q0=4kF/H92521. The dispersion relations /H9253−/H20849q/H20850and /H9278˙−/H20849q/H20850obtained by substituting /H9251=/H92523=0 and replacing /H92521by 2/H92521in Eq. /H2084929/H20850are plotted in Fig. 3. Some qualitative fea- tures of the dispersion relations are similar to the SU /H208492/H20850case, in that /H9253−/H20849q/H20850has a global minimum and /H9278˙−/H20849q/H20850crosses zero at q/H11229q0. The most important difference is that the minimum0.2 0.0 -0.2Sz 20 10 0 q0x6 4 2 0Dq02t FIG. 2. /H20849Color online /H20850The space-time evolution of Szwith a normalized /H9254-function injection at x=0, t=0, and drift velocity vd=2Dq0in the SU /H208492/H20850case. The spin polarization develops into a conserved stationary wave with a Gaussian wave packet.3.0 2.5 2.0 1.5 1.0 0.5 0.0γ−/D q02 2.0 1.5 1.0 0.5 0.0 q/q0λ 0 0.5 1 1.5 2 -0.8-0.40.00.40.8φ− 2.0 1.5 1.0 0.5 0.0 q/q0q0vd 0.2 0.4 0.6 0.8 . (a) (b) FIG. 3. /H20849Color online /H20850The dispersion relations for /H20849a/H20850the decay rate and /H20849b/H20850the rate of phase change of the SO enhanced mode in the linear-Dresselhaus-only case. The main features resemble those in the SU /H208492/H20850case, both /H9253−/H20849q/H20850show a minimum and /H9278˙−/H20849q/H20850vanishes atq0, but the lifetime is finite in this case.RANDOM WALK APPROACH TO SPIN DYNAMICS IN A … PHYSICAL REVIEW B 82, 155324 /H208492010 /H20850 155324-5/H9253−/H20849q/H20850does not reach zero, and therefore the spin spiral does decay. In the limit of low electric field, the lifetime of thespin spiral is only about a factor of 2 longer than the q=0 /H20849DP/H20850lifetime. The propagation of a spin packet in the linear- Dresselhaus-only case is illustrated in Fig. 4, using the same initial condition and drift velocity as in SU /H208492/H20850case. We per- formed numerical integration of Eq. /H2084932/H20850to obtain the propa- gator. As we have seen previously, a drifting and diffusingenvelope function modulates a spiral spin wave. However,now the spiral spin fades very quickly. The contrast betweenlinear Dresselhaus only and SU /H208492/H20850is illustrated in Fig. 5, which is a plot of the integral of the envelope as a function oftime. After a rapid initial decay, the integral is constant in theSU/H208492/H20850case, whereas with only the linear Dresselhaus inter- action the integrated amplitude decays exponentially with rate /H11229Dq 02. Figure 6presents another way of visualizing the differ- ence in propagation for the SU /H208492/H20850/H20851Fig. 6/H20849a/H20850/H20852and linear- Dresselhaus-only /H20851Fig.6/H20849b/H20850/H20852Hamiltonians. The zcomponent of spin polarization is shown /H20849with color coded amplitude /H20850asa function of time on the vertical axis and position on the horizontal axis. It is clear, from the vertical orientation of thecontours that the positions of the nodes and antinodes of S z are fixed in space. IV. SUMMARY AND CONCLUSION We have developed a random walk model to describe the time evolution of electron spin in two dimensions in thepresence of Rashba and Dresselhaus interactions. From therandom walk model we derived equations of motion for spinpolarization and obtained dispersion relations for qparallel to one of the symmetry directions of the Rashba/DresselhausHamiltonian. In Sec. II, we showed that the dispersion rela- tions for spin-polarization waves that spiral in the plane con-taining the surface normal and the wave vector are identicalto those obtained from previous analyses. 13,16The random walk approach is instructive in showing, in a simple but ex-plicit way, how anomalous spin diffusion and the persistentspin helix arise from nonvanishing correlations between thevelocity and spin precession vectors. In Sec. III, we obtained dispersion relations for spin- polarization waves that include the effects of an electric fieldparallel to q, to second order in E. The terms linear in Eare equivalent to those obtained from the quantum kineticapproach. 18,19To first order in E, the field introduces a pre- cession vector in the plane of the 2DEG and perpendicular toE. The precession about the yaxis gives rise to an unusual behavior in that the spiral with wave vector q 0is stationary in space despite the motion of electrons in the field; waveswith q/H11022q 0propagate in the same direction as the drifting electrons while those with q/H11021q0propagate “backward.” The terms that are second order in Eaffect the decay rate of spin polarization without changing the velocity. The solutions ob-tained when these terms are included point to the specialproperties of waves with wave vector q 0, whose lifetime turns out to be unchanged by the field. However, the decayrate of the all other waves increases, in proportion to/H20849q−q 0/H208502. We illustrated these results by considering three represen- tative spin-orbit Hamiltonians: SU /H208492/H20850symmetric or /H9251=/H92521 and/H92523=0; SU /H208492/H20850broken by a small but nonzero /H92523; and linear Dresselhaus coupling only or /H9251=/H92523=0. In order to show the nature of spin propagation more clearly, we Fouriertransformed the solutions from wave vector to real space andobtained the dynamics of spin-polarization packets. In all0.2 0.0 -0.2Sz 20 10 0 q0x6 4 2 0Dq02t FIG. 4. /H20849Color online /H20850The space-time evolution of Szin the linear-Dresselhaus-only case with the same initial condition andapplied Efield as in the SU /H208492/H20850case. The features are similar to those in the SU /H208492/H20850case, except the envelope function decays exponentially. 0.012460.12461|Sz|tot 8 6 4 2 0 Dq02tSU(2) Dresselhaus FIG. 5. /H20849Color online /H20850The the absolute value of the spin polar- ization integrated over position as a function of time. In the SU /H208492/H20850 case, /H20841Sz/H20841totis conserved after an initial decay while in the linear- Dresselhaus-only case, /H20841Sz/H20841totdecays exponentially. 8 6 4 2 0Dq02t 16 12 8 4 0 q0x0.6 0.4 0.2 0.0 -0.2Sz 8 6 4 2 0Dq02t 16 12 8 4 0 q0x0.6 0.4 0.2 0.0 -0.2Sz (a) (b) FIG. 6. /H20849Color online /H20850The space-time images of the spin polar- ization in the /H20849a/H20850SU/H208492/H20850and /H20849b/H20850linear-Dresselhaus-only cases, respectively.YANG, ORENSTEIN, AND LEE PHYSICAL REVIEW B 82, 155324 /H208492010 /H20850 155324-6cases the spin packets move at the electron drift velocity. In the SU /H208492/H20850case the integrated amplitude of the spin spiral is conserved while in the linear-Dresselhaus-only case the am- plitude decays with a rate /H11011Dq02. When SU /H208492/H20850is weakly broken by small, but nonzero /H92523, the integrated amplitude decays at a rate /H11011/H20849/H92523//H92521/H208502Dq02. The conclusions reached by our analysis of the RW model are consistent with a recent Monte Carlo study of a specific2DEG system, a /H20849001 /H20850In 1−xGaxAs quantum well with carrier density /H110111012cm−2/H20849Ref.23/H20850. In this study spin-polarization dynamics were calculated under conditions of steady-stateinjection from a ferromagnetic contact. For /H9251//H92521ratios that are close to unity, the spin polarization is conserved overseveral wavelengths of the PSH, despite the fact that trans-port takes place in the diffusive regime. Moreover, the polar-ization is not diminished with increasing electric field. Theauthors point out that the PSH effect can be used to achievea novel variation of the Datta-Das spin-field-effect transistor/H20849Ref. 24/H20850in which a gate electrode modulates the /H9251to/H92521 ratio only slightly away from unity. This has the effect of varying the wavelength of the PSH without significantly re-ducing its lifetime. Thus small changes in gate voltage can inprinciple lead to large changes in source to drain conduc-tance. Whether such a device can actually be realized de-pends on two factors: fabricating ferromagnetic injectors andanalyzers with high figures of merit, and demonstrating thatthe PSH effects that have been observed at temperatures be-low /H11011100 K /H20849Ref. 17/H20850can be realized at room temperature. ACKNOWLEDGMENTS This work was supported by the Director, Office of Sci- ence, Office of Basic Energy Sciences, Materials Sciencesand Engineering Division, of the U.S. Department of Energyunder Contract No. DE-AC02-05CH11231. 1T. Dietl, D. D. Awschalom, M. Kaminska, and H. Ohno, Spin- tronics , Semiconductors and Semimetals Vol. 82 /H20849Academic, New York, 2008 /H20850. 2J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Žuti ć, Acta Phys. Slov. 57, 565 /H208492007 /H20850. 3M. W. Wu, J. H. Jiang, and M. Q. Weng, Phys. Rep. 493,6 1 /H208492010 /H20850. 4F. J. Ohkawa and Y. Uemura, J. Phys. Soc. Jpn. 37, 1325 /H208491974 /H20850. 5Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 /H208491984 /H20850. 6Y. A. Bychkov and E. I. Rashba, JETP Lett. 39,7 8 /H208491984 /H20850. 7M. I. D’yakonov and V. I. Perel’, JETP Lett. 13, 467 /H208491971 /H20850. 8M. I. D’yakonov and V. I. Perel’, Zh. Eksp. Teor. Fiz. 60, 1954 /H208491971 /H20850/H20851Sov. Phys. JETP 33, 1053 /H208491971 /H20850/H20852. 9F. Meier and B. P. Zakharchenya, Optical Orientation , Modern Problems in Condensed Matter Sciences Vol. 8 /H20849North-Holland, Amsterdam, 1984 /H20850. 10M. I. D’yakonov and V. Yu. Kachorovskii, Fiz. Tekh. Polupro- vodn. 20, 178 /H208491986 /H20850/H20851Sov. Phys. Semicond. 20,1 1 0 /H208491986 /H20850/H20852. 11G. Dresselhaus, Phys. Rev. 100, 580 /H208491955 /H20850. 12J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 /H208492003 /H20850.13B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev. Lett. 97, 236601 /H208492006 /H20850. 14A. A. Burkov, A. S. Nunez, and A. H. MacDonald, Phys. Rev. B 70, 155308 /H208492004 /H20850. 15E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys. Rev. Lett. 93, 226602 /H208492004 /H20850. 16T. D. Stanescu and V. Galitski, Phys. Rev. B 75, 125307 /H208492007 /H20850. 17J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C. Zhang, S. Mack, and D. D. Awschalom, Nature /H20849London /H20850458, 610 /H208492009 /H20850. 18P. Kleinert and V. V. Bryksin, Phys. Rev. B 76, 205326 /H208492007 /H20850. 19P. Kleinert and V. V. Bryksin, Phys. Rev. B 79, 045317 /H208492009 /H20850. 20M. M. Glazov and E. Y. Sherman, Phys. Rev. B 71, 241312 /H20849R/H20850 /H208492005 /H20850. 21V. K. Dugaev, E. Y. Sherman, V. I. Ivanov, and J. Barna ś,Phys. Rev. B 80, 081301 /H20849R/H20850/H208492009 /H20850. 22R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems , Springer Tracts in Modern Physics Vol. 191 /H20849Springer, New York, 2003 /H20850. 23M. Ohno and K. Yoh, Phys. Rev. B 77, 045323 /H208492008 /H20850. 24S. Datta and B. Das, Appl. Phys. Lett. 56, 665 /H208491990 /H20850.RANDOM WALK APPROACH TO SPIN DYNAMICS IN A … PHYSICAL REVIEW B 82, 155324 /H208492010 /H20850 155324-7

PhysRevB.103.174432.pdf

PHYSICAL REVIEW B 103, 174432 (2021) Short-range thermal magnon diffusion in magnetic garnet K. An ,1,*R. Kohno ,1N. Thiery,1D. Reitz ,2L. Vila,1V. V. N a l e t ov ,1,3N. Beaulieu,4,5J. Ben Youssef,5 G. de Loubens ,4Y . Tserkovnyak,2and O. Klein1,† 1Université Grenoble Alpes, CEA, CNRS, Grenoble INP , Spintec, 38054 Grenoble, France 2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA 3Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation 4SPEC, CEA-Saclay, CNRS, Université Paris-Saclay, 91191 Gif-sur-Yvette, France 5LabSTICC, CNRS, Université de Bretagne Occidentale, 29238 Brest, France (Received 28 August 2020; revised 21 April 2021; accepted 29 April 2021; published 26 May 2021) Using the spin Seebeck effect (SSE), we study the propagation distance of thermally induced spin currents inside a magnetic insulator thin film in the short-range regime. We disambiguate spin currents driven bytemperature and chemical potential gradients by comparing the SSE signal before and after adding a heat-sinkingcapping layer on the same device. We report that the measured spin decay behavior near the heat source is wellaccounted for by a diffusion model where the magnon diffusion length is in submicron range, in other words, twoorders of magnitude smaller than previous estimates inferred from the long-range behavior. Our results highlightthe caveat in applying a diffusive theory to describe thermally generated magnon transport, where a single decaylength may not capture the behavior on all length scales. DOI: 10.1103/PhysRevB.103.174432 I. INTRODUCTION The generation of pure spin currents by heat [ 1,2]i sa tantalizing subject, which offers a unique opportunity to reachstrong out-of-equilibrium regimes with large spin current den-sities produced inside a magnetic material [ 3]. Interests lie in the prospect of reaching new collective dynamical be-haviors of spin transport such as the hydrodynamic regimeconspicuous by the emergence of turbulences [ 4]. Magnon superfluidity may also establish when the density exceeds theBose-Einstein condensation threshold under large temperaturegradients applied to low damping magnetic insulators, such asyttrium iron garnets (YIG) [ 5,6], where local heating can be provided by injecting a large electrical current density throughan adjacent metal, advantageously in Pt [ 7–9], or by optically heating with a laser [ 10–12]. The spin-transport properties are governed by λ, the char- acteristic length over which spin is conserved. Previous reports on measuring λin YIG at room temperature by the spin Seebeck effect (SSE) indicates that for distances largerthan∼10μm (long-range regime), the SSE signal follows an exponential decay with a characteristic length of the orderofλ 0≈10μm[13,14]. If this large value exceeds /lscript,t h e magnon mean-free path [ 15,16], the magnons can behave as a diffusive gas at long distances [ 17]. However, such a large value of λmay seem surprising for thermal magnons [18]. If one extends the magnon dispersion up to the THz range,the extrapolated ballistic decay length for thermal magnons isλ bal=λex/(2α√ωT/ωM)=2μm, where ωT=kBT0/¯h= 2π×6.25 THz, T0is room temperature, λex≈15 nm is the *Current address: Quantum Spin Team, Korea Research Institute of Standards and Science, Daejeon, Republic of Korea. †Corresponding author: oklein@cea.frexchange length in YIG, α≈10−4[19] is the Gilbert damp- ing, and ωM=γμ 0M(T0)=2π×4.48 GHz, with Mbeing the saturation magnetization. This estimate λbal, which is al- ready smaller than λ0, should be considered as an upper bound because (i) the Gilbert damping is expected to be increased inthe THz-range, [ 20] (ii) the group velocity is reduced toward the edge of the Brillouin zone [ 21,22], and (iii) it does not account for the√ /lscript/λ balreduction of the characteristic propa- gation distance due to diffusion. In fact, the distance range of the transport study is also a potent means to select a very specific part of the magnonspectrum. In experiments focusing on the long-range behav-ior, one has in essence efficiently filtered out any short decaymagnons. Behind this debate lies a fundamental question ofhow well magnon transport can be described by a diffusivemodel forming one gas with a single λ, whose value would govern SSE on all length scales. Submicron lengths have beeninferred from several longitudinal SSE measurements in thespatial [ 23,24] and temporal domains [ 11,25]. In nonlocal SSE measurements, where two different Pt strips are usedfor the spin injection and detection, only longer spin decaylengths have been reported. The existence of shorter decaylengths has been difficult to observe because the voltage in-duced by SSE shows a nontrivial spatial decay as a functionof the Pt detector position near the heat source [ 4,26]. The complex decay profile can be attributed to the competitionbetween magnons driven by the gradients of temperature andmagnon chemical potential [ 17,27,28]. It has been difficult to control these two sources of spin excitation in experi-ments, which hinders a correct extraction of a characteristicdecay length near the heat source. In this paper we developa way to disambiguate these two contributions after alteringthe temperature profile. We monitor on the same devices theshort-range SSE signal before (case A) and after (case B) 2469-9950/2021/103(17)/174432(9) 174432-1 ©2021 American Physical SocietyK. AN et al. PHYSICAL REVIEW B 103, 174432 (2021) FIG. 1. (a) Comparison of the measurements of the local ( V1) and nonlocal ( V2) voltages generated in YIG |Pt|Si3N4(case A) and YIG|Pt|Si3N4|Al (case B) stacks. Experiments are performed on the same devices before and after the deposition of an Al cappinglayer (left and right schematics). Two Pt electrodes deposited on top of YIG film monitor the spin-transconductance when an exter- nal magnetic field H 0rotates in-plane in the azimuthal direction ϕ. The center-to-center distance dbetween the two electrodes is varied between 0.5 and 6.3 μm. (b) Calculated vertical temperature gradient profiles at the top YIG surface at 2 mA. The light blue-shaded regionindicates the inverted gradient in case B. Beyond this region, ∂ zTis about three orders of magnitude larger in case B. The insets are the calculated temperature profiles for both cases. capping it with a nonmagnetic aluminum layer. The capping allows to change the vertical thermal gradient without alteringthe YIG interface. We observe that the sign of SSE volt-age inverts twice within a distance of 1 μm from the heat source for case B. The corresponding sign reversal of SSEsuggests that the magnons clearly sense the change in localtemperature gradient taking place for case B. With a simplediffusive transport model, the measured SSE decay profilefor both cases can be reproduced if one introduces a thermalmagnon diffusion length λ≈300±200 nm. The extracted shortλfrom our measurement fills the gap between different length scales reported in the longitudinal and nonlocal SSEmeasurements. II. EXPERIMENT We use a 56-nm thick YIG(111) film grown on a 500- μm gadolinium gallium garnet (GGG) substrate by liquid-phaseepitaxy. Ferromagnetic resonance experiments have shown adamping parameter of 2 ×10 −4revealing an excellent crystal quality of the YIG film [ 29]. The sample structure and mea- surement configuration are shown in Fig. 1(a). In our notation, subscripts 1 and 2 refer to the voltages measured by the Pt 1 and Pt 2, respectively. We show the data for both YIG |Pt|Si3N4 (case A, red) and YIG |Pt|Si3N4|Al (case B, blue). The color conventions will be used consistently throughout the paper. FIG. 2. (a), (c) Angular dependence of the background sub- tracted local voltage δVϕmeasured in the Pt injector strip for the current of I=± 0.8 mA and external magnetic field of μ0H0= 200 mT. We subtract the reference voltage Vyfrom the raw signal to remove any contributions not associated with magnons. In (b) and(d) we decompose the measured magnetoresistive voltage into two components, /Sigma1and/Delta1, the even and odd contributions of the signal with respect to the mirror symmetry about the yzplane (see the text). Two Pt strips (Pt 1and Pt 2) with a width of 300 nm, a length of 30μm, and a thickness of 7 nm have been evaporated directly on top of the YIG film. The center-to-center distancedbetween two Pt strips varies from 0.5 to 6.3 μm. The sample is then covered by a 20-nm thick Si 3N4protection film. A local Joule annealing was used to enhance the spinconductance [ 30]. After full characterization of the different devices, a 105-nm thick aluminum layer, with a length of 30μm and a width of 10 μm, is deposited on the top of the Si 3N4 film, and the same devices are measured again. The sample is submitted to an external field of μ0H0=200 mT rotating within the xyplane (in-plane configuration). We first show the expected change in temperature profile by the Al cappingin Fig. 1(b). The temperature rise, /Delta1T, is about 40 K lower in case B at the Pt injector. The reduced temperature rise isexperimentally confirmed by measuring the Pt 1resistance (see Appendix A). Besides the change in the temperature profile, the gradient profile also shows a dramatic change. While incase A the thermal gradient is always directed downward (intothe substrate), in case B, a large thermal gradient directedupward (into Al) is created half a micron away from thesource. The shaded region highlights the effect. Since thevertical thermal gradient drives the SSE, this feature gives riseto an additional signal at Pt 2. We use the same method demonstrated in our previous work to extract the SSE voltage [ 8]. As an illustration, we dis- play the background-subtracted local voltage δVϕ=Vϕ−Vy measured with the Pt injector on YIG |Pt|Si3N4at±0.8m Aa s a function of the in-plane magnetic field angle ϕin Figs. 2(a) and2(c). The offset Vy, measured when the magnetic field is applied along the yaxis (ϕ=90◦), takes account of all the spurious contributions to the spin transport [ 31]. To dis- tinguish the SSE from the spin-orbit torque, we define twoquantities based on the yzmirror symmetry: /Sigma1 ϕ,I,/Delta1ϕ,I≡ 174432-2SHORT-RANGE THERMAL MAGNON DIFFUSION IN … PHYSICAL REVIEW B 103, 174432 (2021) FIG. 3. (a) Current dependence of measured local /Delta11for case A (YIG |Pt|Si3N4) and case B (YIG |Pt|Si3N4|Al). The solid red line shows that the local /Delta11follows the expected behavior based on the temperature rise with increasing current. (b) Measured current dependence of nonlocal /Delta12for three different d’s for case A (red) and case B (blue). (δVϕ,I±δVϕ,I)/2, where ϕ=π−ϕ[32]. Figures 2(b) and 2(d) show the evolution of the extracted /Sigma1and/Delta1as a function ofϕfor both polarities of the current I./Sigma1is antisymmetric with respect to the current and evolves as cos 2 ϕ, as expected from the spin Hall magnetoresistance effect [ 33–35]./Delta1shows a cosϕangular dependence and is symmetric with respect to the current, consistent with the SSE [ 36]. In the following, we shall exclusively focus on the SSE voltage ( /Delta1). Next, we compare the full current dependence of the SSE voltage /Delta11(I)(/Delta1ϕ=0,Iin Pt 1) for both case A (red) and case B (blue) in Fig. 3(a). We clearly see that the voltage is negative for both cases over the entire current range. Theparabolic curvature observed at low currents decreases whenthe Al heat sink is introduced, which agrees with the reducedtemperature rise. We observe that /Delta1 1(I) reaches a minimum at 2 mA for case A with the minimum shifting to a highercurrent for case B. We attribute this reversal of the slope asthe growing influence of the vanishing YIG magnetization asone approaches the Curie temperature T c=544 K determined on our garnet film, which is close to the literature value of bulkYIG [ 37]. The current dependence of /Delta1 1(I) for case A can be reproduced by an empirical formula /Delta11∝SM(T)(T−T0), where Sis the spin Seebeck coefficient, M(T) is the saturation magnetization at the temperature T, while T0=300 K is the temperature of the substrate. From the fit as shown in Fig. 3(a), one can extract S≈0.08μVK−1in good agreement with pre- vious estimates [ 38] (see Appendix C). In case A [red curves in Fig. 3(b)], the sign of the measured voltage changes from negative to positive when going from the local voltage /Delta11 to the nonlocal voltage /Delta12, respectively. This observation is consistent with the previous works that reported a single signreversal of the SSE voltage measured as a function of distancefrom the heat source [ 26,39,40]. It has been reported that the characteristic distance at which the magnon accumulationchanges the sign can be tuned by varying the magnetic filmthickness [ 26] or magnon diffusion length [ 39]. The situation is quite different for case B, as shown with the blue dots inFig. 3(b). In the vicinity of the injector, the nonlocal /Delta1 2is still positive at 0.7 μm but much smaller than case A. The sign of /Delta12for case B eventually becomes negative when the Pt detectors are positioned at 1 and 2.3 μm away from the injector. Our understanding of the sign changes as a functionofdis as follows: At d=0, heat drags magnons from Pt 1down into YIG (negative sign for SSE); at larger d, magnons can transfer spin from the YIG bulk into Pt 2(positive sign); adding Al creates a region of inverted heat flow over a shortrange (positive sign within 0.2 and 0.5 μm) and significant heat flow down into YIG over a longer range (negative sign).The fact that we see a negative signal after the second crossingsignifies that the positive SSE driven by the magnon diffusionalready becomes diminished and the local temperature-drivenSSE dominates. In this perspective, the second crossing pointmay put an upper bound on the estimate of λ. III. MODELING To model our experiment, we treat the magnons in YIG as a diffusive gas with temperature Tand chemical potential μ described by a set of transport coefficients [ 17]. The measured /Delta1signal is proportional to the magnon chemical potential at the interface between YIG and Pt [ 41]. The continuity equation for spin current density Jsin the steady state is −∇·Js=gμ, (1) where gis spin relaxation coefficient. In linear response, the transport equation is Js=−σ(∇μ+ς∇T), (2) where σis the spin conductivity and ςis the bulk spin Seebeck coefficient. The two equations are combined andlead to μ λ2=∇2μ+ς∇2T, (3) where λ=√σ/gis the thermal magnon diffusion length. We use a finite element method, COMSOL, to calculate the temperature profile and the magnon chemical potentialμin 3D assuming translational invariance along the yaxis (see Appendix B). The second crossing point d 2observed in the case B, as indicated in Fig. 4(b), is an important feature that reveals the inverted heat flow near the heat source asshown in Fig. 1(b). The calculated spatial profiles of μfor three different values of λare compared with the experimental data in Figs. 4(a)and4(b) after normalization to the measured values at d=0μm. We find that the second sign reversal atd 2can be reproduced with a short λ=100 nm. Increasing 174432-3K. AN et al. PHYSICAL REVIEW B 103, 174432 (2021) FIG. 4. Comparison of the experimentally measured /Delta1’s at 2 mA with the numerically calculated chemical potential profile for (a) case A and (b) case B on a symmetric log scale. The result with λ=0.1μm reproduces the measured double crossing as shown in (b). The second crossing does not appear anymore when increasing λto 0.5 or 1 μm. Insets show the data taken at 1 mA compared with the calculations with λ=0.2μm. (c) Contour plot of the second crossing position d2as a function of λand thermal conductivity of Si 3N4. The black lines represent the iso-lines for different d2’s. The arrow points to the value of κSi3N4used in (a) and (b). it to 0.5 or 1 μm, however, no longer reproduces the second crossing in case B [compare solid line with dashed and dottedlines in Fig. 4(b)]. In the insets of Fig. 4, we present the best fit obtained with λ=0.2μm for the data taken at 1 mA. Although the /Delta1signal seems to depend on the current value, the extracted λis not significantly modified (see Appendix E). The fit value of λdepends on other parameters in the model. The second crossing point, d 2, can be affected by the temperature profile at the interface, which is sensitive, forexample, to the thermal conductivity of Si 3N4.W es h o wa contour plot for d2as a function of κSi3N4andλin Fig. 4(c). The measured second crossing constrains the parameter spaceto the line along d 2=1μm. The second crossing is not observed for λ> 500 nm regardless of κSi3N4. These consid- erations can be further modified by the interfacial SSE dueto the magnon-phonon temperature difference at the interface[42,43]. Enhanced local temperature-driven contribution to the measured signal can increase λfor a given d 2.T oh a v e quantitative agreement with the data, we also get an upper bound of λ∼500 nm, although it is worth mentioning that in the limit of large temperature mismatch between magnonsin YIG and the Pt one can qualitatively reproduce the single crossing in case A and double crossing in case B for muchlarger λ(see Appendix D). Although our proposed fit with a diffusive equation parametrized by λ∈[100,500]≈300±200 nm captures well the short-range behavior in both cases A and B, weemphasize that this does not contradict earlier works [ 13,44]. The data observed for case A are similar to the ones alreadyobserved in other YIG devices, where the fit of the long-rangedecay behavior has led to the larger λ 0≈30×λ[45]. We note that the magnon diffusion equation [Eq. ( 3)] is constructed under the assumption of a long-distance behavior, where all the magnons are equilibrated to a common chemical potential. It is not surprising, however, to have a significant departure inthe spin-transport behavior at short distances, where magnons are not yet internally equilibrated or thermally equilibratedwith the surrounding phonons [ 24]. In certain special cases nonetheless, this may be modeled by introducing effectivelength scales, which is illustrated in our analysis. Specifically,we suggest a possibility that a subset of out-of-equilibriummagnons, from the thermal energy range, is locally decayingon a shorter length scale ( λ) than the asymptotic long-distance decay ( λ 0). We note that our extracted λis close to the value for the magnon diffusion length predicted in a previ-ous work [ 46], while λ 0is comparable to the upper-bound estimate of Ref. [ 47]. The previously reported energy re- laxation length of magnons [ 11,23–25] is also seen to be similar to λ, where the possible connection needs to be further explored. IV . CONCLUSIONS In summary, we measured the spatial distribution of ther- mally generated magnons in a thin YIG film. We altered thetemperature profile across the YIG film with an aluminumlayer. The results are that the nonequilibrium thermal magnonprofile deviates from an exponential decay and shows a doublesign reversal. We use a linear response magnon transport the-ory to obtain the short-range thermal magnon diffusion lengthof a submicron range, which is about two orders of magnitudesmaller than the value found in previous reports focused on thelong-range measurements. Our results suggest that the localeffect of heating is to produce magnons which decay on ashort-length scale near the source. The experimental approachusing a heat sink to reveal a short magnon diffusion lengthmay find applications to other systems, especially when thelength scale of the diffusion and temperature gradient arecomparable. 174432-4SHORT-RANGE THERMAL MAGNON DIFFUSION IN … PHYSICAL REVIEW B 103, 174432 (2021) ACKNOWLEDGMENTS This work was supported in part by the Grants No. 18-CE24-0021 from the ANR of France. K.A. acknowl-edges support from the National Research Foundation ofKorea (NRF; Grant No. 2021R1C1C201226911) funded bythe Korean government (MSIT). V .V .N. acknowledges sup-port from UGA through the invited Prof. program and fromthe Russian Competitive Growth of KFU. The work at UCLAwas supported by the US Department of Energy, Office ofBasic Energy Sciences under Award No. DE-SC0012190. APPENDIX A: TEMPERATURE CHARACTERIZATION We characterize the temperature rise induced by Joule heat- ing using the Pt resistance as a temperature sensor. The Ptinjector is connected to a 6221 Keithley, which generates a10 -ms pulse current with a duty cycle of 10%. The voltagesare measured with a 2182A Keithley nano-voltmeter [ 8]. In the inset of Fig. 5, we plot the rational increase of Pt resis- tance as a function of ambient temperature between 220 and300 K. The change in resistance /Delta1Ris linearly proportional to the temperature rise /Delta1T=T−T 0, i.e.,/Delta1R(T)/R0=ζ/Delta1T, where R0is the initial resistance, ζis the thermal coefficient of resistance, and T0=300 K is room temperature. We obtain ζ=(2.1±0.3)×10−3K−1for our Pt strip from the fit. The increase of resistance and corresponding temperature rise asa function of current is plotted in Fig. 5. The Pt tempera- ture increases quadratically with applying current owing tothe Joule heating ( ∝I 2). The temperature rise in Pt is about 45 K lower after the Al deposition at 2 mA (current densityof∼10 12A/m2). This indicates that the Al layer effectively spreads the heat from the Pt injector. APPENDIX B: DETAILS OF THEORETICAL CALCULATION The temperature profiles and the chemical potential are calculated in a 2D geometry using a finite element method,COMSOL. We choose a boundary condition that the top and FIG. 5. Resistance increase and the corresponding temperature elevation in the Pt strip as a function of the injected current with and without Al capping (red and blue dots, respectively). The solidred and blue lines are quadratic fits to the data. The inset shows the temperature dependence of Pt resistance. The yellow line is a linear fit to the data. FIG. 6. Calculated temperature rise profiles for two cases with and without the Al layer at 2 mA. side surfaces are thermally isolated and the bottom is held at room temperature, and the normal component of the spincurrent is zero at the boundary. The geometry was chosen to bethe same as the actual sample size, except that the lateral sizeof sample and the thickness of GGG are reduced to 30 μmt o facilitate the calculation. The thermal conductivity parametersare 9, 7.4, 29, 220, and 0.5 Wm −1K−1for GGG [ 48], YIG [48], Pt [ 49], Al [ 50], and Si 3N4[51], respectively. The value of the spin Seebeck coefficient ςdoes not affect the decay profile. The thermal magnon diffusion length λis varied. The calculated temperature profiles at the top of YIG surface areshown for the two cases in Fig. 6. The temperature difference atd=0μm (the center of Pt injector strip) between two cases is 37 K, which roughly agrees with the measured temperaturedifference at 2 mA as shown in Fig. 5. To check the validity of calculated temperature profile, we measured the temperaturerise at the position of the detector in case A. Our estimationyields a temperature drop of 46% for the detector placed atd=0.5μm away from the injector. This is larger than the simulated temperature drop of 30% over the same distance(red curve in Fig. 6). The discrepancy may arise from (i) the simplification to 2D modeling and (ii) the possible differencein parameters between the simulation and the measurement.Also, the heat could be removed by the Pt detector, which isnot taken into account in this calculation. However, we findthat the temperature change due to the presence of Pt detectoris negligible (see Appendix F). APPENDIX C: TEMPERATURE DEPENDENCE OF THE LOCAL SSE VOLTAGE In Fig. 3(a), we fit the measured current dependence of /Delta11for case A. The measured local SSE voltage follows the analytical expression /Delta11=SLPt/angbracketleft∂zT/angbracketright, where Sis the spin Seebeck coefficient, LPtis the length of the Pt electrode, and/angbracketleft∂zT/angbracketrightis the vertical temperature gradient across the YIG thickness. The latter is proportional to the temperature riseof the Pt injector: /angbracketleft∂ zT/angbracketright=(T−T0)/lT, where T0=300 K is the substrate temperature and lTis the characteristic decay length of temperature from the top surface. By comparing themeasured Pt temperature rise T−T 0=130 K at 2 mA with the expected /angbracketleft∂zT/angbracketright=10 K/56 nm as shown in Fig. 1(b),w e obtain lT∼730 nm. Assuming that the temperature depen- dence of Sis simply due to μ0M(T) (in contrast with the fitted temperature dependence used in a previous work [ 52]), the 174432-5K. AN et al. PHYSICAL REVIEW B 103, 174432 (2021) FIG. 7. Measured temperature dependence of magnetization. The solid red line is a fit with Eq. ( C2). expression for /Delta11becomes /Delta11(T)=CLPt lT(T−T0)μ0M(T), (C1) where C≡S/(μ0M(T)). The temperature dependence of magnetization follows an empirical formula, μ0M(T)=μ0M0(1−(T/Tc)a)b, (C2) where μ0M0=0.217 T is the YIG saturation magnetization atT=0 K, while the exponents a=2.0 and b=0.6a r e extracted from the fit as shown in Fig. 7. The expression for /Delta11(T) is converted to /Delta11(I) using the temperature-current calibration curve in Fig. 5. Then we use Cas a single fit- ting parameter to reproduce the observed behavior [solid redline in Fig. 3(a)]. The fit yields C=0.43μVK −1T−1.A t room temperature, where the magnetization of YIG is about0.178 T, the spin Seebeck coefficient of our YIG |Pt system is about S≈0.08μVK −1, which agrees with a previous work [ 38]. APPENDIX D: INTERFACIAL EFFECT Another potential source of spin currents to consider are interfacial effects at the Pt strips arising from the Kapitza re-sistance, which creates a temperature discontinuity, δTacross the YIG |Pt interface [ 43,53]. One can assume that δTis pro- portional to the temperature gradient at the interface, ∂ zT.T h e measured SSE voltage including the interfacial contributioncan be written as V SSE=C1(μ−C2kB∂zT), (D1) where μis the magnon chemical potential obtained by solving Eq. ( 3). Here we are interested in the spin flow to leading order in the spin-exchange coupling across the YIG |Pt interface. In this case, Tandμare calculated in the absence of spin flow into Pt. The two are evaluated along the top YIG surface. C1 is a constant, which normalizes the simulation results to the experimental data at d=0μm.C2is a parameter proportional to the Kapitza length, which represents the contribution of theinterfacial term. The negative sign implies that the heat flow isalong the opposite direction of the temperature gradient. Onecan assume that C 2is the same for both case A and case B for fixed dbecause the Al layer does not touch either the YIG or Pt directly. We recall also that in our Cartesian frame, zis the direction normal to the film. FIG. 8. The calculated spatial profile of μafter normalization to the measured local /Delta11. The results are compared to the measured /Delta1’s with varying λfor (a), (c), (e) case A and (b), (d), (f) case B with C2=0, 0.5, 10 μm. Only λ=100 and 300 nm show qualitative agreements for case B with C2=0. With an increased C2=0.5 μm,λ=500 nm can roughly fit for both cases (c and d). However, λ=1μm fits neither the first crossing in case A nor the second crossing in case B well. The fit does not work for case A anymore with C2=10μm even though the double crossing in case B can be reproduced (e and f). Figure 8shows the effect of adding a finite C2for different values of λ.F o r C2=0.5μm, the calculation can reproduce the observed double crossing in case B for all four values ofλ. However, λ=1μm case does not predict well either the observed first crossing in case A or the second crossing incase B. It is important to also point out that V SSEeventually follows the temperature gradient profile in Fig. 1(b) when the C2term is dominant [Fig. 8(f)] even for large values of λ. Thus in the limit of very large C2, the observation of a double SSE 174432-6SHORT-RANGE THERMAL MAGNON DIFFUSION IN … PHYSICAL REVIEW B 103, 174432 (2021) FIG. 9. Experimental data at 1 and 2 mA are plotted. (a) Red (orange) line is the calculated decay profile for 2 mA (1 mA) withλ=0.4μma n d C 2=0. (b) Similar plots are shown for case B, where blue and cyan solid lines are the calculated profiles for 2 and 1 mA, respectively, with λ=0.1μma n d C2=0. The calculations are normalized to the measured local /Delta11. sign crossing in case B is not anymore conspicuous of a short decay length of thermal magnons. Another consequence ofassuming that the interfacial effects are dominant is to reducestrongly (more than three orders of magnitude) the amplitudeof the signal after the first crossing in case A. The fact thatwe observe experimentally only an order of magnitude re-duction of the SSE signal for case A thus points to a smallvalue of C 2/lessmuch0.5μm( s e eF i g . 3between d=0 and d= 0.7μm). Experimentally, we have performed an estimation of the Kapitza resistance by comparing the increase of the Pttemperature inferred from the variation of its resistance andthe temperature increase of YIG inferred from the decrease ofthe Kittel frequency due to a change of M s(T), whose slope is about 0.4 mT /K at room temperature [ 8]. We have found no temperature difference between the Pt and the YIG underneathwithin the uncertainty of 2 K when the increase of temperaturerise is T−T 0=70 K. At 2 mA, where T−T0=130 K, the temperature gradient is 0.2 K /nm. From this, we estimate an upper bound of Kapitza length of about 20 nm. APPENDIX E: CURRENT DEPENDENCE OF λ Experimentally we find that the /Delta1signal seems to increase with current (see the data points in Fig. 9). However, when FIG. 10. Comparison of calculated temperature profiles at 2 mA with and without Pt detector at 1 μm for (a) case A and (b) case B. quantitatively checking if λdepends on the value of current, we find that the best-fitting λis not significantly modified with current. In Fig. 9(a), we show that λ=0.4μm fits very well both data sets taken at 1 and 2 mA. In case B, λ=0.1μmfi t s both data sets reasonably [Fig. 9(b)]. We note that all these λ’s are within our uncertainty range, i.e., λ∈[100,500] nm. Also, the /Delta1signal in case B becomes more negative with in- creasing current. We believe this is because the /Delta1contribution by the temperature gradient prevails over the magnon diffu-sion process at higher currents leading to the more negativesignal. APPENDIX F: EFFECT OF PT DETECTOR ON TEMPERATURE PROFILE Finally, we consider the effect of Pt detector on the temperature profile. In Fig. 10, we compare the calculated temperature profiles with /without the Pt detector at d= 1μm. There are less than 1 K variations of temperature due the Pt detector in both cases, which amount to less than a 1%change in the temperature rise. This is because (i) the Pt stripcovers only a small fraction of YIG surface and (ii) thermalconductivity of our 7-nm thick Pt is estimated to be as lowas about 29 W m −1K−1expected from the high resistance of 3.8 k/Omega1, consistent with Wiedemann-Franz law. Compared to the temperature change induced by the Al layer, this temper-ature change is negligible. 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PhysRevB.98.220411.pdf

PHYSICAL REVIEW B 98, 220411(R) (2018) Rapid Communications Cavity optomechanics of topological spin textures in magnetic insulators Igor Proskurin,1,2,*Alexander S. Ovchinnikov,2,3Jun-ichiro Kishine,4and Robert L. Stamps1 1Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 2Institute of Natural Sciences and Mathematics, Ural Federal University, Ekaterinburg 620002, Russia 3Institute for Metal Physics, Ural Division of the Russian Academy of Sciences, Ekaterinburg 620137, Russia 4Division of Natural and Environmental Sciences, The Open University of Japan, Chiba 261-8586, Japan (Received 5 October 2018; revised manuscript received 22 November 2018; published 26 December 2018) Collective dynamics of topological magnetic textures can be thought of as a massive particle moving in a magnetic pinning potential. We demonstrate that inside a cavity resonator this effective mechanical system canfeel the electromagnetic radiation pressure from cavity photons through the magneto-optical inverse Faraday andCotton-Mouton effects. We estimate values for the effective parameters of the optomechanical coupling for twospin textures: a Bloch domain wall and a chiral magnetic soliton lattice. The soliton lattice has magnetic chirality,so that in circularly polarized light it behaves like a chiral particle with the sign of the optomechanical couplingdetermined by the helicity of the light and chirality of the lattice. Most interestingly, we find a level attractionregime for the soliton lattice, which is tunable through an applied magnetic field. DOI: 10.1103/PhysRevB.98.220411 Introduction. Cavity optomechanics is an established area in which the effect of radiation pressure on mechanical objectsinside microwave cavity resonators is studied [ 1]. The scope includes an impressive variety of phenomena including lasercooling [ 2], parametric instability [ 3,4] and chaotic dynamics [5], optomechanical entanglement [ 6], and nonclassical pho- ton states [ 7]. There are a number of applications ranging from high-accuracy sensors for gravitational wave detectors[8,9] to quantum information processing protocols based on the fundamental principles of the quantum mechanics [ 1,10]. It was demonstrated recently, both experimentally [ 11,12] and theoretically [ 13,14], that magnons in magnetic insulators can also feel electromagnetic radiation pressure forces owingto magneto-optical interactions such as the inverse Fara-day and Cotton-Mouton effects. These findings establisheda new direction—cavity optomagnonics which has developed rapidly during the last few years [ 15–20]. On a quantum level, optomagnonics describes systems of coupled photons andmagnons using an optomechanical Hamiltonian with whichvarious optomechanical effects in magnetic insulators arepredicted and described [ 21]. Cavity optomagnonics, together with spin optodynamics of cold atoms [ 22,23], shapes the basis of modern optospintronics [ 24], which targets ultrafast optical control of spin states utilizing cavity resonators tocondense the electromagnetic energy [ 25–32]. In this Rapid Communication, we propose a realization of coupled spin-photon dynamics that connects cavity op-tomechanics with optomagnonics. It is well established thatlow-energy dynamics of modulated spin textures, such asdomain walls in ferromagnets [ 33–38] or magnetic soliton lattices in helimagnets [ 39–42], can be described in terms of a few collective coordinates. A canonical example is the *igor.proskurin@umanitoba.caequation of motion for a pinned domain wall (DW), whichcan be expressed in terms of the harmonic oscillator equation m eff¨X+meff/Gamma1m˙X+meff/Omega12 mX=Ftorque, (1) where X(t) is the position of DW, meffis the effective mass determined by the spin configuration, /Gamma1mis the dissipation pa- rameter, which is proportional to the Gilbert damping, and theoscillator frequency, /Omega1 m, determined by the external pinning of DW to impurities or defects [ 37]. The force on the right- hand side is assigned to arise from a spin torque acting uponthe spin texture. In magnetic insulators, spin torque can be ofmagneto-optical origin [ 13,14]. For example, in a circularly birefringent medium the electric field E(ω) is able to generate an effective magnetic field B eff∼E(ω)×E(ω)∗, which is able to excite collective motion of a DW, thus creating aradiation pressure force on the right-hand side of Eq. ( 1). It should be mentioned that our approach is different from those in Refs. [ 13,14], where magneto-optical coupling was applied to (macro)spin dynamics. The collective motionof modulated spin textures makes their behavior similar toactual massive particles moving in the real-space potentialenergy profile, which means that one can realize a variety ofoptomechanical applications using spin textures as effectivemechanical objects, for which parameters can be manipulatedby applying external fields. High tunability of spin textures in combination with low magnetic damping in such materials as iron garnets makesthem suitable for applications. In what follows, we discusshow this scenario can be realized for two magnetic textures:a Bloch DW and a periodic chiral soliton lattice (CSL),which is typical for uniaxial chiral helimagnets [ 43,44]. We demonstrate that the latter can be used for the realization oflevel attraction in optomechanical systems [ 45], which was also demonstrated recently for cavity magnon polaritons [ 46]. Our approach can be further generalized to two-dimensionalspin textures (e.g., magnetic skyrmions), which has attracted 2469-9950/2018/98(22)/220411(6) 220411-1 ©2018 American Physical SocietyPROSKURIN, OVCHINNIKOV , KISHINE, AND STAMPS PHYSICAL REVIEW B 98, 220411(R) (2018) FIG. 1. (a) Coordinate system showing spin S(r,t) parametrized by the azimuthal angle ϕand the polar angle θ. (b) Bloch domain wall pinned at z=0 by the pinning interaction Vpinand the effective mechanical model: the massive particle meffmoving in the potential energy profile Upin(X). (c) Bloch domain wall inside the cavity resonator: circularly polarized cavity mode generates the effectivemagnetic B effthrough the inverse Faraday effect, which creates the optical pressure force /angbracketleftF/angbracketrighton the domain wall in the direction perpendicular to Beff. attention in recent studies of the optomagnonics of a mi- crodisk with a vortex magnetization pattern [ 47]. The model. We begin by considering a DW oscillator in the optical field. Collective dynamics of a DW can be obtainedfrom the spin Lagrangian density [ 37] L=¯hSa −3(cosθ−1)∂tϕ−HM−Vpin, (2) which describes the semiclassical motion of the spin S= S(cosϕsinθ,sinϕsinθ,cosθ) parametrized by the angles ϕ(z,t) and θ(z,t), as shown in Fig. 1(a). The second term in Eq. ( 2) is the magnetic energy density of the DW HM=JS2a−1[(∂zθ)2+sin2θ(∂zϕ)2]+K⊥S2a−3cos2θ −K/bardblS2a−3cos2ϕsin2θ], (3) where Jis the ferromagnetic exchange constant, K/bardblandK⊥ are anisotropy parameters, and ais the lattice constant. In the following, we set S=1 and a=1 and restore these factors whenever necessary. The last term in Eq. ( 2) is the pinning potential, which is modeled through a local anisotropy field atz=0[48], with V pin=−Kpinδ(z)s i n2ϕsin2θ. The static Bloch DW configuration, stabilized by compe- tition of the exchange interaction with the anisotropy energy,is characterized by θ0=π/2 and tan ϕ0(z)/2=exp(z/λ DW), where λDW=/radicalbigJ/K /bardbldenotes the domain wall width [ 49]. The lowest-energy dynamics of DW can be described by the collective coordinate method [ 33]. For this purpose, we introduce two dynamical variables: the position of thewall,X(t), responsible for its translational motion, ϕ(z,t)= ϕ 0[z−X(t)], and the amplitude ξ(t) of the out-of-plane com- ponent, θ(z,t)=π/2+ξ(t)u0[z−X(t)]. The spatial profile ofθ(z,t) is determined by the Pöschl-Teller equation, which givesu0(z)=sech(z/λ DW)/√ 2[33]. In terms of collective variables, the effective Lagrangian for the translational motion of the Bloch domain wall motiontakes the following form: L eff=¯hMξ(t)˙X(t)−/Delta1ξ2(t)−Upin(X), (4) where M=/integraltext u0∂zϕ0dzand/Delta1=2K⊥λDW+π2Kpin/4, and we have kept only leading order terms in ξ(t). The last term in this equation is the potential energy of the pinning givenbyU pin(X)=−Kpinsech2(z/λ DW). Spin relaxation in DW dynamics can be taken into account by a Rayleigh dissipationfunction, W=¯hα/(2Sa)/integraltext (∂ tS)2dz, where αis the Gilbert damping parameter [ 37]. Dissipative Euler-Lagrange equations for Leffcan be re- duced to the second-order equation of motion for a massiveparticle moving in viscose medium in the pinning potentialU pin: meff¨X+meff/Gamma1m˙X=−∂Upin ∂X, (5) where meff=¯h2M2/(2/Delta1) is the effective mass of DW and /Gamma1m=2α¯hK/meffis the effective mechanical damping, where K=/integraltext (∂zϕ0)2dz[see Fig. 1(b)]. For strong pinning, this equation reduces to Eq. ( 1) for a damped harmonic oscillator with/Omega12 m=1 2m−1 eff∂2Upin/∂z2[37]. The Hamiltonian formulation for the Lagrangian in Eq. ( 4) can be found using the general formalism of Ref. [ 50] that treats the canonical momentum PX=∂Leff/∂˙Z=¯hMξas a constraint. This allows us to exclude ξ(t) and find the effective mechanical Hamiltonian for X(t)[39]: Hm=P2 X 2meff+Upin(X). (6) This Hamiltonian is helpful for analyzing a quantum regime of DW motion, achieved by replacing dynamic variables withoperators ˆx=x ZPF(b+b†) and ˆpx=−imeff/Omega1mxZPF(b− b†), where xZPF=(¯h/2meff/Omega1m)1/2, andbandb†satisfy boson commutation relations. The effective damping /Gamma1min this case corresponds to the decoherence rate of the quantum oscillatorstates. Magneto-optical coupling. The central idea of this Rapid Communication is that inside a cavity resonator, a DW canfeel radiation pressure forces from the electromagnetic fieldsimilar to that of a suspended mirror in standard optome-chanical applications. The microscopic mechanism behindthis analogy is the magneto-optical coupling between theelectromagnetic field and the spin system [ 51]. Local spin oscillations modulate the electric permittivity tensor resulting 220411-2CA VITY OPTOMECHANICS OF TOPOLOGICAL SPIN … PHYSICAL REVIEW B 98, 220411(R) (2018) in an interaction energy [ 52] Hmo=−ε0 4/integraldisplay δεij(S)Ei(r,t)E∗ j(r,t)d3r, (7) where E(r,t) denotes the complex amplitude of the electric field, E(r,t)=Re[E(r,t)e x p (−iωt)] [53,54]. The electric permittivity can be expanded in a power series of the spin den-sity,δε ij(r)=ifijkSk(r)+βijklSk(r)Sl(r)+..., where the magneto-optical coupling tensors fijk=−fjikandβijkl= βjikl=βijlk=βjilkare related to the Faraday and Cotton- Mouton effects, respectively [ 51]. Inside the cavity, the electric permittivity determines the frequency of cavity modes, ωcav(X), which becomes de- pendent on the position of the DW [ 55]. This is similar to how a suspended mirror modulates the frequency in op-tomechanics [ 1]. Expanding ω cav(X) near the local min- imum, ωcav(X)a†a=[ωcav+(∂ωcav/∂X)X+...]a†a,w e obtain the magneto-optical interaction −GX(t)a†a, between the DW and the cavity photons described by the a†anda operators, with microscopic details contained in the couplingparameter G=−(∂ω cav/∂X). In order to illustrate the microscopic mechanism, we consider a possible experimental setup in Fig. 1(c), where electromagnetic-field standing waves along the xdirection interact with Bloch DW along the zaxis. For illustration, let us consider only the inverse Faraday effect, so that δεij= if /epsilon1ijkSk, where /epsilon1ijkis the Levi-Civita symbol. In this case, the coupling energy in Eq. ( 7) can be expressed in terms of the spatially uniform effective magnetic field Bx eff∼i/integraltext ex· (E×E∗)dxapplied parallel to the magnetization direction of magnetic domains connected by DW. As is well known [ 56], the uniform magnetic field in such configuration can move DW, so that Bx effcouples di- rectly to the domain wall position. Quantizing the elec-tric field inside the cavity, E(x,t)=−i/summationtext nλ(¯hωn/ε0V)1/2 sin(πnx/L x)eλanλ, where ωnare the frequencies of cavity eigenmodes, and eλ=(0,λ,−i)/√ 2(λ=±1) are the po- larization vectors in the helicity basis, we find an explicitexpression for the magneto-optical coupling between DW andthe cavity photons: H mo=− ¯hg0(b+b†)(a† RaR−a† LaL). (8) Hereg0=1 4fSeffωcavis the single-photon coupling for the nth mode with ωn=ωcav, which is related to Gasg0= GxZPF, andSeff=xZPFA⊥/Vwhere A⊥is the cross section of the sample, and Vis the volume of the cavity. The dimen- sionless parameter Seffis proportional to the total number of spins involved into collective motion. Since this number ismacroscopic, g 0can reach the same orders of magnitude as estimated for macrospin fluctuations in optomagnonics [ 14]. We now estimate values for the mechanical parame- ters of the DW oscillator coupled to the optical field. Foriron garnet ferromagnetic insulators assuming K ⊥≈0.1K , λDW≈100 nm, a≈1n m[ 57]. The effective mass meff≈ ¯h2/(K⊥λDWa) is estimated as 10−27kg; and the oscillator fre- quency /Omega1m≈(2KpinK⊥a/¯h2λDW)1/2≈109s−1. For yttrium iron garnet, the damping parameter can be as low as 3 × 10−5[58], which gives the quality factor Qm=/Omega1m//Gamma1m≈ α−1/radicalbigKpina/(2K⊥λDW)≈104. For the single-photon cou- FIG. 2. CSL configuration for several Hx:H0≈0,H1>H 0,a n d H2>H 1(a); parameters of the effective optomechanical model as functions of Hx/Hc(b)–(d); CSL inside a cavity resonator: cavity modes generate Beff, which induces a sliding motion of CSL, and the corresponding effective model of mass mCSLmoving inside the potential energy profile Upin(X) determined by Vpin. pling, we use f=2cφF√ε/ω cavwithφF=240◦cm−1, andε=5[14,57], which gives g0=c 2φF√εSeff=105s−1 forSeff=10−6. To drive the DW oscillator, the full cou- pling strength should be comparable to the oscillator energy,g 0√ncav/lessorsimilar/Omega1m, where ncavis the number of coherent cavity photons. From this relation, we estimate ncav/lessorsimilar(/Omega1m/g0)2≈ 108. The strength of the cavity electric field can be estimated asE/lessorsimilar√ncav¯hωcav/(ε0V)≈100 V/m for terahertz photons in a centimeter-sized cavity. Magnetic soliton lattice. Another topological structure that may have potential applications in the context of cavityoptomechanics is the chiral soliton lattice (CSL), which istypically found in uniaxial chiral helimagnets [ 43]. Similar to a DW in that it has a twisted structure, a CSL is determinedby competition between the exchange, the Dzyaloshinskii-Moriya interaction (DMI), and the Zeeman energy in anexternal static magnetic field applied perpendicularly to thechiral axis. The equilibrium spin configuration of a CSL canbe thought of as a periodic array of equally spaced 360 ◦do- main walls with period determined by applied magnetic field[see Fig. 2(a)]. This is formally described by the Jacobi am- plitude elliptic function ϕ 0(z)=π+2a m( 2 KL−1 CSLz,/kappa1) and is characterized by the topological winding number, nkink= (2π)−1/integraltext ∂zϕ0(z)dz, where LCSL=8K(/kappa1)E(/kappa1)J/(πD)i s the lattice period, K(E) is the complete elliptic integral of the first (second) kind with the elliptic modulus /kappa1, andDis DMI constant. The elliptic modulus /kappa1is determined by the 220411-3PROSKURIN, OVCHINNIKOV , KISHINE, AND STAMPS PHYSICAL REVIEW B 98, 220411(R) (2018) external field Hxvia the transcendent equation,√Hx/Hc= /kappa1/E (/kappa1), which has a solution for Hx<H c. The critical field 2 μBHc=π2D2/(16J) marks the incommensurate-to- commensurate transition to the forced ferromagnetic state,where L CSLdiverges and nkinkvanishes [ 59]. At zero magnetic field, the CSL is reduced to a helical spin ordering. The collective dynamics of CSL can be described in the same way as the dynamics of Bloch DW. We intro-duce the position X(t) and the out-of-plane component δθ(z,t)=ξ(t)u 0[z−X(t)], where the spatial profile u0(z)= L−1/2 z√K/E dn(2KL−1 CSLz,/kappa1) is now determined by the Lamé equation. The resulting equations of motion for the CSLare identical to Eqs. ( 5) and ( 6) with new mechanical parame- tersm CSL=¯h2nkink/(a2D)Q−1 1and/Gamma1m=2π2αD2Q1/(¯hJ), where Q1=π/(12E3)[(2−/kappa12)E+(1−/kappa12)K][59]. The effective mass, mCSL, is proportional to the density of kinks, which shows a strong dependence on Hxin the vicinity of Hc [see Fig. 2(b)]. In contrast, the magnetic-field dependence of /Gamma1mis relatively weak. The low-energy dynamics of a pinned CSL depends on the position of the pinning site. We choose the pinning en-ergy in the form of a local easy x-axis anisotropy field at the center of the ferromagnetically ordered domain, V pin= −KpinS2 x(z)δ(z−1 2LCSL)[ s e eF i g . 2(e)]. In this case, in the harmonic approximation the oscillator frequency /Omega1CSL= [16K2KpinL−2 CSLm−1 CSL(1−/kappa12)]1/2decreases to zero as ferro- magnetically ordered regions grow with magnetic field, asshown in Fig. 2(c). The magneto-optical coupling mechanism for CSL is dif- ferent from those for Bloch DW. In order to excite collectivemotion of CSL, the magnetic field should be switched alongthe direction of the CSL axis, rather than perpendicularly toit, as for the DW, since the transverse magnetic field onlymodifies the period of CSL and has no impact on collectivedynamics. In contrast, the magnetic-field pulse applied paral-lel to the chiral axis couples directly to the momentum P X= ξ/(¯hM) of the CSL and induces sliding motion [ 41,42]. To couple CSL with the optical field, we use the cavity configuration shown in Fig. 2(e) where the cavity standing waves generate Beff=Bz effˆzalong the zdirection, so that Bz eff couples to Sz∼ξ(t). Equation ( 7), in this situation, gives the following coupling strength between the CSL and the cavitymodes: H mo=i¯hg0(b−b†)(a† RaR−a† LaL), (9) where g0=1 4fSeffωcavmCSL/Omega1CSLa2Q2/¯h and Q2= M−1/integraltext sin2(πnL−1 zz)u0(z)dz [60]. The coefficient Q2 does not show a strong dependence on magnetic field and can be estimated as J/(2D). In small magnetic fields, the single-photon coupling strength is given byg 0=fSeffmCSLωcav/Omega1CSLa2J/(8¯hD). The sign of g0in Eq. ( 9) is related to magnetic chirality of CSL via the sign of DMI constant D. This means that in circularly polarized light, CSL behaves like a chiral me-chanical particle with the sign of the radiation pressure forceproportional to helicity of the light and chirality of the spinstructure. We can estimate the effective mass of CSL in zero applied magnetic field as m CSL≈nkink×10−26kg for D≈0.1K . FIG. 3. Level attraction between the CSL mode /Omega1CSL(Hx) and the cavity mode with the detuning parameter /Delta1cavfor /Omega1CSL(0)//Delta1cav=2a n d g(0)//Delta1cav=0.2. Dashed lines show both modes without interaction. This is approximately nkinktimes larger than the mass of a single domain wall [ 39]. Typically, in millimeter-size samples nkink≈104, which gives mCSL≈10−22kg. Taking Kpin≈ 0.1 K and D/J≈10−3, we obtain the effective mechanical frequency in zero field /Omega1CSL=√ πK pinD3/(¯h2J2nkink)≈ 0.1×106s−1, i.e., in the megahertz range, and the ef- fective damping /Gamma1m=2παD2/(¯hJ) gives a quality factor Qm=α−1/radicalbigKpin/(4πnkinkD)≈10−2α−1forKpin≈10−3J. For CSL, we estimate g0≈1.2×106s−1using the same optical parameters as for DW. Discussion. The optomechanical Hamiltonian with cou- pling terms in Eqs. ( 8) and ( 9) can be useful for realizing various optomechanical applications [ 1], such as, for example, optical cooling of the domain wall motion by analogy withoptical cooling of magnons proposed recently in Ref. [ 21]. For illustration, we suggest a CSL for realization of the level attraction picture, proposed recently in Ref. [ 45], using the applied static magnetic field as a control parameter to drivethe coupled system toward instability. To realize level attraction, we use the linearized optome- chanical Hamiltonian in the rotating wave approximation fora blue detuning regime [ 1] H=− ¯h/Delta1 cava†a+¯h/Omega1CSL(Hx)b†b+i¯hg(Hx)(ab−a†b†), (10) where /Delta1cav>0 is the detuning parameter, g(Hx)= g0(Hx)√ncavdenotes the full optomechanical coupling strength, and aanda†denote the fluctuating part of the cavity field, such as aR=√ncav+a. We consider only the right-polarized mode. The eigenfrequencies of the Hamiltonian in Eq. ( 10)a r eg i v e nb y[ 45]ω1,2(Hx)=1 2(/Delta1+/Omega1CSL)±√1 4(/Delta1−/Omega1CSL)−g2. This shows level attraction in the region 2 g(Hx)<|/Delta1−/Omega1CSL(Hx)|bounded by two exceptional points where the real parts of the frequenciescoalesce, as shown in Fig. 3. Inside this region, an instability develops that resembles synchronization of two oscillators,where the amplitude of one mode shows exponential growthwhile the other is suppressed [ 45]. Summary. We propose to use collective dynamics of spin textures in ferromagnetic insulators as a model of mechanicalsubsystems in optomechanical applications using the inverseFaraday effect as a coupling mechanism. When collective 220411-4CA VITY OPTOMECHANICS OF TOPOLOGICAL SPIN … PHYSICAL REVIEW B 98, 220411(R) (2018) dynamics are excited by cavity electromagnetic modes, spin textures move in real space similar to actual mechanicalparticles. Our approach is illustrated on two topological spinstructures: the Bloch domain wall and the chiral solitonlattice. The latter is a highly tunable structure with the ef-fective mechanical parameters that depend strongly on anapplied magnetic field. This fact allows us to propose it asa realization of level attraction as proposed for microwaveresonators. Acknowledgments. This work was supported by a Grant-in- Aid for Scientific Research (B) (Grant No. 17H02923) and (S)(Grant No. 25220803) from the MEXT of the Japanese Gov-ernment, JSPS Bilateral Joint Research Projects (JSPS-FBR),and the JSPS Core-to-Core Program, A. Advanced ResearchNetworks. I.P. acknowledges financial support by Ministry of Education and Science of the Russian Federation, GrantNo. MK-1731.2018.2 and by Russian Foundation for BasicResearch (RFBR), Grant No. 18-32-00769(mol_a). A.S.O.acknowledges funding by the RFBR, Grant 17-52-50013, theFoundation for the Advancement to Theoretical Physics andMathematics BASIS, Grant No. 17-11-107, by the Govern-ment of the Russian Federation Program 02.A03.21.0006, andby the Ministry of Education and Science of the Russian Fed-eration, Project No. 3.2916.2017/4.6. R.L.S. acknowledgesthe support of the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC). Cette recherche a été financéepar le Conseil de recherches en sciences naturelles et en géniedu Canada (CRSNG). [1] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86,1391 (2014 ). [2] S. Stenholm, The semiclassical theory of laser cooling, Rev. Mod. Phys. 58,699(1986 ). [3] V . B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Parametric oscillatory instability in Fabry-Perot interferometer, Phys. Lett. A287,331(2001 ). [4] T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. 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Boer, Physics of Magnetism and Magnetic Materials (Springer, New York, 2003), V ol. 92. [50] D. M. Gitman and I. V . Tyutin, Quantization of Fields with Constraints (Springer Verlag, Berlin, 1990). [51] M. G. Cottam and D. J. Lockwood, Light scattering in magnetic solids (Wiley, New York, 1986). [52] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electro- dynamics of Continuous Media (Pergamon, New York, 1984), Vo l . 8 . [53] A. M. Kalashnikova, A. V . Kimel, R. V . Pisarev, V . N. Gridnev, P. A. Usachev, A. Kirilyuk, and Th. Rasing, Impulsive excita-tion of coherent magnons and phonons by subpicosecond laserpulses in the weak ferromagnet FeBo 3,P h y s .R e v .B 78,104301 (2008 ). [54] C. Tzschaschel, K. Otani, R. Iida, T. Shimura, H. Ueda, S. Günther, M. Fiebig, and T. Satoh, Ultrafast optical excitationof coherent magnons in antiferromagnetic NiO, P h y s .R e v .B 95,174407 (2017 ). [55] In general, ω cavshould depend on both the position and the momentum of the wall. 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PhysRevB.101.014438.pdf

PHYSICAL REVIEW B 101, 014438 (2020) Analytical description of the topological interaction between magnetic domain walls in nanowires A. Pivano*and V . O. Dolocan† Aix-Marseille Université, CNRS, IM2NP UMR7334, F-13397 Marseille Cedex 20, France (Received 22 July 2019; revised manuscript received 22 November 2019; published 27 January 2020) Magnetic domain walls in nanowires behave as particles interacting through the exchange field. As topological objects, their interaction is determined by their chirality or topological charge. We investigate analytically thetopological repulsion between magnetic domain walls with the same topological charge in nanostripes (witheasy-plane magnetization) and show that it decays algebraically as r −2, being part of a larger class of interactions that produce topological long-range order in two dimensions. We compare the topological repulsion between thewalls with other types of fundamental interactions with exponential spatial decay, such as the Yukawa-Reidpotential, and with micromagnetic simulations. We determine that trains of such walls can be well describedanalytically and can be displaced regularly in nanowires leading to practical applications. DOI: 10.1103/PhysRevB.101.014438 I. INTRODUCTION The interaction forces of nature are few and their exact spatial variation is difficult to determine from first prin-ciples. In quantum field theory, the fundamental interac-tions are mediated by massless spin one particles, suchas gluons for the strong interaction or photons for theelectromagnetic interaction. In the nonrelativistic case, theseinteractions are described by an interaction potential. Inpractice, some phenomenological model is often employedas in the case of nucleon-nucleon interaction where a Reid-type (Yukawa) potential is frequently used and comparedwith experimental results [ 1]. The Yukawa-type potential is equally used to describe the interparticle interaction instrongly coupled systems, such as ultracooled neutral plasmas[2] as well as in colloidal suspensions [ 3] (the so-called Yukawa systems). In several condensed-matter areas and in field theories, the more localized excitations of nonlinear systems are con-sidered as quasiparticles and described within the collective-variable approach [ 4]. The interactions between these quasi- particles, such as vortices in superconductors, which have anequivalent in cosmology (global strings) [ 5,6], decay mono- tonically as r −2in thin superconducting films or exponentially in the bulk [ 7]. The exponential repulsive interaction is also found in more mundane interactions as the one between twopedestrians [ 8]. In magnetic systems, the interaction potential is domi- nated at short range by the exchange interaction. The ex-change interaction is model dependent, the most commonmodel being based on the Heisenberg Hamiltonian, fromwhich is derived the semiclassical exchange interaction pro-portional to the square of the magnetization gradient. This *Present address: CEA, DEN, DTN/SMTA/LEAG, Cadarache F-13108 Saint Paul-lez-Durance, France. †voicu.dolocan@im2np.frHeisenberg-derived dependence is heavily used in numerical calculations of ferromagnets (micromagnetics) [ 9]. Domain walls (DWs) confined in magnets on the nanoscale can be considered as particles (macrospins) which interactthrough the exchange field. The DWs in the confined struc-tures are transverse or vortex DWs depending on the samplesdimensions [ 10]. The DWs are formed from two or more ele- mentary topological defects with an integer winding number,such as vortices in the bulk or fractional winding numberwhich are half-vortices confined to the edges [ 11,12]. In a planar nanowire with in-plane magnetization (nanostrip), thechirality of the DW is protected by topology and is also calleda topological charge [ 13]. A pair of in-plane DWs with op- posite topological charges (opposite fractional edge defects)can be created or annihilated spontaneously, but a pair ofDWs with the same topological charge form a stable magnetictexture—a soliton-soliton pair [ 14] (due to the “topological repulsion,” see Fig. 1(a) where four transverse DWs with the same topological charge are pinned in a nanostrip). In athree-dimensional (3D) cylindrical nanowire, it was shownthat a pair of DWs with the same initial topological chargeform a metastable state that anihilates after a finite time due tothe relative rotation of the walls [ 15] and the nonconservation of the total topological charge. The injection of DWs in ananowire with reliable chirality control has been demonstratedexperimentally [ 16]. The total topological charge is conserved during DW interaction, and a train of this type of DWs canbe displaced jointly in the nanostrip by a polarized currentleading to practical applications [ 17,18]. The interaction potential between the DWs can also be viewed as mediated by the topological defects. The inter-action between vortex DWs was studied analytically basedon Thiele’s approach in a two-dimensional (2D) anisotropicHeisenberg model [ 19,20] and experimentally [ 21]. In the majority of cases, only the dynamics of one DW is stud-ied and simulated micromagnetically or the dynamics ofwell-separated DWs (different nanowires or nanolayers) thatinteract through the dipolar field [ 22–31]. In a few cases, the DWs interaction in the same nanowire was studied 2469-9950/2020/101(1)/014438(10) 014438-1 ©2020 American Physical SocietyA. PIV ANO AND V . O. DOLOCAN PHYSICAL REVIEW B 101, 014438 (2020) (b) (c)(a) -1 1mx+1/2 +1/2 +1/2 +1/2 -1/2 -1/2 -1/2 -1/2 FIG. 1. (a) Simulated structure with four magnetic domain walls with the same topological charge are pinned at symmetric notches: two head-to-head and two tail-to-tail DWs. The fractional winding numbers of the edge defects are indicated for each DW. (b) Current pulse shape used to displace the domain walls. (c) Interaction energyof two domain walls in a nanowire without notches (symbols) as determined from micromagnetic simulations. The curves represent modeling with different trial functions. experimentally and micromagnetically [ 32,33]. Analytically, the interaction potential between transverse DWs in nanos-tripes was considered only based on a the multipole expansion[34], which was tested numerically for DWs pinned at artifi- cial constrictions situated at long distances [ 15,29,35] where the dipolar interaction dominates. To be able to calculateanalytically the dynamics of a train of transverse DWs ina nanowire, a pertinent model should take into account therepulsive (topological) interaction which is important at shortrange. In this paper, we address this issue, establishing the trans- verse DW interaction potential using trial models. We test ourphenomenological model numerically on the fast dynamicsof two–four transverse DWs initially pinned at symmetricnotches along a nanostrip and submitted to ultrashort currentpulses. Our analysis is similar to the classical model of a one-dimensional (1D) chain of interacting particles. In our model,we take into account only nearest-neighbor DW exchange in-teraction and the dipolar interactions up to the third neighbor.We determine that a power-law spatial variation of the DWexchange interaction of r −2type gives quantitatively good results when compared to the micromagnetic calculations,similar with the Heisenberg exchange and the interaction ofsuperconducting vortices in thin films. In a 2D XY model, theinteraction between charged particles (vortices) was shown to decrease logarithmically rather than exponentially belowa topological phase transition [ 36]. We compare the obtained power-law behavior with an exponential or a Yukawa potentialand discuss the observed differences in the DWs dynamicsand depinning currents. We also determine that the transienteffects related with the DW inertialike behavior [ 37–41] due to deformation of the DW diminish when the interactionamong several DWs is considered as symmetric interactionson both sides annihilate the deformations of the walls. Ourpaper shows that a simple analytical model gives good quan-titative results even when the interaction among several DWsis considered, paving the way for calculating phase diagramsin larger memory racetracks. This article is organized as follows. In Sec. II, we present the stochastic 1D model used to calculate the interactionbetween DWs. In Sec. III, we compute and investigate the phase diagram of the DW dynamics in an infinite nanostripatT=0 K and at room temperature. Concluding remarks are presented in Sec. IV. II. MODEL To determine the interaction potential between transverse magnetic DWs, it is necessary to control the position andthe topological charge of the DWs. In the following, theposition is controlled by pinning at notches, and the topo-logical charge is fixed initially as the injection with chiralitycontrol has already been proven experimentally [ 16]. The demagnetizing energy keeps the topological charge fixed forthe DWs up to reasonable high external applied current.We consider several pinned transverse walls with the sametopological charge in an infinite Ni nanostrip (saturationmagnetization M s=477 kA /m, exchange stiffness parameter A=1.05×10−11J/m, and spin polarization P=0.7) with a cross section of Ly×Lz=60×5n m2. No magnetocrys- talline anisotropy is considered, the shape anisotropy ensuresthat the easy axis is in plane. The strip has rectangularsymmetric double notches with dimension 20 ×9×5n m 3 and separated by 80 nm. Figure 1(a) shows the equilibrium position of a train of four neighboring (situated in neighboringnotches at 80-nm distance) transverse DWs: two pairs ofhead-to-head (HH) DWs and tail-to-tail (TT) DWs of the samechirality (and inverse polarity at their centers) to ensure thetopological stability and repulsion between them. Each DWsits in a potential well created by the notches [ 42,43]. The form of the pinning potential was determined from micromag-netic simulations and is presented elsewhere [ 44] (harmonic at the notches and sinusoidal between them). The DWs are displaced simultaneously by a series of periodic spin-polarized current pulses applied along the stripelong axis ( xdirection). The geometry of the current pulse is described in Fig. 1(b):t r,ts,tf, and tzare the rise, stable, fall time, and zero-current time, respectively. The nonadiabaticparameter is set to β=2α, if not specified otherwise. The DW dynamics was computed using the one- dimensional DW model [ 22,45] considering the DWs interac- tion and by 3D micromagnetic simulations with the MUMAX 3 package [ 46]. In both cases, the magnetization dynamics is de- termined from the Landau-Lifschitz-Gilbert (LLG) equation 014438-2ANALYTICAL DESCRIPTION OF THE TOPOLOGICAL … PHYSICAL REVIEW B 101, 014438 (2020) with adiabatic and nonadiabatic spin-transfer torques [ 47], ˙m=−γ0m×Heff+α(m×˙m)−(u·∇)m +βm×(u·∇)m, (1) where mis the normalized magnetization, γ0is the gyro- magnetic ratio, u=jePμB/eMsis the spin drift velocity, Pis the spin polarization of conduction electrons, μBis the Bohr magneton, and jeis the applied current density. No additional exotic torques (such as the ones due to the spin-Hall or Rashbaeffect) were considered. The temperature is considered in theLLG equation as a thermal field added to the effective field.The thermal field has zero average and is uncorrelated in timeand space, and its magnitude is the same as the Gaussian noiseintroduced in the 1D model below. The analytical equations of motion used are based on the 1D model of the DW (collective coordinates: average DWcenter position Xand azimuthal angle ψ)[48,49], (1+α 2)˙X=−αγ/Delta1 2μ0MsS∂E ∂X+γ/Delta1 2Hksin 2ψ +qpγ 2μ0MsS∂E ∂ψ+(1+αβ)u+ηψ−αηX, (1+α2)˙ψ=−qpγ 2μ0MsS∂E ∂X−γα 2Hksin 2ψ −αγ 2/Delta1μ 0MsS∂E ∂ψ+qpβ−α /Delta1u+ηX+αηψ, (2) with/Delta1(t)=/Delta1[/Psi1(t)]=π/radicalBig 2A μ0MS2sin2ψ+μ0MSHkthe DW width, Hkas the DW demagnetizing field, ηXandηψrepresent stochastic Gaussian noise with zero mean value and correla-tions/angbracketleftη i(t)ηj(t/prime)/angbracketright=(2αkBT)/(μ0Ms/Delta1S)δijδ(t−t/prime).Eis the potential energy of the DW that includes the internal en-ergy, the interaction energy with other DWs, and the pinningenergy. The azimuthal angle of the DW ψrepresents the conjugate momentum in the Lagrangian formulation. The in-teraction energy between DWs separated by r ijwas modeled asEint=Eexch+Emm+Edd, where Eij mm=a2D2 rijQiQj,Eij dd=a3/parenleftbiggD3 rij/parenrightbigg3 cos(ψi−ψj), EH exch=a1/parenleftbiggD1 rij/parenrightbigg2 ,Eexp exch=a1e−rij/D1, EY exch=a1D1 rije−rij/D1(3) represent the monopole-monopole ( mm), the dipole-dipole (dd) interaction, and the DW exchange interaction (topolog- ical repulsion). The topological charge of the DW is q= 1 π/integraltext dx∂xψ=±1 and is related with the direction of rotation of the in-plane magnetization when traversing the DW, and p=±1 represents the direction of the magnetization at the DW center along the yaxis (width). Although both HHDW and TTDW can have a positive or negative topological chargeand direction p, the product Q=qpis always equal to +1f o r a HHDW and to −1 for a TTDW. Therefore, the mmanddd interactions between nearest-neighbor HHDW and TTDW ofsame topological charge but opposite pdirections are always negative, meaning attractive. We introduce a repulsion term,in the form of a topological or DW exchange interaction ina phenomenological manner as shown by the E exchterms of Eq. ( 3). Several trial functions were used and compared based on the asymptotic behavior of fundamental potentials, andthe interaction potential which correlates best with the micro-magnetic simulations is the r −2decay. We only considered nearest-neighbor DW exchange interaction, but mmanddd interactions were considered up to the third neighbor (see thediscussion on the displacement of the four DWs below). The parameters a iandDiwere determined by comparing the obtained phase diagrams with the micromagnetic simula-tions. As the number of parameters is large, the starting valueswere chosen by fitting the micromagnetic results obtainedfor two DWs (a HHDW and a TTDW) initially situated at80-nm distance in a very long nanowire of the same sectionand without notches [Fig. 1(c)]. The two DWs repel each other at a closer distance until around an equilibrium positionof 140 nm, beyond which the interaction becomes attractivedue mainly to the long-range dipolar interaction. These initialparameters were modified in the case of the pinned DWs asto follow closely the micromagnetic phase diagrams, but theorder of magnitude was maintained. For the micromagnetic computations, the strip was dis- cretized into a mesh with a cell size of 2 ×3×2.5n m 3, inferior to the exchange length ( ∼5 nm). The DW dynamics is studied in an infinity long wire where the magnetic chargesat the ends of the nanostrip are compensated. III. RESULTS Our analysis of the DWs’ dynamical interaction begins with the study of the impact of the different interactionpotential trial functions on the phase diagram obtained whena symmetric pulse (stable time t s—current amplitude je)o ra n asymmetric pulse (rise time tr−je) are applied to the pinned DWs at T=0 K. Afterwards, the particularities of the DWs’ motion at room temperature are discussed for the differentinteraction terms. The last subsection details the influence ofthe transient displacement on the DW dynamics. A. Influence of the DW exchange energy on the phase diagrams at T=0K To evaluate the impact of the different DW exchange terms on the DWs’ coupled dynamics, we computed 400 ×300 point-by-point analytical phase diagrams integrating Eqs. ( 2) with a fourth-order Runge-Kutta scheme. The phase diagramsrepresent the relative position of the train of DWs afterperiodic spin-polarized current pulses are applied to them.The current pulses are varied in length, amplitude, or shapeand the correlated displacement of the DWs is extracted afterseveral periodic current pulses. When the DWs are displacedcollectively keeping the same relative distance between them,we consider that an expected and desired state is realized.These collective regular displacements form bands dependingon the pulse characteristics and the interaction potential be-tween the DWs. The analytical diagrams are compared withthe micromagnetic ones (24 ×31 points). As previously de- 014438-3A. PIV ANO AND V . O. DOLOCAN PHYSICAL REVIEW B 101, 014438 (2020) FIG. 2. Contour plots of the different bands obtained for a train of two neighboring DWs with different types of a DW exchange interaction atT=0 K using the 1D model are represented in the upper panels (a), (c), (e), and (g). In the lower panels, the 1D results (colored regions) obtained for a Heisenberg-type DW exchange interaction are compared with the micromagnetic calculations (scattered symbols) for the same pulse characteristics and αandβparameters as the upper panels. The numbering of the bands is as follows: Positive bands correspond to the DWs moving collectively in the direction of the electron flow with the same initial relative distance between them, negative numbers to the DWs moving collectively in the opposing direction with the same initial relative distance between them, the zero state corresponds to the DWs staying pinned at initial positions and “ u” corresponds to the unintended states in which the DWs do not move synchronously in either direction. The parameters used are as follows: (a) and (b) tsvariable, tr=tf=5p s,α=0.02,β=0.04, (c) and (d) tsvariable, tr=tf=5p s,α= 0.05,β=0.1, (e) and (f) trvariable, ts=tf=5p s,α=0.02,β=0.04, (g) and (h) trvariable, ts=tf=5p s,α=0.05,β=0.1. termined [ 44], the range of the current amplitude was chosen (/lessorequalslant10 A/μm2) as to to have only viscous motion (no preces- sion) for the pulse duration used ( /lessorsimilar1.5 ns), which is on the same order of magnitude with access or reading/writing timein possible magnetic memories based on DWs. At high currentamplitude or longer pulse duration, an antivortex appearswhen a DW depins from a notch [ 50,51]. The antivortex will perturb the systematic motion of the DWs and theirmutual interaction. In the results shown below, an antivortexappears only in a few points in the upper right quadrant of themicromagnetic phase diagrams (detailed in Ref. [ 44]) where the symbols are missing and does not influence our results. Our analysis starts with a train of a HHDW and TTDW having the same topological charge and situated in neighbor-ing pinning centers (separated by 80 nm). The initial distancebetween the DWs ensures that their repulsive interaction isstill important as determined from their equilibrium positionsand Fig. 1(c).I nF i g . 2, we present the results for various αandβparameters (corresponding to the Ni values at 0 K and room temperature [ 52]) and several pulse shapes. In the upper panels, contour plots for the different bands are shownobtained with the 1D model, whereas the lower panels presenta superposition of the 1D model diagrams (represented bycolors) with the micromagnetic ones (symbols). The two DWsmove together after a pulse application due to the spin transfertorque (STT) in the direction of the electron movement, butthe final DW position can be in the opposite direction due tothe transitory motion (automotion) [ 37–41]. We indexed the different regions in the phase diagram based on the relativeposition of the two DWs as follows: We call the state 0when the DWs stayed in their initial notch (position) after theapplication of the pulse (pinned case), state +1i ft h et w oD W s went to the next notch in the direction of the electron flow (ofthe STT) keeping the same distance between them or state −1 if the two DWs went to the next notch in the direction opposite the electron flow. The higher number states wereindexed in the same way ( +2 means displacement of both DWs to the second next notch in the STT direction). State uis an unintended state (such as depinning of one DW) where theDWs do not keep the initial relative distance between them.This state appears generally as a transition region betweenthe other states. As our calculation is performed on a finitesample of an infinite nanostrip, to be able to compare tothe micromagnetic simulations, the number of bands is finiteand the upper right region, that shows an unintended state,corresponds to the DW reaching the nanowire (finite sample)end. The states were determined after the application of, atleast, four periodic pulses that displace the DWs between theirinitial position and the desired position back and forth. As observed in the lower panels of Fig. 2, the 1D model DW repulsive interaction varying in r −2(called the DW Heisenberg exchange) agrees quantitatively with the micro-magnetic simulations up to the third band, afterwards a smallshift appears. In the upper panels, the contour plots of onlythe first bands are shown for different repulsive interactionand different material and pulse parameters. In panel (a), fora symmetric pulse shape ( t r=tf=5 ps) and α=0.02 (β= 2α), the contour plots obtained with the three types of DW exchange interaction are superposed with the results obtainedwith no repulsive interaction (dashed-dotted line). As can beobserved, even in the absence of the repulsive interaction,the two DWs can still be displaced synchronously due to theperiodic pinning potential and the ultrashort pulses, but the de-pinning current increases to 3 .05 A/μm 2from 2 .60 A/μm2 above ts=0.6 ns, and the bands increase and are more de- formed. This situation is equivalent with the case of two 014438-4ANALYTICAL DESCRIPTION OF THE TOPOLOGICAL … PHYSICAL REVIEW B 101, 014438 (2020) DWs initially separated by a longer distance than the range of the repulsive interaction (Supplemental Material [ 53]). The depinning current diminishes when an exponential orYukawa-type DW exchange is used to 2.21 and 1 .87 A/μm 2, respectively, above ts=0.6 ns. The variation of the repulsive interaction impacts slightly the shape and surface of the upperbands, the most important change is on the depinning currentfor the symmetric pulse [panels (a) and (c)]. For asymmetricpulses [panels (e) and (g)], where t s=tf=5 ps and the rise time tris varied, the change in form and surface of the bands is more important as compared to the symmetric pulses.Increasing the damping parameter αto 0.05 (with β=2α) as shown in panels (c) and (g) shifts all the bands to lowercurrents, including the depinning value. We used in all thecalculations the same parameters for the mmandddinterac- tion: a 2=0.2,a3=0.02 eV,D2=D3=500 nm. For the different DW exchange interactions, the parameters used areas follows: a 1=1.2 eV and D1=350 nm for EH exch,a1= 20 eV and D1=150 nm for Eexp exchanda1=90 eV and D1= 150 nm for EY exch. These parameters were chosen to fit best the micromagnetic depinning line of shortest pulse length.In-depth details about the comparison between analytic andmicromagnetic calculations are given in the SupplementalMaterial [ 53]. The importance of the pulse shape and length was inferred by decoupling Eqs. ( 2)[54,55], ¨X=−˙X τd−1 mdE dX+β ατdu+1+αβ 1+α2˙u, (4) with m=2αSμ0Msτd /Delta1γ0as the DW mass and τd=1+α2 αγ0Hkas the damping time of the wall in the pinning potential. Here, the damping time is 0.27 ns for α=0.05 and 0.68 ns for α= 0.02, so the third term in Eq. ( 4) is more important for higher damping parameter α, resulting in a lower depinning current as observed from Figs. 2(a) and2(c) (1.85 A/μm2compared to 2.60 A/μm2for the Heisenberg DW exchange). The de- pinning current increases to 3 .68 A/μm2(lowest value) in panel (e) and 3 .39 A/μm2in panel (g) for a longer rise time as the last term of Eq. ( 4) is directly proportional to the current derivative. The second term of Eq. ( 4)g i v e sah i n t to the different depinning currents obtained for various DWexchange forms used. Case of four interacting DWs The influence of the repulsive interaction between nearest- neighbor DWs was further studied by extending the analyticalcalculation up to four DWs of same topological charge. Wedescribe, here, the case of a chain of four consecutive inter-acting DWs: We consider the topological repulsive interactionbetween first neighbors in the forms presented above, alongwith the monopole-monopole and dipole-dipole interactionbetween each pair of DWs. The mmandddinteractions are attractive between first neighbors, repulsive between secondneighbors, and attractive between third neighbors. The pa-rameters a iandDiwere kept constant for first-, second-, and third-neighbor mmandddinteractions with the values given above. Figure 3displays the influence of the magnitude of the DW exchange interaction between nearest neighbors for FIG. 3. Influence of the Heisenberg DW exchange energy magni- tude on the bands for a train of four neighboring DWs for α=0.02 andβ=0.04: (a) a 1=0 eV (no DW exchange), (b) a1=1.0e V and (c) a1=1.4 eV. The 1D results (colored regions) obtained for a Heisenberg-type DW exchange interaction are compared with the micromagnetic calculations (scattered symbols) at T=0K . T h e pulse stable time was varied with tr=tf=5p sa n d tz=10 ns. α=0.02 using a Heisenberg-type DW exchange. The dif- ference between no DW exchange [panel (a)] and a DWexchange of the same order of magnitude as used for a train oftwo DWs is much more drastic as the depinning current andthe bandwidth diminish strongly when the DW exchange isturned on [panel (b), a 1=1.0 eV]. A further increase in the DW exchange interaction will lead to the quasisuppression ofthe depinning current (displaced to lower values), but also ofthe bands [panel (c), a 1=1.4 eV]. The analytic results follow very well the micromagnetic ones for the depinning currentline [panel (b)] and semiquantitatively the band form, whichvalidates the model. To further investigate the consequences of the DW ex- change interaction type on the DW dynamics, we presentthe evolution of the phase diagrams in Fig. 4for different pulse shapes and damping parameters. Panels (a) and (b)show the contour plots of the first bands due to a symmetriccurrent pulse shape ( t r=tf=5 ps) and for α=0.02 and 0.05, respectively, whereas panels (c) and (d) display the caseof asymmetric pulse shape ( t s=tf=5 ps). In panel (a), the contour plots obtained with the three types of DW exchangeinteraction are superposed with the results obtained with norepulsive interaction [shown in Fig. 3(a)]. The influence of the different DW exchange forms is more marked for the four 014438-5A. PIV ANO AND V . O. DOLOCAN PHYSICAL REVIEW B 101, 014438 (2020) FIG. 4. Contour plots of the different bands obtained for a train of four neighboring DWs (separated by 80 nm) with different types of the DW exchange interaction at T=0 K using the 1D model: (a) and (c) α=0.02 and β=0.04, (b) and (d) α=0.05 and β= 0.1. In (a) and (b), the pulse stable time was varied with tr=tf= 5 ps, whereas in (c) and (d), the rise time was varied with ts=tf= 5p sa n d tz=10 ns. DWs as the depinning current decreases as compared with t h et w oD W sc a s et o1 .54 A/μm2for the Heisenberg DW exchange and below 1 A /μm2for an exponential or Yukawa- type DW exchange. At the same time, the superior bands aredisplaced to higher currents as compared to the two-DW case,for example, the beginning of the band +1t o4.3A/μm 2from 3.5A/μm2(the Heisenberg DW exchange). This means that higher currents are needed to achieve a synchronous move-ment of the DWs and a larger unintended zone. The surfaceof the bands is also strongly reduced when a Yukawa-typeinteraction is used, which is the most unfavorable scenario.In the case of the asynchronous current pulse [panels (c) and(d)], the same shift of the depinning current and of the bands is observed with a clear difference between the different DWexchange schemes. B. Temperature dependence The effect of temperature was computed with the stochastic 1D model [Eqs. ( 2)] for the first bands and micromagnetically only on several points that corresponded to the highest prob-ability obtained with the 1D model. A more detailed compar-ison between the analytic and the micromagnetic calculatedprobabilities for the first band in Fig. 5(a) is shown in the Supplemental Material [ 53]. The results obtained analytically atT=293 K are presented in Fig. 5for a train of two or four DWs. A symmetric current pulse ( t r=tf=5 ps) was applied after an initial relaxation time of 10 ns followed by anotherrelaxation time of 10 ns. The bands shown in panels (a) and(b) are the bands of Figs. 2(a)and2(b) for the Heisenberg DW exchange, whereas the bands displayed in panels (c) and (d)are the ones from Figs. 4(a) and4(b) for the Heisenberg DW exchange. We computed 1000 realizations for the +1 band and 500 realizations for the +2 and+3 bands. The realizations were calculated for half of the points in each band for the trainof two DWs [panels (a) and (b)] and for all the band points forthe train of four DWs (less total points in the bands). In Fig. 5(a), the maximum of the probability distribution for the positioning of a train of two DWs to the nearestnotch ( +1 band) is of 100% obtained for seven states (points) (α=0.02) out of 3093 calculated points with 29.9% of the states having a probability superior of 95%. The states thathave 100% probability of desired displacement are obtainedfor a pulse with t s=100 ps and current amplitude superior to 9.1A/μm2orts=110 ps and je/greaterorequalslant8.5A/μm2. The max- imum of probability decays in the superior bands, being of95.6% on the +2 band and 68.8% on the +3 band. These FIG. 5. Probability of DWs motion in different bands at T=293 K for a train of two neighboring DWs and a damping parameter α=0.02 in (a) or α=0.05 in (b) or a train of four neighboring DWs with α=0.02 in (c) or α=0.05 in (d). A Heisenberg-type DW exchange interaction is used along with a nonadiabatic parameter β=2α. The vertical dotted lines correspond to the probability profile represented in the figures underneath. The profiles are compared for different types of DW exchange interaction and are chosen in each case as to pass through the maximum probability of the first band (the lowest branch in the figures). 014438-6ANALYTICAL DESCRIPTION OF THE TOPOLOGICAL … PHYSICAL REVIEW B 101, 014438 (2020) probabilities are comparable with the ones when a single DW is displaced by current pulses [ 44]. Micromagnetically, the maximum of probability is of 98% (on 100 realizations)obtained for the same pulse characteristics that give max-imum probability with the 1D model. The discrepancy isprobably due to the small shift of the bands between the twomodels. For a damping parameter α=0.05 [Fig. 5(b)], the maximum of the probability distribution is 99.9% in the +1 band obtained for a lower current amplitude of 7 .8A/μm 2 andts=90 ps. The percentage of states having a probability superior to 95% is of 30.6% of the 3310 calculated states. Themaximum of probability is 91% and 77% for the +2 and +3 bands. For a train of four DWs, the maximum of the probabil- ity distribution in the +1 band decreases slowly to 97.4% [α=0.02, panel (c)] and 96.2% [ α=0.05, panel (d)]. The probability maximum in the +2 and +3 bands is 76.8% and 28.2%, respectively, for α=0.02 and 70% and 47.8% for the α=0.05 case. The current pulse characteristics for which the probability maximum is obtained are ( t s=90 ps,je= 9.9A/μm2)f o rα=0.02 and ( ts=90 ps,je=7.8A/μm2) forα=0.05. The probabilities when an asymmetric pulse is applied are almost equal with the ones obtained for symmetricpulses for all the cases presented above. In the Figs. 5(e)–5(h), we compare the profiles of the probability distribution when passing through the maximumof the probability in the +1 band for the different DW exchange energies considered. The profiles corresponding tor −2DW exchange are represented by a dotted line in the panels directly above them. There is a considerable differencein the probabilities of a train of two DWs and a train of fourDWs: For the two DWs [panels (e) and (f)], the probabilitymaximum is almost the same in the three bands for thedifferent DW exchange interactions with only a shift of thebands along the t saxis. For the train of four DWs [panels (g) and (h)], the probabilities depend strongly on the spatialvariation of the DW exchange interaction. For the α=0.02 case, the maximum probability in the +1 band decreases to 64.1% for the exponential DW exchange and to 17.5% for theYukawa-type interaction. For α=0.05, the maximum proba- bility is 79.1% for the exponential DW exchange and 46.3%for the Yukawa-type interaction. This difference can be relatedto the first two terms in Eq. ( 4), to the damping parameter through the different damping time, and to the force exertedon the walls due to the interaction energy between them. Thelarge difference in probability of the +1 state between the Heisenberg DW exchange and the Yukawa DW exchange isdirectly imputable to the type of the repulsive energy betweenthe DWs (Supplemental Material [ 53]), generally the first DW depins even before the application of the current due to thelarge angular variation and, therefore, large transient effectsdirectly related with the oscillation of the second DW (andtheir mutual interaction). C. Influence of transient effects on the DW dynamics Large transient effects were predicted and observed in the movement of one DW in a nanowire [ 37,38,44,56]. These transient effects were related to the deformation of the wall inthe periodic potential and produced a displacement of the wallin the direction opposite to the STT (opposite to the electron flow), corresponding to negative bands in our phase diagrams.The transient movement was determined to be proportional tothe wall angle, δX=−/Delta1 α/parenleftbigg 1−β α/parenrightbigg δψ. (5) The transient displacement was predicted to appear for a value of the nonadiabatic parameter β=0,β=αand even β=2αfor a single DW submitted to ultrashort current pulses [44] comparable with the DW damping time τd. In the case of interacting DWs, these transient effects still appear as shownin Fig. 6, but they are greatly reduced (which seem to agree with a quantum-classical hybrid approach [ 57]). For a train of two DWs, the transient effects appear only for β=0( o r close to) when a symmetric pulse is applied (details in theSupplemental Material Ref. [ 53]) and even for β=2α(α= 0.02) for an asymmetric pulse when the rise time is larger than 0.35 ns. However, for a train of four DWs, the transient effectsappear only in the case of β=0 and a rise time superior of 0.5 ns forming a reduced −1 band. These effects still appear even for a train of five DWs [Fig. 6(i)] with the −1 band shrinking rapidly. The transient effect appear due to a combination of factors [44]: The presence of the periodic pinning potential which distorts the DWs, restoring force in the potential well, positionof the DWs in the potential well at the pulse end, and a lowdamping value. For the train of four DWs, the walls that aresituated at the interior of the train are less distorted than theones which are situated at the beginning and the end of thesequence as the interior walls fill symmetric interaction forcesfrom both neighboring walls and are situated at the center ofthe potential well. The exterior walls are more deformed asthey are pushed from the equilibrium position of the potentialwell, and they escape first from the train creating unintendedstates. The results obtained with the analytical model for a train of two DWs are displayed in Figs. 6(a)–6(c) for the differ- ent DW exchange interactions and different initial distancesbetween the DWs ( α=0.05,β=0). When the two DWs are initially pinned in nearest-neighboring notches situated80 nm apart, the −1 band is obtained only for Heisenberg DW exchange and only for currents inferiors to 7 .7A/μm 2. There is a discrepancy with the micromagnetic result shownin panel (d) where the −1 band continue to higher currents for shorter pulse length. If the two DWs are initially pinned atsecond-neighboring notches 160 nm apart [panel (b)], the −1 band obtained analytically follows closely the micromagneticone [panel (e)] even though somewhat larger. In this case,the−1 band is obtained also for exponential DW exchange. Furthermore, if the two DWs are pinned initially even furtheraway at third-neighboring notches 240 nm apart [panel (c)],the−1 band is obtained for all three types of DW exchange interaction with almost same bandwidth and form and veryclose to the micromagnetic result [panel (f)]. As the two DWsare further away, the DW exchange interaction have onlya limited influence, and the dipolar interaction determinesthe form of the bands. We observe that the DW exchangeinteraction at shorter distances modifies the width and form ofthe band. For a train of three, four, or five DWs, the −1 band 014438-7A. PIV ANO AND V . O. DOLOCAN PHYSICAL REVIEW B 101, 014438 (2020) FIG. 6. Influence of the pulse rise time tron the phase diagram for a train of several DWs at T=0Kf o r α=0.05 and β=0. The parameter space is the rise time vs current amplitude. In all cases, ts=tf=5p sa n d tz=10 ns. The 1D model results for different DW exchange energies are shown in panels (a)–(c) for a train of two DWs separated by (a) 80 nm, (b) 160 nm, and (c) 240 nm. Only the band −1 is visible in the center of the diagram together with the depinning boundary. The micromagnetic results are presented in panels (d)–(i) whereband−1 is represented by a continuous line and the limit of band 0 is represented by the dotted line. The micromagnetic simulations are for a train of two DWs separated by: (d) 80 nm, (e) 160 nm, and (f) 240 nm and a train of (g) three DWs, (h) four DWs, and (i) five DWs separated by 80 nm. is still obtained micromagnetically as shown in the panels (g)–(i). Analytically, we did not obtain the −1 band for neither of the DW exchange interactions for trains of DWs superior oftwo and the parameters used above. This can be related to thesmallness of the bands width and to the values of the DW ex-change parameters, but also to the pinning potential analyticaldescription (harmonic periodic potential). Changing the DWexchange parameters allows to obtain the negative bands, butthe depinning line no longer follows the micromagnetic resultand differences are obtained for the others values of αandβ. The values for the DW exchange and dipolar parameters werechosen to follow closely the depinning line and the first bandsforβ=2α. These parameters give semiquantitative results even for β=0 (the depinning line, for example), but the limit of the 1D model is reached. IV . DISCUSSION AND CONCLUSION We investigated the repulsive interaction between trans- verse DWs with the same topological charge, pinned at con-strictions in a magnetic nanowire. Our analytical study ofthe DW interaction shows that a r −2decay describes best the micromagnetic results. The same power-law variationwas found to describe the vortex-vortex interaction in su- perconducting films [ 7] but differs from the vortex-vortex interaction in bulk superconductors or the magnetic vortex-vortex interaction calculated in a 2D anisotropic film (whichdecays logarithmically). Our trial functions for the interactionpotential also included an exponential or Yukawa potential,which describe a large number of interaction forces in manyareas of condensed-matter physics (in discrete or continuummodels). The analytical phase diagrams for a train of up to four DWs follow closely the ones calculated with micromagneticsimulations when the r −2decay is used. If the repulsive inter- action would decay with an Yukawa potential, the changes tothe phase diagrams are important starting from the depinningcurrent and the form of the bands to the large decrease inthe maximum jump probability to the nearest pinning position(the+1 band) at room temperature and the suppression of the transient effects. The Yukawa-type repulsive interaction be-tween DWs is the most unfavorable scenario for the collectivemotion of DWs in a nanostrip at room temperature. A train of four DWs is shown to be displaced regu- larly between pinning centers with ultrashort current pulses(100 ps), leading to practical applications, such as magnetic 014438-8ANALYTICAL DESCRIPTION OF THE TOPOLOGICAL … PHYSICAL REVIEW B 101, 014438 (2020) memories. The lowest depinning currents are found for the ultrashort rise time of the pulse as described before [ 44]a s long as the largest bandwidth. When going from a train oftwo nearest-neighboring DWs to a train of four, the mainimpact is the decrease in the depinning current, but, at thesame time, a decrease in the bandwidth and an increasein the unintended region with larger transition regions be-tween the bands. The transient effects are also severely di-minished due to the mutual interaction and are eventuallysuppressed. We would like to discuss our results on the repulsive interaction between DWs from the spin waves perspective.The spin waves or magnons are the elementary excitationsof the electronic magnetic system [ 58], quasiparticles with a¯hangular momentum, and ¯ hklinear momentum. It was shown both theoretically [ 59–64] and experimentally [ 65] that the spin waves can induce DW motion in both directionsdue to the angular or linear momentum transfer. The motionis directly related with the transmission coefficient of thespin waves passing through the wall. It was also shownnumerically that the rigid DW motion is not stable againstspin wave emission [ 66–69]. In principle, a DW submitted to an ultrashort current pulse could emit spin waves thatinteract with other DWs. This interaction will be attractive ifonly the angular momentum is transferred to the second DW(magnonic spin torque) or more complex if linear momentumis also transferred. The repulsive interaction considered here isthought to be mediated through the exchange of gauge bosonsof integer spin which could be virtual magnons. A DW would emit a magnon that is absorbed by another DW of oppositetopological charge, therefore, inducing the repulsive interac-tion which is a fundamental interaction that exists with orwithout the presence of notches or an applied external current.If the DWs could rotate out of plane, one of the DWs couldchange its topological charge, and the interaction can becomeattractive as observed in cylindrical nanowires [ 15]. In our view, this can be demonstrated exactly only in a microscopictheory and cannot be proved in a continuum theory. In ourmicromagnetic simulation, this interaction arises due to theexchange energy term and is described analytically, such as anexchange interaction between DWs (which can be describedas a magnon spin current). 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PhysRevB.99.024405.pdf

PHYSICAL REVIEW B 99, 024405 (2019) Micromagnetic modeling of terahertz oscillations in an antiferromagnetic material driven by the spin Hall effect V . Puliafito,1R. Khymyn,2M. Carpentieri,3B. Azzerboni,1V . Tiberkevich,4A. Slavin,4and G. Finocchio5 1Department of Engineering, University of Messina, 98166 Messina, Italy 2Department of Physics, Gothenburg University, 40530 Gothenburg, Sweden 3Department of Electrical and Information Engineering, Politecnico of Bari, 70125 Bari, Italy 4Department of Physics, Oakland University, Rochester, Michigan 48309, USA 5Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, 98166 Messina, Italy (Received 4 August 2018; published 7 January 2019) The realization of terahertz (THz) sources is a fundamental aspect for a wide range of applications. Over dif- ferent approaches, compact THz oscillators can be realized, taking advantage of dynamics in antiferromagneticthin films driven by the spin Hall effect. Here we perform a systematic study of these THz oscillators within afull micromagnetic solver based on the numerical solution of two coupled Landau-Lifshitz-Gilbert-Slonczewskiequations, considering ultrathin films. We find two different dynamical modes depending on the strength of theDzyaloshinskii-Moriya interaction (DMI). At low DMI, a large-amplitude precession is excited, where both themagnetizations of the sublattices are in a uniform state and rotate in the same direction. At large enough DMI,the ground state of the antiferromagnet becomes nonuniform and the antiferromagnetic dynamics is characterizedby ultrafast domain-wall motion. DOI: 10.1103/PhysRevB.99.024405 I. INTRODUCTION Terahertz (THz) radiation covers the range of frequencies from 300 GHz (gigahertz) to 3 THz, between microwaves andinfrared, corresponding to wavelengths ranging from 1000 to100μm[1,2]. Since a wide variety of lightweight molecules emits in this range of the electromagnetic spectrum, THzwere intensely investigated by astronomers and chemists inthe past [ 3,4]. However, THz oscillations have turned out to be very promising in many other fields, such as biomedicine [5], defense and security [ 6], material science [ 7], industrial nondestructive testing [ 8], and information and communica- tion technology [ 9,10]. THz sources can be realized with quantum cascade lasers [ 11] and solid-state devices [ 12]; however, the development of compact nanosized electricalgenerators and receivers of THz signals represents a keychallenge of the modern technology. With the experience maturated after decades of research on the generation and manipulation of GHz-frequency dynamics in ferromagneticmaterials [ 13–18], the development of high-quality antiferro- magnetic materials (AFMs) for several applications [ 19–23], and proof of concept of antiferromagnetic memories[24–27] driven by the spin Hall effect (SHE) [ 28], research is now combining this know-how focusing on the develop-ment of AFM-based oscillators for application in 4G and 5G telecommunication systems [ 29–35]. Up to now, there is no experimental proof of AFM-based oscillators, and allthe theoretical studies are considering two sublattices withtheir magnetizations antiferromagnetically coupled [ 36] and their dynamics is studied by solving two Landau-Lifshitz-Gilbert-Slonczewski equations [ 37] within the macrospin approximation [ 29,31,32].The first motivation of this work is to extend the study of AFMs to a full micromagnetic framework for consideringpossible nonuniformities of the magnetization. Second, wewant to move a step forward to the understanding of THzAFM dynamics driven by a dampinglike torque originatingfrom the spin Hall effect in a typical bilayer AFM heavymetal. We focus, in particular, on 1–5-nm-thick film of nickeloxide and, although we assume a small exchange stiffnessconstant as compared to the bulk values, THz dynamics canbe excited at large enough current. We show a systematicstudy of the threshold currents and the output frequency asa function of spin-polarization direction, exchange constant, Gilbert damping, AFM thickness, and Dzyaloshinskii-Moriya interaction (DMI) coming from the interface between theAFM and the heavy metal [ 38]. We find that the DMI is the most influent parameter in controlling the type of AFM dynamics. At low DMI, thethreshold current is a subcritical Hopf bifurcation [ 39] and the dynamics is related to a large-amplitude uniform preces-sion of the magnetization of the two sublattices in the samedirection with an angle between the magnetization and the precession axis that depends on the applied current. As the DMI increases, the ground state becomes nonuniform and theexcited dynamics changes qualitatively since it is related to acontinuous domain-wall nucleation, propagation, and annihi-lation. In addition, the threshold current is a supercritical Hopfbifurcation. Our results highlight that a full micromagneticmodel can be used for the description of all the scenarios where AFM oscillations occur. This paper is organized as follows. Section IIis devoted to the micromagnetic model developed for the analysis. Results 2469-9950/2019/99(2)/024405(7) 024405-1 ©2019 American Physical SocietyV . PULIAFITO et al. PHYSICAL REVIEW B 99, 024405 (2019) FIG. 1. Schemes of the device under investigation with the in- dication of the Cartesian reference systems. (a) A schematic of the bilayered ASHO. The four terminals can be used for the applica-tion of charge currents, and for the measurement of the spin Hall resistance. (b) Top view of the antiferromagnet. m 1andm2represent the initial configuration of the magnetizations of the two sublatticeswhile pis the spin polarization. (c) Sketch of the idea at the basis of the continuous modeling of antiferromagnetic sublattices. For a given computational cell we consider that the average magnetizationis given by the two vectors m 1andm2. are given in Sec. IIIin detailed paragraphs, then the conclu- sions are summarized in Sec. IV. II. MODEL The device under examination is an AFM-based spin Hall oscillator (ASHO), consisting of an antiferromagnetic layercoupled to a four-terminal heavy metal layer, representingboth electrodes and source of spin current [ 32] [see Figs. 1(a) and 1(b), where the Cartesian coordinate systems are also shown]. The AFM has a square cross section with dimensions40×40 nm 2, whereas the thickness dvaries from 1 to 5 nm. We use a continuous micromagnetic formalism, whichextends the one of ferromagnets, considering the macroscopicproperties of an AFM as computed from averaging the spinvectors [ 40]. In detail, starting from the atomistic model, the magnetization at each point is modeled by means of twovectors m 1andm2[Fig. 1(c)] that are the average magnetic effect of the spins pointing parallel or antiparallel to a specificdirection. AFM dynamics of m 1andm2is obtained by solving two coupled Landau-Lifshitz-Gilbert equations, where theSHE-driven spin transfer torque is taken into account bymeans of an additional Slonczewski-like torque term [ 36]: dm 1 dτ=−(m1×heff-1)+αm1×dm1 dτ+dJ(m1×m1×p) dm2 dτ=−(m2×heff-2)+αm2×dm2 dτ+dJ(m2×m2×p). (1) On the left-hand side of Eq. ( 1),m1andm2are therefore the magnetizations of the two sublattices, normalized withrespect to the saturation magnetization M S, and τis thedimensionless time τ=γ0MSt, where γ0is the gyromagnetic ratio [ 41]. On the right-hand side, heff-1 andheff-2 are the normalized effective fields acting on the two sublattices, andαis the Gilbert damping. The third term represents the SHE- driven torque, where d J=gμBJS 2γ0eM2 Sd,gis the Landé factor, μB is the Bohr magneton, eis the electron charge, JSis the spin current, which is proportional to the charge current J through the so-called spin Hall angle θSH,JS=θSHJ.T h e spin Hall effect creates a Néel torque that is assumed to havethe same form for each magnetic sublattice. The vector pis the direction of the spin Hall polarization, given by p=ˆz×j, where ˆzandjare the directions of the spin and electric currents, respectively. By a proper combination of the currentat the source terminals, it is possible to manage the directionof the spin Hall polarization. In our case, pcan be fixed in thex−yplane with an angle θ pbetween 0° and 90°: If the electric current is applied only at the terminals B-B’ (A-A’),thenθ p=0◦(θp=90◦), resulting in a polarization collinear (normal) to the easy axis; see Fig. 1(b). The effective fields include the standard contributions from exchange, anisotropy, and demagnetizing field, together withthe interfacial DMI and the thermal field: h eff-1=hexch-1+hani-1+hdemag-1 +hdmi-1+hth-1 heff-2=hexch-2+hani-2+hdemag-2 +hdmi-2+hth-2. (2) The exchange fields take into account both ferromagnetic coupling between neighbors in each sublattice (this is thesame as in the standard model for the ferromagnets) andthe antiferromagnetic coupling between the two sublattices.The latter is considered of atomistic origin because the twomagnetization vectors are at the same point and it is modeledconsidering only the homogeneous part, h exch-1=αexch-FM ∇2m1+λAFMm2 hexch-2=αexch-FM ∇2m2+λAFMm1, (3) where αexch-FM =2AFM/μ0M2 SandλAFM=4AAFM/μ0a2M2 S ponder the two main contributions, AFMandAAFM are the ferromagnetic and antiferromagnetic exchange constant, re-spectively, ais the lattice constant, and μ 0is the vacuum permeability. We consider anisotropy fields originating from uniaxial material: hani-1=αanim1·uk hani-2=αanim2·uk(4) where αani=2KU/μ0M2 S,KUis the uniaxial anisotropy con- stant, and uKis the direction of the easy axis that is the xaxis in our study [ 42]. The demagnetizing field is calculated by solving the mag- netostatic problem for the total magnetization ( m1+m2)/2. We have included this field in our simulations because fromthe theory a small, but not zero, total magnetization is ex-pected. However, some simulations performed without thisterm of the effective field have provided the same qualitativeresults with a slight quantitative difference. The additional contribution to the effective field for considering the interfacial DMI is given by the following 024405-2MICROMAGNETIC MODELING OF TERAHERTZ … PHYSICAL REVIEW B 99, 024405 (2019) expression: hdmi-1=−2D μ0MS[(∇·m1)ˆz−∇mz−1] hdmi-2=−2D μ0MS[(∇·m2)ˆz−∇mz−2], (5) with Dbeing the parameter accounting for the intensity of DMI. The boundary conditions now hold,dmi dn=1 χ(ˆz×n)× mi(i=1,2), where nis the unit vector perpendicular to the edge and χ=2AFM Dis a characteristic length in the presence of DMI. The thermal field is considered as a stochastic contribution added to the deterministic effective field: /bracketleftbigghth-1 hth-2/bracketrightbigg =ξ MS/radicalBigg 2αkBT μ0γ0/Delta1VMS/Delta1t, (6) where kBis the Boltzmann constant, /Delta1Vand/Delta1tare the discretization volume and integration time step, respectively,while Tis the temperature. ξis a six-dimensional white Gaussian noise with zero mean and unit variance, uncorrelatedfor each computational cell [ 43,44]. As we are interested in the dynamics of ultrathin antifer- romagnetic films, we assume here a substantially low valueof the homogeneous intersublattice exchange A AFM/a2= 1.25 MJ/m3, where a=0.5 nm. The discretization cell used for the simulations is 2 nm ×2n m ×d. When not speci- fied, we have used the following parameters for the ASHO:d=5nm,M S=350×103A/m,α=0.05,KU=105J/m3, θSH=0.2, and AFM=0.5×10−11J/m. III. RESULTS A. Role of spin-polarization direction We consider three experimental realizable spin- polarization directions p1,p2, and p3: (1)p1is obtained if the current is applied at the terminals B-B/primealong the −y direction θp=0◦, the spin polarization is collinear with the equilibrium magnetization of the twosublattices [ 30]; (2)p 2is obtained if the same current is applied simultane- ously at both A-A/primeand B-B/prime,θp=45◦in the region where the AFM is positioned; (3)p3is obtained if the electric current is applied at the terminals A-A/primealong the xdirection; hence θp=90◦ and the spin polarization is perpendicular to the equilibrium magnetization of the two sublattices [ 32]. Figures 2(a) and2(b) show the threshold currents and the oscillation frequencies as a function of current density forthe three different spin polarizations without DMI. In all thecases, the self-oscillation is a subcritical Hopf bifurcationcharacterized by hysteresis with J ONandJOFFswitching-on and switching-off current densities, respectively. This hys-teretic excitation has been already predicted by an analyticaltheory for the p 3configuration [ 32] and can be understood qualitatively by considering that at JONthe precession of the magnetization of the two sublattices has a finite largeamplitude. Differently from subcritical Hopf bifurcation inferromagnet-based spin transfer torque oscillators [ 17,45–47], here also at J OFFthe amplitude of the oscillation of sublattices FIG. 2. (a) Threshold current densities ( JONandJOFF) for the excitation of the antiferromagnetic dynamics for three different directions of spin polarization. (b) Oscillation frequency as a function of the applied current with a zoom near the threshold current. (c), (d) Amplitude of the ycomponent of the magnetization as a function of the current density for θp=0◦andθp=90◦. magnetization is finite and even larger than the one at JON[see Figs. 2(c) and2(d), where the amplitude of the ycomponent of the magnetization for θp=0◦and 90° as a function of current density is displayed—-see also Supplemental Mate-rial, Note 1 [ 48], where the differences between subcritical and supercritical Hopf bifurcation for AFMs and FMs arehighlighted]. This result is relevant from a technological pointof view because the AFM-based oscillator can also work at acurrent density below J ONas already pointed out in Ref. [ 32]. The width of the hysteretic region depends on the polarizationdirection as for θ p=0◦it is very narrow (0 .4×108A/cm2), whereas it increases with the increase of θp(3.2×108A/cm2 forθp=90◦). This result can be directly linked to the fact that the precession axis is parallel to the spin polarization,then at θ p=0◦it coincides with the equilibrium axis, while atθp=90◦the precession axis is perpendicular to it (see top right inset of Fig. 3). The AFM magnetization dynamics is characterized by the rotation of the magnetization of both sublattices m1andm2 in the same direction with an angle ψwith respect to the os- cillation axis (top left inset of Fig. 3). The rotation frequency [Fig. 2(b)] exhibits blueshift tunability [21 GHz /(108A/cm2)] and is basically independent of the spin-polarization directionat high currents, which is associated with the high energyof the rotation of the Néel vector, defined as ( m 1−m2)/2. The anisotropy of the AFM defines the potential profile forthe magnetizations m 1andm2and, thus, the ground state of the AFM. However, at high currents, the kinetic energy ofthe magnetizations rotation significantly exceeds the potentialenergy of anisotropy [ 32], and consequently the angular ve- locity does not depend on the anisotropy profile and directionof spin polarization. In this case, the frequency is definedonly by the spin torque to damping ratio [ 32]. For a fixed current density, the trajectory is characterized by the sameψaround the oscillation axis fixed by the spin-polarization 024405-3V . PULIAFITO et al. PHYSICAL REVIEW B 99, 024405 (2019) FIG. 3. Trajectories of the magnetizations of the two sublattices in the three different cases of spin Hall polarization, around its di- rection, for J=30×108A/cm2. Left inset: sketch of the precession of the two magnetizations around the spin polarization. Right inset: directions of the spin polarization in the three cases. direction. This fact is preserved also at very large current; see for example the main panel of Fig. 3for the trajectories at J= 30×108A/cm2. As expected from analytical computations the frequency is proportional to the current density (see Eq. (7)of Ref. [ 32]). For the simulation parameters of this study, a maximum frequency of 0.6 THz at J=30×10 8A/cm2is observed. Atθp=90◦, we have performed a comparison with the analytical model developed in Ref. [ 32], finding an agreement described below in the paper (see also Supplemental Material,Note 2 [ 48]).B. Output signal The first challenge to face is the conversion of the AFM dynamics in a measurable THz signal. Some proposed strate-gies are based on the inverse spin Hall effect [ 32] or dipolar radiation [ 49]. Those two approaches need tilting of the mag- netization of the two sublattices for originating a net rotatingmagnetic vector or a time-varying phase angle between thetwo sublattices; however, for realistic parameters the outputpower should be very small. On the other hand, our four-terminal scheme can be used biasing the device with a propercurrent in order to have p 1,p2, and p3, and reading the mag- netoresistance at one of the couples of terminals AA/primeor BB/prime [50,51]. For example, when the bias current is applied through the AA/primeterminals and hence the spin polarization is p3,t h e THz signal should be read out via the BB/primeterminals and it is mainly originated by the oscillation of the ycomponent of the magnetization of the two sublattices; such an oscillationhas a frequency that is two times the precession frequency(see Supplemental Material, Note 3 [ 48]). Alternatively, the THz signal can be read via the same AA /primeterminal via the magnetoresistance that originates from the oscillation of the x component of the magnetization of the two sublattices [ 52]. C. Systematic study for p3spin polarization Figures 4(a)–4(c) show the switching-on JON and switching-off JOFFcurrent density as a function of d,α, and Awhile maintaining the other two parameters constant. The threshold currents clearly increase with the increase of boththe AFM thickness and the damping [Figs. 4(a) and4(b)]. On the other hand, our simulations confirm that the exchangecontribution plays an important role mainly in the switching-off current density, which slightly increases with the value ofA, whereas the switching-on current density is almost constant FIG. 4. Summary of micromagnetic simulations for a current applied along the xaxis, so that the spin Hall polarization is along the yaxis (θp=90◦). (a)–(c) Switching-on and -off current densities as a function of AFM thickness d(a), damping α(b), and exchange constant A (c). (d)–(f) Oscillation frequency of the ycomponent of the magnetization of the AFM as a function of the current density, for different values of the thickness d(d), the damping α(e), and the exchange constant A(f). 024405-4MICROMAGNETIC MODELING OF TERAHERTZ … PHYSICAL REVIEW B 99, 024405 (2019) FIG. 5. Comparison between micromagnetic simulations and analytical models in the case θp=90◦: (a) threshold currents, (b) oscillation frequency of the ycomponent of the magnetization, and (c) oscillation frequency of the ycomponent of the magnetization in the case of high intersublattice exchange ( AAFM/a2=20 MJ/m3). [Fig. 4(c)]. The hysteresis width increases with the thickness, decreases with the damping, and slightly decreases with thevalue of A. Such results agree with the theoretical predictions (see Eqs. (4) and (5) of Ref. [ 32]). Within the same parametric study, Figs. 4(d)–4(f) show the oscillation frequency (as computed from the ycomponent of the magnetization) of the excited dynamics as a functionof the applied current J, for different values of thickness, damping, and exchange constant. The frequency tunabilityis blueshift on current with frequency values ranging fromhundreds of gigahertz up to several terahertz. In particular,the frequencies increase with either the decrease of thickness[Fig. 4(d)] or damping [Fig. 4(e)]. In conclusion, full numer- ical micromagnetic simulations are in qualitative agreementwith the theoretical predictions that hence can be used as atool to identify the parameter region where to optimize theTHz AFM-based oscillators [see Fig. 4(f), and Eqs. (6) and ( 7 )o fR e f .[ 32]). The oscillation frequencies in Fig 4(e), computed for a= 0.01, shows a jump to zero at J=5×10 8A/cm2where the dynamics of the ycomponent of the magnetization is off (m y is constant) and the trajectory is in the x-zplane. This is a direct consequence of the reduced exchange stiffness or, inother words, the low thickness of the AFM film. As can beobserved, the damping is a critical parameter either for theoscillation frequency or for the range of current tunability.This brings us to the conclusion that the THz dynamics inultralow-damping AFMs will be observable in a narrow rangeof current density, at least if we read out the signal via the spinHall resistance. We have also performed simulations of a smaller (30 × 30 nm 2) and a larger (80 ×80 nm2) AFM sample, with the default values for d,A, and α, to reveal the possible role of the dimensions in the magnetization dynamics. However,those simulations have shown that both the current needed toswitch on the dynamics and the frequency of oscillation turnout to be equal to the case of 40 ×40 nm 2sample. Actually, this outcome was expected, considering that the volume of theactive layer does not appear in the analytical model. D. ASHO linewidth Together with frequency tunability and threshold current, the linewidth is another fundamental property of an oscillator.In order to calculate the linewidth for the AFM oscillator, wehave performed micromagnetic simulations at room tempera-ture (T=300 K).We have computed the linewidth for different values of current density, T=300 K, θ p=90◦, and the default values ford,α, and A. Our results point out that it is smaller than 10 MHz (our simulations are 100 ns long), corresponding to aquality factor of Q=f//Delta1f=41 000 at least. E. Comparison with analytical model As already cited, our main numerical results agree with recently published theoretical predictions [ 32]. For this rea- son, we focused on a direct comparison between micromag-netic simulations and those analytical models, finding a goodagreement for both the threshold currents and the outputfrequencies. Figure 5summarizes this comparison. In the first graph, numerical threshold currents, as a function of the AFMlayer thickness, are compared with the analytical formulas(Eqs. (4) and (5) in Ref. [ 32]): J ON=ωani 2σ JOFF=2α πσ√ωexchωani, (7) where ωani=γ0(2KU/MS),σ=(gμBθSH/2eMSd),ωexch= γ0(4AAFM/a2MS). Figure 5(b) shows the comparison concerning the output frequency of the oscillator for the default set of parameters.The analytical formula corresponds to Eq. (7) of Ref. [ 32]: ω=σJ α, (8) where, however, we are referring to the double frequency of theycomponent of the magnetization. We also performed numerical and analytical calculations in the case of higher exchange, considering AAFM/a2= 20MJ/m3. Again, the comparison is convincing [see Fig. 5(c)], and we can state that, from the qualitative point of view, there is no significant change in the dynamics and inthe inertial nature of their excitation. F. Effect of the DMI The need of the full micromagnetic framework to analyze the magnetization dynamics in an AFM driven by SHE isclear in the presence of the interfacial DMI. The first effectof the interfacial DMI is on the ground state. In particular,Fig.6shows the evolution of the equilibrium configuration of the magnetization for different D. Starting from the uniform state [Fig. 6(a)], Néel-type domain walls (DWs) are stabilized 024405-5V . PULIAFITO et al. PHYSICAL REVIEW B 99, 024405 (2019) FIG. 6. (a)–(e) Equilibrium configurations of the magnetization in the two sublattices as a function of the interfacial DMI parameter D. (f) Switching-on and -off current densities as a function of D. (g) Oscillation frequency of the spin Hall magnetoresistance as a function of current density for different values of D. starting from D=1.5mJ/m2[see Figs. 6(c)–6(e)][53–55]. The second effect is the change of the bifurcation at D= 1.0mJ/m2from subcritical to supercritical and hence with the disappearing of the hysteretic excitation ( JON=JOFF)a s displayed in Fig. 6(f). The third effect is the qualitative change of the magnetization dynamics that now it is characterized bya continuous nucleation, shifting, and annihilation of DWsalong a direction that depends on the applied current (seeSupplemental Material [ 48], Movies 1 and 2, to compare the dynamics at D=0.0m J/m 2and 2.0m J/m2)[56]. Figure 6(g) summarizes the output frequency as a function of current density for different DMI parameters and it turns out that DMIdoes not play a very important role in this case. This result isdue to the fact that the main role of the DMI is the stabilizationof the domain-wall chirality. We performed micromagnetic simulations considering a high intersublattice exchange, also in the case of interfacialDMI. The magnetic configuration of the AFM sublattices isstill characterized by nonuniform DWs, which translate alongthe current as in the case of low exchange [see Fig. 7(a) with the equilibrium configuration of sublattices obtained FIG. 7. (a) Equilibrium configuration of the magnetization in the two sublattices in the case of high interlayer exchange ( AAFM/a2= 20 MJ/m3)f o r D=1.5m J/m2. (b) Comparison of the output frequency with low and high intersublattice exchange for D= 1.5m J/m2. forD=2.0mJ/m2. The high exchange, moreover, does not influence significantly the frequency of dynamics, as shownin Fig. 7(b). Nucleation and dynamics of DWs, in fact, are strictly connected with nonlocal terms. These contributionsgenerally come from magnetostatic and nonhomogeneousexchange fields, including DMI. Intersublattice homogeneousexchange, instead, is a local term, due to the interaction of themagnetization of the two sublattices in the same cell. IV . CONCLUSIONS AFM materials are promising for the realization of a compact submicrometer-scale THz oscillator tunable witha current in a wide range of frequency ranging from fewhundreds of GHz up to 1–2 THz. Actually, this idea is still notdemonstrated experimentally; this paper contributes to furnisha more detailed numerical understanding of the THz dynamicsdriven by spin Hall effect. We find that the macrospin-basedtheoretical model can be used for a qualitative study at verylow DMI while a full micromagnetic approach is necessaryin the presence of DMI, which is an energy contribution thatarises in most of the experimental promising solutions forAFM-based oscillators. ACKNOWLEDGMENT The authors thank Takahiro Moriyama, Oksana Fesenko, and Pedram Khalili Amiri for the fruitful discussions. [ 1 ]P .H .S i e g e l , IEEE Trans. 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PhysRevB.72.054431.pdf

Current-induced switching in Co/Cu/Co spin valves: The effect of interdiffusion C. Sommers Laboratoire de Physique des Solides, Université de Paris-Sud, 91405 Orsay Cedex, France P. Weinberger Center for Computational Materials Science, Gumpendorferstrasse 1a, A-1060 Wien, Austria /H20849Received 8 March 2005; revised manuscript received 9 June 2005; published 19 August 2005 /H20850 The effect of interdiffusion at the Co/Cu interfaces on current-induced switching in Co/Cu n/Co spin valves, n=21, 25, and 33, with the interdiffusion concentration varying between 0 and 10%, is described theoretically in terms of ab initio calculations using the relativistic screened Korringa-Kohn-Rostoker method and the Landau-Lifshitz-Gilbert equation. It is found that interdiffusion forces the system to form a noncollinearground state such that switching to both kinds of collinear final states is possible. Furthermore, it is shown that/H20849i/H20850this behavior is caused by magnetic anisotropy effects, and /H20849ii/H20850by decreasing the interdiffusion, the current necessary to achieve switching to such a final state /H20849critical current /H20850can be reduced substantially. DOI: 10.1103/PhysRevB.72.054431 PACS number /H20849s/H20850: 75.30.Gw, 75.70.Ak, 75.70.Cn I. INTRODUCTION Although suggested theoretically by Slonczewski,1 current-induced switching is now thought to be of enormous technological interest,2,3since in principle it is much easier to switch the orientation of the magnetization in the freelayer of a spin-valve-type system by a current applied in theCPP /H20849current perpendicular to the planes of atoms /H20850geometry than by an external magnetic field. Ultimately, current-induced switching can perhaps replace most giant magne-toresistance /H20849GMR /H20850devices, now used in many commercial applications, provided, however, that the critical current,namely, the current that has to be applied to perform switch-ing, can be reduced by at least one order of magnitude. At present, mostly nanopillars are investigated experimentally,i.e., systems that because of the preparation techniques used,necessarily show macroscopic roughness and chemical dis-order /H20849interdiffusion at interfaces /H20850, a fact that has to be taken into account also in theoretical descriptions. In the present paper, the effect of interdiff- usion on current-induced switching has been studiedtheoretically by investigating systems of thetype Co /H20849100/H20850/Co m1/Cu 1−cCoc/Cu 1−cCoc/Cu n/Cu 1−cCoc /Cu 1−cCoc/Co m2/Co/H20849100/H20850, with m1,m1/H3335611 serving as buffer layers to the semi-infinite leads and cvarying between 0 and 0.10, i.e., by assuming at the Co/Cu interfaces aninterdiffusion profile extending over two adjacent atomiclayers. The n=19, 23, and 31 interdiffused Cu spacer layers correspond in turn to a spacer thickness of 36.4, 43.24, and57.18 Å. It should be noted that for c=0, the composition of the investigated spin-valve systems is simply of the formCo/H20849100/H20850/Co m1+1/Cu n+2/Co m2+1/Co/H20849100/H20850. The orientation of the magnetization in the left Co lead and the left half of the Cu spacer is kept fixed to point alongthe surface normal /H20849see Fig. 1 /H20850, whereas that of the right Co lead and the remaining spacer is rotated continuously ar-round an axis perpendicular to the surface normal until theground state /H9008 0/H33528/H208510,/H9266/H20852is reached. If /H90080/HS110050o r/H9266/H20849collinear magnetic configurations /H20850then a noncollinear magnetic con- figuration characterizes the ground state.II. CONCEPTUAL AND COMPUTATIONAL DETAILS In defining the twisting energy4/H9004E/H20849/H9008;c,N/H20850as /H9004E/H20849/H9008;c,N/H20850=E/H20849/H9008;c,N/H20850−E/H20849/H90080;c,N/H20850, /H208491/H20850 /H9008=/H90080+/H9004/H9008, N=m1+m2+n, /H208492/H20850 /H90080serves as the zero point of eventual /H20849further /H20850rotations. /H9004/H9008=−/H90080corresponds then to the parallel configuration, and /H9004/H9008=180− /H90080to the antiparallel configuration /H20849see Fig. 1 /H20850. In principle, cis an N-dimensional vector that contains lay- erwise the concentrations of Co and Cu. It should be notedthat only by applying an external magnetic field or a currentdoes the system assume a magnetic configuration /H9008other than/H9008 0, since /H9004E/H20849/H9008;c,N/H20850/H333560. FIG. 1. Noncollinear ground state of two magnetic slabs sepa- rated by a nonmagnetic spacer. The orientation of the magnetizationMin the thick magnetic layer is pointing along the surface normal n. In the so-called free layer, the orientation of the magnetization M /H11032forms an angle /H90080with n.PHYSICAL REVIEW B 72, 054431 /H208492005 /H20850 1098-0121/2005/72 /H208495/H20850/054431 /H208495/H20850/$23.00 ©2005 The American Physical Society 054431-1Provided that in a CPP geometry the corresponding sheet resistance5r/H20849/H9008;c,N/H20850is also evaluated, a current I/H20849/H9008;c,N/H20850 can be defined4,6as I/H20849/H9008;c,N/H20850=/H20881A0//H20841/H9270/H20849/H9008;c,N/H20850/H20841I0/H20849/H9008;c,N/H20850, /H208493/H20850 I0/H20849/H9008;c,N/H20850= sgn /H20851/H9270/H20849/H9008;c,N/H20850/H20852/H20881/H9004E/H20849/H9008;c,N/H20850/r/H20849/H9008;c,N/H20850,/H208494/H20850 where /H9270/H20849/H9008;c,N/H20850is the time needed to accomplish a rotation by/H9004/H9008, and A0is the unit area in the relation r/H20849/H9008;c,N/H20850 =A0R/H20849/H9008;c,N/H20850, with R/H20849/H9008;c,N/H20850being the resistance. In the following, I0/H20849/H9008;c,N/H20850will be referred to as reduced current . The /H20849positive definite /H20850twisting energy /H9004E/H20849/H9008;c,N/H20850can be expressed in terms of a power series in cos /H20849/H9008/H20850, /H9004E/H20849k/H20850/H20849/H9008;c,N/H20850=/H20858 s=0k ass/H20849c,N/H20850cos/H20849/H9008/H20850s. /H208495/H20850 The expansion coefficients thereof are then used to solve the Landau-Lifshitz-Gilbert equation4in order to obtain for a given/H9008the corresponding characteristic time /H9270/H20849/H9008;c,N/H20850.I n choosing a Gilbert damping factor of one, so-called minimal switching times are obtained.4Clearly enough, the unit area A0in Eq. /H208493/H20850is an experiment-dependent parameter. For all systems investigated, the parallel configuration /H20849the orientation of the magnetization points uniformly along the surface normal /H20850was calculated self-consistently by using the fully relativistic screened Korringa-Kohn-Rostokermethod 7and the density functional parametrization of Vosko et al.8The problem of interdiffusion was dealt with using the /H20849inhomogeneous /H20850coherent potential approximation.9The twisting energies were then obtained via the magnetic forcetheorem 10by calculating the grand potentials E/H20849/H9008;c,N/H20850in Eq. /H208491/H20850using a sufficient number of kpoints in the surface Brillouin zone in order to guarantee stable convergence withrespect to k. The sheet resistances r/H20849/H9008;c,N/H20850were evaluated in terms of the fully relativistic Kubo-Greenwood equation, 5,9using again a sufficiently large enough kset. In both types of calculations, the angle /H9008was varied between 0° and 180° in steps of at most 20°. The expansion in Eq. /H208494/H20850 was restricted to k=3, and the coefficients thereof were de- termined numerically in terms of a least-squares fittingprocedure, 11the fitting errors being typically of the order of 10−5meV. III. RESULTS A. Probability for interdiffusion at the interfaces It is well known that in the binary bulk system Co/Cu, the solubility of Co in Cu /H20849and oppositely /H20850is at best 1–2%. As this percentage not necessarily also applies for a possibleinterdiffusion at Co/Cu interfaces, total-energy calculationswithin the atomic sphere approximation /H20849ASA /H20850were per- formed for n=19 /H20849spacer thickness 36.4 Å /H20850, with the orien- tations of the magnetizations in the magnetic parts beingaligned parallel and pointing along the surface normal /H20849see Fig. 1 /H20850in order to determine a realistic range of interdiffu- sion concentrations. In Fig. 2, the following difference intotal energies:/H9004E tot/H20849c,N/H20850=Etot/H20849c,N/H20850−Etot/H20849c=0 ,N/H20850/H20849 6/H20850 is displayed, since by forming total-energy differences, most of the inherent errors in the ASA can be avoided. As can beseen in this figure, by assuming an error of about±0.025 mryd /H20849indicated by horizontal dashed lines /H20850, /H9004E tot/H20849c,N/H20850/H110110 for all interdiffusion concentrations below about 1.5%. Figure 2 clearly indicates that in Co/Cu/Co spin valves interdiffusion at the interfaces definitely has to beconsidered with interdiffusion concentrations between aboutzero and 2%. It should be noted that Fig. 2 can only serve asan argument that interdiffusion at the interfaces is very likelyto occur also in cases of noncollinear ground states. B. Orientation dependence of the magnetization In the top part of Fig. 3, it is shown that for the present purposes, the orientation of the magnetization was correctlyassumed to point along the surface normal. As the energydifference between a uniformly perpendicular and a uni-formly parallel to the surface normal magnetic configuration/H20849the so-called band-energy part in a magnetic anisotropy en- ergy calculation 7/H20850is very small, this quantity was analyzed with respect to the kconvergence. As can be seen in this figure, if the number of kpoints used in the irreducible part of the surface Brillouin zone is above about 2000, this en-ergy difference settles down at 0.05 meV, indicating that inthe noninterdiffused system, indeed a magnetic configurationis preferred with the magnetization pointing uniformly alongthe surface normal. In the lower half of this figure, this en-ergy difference is displayed with respect to the interdiffusionconcentration. The very meaning of this dependency on theinterdiffusion concentration and of the second curve dis-played will be discussed in the Sec. III D. C. Reduced currents and magnetoresistance It is quite well known that the interlayer exchange cou- pling energy—and, therefore, also the more general twistingenergy—is mostly determined by contributions from the in-terfaces. It is, therefore, not at all surprising that interdiffu-sion produces an almost dramatic effect on /H9004E/H20849/H9008;c,N/H20850and consequently on the reduced current I 0/H20849/H9008;c,N/H20850/H20851see Eq. /H208494/H20850/H20852. FIG. 2. Total energy difference with respect to the interdiffusion concentration for n=19 /H20849spacer thickness 36.4 Å /H20850; see also Eq. /H208496/H20850.C. SOMMERS AND P. WEINBERGER PHYSICAL REVIEW B 72, 054431 /H208492005 /H20850 054431-2This is depicted in Fig. 4. While in the absence of interdif- fusion for n=21 /H20849spacer thickness 36.4 Å /H20850the parallel mag- netic configuration corresponds to the ground state, with in-creasing interdiffusion, a perpendicular arrangement of theorientations of the magnetization in the magnetic slabs ispreferred, i.e., a noncollinear ground state is formed. Ascompared to the twisting energy, the changes in the sheetresistance caused by interdiffusion /H20849not shown here /H20850are much less spectacular— r/H20849/H9008;c/H20850is predominantly propor- tional to /H208491−cos /H9008/H20850for all concentrations investigated. In order to recover the “traditionally” well-known defini- tion of the magnetoresistance, MR/H20849c,N/H20850=/H20851r/H20849 /H9266;c,N/H20850−r/H208490;c,N/H20850/H20852/r/H20849/H9266;c,N/H20850, /H208497/H20850 in Fig. 4 also the magnetoresistance, defined as MR/H20849/H9008;c,N/H20850=/H20851r/H20849/H9008;c,N/H20850−r/H208490;c,N/H20850/H20852/r/H20849/H9008;c,N/H20850, is displayed in this particular case, however, as an implicit function of the applied reduced current MR/H20849/H9008;c,N/H20850=f/H20851I0/H20849/H9008;c,N/H20850/H20852. It should be noted that in the left half of Fig. 4, the reduced current is displayed with respect to /H9004/H9008 /H20851see Eq. /H208492/H20850/H20852, since /H9004/H9008=0 refers to the ground state. It is worthwhile to mention that MR/H20849c,N/H20850/H20851see Eq. /H208497/H20850/H20852, does decrease with increasing interdiffusion concentration and also slightly with respect to the thickness of the spacer—forn=19, the magnetoresistance MR/H20849c,N/H20850, changes linearlyfrom 36.3% at 5% interdiffusion to 42.9% for the noninter- diffused system. Forn=23 and 5% interdiffusion, MR/H20849c,N/H20850drops to about 32.8%, i.e., at a constant interdiffusion concentration of 5%, by increasing the spacer thickness by about 7 Å, the magne-toresistance decreases by about 3.5%. In Fig. 5, again the magnetoresistance is shown vs the reduced current; however, this time for n=31 /H20849spacer thick- ness 57.18 Å /H20850. In this figure, the switching from the ground state /H20849I 0=0/H20850to the parallel final state is indicated by open symbols, and the switching to the antiparallel final state by solid symbols. One can easily see /H20849i/H20850that the current needed to switch the system is bigger in the first case than in thesecond one and /H20849ii/H20850that with decreasing interdiffusion, this current is decreasing. A switching to the parallel state /H20849nega- tive current /H20850yields a change in the magnetoresistance of about 12.5%, and to the antiparallel state /H20849positive current /H20850,a change of about 22.5%. Figure 5 proves that the formation of a noncollinear ground state by interdiffusion effects is not theproperty of a particular spacer thickness. For n=23 /H20849spacer thickness 43.24 Å /H20850, very similar results /H20849not shown here /H20850are obtained. FIG. 3. /H20849a/H20850Band energy part of the magnetic anisotropy energy forn=23 /H20849spacer thickness 43.24 Å /H20850with respect to N−2/3, where N is the number of kpoints used in the irreducible part of the surface Brillouin zone. /H20849b/H20850Band-energy part of the magnetic anisotropy en- ergy /H20849squares /H20850and twisting energy for a perpendicular arrangement of the orientations of the magnetization /H20849circles /H20850with respect to the interdiffusion concentration /H20849see also Fig. 1 /H20850. FIG. 4. Reduced current with respect to /H9004/H9008 /H20849left column /H20850, and magnetoresistance with respect to the reduced current /H20849right col- umn /H20850forn=19 /H20849spacer thickness 36.4 Å /H20850. In each row, the interdif- fusion concentration is marked explicitly. It should be noted that formatters of comparison in both columns, the scale on the ordinate iskept constant.CURRENT-INDUCED SWITCHING IN Co/Cu/Co SPIN … PHYSICAL REVIEW B 72, 054431 /H208492005 /H20850 054431-3D. Expansion coefficients and switching times The explanation for the formation of a noncollinear state in the interdiffused systems can be read directly from Fig. 6.As can be seen, there the coefficient of cos 2/H20849/H9008/H20850, the so-called anisotropy term, increases sharply with increasing interdiffu- sion, while all other coefficients are very close to zero. Onlyfor vanishing interdiffusion /H20849c=0/H20850do the first two coeffi-cients start to grow and the anisotropy term changes sign. This then yields the results shown in Ref. 4, namely, that thetwisting energy has a maximum for a perpendicular arrange-ment of the orientations of the magnetization. Going backnow to the lower part of Fig. 3, it is evident that it is indeedonly the anisotropy term that causes the existence of a non-collinear ground state—the band-energy part of the aniso-tropy energy /H20849difference in grand potentials between a uni- form perpendicular and a uniform in-plane orientation of themagnetization /H20850is nearly twice as big as the twisting energy forM /H11032in Fig. 1, being perpendicular to the surface normal n. Figures 3 and 6 prove that in the presence of interdiffusion at the interfaces, anisotropy effects not only change the twist-ing energy dramatically, but they in turn change the size ofthe reduced current. In Fig. 7, the minimal switching times are depicted vs the interdiffusion concentration by displaying the more interest-ing low-interdiffusion regime. The squares in this figure referto a switching from a perpendicular arrangement of the ori-entations of the magnetization /H20849ground state in the presence of interdiffusion /H20850to the parallel magnetic configuration, the circles refer to the antiparallel configuration, and the dia-monds to the sum of both, which only in the noninterdiffusedcase yields the correct /H20849minimal /H20850switching time. It is inter- esting to note that with decreasing interdiffusion, the switch-ing times increase strongly. For n=19 /H20849spacer thickness 36.4 Å /H20850and 2% interdiffusion, the minimal switching time from the ground state to either of the two collinear finalstates is only about 1 ps, while in the noninterdiffused sys-tem, the switching times is larger by at least one order ofmagnitude. IV . DISCUSSION From the entry for the reduced currents in Fig. 5, one can immediately determine that in the interdiffused systems, the FIG. 5. Magnetoresistance vs reduced current for n=31 /H20849spacer thickness 57.18 Å /H20850and 1% /H20849circles /H20850and 2% /H20849squares /H20850interdiffu- sion. Open symbols refer to a switching to the parallel final state,and solid symbols to the antiparallel final state. FIG. 6. Expansion coefficients for the twisting energy for n =19 /H20849spacer thickness 36.4 Å /H20850vs the interdiffusion concentration /H20851see also Eq. /H208495/H20850/H20852. FIG. 7. Minimal switching times for n=19 /H20849spacer thickness 36.4 Å /H20850. Squares denote switching from the ground state to the parallel, circles denote switching to the antiparallel magnetic con-figuration, and diamonds refer to the sum of both. Only in the caseof a collinear ground state does this sum reflect the correct switch-ing time.C. SOMMERS AND P. WEINBERGER PHYSICAL REVIEW B 72, 054431 /H208492005 /H20850 054431-4current needed to switch from the ground state to the anti- parallel alignment is smaller than that needed to switch to theparallel alignment. This was the case for all interdiffusionconcentrations and spacer thicknesses investigated andseems to confirm recent experimental evidence. 3It is also evident in this figure that in the presence of interdiffusion,the reduced currents are larger by one order of magnitudethan in the absence of interdiffusion. Assuming an interdif-fusion concentration of 0.5%, which according to Fig. 2 isquite realistic, a unit area of 100 nm /H11003100 nm, and taking into account for n=19, the corresponding calculated reduced critical currents and switching times, then according to Eq. /H208493/H20850, the current needed to switch to the parallel /H20849antiparallel /H20850 magnetic configuration amounts to 0.21 /H208490.17 /H20850mA. For a unit area of 500 nm /H11003500 nm, one would get 1.05 and 0.85 mA, respectively, which is already within the scale ofexperimentally observed critical currents. This simple ex-ample suggests strongly that in all experimental studiesbased on Co/Cu-related spin valves, interdiffusion at the in-terfaces /H20849of unknown degree /H20850was present. The results displayed in Figs. 4 and 5 not only indicate that current-induced switching /H20849formation of noncollinear ground-states induced by interdiffusion /H20850is perhaps even more complicated than originally thought, but also that mod-els based only on spin-up and spin-down electrons /H20849strict collinearity /H20850most likely are not suitable, to describe this kind of situation, since quite obviously strong anisotropy effectshave to be taken into account. This was also found forcurrent-induced switching in spin valves with Permalloy serving as magnetic slabs. 6 Clearly, different interdiffusion profiles can be assumed, extending over several atomic layers, and different spacerthicknesses can be investigated. The main conclusions fromthe present results, however, are that /H20849i/H20850in principle, a well- defined noncollinear ground state is formed by interdiffusioneffects, and /H20849ii/H20850the reduced current and the switching time /H20849s/H20850 depend crucially on the amount of interdiffusion. Applying,e.g., a small external magnetic field as proposed in Ref. 3simultaneously with the current automatically changes /H9008 0 and therefore the critical current /H20849s/H20850. Clearly, the present re- sults also show that in principle the critical current can bereduced by reducing interdiffusion effects either by usingsuitable thin metallic layers /H20849Ru, Ta /H20850between the magnetic slabs and the nonmagnetic spacer or by means of other ex-perimental “tricks” in order to prevent interdiffusion. Lowercritical currents, on the other hand, imply slower switchingtimes. It seems, therefore, that a technologically relevantcompromise between these two aspects of current-inducedswitching is needed. ACKNOWLEDGMENTS Financial support from the Austrian Ministry for Econom- ics and Labour /H20849Zl 98.366 /H20850and the TU Vienna are gratefully acknowledged. We also want to thank the computing centerIDRIS at Orsay, since part of the calculations were per-formed using their facilities. 1J.. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2J. Grollier, P. Boulenc, V. Cros, A. Hamzi ć, A. Vaurès, and A. Fert, Appl. Phys. Lett. 83, 509 /H208492003 /H20850. 3Y. Jiang, T. Nozaki, S. Abe, T. Ochiai, A. Hirohata, N. Tezuka, and K. Inomata, Nat. Mater. 3, 361 /H208492004 /H20850. 4P. Weinberger, A. Vernes, B. L. Györffy, and L. Szunyogh, Phys. Rev. B 70, 094401 /H208492004 /H20850. 5P. Weinberger, Phys. Rep. 377, 281 /H208492003 /H20850. 6A. Vernes, P. Weinberger, and L. Szunyogh, Phys. Rev. B 72, 012401 /H208492005 /H20850. 7J. Zabloudil, R. Hammerling, L. Szunyogh, P. Weinberger, Elec-tron Scattering in Solid Matter /H20849Springer-Verlag, Heidelberg, 2004 /H20850. 8S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 /H208491980 /H20850. 9P. Weinberger, P. M. Levy, J. Banhart, L. Szunyogh, and B. Úfalussy, J. Phys.: Condens. Matter 8, 7677 /H208491996 /H20850. 10H. J. F. Jansen, Phys. Rev. B 59, 4699 /H208491999 /H20850. 11W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetter- ling, Numerical Recipes in Fortran: The Art of Scientific Com- puting /H20849Cambridge University Press, Cambridge, England, 1992 /H20850.CURRENT-INDUCED SWITCHING IN Co/Cu/Co SPIN … PHYSICAL REVIEW B 72, 054431 /H208492005 /H20850 054431-5

PhysRevLett.123.167201.pdf

Ferromagnetic Resonance with Magnetic Phase Selectivity by Means of Resonant Elastic X-Ray Scattering on a Chiral Magnet S. Pöllath,1A. Aqeel,2A. Bauer,2C. Luo,3,2H. Ryll,3F. Radu,3C. Pfleiderer,2,4 G. Woltersdorf,5and C. H. Back1,2,4 ,* 1Institut für Experimentelle Physik, Universität Regensburg, D-93040 Regensburg, Germany 2Physik-Department, Technische Universität München, D-85748 Garching, Germany 3Helmholtz-Zentrum Berlin für Materialien and Energie, D-12489 Berlin, Germany 4Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, D-80799 München, Germany 5Institut für Physik, Universität Halle-Wittenberg, D-06120 Halle (Saale), Germany (Received 24 May 2019; revised manuscript received 24 July 2019; published 14 October 2019) Cubic chiral magnets, such as Cu 2OSeO 3, exhibit a variety of noncollinear spin textures, including a trigonal lattice of spin whirls, the so-called skyrmions. Using magnetic resonant elastic x-ray scattering (REXS) on a crystalline Bragg peak and its magnetic satellites while exciting the sample with magneticfields at gigahertz frequencies, we probe the ferromagnetic resonance (FMR) modes of these spin textures by means of the scattered intensity. Most notably, the three eigenmodes of the skyrmion lattice are detected with large sensitivity. As this novel technique, which we label REXS FMR, is carried out at distinctpositions in reciprocal space, it allows us to distinguish contributions originating from different magnetic states, providing information on the precise character, weight, and mode mixing as a prerequisite of tailored excitations for applications. DOI: 10.1103/PhysRevLett.123.167201 Ferromagnetic resonance (FMR) measurements represent a well-established technique for the study of systems withcollinear magnetization [1], allowing us to extract informa- tion on the magnetic energy landscape and material-specificparameters such as the effective magnetization, the Land´ eg factor, or the magnetic damping constant α. In systems with noncollinear spin textures, measurements of resonant micro- wave excitations exhibit very complex spectra, where theidentification of specific modes proves to be prohibitivelydifficult. However, in view of new technological develop-ments such as antiferromagnetic spintronics and the use ofquantum magnetism, the precise identification of specificmodes will be of great importance [2]. The cubic chiral magnets MnSi, Fe 1−xCoxSi, and Cu2OSeO 3represent excellent showcases for the inherent complexity of materials with technological potential. Thesematerials host long-wavelength helimagnetic order includ- ing a trigonal lattice of topologically nontrivial spin whirls, the so-called skyrmion lattice [3–7]. Their microwave excitations have been studied by means of coplanar wave-guides and cavities [8–12]. In the helimagnetic states, two collective spin-precessional modes, denoted þqand−q, are observed. In the skyrmion lattice state, three eigenm-odes exist —namely, a clockwise and a counterclockwise gyration mode as well as a breathing mode [13]. The remarkably detailed understanding of the cubic chiral magnet allows us to assign these modes in the experimentalspectra by comparing their resonance frequency andspectral weight, as well as the evolution of the latter asa function of temperature and field, to results of analytic calculations and micromagnetic simulations solving theLandau-Lifshitz-Gilbert equation taking into account dipo-lar interactions [12]. Such an in-depth understanding, however, may not be available when investigating novel materials [14–16]or when phenomena such as metastable states, glassy textures, phase coexistence, topologicaltransitions, and pronounced history dependencies play arole[17–25]. In this Letter, we present a novel technique, called REXS FMR, which combines the excitation of collectivemodes by means of a coplanar waveguide with thedetection by means of the scattered intensity in magnetic resonant elastic x-ray scattering (REXS). The intensity of a crystalline Bragg peak or magnetic satellites may bestudied, permitting one to clearly identify the hostingmagnetic state of a given excitation via the scatteringpattern in reciprocal space. As a point of reference, wedemonstrate the potential of REXS FMR using the insulat- ing cubic chiral magnet Cu 2OSeO 3, which was studied previously both by means of REXS [26–31]and standard microwave spectroscopy [11,12,32 –36]. Our present study was carried out on the beam line PM2 at BESSY II with the VEKMAG end station [37]. The sample was a single-crystal cuboid of Cu 2OSeO 3, cut from an ingot grown by means of chemical vapor transport, with dimensions of 1.8×0.5×0.5mm3and edges oriented parallel to [110], [001], and ½¯110/C138. One of the surfaces normal to [001] was mechanically polished. The samplePHYSICAL REVIEW LETTERS 123, 167201 (2019) Editors' Suggestion 0031-9007 =19=123(16) =167201(7) 167201-1 © 2019 American Physical Societywas placed into the 1 mm wide gap of a coplanar wave- guide with the polished surface being on top. The sample slightly protrudes the top surface of the waveguide resultingin microwave excitation that comprises both in-plane and out-of-plane components. Typical excitation fields are of the order of 3 to 10μT, depending on the excitation frequency. For the REXS measurements, the energy ofthe circularly polarized photons is tuned to the Cu L 3edge (931 eV). Note that the element specific character of REXS is also inherent to REXS FMR. Further note that only inresonant x-ray scattering is the crystallographically for- bidden Bragg peak at 2θ≈96.5° observed [27,28,38,39] . The geometry of the experiment is illustrated in Fig. 1(a). The sample surface is illuminated by a x-ray spot of 100μm diameter. The scattered intensity of the structural (001) Bragg peak and its magnetic satellites is captured using a photodiode. Initially the diode angle ð2θÞis adjusted such that the intensity of the Bragg peak ismaximized. Subsequently, ð2θÞremains fixed, and thereciprocal space around the Bragg peak is mapped by varying the sample angle ωand the vertical diode position z d. The diode is located behind a pinhole with a diameter of 300μm that convolutes the measured signal, leading to an elongation of the peaks in the linearly scanned zddirection. Despite the rather small sample-detector distance of 20 mm, the Bragg peak and its magnetic satellites may be separated clearly. Using the superconducting vectormagnet at VEKMAG, the magnetic field is appliedparallel to the [110] axis, the [001] axis, or the incoming x-ray beam. The magnetic phase diagram of Cu 2OSeO 3is schemati- cally depicted in Fig. 1(b) [6,40] . Starting in the para- magnetic (pm) state at high temperatures and low fields, long-wavelength helimagnetic order (wavelength λ¼620Å) is observed below the transition temperature Tc¼58K. In the helical state, macroscopic domains of helices propagatealong one of the easy h001iaxes. In the corresponding reciprocal space map, shown in Fig. 1(c), the structural (001) Bragg peak is surrounded by four magnetic satellite peaks ðδ01Þ,ð¯δ01Þ,ð0δ1Þ, and ð0¯δ1Þ. Although after zero-field cooling the equivalent domains are expected to be populated equally across a bulk sample, the present experiment mapsonly a few domains due to the small x-ray spot size,rendering asymmetric intensity distributions possible. Under applied magnetic field, the propagation directions of the helices reorient into the field direction, and finitenet magnetization emerges as the magnetic moments increasingly tilt towards the field direction. In REXS, this conical state is associated with magnetic satellites along thefield direction that are observed when the field is appliedperpendicular to the [001] direction, as shown in Fig. 1(d). Further increasing the magnetic field to the critical field H c2results in a field-polarized state in which the moments are aligned along the field and no long-wavelength modu-lation is observed (not shown). At intermediate magnetic field just below T c, a pocket of the skyrmion lattice state is observed. The trigonal order of the spin whirls in the plane perpendicular to the field translates to the characteristic sixfold pattern of magneticsatellites in both small-angle neutron scattering [41–45] and REXS [26–31], as shown in Fig. 1(e). Under field cooling, the skyrmion lattice may be frozen-in to lower temperatures as a metastable state [17,20,23,46] .A s depicted in Fig. 1(f), the intensity of the magnetic satellites increases by an order of magnitude due to the increase ofthe magnetic moment with decreasing temperature. Typical REXS-FMR data are shown in Fig. 2for the field-polarized state at T¼15K, when a field larger than μ 0Hc2≈125mT is applied parallel to the beam direction [47]. Because of the lack of a long-wavelength modulation, there are no magnetic satellites around thestructural (001) Bragg peak. This peak, however, comprises a magnetic contribution that depends on the magnitude and orientation of the magnetization ⃗Mwith respect to the (a) (c) (d) (e) (f)(b) FIG. 1. Setup and typical REXS data. (a) Schematic view of the experimental setup. (b) Schematic zero-field cooled magneticphase diagram of Cu 2OSeO 3. (c) Reciprocal space map around the (001) Bragg peak in the helical state. (d) Reciprocal spacemap in the conical state. The magnetic field is applied along the[110] axis, i.e., within the sample plane. (e),(f) Reciprocal spacemap in the skyrmion lattice state. The magnetic field is appliedalong the [001] axis, i.e., perpendicular to the sample plane.Under field cooling, the skyrmion lattice may be metastablefrozen-in to lower temperatures.PHYSICAL REVIEW LETTERS 123, 167201 (2019) 167201-2incident and scattered x-ray wave vector. In turn, the field dependence of the magnetization may be inferredby tracking the intensity of the Bragg peak, shown inFig. 2(a). Clear kinks at /C6H c2and saturated behavior at larger fields are observed when the microwave excitation is switched off (the gray curve). Subtle changes in the curve ’s slope indicate phase transitions which are discussed inmore detail in Refs. [41,48,49] . As the magnetic contri- bution includes terms linear and quadratic in ⃗M, the intensity curve is not symmetric with respect to zero field.We refer the reader to the Supplemental Material [50]and Refs. [51,52] for information on the determination of the absolute magnetization values. Resonant excitation is studied by repeatedly switching on and off the microwave excitation while stepping the magnetic field, starting at high positive fields. At each field point, the REXS intensity under excitation is integrated for3 s before the excitation is switched off and the intensity isintegrated again for 3 s. For an excitation frequency of4.5 GHz (the red curve), a minimum of the magnetic intensity contribution emerges in the field-polarized state above H c2. As shown in Fig. 2(b), the normalized signal difference, ðIon−IoffÞ=ðIonþIoffÞ, at this minimum is of the order of 2%, which translates to a reduction of the magnetization by about 6.5%. Note that the way how the excitation is applied means that the magnetization repro-ducibly switches from its reduced to its regular value ateach field step. When assuming that precessional motion of the moments causes the reduction, a precession angle of 21° is required. This value is large but plausible considering the rather low effective damping of α≈10 −4observed in the insulator Cu 2OSeO 3[53,54] . In contrast, a change of the moment of 6.5% by heating effects requires an increase(decrease) of the sample temperature by ∼13K after switching the excitation on (off) in less than the time frame of 1 s resolvable in the present REXS experiment.Such drastic heating effects, however, would interfere when studying the skyrmion lattice state with its rather narrow temperature width of ∼2K close to T c(see below) and can be excluded. Simultaneously with the REXS measurements, the reflected microwave power S11of the coplanar waveguide was recorded using a Schottky diode detector. As shown in Fig.2(c), minima in the field-polarized state above /C6Hc2 are observed at the same field values as in REXS. An additional broad signature around zero field is attributed to resonant excitations in the helimagnetic state, notably of the /C6qmodes. The absence of these resonances in the REXS data highlights the potential of REXS FMR to selectively study individual magnetic phases and determine the origin of specific excitations. Figure 2(d)shows that with increas- ing excitation frequency the resonance field in REXS data increases linearly, consistent with Kittel behavior in the field-polarized state. The resonance field values in the field-polarized state are also in excellent agreement with reso-nance frequencies inferred from conventional microwave spectroscopy using a vector network analyzer (spectra not shown). As one of its decisive advantages, REXS FMR may be carried out not only on structural Bragg peaks but also onmagnetic satellites, allowing us to unambiguously make the connection between the underlying magnetic phase and the resonant mode. In Fig. 3(a), the intensity at a helical satellite position is shown as a function of field for different excitation frequencies at T¼15K. Similar to before, data are recorded with microwave excitation switched on and offat each magnetic field step. The integration time was increased to 10 s. Finite intensity arises only around zero field, where the helical state is observed in the magneticphase diagram after zero-field cooling; see Fig. 1(b). Note, however, that the field is applied along [001], i.e., an easy axis for the helices in Cu 2OSeO 3, and the measurement starts in the field-polarized state. Therefore, when decreas-ing the field to zero through the conical state, the helices may be expected to remain in the helical domain oriented along the field direction even at zero field [55], resulting in the absence of magnetic satellites in the present scattering geometry at temperatures well below T c. This putative contradiction connects to the recent dis- covery that Cu 2OSeO 3hosts not only a skyrmion lattice at high temperatures, common to all cubic chiral magnets,but also an independent second skyrmion phase at low(a) (b) (c) (d) FIG. 2. REXS FMR on the structural (001) peak. (a) Intensity of the Bragg peak as a function of magnetic field applied parallelto the incoming x-ray beam, tracking the magnetization of thesample. Data are recorded at each field step with microwaveexcitation on (red curve) and off (gray curve). (b) Relative changeof the intensity under excitation. (c) Reflected microwave power,S 11, as a function of field. The signature around zero field is attributed to the helimagnetic states. (d) Resonant modes for themagnetic field along the [001] axis at low temperature. Frequen-cies are normalized to their value at H c2. Resonance fields inferred from REXS FMR (solid symbols) are compared toresonance frequencies inferred from microwave spectroscopy(open symbols).PHYSICAL REVIEW LETTERS 123, 167201 (2019) 167201-3temperatures [25,49] . In contrast to the high-temperature phase, the low-temperature phase is stabilized by magneto-crystalline anisotropies and exists only for field values around H c2applied along [001]. Because of the topological protection inherent to skyrmions and the rather low temper-ature, the energy barrier of the low-temperature skyrmionphase is comparatively high. As a result, the skyrmion state exhibits a rather glassy texture without well-defined long- range order. When this state decays at lower fields bymeans of coalescence of neighboring skyrmions, a texture resembling poorly ordered helices with propagation perpendicular to the field forms [56,57] . The weak mag- netic satellites associated with such a helical state aredetected in our REXS experiment. Perhaps most strikingly, the glassy texture is highly susceptible to changes induced by resonant microwave excitation, as explained in thefollowing. At low and high frequencies, i.e., when the excitation is off resonance, data with microwave switched on and off agree with each other, corroborating that heating effects are negligible. In resonance, the excitation distinctly reducesthe satellite intensity. Consistent with the behavior in the field-polarized state, the intensity reproducibly switches between its low and high value at each field step and thereduction is attributed to a precessional motion of the magnetic moments. Note, however, that data without excitation (gray curves) are expected to track each otheras long as the same magnetic texture is probed, which isclearly not the case. The discrepancy becomes especially pronounced for an excitation frequency of 3.5 GHz, for which the satellite intensity increases by an order of magnitude while the field range shrinks by a factor of 2. This finding suggests that themicrowave excitation interacts with the glassy magnetic texture described above, improving its long-range order. In resonance, this pumping effect is particularly effective, andthe corresponding magnetic satellite in reciprocal space not only increases in intensity but also decisively sharpens. When also taking into account a small misalignment between sample plane and field direction, as a result, the scattering condition is fulfilled only in a reduced fieldinterval around zero field. In Fig. 3(b), the relative reduction of the helical satellite intensity at zero field is shown as a function of the excitation frequency. Two sharpmaxima are observed that are unambiguously attributed tothe−qandþqcollective modes of the helical state. Both frequency values and heights of the maxima are in excellent agreement with the literature [11,12] . Finally, REXS FMR on the high-temperature skyrmion lattice is presented in Fig. 3(c), showing the intensity at a skyrmion satellite position as a function of field at T¼ 56K for different excitation frequencies. Starting the description off resonance at low excitation frequency(bottom), measurements with and without microwave excitation track each other. The intensity maxima in finite fields around /C630mT are attributed to the skyrmion lattice state. In addition, finite intensity emerges around zero field as the tail of the broad helical satellite close by in reciprocal space reaches the position of the skyrmion satellite. Athigher frequencies, a distinct reduction of the skyrmionsatellite intensity is observed under microwave excitation,(a) (b) (d)(c) FIG. 3. REXS FMR on the magnetic satellites. (a) Intensity of a helical satellite as a function of magnetic field for differentexcitation frequencies. Data are recorded at each field step withmicrowave excitation on (red curves) and off (gray curves). Thelarge value at 3.5 GHz is attributed to changes of the spin texture;see the text for details. (b) Relative reduction of the helicalsatellite intensity at zero field as a function of frequency. Thesketch illustrates the position of the detector diode. Statisticalerror bars are typically much smaller than the symbol size.(c) Intensity at the reciprocal space position of a skyrmion latticesatellite. Around zero field, the tail of the broad helical satellite isobserved at the skyrmion satellite position. (d) Relative reductionof the skyrmion satellite intensity at a field of 35 mT as a functionof frequency.PHYSICAL REVIEW LETTERS 123, 167201 (2019) 167201-4again attributed to a precessional motion of the magnetic moments. At frequencies above 1.9 GHz, resonance effects associated with the helical state are observed aroundzero field. In order to further analyze the skyrmion resonances, the relative reduction of the satellite intensity at 35 mT, i.e., in the skyrmion lattice state, is depicted in Fig. 3(d) as a function of the excitation frequency. Three distinct maximaare observed and are associated with the counterclockwise (CCW) gyration, breathing (bre), and clockwise (CW) gyration modes, in excellent agreement with the literature[11,12] . The relative heights of the maxima are also perfectly consistent with calculated spectral weight distri- butions, as the coplanar waveguide combines in-plane excitation, driving the two gyration modes, and out-of- plane excitation, driving the breathing mode [35]. Owing to a photon penetration depth of about 30 nm, the REXS- FMR measurements probe the recently discovered surface states of the skyrmion lattice in Cu 2OSeO 3[58,59] . Although roughly 25% of the probed volume is expected to contain such surfaces states, no contributions distin- guishable from the bulk resonance modes are observed,indicating that the resonance frequencies of bulk and surface states are very similar or equal. Note that above and below 2 GHz, different circulators are used, leading toa small offset in the applied microwave power. Furthermore, the diode position was changed, as indicated in the sketch in Fig. 3(d), resulting in quantitative discrep- ancies in the measured intensity profiles at that frequency. In summary, we established a novel x-ray scattering technique, referred to as REXS FMR, that combines micro- wave excitation by means of a coplanar waveguide withdetection in reciprocal space by means of magnetic REXS. In the cubic chiral magnet Cu 2OSeO 3, we identified the resonant modes in the field-polarized, helimagnetic, andskyrmion lattice state by tracking the intensity of the structural (001) Bragg peak and its magnetic satellites under microwave excitation. The surface sensitive measurementindicates equal magnetic resonance frequencies for sky- rmionic surface and bulk states. REXS FMR also allows for stroboscopic measurements in order to determine the char-acter of eigenmodes microscopically, e.g., to distinguish between gyration and breathing modes. Also note that the selectivity to certain magnetic phases may prove particularly useful in complex magnetic environments for which the clear identification of eigenmodes in conventional micro-wave spectroscopy is challenging, such as for multidomain skyrmion states in lacunar spinels [14,60,61] , glassy sky- rmionic textures in Co-Mn-Zn compounds [62–64], or the low-temperature skyrmion state in Cu 2OSeO 3with its concomitant tilted conical state [25,49,65] . We wish to thank W. Simeth for the fruitful discussions and the assistance with the experiments. C. H. B., G. W.,and F. R. acknowledge funding from the BMBF via the VEKMAG project. A. B., C. P., S. P., G. W., and C. H. B.acknowledge funding by the German Research Foundation via Project No. SPP2137. This project has received fundingfrom the European Metrology Programme for Innovationand Research (EMPIR) programme co-financed by theParticipating States and from the European Union ’s Horizon 2020 research and innovation programme. 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PhysRevLett.123.117204.pdf

Exchange-Enhanced Ultrastrong Magnon-Magnon Coupling in a Compensated Ferrimagnet Lukas Liensberger,1,2,*Akashdeep Kamra,3,†Hannes Maier-Flaig,1,2Stephan Geprägs,1Andreas Erb,1 Sebastian T. B. Goennenwein,4Rudolf Gross,1,2,5,6Wolfgang Belzig,7 Hans Huebl,1,2,5,6and Mathias Weiler1,2,‡ 1Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 2Physik-Department, Technische Universität München, 85748 Garching, Germany 3Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, 7491 Trondheim, Norway 4Institut für Festkörper- und Materialphysik, Technische Universität Dresden, 01062 Dresden, Germany 5Nanosystems Initiative Munich, 80799 Munich, Germany 6Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany 7Department of Physics, University of Konstanz, 78457 Konstanz, Germany (Received 25 March 2019; revised manuscript received 15 July 2019; published 13 September 2019) We experimentally study the spin dynamics in a gadolinium iron garnet single crystal using broadband ferromagnetic resonance. Close to the ferrimagnetic compensation temperature, we observe ultrastrong coupling of clockwise and counterclockwise magnon modes. The magnon-magnon coupling strength reaches almost 40% of the mode frequency and can be tuned by varying the direction of the externalmagnetic field. We theoretically explain the observed mode coupling as arising from the broken rotational symmetry due to a weak magnetocrystalline anisotropy. The effect of this anisotropy is exchange enhanced around the ferrimagnetic compensation point. DOI: 10.1103/PhysRevLett.123.117204 The strong and ultrastrong interaction of light and matter is foundational for circuit quantum electrodynamics [1–3]. The realizations of strong spin-photon [4–6]and magnon- photon [7–12]coupling have established magnetic systems as viable platforms for frequency up-conversion [13,14] and quantum state storage [15]. Antiferromagnets and ferrimagnets further host multiple magnon modes. Their coupling allows for coherent control and engineering of spin dynamics for applications in magnonics [16,17] and antiferromagnetic spintronics [18,19] . Recently, it has been shown [20–22]that the weak interlayer exchange interaction between two magneticmaterials can cause magnon-magnon coupling. However,the much stronger intrinsic exchange has not yet beenleveraged for coupling phenomena. While the THz-fre-quency dynamics in antiferromagnets is challenging toaddress experimentally [23], the sublattice magnetizations in compensated ferrimagnets can be tuned to achieveGHz-frequency quasiantiferromagnetic dynamics. Here,we report the experimental observation of ultrastrongexchange-enhanced magnon-magnon coupling in a com-pensated ferrimagnet with the coupling rate reaching up to37% of the characteristic magnon frequency. We further- more demonstrate that the coupling strength can be con- tinuously tuned from the ultrastrong to the weak regime. We investigate spin dynamics, or equivalently the magnon modes, in a compensated, effectively two-sublattice ferri-magnet in the collinear state. Around its compensationtemperature, this system can be viewed as a “quasiantiferro- magnet ”due to its nearly identical sublattice magnetizations M A≳MB.F i g u r e 1schematically depicts the typical spa- tially uniform spin dynamics eigenmodes of the system [24]. Within the classical description, these become clockwise(CW) and counterclockwise (CCW) precessing modeswhich correspond to spin-down and spin-up magnons, respectively, in the quantum picture. The key physics under- lying our experiments is the tunable exchange-enhancedcoupling, and the concomitant hybridization, between thesestwo modes. The essential ingredients —mode coupling and exchange enhancement —are both intuitively understood within the quantum picture as follows. First, due to theiropposite spins, a spin-up magnon can only be coupled to itsspin-down counterpart by a mechanism that violates theconservation of spin along the sublattice magnetization, and thus magnon spin, direction [25]. Since angular momentum conservation is a consequence of rotational invariance orisotropy, an anisotropy about the magnon spin axis providessuch a coupling mechanism. Achieving the equilibriumsublattice magnetizations, or equivalently the magnon spin axis, to lie along directions with varying degrees of local axial anisotropy allows us to correspondingly vary the resultantmagnon-magnon coupling. This explains the nonzero modecoupling along with its tunability. However, the typically weak magnetocrystalline anisotropy may not be expected to yield observable effects and, therefore, has typically beendisregarded. This is where exchange enhancement in aPHYSICAL REVIEW LETTERS 123, 117204 (2019) 0031-9007 =19=123(11) =117204(7) 117204-1 © 2019 American Physical Societyquasiantiferromagnet makes the crucial difference. The anti- ferromagnetic magnons, despite their unit net spin, are formed by large, nearly equal and opposite spins on the twosublattices [26]. The anisotropy-mediated mode coupling results from, and is proportional to, this large sublattice spin instead of the unit net spin, and is therefore strongly amplified. This amplification effect is termed exchange enhancementwithin the classical description [26–28]. In our corresponding experiments, we study the mag- netization dynamics of a (111)-oriented single crystal Gd 3Fe5O12(gadolinium iron garnet, GdIG) disk by broad- band magnetic resonance (BMR) [29]. A schematic depic- tion of the setup is shown in Fig. 2(a). We use a vector network analyzer to record the complex transmission S21 as a function of the microwave frequency fand the external magnetic field H0applied in the (111) plane. Our experi- ments are performed at T¼282K, slightly below the ferrimagnetic compensation point Tcomp ¼288K, as deter- mined by SQUID magnetometry [30]. At this temperature, the resonance frequencies of the spin-up and spin-downmodes are in the microwave frequency range. In Fig. 2(b), we show the normalized background- corrected field derivative of S 21[42] recorded at fixed magnetic field magnitude μ0H0¼0.58T applied at φ¼90°. As discussed later, this is a situation in which the magnetocrystalline anisotropy energy has axial sym-metry about the magnetic field direction. We refer to this case as an effectively axially symmetric (EAS) direction. By fitting the data to Eq. (S7) [30], we extract the resonancefrequencies f 1andf2of the two observed resonances, their difference Δfres, and their linewidths κ1andκ2. In Fig. 2(c) we show corresponding data and fits for φ¼0° and μ0H0¼0.65T, which corresponds to a situation in which the magnetocrystalline anisotropy energy is anisotropic about the applied magnetic field direction, which we referto as an axial symmetry broken (ASB) direction, asexplained below. Again, two resonances are observed. In contrast to the data in Fig. 2(b), the resonances are now clearly separated. We repeat these experiments for a range of magnetic field magnitudes H 0applied along the two directions (EAS and ASB) of interest. The obtained resonance frequenciesare shown as symbols in Figs. 2(d) and2(e). In the EAS case shown in Fig. 2(d), we clearly observe two resonance modes. The first one follows ∂f res=∂H0>0and is the spin-up mode f↑, and the second resonance with ∂fres=∂H0<0is the spin-down mode f↓. The vertical dashed line corresponds to μ0H0¼0.58T, where Δfresis minimized and the data shown in Fig. 2(b) are obtained. The resonance frequencies are in excellent agreement withthose obtained from numerical (see Supplemental Material[30]) and analytical (see below) solutions to the Landau- Lifshitz equation. When applying H 0along the ASB axis, we obtain the resonance frequencies shown in Fig. 2(e). Here, we observe a more complex evolution of the resonance frequencies fortwo reasons. First, for μ 0H0⪅0.4T, the equilibrium net magnetization is titled away from H0and varies with H0. Second, and crucially, f↑andf↓exhibit a pronounced avoided crossing. The dashed vertical line indicates the value of H0of minimal Δfres[cf., Fig. 2(e)]. We plot Δfresand the half width at half maximum linewidths κ↑andκ↓as a function of the magnetic field H0 in Figs. 2(f) and 2(g) for the EAS and ASB cases, respectively. We find the mutual coupling strength gc=2π¼ minjΔfres=2j¼0.92GHz for the EAS case and gc=2π¼ 6.38GHz for the ASB configuration. In the former case, gc≲κ↑;κ↓[cf., Fig. 2(f)]. Thus, the system is in the weak to intermediate coupling regime. For the ASB case, the linewidths κare at least 3 times smaller than gc. Hence, the condition for strong coupling gc>κ↑;κ↓is clearly satisfied. Furthermore, the extracted coupling rate of gc=2π¼6.38GHz is comparable to the intrinsic excitation frequency fr¼ðf1þf2Þ=2¼17.2GHz. The normalized coupling rate η¼gc=ð2πfrÞ[8,43] evaluates to η¼0.37. Consequently, we observe magnon-magnon hybridizationin the ultrastrong coupling regime [1]. Importantly, the measured g cis the intrinsic coupling strength between spin-up and spin-down magnons and is independent of geometrical factors, in particular, sample volume or fillingfactor. This is in stark contrast to the magnon-photon orcavity-mediated magnon-magnon coupling typicallyobserved in spin cavitronics [8,44 –48].FIG. 1. Classical and quantum representations of the magneti- zation dynamics in a two-sublattice compensated ferrimagnet.The eigenmodes of the compensated ferrimagnet close to itscompensation temperature are similar to that of an antiferro-magnet since the sublattice magnetizations are almost identical(we choose M A≳MB). In the quantum picture, the classical modes with counterclockwise (CCW) and clockwise (CW)precession are identified as spin-up and spin-down magnons.The hybridized modes with linear polarization correspond tospin-zero magnons [25]. The angles between the two sublattice magnetizations have been exaggerated for clarity.PHYSICAL REVIEW LETTERS 123, 117204 (2019) 117204-2To demonstrate that the coupling is continuously tunable between the extreme cases discussed so far, we rotated H0 with fixed magnitude in the (111) plane at T¼280K. The background-corrected transmission parameter (see Supplemental Material [30]) as a function of the angle φ is shown in Figs. 3(a) and3(b) forμ0H0¼0.5T and μ0H0¼0.8T, respectively. These magnetic field magni- tudes correspond to H0slightly below and above the hybridization point at T¼280K (see Fig. S2 [30]). For bothH0values, we observe two resonances for each value ofφ, where the lower resonance frequency depends strongly on φwhile the upper one is nearly independent ofφ. Overall, these results strongly indicate a φ-dependent level repulsion that allows us to continuously adjust thecoupling strength. To understand the coupling strength variation with φ,w e analyze the cubic anisotropy landscape of our GdIG disk byplotting its magnetic free-energy density F[cf., Eq. (S9) [30]]i nF i g . 3(c). The applied field directions for the EAS and ASB cases are indicated by the two gray dots in Fig. 3(c). The sublattice magnetizations as well as the magnon spinaxis are collinear with the applied field in our considerations. As derived rigorously below, coupling between the opposite- spin magnons is proportional to the degree of anisotropy inthe free energy about the magnon spin axis [25]. Moreover, since they represent small and symmetric deviations ofmagnetization about the equilibrium configuration, themagnons can only sense anisotropy variations that are local and averaged over antiparallel directions. Considering the ASB configuration first, if the magnetization deviates fromequilibrium along the orange (white) arrows, it experiences an increase (a decrease) in energy. Therefore, the free-energy change depends on the direction of deviation, and thesymmetry about the magnon spin axis in this configurationis clearly broken by anisotropy. This causes a nonzero mode coupling in the ASB configuration. In contrast, for the EAS configuration, an averaging over the two antiparallel direc-tions results in a nearly vanishing and direction-independentchange in the free energy, thereby effectively maintaining axial symmetry. This is prominently seen when considering the direction collinear with the orange and white arrows,which nearly cancel each other ’s effect on averaging. This configuration is thus named effectively axially symmetric (EAS). The corresponding degree of axial anisotropy, andthus mode coupling, varies smoothly with φbetween these two extreme cases. The two key ingredients in the physics observed herein are (i) nonzero mode coupling arising from violation of H0(a) 12 15 18 21 24-0.30.00.30.6= 90°, EAS∂DS21/∂H0(1/T) f(GHz)(b) ΔfresRe Im 0 5 10 15 20 25-0.2-0.10.00.10.2= 0°, ASB f(GHz)(c) Δfres0.0 0.5 1.0 1.5 2.00510152025 0 . 00 . 51 . 01 . 52 . 00510152025 0.0 0.5 1.0 1.5 2.00123 0 . 00 . 51 . 01 . 52 . 00246numerical analyticalf(GHz) 0H0(T)(d) = 90°, EAS f↓f↑ f↑f↓ 0H0(T)(e) = 0°, ASB f↓f↑ gc/2π/2π(GHz) 0H0(T)↓/2π(f) ↑/2πfres/2 gc/2π 0H0(T)↑/2π ↓/2π(g) fres/2 FIG. 2. (a) Schematic broadband ferromagnetic resonance (BMR) setup, with the GdIG disk on the coplanar waveguide (CPW). The angle φdefines the in-plane direction of the magnetic field H0. (b),(c) BMR spectra obtained for fixed magnetic field strengths applied along the (b) effectively axially symmetric (EAS) direction in the (111) plane at φ¼90°(μ0H0¼0.58T) and along the (c) axial symmetry broken (ASB) axis φ¼0°(μ0H0¼0.65T) recorded at T¼282K(Tcomp ¼288K). The solid lines are fits to Eq. (S7) [30]. The resonance frequencies are indicated by the red arrows and their difference is denoted as Δfres. (d),(e) Mode frequencies versus applied magnetic field strength measured at T¼282K, where MGd≳MFe. Open circles and triangles denote measured resonance frequencies. The red dotted curves depict results of our analytical model and the blue dashed lines are obtained by numerical simulation. Along the EAS direction φ¼90° (d), weak coupling is observed, whereas along the ASB direction φ¼0° (e), we find ultrastrong coupling (see text). The solid gray lines in (e) indicate the uncoupled case taken from the analytical solution of (d). (f),(g) Linewidthsκ=2πof the spin-up κ ↑and spin-down κ↓modes, and resonance frequency splitting Δfres=2as a function of H0. The coupling strength gc=2πis given by the minimum of Δfres=2.PHYSICAL REVIEW LETTERS 123, 117204 (2019) 117204-3spin conservation by an axial anisotropy [25] and (ii) a strong amplification of the otherwise weak coupling via an exchange-enhancement effect characteristic of (quasi)anti-ferromagnetic magnons [26]. We now present a minimal- istic, analytically solvable model that brings out both these pillars underlying our experiments, and yields results ingood agreement with our data [Figs. 2(d)and2(e)]. To this end, we employ a two-sublattice model, which corresponds to the net Fe and Gd sublattice in GdIG, within the Landau-Lifshitz framework and macrospin approximation, treating anisotropies as uniaxial to enable an analytical solution. In our experiments, both of the distinct anisotropy contribu-tions considered here are provided by the cubic crystalline anisotropy of the material. Parametrizing the intersublattice antiferromagnetic exchange by J(>0) and uniaxialanisotropies by K(>0) and K a, the free-energy density Fmis expressed in terms of the sublattice AandB magnetizations MA;B, assumed spatially uniform, as Fm¼−μ0H0ðMAzþMBzÞ∓KðM2 AzþM2 BzÞ þKaðM2 AxþM2 BxÞþJMA·MB; ð1Þ where the first term is the Zeeman contribution due to the applied field H0ˆz. We further assume an appropriate hierarchy of interactions J≫K≫jKaj, such that Ka terms do not influence the equilibrium configurations. The upper and lower signs in Eq. (1)above represent the cases of an applied field along easy and hard axes,respectively. The EAS (ASB) direction is magnetically easy(hard) [30]. The axial symmetry is broken by the term proportional to K a, with Ka≈0for the EAS case and Ka≠0to the ASB case. We have chosen coordinate systems for treating the two configurations with the zdirection always along the applied field. The equilibrium configuration is obtained by minimizing Eq. (1)with respect to the sublattice magnetization directions (see Supplemental Material [30]). The dynamics are captured by the Landau-Lifshitz equations for the two sublattices: ∂MA;B ∂t¼−jγA;Bj/C20 MA;B×/C18 −∂Fm ∂MA;B/C19/C21 ; ð2Þ where γA;Bare the respective sublattice gyromagnetic ratios, assumed negative. It is convenient to employ a new primed coordinate system with equilibrium magnetizations collinearwith ˆz 0. The ensuing dynamical equations are linearized about the equilibrium configuration which, on switching to Fourier space (i.e., MAx0¼mAx0eiωtand so on), lead to the coupled equations describing the eigenmodes expressed succinctly as a 4×4matrix equation: ð˜P0þ˜PaÞ˜m¼0; ð3Þ where ˜m⊺¼½mAþmBþmA−mB−/C138,w i t h mA/C6≡mAx0/C6imAy0, and so on. The matrix ˜P0is block diagonal in 2×2 submatrices and describes the uncoupled spin-up and spin-down modes, distributed over both sublattices. The matrix ˜Pacaptures axial-symmetry-breaking anisotropy effects, and provides the spin-nonconserving, off-diagonal terms that couple the two modes and underlie the hybridi- zation physics at play. The detailed expressions for thematrices are provided in the Supplemental Material [30]. For applied fields along the easy axis (EAS), the equilibrium configuration is given by M A¼MA0ˆzand MB¼−MB0ˆz, with MA0;B0the respective sublattice satu- ration magnetizations and MA0≳MB0. For the case of a sufficiently small field applied along the hard axis (ASB), the equilibrium orientation of MAis orthogonal to the hard axis. With increasing field strength, MAmoves to align-30 0 30 60 90 1200306090120150180 A(°)A(°) -315-287-258-230-202-173-145 F(H0=0)(J/m3)90 45 0 -45 -900510152025f(GHz) (°)0H0=0.5T 90 45 0 -45 -90 (°)-707 Re(∂DS21/∂)( 1 0-3/°) 0H0=0.8T (c) EASASB [001], HAxx(a) (b) [111], EA FIG. 3. Tunable coupling strength and anisotropy landscape. (a),(b) BMR data obtained with fixed magnetic field magnitudeswith (a) μ 0H0¼0.5T (below the hybridization point) and (b)μ0H0¼0.8T (above the hybridization point) as a function of theH0orientation φin the (111)-disk plane at T¼280K. The blue dashed lines are the results from the numerical simulation.(c) Color map of the free-energy density FforH 0¼0. The angles φAand θAdenote the orientation of MA, defined analogously to φandθin Fig. 2(a). The dashed horizontal line atθA¼90° corresponds to the (111)-disk plane. The orange and white arrows at the EAS ( φA¼90°) and ASB ( φA¼0°) orientations point towards increasing and decreasing free-energydensity, respectively. The [001] direction denotes a crystalline hard axis (HA) and ½¯111/C138a crystalline easy axis (EA).PHYSICAL REVIEW LETTERS 123, 117204 (2019) 117204-4with the applied field. In the considered temperature and field range, MBalways remains essentially anticollinear to MA[49]. The initial decrease of the resonance mode with lower frequency [Fig. 2(e)] is associated with this evolution of the equilibrium configuration. The frequency dip signi- fies alignment of equilibrium MAwith the zaxis. Only the Kaanisotropy term breaks axial symmetry about the equilibrium magnetization direction ( zaxis) and leads to off-diagonal terms in ˜Pa, which couples the two modes. The coupling-mediated frequency splitting Δfres, where uncoupled eigenmode frequencies would cross, is evalu-ated employing Eq. (3)as 2πΔf res¼ωcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16JM2 0 JðMA0−MB0Þ2þFeqs ; ð4Þ where ωc≡jγjjKajM0is the bare coupling rate, consider- ingγA≈γB≡γandMA0≈MB0≡M0near the compen- sation point. Feq, given by 16KM2 0forH0along an easy axis, is an equivalent free-energy density comparable to the anisotropy contribution, parametrized by K. The bare coupling rate is thus enhanced by a maximum value offfiffiffiffiffiffiffiffiffi J=Kp at the compensation point yielding a greatly enlarged coupling. Hereby a small coupling of ωc¼ 2π×160MHz originating from a weak cubic anisotropy present in GdIG is greatly enhanced as demonstrated by Eq. (4)and the analytical model results displayed in Fig.2(e), quantitatively describing our experimental obser- vations. The amplification of coupling from 160 MHz to several GHz is an exchange-enhancement effect [26–28,50] . This (exchange) enhancement is an embodiment of anti- ferromagnetic quantum fluctuations [26]predicted similarly to amplify magnon-mediated superconductivity [51]. Our findings demonstrate that previously typically neglected details of the magnetocrystalline anisotropycan lead to giant effects on spin dynamics if they have the appropriate symmetry and are exchange enhanced. The ultrastrong and size-independent magnon-magnon coupling reported here opens exciting perspectives for studying ultrastrong coupling effects in nanoscale devicesand exploring quantum-mechanical coupling phenomena beyond classical electrodynamics. The reported effect also enables the tuning and tailoring of quasiantiferromagneticdynamics and magnons. We thank A. Habel, K. Helm-Knapp, and K. Danielewicz for technical support. We gratefully acknowledge the finan-cial support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Germany ’sE x c e l l e n c e Strategy EXC-2111-390814868 (R. G. and H. H.) and theProjects No. WE5386/4 and No. WE5386/5 (L. L. and M. W.). A. K. acknowledges financial support from the Research Council of Norway through its Centers of Excellence funding scheme, Project No. 262633, “QuSpin. ”W. B. was supported by the DFG through SFB767 and thanks the Center of Excellence QuSpin by the Research Council of Norway and Arne Brataas (NTNUTrondheim) for hospitality. Note added. —Recently, we became aware of a related study showing magnon-magnon coupling in the canted antifer- romagnet CrCl 3[52]. *Lukas.Liensberger@wmi.badw.de †Akashdeep.Kamra@ntnu.no ‡Mathias.Weiler@wmi.badw.de [1] A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between lightand matter, Nat. Rev. Phys. 1, 19 (2019) . [2] X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S.-i. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K.Nemoto, M. Kasu, N. Mizuochi, and K. Semba, Coherentcoupling of a superconducting flux qubit to an electronspin ensemble in diamond, Nature (London) 478, 221 (2011) . [3] J. J. Viennot, M. C. Dartiailh, A. Cottet, and T. Kontos, Coherent coupling of a single spin to microwave cavityphotons, Science 349, 408 (2015) . [4] D. I. Schuster, A. P. 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PhysRevB.91.134411.pdf

PHYSICAL REVIEW B 91, 134411 (2015) Realization of the thermal equilibrium in inhomogeneous magnetic systems by the Landau-Lifshitz-Gilbert equation with stochastic noise, and its dynamical aspects Masamichi Nishino1,*and Seiji Miyashita2,3 1Computational Materials Science Center, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan 2Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-Ku, Tokyo, Japan 3CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan (Received 28 May 2014; revised manuscript received 6 March 2015; published 9 April 2015) It is crucially important to investigate the effects of temperature on magnetic properties such as critical phenomena, nucleation, pinning, domain wall motion, and coercivity. The Landau-Lifshitz-Gilbert (LLG)equation has been applied extensively to study dynamics of magnetic properties. Approaches of Langevinnoises have been developed to introduce the temperature effect into the LLG equation. To have the thermalequilibrium state (canonical distribution) as the steady state, the system parameters must satisfy some conditionknown as the fluctuation-dissipation relation. In inhomogeneous magnetic systems in which spin magnitudes aredifferent at sites, the condition requires that the ratio between the amplitude of the random noise and the dampingparameter depend on the magnitude of the magnetic moment at each site. Focused on inhomogeneous magneticsystems, we systematically showed agreement between the stationary state of the stochastic LLG equation andthe corresponding equilibrium state obtained by Monte Carlo simulations in various magnetic systems includingdipole-dipole interactions. We demonstrated how violations of the condition result in deviations from the trueequilibrium state. We also studied the characteristic features of the dynamics depending on the choice of theparameter set. All the parameter sets satisfying the condition realize the same stationary state (equilibriumstate). In contrast, different choices of parameter set cause seriously different relaxation processes. We show tworelaxation types, i.e., magnetization reversals with uniform rotation and with nucleation. DOI: 10.1103/PhysRevB.91.134411 PACS number(s): 75 .78.−n,05.10.Gg,75.10.Hk,75.60.Ej I. INTRODUCTION The Landau-Lifshitz-Gilbert (LLG) equation [ 1] has been widely used in the study of dynamical properties of magneticsystems, especially in micromagnetics. It contains a relaxationmechanism by a phenomenological longitudinal dampingterm. The Landau-Lifshitz-Bloch (LLB) equation [ 2] con- tains, besides the longitudinal damping, a phenomenologicaltransverse damping, and the temperature dependence of themagnetic moment is taken into account with the aid of the mean-field approximation. Those equations work well in the region of saturated magnetization at low temperatures. Thermal effects are very important to study properties of magnets, e.g., the amount of spontaneous magnetization,hysteresis nature, relaxation dynamics, and the coercive forcein permanent magnets. Therefore, how to control temperaturein the LLG and LLB equations has been studied extensively. Tointroduce temperature in equations of motion, a coupling with athermal reservoir is required. For dynamics of particle systemswhich is naturally expressed by the canonical conjugatedvariables, i.e., ( q,p), molecular dynamics is performed with aN o s ´e-Hoover (NH) type reservoir [ 3–5] or a Langevin type reservoir [ 6]. However, in the case of systems of magnetic moments, in which dynamics of angular momenta is studied,NH type reservoirs are hardly used due to complexity [ 7]. On the other hand, the Langevin type reservoirs have been rathernaturally applied [ 2,8–18] although multiplicative noise [ 19] requires the numerical integration of equations depending onthe interpretation, i.e., Ito or Stratonovich type. *Corresponding author: nishino.masamichi@nims.go.jpTo introduce temperature into a LLG approach by a Langevin noise, a fluctuation-dissipation relation is used,where the temperature is proportional to the ratio betweenthe strength of the fluctuation (amplitude of noise) and thedamping parameter of the LLG equation. For magnetic systemsconsisting of uniform magnetic moments, the ratio is uniquelygiven at a temperature and it has been often employed to studydynamical properties, e.g., trajectories of magnetic momentsof nanoparticles [ 8] and relaxation dynamics in a spin-glass system [ 20] or in a semiconductor [ 21]. The realization of the equilibrium state by stochastic LLG approaches by numericalsimulations is an important issue, and it has been confirmedin some cases of the Heisenberg model for uniform magneticmoments [ 22,23]. In general cases, however, magnetic moments on the atomic scale have various magnitudes of spins. This inhomogeneity of magnetization is important to understand the mechanismsof nucleation or pinning [ 24–28]. To control the temperature of such systems, the ratio between the amplitude of noiseand the damping parameter depends on the magnetic momentat each site. In order to make clear the condition for therealization of the canonical distribution as the stationary state in inhomogeneous magnetic systems, we review the guidelines of the derivation of the condition in the Fokker-Planck equationformalism in the Appendix A. Such a generalization of the LLG equation with a stochastic noise was performed to study properties of the alloy magnetGdFeCo [ 29], in which two kinds of moments exist. They exploited a formula for the noise amplitude, which is equiv-alent to the formula of our condition A (see Sec. II). They found surprisingly good agreement of the results betweenthe stochastic LLG equation and a mean-field approximation. 1098-0121/2015/91(13)/134411(13) 134411-1 ©2015 American Physical SocietyMASAMICHI NISHINO AND SEIJI MIY ASHITA PHYSICAL REVIEW B 91, 134411 (2015) However, the properties in the true canonical distribution are generally different from those obtained by the mean-fieldanalysis. The LLG and LLB equations have been often applied for continuous magnetic systems or assemblies of blockspins in the aim of simulation of bulk systems, but suchtreatment of the bulk magnets tends to overestimate theCurie temperature [ 11], and it is still under development to obtain properly magnetization curves in the whole temperatureregion [ 2,11,17,18]. The influence of coarse graining of block spin systems on the thermal properties is a significant theme,which should be clarified in the future. To avoid such adifficulty, we adopt a lattice model, in which the magnitude ofthe moment is given at each magnetic site. Within the condition there is some freedom of the choice of parameter set. In the present paper, in particular, weinvestigate the following two cases of parameter sets, i.e.,case A, in which the LLG damping constant is the same inall the sites and the amplitude of the noise depends on themagnitude of the magnetic moment at each site, and case B,in which the amplitude of the noise is the same in all thesites and the damping constant depends on the magnitude ofthe moment (see Sec. II). We confirm the realization of the equilibrium state, i.e., the canonical distribution in variousmagnetic systems including critical region by comparisonof magnetizations obtained by the LLG stochastic approachwith those obtained by standard Monte Carlo simulations, notby the mean-field analysis. We study systems with not onlyshort-range interactions but also dipole-dipole interactions,which causes the demagnetizing field statically. We find thatdifferent choices of the parameter set which satisfies thefluctuation-dissipation relation give the same stationary state(equilibrium state) even near the critical temperature. We alsodemonstrate that deviations from the relation cause systematicand significant deviations of the results. In contrast to the static properties, we find that different choices of parameter set cause serious difference in thedynamics of the relaxation. In particular, in the rotationtype relaxation in isotropic spin systems, we find that thedependencies of the relaxation time on the temperature in casesA and B show opposite correlations as well as the dependenciesof the relaxation time on the magnitude of the magneticmoment. That is, the relaxation time of magnetization reversalunder an unfavorable external field is shorter at a highertemperature in case A, while it is longer in case B. On theother hand, the relaxation time is longer for a larger magneticmoment in case A, while it is shorter in case B. We alsoinvestigate the relaxation of anisotropic spin systems andfind that the metastability strongly affects the relaxation at low temperatures in both cases. The system relaxes to the equilibrium state from the metastable state by the nucleationtype of dynamics. The relaxation time to the metastable stateand the decay time of the metastable state are affected by thechoice of the parameter set. The outline of this paper is as follows. The model and the method in this study are explained in Sec. II. Magnetization processes as a function of temperature in uniform magneticsystems are studied in Sec. III. Magnetizations as a function of temperature for inhomogeneous magnetic systems are in-vestigated in Sec. IV, in which not only exchange interactions(short-range) but also dipole interactions (long-range) are taken into account. In Sec. Vdynamical aspects with the choice of the parameter set are considered, and the dependencies of therelaxation process on the temperature and on the magnitude ofmagnetic moments are also discussed. The relaxation dynam-ics via a metastable state is studied in Sec. VI. Section VIIis de- voted to summary and discussion. In Appendix Athe Fokker- Planck equation for inhomogeneous magnetic systems is givenboth in Stratonovich and Ito interpretations, and Appendix B presents the numerical integration scheme in this study. II. MODEL AND METHOD As a microscopic spin model, the following Hamiltonian is adopted, H=−/summationdisplay /angbracketlefti,j/angbracketrightJi,jSi·Sj−/summationdisplay iDA i/parenleftbig Sz i/parenrightbig2−/summationdisplay ihi(t)Sz i +/summationdisplay i/negationslash=kC r3 ik/parenleftbigg Si·Sk−3(rik·Si)(rik·Sk) r2 ik/parenrightbigg . (1) Here we only consider a spin angular momentum Sifor a magnetic moment Miat each site ( iis the site index) and regard Mi=Siignoring the difference of the sign between them and setting a unit gμ B=1 for simplicity, where gis the gfactor andμBis the Bohr magneton [ 30]. Interaction Ji,jbetween theith and jth magnetic sites indicates an exchange coupling, /angbracketlefti,j/angbracketrightdenotes a nearest-neighbor pair, DA iis an anisotropy constant for the ith site, hiis a magnetic field applied to the ith site, and the final term gives dipole interactions between theith and kth sites whose distance is rik, where C=1 4πμ 0is defined using the permeability of vacuum μ0. The magnitude of the moment Miis defined as Mi≡|Mi|, which is not necessarily uniform but may vary from site tosite. In general, the damping parameter may also have sitedependence, i.e., α i, and thus the LLG equation at the ith site is given by d dtMi=−γMi×Heff i+αi MiMi×dMi dt, (2) or in an equivalent formula, d dtMi=−γ 1+α2 iMi×Heff i−αiγ/parenleftbig 1+α2 i/parenrightbig MiMi ×/parenleftbig Mi×Heff i/parenrightbig , (3) where γis the gyromagnetic constant. Here Heff iis the effective field at the ith site and described by Heff i=−∂ ∂MiH(M1,...,MN,t), (4) which contains fields from the exchange and the dipole interactions, the anisotropy, and the external field. We introduce a Langevin-noise formalism for the thermal effect. There have been several ways for the formulationto introduce a stochastic term into the LLG equation. Thestochastic field can be introduced into the precession termand/or damping term [ 8,9,11]. Furthermore, an additional noise term may be introduced [ 10,12]. In the present study we 134411-2REALIZATION OF THE THERMAL EQUILIBRIUM IN . . . PHYSICAL REVIEW B 91, 134411 (2015) add the random noise to the effective field Heff i→Heff i+ξi and we have d dtMi=−γ 1+α2 iMi×/parenleftbig Heff i+ξi/parenrightbig −αiγ/parenleftbig 1+α2 i/parenrightbig MiMi ×/bracketleftbig Mi×/parenleftbig Heff i+ξi/parenrightbig/bracketrightbig , (5) where ξμ iis theμ(=1, 2, or 3 for x,y,o rz) component of the white Gaussian noise applied at the ith site and the following properties are assumed: /angbracketleftbig ξμ k(t)/angbracketrightbig =0,/angbracketleftbig ξμ k(t)ξν l(s)/angbracketrightbig =2Dkδklδμνδ(t−s). (6) We call Eq. ( 5) the stochastic LLG equation. We derive a Fokker-Planck equation [ 6,8] for the stochastic equation of motion in Eq. ( 5) in the Stratonovich interpretation, as given in Appendix A, ∂ ∂tP(M1,...,MN,t)=/summationdisplay iγ 1+α2 i∂ ∂Mi ·/braceleftbigg/bracketleftbiggαi MiMi×/parenleftbig Mi×Heff i/parenrightbig −γDiMi×/parenleftbigg Mi×∂ ∂Mi/parenrightbigg/bracketrightbigg ×P(M1,...,MN,t)/bracerightbigg . (7) Here we demand that the distribution function at the stationary state (t→∞ ) of the equation of motion [Eq. ( 7)] agree with the canonical distribution of the system [Eq. ( 1)] at temperature T, i.e., Peq(M1,...,MN)∝exp[−βH(M1,...,MN)],(8) where β=1 kBT. Considering the relation ∂ ∂MiPeq(M1,...,MN)=βHeff iPeq(M1,...,MN),(9) we find that if the relation αi Mi−γDiβ=0 (10) is satisfied at each site i, the canonical distribution in the equilibrium state is assured. When the magnetic moments are uniform, i.e., the magni- tude of each magnetic moment is the same and Mi=|Mi|= M, the parameters αiandDiare also uniform αi=αand Di=Dfor a given T. However, when Miare different at sites, the relation ( 10) must be satisfied at each site independently. There are several ways for the choice of the parameters αiand Dito satisfy this relation. Here we consider the following two cases: A and B. A: We take the damping parameter αito be the same at all sites, i.e., α1=α2=···= αN≡α. In this case the amplitude of the random field at the ith site should be Di=α MikBT γ∝1 Mi. (11) B: We take the amplitude of the random field to be the same at all sites, i.e., D1=D2=···= DN≡D. In this case the00.20.40.60.81 0123456m T FIG. 1. (Color online) Comparison of the temperature depen- dence of min the stationary state between the stochastic LLG method and the Langevin function (green circles). Crosses and boxes denotemin case A ( α=0.05) and case B ( D=1.0), respectively. In the stochastic LLG simulation /Delta1t=0.005 was set and 80 000 time steps (40 000 steps for equilibration and 40 000 steps for measurement)were employed. The system size N=L 3=103was adopted. damping parameter at the ith site should be αi=DγM i kBT∝Mi. (12) We study whether the canonical distribution is realized in both cases by comparing data obtained by the stochastic LLGmethod with the exact results or with corresponding dataobtained by Monte Carlo simulations. We set the parametersγ=1 andk B=1 hereafter. III. REALIZATION OF THE THERMAL EQUILIBRIUM STATE IN HOMOGENEOUS MAGNETIC SYSTEMS A. Noninteracting magnetic moments As a first step, we check the temperature effect in the simplest case of noninteracting uniform magnetic moments,i.e.,J i,j=0,DA i=0,C=0i nE q .( 1) and Mi=M(or Si=S), where αandDhave no site i-dependence. In this case the magnetization in a magnetic field ( h) at a temperature (T) is given by the Langevin function: m=1 N/angbracketleftBiggN/summationdisplay i=1Sz i/angbracketrightBigg =M/parenleftbigg coth/parenleftbigghM kBT/parenrightbigg −kBT hM/parenrightbigg . (13) We compare the stationary state obtained by the stochastic LLG method and Eq. ( 13). We investigate m(T)a th=2f o r M=1. Figure 1shows m(T) when α=0.05 is fixed (case A) and when D=1.0 is fixed (case B). We find a good agreement between the results of the stochastic LLG method and theLangevin function in the whole temperature region as long asthe relation ( 10) is satisfied. The numerical integration scheme is given in Appendix B. The time step of /Delta1t=0.005 and a total of 80 000 time steps (40 000 steps for equilibration and40 000 steps for measurement) were adopted. 134411-3MASAMICHI NISHINO AND SEIJI MIY ASHITA PHYSICAL REVIEW B 91, 134411 (2015) 00.511.52 0 5 10 15 20 25m T FIG. 2. (Color online) Comparison of temperature ( T) depen- dence of mbetween the Monte Carlo method (green circles) and the stochastic LLG method in the homogeneous magnetic system withM=2. Crosses and boxes denote case A with α=0.05 and case B with D=1.0, respectively. B. Homogeneous magnetic moments with exchange interactions Next, we investigate homogenous magnetic moments (Mi=|Mi|=M) in three dimensions. The following Hamil- tonian [ C=0,Ji,j=J,DA i=DA, andh(t)=hin Eq. ( 1)], H=−/summationdisplay /angbracketlefti,j/angbracketrightJSi·Sj−/summationdisplay iDA/parenleftbig Sz i/parenrightbig2−/summationdisplay ihSz i, (14) is adopted. There is no exact formula for magnetization ( m)a sa function of temperature for this system, and thus a Monte Carlo(MC) method is applied to obtain reference magnetizationcurves for the canonical distribution because MC methodshave been established to obtain finite-temperature propertiesfor this kind of systems in the equilibrium state. Here weemploy a MC method with the Metropolis algorithm to obtainthe temperature dependence of magnetization. In order to check the validity of our MC procedure, we investigated magnetization curves as functions of temperature(not shown) with system-size dependence for the three-dimensional classical Heisenberg model [ D A=0 and h=0 in Eq. ( 14)], and confirmed that the critical temperature agreed with past studies [ 31], where kBTc=1.443Jfor the infinite system size with M=1. We give m(T) for a system of M=2 with the parameters J=1,h=2, andDA=1.0 for cases A and B in Fig. 2.T h e system size was set N=L3=103and periodic boundary conditions (PBC) were used. Green circles denote mobtained by the Monte Carlo method. At each temperature ( T) 10 000 MC steps (MCS) were applied for the equilibration andfollowing 10 000–50 000 MCS were used for measurement toobtain m. Crosses and boxes denote min the stationary state of the stochastic LLG equation in case A ( α=0.05) and in case B (D=1.0), respectively. Here /Delta1t=0.005 was set and 80 000 steps (40 000 for transient and 40 000 for measurement) wereused to obtain the stationary state of m.T h em(T) curves showgood agreement between the MC method and the stochastic LLG method in both cases. We checked that the choice of theinitial state for the MC and the stochastic LLG method doesnot affect the results. The dynamics of the stochastic LLGmethod leads to the equilibrium state at temperature T. IV . REALIZATION OF THE THERMAL EQUILIBRIUM STATE IN INHOMOGENEOUS MAGNETIC SYSTEMS A. Inhomogeneous magnetic moments with exchange interactions Here we study a system which consists of two kinds of magnitudes of magnetic moments. The Hamiltonian ( 14)i s adopted but the moment Mi=|Mi|hasidependence. We investigate a simple cubic lattice composed of alternatingM=2 andM=1 planes [see Fig. 3(a)], where J=1,h=2, andD A=1.0 are applied. We consider two cases A and B mentioned in Sec. II. The reference of m(T) curve was obtained by the MC method and is given by green circles in Figs. 3(b) and3(c).I n the simulation, at each temperature ( T) 10 000 MCS were applied for the equilibration and following 10 000–50 000MCS were used for measurement. The system size N=L 3= 103was adopted with PBC. In case A, α(=0.05) is common for all magnetic moments in the stochastic LLG method and Mi (orSi) dependence is imposed on DiasDi=D(Mi)≡α MikBT γ. In case B, D=1.0 is common for all magnetic moments in the stochastic LLG method and αi=α(Mi)≡DγM i kBT. Crosses in Figs. 3(b) and3(c) denote mby the stochastic LLG method for cases A and B, respectively. For those simulations /Delta1t=0.005 and 80 000 steps (40 000 for transient time and 40 000 formeasurement) were employed at each temperature. In bothFigs. 3(b) and 3(c), we find good agreement between m(T) by the stochastic LLG method (crosses) and m(T)b yt h eM C method (green circles). Next, we investigate how the results change if we take wrong choices of parameters. We study m(T) when a uniform valueD i=Dfor case A ( αi=αfor case B) is used for all spins, i.e., for both Mi=1 andMi=2. IfD(Mi=2)=α 2kBT γ is used for all spins, m(T) is shown by diamonds in Fig. 3(b), while if D(Mi=1)=αkBT γis applied for all spins, m(T)i s given by triangles in Fig. 3(b). In the same way, we study m(T) for a uniform value of α.I nF i g . 3(c) triangles and diamonds denote m(T) when αi=α(Mi=1) and αi=α(Mi=2) are used, respectively. We find serious difference in m(T) when we do not use correct Mi-dependent choices of the parameters. The locations of the triangle (diamond) at each temperature T are the same in Figs. 3(a) and 3(b), which indicates that if the ratio α/D is the same in different choices, the same steady state is realized although this state is not the true equilibriumstate for the inhomogeneous magnetic system. Thus weconclude that to use proper relations of M idependence of Di orαiis important for m(T) curves of inhomogeneous magnetic systems and wrong choices cause significant deviations. B. Critical behavior of inhomogeneous magnetic moments In this subsection, we examine properties near the critical temperature. Here we adopt the case of h=0 andDA=0 in the same type of lattice with M=1 and 2 as Sec. IV A . 134411-4REALIZATION OF THE THERMAL EQUILIBRIUM IN . . . PHYSICAL REVIEW B 91, 134411 (2015) 00.511.5 0 5 10 15 20m T(c) 00.511.5 0 5 10 15 20m T(b) (a) FIG. 3. (Color online) (a) A part of the system composed of alternating M=2 (red long arrows) and M=1 (short blue arrows) layers. (b) Comparison of temperature ( T) dependence of mbetween the Monte Carlo method (green circles) and the stochastic LLG method for α=0.05./Delta1t=0.005 and 80 000 steps (40 000 for transient time and 40 000 for measurement) were employed. Crosses denote mwhen Di=D(Mi)≡α MikBT γwas used. Triangles and diamonds are mforDi=D(1)=αkBT γfor all iandDi=D(2)=α 2kBT γfor all i, respectively. (c) Comparison of temperature ( T) dependence of mbetween the Monte Carlo method (green circles) and the stochastic LLG method for D=1.0./Delta1t=0.005 and 80 000 steps (40 000 for transient time and 40 000 for measurement) were employed. Crosses denote mwhen αi=α(Mi)≡DγM i kBTwas used. Triangles are mforαi=α(Mi=1)=Dγ×1 kBTfor all iand diamonds are mforαi=α(2)=Dγ×2 kBTfor all i. We investigate both cases of the temperature control (A and B). The Hamiltonian here has O(3) symmetry and mis not a suitable order parameter. Thus we define the following quantityas the order parameter [ 31]: m a=/radicalBig m2x+m2y+m2z, (15) where mx=1 N/angbracketleftBiggN/summationdisplay i=1Sx i/angbracketrightBigg ,m y=1 N/angbracketleftBiggN/summationdisplay i=1Sy i/angbracketrightBigg , mz=m=1 N/angbracketleftBiggN/summationdisplay i=1Sz i/angbracketrightBigg . (16) In Fig. 4, green circles denote temperature ( T) dependence ofmagiven by the MC method. The system size N=L3=203 with PBC was adopted and in MC simulations 10 000 MCS and following 50 000 MCS were employed for equilibrationand measurement, respectively, at each temperature. Themagnetizations of m aobtained by the stochastic LLG method for case A (crosses) and case B (diamonds) are given inFig. 4.H e r e α=0.05 and D=1.0 were used for (a) and (b),respectively. /Delta1t=0.005 was set and 240 000 steps (40 000 for transient and 200 000 for measurement) were applied. In both cases the m a(T) curve given by the stochastic LLG method shows good agreement with that obtained by the MCmethod. Thus, we conclude that as long as the relation ( 10)i s satisfied, the temperature dependence of the magnetization isreproduced very accurately even around the Curie temperature,regardless of the choice of the parameter set. C. Inhomogeneous magnetic moments with exchange and dipole interactions We also study thermal effects in a system with dipole interactions. We use the same lattice as in the previoussubsections. The system is [ J i,j=J,DA i=DA, andhi(t)=h in Eq. ( 1)] given by H=−/summationdisplay /angbracketlefti,j/angbracketrightJSi·Sj−/summationdisplay iDA/parenleftbig Sz i/parenrightbig2−/summationdisplay ihSz i +/summationdisplay i/negationslash=kC r3 ik/parenleftbigg Si·Sk−3(rik·Si)(rik·Sk) r2 ik/parenrightbigg .(17) 134411-5MASAMICHI NISHINO AND SEIJI MIY ASHITA PHYSICAL REVIEW B 91, 134411 (2015) 00.511.5 0123456ma T FIG. 4. (Color online) Comparison of temperature ( T) depen- dence of mabetween the MC method (green circles) and the stochastic LLG method for the system of inhomogeneous magnetic moments.N=L 3=203. PBC were used. In the MC method 10 000 MCS and following 50 000 MCS were used for equilibration and measurement at each temperature, respectively. The stochastic LLG method wasperformed in case A with α=0.05 (crosses) and in case B with D=1.0 (diamonds). Here /Delta1t=0.005 was applied and 240 000 steps were used (40 000 for transient and 200 000 for measurement). Here a cubic lattice with open boundary conditions (OBC) is used. Since Jis much larger than C/a3(J/greatermuchC/a3)f o r ferromagnets, where ais a lattice constant between magnetic sites. However, we enlarge dipole interaction as C=0.2 with a=1f o rJ=1 to highlight the effect of the noise on dipole interactions. We set other parameters as h=0.1,DA=0.1. Studies with realistic situations will be given separately. We study cases A ( α=0.05) and B ( D=1.0) for this system. We depict in Fig. 5the temperature ( T) dependencies ofmwith comparison between the MC (green circles) and stochastic LLG methods. Crosses and diamonds denote m(T) for cases A and B, respectively. Dipole interactions arelong-range interactions and we need longer equilibration steps,and we investigate only a small system with N=L 3=63.I n the MC method 200 000 MCS were used for equilibrationand 600 000 steps were used for measurement of m, and for the stochastic LLG method /Delta1t=0.005 was set and 960 000 steps (160 000 and 800 000 time steps for equilibration andmeasurement, respectively) were consumed. A reduction of m from fully saturated magnetization is observed. As a reference,mby the MC method without the dipole interactions ( C=0) is given by open circles in Fig. 5. This reduction of mis caused by the dipole interactions. We find that even when dipole interactions are taken into account in inhomogeneous magnetic moments, suitablechoices of the parameter set lead to the equilibrium state.Finally, we comment on the comparison between the LLGmethod and the Monte Carlo method. To obtain equilibriumproperties of spin systems, the Monte Carlo method is moreefficient and powerful in terms of computational cost. It ismuch faster than the stochastic LLG method to obtain theequilibrium m(T) curves, etc. For example, it needs more than00.511.5 0123456m T FIG. 5. (Color online) Comparison of temperature ( T) depen- dence of mbetween the Monte Carlo method (green circles) and the stochastic LLG method. Crosses and diamonds denote case Awithα=0.05 and case B with D=1.0, respectively. A reduction ofmfrom fully saturated magnetization is observed at around T=0 due to the dipole interactions. As a reference, mby the MC method without the dipole interactions ( C=0) is given by open circles. 10 times of CPU time of the MC method to obtain the data for Fig. 5. However, the MC method has little information on the dynamics and the stochastic LLG method is used to obtaindynamical properties because it is based on an equation ofmotion of spins. Thus, it is important to clarify the nature ofstochastic LLG methods including the static properties. Forstatic properties, as we saw above, the choice of the parameterset, e.g., cases A and B, did not give difference. However, thechoice gives significant difference in dynamical properties,which is studied in the following sections. V . DEPENDENCE OF DYNAMICS ON THE CHOICE OF THE PARAMETER SET IN ISOTROPIC SPIN SYSTEMS ( DA=0) Now we study the dependence of dynamics on the choice of parameter set. The temperature is given by kBT=γDiMi αi, (18) which should be the same for all the sites. In general, if the parameter D(amplitude of the noise) is large, the system is strongly disturbed, while if the parameter α(damping parameter) is large, the system tends to relax fast. Therefore,even if the temperature is the same, the dynamics changeswith the values of Dandα. When the anisotropy term exists, i.e.,D A/negationslash=0, in homogeneous systems ( Mi=M) given by Eq. ( 14), the Stoner-Wohlfarth critical field is hc=2MDA atT=0. If the temperature is low enough, the metastable nature appears in relaxation. On the other hand, if Tis rather high or DA=0, the metastable nature is not observed. In this section we focus on the dynamics of isotropic spin systems,i.e.,D A=0. 134411-6REALIZATION OF THE THERMAL EQUILIBRIUM IN . . . PHYSICAL REVIEW B 91, 134411 (2015) -2-1012 0 50 100 150 200m time-2.2-2-1.8-1.6-1.4-1.2-1 012345678(b) -2-1012 0 50 100 150 200m time(a) -2.2-2-1.8-1.6-1.4-1.2-1 012345678 FIG. 6. (Color online) (a) Time dependence of the magnetization [ m(t)] in case A, where α=0.05 for a homogeneous system with M=2. Red dash-dotted line, blue dotted line, green solid line, and black dashed line denote T=0.2,T=1,T=2, and T=10, respectively. Inset shows the time dependence of m(t) in the initial relaxation process. (b) Time dependence of the magnetization [ m(t)] in case B, where D=0.05 for a homogeneous system with M=2. Correspondence between lines and temperatures is the same as (a). A. Relaxation with temperature dependence In this subsection we investigate the temperature depen- dence of magnetization relaxation in cases A and B. We adopt ahomogeneous system ( M i=M=2) withDA=0i nE q .( 14). Initially all spins are in the spin down state and they relax undera unfavorable external field h=2. The parameter set M= 2,α=0.05,D=0.05 gives T=2 by the condition ( 10). Here we study the system at T=0.2,1,2, and 10. We set α=0.05 in case A and the control of the temperature is performed by D; i.e.,D=0.005,0.025,0.05, and 0 .25, respectively. In case B we set D=0.05, and the control of the temperature is realized byα; i.e.,α=0.5,0.1,0.05, and 0 .01, respectively. We depict the temperature dependence of m(t) for cases A and B in Figs. 6(a) and 6(b), respectively. Here the same random number sequence was used for each relaxation curve.The red dash-dotted line, blue dotted line, green solid line, andblack dashed line denote T=0.2,T =1,T=2, and T=10, respectively. Relaxation curves in the initial short time aregiven in the insets. In case A, as the temperature is raised, the initial relaxation speed of mbecomes faster and the relaxation time to the equilibrium state also becomes shorter. This dependenceis ascribed to the strength of the noise with the depen-dence D∝T, and a noise with a larger amplitude disturbs more the precession of each moment, which causes fasterrelaxation. On the other hand, in case B, the relaxation time to the equilibrium state is longer at higher temperatures althoughthe temperature dependence of the initial relaxation speedofmis similar to the case A. In the initial relaxation process all the magnetic moments are in the spin-down state(S z i/similarequal− 2). There the direction of the local field at each site is given by Heff i/similarequalJ/summationtext jSz j+h=− 2×6+2=− 10, which is downward and the damping term tends to fix moments tothis direction. Thus, a large value of the damping parameterat a low temperature T(α∝ 1 T) suppresses the change of the direction of each moment and the initial relaxation speed issmaller. However, in the relaxation process thermal fluctuationcauses a deviation of the local field and then a rotation ofmagnetic moments from −ztozdirection advances (see alsoFig.11). Once the rotation begins, the large damping parameter accelerates the relaxation and finally the relaxation time isshorter. B. Relaxation with spin-magnitude dependence Next we study the dependence of relaxation on the magnitude of magnetic moments in cases A and B. Here weadopt a homogeneous system ( M i=M) without anisotropy (DA=0) atT=2 and h=2. The initial spin configuration is the same as the previous subsection. Because D∝T M,α∝M T, (19) raising the value of Mis equivalent to lowering the temperature in both cases A and B and it causes suppression of relaxationin case A, while it leads to acceleration of relaxation in caseB. Because Maffects the local field from the exchange energy at each site, changing the value of Munder a constant external fieldhis not the same as changing Tand it may show some modified features. In the relation ( 19),T=0.2, 1, 2, 10 at M=2 [Figs. 6(a) and 6(b)]a r et h es a m ea s M=20, 4, 2, 0.4 at T=2, respectively. We studied the relaxation ratio defined as m(t)/M withMdependence at T=2 for these four values of M, and compared with the relaxation curves of Figs. 6(a) and6(b). We found qualitatively the same tendency between relaxation curves with Mdependence and those with 1 /T dependence in both cases. A difference was found in the initial relaxation speed (not shown). When M> 2, the initial relaxation at T=2 is slower than that of the corresponding TatM=2. The downward initial local field at each site is stronger for larger Mdue to a stronger exchange coupling, which also assists the suppression of the initialrelaxation. It is found that the relaxation time under a constant external filed becomes longer as the value of Mis raised in case A, while it becomes shorter in case B. This suggests that differentchoices of the parameter set lead to serious difference in therelaxation dynamics with Mdependence. 134411-7MASAMICHI NISHINO AND SEIJI MIY ASHITA PHYSICAL REVIEW B 91, 134411 (2015) -1.5-1-0.500.511.5 02468 1 0m time(b) -1.5-1-0.500.511.5 0 1 02 03 04 05 0m time(a) FIG. 7. (Color online) Comparison of the time dependence of mbetween cases A and B by the stochastic LLG method. Red and blue lines denote cases A and B, respectively. (a) α=0.05 for case A and D=1.0 for case B, (b) α=0.2 for case A and D=1.0 for case B. VI. DEPENDENCE OF DYNAMICS ON THE CHOICE OF THE PARAMETER SET IN ANISOTROPIC SPIN SYSTEMS ( DA/negationslash=0) A. Different relaxation paths to the equilibrium in magnetic inhomogeneity If the anisotropy term exists DA/negationslash=0 but the temperature is relatively high, metastable nature is not observed inrelaxation. We consider the relaxation dynamics when M ihas idependence in this case. We study the system (alternating M=2 and M=1 planes) treated in Sec. IV A .W es e ta configuration of all spins down as the initial state and observerelaxation of min cases A and B. In Sec. IV A we studied cases A ( α=0.05) and B ( D=1.0) for the equilibrium state and the equilibrium magnetization is m/similarequal0.95 atT=5. We give comparison of the time dependence of mbetween the two cases in Fig. 7(a), with the use of the same random number sequence. The red and blue curves denote cases A and B,respectively. We find a big difference in the relaxation time ofmand features of the relaxation between the two cases. The parameter values of αandDare not so close between the two cases at this temperature ( T=5); i.e., D(M=1)= 0.25 and D(M=2)=0.125 for case A and α(M=1)=0.2 andα(M=2)=0.4 for case B. Thus, to study whether thereis a difference of dynamics even in close parameter values ofαandDbetween cases A and B at T=5, we adopt a common α =0.2, where D(M=1)=1 and D(M=2)= 0.5, as case A, and a common D=1.0, where α(M=1)= 0.2 and α(M=2)=0.4, as case B. We checked that this case A also gives the equilibrium state. In Fig. 7(b), the time dependence of mfor both cases is given. The red and blue curves denote cases A and B, respectively. There is also adifference (almost twice) of the relaxation time of mbetween cases A and B. Thus, even in close parameter region of αand D, dynamical properties vary depending on the choice of the parameters. B. Relaxation with nucleation mechanism In this subsection we study a system with metastability. We adopt a homogeneous system ( M=2) with J=1,DA=1, andh=2. Here the Stoner-Wohlfarth critical field is hc= 2MDA=4, and if the temperature is low enough, the system has a metastable state under h=2. At a high temperature, e.g., T=10 (α=0.05,D=0.25), the magnetization relaxes without being trapped as depictedin Fig. 8(a) with a black dotted line. When the temperature is lowered, the magnetization is trapped at a metastable state. We (a) -2-1012 0 80 160 240 320m time-2-1012 0 50 100 150 200 250 300 350m time(b) -2-1012 0 50 100 150 200 250 300 350m time(c) FIG. 8. (Color online) (a) Dashed line shows m(t)f o rα=0.05,D=0.25, and T=10. Blue and green solid lines give m(t)f o rα=0.05 atT=3.5 (case A) and D=0.25 atT=3.5 (case B), respectively. These two lines were obtained by taking average over 20 trials with different random number sequences. The 20 relaxation curves for cases A and B are given in (b) and (c), respectively. 134411-8REALIZATION OF THE THERMAL EQUILIBRIUM IN . . . PHYSICAL REVIEW B 91, 134411 (2015) -2-1012 0 200 400 600 800m time(b) -2-1012 0 200 400 600 800m time(a) FIG. 9. (Color online) (a) and (b) illustrate 20 relaxation curves for α=0.05 atT=3.1 (case A) and D=0.25 atT=3.1 (case B), respectively. Metastability becomes stronger than T=3.5. No relaxation occurs in all 20 trials in (a), while five relaxations take place in 20 trials in (b). observe relaxations in cases A and B, where α=0.05 for case A and D=0.25 for case B are used. In Figs. 8(b) and8(c),w e show 20 samples (with different random number sequences)of relaxation processes at T=3.5 for case A ( α=0.05,D= 0.0875) and case B ( D=0.25,α=0.143), respectively. The average lines of the 20 samples are depicted in Fig. 8(a) by blue and green solid lines for cases A and B, respectively. Inboth cases, magnetizations are trapped at a metastable statewith the same value of m(m/similarequal− 1.55). This means that the metastability is independent of the choice of parameter set.Relaxation from the metastable state to the equilibrium is theso-called stochastic process and the relaxation time distributes.The relaxation time in case A is longer. If the temperature isfurther lowered, the escape time from the metastable statebecomes longer. In Figs. 9(a) and9(b), we show 20 samples of relaxation at T=3.1 for cases A and B, respectively. There we find the metastable state more clearly. Here we investigate the initial relaxation to the metastable state at a relatively low temperature. In Figs. 10(a) and10(b) , we depict the initial short-time relaxation of 20 samples atT=2 in cases A ( α=0.05,D=0.05) and B ( D=0.25,α=0.25), respectively. The insets show the time dependence of the magnetization in the whole measurement time. We findthat the relaxation is again faster in case B. The metastability also depends on Mas well as D Aand largeMgives a strong metastability. Here we conclude that regardless of the choice of the parameter set, as the temperatureis lowered, the relaxation time becomes longer due to thestronger metastability, in which larger D(larger α) gives faster relaxation from the initial to the metastable state and fasterdecay from the metastable state. Finally we show typical configurations in the relaxation process. When the anisotropy D Ais zero or weak, the magnetization relaxation occurs with uniform rotation from−ztozdirection, while when the anisotropy is strong, the magnetization reversal starts by a nucleation and inhomo-geneous configurations appear with domain wall motion. InFig. 11we give an example of the magnetization reversal of (a) the uniform rotation type (magnetization reversal forD A=0 with D=0.05,T=2,α=0.1,M=4) and of (b) the nucleation type (magnetization reversal for DA=1 with D=0.25,T=3.1,α=0.161,M=2). -2.2-2-1.8-1.6-1.4-1.2-1 012345678m time(b) -2-1012 0 200 400 600 800 time -2.2-2-1.8-1.6-1.4-1.2-1.0 012345678m time(a) -2-1012 0 200 400 600 800 time FIG. 10. (Color online) Initial relaxation curves of magnetization. Insets show m(t) in the whole measurement time. (a) and (b) illustrate 20 relaxation curves for α=0.05 atT=2 (case A) and D=0.25 atT=2 (case B), respectively. 134411-9MASAMICHI NISHINO AND SEIJI MIY ASHITA PHYSICAL REVIEW B 91, 134411 (2015) FIG. 11. (Color online) (a) Typical uniform rotation type relaxation observed in the isotropic spin system. (b) Typical nucleation type relaxation observed in the anisotropic spin system. VII. SUMMARY AND DISCUSSION We studied the realization of the canonical distribution in magnetic systems with the short-range (exchange) andlong-range (dipole) interactions, anisotropy terms, and mag- netic fields by the Langevin method of the LLG equation. Especially we investigated in detail the thermal equilibrationof inhomogeneous magnetic systems. We pointed out that thespin-magnitude dependent ratio between the strength of therandom field and the coefficient of the damping term mustbe adequately chosen for all magnetic moments satisfyingthe condition ( 10). We compared the stationary state obtained by the present Langevin method of the LLG equation with the equilibrium state obtained by the standard Monte Carlosimulation for given temperatures. There are several choicesfor the parameter set, e.g., A and B. We found that aslong as the parameters are suitably chosen, the equilibriumstate is realized as the stationary state of the stochastic LLGmethod regardless of the choice of the parameter set, and thetemperature dependence of the magnetization is accurately produced in the whole region, including the region around the Curie temperature. We also studied dynamical properties which depend on the choice of the parameters. We showed that the choice ofthe parameter values seriously affects the relaxation processto the equilibrium state. In the rotation type relaxation inisotropic spin systems under an unfavorable external field, thedependencies of the relaxation time on the temperature in casesA and B exhibited opposite correlations as well as the depen-dencies of the relaxation time on the magnitude of the magneticmoment. The strength of the local field in the initial statestrongly affects the speed of the initial relaxation in both cases. We also found that even if close parameter values are chosen in different parameter sets for inhomogeneous magneticsystems, these parameter sets cause a significant difference ofrelaxation time to the equilibrium state. In the nucleation typerelaxation, the metastability, which depends on D AandM,strongly affects the relaxation in both cases A and B. Lowering temperature reinforces the metastability of the system andcauses slower relaxation. The relaxation to the metastable stateand the decay to the metastable state are affected by the choiceof the parameter set, in which larger Dcauses fast relaxation at a fixed T. In this study we adopted two cases, i.e., A and B, in the choice of the parameter set. Generally a more complicateddependence of M iorTon the parameters is considered. How to chose the parameter set is related to the questfor the origin of these parameters. It is very important forclarification of relaxation dynamics but also for realization ofa high speed and a low power consumption, which is requiredto development of magnetic devices. Studies of the originofαhave been intensively performed [ 32–41]. To control magnetization relaxation at finite temperatures, investigationsof the origin of Das well as αwill become more and more important. We hope that the present work gives someuseful insight into studies of spin dynamics and encouragesdiscussions for future developments in this field. ACKNOWLEDGMENTS The authors thank Dr. S. Hirosawa and Dr. S. Mohakud for useful discussions. The present work was supported bythe Elements Strategy Initiative Center for Magnetic Materialsunder the outsourcing project of MEXT and a Grant-in-Aid forScientific Research (C) 26400324 from MEXT. The authorsalso thank the Supercomputer Center, the Institute for SolidState Physics, the University of Tokyo for the use of thefacilities. APPENDIX A: FOKKER-PLANCK EQUATION The LLG equation with a Langevin noise [Eq. ( 5)] is rewritten in the following form for the μcomponent ( μ=1,2, 134411-10REALIZATION OF THE THERMAL EQUILIBRIUM IN . . . PHYSICAL REVIEW B 91, 134411 (2015) or 3 for x,y,o rz)o ft h e ith magnetic moment, dMμ i dt=fμ i(M1,...,MN,t)+gμν i(Mi)ξν i(t).(A1) Herefμ iandgμν iare given by fμ i=−γ 1+α2 i/bracketleftbigg /epsilon1μνλMν iHeff,λ i +αi Mi/epsilon1μνλ/epsilon1λρσMν iMρ iHeff,σ i/bracketrightbigg , (A2) gμλ i=−γ 1+α2 i/bracketleftbigg /epsilon1μνλMν i+αi Mi/parenleftbig −M2 iδμ λ+Mμ iMλ i/parenrightbig/bracketrightbigg ,(A3) where Heff,λ i can have an explicit time ( t) dependence, and /epsilon1μνλdenotes the Levi-Civita symbol. We employ the Einstein summation convention for Greek indices ( μ,ν,... ). We consider the distribution function F≡F(M1,..., MN,t)i nt h e3 N-dimensional phase space ( M1 1,M2 1,M3 1, ..., M1 N,M2 N,M3 N). The distribution function F(M1,..., MN,t) satisfies the continuity equation of the distribution: ∂ ∂tF(M1,...,MN,t)+N/summationdisplay i=1∂ ∂Mα i/braceleftbigg/parenleftbiggd dtMα i/parenrightbigg F/bracerightbigg =0. (A4) Substituting the relation ( A1), the following differential equation for the distribution function Fis obtained: ∂ ∂tF(M1,...,MN,t)=−N/summationdisplay i=1∂ ∂Mα i/braceleftbig/parenleftbig fi+gαβ iξβ i/parenrightbig F/bracerightbig . (A5) Regarding the stochastic equation ( A1) as the Stratonovich interpretation, making use of the stochastic Liouville ap-proach [ 42], and taking the average for the noise statistics [Eq. ( 6)], we have a Fokker-Planck equation, ∂ ∂tP(M1,...,MN,t) =−N/summationdisplay i=1∂ ∂Mα i/braceleftbigg fα iP−Digαβ i∂ ∂Mσ i(gσβ iP)/bracerightbigg , (A6) where P≡P(M1,...,MN,t) is the averaged distribution function /angbracketleftF/angbracketright. Substituting the relation ∂ ∂Mσ igσβ i=−γαi Mi/parenleftbig 1+α2 i/parenrightbig4Mβ i (A7) and Eq. ( A3)i n t ogαβ i(∂ ∂Mσ igσβ i), we find gαβ i/parenleftbigg∂ ∂Mσ igσβ i/parenrightbigg =0. (A8)Thus Eq. ( A6) is simplified to ∂ ∂tP(M1,...,MN,t) =−N/summationdisplay i=1∂ ∂Mα i/braceleftbigg/parenleftbigg fα i−Digαβ igσβ i∂ ∂Mσ i/parenrightbigg P/bracerightbigg . (A9) Substituting Eqs. ( A2) and ( A3), we have a formula in the vector representation, ∂ ∂tP(M1,...,MN,t) =/summationdisplay iγ 1+α2 i∂ ∂Mi ·/braceleftbigg/bracketleftbigg Mi×Heff i+αi MiMi×/parenleftbig Mi×Heff i/parenrightbig −γDiMi×/parenleftbigg Mi×∂ ∂Mi/parenrightbigg/bracketrightbigg P(M1,...,MN,t)/bracerightbigg . (A10) Since∂ ∂Mi·(Mi×Heff i)=0, it is written as ∂ ∂tP(M1,...,MN,t) =/summationdisplay iγ 1+α2 i∂ ∂Mi·/braceleftbigg/bracketleftbiggαi MiMi×/parenleftbig Mi×Heff i/parenrightbig −γDiMi×/parenleftbigg Mi×∂ ∂Mi/parenrightbigg/bracketrightbigg P(M1,...,MN,t)/bracerightbigg . (A11) In the case that Eq. ( A1) is given under the Ito definition, we need an Ito-Stratonovich transformation, and the correspond-ing equation of motion in the Stratonovich interpretation is dMμ i dt=fμ i(M1,...,MN,t)−Digλν i(Mi)∂gμν i(Mi) ∂Mλ i +gμν i(Mi)ξν i(t). (A12) Then the Fokker-Planck equation in the Ito interpretation is ∂ ∂tP(M1,...,MN,t) =−N/summationdisplay i=1∂ ∂Mα i/braceleftbigg/parenleftbigg fα i−Digλν i∂gαν i ∂Mλ i−Digαβ igσβ i∂ ∂Mσ i/parenrightbigg P/bracerightbigg . Sincegλν i∂gαν i ∂Mλ i=−2γ2 1+α2 iMα i, the vector representation is given by ∂ ∂tP(M1,...,MN,t) =/summationdisplay iγ 1+α2 i∂ ∂Mi·/braceleftbigg/bracketleftbiggαi MiMi×/parenleftbig Mi×Heff i/parenrightbig −2γDiMi−γDiMi×/parenleftbigg Mi×∂ ∂Mi/parenrightbigg/bracketrightbigg ×P(M1,...,MN,t)/bracerightbigg . (A13) 134411-11MASAMICHI NISHINO AND SEIJI MIY ASHITA PHYSICAL REVIEW B 91, 134411 (2015) APPENDIX B: NUMERICAL INTEGRATION FOR STOCHASTIC DIFFERENTIAL EQUATIONS In stochastic differential equations, we have to be careful to treat the indifferentiability of the white noise. In the present paper we regard the stochastic equation, e.g., Eq. ( 5), as a stochastic differential equation in Stratonovich interpretation: dMμ i=fμ i(M1,...,MN,t)dt+gμν i/parenleftbig1 2[Mi(t)+Mi(t+dt)]/parenrightbig dWν i(t), (B1) where dWν i(t)=/integraltextt+dt tdsξν i(s), which is the Wiener process. This equation is expressed by dMμ i=fμ i(M1,...,MN,t)dt+gμν i(Mi(t))◦dWν i(t), (B2) where ◦indicates the usage of the Stratonovich definition. A simple predictor-corrector method called the Heun method [ 8,19], superior to the Euler method, is given by Mμ i(t+/Delta1t)=Mμ i(t)+1 2/bracketleftbig fμ i(ˆM1(t+/Delta1t),..., ˆMN(t+/Delta1t),t+/Delta1t)+fμ i(M1(t),...,MN(t),t)/bracketrightbig /Delta1t +1 2/bracketleftbig gμν i(ˆMi(t+/Delta1t))+gμν i(Mi(t))/bracketrightbig /Delta1Wν i, (B3) where /Delta1Wν i≡Wν i(t+/Delta1t)−W(t) and ˆMμ i(t+/Delta1t) is chosen in the Euler scheme: ˆMμ i(t+/Delta1t)=Mμ i(t)+fμ i(M1(t),...,MN(t),t)/Delta1t+gμν i(Mi(t))/Delta1Wν i. (B4) This scheme assures an approximation accuracy up to the second order of /Delta1W and/Delta1t. Several numerical difference methods [ 19] for higher-order approximation, which are often complicated, have been proposed. Here we adopt a kind of middle point method equivalent to the Heun method, Mμ i(t+/Delta1t)=Mμ i(t)+fμ i(M1(t+/Delta1t/2),...,MN(t+/Delta1t/2),t+/Delta1t/2)/Delta1t +gμν i(Mi(t+/Delta1t/2))/Delta1Wν i, (B5) where Mμ i(t+/Delta1t/2) is chosen in the Euler scheme: Mμ i(t+/Delta1t/2)=Mμ i(t)+fμ i(M1(t),...,MN(t),t)/Delta1t/2+gμν i(Mi(t))/Delta1˜Wiν, (B6) where /Delta1˜Wiν≡Wν i(t+/Delta1t/2)−Wν i(t). Considering the following relations, /angbracketleftbig /Delta1˜Wiν/Delta1Wν i/angbracketrightbig =/angbracketleftbig/bracketleftbig Wν i(t+/Delta1t/2)−Wν i(t)/bracketrightbig/bracketleftbig Wν i(t+/Delta1t)−Wν i(t)/bracketrightbig/angbracketrightbig =Di/Delta1t, (B7) /angbracketleft/Delta1Wν i/angbracketright=0, and /angbracketleft/Delta1˜Wiν/angbracketright=0, this method is found equivalent to the Heun method. We can formally replace /Delta1˜Wiνwith/Delta1Wν i/2 in Eq. ( B6) in numerical simulations. [1] H. Kronm ¨ullar and M. 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PhysRevB.87.094509.pdf

PHYSICAL REVIEW B 87, 094509 (2013) Band structure of magnetic excitations in the vortex phase of a ferromagnetic superconductor A. A. Bespalov1,2and A. I. Buzdin2 1Institute for Physics of Microstructures, Russian Academy of Sciences, GSP-105, 603950, Nizhny Novgorod, Russia 2Universit ´e Bordeaux, LOMA, UMR 5798, F-33600 Talence, France (Received 16 January 2013; published 18 March 2013) Magnetic excitations in a ferromagnetic superconductor in the presence of an Abrikosov vortex lattice have been studied using the phenomenological London and Landau-Lifshitz equations. Due to the periodicity of the vortexfield the magnon spectrum has a band structure, similar to the structure of the electon spectrum in a crystal lattice.The gaps between adjacent bands have been calculated using an analog of the weak-binding approximation.When the applied magnetic field is altered the band structure undergoes a qualitative transformation due tocommensurability effects, connected with the nonmonotonicity of the magnon spectrum in the Meissner state. Indirty samples the energy gaps may be smeared out because of the dissipation connected with vortex motion. Insufficiently clean samples the gaps manifest themselves as maxima in the frequency dependence of the microwavereflectivity coefficient. DOI: 10.1103/PhysRevB.87.094509 PACS number(s): 75 .30.Ds, 74 .25.Uv, 74 .20.De I. INTRODUCTION The discovery of ferromagnetic superconductors1–5pro- vided a unique opportunity to study the interplay betweenmagnetism and triplet superconductivity in one compound.However, the investigation of magnetic properties in thesuperconducting state of these materials is hampered by theMeissner effect, consisting in the screening of static magneticfields. Still, dynamical measurements can be applied, forexample, microwave probing of the spin wave spectrum. 6 Spin waves in magnetic superconductors in the Meissnerstate have been studied theoretically, using different phe-nomenological approaches, in several papers. 6–10Buzdin7 determined the magnon spectrum in an antiferromagnetic superconductor with an easy-axis anisotropy. Braude andSonin 6,9calculated the magnon spectrum and the microwave response of a ferromagnetic supercondutor. In Refs. 8and10 two-dimensional magnons on domain walls and on the surface of the ferromagnet have been studied. Experimental measurements of the ac magnetic suscepti- bility of superconducting ferromagnets revealed that the idealdiamagnetic response is not reached in the superconductingphase. 2–4This observation allows to suggest that these materi- als are always found in the spontaneous mixed state due to thepresence of the intrinsic field created by the magnetization.Thus, the theoretical study of magnetic excitations in thevortex phase is also relevant. The influence of Abrikosovvortices on magnetization dynamics in magnetic supercon-ductors has been considered in a number of theoretical papers.In Refs. 11–13it has been demonstrated that the magnon spectrum can be examined using the vortex motion inducedby a dc or ac transport current. The contribution to the vortex viscosity connected with dissipation due to the Cherenkov radiation of magnons has been determined. In Ref. 14it has been predicted that the flux flow should lead to the creation ofdomain walls in systems with slow relaxation of the magneticmoments. Ng and Varma 15considered coupled spin and vortex dynamics in ferromagnetic superconductors in the absence ofa transport current. The authors used the continuous mediumapproximation, which is valid in the limit of long wavelengthλ w>a, where ais the intervortex distance. In this limit it doesnot matter whether the vortices form a regular or disordered array, hence, the effects connected with the periodicity of thevortex field can not be detected. In the present paper we investigate theoretically the magnon spectrum in the mixed state of a superconducting ferromagnetby solving the phenomenological Landau-Lifshitz and Londonequations. V ortex motion is taken into account using a viscousdamping equation (see Ref. 15). Our analysis extends to the case of short wavelengths λ w/lessorsimilara, where Bragg scattering of magnons by the vortex lattice becomes important. The magnonspectrum has a Bloch-like band structure with frequency gapsbetween adjacent bands. These gaps manifest themselves asanomalies in the frequency dependence of the reflectivitycoefficient of electromagnetic radiation. The outline of the paper is as follows. In Sec. II A we present a model of the ferromagnetic superconductor andreview the result for the magnetic excitations spectrum in theMeissner state (see Ref. 8). In Sec. II B we derive the basic equations for the collective vortex-magnetization dynamics.In Secs. II C and II D the weak-binding approximation is developed and the frequency gaps between adjacent bandsof the magnon spectrum are determined analytically. Thenumerical spectra, obtained using realistic parameters, arepresented in Sec. II E. In Sec. II F the role of dissipation connected with viscous vortex motion is discussed. Finally,in Sec. IIIwe consider a boundary problem for an electromag- netic wave incident at a ferromagnetic superconductor. Thefrequency-dependent reflectivity coefficient is examined forfrequencies lying within and close to the gaps of the magnonscpectrum. II. MAGNON SPECTRUM IN A FERROMAGNETIC SUPERCONDUCTOR IN THE MIXED STATE A. Model of the medium and magnetic excitations in the Meissner state Let us consider a superconducting ferromagnet with an easy-axis magnetocrystalline anisotropy. At the beginningwe assume the exchange interaction and superconductingproperties of the material to be isotropic. A generalization 094509-1 1098-0121/2013/87(9)/094509(12) ©2013 American Physical SocietyA. A. BESPALOV AND A. I. BUZDIN PHYSICAL REVIEW B 87, 094509 (2013) for the case of uniaxial anisotropy is briefly discussed in the end of Sec. II B. Within the London approximation the Gibbs free energy of the ferromagnet is6 F=/integraldisplay/bracketleftbiggα 2/parenleftbigg∂M ∂xi∂M ∂xi/parenrightbigg +KM2 ⊥ 2+1 8πλ2/parenleftbigg A−/Phi10 2π∇θ/parenrightbigg2 +(rotA−4πM)2 8π−HerotA 4π/bracketrightbigg d3r. (1) Here Mis the magnetization, αis the exchange constant, λis the London length, Kis the anisotropy coefficient, /Phi10is the flux quantum, θis the superconducting phase, Heis an external magnetic field, and M⊥is the component of Mperpendicular to the anisotropy axis which we direct along the zaxis. By setting the variational derivative of Fwith respect to Aequal to zero we obtain the London equation rot rot B+B λ2=4πrot rot M+1 λ2κ, (2) where B=rotA, and κis the vorticity κ=/Phi10 2πrot∇θ. (3) It has been assumed that the external field has no sources inside the material. The magnetization vector obeys the Landau-Lifshitz equation ∂M ∂t=−γδF δM×M, (4) or ∂M ∂t=γ(α∇2M−KM⊥+B)×M. (5) We will determine the spectrum of low-energy excitations in our system. First, we recall the spectrum for the Meissnerstate. In equilibrium A=0,M=M 0=Mz0, and κ=0. For small perturbations we can linearize Eqs. (2)and (5)with respect to the deviation from equilibrium rot rot B+B λ2=4πrot rot m, (6) ∂m ∂t=γ(α∇2m−Km+B)×M0, (7) where m=M⊥. Assuming B,m∼e−iωt+ikzz+iqr, where q= (qx,qy,0), we obtain the magnon spectrum (see Ref. 8) ω=ω0(q)=γM/radicalbig K1K2, (8) K1(q,kz)=˜K+αq2−4πk2 z λ−2+k2z+q2, (9) K2(q,kz)=˜K+αq2−4π/parenleftbig k2 z+q2/parenrightbig λ−2+k2z+q2, (10) where ˜K=K+αk2 z. The parameters of some U-based compounds are listed in Table I. It can be seen that these supercondutorsTABLE I. Parameters of ferromagnetic superconductors. The data have been taken from Refs. 1,16–18. Effective domain wall Anisotropy thickness, field, Compound ˜ w∼√α/K ,n m Han,T K=Han/M 0λ,n m UGe 2 13,6 ∼100 ∼1041000 UCoGe 45 ∼10 ∼1041200 URhGe 3450 ∼10 ∼103–104900 possess a rather high magnetic anisotropy. This fact allows to simplify the expression for the frequency by expandingthe root in Eq. (8)in the powers of the small quantity (K 1−K2)/K 1 ω0(q,kz)≈γM/bracketleftbigg ˜K+αq2−2π/parenleftbig 2k2 z+q2/parenrightbig λ−2+k2z+q2/bracketrightbigg . (11) A characteristic feature of this spectrum is the presence of a minimum at q=qmin=/radicaltp/radicalvertex/radicalvertex/radicalbt/radicalBigg 2πλ−2−k2z α−λ−2−k2z (12) for sufficiently small kz.T h eω0vs.qdependence for kz=0 is depicted in Fig. 1. B. Magnetic exitations in the mixed state: Basic equations Now we consider a more realistic case of a ferromagnetic superconductor in the mixed state. If the external magneticfield is absent or parallel to the easy axis, the Abrikosovvortices in equilibrium are directed along the magnetizationvector. We assume the vortex lattice to be triangular with thepositions of the vortices given by the vectors R i=ay0p+a/parenleftbigg√ 3 2x0+1 2y0/parenrightbigg n, (13) where ais the distance between neighboring vortices and pand nare integers. Then, the equilibrium vorticity and magnetic FIG. 1. The magnon spectrum in the Meissner state. 094509-2BAND STRUCTURE OF MAGNETIC EXCITATIONS IN THE ... PHYSICAL REVIEW B 87, 094509 (2013) field are κ0=/Phi10z0/summationdisplay iδ(2)(ρ−Ri), (14) B0=z0/summationdisplay G<ξ−1B0(G)eiGr,B 0(G)=/Phi10 1+G2λ2·2√ 3a2, (15) where ρ=(x,y),ξis the coherence length, and Gare the vectors of the reciprocal lattice G=pG1+nG2,G1=4π√ 3ax0,G2=2π√ 3ax0+2π ay0. (16) Following Ng and Varma,15we will consider the magnetization dynamics, taking into account vortex motion as well. Thelinearized equations (2)and(4)read ∂m ∂t=γM/bracketleftbigg α∇2m−/parenleftbigg K+B0(r) M/parenrightbigg m+b/bracketrightbigg ×z0, (17) −∇2b+b λ2=4πrot rot m+1 λ2κ1, (18) where b=B−B0andκ1=κ−κ0. The linear deviation from equilibrium of the vorticity is given by κ1=/Phi10/summationdisplay i/braceleftbigg δ(2)(ρ−Ri)d/Delta1Ri dz −z0[/Delta1Ri·∇ρδ(2)(ρ−Ri)]/bracerightbigg , (19) where /Delta1Ri(z) is the local displacement of the vortex with respect to its equilibrium position Ri. To determine the quantities /Delta1Riwe use the phenomonological equation of dissipative vortex dynamics15 ηd dt/Delta1Ri(z)=−δF δ/Delta1Ri, (20)where ηis a viscosity coefficient. After the evaluation of the variational derivative, Eq. (20) transforms into −ηd dt/Delta1Ri(z) =/Phi10 (2π)3/2/integraldisplay k<ξ−1−ikz(κ1k−4πmk)+ik(z0κ1k) 4π(λ2k2+1) ×exp (ikRi+ikzz)d3k−/summationdisplay G<ξ−1B0(0)/Phi10(G/Delta1Ri)G 4π(λ2G2+1), (21) where ξis the superconducting coherence length, and Xkfor any quantity Xdenotes its Fourier transform Xk=1 (2π)3/2/integraldisplay X(r)e−ikrd3r. We may rewrite Eqs. (17),(18), and (21) in the Fourier representation. If we do so, we will find that these equationsconnect the Fourier components of the functions m,b, and κ corresponding to wave vectors satisfying the condition k=G+k 0, (22) where k0is a fixed arbitrary vector and the vector Gruns over the whole reciprocal lattice (16). Hence, the general solution of Eqs. (17),(18), and (21) can be presented as a superposition of particular solutions having the form m=e−iωt+ikzz+iqr/summationdisplay Gm(G)eiGr, (23) b=e−iωt+ikzz+iqr/summationdisplay Gb(G)eiGr, (24) κ1=e−iωt+ikzz+iqr/summationdisplay Gκ1(G)eiGr, (25) where qis the quasi-wave-vector in the xyplane. The fact that the functions (23) to(25) satisfy our equations represents a simple generalization of the Bloch theorem. The condition(25) is equivalent to the following one: /Delta1R i(z)=/Delta1Re−iωt+ikzz+iqRi. (26) If we substitute Eqs. (23),(24), and (26) into Eqs. (17),(18), and(21), we obtain the system iωη /Phi10/Delta1R=/summationdisplay Gi<ξ−14πikzm(Gi)+B0(0)k2 z/Delta1R 4π/bracketleftbig 1+λ2/parenleftbig k2z+q2 i/parenrightbig/bracketrightbig+/summationdisplay Gi<ξ−1B0(0) 4π/bracketleftbiggqi(qi/Delta1R) 1+λ2(q2 i+k2z)−G(G/Delta1R) 1+λ2G2/bracketrightbigg , (27) −iω γMm(Gi)=/bracketleftBigg −/parenleftbig˜K+αq2 i/parenrightbig m(Gi)+4π/parenleftbig k2 z+q2 i/parenrightbig m(Gi)−4πqi(qim(Gi))+B0(0)λ−2ikz/Delta1R k2z+q2 i+λ−2 −1 M/summationdisplay G/prime/negationslash=Gim(G/prime)B0(Gi−G/prime)/bracketrightBigg ×z0, (28) where qi=q+Gi, and ˜K=K+B0(0)/M+αk2 z. (29) By solving Eqs. (27) and (28) the dispersion relation may be found.First, we restrict ourselves to the case when the dissipation due to vortex motion is negligible, i.e., η→∞ and/Delta1R=0. The role of thermal losses will be discussed in Sec. II F. It has been mentioned that in the U-based compounds the magnetic anisotropy is rather large. Using this fact 094509-3A. A. BESPALOV AND A. I. BUZDIN PHYSICAL REVIEW B 87, 094509 (2013) one can make an approximation which will considerably simplify the problem. Let the vectors qihave the components (qicosαi,qisinαi). For qi=0 the angle αiis arbitrary. We introduce the new variables m/prime ix=[cosαimx(Gi)+sinαimy(Gi)]4/radicalBigg K1(qi) K2(qi), m/prime iy=[cosαimy(Gi)−sinαimx(Gi)]4/radicalBigg K2(qi) K1(qi), where K1andK2are given by Eqs. (9)and (10). Here and further we omit kzin the list of arguments of K1,K2, andω0 for brevity. The quantities m/prime ixandm/prime iysatisfy the equations −iω γMm/prime ix=−ω0(qi) γMm/prime iy−/summationdisplay j/negationslash=ibij/bracketleftbigg m/prime jy4/radicalBigg K1(qi)K1(qj) K2(qi)K2(qj) ×cos (αi−αj)−m/prime jx4/radicalBigg K1(qi)K2(qj) K2(qi)K1(qj) ×sin (αi−αj)/bracketrightbigg , (30) −iω γMm/prime iy=ω0(qi) γMm/prime ix+/summationdisplay j/negationslash=ibij/bracketleftbigg m/prime jx4/radicalBigg K2(qi)K2(qj) K1(qi)K1(qj) ×cos (αi−αj)+m/prime jy4/radicalBigg K2(qi)K1(qj) K1(qi)K2(qj) ×sin (αi−αj)/bracketrightbigg , (31) where bij=B0(Gi−Gj)/M, and ω0(q) γM≈K+α/parenleftbig q2+k2 z/parenrightbig +B0(0) M−2π/parenleftbig 2k2 z+q2/parenrightbig λ−2+k2z+q2. (32) The main assumption of our approximation is that all fourth roots in Eqs. (30) and(31) can be replaced by unity. Indeed, 4/radicalBigg K1(qi)K1(qj) K2(qi)K2(qj)−1 ≈1 4/bracketleftbiggK1(qi)−K2(qi) K2(qi)+K1(qj)−K2(qj) K2(qj)/bracketrightbigg /lessmuch1. It is convenient to introduce the variables m+ i=(m/prime ix− im/prime iy)e−iαiandm− i=(m/prime ix+im/prime iy)eiαi. Equations (30) and (31) yield ω γMm+ i=ω0(qi) γMm+ i+/summationdisplay j/negationslash=ibijm+ j, (33) −ω γMm− i=ω0(qi) γMm− i+/summationdisplay j/negationslash=ibijmj. (34) It can be seen that the solutions of Eq. (34) coincide with those of Eq. (33), but the frequencies have the opposite sign. A great advantage of Eq. (33) over Eq. (28) is that it represents an eigenvalue problem for a real symmetric matrix, so simplenumerical and analytical procedures may be applied to solve it.Equation (33) can be also derived if the uniaxial ( z axis) superconducting and exchange interaction anisotropy istaken into account. The only modification is that the Fouriercomponents of the field and the unperturbed frequencies aregiven by B 0(G)=/Phi10 1+G2λ2 ⊥·2√ 3a2, (35) ω0(q) γM≈K+α||k2 z+α⊥q2+B0(0) M −2πk2 zλ2 ⊥ 1+λ2 ⊥/parenleftbig k2z+q2/parenrightbig−2π/parenleftbig k2 zλ2 ⊥+q2λ2 ||/parenrightbig 1+k2zλ2 ⊥+q2λ2 ||,(36) where the quantities α||,λ||are related to the zaxis, and α⊥, λ⊥are related to the perpendicular plane. C. Weak-binding approximation To determine the eigenvalues of the system (33) we are going to use a method, which is equivalent to the weak-bindingapproximation for electrons in a crystal. We assume thatseveral eigenvalues ωfor a given vector qare close to ω 0(q), and the deviations ω−ω0(q) can be determined using the degenerate state perturbation theory. Let us find the applicability conditions for this approxima- tion. Consider the case when ω≈ω0(qi),|m+ i|/greaterorequalslant|m+ j|and all the quantities ω0(qj) are not close to ω0(qi)f o rj/negationslash=i. Then the perturbation theory in its simplest form can be applied. Itfollows from Eq. (33) that for i/negationslash=j m j≈bijmiγM ω0(qi)−ω0(qj). Here and further we omit the upper index “ +” for brevity. The correction to the unperturbed frequency ω0(qi) due to the fact that mj/negationslash=0 equals δω=b2 ijγ2M2 ω0(qi)−ω0(qj). For the perturbation theory to be valid we have to demand at least|δω|/lessmuch|ω0(qi)−ω0(qj)|,o r ⎧ ⎨ ⎩bij (qi−qj)(qi+qj)/bracketleftbig α−2πλ−2 (λ−2+q2 i)(λ−2+q2 j)/bracketrightbig⎫ ⎬ ⎭2 /lessmuch1 (37) for all j/negationslash=i. Here we assumed kz=0 for simplicity. If we apply the perturbation theory for a degenerate state, we can permit the condition (37) to be violated for Nj>1 different indices j. The number Njcan be estimated as Nj∼Sqa2, (38) where Sqis the area in the qplane occupied by the vectors qj for which the condition bij |qi−qj|(qi+qj)/vextendsingle/vextendsingleα−2πλ−2 (λ−2+q2 i)(λ−2+q2 j)/vextendsingle/vextendsingle/greaterorequalslant1 (39) holds. To avoid solving secular equations for large matrices, we demand Nq∼1. Hence, the area Sqshould not be too large. Restrictions on Sqare the most strong in two cases: (i)qiis close to the value corresponding to the minimum of 094509-4BAND STRUCTURE OF MAGNETIC EXCITATIONS IN THE ... PHYSICAL REVIEW B 87, 094509 (2013) ω0(qi) and (ii) qiis sufficiently large. To estimate Sqin the first case, we take qiequal to qmin. Assuming α∼λ2(which seems to be realistic, according to Table I),qmin∼λ−1and bij∼/Phi10/(a2M) we obtain from Eq. (39) |q−qmin|/lessorsimilar1 a/radicalbigg /Phi10 λ2M. (40) Hence, Sq≈4πqmin1 a/radicalbigg /Phi10 λ2M, and Nq∼a λ/radicalbigg /Phi10 λ2M. SinceNqshould be of the order of unity, we have the limitation a/lessorsimilarλ/radicalBigg λ2M /Phi10. (41) Typically, the ratio λ2M//Phi1 0is not very large. Hence, Eq. (41) means that the intervortex distance should be not much largerthan the London length. In further consideration we will implythat the inequality (41) holds. At the large q ilimit we obtain the constraint /Phi10/Mα/lessorsimilar1, which is satisfied for the materials listed in Table I. Similar to the electron spectrum in solid matter, the magnon spectrum in our system consists of bands separated by the gaps.We will use the reduced zone scheme, i.e., the bands are foldedin the first Brillouin zone (see Fig. 2). Now we calculate the gap between two neighboring bands using the weak-binding approximation. Obviously, the gap isthe smallest on the lines of the qplane where the bands would intersect if the matrix elements b ijwere negligible. On these linesω≈ω0(qi)=ω0(qj) for some different indices iandj. To find small corrections to the unperturbed frequency ω0(qi) FIG. 2. First Brillouin zone. It is sufficient to calculate the spectrum in the shaded area to determine the spectrum in the whole zone using symmetry relations.one should solve the simple secular equation /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleω0(qi)−ω γMbij bijω0(qi)−ω γM/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=0. (42) The frequency gap /Delta1is /Delta1=2γB 0(Gi−Gj). (43) It is remarkable that the gap does not change along the line where the two bands would intersect, if B0(Gi−Gj) was equal to zero. The dimensionless parameter characterizing the modifica- tion of the unperturbed spectrum (11) by the vortex lattice is the ration of the gap /Delta1to the frequency. For the compounds listed in Table Ithis ratio is the largest in URhGe, where /Delta1 ω∼/Phi10 λ2Han∼10−2−10−3. (44) This parameter appears to be small due to the large magnetic anisotropy of the compound. According to Eq. (44), the effects connected with the band structure of the magnon spectrum willbe more pronounced in materials with low anisotropy. D. Magnon spectrum in points of high symmetry: Intersections of bands In previous sections we made calculations assuming that the ferromagnet has isotropic superconducting properties, but theresults derived there are also qualitatively valid for anisotropic(uniaxial and biaxial) superconductors. In this section weconsider the properties of the band structure which appearonly in materials with uniaxial symmetry. We will study the magnon spectrum in points of the Brillouin zone where the system (33) admits a nontrivial symmetry group. We restrict ourself to the case of a relativelystrong magnetic field, when ω 0(qi)=ω0(qj) implies qi=qj, i.e., the nonmonontonicity of the unperturbed spectrum (36) is irrelevant. The point with the highest symmetry is the center of the Brillouin zone, where q=0. The corresponding symmetry group is G0={Ri,σi},i=0..5, (45) which is isomorphic to the group C6v.H e r e σi=Riσ0, andR andσ0are defined as follows: R:mi=m(Gi)→m(ˆTGi), ˆT=/parenleftbiggcosπ 3−sinπ 3 sinπ 3cosπ 3/parenrightbigg ; (46) σ0:mi=m(Gix,Giy)→m(Gix,−Giy). The characters for the irreducible representations of the group G0are listed in Table II. Due to the presence of the two-dimensional representations B1andB2intersections of two bands appear in the center of the Brillouin zone.Indeed, consider the bands with the numbers from 2 to 7.In the zero-order perturbation theory ω(q=0)=ω 0(G1)i n all these bands. In the first-order perturbation theory we haveto take into account the six components m(ˆT iG1),i=0..5, where the matrix ˆThas been introduced in Eq. (46).T h e solutions of the sixth-order eigenvalue problem can be found 094509-5A. A. BESPALOV AND A. I. BUZDIN PHYSICAL REVIEW B 87, 094509 (2013) TABLE II. Character table of the group G0. R0R,R5R2,R4R3σ0,σ2,σ4σ1,σ3,σ5 A1 11 1 1 1 1 A2 11 1 1 −1 −1 A3 1 −11 −11 −1 A4 1 −11 −1 −11 B1 2 −1 −12 0 0 B2 21 −1 −20 0 in Table III.H e r e b12=B0(G1)/M,b13=B0(G1+G2)/M, b14=B0(2G1)/M. The pairs of bands 3,4 and 5,6 have a point of contact at q=0. Another point of high symmetry is the Bpoint (see Fig. 2), where q=qB=(G1+G2)/3. The corresponding symmetry group is GB=/braceleftbig Ri B,σBi/bracerightbig ,i=0,1,2, (47) where RB:m(Gi)→m(ˆT2Gi−G1), σB0: m(Gi)→m(ˆσOBGi), σBi=Ri BσB0, and ˆσOBis the reflection operator with respect to the OB axis. The group GBis isomorphic to the group C3v. The characters of its irreducible representations are listed in Table IV. The intersection of bands in the Bpoint occurs at frequen- cies close to ω0(qB). In the first-order perturbation theory we take into account the elements m(0),m(−G1), andm(−G2)i n Eq.(33). The solutions of the eigenvalue problem are given in Table V. At this time, the first and second band have a point of contact. Finally, we want to make a remark concerning the symmetry of the initial system (26) to(28). In Sec. II B we derived the approximate equation (33) using the fact that the quantity ˜Kis typically very large. As a by-product of this simplification wegained the reflection symmetry operations σ iandσBi, which are not present in the initial system (to be more accurate,the analogs of σ iandσBiinvolve complex conjugation, so these operations are not linear). The system (26) to(28) forq=0 and q=qBadmits symmetry groups which are isomorphic to the groups C6andC3, respectively, which have only one-dimensional irreducible representations. As a result,a small gap exists between the the bands which had a pointof contact within Eq. (33). However, this gap is negligible (/Delta1∼γMb ij/˜K) for materials with large magnetocrystalline anisotropy, or at large average magnetic fields B0(0). TABLE III. The solutions of Eq. (33) withq=0 in the first-order perturbation theory. Represen- Band Solutions tation numbers m(ˆTkG1)=...ω−ω0(G1) γM A3 2( −1)km(G1) −2b12+2b13−b14 B1 3,4 e±2ikπ/ 3m(G1) −b12−b13+b14 B2 5,6 e±ikπ/ 3m(G1) b12−b13−b14 A1 7 m(G1)2 b12+2b13+b14TABLE IV . Character table of the group GB. R0 B RB,R2 B σBi,i=0,1,2 D1 11 1 D2 11 −1 E 2 −10 E. Numerical calculation of the magnon spectra In this section we give numerical magnon spectra for different average magnetic fields B0(0). We solved Eq. (33) numerically neglecting all Fourier components miexcept for those corresponding to the 31 vectors Giwith the smallest lengths. In the weak-binding approximations, this is sufficientto calculate the spectra in the six lowest frequency bands.As the parameters we used those of UGe 2(Refs. 1and 16): λ=1μm,α=(13.6×10−5)2cm2,K=104, and M= 150 emu /cm3. In all calculations kz=0 is assumed. In Fig. 3we depict the magnon spectra in the lowest three bands for the average magnetic field equal to β=/Phi10√ 3 2π2/parenleftbigg2π α−λ−2/parenrightbigg . (48) At this field ω0(0)=ω0(G1/2). It may be seen that the spectra in the second and third bands have corners. Thesecorners correspond to band intersection lines in zero-orderperturbation theory. In fact, the corners are smoothed out, butthis may be visible only on a small-scale graph. In Fig. 4 two cross sections of the six lowest bands are shown. In thevertical axis label ω FM=γMK is the ferromagnetic resonance frequency. The gaps between some bands are so small thatthese bands are indistinguishable on the graphs, so they arerepresented by one curve. In Figs. 5to8the same spectra for lower magnetic fields are shown. As the field decreases, the shape of the bandstructure changes qualitatively. For example, at the field 0 .25β the smooth maximum in the center of the first band transformsinto a peak [see Fig. 7(a)]. This peculiar behavior the first band is a consequence of the nonmonotonicity of the unperturbedspectrum (11) and is connected with a commensurability effect: The peak appears when ω 0(0)=ω0(G1). F. Taking into account dissipation In this section we will discuss how the magnon spectrum is modified when dissipation is taken into account. First, we TABLE V . The solutions of Eq. (33) withq=qBin the first-order perturbation theory. Represen- Bandtation numbers Solutions ω−ω0(qB) γM E 1,2 m(−G1)=e±2πi/3m(0), −b12 m(−G2)=e∓2πi/3m(0) D1 3 m(−G1)=m(−G2)=m(0) 2 b12 094509-6BAND STRUCTURE OF MAGNETIC EXCITATIONS IN THE ... PHYSICAL REVIEW B 87, 094509 (2013) (a) (c)(b) FIG. 3. (Color online) The magnon spectra in the (a) first, (b) second, and (c) third bands. B0(0)=β. consider losses due to vortex motion. Generally, it is rather difficult to express the displacement amplitude /Delta1Rin terms of m(Gi), using Eq. (27). However, a simplification is possible (a) (b) FIG. 4. The cross sections of the lowest six bands along the lines (a)OC and (b) OB.B0(0)=β.(a) (c)(b) FIG. 5. (Color online) The magnon spectra in the (a) first, (b) second, and (c) third bands. B0(0)=0.3β. in the high- ηlimit, when all terms in the right-hand side of Eq.(27) containing /Delta1Rcan be neglected as compared to the relaxational term in the left-hand side. For this approximation (a) (b) FIG. 6. The cross sections of the lowest six bands along the lines (a)OC and (b) OB.B0(0)=0.3β. 094509-7A. A. BESPALOV AND A. I. BUZDIN PHYSICAL REVIEW B 87, 094509 (2013) (a) (c)(b) FIG. 7. (Color online) The magnon spectra in the (a) first, (b) second, and (c) third bands. B0(0)=0.25β. to be valid it is sufficient to demand ωη /Phi10/greatermuchB0(0)λ−2,H c1k2 z,H c1k2, (49) (a) (b) FIG. 8. The cross sections of the lowest six bands along the lines (a)OC and (b) OB.B0(0)=0.25β.where Hc1=/Phi10 4πλ2lnλ ξ is the first critical field. The conditions (49) can be satisfied in the clean limit: It is known that the viscosity increases withincreasing normal state conductivity. 19 Within our approximation /Delta1R=/summationdisplay Gi<ξ−1/Phi10kzm(Gi) ωη/bracketleftbig 1+λ2/parenleftbig k2z+q2 i/parenrightbig/bracketrightbig. (50) After substituting this into Eq. (28) we can repeat the calculations from Sec. II B and obtain the system (33) with ω0(q,kz)=γM/parenleftbigg ˜K+αq2−2π/parenleftbig 2k2 z+q2/parenrightbig λ−2+k2z+q2 −iB0(0)/Phi10k2 z ωη/parenleftbig 1+λ2/parenleftbig k2z+q2/parenrightbig/parenrightbig2/parenrightbigg , bij=B0(Gi−Gj) M−iB0(0)/Phi10k2 z ωη ×1/bracketleftbig 1+λ2/parenleftbig k2z+q2 i/parenrightbig/bracketrightbig/bracketleftbig 1+λ2/parenleftbig k2z+q2 j/parenrightbig/bracketrightbig. Equation (33) now represents an eigenvalue problem for a symmetric non-Hermitian matrix. Due to dissipation themagnetic excitation levels are broadened, which can leadto the smearing of the gaps between the energy bands. Toobserve the effects connected with the presence of the gap/Delta1 ijforqi≈qjwe have to provide that B0(Gi−Gj) M/greaterorsimilarB0(0)/Phi10k2 z ωη/bracketleftbig 1+λ2/parenleftbig k2z+q2 i/parenrightbig/bracketrightbig2. (51) This means that the viscosity should be sufficiently large, or the longitudinal wave number kzshould be small so that vortex motion is not excited. In metallic ferromagnets another important mechanism of dissipation exists, which is due to magnon-conduction electronscattering. 20This kind of dissipation can be qualitatively taken into account by introducing a phenomenological damping termin the right-hand side of the Landau-Lifshitz equation (4) 21 /parenleftbigg∂M ∂t/parenrightbigg damp=−γνM×∂M ∂t, (52) where νis a relaxation constant defining the magnetization relaxation time τ=(γνMω )−1. The mentioned mechanism of dissipation does not smear out the gaps in the magnonspectrum if τ −1/lessorsimilarγB0(Gi−Gj). (53) Data on the relaxation time τin the U-based ferromagnetic superconductors are not available yet. The typical theoreticaland experimental values for this quantity in ordinary metallicferromagnets are 10 −9–10−8s(see Chap. 5 in Ref. 20and references therein). We can estimate the Fourier component ofthe vortex field as B 0(Gi−Gj)∼/Phi10/λ2. 094509-8BAND STRUCTURE OF MAGNETIC EXCITATIONS IN THE ... PHYSICAL REVIEW B 87, 094509 (2013) FIG. 9. An electromagnetic wave ( k) incident on the flat surface of a ferromagnetic superconductor is partially reflected back as a wave with the wave vector k1. Inside the material one propagating ( q1)a n d two decaying magnon modes ( q2andq/prime 2) are excited. The mode q1 undergoes Bragg reflection on the vortex lattice (represented by dots) and transforms into the mode q3propagating towards the sample surface. Hence, the frequency gap /Delta1=2γB0(Gi−Gj) is of the order of 10−8s−1. Thus, the relaxation rate τ−1and the gap /Delta1appear to be of the same order of magnitude. III. MICROWA VE PROBING OF THE BAND STRUCTURE In this section we will demonstrate how the evidence of the gaps in the magnon spectrum can be found using microwaveprobing. Consider an electromagnetic Transverse Electric (TE)wave with the wave vector kand amplitude H 0incident on a ferromagnetic superconductor occupying the half-space x> 0 (see Fig. 9). For simplicity, we assume kz=0. Note that in a Transverse Magnetic (TM) wave the field Hwould oscillate along the direction of the uniform magnetization, hence, thiswave does not excite magnons and is totally reflected. For thisreason, we consider further a TE wave. Such a wave excitesthree magnon modes inside the ferromagnet: one propagating(q 1) and two decaying modes ( q2andq/prime 2). The wave vectors of these modes are determined from the two equations ω2=ω2 0(q)=γ2M2/parenleftbigg K+B0(0) M+αq2/parenrightbigg ×/parenleftbigg K+B0(0) M+αq2−4πq2 q2+λ−2/parenrightbigg , (54) q2 x=q2−k2 y, (55) The propagating mode can be reflected back to the surface of the ferromagnet due to Bragg scattering on vortices, if twoconditions are fulfilled for some wave vector q 3 q3=q1+G,ω 0(q3)≈ω0(q1), where G=−Gx0is a vector of the reciprocal lattice (16).We will determine the amplitude H1of the reflected electromagnetic wave. For the evaluation of this amplitude theequilibrium field distributiton B eq(r) in the material is required Beq=4πMz0e−x/λ+B0(r−xvx0)+B/prime 0(r). (56) Here, the first term represents the screened intrinsic magnetic field (we assume that there is no constant external field He= 0),B0is the vortex field given by Eq. (15),xvspecifies the shift of the vortex lattice with respect to the surface, and theterm B /prime 0(r) is responsible for the vortex lattice distortion in a surface layer with a thickness of the order of λ. We consider a dense vortex lattice, so that a/lessmuchλ.T o observe the effects connected with Bragg reflection ofmagnons we have to demand q 1∼a−1, hence, αq2 1/greatermuch1. The nonstationary component of the magnetization can bepresented in the form m≈m 1(x)eiq1r+m2eiq2r+m/prime 2eiq/prime 2r+m3(x)eiq3r, (57) where m1(x) and m3(x) vary slowly in space. In the Appendix, using a simple perturbation theory, we demonstrate that theinfluence of the screened intrinsic field on the magnon modesis not essential. By similar reasons, the distortion field B /prime 0(r) also does not affect significantly the spin wave amplitudes.Hence, we can consider m 2andm/prime 2to be constant. Now we write down the boundary conditions. Directly from Eq.(4)we obtain ∂m ∂x(x=0)=0, (58) The continuity condition for the tangential component of the magnetic field Hreads (H0+H1) cosβ=−4π/summationdisplay i/parenleftbig λ−2+k2 y/parenrightbig miy(0)+kyqixmix(0) q2 i+λ−2, (59) where summation is performed over all four modes. The electric field inside the material is e=− 4πk/summationdisplay iqi×mi q2 i+λ−2eiqir. (60) The continuity condition for the electric field reads H1−H0=4πk/summationdisplay ikymix(0)−qixmiy(0) q2 i+λ−2. (61) The wave number q/prime 2has a large modulus ( q/prime2 2≈− 2ω/γMα ) as compared to the other wave numbers, and the correspondingmagnetization component m /prime 2is small. It can be neglected in Eqs. (59) and(61). To exclude m/prime 2from Eq. (58), we note that m/prime 2x≈im/prime 2y, hence 3/summationdisplay i=1qix(mix−imiy)=0, or 3/summationdisplay i=1qixmiy=0, (62) 094509-9A. A. BESPALOV AND A. I. BUZDIN PHYSICAL REVIEW B 87, 094509 (2013) since mix≈−imiyfori=1,2,3. Now we have to find a connection between m1and the amplitude of the Bragg- reflected mode m3. In these modes the magnetic field is small as compared to α∇2m, so the linearized Landau-Lifshitz equation can be simplified as follows: ω γMm=˜Km−α∇2m+1 MB0(r−xvx0)m. (63) To find the link between the mentioned modes it is sufficient to conserve only two terms in the Fourier series of the vortexfield B 0(G)(eiG(x−xv)+e−iG(x−xv)). By substituting m=m1(x)eiq1r+m3(x)eiq3rinto Eq. (63) and neglecting the second derivatives of m1(x) and m3(x), we obtain ivgx∂m1 ∂x=/Delta1 2e−iϕ−iδxm3(x), (64) ivgx∂m3 ∂x=−/Delta1 2eiϕ+iδxm1(x), (65) where vgx=∂ω0 ∂qx(q1)=2γMαq 1x, /Delta1=2γB(G) is the frequency gap [see Eq. (43)],ϕ=Gxv, andδ=2q1x−G. The two linearly independent solutions of Eqs. (64) and(65) are m1(x)=(x0+iy0)e(/epsilon1−iδ/2)x, m3(x)=2ivgxeiϕ /Delta1/parenleftbigg /epsilon1−iδ 2/parenrightbigg (x0+iy0)e(/epsilon1+iδ/2)x,(66) /epsilon1=±1 2/radicalBigg /Delta12 v2gx−δ2. For|δ|</Delta1 / v gxwe reject the growing solution, selecting the minus sign in Eq. (66).F o r|δ|>/Delta1 / v gxwe select the solutionwhere |m1|/greatermuch|m3|when|δ|/greatermuch/Delta1/v gx /epsilon1=i 2/radicalBigg δ2−/Delta12 v2gxforδ> 0, (67) /epsilon1=−i 2/radicalBigg δ2−/Delta12 v2gxforδ< 0. This choice of the sign allows to reject the solution with the negative xcomponent of the group velocity. At x=0w e have m3(0)=Am1(0), (68) A=2ivgxeiϕ /Delta1/parenleftbigg /epsilon1−iδ 2/parenrightbigg . (69) Now we are ready to write the system of linear equations which will allow us to determine the amplitude of the reflected waveH 1. Equations (59),(61),(62), and (68) yield ˜q1x˜m1y+q2xm2y=0, (70) 4πiky˜q1x−λ−2−k2 y q2 1+λ−2˜m1y+4πikyq2x−λ−2−k2 y q2 2+λ−2m2y =(H0+H1) cosβ, (71) −4πkiky+˜q1x q2 1+λ−2˜m1y−4πkiky+q2x q2 2+λ−2m2y=H1−H0,(72) where ˜m1y=(A+1)m1y,˜q1x=1−A 1+Aq1x. (73) Note that the problem of electromagnetic wave reflection from a superconductor in the mixed state is formally equivalentto the same problem for a superconductor without vorticeswith the only difference that q 1xis replaced by ˜q1x.F r o mt h e systems (70) to(72) we find the reflection coefficient R=H1 H0=λ−2q2x q2 1+λ−2−α(q2 1+λ−2) 2π˜q1x+e−iβ/parenleftbigg kq2xiky+˜q1x q2 1+λ−2−k˜q1x(iky+q2x)α(q2 1+λ−2) 2πλ−2/parenrightbigg λ−2q2x q2 1+λ−2−α(q2 1+λ−2) 2π˜q1x−eiβ/parenleftbigg kq2xiky+˜q1x q2 1+λ−2−k˜q1x(iky+q2x)α(q2 1+λ−2) 2πλ−2/parenrightbigg, (74) w h e r ew eu s e dt h er e l a t i o n αq2 1−2πq2 1 q2 1+λ−2=αq2 2−2πq2 2 q2 2+λ−2, which is valid in the large anisotropy limit. The expression for the reflectivity coefficient can be simplified, if we take intoaccount that q 1/greatermuchλ−1andq2x≈iλ−1 R=1+ik2λ2e−iβsinβ+Q(i−kλe−iβ) 1−ik2λ2eiβsinβ+Q(i+kλeiβ), (75) where Q=αλ2q4 1 2πq1xλ1−A 1+A. (76)In Eq. (75) we dropped small terms which have a negligible effect on the modulus of the reflectivity coefficient. Since thequantity Acan take any value within the circle |A|/lessorequalslant1, so the only restriction on Qis ReQ/greaterorequalslant0. When |A|=1,Qis purely imaginary, and |R|=1, i.e., the wave is totally reflected. This is explained by the factthat in this range of parameters the frequency ωis within the frequency gap, and magnons cannot propagate in the sample. Consider now frequencies far from the gap: δ/greatermuch/Delta1/v gx.I n this case the magnons do not interact with the vortex lattice,and the quantity Qis real and large: Q/greatermuch1. From Eq. (75) we obtain 1−|R| 2=4kλcosβ Q/lessmuch1. (77) 094509-10BAND STRUCTURE OF MAGNETIC EXCITATIONS IN THE ... PHYSICAL REVIEW B 87, 094509 (2013) An interesting effect which follows from Eq. (75) is the complete transmission of the wave for a frequency close to thefrequency gap. Let us put R=0. Then Q≈i+kλe −iβ. (78) This is possible when A≈1 and |A|<1, i.e., the detuning from the gap must be very small. For example, if kαλ4q5 1/greatermuch1 |δ|vgx /Delta1−1≈8π2k2cos2β α2λ4q10 1. (79) Note that for A≈1m1(0)≈m3(0). The effect of complete transmission is related to a similar effect in a Fabri-Perrotresonator: In our system, the surface of the material and thevortex lattice play the roles of the first and second mirrors,respectively. We need to stress that the system must be finely tuned to make the dip in the R(ω) dependence observable. Indeed, the parameter Amust be equal to unity on the border of the gap, which imposes a constraint on the parameters Gandx v: eiGxv=± 1. (80) This condition may be satisfied by applying an external magnetic field. IV . CONCLUSION By solving the London and Landau-Lifshitz equations we investigated the magnon spectrum of a ferromagneticsuperconductor in the mixed state. The case of a large easy-axismagnetocrystalline anisotropy has been considered, whichis relevant to the U-based compounds. 1,2,4We proved that the magnon spectrum has a Bloch-like band structure dueto the presence of the periodic vortex field. For sufficientlysmall intervortex distances a/lessorsimilarλ, the gaps between adjacent bands can be calculated using an analog of the weak-bindingapproximation. These gaps are proportional to the Fouriercomponents B 0(G) of the unperturbed vortex field [see Eq.(43)]. If the material has isotropic properties in the plane perpendicular to the easy zaxis, some bands may intersect in points of high symmetry of the Brillouin zone. For spinwaves having a nonzero zcomponent of the wave vector the band structure is smeared out in dirty materials because ofdissipation connected with viscous vortex motion. Using numerical calculations, we demonstrated that the band structure changes qualitatively with varying applied field.For example, when the average field becomes smaller than β/4 [see Eq. (48)] the smooth maximum in the center of the lowest band transforms into a rather sharp peak (see Figs. 3to7). This effect appears due to the nonmonotonicity of the spectrum inthe Meissner state.We propose to probe the energy gaps by measuring the frequency dependence of the reflectivity coefficient R(ω)f o r an electromagnetic wave incident on the flat surface of aferromagnetic superconductor. For frequencies lying insidethe gap a maximum of |R(ω)|should be observable. Also, for a very small detuning from the gap, the reflectivitycoefficient may exhibit a narrow dip. The knowledge of thegap frequencies allows to determine with high accuracy suchparameters as the anisotropy coefficient Kand the exchange constant α. Finally, note that the self-induced vortex state in ferromag- netic superconductors may provide a negative permeability atfrequencies near the ferromagnetic resonance, thus makingthe ferromagnetic superconductors potential candidates formetamaterials design. ACKNOWLEDGMENTS We are thankful to A. S. Mel’nikov and L. N. Bulaevskii for helpful discussion. This work was supported, in part, byEuropean IRSES program SIMTECH (Contract No. 246937),the Russian Foundation for Basic Research, FTP “Scientificand educational personnel of innovative Russia in 2009-2013”,the French ANR program MASH, and LabEx “Amadeus”program. APPENDIX In this Appendix we consider the modification of the mode with the wave vector q2in the vicinity of the surface. This consideration can be simply generalized for all other modes. First, we note that q2 2≈−λ−2andq2x≈iλ−1since|ky|/lessmuch λ−1(this inequality holds for frequencies, which are much smaller than the plasma frequency). We substitute into Eq. (17) B0=4πMe−x/λ, andm≈m2e−x/λ+m(1) 2e−2x/λ, where m(1) 2 is the amplitude of the first-order correction, which is to be estimated. This correction is determined from the followingequations: −iω γMm(1) 2+(K−4αλ−2)m(1) 2×z0=/parenleftbig b(1) 2−4πm2/parenrightbig ×z0, −3 λ2b(1) 2=−16π λ2m(1) 2yy0.(A1) Forω≈γM(K+αq2 1),αq2 1/greatermuch1 we find m(1) 2≈4π αq2 1m2. (A2) Hence, |m(1) 2|/lessmuch|m2|, so the small correction can be neglected. 1S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R. Julian,P. Monthoux, G. G. Lonzarich, A. Huxley, I. Sheikin,D. Braithwaite, and J. Flouquet, Nature (London) 406, 587 (2000). 2Dai Aoki, Andrew Huxley, Eric Ressouche, Daniel Braithwaite, Jacques Flouquet, Jean-Pascal Brison, Elsa Lhotel, and CarleyPaulsen, Nature (London) 413, 613 (2001).3T. C. Kobayashi, S. Fukushima, H. Hidaka, H. Kotegawa, T. Akazawa, E. Yamamoto, Y . Haga, R. Settai, and Y . Onuki,Physica B 378-380 , 355 (2006). 4N. T. Huy, A. Gasparini, D. E. de Nijs, Y . Huang, J. C. P. Klaasse, T. Gortenmulder, A. de Visser, A. Hamann, T. G ¨orlach, and H. v. L¨ohneysen, Phys. Rev. Lett. 99, 067006 (2007). 5J. Floquet and A. Buzdin, Phys. World 15, 41 (2002). 094509-11A. A. BESPALOV AND A. I. BUZDIN PHYSICAL REVIEW B 87, 094509 (2013) 6V . Braude and E. B. Sonin, P h y s .R e v .L e t t . 93, 117001 (2004). 7A. I. Buzdin, JETP Lett. 40, 956 (1984) [Pis’ma v ZhETF 40, 193 (1984)]. 8V . Braude and E. B. Sonin, Europhys. Lett. 72, 124 (2005). 9V . Braude, P h y s .R e v .B 74, 054515 (2006). 10N. A. Logoboy and E. B. Sonin, P h y s .R e v .B 75, 153206 (2007). 11L. N. Bulaevskii, M. Hru ˇska, and M. P. Maley, Phys. Rev. Lett. 95, 207002 (2005). 12A. Shekhter, L. N. Bulaevskii, and C. D. Batista, Phys. Rev. Lett. 106, 037001 (2011). 13Shi-Zeng Lin and Lev N. Bulaevskii, Phys. Rev. B 85, 134508 (2012). 14S.-Z. Lin, L. N. Bulaevskii, and C. D. Batista, Phys. Rev. B 86, 180506(R) (2012).15T. K. Ng and C. M. Varma, P h y s .R e v .B 58, 11624 (1998). 16Vu Hung Dao, Sebastien Burdin, and Alexandre Buzdin, Phys. Rev. B84, 134503 (2011). 17A. B. Shick, Phys. Rev. B 65, 180509 (2002). 18N. T. Huy, D. E. de Nijs, Y . K. Huang, and A. de Visser, Phys. Rev. Lett.100, 077002 (2008). 19V . V . Schmidt, The Physics of Superconductors: Introduc- tion to Fundamentals and Applications (Springer-Verlag, Berlin, 1997). 20S . V. Vo n s ov s k i i , Ferromagnetic resonance: The phenomenon of resonance absorption of HF electromagnetic field in ferromagneticmaterials (Israel Program for Scientific Translations, Jerusalem, 1964). 21T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 094509-12

PhysRevE.92.012923.pdf

PHYSICAL REVIEW E 92, 012923 (2015) Current-driven electromagnetic soliton collision in a ferromagnetic nanowire M. Saravanan* Department of Physics, Saveetha School of Engineering, Saveetha University, Chennai-602 105, Tamilnadu, India (Received 4 December 2014; revised manuscript received 14 May 2015; published 31 July 2015) The propagation of an electromagnetic wave in a uniaxial ferromagnetic nanowire under the spin transfer torque effect is widely investigated in the soliton frame. The magnetization dynamics of the ferromagneticnanowire is governed by the Landau-Lifshitz-Gilbert (LLG) equation coupled to the Maxwell equation for theelectromagnetic wave propagation. A nonuniform multiscale analysis is invoked for the coupled LLG-Maxwellequations and obtains the extended derivative nonlinear Schr ¨odinger (DNLS) equation for the magnetization and external magnetic field. The effect of electric current is explored by constructing multisoliton solutions to theextended DNLS equation and the possibility of the soliton collision is exploited using the Hirota bilinearizationprocedure. DOI: 10.1103/PhysRevE.92.012923 PACS number(s): 05 .45.Yv,03.75.Lm,72.25.Ba,02.70.−c I. INTRODUCTION Current-driven magnetic materials are highly important in constructing technological devices [ 1,2]. The proper manipu- lation of magnetization direction driven by electric current ishighly promising for ultrahigh-density storage devices [ 3,4]. The magnetization dynamics in ferromagnetic materials as-sumes more importance and the magnetic property can beconsiderably enhanced in the environment of electric currentdensity. Current-induced magnetic materials have been foundto be a good alternative for investigating materials when exposed to a magnetic field alone. Magnetization reversal under electric current shows an effective response in additionto the applied magnetic field to construct compact andreliable memory devices [ 5–7]. The magnetization dynamics in the presence of an electric current was first predicted byBerger and Slonczewski in a spin valve structure [ 8,9]. The current-induced magnetization dynamics occurs in magneticconductors through the transfer of spin angular momentumbetween the conducting electrons and the local magnetizationmediated by the exchange interaction. Due to the transferof spin angular momentum between the electrons and localmagnetic moments, an effective torque is developed, namely,spin transfer torque (STT) [ 10]. Tatara et al. proposed a metamaterial active for the electromagnetic (EM) wavepropagation based on the spin torque oscillators [ 11]. It is shown that the spin torque oscillators act as an active filter toobtain the circularly polarized radiation and electromagneticmetamaterial admitting a negative refractive index controlledby the electric current. By considering the work of Tataraet al. , it is believed that the propagating EM wave in the spin transfer torque ferromagnet deserves considerable attention inthe construction of magnetic memory devices. The propagation of the EM wave in the ferromagnetic medium has been studied for the past two decades [ 12–16] particularly for soliton excitations. The phenomenologicalLandau-Lifshitz-Gilbert (LLG) equation is studied with theeffect of EM wave propagation in the ferromagnetic medium.By invoking a multiscale analysis, typical Korteweg–de Vries(KdV) soliton modes for the EM wave modulation in the *saravanan_manickam@yahoo.comferromagnetic medium is observed [ 12]. A blowup solution, stabilization of the EM pulse, and the pulse’s deformationhave been demonstrated in [ 13] through numerical compu- tation for the EM wave governed by the (3 +1)-dimensional Davey-Stewartson system. The pivotal role of the nonlinearSchr ¨odinger equation and its extended forms shows that the generating solitons cancel the effect of damping for the case oflow damping and for higher values of the damping parameterit is observed that the propagating EM wave has exponentialdecay and no nonlinear modulation is observed [ 15]. In the presence of free charges, the current density createsan effective damping in the ferromagnetic material and theMaxwell equations include the current density effect onthe propagating EM wave [ 16]. The damping considerably reduces the amplitude of the soliton that is governed bythe perturbed KdV equation for the strong damping caseand the usual KdV equation in the case of weak damping.Recently the present author has rigorously solved the coupledLandau-Lifshitz and Maxwell equations for the helimagneticspin system witnessing the propagation of a kink soliton forthe magnetization and EM wave [ 17]. The spin torque effects in broken centrosymmetric crystals such as the cubic B20 type are computed for current-induced magnetization dynamics ofspin spirals and Doppler shifts in spin waves. The torquedeveloped tilts the helimagnetic structure of the crystal anddisplaces the spin waves of the spiral structure with no changesin the dispersion relation [ 18]. Motivated by the above complex scenario arising in the EM wave dynamics in the ferromagnet,in this paper we extend the idea of a propagating EM wavein the ferromagnetic medium driven by the electric currentdensity. In the present investigation the evolution equation issolved for the perturbation due to the propagating EM wave inthe absence of Gilbert damping and the collision scenario ofthe magnetization in the concerned system is exploited. Thecollision dynamics is studied with the real material parametersof CoPt 3. The Gilbert damping parameter is excluded because it is a phenomenological damping and a choice of study to lookfor localized excitations. He and Liu [ 19] explored the soliton dynamics in the STT system under the effect of an externalmagnetic field using the stereographic projection method. Thesoliton solution of the LLG equation without damping showsthat the current can change the velocities of the magneticsolitons and affect the soliton collision with appreciable phase 1539-3755/2015/92(1)/012923(8) 012923-1 ©2015 American Physical SocietyM. SARA V ANAN PHYSICAL REVIEW E 92, 012923 (2015) shift. The scheme of this paper is organized as follows. In Sec. IIthe dynamical model is briefly introduced. In Sec. III a multiscale analysis is performed on the dynamical modelderived. The magnetization collision in the ferromagneticnanowire is illustrated in Sec. IV. The results are summarized in Sec. V. II. LANDAU-LIFSHITZ-GILBERT AND MAXWELL EQUATIONS The dynamics of localized magnetization under the influ- ence of electric current density is described by the celebratedLLG equation with the spin transfer torque term as given by ∂M ∂t=−γM×Heff+α MSM×∂M ∂t+τb, (1a) τb=bJ∂M ∂x, (1b) where the magnetization M≡M(x,t),αis the Gilbert damp- ing, and γis the gyromagnetic ratio. The model parameter bJis defined as bJ=PjeμB/eM S, where Pis the spin polarization of the current, μBis the Bohr magneton, eis the charge of the electron, and jeis the electric current density. The parameter bJhas the unique property that it constitutes the velocity [ 20], thus leading τbto associate with the domain wall velocity, and Heffis the effective field in the ferromagnetic nanowire, which can be written as Heff=2A M2 S/parenleftbigg∂2M ∂x2/parenrightbigg +[(HK/MS−4π)Mx]ex+H, (2) where exis the unit vector along the easy axis, HKis the uniaxial anisotropy parameter, 4 πMxis the demagnetization field,Ais the exchange integral, and the external magnetic field Hacts via the magnetic field component of the propagating EM wave. The external magnetic field satisfies the followingMaxwell equation. Let the ferromagnetic wire be driven byelectric current along the xdirection, which is also the direction of the easy axis with a constant velocity V=v eex, with vea constant that indicates the speed of moving electric charges.The geometry of the model considered is shown in Fig. 1.T h e Maxwell equation governing the propagation of the EM wavein the ferromagnetic medium for the present case is given by ∇×E=−∂B ∂t, (3a) ∇×H=J+/epsilon10∂E ∂t, (3b) where J=σ[E+(V×B)] is the electron current density, /epsilon10 is the dielectric constant, and μ0is the magnetic permeability of the medium. Equation ( 3b) can be modified by taking the curl on both sides and using Eq. ( 3a) and velocity vector Vto get −∇(∇·H)+∇2H=σ/bracketleftbigg μ0∂ ∂t(H+M)+(V·∇)(H+M)/bracketrightbigg +/epsilon10μ0∂2 ∂t2(H+M). (4)Thus, the set of coupled equations ( 1a) and ( 4) describes the magnetization dynamics of the ferromagnetic nanowire andpropagating EM wave with spin transfer torque. In the nextsection, the set of coupled equations ( 1a) and ( 4) is solved for the perturbation with zero damping. III. MULTISCALE ANALYSIS AND EXTENDED DERIVATIVE NONLINEAR SCHR ¨ODINGER EQUATION The dynamics of EM wave modulation and the magneti- zation in the ferromagnetic nanowire with spin torque can bededuced by solving the set of coupled equations ( 1a) and ( 4) through the multiscale analysis. The multiscale analysis orreductive perturbation method is the generalized asymptoticmethod to reduce a highly nonlinear vector or scalar equation toa more standard nonlinear equation comprising the integrabil-ity conditions [ 21,22]. It is a standard procedure in perturbation technique to transform the original coordinate variables intoslow variables by making use of the perturbation parameter.In that new coordinate system, the effect of perturbation canbe well understood and one can determine the perturbationeffect in an appropriate way. Let us seek the dynamicalevolution in a slow space and time coordinate frame bystretching the wave variable as ζ=/epsilon1(x−V st) andτ=/epsilon12t, where /epsilon1is a small perturbation that measures the weakness of the perturbation [ 12] and ζsignifies the evolution of the propagating magnetization pulse with velocity Vs. Before employing the multiscale analysis, the following assumptionsare made. (i) The ferromagnetic nanowire is immersed in a constant exterior field H extrelevant to the ferromagnetic resonance experiment. (ii) The external field Hextis strong enough to magnetize the ferromagnetic nanowire to saturation, which allows us toexclude the existence of domain walls. This saturation alsoexcludes the effects of the finite size of the nanowire. (iii) The static field M 0andH0given in the expansion below represents the field created by the Hextinside the medium. FIG. 1. Schematic representation of the propagating EM wave in the ferromagnetic nanowire. The propagating magnetization pulse is along the xdirection, with velocity Vsand electric charges in the −x direction. 012923-2CURRENT-DRIVEN ELECTROMAGNETIC SOLITON . . . PHYSICAL REVIEW E 92, 012923 (2015) (iv) In the present study, the analysis is restricted up to the lowest order, i.e., at O(/epsilon11), even though at higher order the dynamics would be more interesting. Let the magnetization Mand magnetic field Hbe expanded with the perturbation parameter /epsilon1: Mx=M0+/epsilon1Mx 1+/epsilon12Mx 2+··· , (5a) My=/epsilon11/2/bracketleftbig My 1+/epsilon1My 2+/epsilon12My 3+···/bracketrightbig , (5b) Mz=/epsilon11/2/bracketleftbig Mz 1+/epsilon1Mz 2+/epsilon12Mz 3+···/bracketrightbig , (5c) Hx=H0+/epsilon1Hx 1+/epsilon12Hx 2+··· , (6a) Hy=/epsilon11/2/bracketleftbig Hy 1+/epsilon1Hy 2+/epsilon12Hy 3+···/bracketrightbig , (6b) Hz=/epsilon11/2/bracketleftbig Hz 1+/epsilon1Hz 2+/epsilon12Hz 3+···/bracketrightbig . (6c) In these equations M0andH0are unperturbed states of the corresponding fields about which the magnetization andexternal fields are expanded. The perturbation expansion isnonuniform due to the anisotropic electric current alongthexdirection in which the magnetization is nonuniform. We report the above perturbation expansion in Eqs. ( 1a) and ( 4) and collect the different orders of the perturbationparameter without damping. We also consider the conductivity of the nanowire to be small, say, σ=/epsilon1 2σ. Let us assume the exchange coupling to be stronger than the anisotropyand demagnetization fields that can be satisfied by rescalingthe respective fields by A→/epsilon1 −1Aand (HK/MS−4π)→ /epsilon1(HK/MS−4π). Thus, at order /epsilon10we have Hr 1=λMr 1 withr=y,z,λ=V2 s/(c2−V2 s), and H0=−M0from the Maxwell equation ( 4) and at the corresponding order in Eq. ( 1a) perturbed fields are identically satisfied. At the next order/epsilon11from the Maxwell equation ( 4) the perturbed evolution of magnetization and external magnetic field is given by ∂ ∂ζ/bracketleftbig Hr 2−λMr 2/bracketrightbig =∂Mr 1 ∂τ−λ1Mr 1, (7) while writing Eq. ( 7), the τis rescaled as τ→[2Vs(1+ λ)/(V2 s−c2)]τandλ1is given by λ1=σμ 0(1+λ)(ve−Vs)c2 /parenleftbig V2s−c2/parenrightbig. (8) Collecting at the same order in Eq. ( 1a) the solvability conditions leads to −Vs∂Mx 1 ∂ζ=g1/parenleftbigg My 1∂2Mz 1 ∂ζ2−Mz 1∂2My 1 ∂ζ2/parenrightbigg −γ/bracketleftbigg My 1/integraldisplayζ −∞/parenleftbigg∂Mz 1 ∂τ−λ1Mz 1/parenrightbigg dζ/prime−Mz 1/integraldisplayζ −∞/parenleftbigg λ2∂My 1 ∂τ−λ1My 1/parenrightbigg dζ/prime/bracketrightbigg ,(9a) −Vs∂My 1 ∂ζ=−g1M0∂2Mz 1 ∂ζ2+g2M0Mz 1+γ/bracketleftbigg (1+λ)Mz 1Mx 1+M0/integraldisplayζ −∞/parenleftbigg∂Mz 1 ∂τ−λ1Mz 1/parenrightbigg dζ/prime/bracketrightbigg +bJ∂My 1 ∂ζ, (9b) −Vs∂Mz 1 ∂ζ=g1M0∂2My 1 ∂ζ2−g2M0My 1−γ/bracketleftbigg (1+λ)My 1Mx 1+M0/integraldisplayζ −∞/parenleftbigg∂My 1 ∂τ−λ1My 1/parenrightbigg dζ/prime/bracketrightbigg +bJ∂Mz 1 ∂ζ, (9c) where we have defined g1=2A/M2 Sandg2=HK/MS−4π. In the low-temperature and long-wavelength limit, the magne-tization of the ferromagnetic nanowire is constant and from thefirst integral of motion, i.e., |M| 2≡M2 S≡M2 0, we define new complex functions u=My 1−iMz 1andMx 1=− |u|2, With the transformations ξ=ζ+g2τandτ/prime=τ, we obtain i∂u ∂τ/prime+a1∂2u ∂ξ2+ia2∂ ∂ξ|u|2u=iR(u), (10) where R(u)=a3∂3u ∂ξ3+a4u, (11a) a1=Vs+bJ γM 0,a 2=−(1+λ) M0, (11b) a3=−g1,a 4=λ1. Equation ( 10) represents the extended derivative nonlinear Schr ¨odinger (DNLS) equation that governs the lowest-order magnetization of the ferromagnetic nanowire. The effect ofSTT appears in the usual dispersion term of the DNLS equationas it contains the spin polarization factor P, a similar DNLS equation already obtained in [ 23] without the term ia 4u.I t is interesting to note when R(u)=0, we have the exact DNLS equation derived by Kaup and Newell [ 24]. Whena4=0, Eq. ( 10) reduces to a DNLS equation with third-order dispersion solved by the present author recently [ 23]. More- over,a1=a4=0 reduces the DNLS equation to a complex modified Korteweg–de Vries equation for the effective field u governing the propagation of nelectromagnetic soliton pulses as signals with different frequencies [ 25,26]. IV . ELECTROMAGNETIC SOLITON COLLISION UNDER ELECTRIC CURRENT DENSITY Hirota bilinearization is a direct method for constructing multisoliton solutions [ 27]. The primary advantage of this method is that one can easily derive the soliton solutions bytransforming the original equation into a bilinear form andfurther applying the perturbation technique. The complexityinvolved in Eq. ( 10) is the presence of ia 4u. The electric current density induces the relevant spin transfer torque thatappears in the coefficient of the linear dispersion term and theconductivity of the nanowire appears as the coefficient in thesecond term on the right-side of Eq. ( 10) through Ampere’s correction in Maxwell’s equations ( 3b). The role played by ia 4uis crucial in resolving the dynamics as it is complex and cannot be eliminated through the usual rescaling of the timevariable. In order to exploit the dynamics in a more significantway we recast Eq. ( 10) into a more generalized nonlinear equation that is more compact to apply bilinearization. The 012923-3M. SARA V ANAN PHYSICAL REVIEW E 92, 012923 (2015) modified equation can be written as i/Psi1T+a0/Psi1XX+a2|/Psi1|2/Psi1+ia2∂ ∂X|/Psi1|2/Psi1 =ia3/Psi1XXX+ia4/Psi1, (12) where the solutions u(ξ,τ/prime) and/Psi1(X,T) are connected through the transformation u(ξ,τ/prime)=/Psi1(X,T)e−i[ξ−(a3−a1)τ/prime], (13) X=ξ+(3a3−2a1)τ/prime,T=τ/prime and the coefficient a0=a1−3a3. Equation ( 13)i st h e higher-order nonlinear Schr ¨odinger (HONLS) equation that represents the dynamics of magnetization in the nanowirewhen the EM wave propagates through it and admits thesoliton solution as given in [ 28]. In another context without the conductivity term, i.e., a 4=0, Eq. ( 12) represents the soliton propagation in nonlinear fiber optics and in a particular caseit is integrable and possesses N-soliton solutions [ 29]. For instance, the one-soliton solution is given by /Psi1=/radicalBigg 2a3 a2bsech[b(X−VsT)]ei(κX−/Omega1T), (14a) Vs=κ−3a3κ2+a3b2,κ=a2−2a3 4a2a3, (14b) /Omega1=(1/2)κ2−a3κ2−(1−6κa3)b 2, (14c) where bis the inverse width of the soliton. Thus, it is expected that in the present case the magnetization dynamics is governedby soliton modes under the effect of EM wave propagation.To construct the multisoliton solutions of Eq. ( 12), we make an ansatz in the form of a dependent variable transformationas given by /Psi1(X,T)≡e a4Tg(X,T) f(X,T). (15) In the above ansatz g(X,T) is complex and f(X,T) is a real function. We substitute the above ansatz in Eq. ( 12) to obtain the bilinear form /bracketleftbig iDT+a0D2 X−ia3D3 X/bracketrightbig (g·f)=0, (16a) a0D2 X(f·f)=a2ea4Tgg∗,(16b) [2a2e2a4Tgg∗DX(g·f)+a2e2a4Tg2DX(g∗·f)] +3a3DX(g·f)D2 X(f·f)=0, (16c) where g∗is the complex conjugate and Dis the well known Hirota bilinear operator defined as Dm TDn X(a·b)=/parenleftbigg∂ ∂T−∂ ∂T/prime/parenrightbiggm/parenleftbigg∂ ∂X−∂ ∂X/prime/parenrightbiggn ×a(X,T)b(X/prime,T/prime)|X=X/prime,T=T/prime (17)and the functions g(X,T) andf(X,T) are expanded in a series form with a formal expansion parameter χ: g=χg1(X,T)+χ3g3(X,T)+χ5g5(X,T)+··· ,(18a) f=1+χ2f2(X,T)+χ4f4(X,T)+χ6f6(X,T)+··· . (18b) TheN-soliton solutions of the HONLS equation can be obtained by substituting the expansions ( 18a) and ( 18b)i nt h e bilinear forms ( 16a)–(16c) and solving the recursive equations obtained at different orders of χ. A. One-soliton solution The soliton solutions of the system ( 12) can be obtained by terminating the expansions ( 18a) and ( 18b) at various orders ofχ. Thus, the one-soliton solution of the HONLS equation can be written using Eq. ( 15)a s g=χg1, (19a) f=1+χ2f2, (19b) /Psi1=ea4Tg f=ea4Tg1 1+f2, (19c) where g1=eη1, (20a) f2=G 2a0(k1+k∗ 1)2, (20b) G=a2e2a4t, (20c) η1=k1X+ω1T+η1(0), (20d) ω1=ia0k2 1+a3k3 1, (20e) withη∗ 1andω∗ 1complex conjugates and k1andη1(0) arbitrary complex numbers. The solution ( 19c) is characterized by complex numbers k1andη1(0) and relative variations of intensity of the propagating magnetization pulse can be carriedout for different values of these complex numbers. B. Two-soliton solution We truncate the expression for the functions g(X,T) and f(X,T) as given by g=χg1+χ3g3, (21a) f=1+χ2f2+χ4f4, (21b) /Psi1=ea4Tg f=ea4Tχg1+χ3g3 1+χ2f2+χ4f4(21c) and substitute into the bilinear equations ( 16a)–(16c). Thus, the explicit forms of various functions at different orders ofperturbation are given by g 1=eη1+eη2, ηj=kjX+/parenleftbig ia0k2 j+a3k3 j/parenrightbig T+ηj(0),j=1,2; (22) 012923-4CURRENT-DRIVEN ELECTROMAGNETIC SOLITON . . . PHYSICAL REVIEW E 92, 012923 (2015) g3=2/summationdisplay j=1Bjeη1+η2+η∗ j,B j=G 2a0(k1−k2)2 (k1+k∗ j)2(k2+k∗ j)2; (23) f2=/summationdisplay 1/lessorequalslantl,j/lessorequalslant2Fljeηl+η∗ j,F lj=G 2a01 (kl+k∗ j)2; (24) f4=Feη1+η2+η∗ 1+η∗ 2, F=G2 2a0|k1−k2|4 (k1+k∗ 1)2(k2+k∗ 2)2|k1+k∗ 2|4. (25) It is interesting to note that the two-soliton solution is characterized by the four complex numbers kjandηj(0),j= 1,2. The intensity variations of the colliding solitons can be elucidated using the complex numbers kjandηj(0) in addition to the inherent system parameters. In the limit t→± ∞ ,t h etwo-soliton solution given above is asymptotically reduced to two single-soliton solutions represented in ( 19c). C. Three-soliton solution As a follow-up, the three-soliton solutions can be obtained by the series g=/epsilon1g1+/epsilon13g3+/epsilon15g5, (26) f=1+/epsilon12f2+/epsilon14f4+/epsilon16f6, (27) /Psi1=ea4Tg f=ea4T/epsilon1g1+/epsilon13g3+/epsilon15g5 1+/epsilon12f2+/epsilon14f4+/epsilon16f6.(28) After substituting Eq. ( 28)i nE q s .( 16a)–(16c) and solving recursively, we obtain the explicit forms of the unknownsg 1,g3,g5andf2,f4,f6as given by g1=eη1+eη2+eη3, (29) g3=G 2a0[eη1+η2+η∗ 1+c1+eη1+η2+η∗ 2+c2+eη1+η2+η∗ 3+c3+eη3+η2+η∗ 1+c4+eη3+η2+η∗ 2+c5+eη3+η2+η∗ 3+c6 +eη1+η3+η∗ 1+c7+eη1+η3+η∗ 3+c8+eη1+η3+η∗ 3+c9], (30) g5=G2 2a0[eη1+η2+η∗ 1+η∗ 2+η3+d1+eη1+η3+η∗ 1+η∗ 3+η2+d2+eη2+η3+η∗ 2+η∗ 3+η1+d3], (31) f2=G 2a0[eη1+η∗ 1+b1+eη∗ 2+η1+b2+eη1+η∗ 3+b3+eη2+η∗ 1+b4+eη∗ 2+η2+b5+eη∗ 3+η2+b6+eη∗ 1+η3+b7+eη∗ 2+η3+b8+eη∗ 3+η3+b9],(32) f4=G2 2a0[eη1+η2+η∗ 1+η∗ 2+R1+eη1+η3+η∗ 1+η∗ 3+R2+eη2+η3+η∗ 2+η∗ 3+R3+eη1+η2+η∗ 1+η∗ 3+R4+eη1+η3+η∗ 1+η∗ 2+R5+eη1+η2+η∗ 3+η∗ 2+R6 +eη3+η2+η∗ 2+η∗ 1+R7+eη1+η3+η∗ 3+η∗ 2+R8+eη3+η2+η∗ 1+η∗ 3+R9], (33) f6=G3 2a0[eη1+η2+η3+η∗ 1+η∗ 2+η∗ 3+M1], (34) where ηj=kjX+ωjT+ηj(0), (35) ωj=ia0k2 j+a3k3 j,j=1,2,3. (36) The exponential factors in Eqs. ( 29)–(34) are given in the Appendix. For the specific choices of the arbitrary complexnumbers k jandηj(0),j=1,2,3, the colliding magnetic soliton shows intensity variations through the amplitudes ofthe respective solitons. Having obtained the classes of solitonsolutions, the evolution of the same can be interpreted bychoosing the physical parameters involved in the system. For aconcrete analysis, the physical parameters relevant to the CoPt 3 alloy films [ 30] have been used for the graphic description of the soliton evolution. For CoPt 3alloy films MS=1125 G, γ=1.75×107Oe−1s−1,A=1×10−6erg cm−1,je=1× 109Ac m−2, and P=0.35. With these choices the model parameter bJis found to be 18 cm s−1. Let the velocity of the propagating soliton be VS=1.5,λ=1.0, and λ1=5.0. The one-soliton solution is pictorially represented in Fig. 2 for the parametric choices k1=1.51+2.39iandη1(0)= 2.24+3.15i. This soliton solution of /Psi1was predicted byHe and Liu [ 19] through Hirota bilinearization with identical boundaries such as /Psi1=0 and /Psi1=2πcalled 2 πkinks, which are stable traveling wave solutions. These localizedsolutions continue to preserve average magnetization along thedirection of propagation. In the case of nonidentical boundaryconditions, the nonlocalized solutions in the form of kinksindicate a separation of two neighboring domain walls withdifferent values of the field called πkinks or true domain walls, which are also important since they have experimental applica- tions [ 31–33]. The one-soliton solution presented in Fig. 2can be interpreted as propagation of the 2 πdomain wall, which is a bound state of two πwalls driven by the electric current density. The soliton solution becomes more interesting whenone generates the N-soliton solutions. Thus, the interesting feature of the soliton system is the collision of solitons drivenby physical parameters at higher orders of χ. In the present case the two solitons obtained in Eq. ( 21c) admit a collision scenario for the effective material parameter of CoPt 3.T h e collision dynamics is characterized by the complex parametersk 1,2andω1,2. The characteristic feature of the collision is the enhancement (suppression) of the intensity. This feature wasrecently described in [ 23] for a ferromagnetic spin chain with 012923-5M. SARA V ANAN PHYSICAL REVIEW E 92, 012923 (2015) −10 −5 0 5 10 −10−50510 0510152025 time (T) space (X)|Ψ| FIG. 2. (Color online) Intensity plot of the one-soliton solution expressed via ( 19c). The parametric choices are k1=1.51+2.39i andη1(0)=2.24+3.15i. a Dzyaloshinskii-Moriya interaction under the EM wave prop- agation. In [ 23] for an appropriate choice of the antisymmetric interaction due to the spin-orbit coupling the two solitons collide with intensity enhancement and suppression before and after the collision. In the present case, for the specificchoices k 1=1.586+0.182i,k2=1.129−2.93i,η1(0)= 2.95+1.5i, andη2(0)=0.531−2.50i, it is found that the system admits two solitons colliding with each other, resultinga complete amplitude suppression after collision. This couldbe exploited in Fig. 3in the negative time scale region, where the amplitudes of the two solitons S 1andS2are approximately at four and six units and as time progresses the two solitonsmutually collide inelastically nearly at the origin and suffer asuppression in their amplitudes. This scenario can be explainedas the intensity redistribution between the colliding solitons.A similar scenario can be also be drawn for the three-soliton FIG. 3. (Color online) Enhancement and suppression of the two- soliton collision expressed via ( 21c) for the parametric choices k1=1.586+0.182i,k2=1.129−2.93i,η1(0)=2.95+1.5i,a n d η2(0)=0.531−2.50i. FIG. 4. (Color online) Collision of three solitons with k1= 1.565+0.192i,k2=1.9−1.15i,k3=1.596−2.0i,η1(0)= 0.154+2.5i,η2(0)=1.582−0.5i,a n dη3(0)=1.149+0.5i. collision for the same material parameters and arbitrary complex parameters take the values k1=1.565+0.192i, k2=1.9−1.15i,k3=1.596−2.0i,η1(0)=0.154+2.5i, η2(0)=1.582−0.5i, andη3(0)=1.149+0.5i.F r o mF i g . 4 it is obvious that the three solitons show a considerableinelastic collision among themselves for the specific choicesgiven above. As time progresses the complete suppression ofthe soliton S 2shows an appreciable increase in its amplitude. However, the solitons S1andS3show marginal changes in their amplitudes. The shape changing nature of the solitons orintensity redistribution between the colliding solitons admits arelative phase shift and amplitude variations. With knowledgeof the one-, two-, and three-soliton solutions through theHirota bilinearization, the magnetic field component of the propagating EM wave can be obtained using the relation ( 13) and using u=M y 1−iMz 1andHr 1=λMr 1withr=y,z. Thus, the magnetization and the magnetic field component of the EMwave are highly localized and appear in the form of solitons asthe EM wave propagates in the ferromagnetic nanowire underthe effect of STT. V . CONCLUSION In this paper the magnetization dynamics of a uniaxial ferromagnetic nanowire under the effect of electromagneticwave propagation and in the presence of an applied electric cur-rent was rigorously analyzed. The dynamical Landau-Lifshitz-Gilbert equation was considerably reduced to the extendedDNLS equation using the multiscale analysis. The current-induced motion of the soliton shows that the nanowire acquiresa collision scenario for the propagating solitons with enhance-ment and suppression in their amplitudes that is due to the in-elastic collision. This inelastic collision results in the intensityredistribution among the colliding solitons under the influenceof current-induced spin torque. As reported earlier by He andLiu [ 19], the soliton collision admits fascinating dynamics in view of magnetization dynamics. However, the propagationof the EM wave in a ferromagnetic nanowire for the solitondynamics controlled electrically is different. The enhancement 012923-6CURRENT-DRIVEN ELECTROMAGNETIC SOLITON . . . PHYSICAL REVIEW E 92, 012923 (2015) and suppression of the soliton amplitude can be viewed as a magnetization state change that can be widely used for theconstruction of magnetic devices for transporting logical infor-mation and quantum computation. The colliding solitons canbe visualized as analogous to a domain wall in bulk materialand separate a region of one stable magnetic state of the logicalbit from another. Thus, it is strongly believed that the collisionof solitons under EM wave propagation and electric currentcreate a possibility for the exploitation of magnetic devices.ACKNOWLEDGMENT The author wishes to thank the anonymous referees for their critical comments and suggestions. APPENDIX In this Appendix the explicit forms of various exponential factors obtained for the three-soliton solution are presented: ec1=(k1−k2)2 (k2+k∗ 1)2(k1+k∗ 1)2,ec2=(k1−k2)2 (k1+k∗ 2)2(k2+k∗ 2)2,ec3=(k1−k2)2 (k1+k∗ 3)2(k2+k∗ 3)2, ec4=(k2−k3)2 (k2+k∗ 1)2(k3+k∗ 1)2,ec5=(k2−k3)2 (k2+k∗ 2)2(k3+k∗ 2)2,ec6=(k2−k3)2 (k2+k∗ 3)2(k3+k∗ 3)2, ec7=(k1−k3)2 (k1+k∗ 1)2(k3+k∗ 1)2,ec8=(k1−k3)2 (k1+k∗ 2)2(k3+k∗ 2)2,ec9=(k1−k3)2 (k1+k∗ 3)2(k3+k∗ 3)2, ed1=(k1−k2)2(k1−k3)2(k2−k3)2(k∗ 1−k∗ 2)2 (k1+k∗ 1)2(k2+k∗ 1)2(k3+k∗ 1)2(k1+k∗ 2)2(k2+k∗ 2)2(k3+k∗ 2)2, ed2=(k1−k2)2(k1−k3)2(k2−k3)2(k∗ 1−k∗ 3)2 (k1+k∗ 1)2(k2+k∗ 1)2(k3+k∗ 1)2(k1+k∗ 3)2(k2+k∗ 3)2(k3+k∗ 3)2, ed3=(k1−k2)2(k1−k3)2(k2−k3)2(k∗ 2−k∗ 3)2 (k1+k∗ 2)2(k2+k∗ 2)2(k3+k∗ 2)2(k1+k∗ 3)2(k2+k∗ 3)2(k3+k∗ 3)2, eb1=1 (k1+k∗ 1)2,eb2=1 (k1+k∗ 2)2,eb3=1 (k1+k∗ 3)2,eb4=1 (k2+k∗ 1)2, eb5=1 (k2+k∗ 2)2,eb6=1 (k2+k∗ 3)2,eb7=1 (k3+k∗ 1)2,eb8=1 (k3+k∗ 2)2, eb9=1 (k3+k∗ 3)2, eR1=(k1−k2)2(k∗ 1−k∗ 2)2 (k2+k∗ 1)2(k1+k∗ 1)2(k1+k∗ 2)2(k2+k∗ 2)2,eR2=(k1−k3)2(k∗ 1−k∗ 3)2 (k3+k∗ 1)2(k1+k∗ 1)2(k1+k∗ 3)2(k3+k∗ 3)2, eR3=(k2−k3)2(k∗ 2−k∗ 3)2 (k2+k∗ 2)2(k3+k∗ 2)2(k2+k∗ 3)2(k3+k∗ 3)2,eR4=(k1−k2)2(k∗ 1−k∗ 3)2 (k1+k∗ 1)2(k2+k∗ 1)2(k1+k∗ 3)2(k2+k∗ 3)2, eR5=(k1−k3)2(k∗ 1−k∗ 2)2 (k1+k∗ 1)2(k1+k∗ 2)2(k3+k∗ 1)2(k3+k∗ 2)2,eR6=(k1−k2)2(k∗ 2−k∗ 3)2 (k1+k∗ 2)2(k2+k∗ 2)2(k1+k∗ 3)2(k2+k∗ 3)2, eR7=(k3−k2)2(k∗ 1−k∗ 2)2 (k2+k∗ 1)2(k2+k∗ 2)2(k3+k∗ 1)2(k3+k∗ 2)2,eR8=(k1−k3)2(k∗ 2−k∗ 3)2 (k1+k∗ 2)2(k3+k∗ 2)2(k1+k∗ 3)2(k3+k∗ 3)2, eR9=(k2−k3)2(k∗ 1−k∗ 3)2 (k2+k∗ 1)2(k2+k∗ 3)2(k3+k∗ 1)2(k3+k∗ 3)2,eM1=|k1−k2|4|k2−k3|4|k3−k1|4 (k1+k∗ 1)2(k2+k∗ 2)2(k3+k∗ 3)2|k1+k∗ 2|4|k2+k∗ 3|4|k3+k∗ 1|4, [1] J. 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PhysRevE.94.042220.pdf

PHYSICAL REVIEW E 94, 042220 (2016) Breathers and rogue waves excited by all-magnonic spin-transfer torque Zai-Dong Li,1,2,*Qiu-Yan Li,1Tian-Fu Xu,3and Peng-Bin He4 1Department of Applied Physics, Hebei University of Technology, Tianjin 300401, China 2International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China 3Department of Physics, Yanshan University, Qinhuangdao 066004, China 4School of Physics and Electronics, Hunan University, Changsha 410082, China (Received 5 June 2016; published 25 October 2016) In terms of Darboux transformation we investigate the dynamic process of spin wave passing through a magnetic soliton. It causes nonlinear excitations, such as Akhmediev breathers solution and Kuznetsov-Ma soliton. Theformer case demonstrates a spatial periodic process of a magnetic soliton forming the petal with four pieces. Thespatial separation of adjacent magnetic petals increases rapidly, while one valley splits into two and the amplitudeof valley increases gradually with the increasing amplitude of spin wave. The other case shows a localized processof the spin-wave background. In the limit case, we get rogue waves and clarify its formation mechanism. DOI: 10.1103/PhysRevE.94.042220 I. INTRODUCTION During the past several decades there has been significant progress in describing dynamics of magnetization in magneticnanostructures. In these studies, self-organization [ 1]i so n e of the most interesting phenomena in nature. In magnetism,this phenomenon has been intensively studied in terms of thespontaneous formation of magnetic domains such as stripedomains, bubble domains, soliton, and magnetic vortex. In addition, the study on two-dimensional magnetic systems of thin films have revealed further interesting magnetic self-organization patterns such as spin waves [ 2] and skyrmion lattices [ 3–5], which can be nucleated as a metastable state in thin films. It opens a path to concepts of magnetic memoriesand contributes to designing memories based on skyrmionmotion in nanotracks. The dynamics of domain wall is of great significance in ferromagnetic nanowires for its potentially technologicalapplications [ 6–11]. For example, a magnetic domain wall forms a spatially localized configuration of magnetizationin ferromagnet, and it can be seen as a potential hill,which separates two generated magnetic states [ 12,13]. The propagation of domain wall with the influence of spin-Halleffect [ 14], Rashba effect [ 15], and Dzyaloshinskii-Moriya interaction [ 16–19] has drawn considerable interest in low- dimensional magnetism. These studies have been inspirednot only by the fundamental physics questions but also bythe potential application for the spintronic memory and logicnanodevices. Recently, considerable attention has been paid tothe dynamics of magnetization associated with spin-polarizedcurrent in layered materials [ 20,21]. The spin-polarized current can cause many unique phenomena [ 22,23], such as spin-wave excitation [ 24,25], magnetization switching [ 20] and reversal [26–28], and enhanced Gilbert damping [ 29,30] in magnetic multilayers. Nowadays, spin-polarized currents are commonlyused to create, manipulate, and control nanoscale magneticexcitations such as domain walls [ 31–34] and vortices [ 35–37]. Nonlinear excitations [ 12,13] are general phenomena in magnetic-ordered materials. In ferromagnet a cluster of *lizd@hebut.edu.cnmagnons tends to self-localization because of attractive inter-action. In a certain sense, the attraction of magnons is criticalfor a one-dimensional ferromagnet because it produces abound state of quasiparticles (magnons), i.e., self-localization.A spin wave may be regarded as a cluster of a macroscopicnumber of coherent magnons. Because of the attractiveinteraction, the magnon cluster tends to be localized, and thusthe spin wave becomes unstable. The developing instabilitycauses magnetization localization and brings about a domainwall and a magnetic soliton. However, the nonlinear excitations have not been well explored. When a spin wave passes through a magneticsoliton, a spin angular momentum can be transferred fromthe propagating magnons to the soliton, which is called byall-magnonic spin-transfer torque [ 43]. This all-magnonic spin-transfer torque can affect the dynamics of magnetizationand magnetic states can occur. In this paper, we will study theexact breather solutions and magnetic states. As an example,we give the exact solutions of bright (dark) rogue waves causedby this magnonic spin-transfer torque. II. EXACT BREATHER SOLUTIONS AND ROGUE WA VES As a simple model, we consider the Landau-Liftshitz equation, ∂m ∂t=−m×∂2m ∂x2, (1) which admits spin-wave and soliton solutions. The exact breather solutions and rogue waves of Eq. ( 1) can be structured by the Darboux transformation. The main idea of the Darbouxtransformation is that it first transforms the nonlinear equationinto the Lax representation, and then by a series of transfor-mations the soliton solution can be constructed algebraicallywith the obvious seed solution of the nonlinear equation. Itis an effective technique to generate a solution for Eq. ( 1) once a seed solution m 0is known. In the following, we take the initial “seed” as spin wave, i.e., m0≡(m01,m02,m03)= (Ascosδ,Assinδ,/radicalbig 1−A2s) with δ=ksx−ωstand the dis- persion relation ω=−k2 sm03. In terms of the developed procedure of Darboux transfor- mation [ 38–42], we obtain the exact solutions of Eq. ( 1)a st h e 2470-0045/2016/94(4)/042220(5) 042220-1 ©2016 American Physical SocietyZAI-DONG LI, QIU-Y AN LI, TIAN-FU XU, AND PENG-BIN HE PHYSICAL REVIEW E 94, 042220 (2016) form m·σ=K(m0·σ)K−1, (2) where σis pauli matrix and the matrix Kis given by K=1 |ξ|2(P+Q)/parenleftbiggξ∗P+ξQ −μR∗e−iδ μReiδξP+ξ∗Q/parenrightbigg , with ξ=iμ/2+ν/2,N=√ (ks−2ξm 03)2+4ξ2A2 s,β= −i2ξ−im0,3ks,P=h11h∗ 11,Q=h12h∗ 12,R=−ie−iδ h∗ 11h12,h11=i(C1eB−C2e−B)e−iδ/2,h12=(C1e−B− C2eB)eiδ/2,C1=/radicalbig (μm 03+i(A2sks−N))/2,C2=/radicalbig (μm 03+i(A2sks+N))/2, and B=−iN(x+iβt)/2. The solution in Eq. ( 2) denotes a soliton solution embedded in a spin wave background. With the increasing of μ,t h e spin wave background is gradually localized and formsbreathers due to the interaction between soliton and spin wavebackground. With the analytical solutions in Eq. ( 2) we can obtain the Akhmediev breathers, Kuznetsov-Ma soliton, andmagnetic rogue waves of magnetization. From the solutionin Eq. ( 2) we find that the critical point |μ|=A sksforms a dividing line between the modulation instability process(|μ|<A sks), the periodization process ( |μ|>A sks), and magnetic states ( |μ|→Asks). A. Modulation instability and Akhmediev breathers Modulation instability has been extensively studied in nonlinear physics [ 44], which is characterized by the periodic energy exchange between a perturbation and a continuouswave background. It can be used to generate the high-repetition-rate pulse trains in optical fibers [ 45] and can be described by near exactly the Akhmediev breathers [ 46] solution of the nonlinear Schr ¨odinger equation. In optical fibers, Akhmediev breathers are temporal periodic and showthe properties of single growth-return cycle in the propagationdirection, namely a visual illustration of the famous Fermi-Pasta-Ulam recurrence [ 47]. Recently, modulation instability has been found to play a central role in the emergence ofhighly localized rogue-wave structures in various contexts ofnonlinear physics. In ferromagnet this ubiquitous process of magnetization dynamics can be realized by the condition |μ|<A sksand ν=ksm03in Eq. ( 2). The parameters are given by P=As(kscoshθ−Nsinhθ)−μcosφ−Nm 03sinφ, Q=As(kscoshθ+Nsinhθ)−μcosφ+Nm 03sinφ, R=μcoshθ+iNm 03sinhθ−As(kscosφ+iNsinφ), (3) where θ=μNT,φ =−N(X+2ksm03T) with N=/radicalbig A2sk2s−μ2. The above result reveals that the solution to Eq. ( 2) is spatial periodic denoted by 2 π/N , and aperiodic in the temporal variable, as shown in Fig. 1. This process can also be seen as the spatial manifestation ofFermi-Pasta-Ulam recurrence realized by the magnetizationdynamics. The spatial periodic distribution of magnetizationshows that the component m 3has two peaks and one valley in each unit distribution. As the spin wave amplitude As increases the connection line of two peaks in the component FIG. 1. Evolution of Akhmediev breathers for magnetization m=(m1,m2,m3)i nE q s .( 2)a n d( 3). The component m3takes the spatial periodic distribution, which is characterized by two peaks and two valleys in each unit distribution. Parameters are given as follows:A s=0.8,ks=1,ν=ks/radicalbig 1−A2 s,a n dμ=0.64. m3rotates clockwise and the two peaks move with the opposite direction, as shown in Fig. 2. Also, the one valley splits into two and the distance of two valleys increases withthe increasing A s. In order to study the asymptotic form of modulation instability of magnetization we consider the case of t→± ∞ . The background of m3approaches to m03(1−4μ2/k2 s)a s t→± ∞ . When As=1o r|μ1|=ks/2 with 1 /2/lessorequalslantAs<1, the magnetization lies in the m1-m2plane and the component m3takes zero background. Under the condition As=1o r |μ1|=ks/2 with 1 /2/lessorequalslantAs<1 the magnon density distri- bution |m+(x,t)|2takes a maximum 1 at t→± ∞ , where m+≡m1+im2. The solution in Eq. ( 2) with the parameters of Eq. ( 3) can be considered as the modulation instability process [ 44]. This instability process can also be expressed by FIG. 2. The formation of magnetic petal in the component m3, shown in (a)–(d). As the spin wave amplitude Asincreases, the connection line of two peaks in the component m3rotates clockwise and the two peaks move with the opposite direction. The one valley splits into two and the distance of two valleys increaseswith the increasing A s. Parameters are given as follows: ks=1,ν= ks/radicalbig 1−A2 s,a n d μ=0.8Asks. The parameters Asis given by (a) As=0.7, (b)As=0.9, (c)As=0.98, and (d) As=1, respectively. 042220-2BREATHERS AND ROGUE W A VES EXCITED BY ALL- . . . PHYSICAL REVIEW E 94, 042220 (2016) linearizing the initial value of corresponding solution as m+(0,t)≈/parenleftbigg −1±i/epsilon14μN k2ssinφ/parenrightbigg eiksx, m3≈±/epsilon14N2 k2ssinφ, (4) where we use the condition As=1,/epsilon1=exp(−x0)i sas m a l l quantity for x0>0. The magnetic Akhmediev breathers in Eq. ( 2) with the parameters of Eq. ( 3) in fact denotes the instability process of spin wave background. Small perturbations that disturbthe spin wave can be amplified exponentially. The spin wavebackground is unstable against small perturbations. At thisinstability process there occurs the spatial periodic distributionof high magnon density, as shown in Fig. 1. A periodic magnon exchange occurs between the magnetic soliton and thespin-wave background. It should be noted that the magneticsoliton will lose this character on the ground-state background.It is worth mentioning that the interaction between spin waveand magnetic soliton causes this very interesting phenomenon. B. Kuznetsov-Ma soliton solution Under the conditions |μ|>A sksandν=ksm03we obtain the magnetic Kuznetsov-Ma soliton solution of Eq. ( 2), which can be proposed as prototypes of hydrodynamic ofrogue waves. This solution is characterized by the followingparameters: P=μcoshθ+ζm 03sinhθ−As(kscosφ+ζsinφ), Q=μcoshθ−ζm 03sinhθ−As(kscosφ−ζsinφ), R=As(kscoshθ+iζsinhθ)−μcosφ+iζm 03sinφ, (5) where ζ=/radicalbig μ2−A2sk2s,θ=ζ(x+2m03kst), and φ=μζt. With the above parameters we see that the main characteristicproperties of magnetic Kuznetsov-Ma soliton solution isspatially aperiodic and temporally periodic, while the solitonpropagates with the velocity −2k sm03and width 1 /ζ. Similar to the discussion in the section of Akhmediev breathers thecomponent m 3shows two peaks and one valley in each periodic distribution, and the connection line of two peaks also rotatesclockwise and the two peaks move with the opposite directionas spin-wave amplitude A sincreases. The illustration of magnetic Kuznetsov-Ma soliton is depicted in Fig. 3. When As=1, the parameter θdepends only on xwhich implies the envelope velocity becomes zero, i.e., the soliton is trapped in space by spin wave background.In order to study the asymptotic form of Kuznetsov-Ma solitonwe consider the limitation case x→± ∞ .F r o mE q s .( 2) and ( 5) we see that the component m 3approaches to (1−4A2 s)m03, while the transverse components denoted by m+approach to m0+(4A2 s−3)(N1∓iks)/(N1±iks) with m0+≡m01+im02asx→± ∞ . This result shows that a spin wave undergoes a phase change 2 arctan [2 Nks/(N2−k2 s)] when it pass across a magnetic soliton. This phase changeof spin wave can affect the propagation velocity of magneticsoliton, which denotes the transfer of spin angular momentumfrom spin wave background to a dynamic soliton called FIG. 3. Evolution of Kuznetsov-Ma soliton for magnetization m=(m1,m2,m3)i nE q s .( 2)a n d( 5). This soliton is spatially aperiodic and temporally periodic, while the component m3shows two peaks and two valleys in each periodic distribution. Parameters are given as follows: As=1,ks=1,ν=ks/radicalbig 1−A2 s,a n dμ=1.3. magnonic spin-transfer torque [ 43]. We also obtain that the zero background of m3can be realized by two cases, i.e., As=1o r|μ1|=ks/2 with 1 /2/lessorequalslantAs<1, while the magnon density distribution attains the maximum value 1 at x→± ∞ . One also finds that the maximum and minimum evolution ofthe component m 3is the same as the propagation direction of soliton. This feature illustrates the characteristic breatherbehavior of the soliton as it propagates on the background ofa periodic solution of magnetization in ferromagnet. Different from the magnetic Akhmediev breathers, the magnetic Kuznetsov-Ma soliton in Eq. ( 2) with the parameters of Eq. ( 5) expresses the localized periodic magnon exchange, which takes the temporal periodic evolution. Also, the highmagnon density shows the temporal periodicity along thepropagation direction of soliton. C. Bright and dark rogue waves The above discussion shows that the condition |μ|=Asks forms a critical point that divides the modulation instability process ( |μ|<A sks) and the periodization process ( |μ|> Asks). It leads to the different physical behavior of how the breather character depends strongly on the modulation param-eterμ, as shown in Fig. 4. Two different asymptotic behaviors are plotted in Fig. 4in the limit processes |μ|→ (A sks)− and (Asks)+under the condition ν=ksm03, respectively. The former case demonstrates a spatial periodic processof a magnetic soliton forming the petal with four pieces.The spatial separation of adjacent magnetic petals increasesrapidly, while the one valley splits in two and the amplitudeof valley increases gradually as the modulation parameter |μ| approaches A sks. The other case shows a localized process of the spin-wave background. In this case, the temporal separationof adjacent magnetic petals also increases rapidly as themodulation parameter μapproaches ( A sks)+. In the limit case of |μ|→Asks, we get the magnetic rogue wave of Eq. ( 1), where the main parameters are given by P=/parenleftbig 2tAsk2 s+Asksm03x±1/parenrightbig2 +A3 sk2 s/parenleftbig Asx2−3Ask2 st2∓6t/parenrightbig , 042220-3ZAI-DONG LI, QIU-Y AN LI, TIAN-FU XU, AND PENG-BIN HE PHYSICAL REVIEW E 94, 042220 (2016) FIG. 4. The asymptotic processes of the magnetic component m3 in the limit processes μ→Asksandν=ks/radicalbig 1−A2 sin Eq. ( 2). As μ→Asksthe spatiotemporal separation between adjacent magnetic petal increases gradually, as shown in (a)–(f), where the parametersareA s=0.8,ks=1, (a)μ=0.9, (b)μ=0.98, (c) μ=0.9999, (d) μ=1.2, (e)μ=1.1, and (f) μ=1.0001, respectively. Q=/parenleftbig 2tAsk2 s+Asksm03x∓1/parenrightbig2 +A3 sk2 s/parenleftbig Asx2−3t2Ask2 s±6t/parenrightbig , R=i2A2 sks(x+3tksm03)+(P+Q)/2−2, (6) where the sign ±denotes the limit case μ→±Asks,r e - spectively. In order to study the asymptotic form of therogue waves in Eqs. ( 2) and ( 6) we consider the case of x→± ∞ (t→± ∞ ) and x→0(t→0). The component m 3approaches to (1 −4A2 s)m03asx→± ∞ (t→± ∞ ) and m03asx→0(t→0) for the case +, while approaches to m03 asx→± ∞ (t→± ∞ ) and (1 −4A2 s)m03asx→0(t→0) for the case −. The transverse components m+approaches tom0+(3−4A2 s)a sx→± ∞ (t→± ∞ ) and −m0+as x→0(t→0) for the case +, while approaches to −m0+as x→± ∞ (t→± ∞ ) and m0+(3−4A2 s)a sx→0(t→0) for the case −. The above analysis shows that the case + expresses the bright rogue wave, while the case −corresponds to dark rogue wave. The graphical representation of bright anddark rogue waves are shown in Fig. 5. Especially, when A s=1 we can get the compact magnetic rogue waves as follows: m+=−eiksx/bracketleftbig 1−/parenleftbig 8x2k2 s−i4xks(F1−2)/bracketrightbig /F2 1/parenrightbig , m3=± 8txk3 s/F2 1, (7) where F1=1+t2k4 s+x2k2 s. The component m3is character- ized by the antisymmetric distribution of two peaks and twovalleys, as shown in Fig. 4. The above results show that there exist two processes of the formation of the magnetic rogue wave: one is the localizedprocess of the spin-wave background, and the other is thereduction process of the periodization of the magnetic brightsoliton. The magnetic rogue wave is exhibited by the strong FIG. 5. The graphical evolution of rogue waves for the mag- netization m=(m1,m2,m3)i nE q s .( 2)a n d( 6), i.e., bright rogue waves (a)–(c) and dark rogue waves (d)–(f). The parameters areA s=√ 3/2,ks=1,ν=ks/radicalbig 1−A2 s,a n dμ=±√ 3/2 with the sign ±corresponding to the bright and dark rogue waves, respectively. temporal and spatial localization of the magnon exchange and high magnon density. Also, the magnetic rogue wavescan be excited by a small localized perturbation of spin-wavebackground, as shown in Fig. 4. It should be interesting to discuss how to detect such breathers and rogue waves in experiment. In spinor Bose-Einstein condensates trapped in optical potentials [ 48–50] the average of m 3component is measured directly by the difference numbers of the population between the spin +1 and −1 Zeeman sublevel. It implies that there exists the temporal or spatial periodic population of atoms for magnetic breathersolutions, while the atoms take the nonuniform populationfor rogue waves. For the fermionic ferromagnet the currentflow is strongly affected by the orientation of the magneticmoments. Therefore, a periodic change of electrical resistancein magnetic layer may occur for magnetic breathers solutions,while a higher electrical resistance for rogue waves. III. CONCLUSIONS In summary, we investigate the dynamics of magnetization in a ferromagnet excited by the all-magnonic spin-transfertorque with the developed Darboux transformation. As anexample, we obtain the exact expressions of Akhmedievbreathers solution, Kuznetsov-Ma soliton, and rogue waves.We also obtain the critical condition between the modulationinstability process, the periodization process, and magneticstates. These results can be useful for the exploration ofnonlinear excitation in Bosonic and fermionic ferromagnet. ACKNOWLEDGMENTS We are grateful to Professor Lu Li and Biao Wu for his helpful discussions. 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PhysRevB.87.054437.pdf

PHYSICAL REVIEW B 87, 054437 (2013) Dynamics of laser-induced spin reorientation in Co/SmFeO 3heterostructure L. Le Guyader,*A. Kleibert, and F. Nolting Swiss Light Source, Paul Scherrer Institut, CH-5232 PSI-Villigen, Switzerland L. Joly IPCMS, 23 rue du Loess, BP 43 F-67034 Strasbourg Cedex 2, France P. M. Derlet Condensed Matter Theory Group, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland R. V . Pisarev Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia A. Kirilyuk, Th. Rasing, and A. V . Kimel Radboud University Nijmegen, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands5 (Received 5 November 2012; published 28 February 2013) Ultrafast control of a ferromagnet (FM) via exchange coupling with an antiferromagnet (AFM) is demonstrated in a Co/SmFeO 3heterostructure. Employing time-resolved photoemission electron microscopy combined with x-ray magnetic circular dichroism, a sub-100-ps change of the Co spins orientation by up to 10◦driven by the ultrafast heating of the SmFeO 3orthoferrite substrate through its spin reorientation phase transition is revealed. Numerical modeling of the ultrafast-laser-induced heat profile in the heterostructure, and the subsequent coupledspins dynamics and equilibration of the spin systems suggest that the localized laser-induced spin reorientation ishindered compared with the static case. Moreover, numerical simulations show that a relatively small Co/SmFeO 3 exchange interaction could be sufficient to induce a complete and fast spin reorientation transition (SRT). DOI: 10.1103/PhysRevB.87.054437 PACS number(s): 75 .30.Kz, 75 .50.Ee, 75 .70.Cn, 75 .78.Jp I. INTRODUCTION Ultrafast control of the magnetization of thin films using femtosecond laser pulses has attracted remarkable interestin the last fifteen years. Beginning with the report of anunexpected ultrafast demagnetization in a nickel thin filmin 1996, 1the research on ultrafast magnetization dynamics quickly developed. While the proper microscopic description of ultrafast demagnetization is still intensely debated,2novel laser-induced magnetic phenomena have been discoveredduring these years in a large variety of materials rangingfrom ferromagnets (FMs) to antiferromagnets (AFMs) andfrom metals to insulators. 3While most of these studies where conducted in single-phase materials, exchange-coupledFM/AFM heterostructures are particularly interesting since novel material properties can there be engineered. Funda- mentally, FM and AFM materials display very differentmagnetic properties. 4For example, in the quasistatic regime, FM materials react to moderate magnetic fields of the order oftheir anisotropy field H A≈1 T while, on the other hand, AFM materials are largely insensitive to magnetic fields up to thespin-flop transition. Combining these two types of materials in FM/AFM heterostructures offers the possibility of enhancing the magnetic anisotropy of the FM layer and creating a shiftedhysteresis loop via the exchange bias effect, with numerousapplications in spin-valve devices. 5,6In the dynamic regime, FM materials display a rather slow magnetic response given bytheir ferromagnetic resonance frequency ω≈γH Aof a few GHz, while AFM materials have a much faster response thanks to their higher antiferromagnetic resonance frequency ω≈ γ√HexHAof several hundreds of GHz. This advantage hasbeen recently employed to trigger ultrafast spin dynamics and spin reorientation in antiferromagnets.7–10Combining these two different classes of materials in a FM/AFM heterostructurecould thus produce a composite material with novel dynamicalproperties. While there is plenty of literature on the static properties of such FM/AFM systems, their dynamical behaviours areoften not considered. In particular, the possibility of speedingup the slow FM dynamics via coupling to the fast AFMdynamics seems to be very intriguing. Among the few studiesthat investigated this question, most of them considered thepossibility of modifying the exchange bias with an opticalpulse, such as in NiFe/NiO by Ju et al. , 11–13in NiFe/FeMn by Weber et al. ,14,15and in Co/IrMn by Dalla Longa et al.16,17In all these cases, a reduction of the exchange bias within a fewpicoseconds followed by a few 100 ps recovery is reported.This sudden change of the exchange field and, in turn, theeffective field, triggers damped spin precessions in the FMlayer which are measured optically via the magneto-opticalKerr effect and which correspond to the relaxation of theFM spins towards the newly created equilibrium. While inprinciple, a sudden quenching of the exchange bias field could lead to a magnetization reversal (see Fig. 1of Ref. 17), only weak effects of a few percent magnetization changes haveso far been reported. Moreover, these triggered dynamics inthe FM are of ferromagnetic nature since the AFM layeris used to pin the FM material and suddenly depin it uponlaser excitation. By using stronger laser pulses, it is possibleto bring the sample around its blocking temperature wherechanges in the AFM at the interface can be seen. 18,19However, the triggered FM dynamics in this case are again small in 054437-1 1098-0121/2013/87(5)/054437(11) ©2013 American Physical SocietyL. LE GUY ADER et al. PHYSICAL REVIEW B 87, 054437 (2013) 0.0 0.0 differenceXMCD −0.05+0.2 +0.05−0.2ca acX−rays // c−axis X−rays // a−axist = −0.4 ns t = +1.1 ns(b)(a) average t >0-t<0 FIG. 1. (Color online) (a) Sketch of various layers and orientation of sample with respect to x-rays and laser-pulse propagation direction. The in-plane magnetization angle θCoof the Co film is given with respect to the SmFeO 3caxis. (b) Time-resolved XMCD asymmetry images at the Co L3edge taken at two different time delays before (t=− 0.4 ns) and after ( t=+ 1.1 ns) the laser-pulse time overlap, as well as the averaged difference between images at positive (t >0) and negative (t <0) time delays of the measured sequences, for x-rays propagating along the aand the caxis of the underlying SmFeO 3 single crystal. In all images, the scale bars are 20 μm long. The gray level values in the XMCD and difference images are given by the gray scales, respectively. The pump pulse was at λ=800 nm wavelength with a pulse length equal to τ=50 fs and fluence of F=10 mJ/cm2. amplitude and show slow relaxations by spin precessions towards the new equilibrium. The question of whether or not it is possible to drive a FM at the speed of an AFM in a heterostructure cannot beaddressed in those systems where the only possible action ofthe laser pulse is to reduce the coupling between the layers.This question can, however, be addressed in the Co/SmFeO 3 heterostructure where the quasistatic 90◦spin reorientation transition (SRT) in the rare-earth orthoferrite substrate inducesa coupled SRT in the Co overlayer as demonstrated in Ref. 20. This coupled SRT in the Co overlayer indicates that theexchange coupling between the layers is preserved across the15 K temperature range in which this SRT occurs. In this paper we show that ultrafast laser heating of an orthoferrite substrate through its SRT results in an inducedsub-100-ps spin reorientation of the exchange-coupled Cooverlayer, this time limit being determined by the timeresolution of the experiment. The amplitude of the observedexchange-induced spin reorientation in the Co layer is,however, limited to about 10 ◦in contrast to the 90◦observed in the static case in this system. Different possible scenariosto explain this discrepancy were thus investigated. Effectsfrom the inhomogeneous heating of the different layers bythe laser pulse could be ruled out by numerical modelingof the laser-induced heat profile in the multilayer and byexperiments at different laser pump wavelengths. The effecton the SRT of the formation of an exchange spring in theorthoferrite substrate between the hot surface and the coldbulk was investigated numerically and found to be significantbut also to be overcome with higher pump fluence. Finally,the coupled dynamics of the orthoferrite and cobalt spins weresimulated by numerical integration of a coupled pendulumequation and Landau-Lifschitz-Gilbert equation and found tobe in good agreement with the experiments at the condition ofa limited reorientation in the orthoferrite substrate. All theseconsiderations suggest that the laser-induced dynamics in theorthoferrite itself is more complex than anticipated. The paper is organized as follows: Section IIis devoted to the description of the sample and of the experiments. InSec. III, the obtained results are presented. In Sec. IV,t h e different simulations undertaken are described and consists ofthe modeling of the laser-heating profile in the multilayer inSec. IV A , of the calculation of the transient quasi-equilibrium state of the heated sample in Sec. IV B , and finally of the simulation of the triggered exchange dynamics in Sec. IV C . Conclusions are presented in Sec. V. II. EXPERIMENTS The sample consists of a 2 nm magnetron sputtered Co film deposited on the [010]-oriented surface of a 1-mm-thickSmFeO 3single crystal substrate and capped with 1 nm Pt and is the same sample as in Ref. 20. The rare-earth orthoferrite SmFeO 3is aG-type canted antiferromagnet with an orthorhombically distorted perovskite structure.21The Fe moments of this compound order antiferromagneticallybelow 673 K, with the spins aligned along the aaxis of the crystal. Due to the Dzialoshinsky-Moria antisymmetricexchange interaction, the Fe spins are slightly canted by asmall angle of 8 .2×10 −3rad, producing a net magnetization moment.22Due to this spin canting, the antiferromagnetic resonance splits in two modes without any applied magneticfield, resulting in a quasiferromagnetic mode at 270 GHz anda quasi-antiferromagnetic mode at 550 GHz, both at roomtemperature. 23The Sm moments remain unordered down to 5 K, below which they order antiferromagnetically with theFe moments and induce a magnetization reversal. 24As the temperature is lowered from the N ´eel temperature, the Sm ions become increasingly polarized, resulting in a stronglytemperature-dependent magnetic anisotropy for the Fe ions.This interaction between the Fe and Sm sublattices induces asecond-order spontaneous spin reorientation transition (SRT),from above 495 K with the Fe spins aligned along the aaxis and the net magnetization along the caxis to under 480 K with the Fe spins aligned along the caxis and the net magnetization along the aaxis. The canting of the spins remains largely constant during the SRT. 22Recently, it has been reported that SmFeO 3is also a room-temperature improper ferroelectric and thus a multiferroic material, with a small ferroelectric 054437-2DYNAMICS OF LASER-INDUCED SPIN REORIENTATION ... PHYSICAL REVIEW B 87, 054437 (2013) polarization along the baxis induced mostly by the inverse Dzialoshinsky-Moria exchange interaction.24Considering that this ferroelectric polarization is induced by the magnetic order,and that its magnitude and direction does not change acrossthe SRT, one can infer that its influence on the laser-inducedspin dynamics investigated here should be negligible. The 2 nm Co film sputtered on top of the SmFeO 3sub- strate is polycrystalline with a hexagonal-close-packed (hcp)structure 25and couples ferromagnetically via an exchange interaction to the net moment of the SmFeO 3substrate. Thus, it displays the same SRT as the orthoferrite, i.e., a spontaneousreorientation of the Co moments from the substrate aaxis at low temperature to the caxis at high temperature. 20Note that a magnetic dipolar coupling between the Co moments and thesmall SmFeO 3moments can be excluded since the observation of the induced Co SRT is very sensitive to the orthoferritesurface preparation. A change of magnetic anisotropy of theCo film induced by magnetostriction from the substrate canalso be ruled out as the main coupling mechanism. Whilethere is a small crystal elongation along the caxis and contraction along the aandbaxes of /Delta1l/l≈10 −6when heating through the SRT,26the Co film with hcp structure displays a negative longitudinal magnetostriction,27–29the effect of which counteracts the Co SRT instead of producingit. On the other hand, since a complete Co SRT is observed inthis sample via slow heating, the magnetostriction is thereforenegligible compared with the exchange coupling between thelayers. To image the magnetic domain configuration of the Co film, the Elmitec photoemission electron microscope (PEEM) at theSurface/Interface: Microscopy (SIM) beamline 30at the Swiss Light Source (SLS) was used. Employing the x-ray magneticcircular dichroism (XMCD) effect at the Co L 3edge at 778 eV , a quantitative determination of the Co spin orientation anglecan be derived. The intensity of a PEEM image of a magneticsample recorded with circularly polarized x-rays is a spatiallyresolved measure of the total electron yield and can be writtenin the form of 31 Iσ±=I0(1+αM·L±), (1) where I0is the image intensity, αaccounts for the magnitude of the XMCD effect, Mis the magnetization vector, and L± is the x-ray polarization vector. An XMCD asymmetry image is obtained by a pixel-wise computation of IXMCD=(Iσ+− Iσ−)/(Iσ++Iσ−) which simplifies to IXMCD=αMcos(β) where βis the angle between the magnetization vector and the x-ray polarization vector. This XMCD asymmetry imagecontains only normalized magnetic contrast information andtypically shows white or black regions which are magnetic do-mains with magnetizations of opposite directions with respectto the x-ray propagation vector. Assuming that the magnitudeof the Co magnetization does not change significantly aroundthe spin reorientation transition because of its much higherCurie temperature, a derivation of the Co magnetization angleis possible. It is convenient to choose the orthoferrite substratecaxis as a reference axis for measuring angles, as shown in Fig. 1(a). The XMCD asymmetry image intensity becomes I XMCD=αMcos(θc−Xc), where θcis the angle of the Co magnetization MandXcis the angle of the x-ray direction. Defining the magnetic contrast ξas the difference between theXMCD asymmetry value for white and black domains, the Co magnetization angle θccan be calculated via θc=arccos/parenleftbiggξ 2αM/parenrightbigg +Xc, (2) where the quantity 2 αM=0.32 is a normalization constant which is determined experimentally from measurements of themagnetic contrast ξbelow the SRT in which case θ cis known. Time-resolved measurements of the Co magnetization configuration were performed by taking advantage of thepulsed nature of the x-rays produced by the SLS synchrotronvia the gating of the detection in synchronization to an isolatedx-ray pulse present in the gap of the filling pattern of thestorage ring. This scheme, presented in detail in Ref. 32,a l l o w s stroboscopic pump-probe imaging of the sample with a timeresolution determined by the 70 ps full width at half maximum(FWHM) temporal x-ray pulse length. To investigate the effectof the laser-induced heat profile on the achieved amount of spinreorientation, the wavelength of the pump laser was varied. For experiments with λ=800 nm or 400 nm laser wavelength, an XL-500 oscillator from Femtolasers Produktions GmbHwas used, producing a τ=50 fs laser pulse with 500 nJ per pulse at a 5.2 MHz repetition rate. This repetition rate isthen reduced by a Pockels cell in combination with a crossedpolarizer to match the 1.04 MHz repetition rate of the isolatedx-ray probe pulses. The 400 nm pump was then obtainedfrom the 800 nm fundamental wavelength by doubling with aβ-barium-borate crystal with a conversion efficiency of 20%. The 532 nm wavelength, τ=10-ps-long pump was produced by a Duetto laser system from Time-Bandwidth Products AGwith a maximum energy per pulse of 120 nJ used in theexperiments. In all cases, the linearly p-polarized laser pumppulses were focused on the sample at a grazing incidence of16 ◦as shown in Fig. 1(a) to a spot size of about 30 ×100μm2 FWHM. The time overlap ( t=0) between the laser and the x-ray pulse is unambiguously determined to better than±15 ps by the sudden space charging 33,34which is induced by the laser pump pulse and which reduces significantly theamount of photoemitted electrons collected by the microscope.Finally, the sample can be heated via a resistive heater and thetemperature measured with a thermocouple attached to thesample holder. III. RESULTS In order to study the Co spin dynamics in a Co/SmFeO 3 heterostructure upon heating the orthoferrite substrate throughits SRT with a laser pulse, element-specific stroboscopic time-resolved XMCD PEEM measurements were conducted. Theobtained time-resolved XMCD asymmetry images at the CoL 3edge taken at two different time delays before ( t=− 0.4n s ) and after ( t=+ 1.1 ns) the laser pulse, and for x-rays propagat- ing along the aandcaxis of the underlying SmFeO 3substrate, are shown in Fig. 1(b). In these images, the Co magnetic domain configuration with typical sizes of the order of fewmicrometers can be seen. It must be noted that the sampleposition is different between the experiments with x-rays alongtheaandcaxis and therefore the magnetic domains do not correlate between these two sets of images. In both cases, thetemperature of the sample was adjusted by resistive heating 054437-3L. LE GUY ADER et al. PHYSICAL REVIEW B 87, 054437 (2013) such that before the laser pulse, the Co/SmFeO 3heterostruc- ture is already within the SRT temperature range. This explainswhy unsaturated magnetic domains are visible for x-rays prop-agating both along the aandcaxis of the underlying SmFeO 3 substrate. It should be noted that a starting point within the SRT temperature range is a necessary requirement for this strobo-scopic pump-probe measurement, since in principle, this SRTcan occur via two equivalent routes as the net orthoferrite mo-ment can reorient in opposite direction from the aaxis towards thecaxis (i.e., +cor−c). Applying an in-plane magnetic field would break this equivalence of the reorientation routes.However, this would possibly alter the measured dynamics andsubstantially deviate the photoemitted electrons away from themicroscope, rendering the measurement difficult. A more ele-gant approach would be to use circularly polarized laser pumppulses to break this equivalence of the two reorientation routesvia the inverse Faraday effect as recently demonstrated. 35 This, unfortunately, seems to be inefficient at the grazingincidence used in our setup. Instead, the equivalence betweenthe two reorientation routes is broken here by heating thesubstrate within its SRT temperature range such that before thelaser pulse, the coupled orthoferrite and Co SRT has alreadystarted. By comparing in both cases the domain configurationand magnetic contrast ξbefore ( t=− 0.4 ns) and after (t=+ 1.1 ns) the laser pulse in Fig. 1(b), it is evident that the domain configuration does not change, as expected, but alsothat the change in magnetic contrast within a domain is small. To highlight this laser-induced change, an averaged image difference is computed, where from the sequences of time-resolved XMCD images measured at different time delays, theimages taken after the laser pulse t >0are counted as positive and the images taken before the laser pulse t<0are counted as negative. In such difference images, as shown in Fig. 1(b),a region where the magnetic domain structures are again visiblecan be seen. The shape of this region is an elongated ellipsewhich is the signature of the focused laser beam impinging atgrazing incidence on the sample surface. Note that, dependingon the field of view used, the orientation of this ellipse is seenwith a different orientation direction in the image frame. Whatis visible within the laser spot in the difference image thuscorresponds to a localized change of magnetic contrast inducedby the ultrafast laser pulse. By convention, the white colorcorresponds to positive values and the black color to negativevalues of I XMCD . In the case of x-rays propagating along the a axis, the magnetic contrast displayed before the time overlapand in the difference image are opposite in sign since whatappears as white domains, i.e., positive values, in the XMCDasymmetry image before the laser overlap turns into blackdomains, i.e., negative values, in the difference image. Thismeans that the magnetic contrast is reduced after the laserheat pulse. On the contrary, in the case of x-rays along the c axis, the magnetic contrast has the same sign before and in thedifference images, meaning an increase of contrast after thelaser heat pulse. This increase of contrast along the caxis and decrease of contrast along the aaxis are better visualized in Fig. 2where the magnetic contrast ξmeasured for domains inside the laser spot are shown as a function of the time delayafter the laser pump pulse. Here again, a clear reduction ofmagnetic contrast along the aaxis and increase along the c axis is observed. While in principle the reduction of magneticFIG. 2. (Color online) Magnetic contrast ξmeasured at Co L3edge for x-rays propagating along the aandcaxes of the underlying SmFeO 3single crystal. The pump pulse was at λ= 800 nm wavelength with τ=50 fs pulse length and a F=10 mJ/cm2 fluence. contrast along the aaxis could be qualitatively explained by various effects other than a coupled SRT, the observation of anincreased magnetic contrast for x-rays along the caxis permits us to exclude them. Therefore, what is shown in Fig. 2is not related with a partial demagnetization of the Co film by heatingtowards its Curie temperature as well as a decoupling betweenthe Co and the SmFeO 3spins since, for these two effects, the same reduction of contrast would be observed with x-rayspropagating along the aandcaxes. This increase of contrast along the caxis and reduction along the aaxis is thus a clear signature of a spin reorientation in the Co film triggered by alaser pulse absorbed in the SmFeO 3substrate. Assuming now that the only source of change in the magnetic contrast arises from the reorientation of the Co spins,it is then possible to use Eq. (2)to derive the Co spin angle θ Co from the orthoferrite caxis as defined in Fig. 1(a).T h eC os p i n angle change /Delta1θCo=θCo(t)−θCo(t<0) induced by the laser from the negative time delay orientation is shown in Fig. 3.A sudden change of the Co spin angle is seen right after the laseroverlap and corresponds to an alignment of the Co spins furtheralong the SmFeO 3caxis in both measurements. Apart from the different amplitudes of reorientation in these two differentmeasurements which are related to different experimentalconditions, the Co spin dynamics is very similar in both casesand could be described as an overdamped spin precessionaround a new equilibrium. Together with the experimentaldata, the calculated response from the simulated coupled spindynamics derived in Sec. IV C is shown as a continuous line where the simulated Co spin dynamics has been projectedon the x-ray wave vector and convoluted by the x-ray pulselength of 70 ps. The similarity between the measured and thecalculated reorientation indicates that the observed dynamicsoccurs within 100 ps. The amplitude of the Co reorientationobtained is, however, much smaller than the observed 90 ◦from the static case.20 To verify whether this initial quick and small-amplitude response of the Co spins is further followed by a larger andslower reorientation dynamics, measurements at a longer timescale were performed as well and the results are shown in Fig. 4 054437-4DYNAMICS OF LASER-INDUCED SPIN REORIENTATION ... PHYSICAL REVIEW B 87, 054437 (2013) FIG. 3. (Color online) Co spin reorientation observed with x-rays propagating along the aandcaxes of the underlying SmFeO 3single crystal. The pump pulse was at λ=800 nm wavelength with τ= 50 fs pulse length and a F=10 mJ/cm2fluence. The dots correspond to the measurement points and the lines to a simulated coupled-spin dynamics response convoluted by the temporal length of the x-ray probe. for delays up to 15 ns. Here, only a slow relaxation towards the initial orientation of the Co film is visible and is compatiblewith a slow cooling down of the SmFeO 3substrate, inducing a coherent rotation of the SmFeO 3and Co spins together back to the initial state before the laser pulse. Thus, there is noindication of a larger SRT occurring at longer time delays. In order to investigate the effect of different levels of absorption of the pump laser pulse in the SmFeO 3substrate, experiments with different pump wavelengths were performed.The results shown in Fig. 5represent the maximum amount of spin reorientation observed in the Co film that couldbe reached for each situation. Here again, despite a visibleimprovement for the case of 400 nm pump wavelength, theobserved reorientation remains small compared to the 90 ◦ obtained in the static case.20 Finally, time-resolved measurements performed as a func- tion of the sample base temperature are shown in Fig. 6while the laser fluence was kept constant. As the thermocouple with FIG. 4. (Color online) Time-resolved Co spin reorientation ob- served with x-rays propagating along the caxis for time delays up to 15 ns. The pump pulse was at λ=800 nm wavelength with τ=50 fs pulse length and a F=10 mJ/cm2fluence.FIG. 5. (Color online) Time-resolved Co spin reorientation mea- sured with x-rays propagating along the caxis as a function of the laser wavelength. which the sample temperature is measured is at a certain distance from the imaged region, the temperature measuredis somewhat lower than the actual sample temperature. Onecan see that, starting closer to the end of the SRT at 383 K, theonly laser-induced effect visible is a reduction of the magneticcontrast. As the x-rays are propagating along the caxis, this would translate in this geometry into a short-lived increase ofthe Co spin angle which would therefore go against the SRT.As the average temperature of the sample decreases, thelaser-induced effect changes slowly to a longer-lived increaseof magnetic contrast, i.e., a reduction of the Co spin angle as theSRT would produce. The short-lived effect is thus interpretedas a Co partial demagnetization which appears on top of theSRT, indicating that the laser-induced heating is significant. The experimental findings regarding the Co spin dynamics subsequent to an ultrafast heating of the Co/SmFeO 3het- erostructure can be summarized as follows: First of all, asub-100-ps reorientation of the Co spins takes place, followedby a slow relaxation back to the initial state. The absence ofadditional long-term dynamics apart from the relaxation meansthat the substrate has reached a transient quasi-equilibriumstate within this 100 ps. In all the experiments, the amount of FIG. 6. (Color online) Time-resolved Co XMCD contrast mea- sured with x-rays propagating along the caxis as a function of the base temperature T0of the sample. The pump pulse was at λ=532 nm wavelength with τ=10 ps pulse length and a F=3m J/cm2fluence. 054437-5L. LE GUY ADER et al. PHYSICAL REVIEW B 87, 054437 (2013) reorientation achieved was significantly smaller than the 90◦ obtained in the static case, while the laser fluences used were enough to induce a transient partial demagnetization of the Cofilm. IV . NUMERICAL MODELLING To better understand our experimental observations and the physics that they contain, detailed simulations were carriedout. These simulations are divided in three different parts,which will be presented in the next sections. First of all,it is necessary to calculate the heat profile created in thesample by the absorption of the pump laser pulse in thevarious layers. After this, the transient quasi-equilibrium stateof the inhomogeneously heated sample is determined. Finally,a simplified dynamics of an exchange-coupled Co-orthoferritesystem is investigated. A. Laser-induced heat profile The laser intensity inside the Pt/Co/SmFeO 3multilayer was calculated using a matrix formalism of light scatteringat the different interfaces and of light propagation inside thelayers based on Abeles’s formulas. 36From this laser intensity, the differential absorbance dA(y) at any given depth yfrom the sample surface can be derived [see Eq. (46) of Ref. 36]. The inhomogeneous temperature change induced by the absorptionof the laser-pulse energy is then determined by the heatdiffusion equation: ρC p∂T(y,t) ∂t−k∇2T(y,t)=I(t)dA(y), (3) where ρis the density, Cpis the heat capacity, and kis the heat diffusion of the materials and all are a function of the depth y within the sample, and I(t) is the time-dependent incoming laser intensity at the center of the laser spot. Neglectingany heat-diffusion effects, i.e., k=0 for all depths y,t h e temperature increase due to laser absorption is simply givenby/Delta1T(y)=FdA(y)/(ρC p), where Fis the incoming laser fluence. The values for the material parameters used aregiven in Table I. Three different cases have been simulated, corresponding to the three different laser wavelengths availablein the experiments. The laser fluence used in these simulationshas been adjusted for each wavelength such that the temper-ature increase at the surface of the SmFeO 3crystal is always 16 K, which should be sufficient to induce a complete SRT.These laser fluences are F=0.35 mJ/cm 2forλ=400 nm, F=3.20 mJ/cm2forλ=532 nm, and F=392 mJ /cm2for λ=800 nm. Comparing these values with those used in the experiments shows that the fluence was around 10 times higherFIG. 7. (Color online) Heat profiles corresponding to a tempera- ture increase of /Delta1T=16 K at the SmFeO 3surface for different laser wavelengths. The incident laser fluences used in the calculations wereF=0.35 mJ /cm2forλ=400 nm, F=3.20 mJ /cm2for λ=532 nm, and F=392 mJ /cm2forλ=800 nm. The heat profiles for the experimental fluences of F=2m J/cm2forλ=400 nm, F=3m J/cm2forλ=532 nm, and F=10 mJ/cm2forλ=800 nm are shown in the inset. for 400 nm wavelength, about the same for 532 nm wavelength, and 40 times lower for 800 nm wavelength than the fluencerequired in the simulation to obtain a /Delta1T=16 K temperature increase at the SmFeO 3surface. Shown in Fig. 7are the heat profiles obtained from these simulations as a function of thedepthyfrom the sample surface. 37 Starting with the 800-nm-wavelength case, the calculation in Fig. 7shows that, due to the low absorption of the orthoferrite at this wavelength,38the fluence required to obtain the proper temperature increase at the SmFeO 3surface in order to induce the full SRT is very high and brings the Co filmway above its Curie temperature and even possibly destroysit. Since the ratio between the temperature increase in theCo film and at the SmFeO 3surface is independent of the incoming laser intensity, it is easy to calculate the maximumSRT achievable without completely demagnetizing the Cofilm. With a Co film heated to less than 10 3K, the SmFeO 3 surface is heated to less than 0.16 K. Considering that a 90◦ SRT occurs within a 10 K temperature change, this leadsto a maximum expected effect of about 1 ◦. It is thus clear that this large heat difference between the Co film and theSmFeO 3surface is a limiting factor in achieving a 90◦SRT with a λ=800 nm laser pump wavelength, which is a direct consequence of the low absorption in the orthoferrite at thiswavelength. TABLE I. Parameters used in the simulations for the layer thickness d, density ρ, heat capacity Cp, and complex refractive index ˜nfor the different wavelength λof the different materials constituting the sample. dρ C p ˜n (nm) (103kg m−3)( J k g−1K−1) λ=400 nm λ=532 nm λ=800 nm Pt 1 21.45 130 1 .718+2.84i 2.074+3.63i 2.839+4.95i Co 2 8.90 420 1 .455+3.00i 2.209+3.9i 3.618+4.71i SmFeO 3 ∞ 7.26 453 2 .5+0.7i 2.4+0.1i 2.3+0.0013i 054437-6DYNAMICS OF LASER-INDUCED SPIN REORIENTATION ... PHYSICAL REVIEW B 87, 054437 (2013) To overcome this limitation, it is necessary to change the laser wavelength to higher photon energy for which theSmFeO 3absorption is increased. The resulting heat profiles obtained with 532 nm and 400 nm are shown in Fig. 7and indeed display an increasingly more balanced heat distributionbetween the various layers as the photon energy is increased.For these wavelengths, it is then possible to heat the SmFeO 3 such that a complete SRT is obtained while the Co overlayerstays well below the Curie temperature. While measurementswith 400 nm clearly show an improvement in the amplitudeof Co spin reorientation obtained, as shown in Fig. 5,i t is still far from being complete. As the absorption in theSmFeO 3increases, the penetration depth of the laser pulse also decreases, leading to more surface rather than bulk heating ofthe SmFeO 3. As shown in Fig. 7, in the case of the 400 nm pump wavelength, the temperature change drops within fewtens of nanometers. One could therefore expect that the heatedspins at the SmFeO 3surface are pinned by the cold bulk, forming an exchange spring inside the crystal which couldseverely hinder the 90 ◦SRT. It is thus necessary to calculate the spin configuration in the SmFeO 3for the case of an inhomogeneous heating created by the absorbed laser pumppulse. B. Transient state Simulation of the spin dynamics induced by the laser-pulse energy absorbed in the SmFeO 3substrate is not a trivial task since, to the best of our knowledge, no micromagneticsimulation code for inhomogeneously excited antiferromagnethas been demonstrated so far. While neglecting the actualdynamical nature of the spin reorientation, the equilibriumconfiguration that the spin would eventually reach, if the heatprofile induced by the laser were permanent, is a much simplerproblem. At the same time, such simulations would givesome insights into the maximum amount of spin reorientationthat can be reached given a certain laser-heat-induced profile.The actual equilibrium spin configuration is mainly the resultof two competing energy terms, which are the temperature-dependent magnetic anisotropy energy, responsible for theactual SRT, and the exchange energy, which should prevent thereorientation to occur due to the coupling with the unheatedcold bulk. For the modeling, we neglect any other energy contributions as well as the Gaussian laser profile in the xzplane, resulting in an effective one-dimensional problem, which is the depthywithin the crystal. In addition, we neglect any effect due to the coupling with the Co layer. We neglect any heatdiffusion as well. This means that the spins have enoughtime to fully reorient to the new equilibrium configuration. All these approximations should maximize the amount of spin reorientation obtained and thus allow us to calculate its upperlimit within these approximations. In SmFeO 3, the anisotropy energy Eacan be written as39 Ea(T)=/integraldisplay0 −∞[K2(T)s i n2(θ)+K4sin4(θ)]dy, (4) where K2(T) is the second-order anisotropy constant, which varies linearly with the temperature in the SRT region, K4 is the fourth-order anisotropy constant, which is independentFIG. 8. (Color online) Anisotropy constant K2andK4and SmFeO 3magnetization angle θwith respect to the caxis as function of the temperature within the SRT. of the temperature in the SRT region, and θ(y) is the angle between the small SmFeO 3magnetization and the caxis at the depth yfrom the SmFeO 3surface. By minimizing the anisotropy energy with respect to θ, one derives the equilibrium orientation of the magnetization as function ofthe temperature for an homogeneously heated sample. Thisgives the SRT shown in Fig. 8, based on the values for K 2(T) andK4reported in the literature.40 In the case of inhomogeneous heating, the equilibrium con- figuration arising from the anisotropy term becomes positiondependent and thus competes with the exchange energy termE ex, which can be written as Eex=/integraldisplay0 −∞A/parenleftbiggdθ dy/parenrightbigg2 dy, (5) where the exchange stiffness constant A=nJm2 0/b=3.33× 10−11Jm−1in which n=4 is the number of Fe ions per unit cell, b=0.5592 nm is the lattice constant along the baxis,41andJm2 0is estimated via the mean-field relation zJm2 0=3kbTNwhere z=6 is the coordination number, TN=674 K is the N ´eel temperature of the orthoferrite, and kb=1.381×10−23JK−1is the Boltzmann constant, giving Jm2 0=4.65×10−21J. Considering only the anisotropy and exchange energy, the thermodynamical potential is /Phi1(θ)=/integraldisplay0 −∞/bracketleftbigg A/parenleftbiggdθ dy/parenrightbigg2 +K2(T)s i n2(θ)+K4sin4(θ)/bracketrightbigg dy. (6) Euler’s equation of the variational problem of this func- tional is given by39 d2θ dy2=K2(T) Acos(θ)s i n (θ)+2K4 Acos(θ)s i n3(θ). (7) Solving Eq. (7)numerically in the case of shallow energy landscape which occurs during the SRT is not trivial andleads to numerical instabilities with the usual approach of theinitial-value problem when θandθ /primeaty=− ∞ are given. An alternative procedure is to solve the equation as a boundary 054437-7L. LE GUY ADER et al. PHYSICAL REVIEW B 87, 054437 (2013) FIG. 9. (Color online) Amount of spin reorientation /Delta1θobtained at the SmFeO 3surface as function of the light-penetration depth δ for two different surface temperature changes /Delta1T. The vertical lines indicate the SmFeO 3penetration depth for the indicated wavelengths. value problem where θ(−∞ ) andθ(0) are given and then to search for the resolved spin configuration minimizing the totalenergy as function of θ(0). 42In those calculations, the laser heating is described by T(y)=T0+/Delta1T eαy, where T0is the SmFeO 3temperature without laser heating, /Delta1T is the laser- induced temperature change at the surface, and α=4πκ/λ is the optical absorption of the SmFeO 3in which κis the extinction coefficient from the complex index of refraction ˜n=n+ik. Given a certain laser-induced temperature change /Delta1T at the surface, the amount of spin reorientation /Delta1θas a function of the light penetration depth δ=1/αcan be calculated and is shown in Fig. 9for two different temperature induced changes. In addition, the amount of spin reorientation/Delta1θas a function of depth from the SmFeO 3surface for the large temperature increase of /Delta1T=80 K is shown in Fig. 10. At very low absorption, such as when λ=800 nm, the penetration depth of the light is long enough to induce aquasihomogeneous heating such that no limitation of theamplitude of the spin reorientation from the exchange springformation is to be expected. The limiting factor in this caseis only the unbalanced heating between the Co film and theSmFeO 3substrate, as demonstrated in the previous section. FIG. 10. (Color online) Spin reorientation /Delta1θ as function of depth from the surface for different wavelengths in the case of a surface temperature change /Delta1T=80 K.At the opposite case of strong absorption, which corre- sponds to experiments with λ=400 nm wavelength, the achievable SRT is found to be limited by the formation ofan exchange spring with the cold bulk. Increasing the laserfluence, i.e., the temperature change at the SmFeO 3surface /Delta1T, partly overcomes this pinning effect by heating deeper into the sample. However, a five-times increase of the laserfluence still leads to exchange spring effects, as shown by the70 ◦reorientation obtained in this case. This is better visualized in the reorientation profile in Fig. 10where this 70◦occurs only within the first 100 nm of the single crystal. Nevertheless, thisachievable reorientation is much larger than what has beenobserved in the experiments shown in Fig. 5. At intermediate absorption corresponding to experiments realized with 532 nm wavelength, a large spin reorientationcan be expected even for moderate heating, and completereorientation should be achieved for stronger heating. Thisfinding is somewhat in conflict with the results shown in Fig. 6 where large heating effects are demonstrated by the transientdemagnetization observed for base temperature at the end ofthe SRT, while small reorientation is observed when coolingdown through the SRT. The comparison of these calculations with the experiments strongly suggests that there must be another effect takingplace which prevent the orthoferrite to fully reorient after laserheating. It should be noted that in fact a 90 ◦laser-induced spin reorientation in any rare-earth orthoferrite has not yetbeen experimentally demonstrated. The few studies done showa maximum 20 ◦reorientation.35,43–45It seems that ultrafast laser heating is not equivalent to slow heating. Such an effecthas been reported in another system has well, 46where it was suggested that the laser pulse brings the system intoa metastable state not accessible otherwise. While furtherexperiments would be required to unveil this possible transientmetastable state, it is nevertheless interesting to investigate thepossible coupled-spin dynamics that could occur with a 90 ◦ SRT. C. Coupled-spin dynamics To study how a thin film of cobalt might react to the spin dynamics of the underlying orthoferrite substrate, a numberof simplifying assumptions are made. It is assumed that theCo moment interacts directly with the Fe moments via anearest-neighbor ferromagnetic exchange mechanism, and theexchange field that the Co moments see is derived from thenet magnetization of bulk orthoferrite—thus the orthoferrite is treated as an external field that is not affected by the presenceof the cobalt and the associated interface geometry. Theinteraction term is, therefore, H Co,exchange =1 2Jcmc·m. (8) In the above, mandmcare the total magnetic moments per unit cell of the orthoferrite and the cobalt, respectively, and Jc is their exchange interaction parameter. The factor of one half arises from the Co moment interacting with only two of thefour sublattice magnetizations of the orthoferrite. Finally, therigid moment approximation is considered for the Co systemand thus spatial fluctuations in the magnetic moment within 054437-8DYNAMICS OF LASER-INDUCED SPIN REORIENTATION ... PHYSICAL REVIEW B 87, 054437 (2013) the Co layer are also not considered. These simplifications, whilst severe, make the problem theoretically tractable. The Landau-Lifschitz-Gilbert (LLG) equation for the Co moment is dmc dt=−γ 1+α2c/bracketleftbigg mc×Bc,eff−αc mcmc×(mc×Bc,eff)/bracketrightbigg , (9) where γis the gyromagnetic ratio and αcis an empirical damping parameter for Co. Here the time-dependent effectivefield is B c,eff(t)=−dH(t) dmc=−1 2Jcm(t), (10) where the temporal evolution of the orthoferrite magnetization m(t) is entirely determined by the evolution of the orthoferrite AFM moment l(t)v i a m(t)/similarequal1 2Jl(t)×D. (11) In the above, Jis the exchange interaction parameter and Dis the Dzyaloshinskii-Moriya interaction vector parameter for the orthoferrite. Equation (11) embodies the “slave” approximation to the magnetization45,47in which the anti- ferromagnetic moment dynamics is well approximated viadamped harmonic motion—the so-called pendulum approach. A derivation for the present context is given as supplementarymaterial, 48which gives l(t) as a solution to a nonlinear damped harmonic oscillator [Eq. (22) in Ref. 48]. Equation (11) is an approximation to Eq. (19) in Ref. 48, which ignores the effect of the precessional dynamics of the AFM moment on thetotal moment. For the present system, this was found to bevalid. To include the expected thin film demagnetization field, an anisotropy term is added to the magnetic energy ofthe form H SA=KSA[1−(ˆmc·ˆe)2], (12) where the reference direction is defined as ˆe=(0,0,1) and KSA=−μ0M2 c/2. Here Mcis the appropriate magnetization for Co. Taking the magnetization of bulk Co to be Mc= 1400×103A/m,KSA=− 1.2×106J/m3, which corre- sponds to an effective demagnetization field magnitude μ0Mc of approximately 1.8 T. Figure 11displays the three-dimensional evolution of the Co moment for the orthoferrite exchange field arising fromthe orthoferrite reorientations at the temperatures 460, 464, and468 K—simulated in Ref. 48and shown in Fig. 1(a) in Ref. 48. The Co moment direction is initially orientated perpendicularto the AFM moment under the assumption that it initially alignswith the total magnetic moment of the orthoferrite. In Fig. 11, the left panels display the in-plane angular evolution and theright panels display the out-of-plane evolution. The dampingtermα cis chosen to be 0.014 which is the known value for bulk Co. The upper panels of Figs. 11(a) and11(b) correspond to an exchange interaction between the orthoferrite and the CoofJ cm0=− 0.1 T, the middle panels (c) and (d), correspond toJcm0=− 10 T, and the lower panels (e) and (f) correspond toJcm0=− 150 T. The lower limit is that used in Sec. IIIto fit the experimental data, whereas the upper limit is comparable to0 200 400 600 800 1000 time (ps)-80-60-40-2002040φ(t) 0 200 400 600 800 1000 time (ps)89.59090.59191.5 θ(t) 0 50 100 150 200 time (ps)-135-90-4504590135φ(t) 0 50 100 150 200 time (ps)0306090120150 θ(t) 460K 464K 468K 0 50 100 150 200 time (ps)-90-4504590φ(t) 0 50 100 150 200 time (ps)6090120 θ(t))b( )a( (c) (d) (e) (f) FIG. 11. (Color online) Angular evolution of the Co moment for the (left panels) in-plane and (right panels) out-of-plane component for three different values of Jcm0, where the upper-two panels are at −0.1 T, the middle two at −10 T, and the lower two at −150 T. In all cases the damping coefficient of the Co moment was set to αc= 0.014. For each figure, the three curves correspond to the anisotropy energy landscape corresponding to the temperatures 460, 464, and468 K. the exchange interaction within the orthoferrite. The remaining value of −10 T is taken as an intermediate value. It is seen that, with an increasing ferromagnetic exchange interaction, the time scale of the Co reorientation decreases,where for J cm0∼− 150 T, the Co has converged to the reorientation time scale of the orthoferrite [compare to Fig. ( 1) of Ref. 48]. This may be partially understood by estimating the relaxation time for the Co moment. Inspection of Eqs. (9) and(11) reveals that 2 γαcJcm0D/J has units of inverse time and corresponds to the relaxation time of the damped LLGdynamics. Using the parameters of Fig. 11, the corresponding relaxation times are approximately 120, 1.2, and 0.08 ns forJ cm0corresponding, respectively, to −0.1,−10, and −150 T. For the case of Jcm0equal to −10 and −150 T, these time scales are quite compatible with the relaxation entailed inFig. 11, however, for the case of J cm0=− 0.1 T it is clearly an overestimate. This is most likely due to the now-dominantdemagnetization field of 1.8 T arising from the thin-filmanisotropy which results in little out-of-plane dynamics of the 054437-9L. LE GUY ADER et al. PHYSICAL REVIEW B 87, 054437 (2013) Co moment, a contribution that has not been included in the above time estimate. It is in this regime, with Jcm0=− 0.1T that this simple model reproduces quite well the measured Coreorientation of 4 ◦as shown in Fig. 3in Sec. IIIas continuous lines, including the initial overshot which corresponds to adamped spin precession. It would be tempting to push thecomparison further; however, one should keep in mind that themodel used is rather simple and that the SmFeO 3dynamics is more complex than initially anticipated. V . CONCLUSIONS Ultrafast-laser-induced spin reorientation in a Co/SmFeO 3 heterostructure was investigated employing time-resolvedXMCD PEEM imaging techniques. It is found that, subsequentto the laser-induced heat pulse, the Co spin direction changeswithin 100 ps to a new orientation under the influence ofthe orthoferrite substrate. However, the amount of changethat can be obtained in these experiments is at most 10 ◦ compared to the 90◦achievable in the static case. Simulations of the heat profile induced in the heterostructure and ofthe resulting equilibrium spin configuration in the orthofer-rite substrate done by considering the competition betweenthe exchange and anisotropy energy, and comparison of thesesimulations with the experiments suggest that the dynamicsof the reorientation in the SmFeO 3is more complex than that driven by an adiabatic heating. Single-shot time-resolvedmeasurement in different orthoferrites showing the SRT couldgive insight into the effect responsible for the limited laser-induced reorientation. Nevertheless, fast laser control of aferromagnet via an antiferromagnet is demonstrated in thissystem without apparent loss of coupling between the twolayers. ACKNOWLEDGMENTS This work was partially supported by de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO),NanoSci-E+ program, Foundation for Fundamental Research(FOM) and the Technology Foundation (STW), the European Union’s Seventh Framework Programme (FP7/2007-2013) Grants No. NMP3-SL-2008-214469 (UltraMagnetron) andNo. 214810 (FANTOMAS), as well as the European ResearchCouncil (FP7/2007-2013)/ERC Grant No. 257280 (Femto-magnetism). Part of this work was performed at the SwissLight Source, Paul Scherrer Institut, Villigen, Switzerland. Wethank C. Milne for the use of the Duetto laser, A. Bullemerand M. 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PhysRevX.7.031014.pdf

Stochastic p-Bits for Invertible Logic Kerem Yunus Camsari,*Rafatul Faria, Brian M. Sutton, and Supriyo Datta† School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA (Received 16 November 2016; revised manuscript received 27 April 2017; published 20 July 2017) Conventional semiconductor-based logic and nanomagnet-based memory devices are built out of stable, deterministic units such as standard metal-oxide semiconductor transistors, or nanomagnets with energy barriers in excess of ≈40–60kT. In this paper, we show that unstable, stochastic units, which we call “p-bits,”can be interconnected to create robust correlations that implement precise Boolean functions with impressive accuracy, comparable to standard digital circuits. At the same time, they are invertible , a unique property that is absent in standard digital circuits. When operated in the direct mode, the input is clamped,and the network provides the correct output. In the inverted mode, the output is clamped, and the network fluctuates among all possible inputs that are consistent with that output. First, we present a detailed implementation of an invertible gate to bring out the key role of a single three-terminal transistorlikebuilding block to enable the construction of correlated p-bit networks. The results for this specific, CMOS- assisted nanomagnet-based hardware implementation agree well with those from a universal model for p-bits, showing that p-bits need not be magnet based: any three-terminal tunable random bit generator should be suitable. We present a general algorithm for designing a Boltzmann machine (BM) with a symmetric connection matrix [ J](J ij¼Jji) that implements a given truth table with p-bits. The [ J] matrices are relatively sparse with a few unique weights for convenient hardware implementation. We then show how BM full adders can be interconnected in a partially directed manner ( Jij≠Jji) to implement large logic operations such as 32-bit binary addition. Hundreds of stochastic p-bits get precisely correlated such that the correct answer out of 233(≈8×109) possibilities can be extracted by looking at the statistical mode or majority vote of a number of time samples. With perfect directivity ( Jji¼0) a small number of samples is enough, while for less directed connections more samples are needed, but even in the former case logical invertibility is largely preserved. This combination of digital accuracy and logical invertibility is enabled by the hybrid design that uses bidirectional BM units to construct circuits with partially directed interunit connections. We establish this key result with extensive examples including a 4-bit multiplierwhich in inverted mode functions as a factorizer. DOI: 10.1103/PhysRevX.7.031014 Subject Areas: Electronics, Magnetism, Spintronics I. INTRODUCTION Conventional semiconductor-based logic and nanomag- net-based memory devices are built out of stable, deter-ministic units such as standard metal-oxide semiconductor(MOS) transistors, or nanomagnets with energy barriers in excess of ≈40–60kT. The objective of this paper is to introduce the concept of what we call “p-bits”representing unstable, stochastic units which can be interconnected to create robust correlations that implement precise Boolean functions with impressive accuracy comparable to standarddigital circuits. At the same time, this “probabilistic spin logic”(PSL) is invertible , a unique property that is absentin standard digital circuits. When operated in the direct mode, the input is clamped, and the network provides the correct output. In the inverted mode, the output is clamped, and the network fluctuates among all possible inputs thatare consistent with that output. Any random signal generator whose randomness can be tuned with a third terminal should be a suitable building block for PSL. The icon in Fig. 1(b)represents our generic building block whose input I icontrols the output mi according to the equation [Fig. 1(a)] miðtÞ¼sgnfrand ð−1;1Þþtanh½IiðtÞ/C138g; ð1Þ where rand ð−1;þ1Þrepresents a random number uni- formly distributed between −1and þ1. It is assumed to change every τseconds, which represents the retention time of individual p-bits. We normalize the time axis to τso that tis dimensionless and progresses in steps ( 0;1;2;…). At each time step, if the input is zero, the output takes on a value of −1orþ1with equal probability, as shown in the middle panel of Fig. 1(d). A negative input Iimakes*kcamsari@purdue.edu †datta@purdue.edu Published by the American Physical Society under the terms of theCreative 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.PHYSICAL REVIEW X 7,031014 (2017) 2160-3308 =17=7(3)=031014(19) 031014-1 Published by the American Physical Societynegative values more likely (left-hand panel) while a positive input makes positive values more likely (right-hand panel). Figure 1(c)shows m iðtÞas the input is ramped from negative to positive values. Also shown is the time-averaged value of m i, which equals tanh ðIiÞ. A possible physical implementation of p-bits could use stochastic nanomagnets with low-energy barriers Δwhose retention time [1], τ¼τ0expðΔ=kTÞ; is very small, on the order of τ0, which is a material- dependent quantity called the attempt time and is exper-imentally found to be ≈10ps−1ns[1]among different magnetic materials. Such stochastic nanomagnets can be pinned to a given direction with spin currents that are atleast an order of magnitude less than those needed to switch40-kT magnets. The sigmoidal tuning curve in Fig. 1(c)describing the time average of a fluctuating signal repre- sents the essence of a p-bit. Purely CMOS implementations of ap-bit are possible [2,3], but the sigmoid seems like a natural feature of nanomagnets driven by spin currents. Indeed, the use of stochastic nanomagnets in the contextof random number generators, stochastic oscillators, andautonomous learning [4–6]has been discussed in the literature. But performing “invertible ”Boolean logic uti- lizing large-scale correlations has not been discussed beforeto our knowledge. Note that we are using the term invertibility in the broader sense of relation inverses and not in thenarrower sense of function inverses. For example, AND,when interpreted as a relation, consists of the set ff1;1→1g;f0;0→0g;f1;0→0g;f0;1→0gg, where each term is of the form fA; B→AND ðA; B Þg. The relation inverse of 0 is the set ff0;0g;f0;1g;f1;0gg even though the corresponding functional inverse is not(d)(c) (a) (b) FIG. 1. Generic building block for PSL. (a) Generic model for PSL described by Eq. (1)with distinct READ and WRITE units represented by the RandWicon shown in (b). Useful functionalities are obtained by interconnecting RandWunits according to Eq. (2), Ii¼I0ðhiþPJijmjÞ, with appropriately designed fhgand [J]. (c) The blue trace shows the “magnetization ”(mi) obtained from Eq.(1)as the current ( Ii) is ramped. The red trace shows the sigmoid response obtained from a RCcircuit which provides a moving average of the time-dependent “magnetization ”that agrees very well with the black curve showing tanh ðIiÞ. The bias terminal could involve a voltage ( V) instead of a current ( I), just as the output could involve quantities other than magnetization. (d) The idealized telegraphic behavior of the model is shown at various bias points along with corresponding distributions.CAMSARI, FARIA, SUTTON, and DATTA PHYS. REV. X 7,031014 (2017) 031014-2defined. What our scheme provides, probabilistically, is the relation inverse [7,8]. Ensemble average versus time average. —A sigmoidal response was presented in Ref. [9]for the ensemble- averaged magnetization of large barrier magnets biased along a neutral state. This was proposed as a building block for both Ising computers as well as directed belief networks and a recent paper [10] describes a similar approach applied to a graph coloring problem. By contrast,low-barrier nanomagnets provide a sigmoidal response for the time-averaged magnetization, and a suitably engineered network of such nanomagnets could cycle through the 2 N collective states at GHz rates, with an emphasis on the “low- energy states ”which can encode the solution to the combinatorial optimization problems, like the traveling salesman problem, as shown in Ref. [11]. Once the time- varying magnetization has been converted into a time- varying voltage through a READ circuit, a simple RC circuit can be used to extract the answer through a moving time average. For example, in Fig. 1(c) the red trace iss obtained from the rapidly varying blue trace using a RC circuit in a SPICE simulation. The central feature underlying both implementations is thep-bit that acts like a tunable random number generator, providing an intrinsic sigmoidal response for the ensemble- averaged or the time-averaged magnetization as a function of the spin current. It is this response that allows us to correlate the fluctuations of different p-bits in a useful manner by interconnecting them according to IiðtÞ¼I0/C18 hiðtÞþX jJijmjðtÞ/C19 ; ð2Þ where hiprovides a local bias to magnet iandJijdefines the effect of bit jto bit i, and I0sets a global scale for the strength of the interactions like an inverse “pseudotemper- ature”giving a dimensionless current Iito each p-bit. The computation of IiðtÞin terms of mjðtÞin Eq. (2)is assumed instantaneous; in hardware implementations there can be interconnect delays that relate mjðtÞto currents at a later timeIiðt0Þ. Equation (1)arises naturally from the physics of low- barrier nanomagnets, as we discuss above. Equation (2) represents the “weight logic ”for which there are many candidates such as memristors [12], floating-gate-based devices [13], domain-wall-based devices [14], and standard CMOS [15]. The suitability of these options will depend on the range of Jvalues and the sparsity of the Jmatrix. Equations (1)–(2)are essentially the same as the defining equations for Boltzmann machines introduced by Hinton and his collaborators [16], which have had enormous impact in the field of machine learning, but they are usually implemented in software that is run on standard CMOS hardware. The primary contributions of this paper are threefold.(i)Hardware implementation. —It may seem obvious that an unstable magnet could provide a naturalhardware for representing a p-bit, but we stress a less obvious point. To the best of our knowledge, simple two-terminal devices are not suitable forconstructing large-scale correlated networks of the type envisioned here. Instead, we need three- terminal building blocks with transistorlike gainand input-output isolation, as shown in Fig. 1(b) [9]. To stress this point, we describe a concrete implementation of a Boolean function using detailednanomagnet and transport simulations that are in good agreement with those obtained by the generic model based on Eq. (1). All other results in this paper are based on Eq. (1)in order to emphasize the generality of the concept of p-bits, which need not necessarily be nanomagnet based [17,18] . (ii)Boltzmann machines (BM) for invertible Boolean logic [Fig. 2(a)]. —Much of the current emphasis on BMs is on “learning ”giving rise to the concept of restricted Boltzmann machines [19]. By contrast, this paper is about Boolean logic, extending an established method for Hopfield networks [20] to provide a mathematical prescription to turn any Boolean truth table into a symmetric Jmatrix [Eq.(2), with J ij¼Jji], in one shot with no learning being involved. This design principle seems quite robust, functioning satisfactorily even when the J-matrix elements are rounded off, so that the required interconnections are relatively sparse andquantized, which simplifies the hardware implemen- tation. The numerical probabilities agree well with those predicted from the energy functional. EðfmgÞ ¼−I 0/C18X i;j1 2ðJijmimjÞþX ihimi/C19 ð3Þ using the Boltzmann law: PðfmgÞ ¼expð−EÞP i;jexpð−EÞ: ð4Þ Most importantly, we show that the resulting Boolean gates are invertible: not only do they provide the correct output for a given input, for a given output they provide the correct input(s). If thegiven output is consistent with multiple inputs, the system fluctuates among all possible answers. This remarkable property of invertibility is absentin standard digital circuits and could help provide solutions to the Boolean satisfiability problem (Fig. 8)[21]. (iii) Directed networks of BM [Fig. 2(b)]. —Finally, we show that individual BMs can be connected to perform precise arithmetic operations, which are theSTOCHASTIC p-BITS FOR INVERTIBLE LOGIC PHYS. REV. X 7,031014 (2017) 031014-3norm in standard digital logic, but quite surprising for BM, which are more like a collection of interactingparticles than like a digital circuit. We show that a 32-bit adder converges to the one correct sum out of 2 33≈8×109possibilities when the interaction parameter is suddenly turned up from, say, I0¼ 0.25toI0¼5. This can be likened to quenching a molten liquid and getting a perfect crystal. What weexpect is plenty of defects, distributed differently every time we do the experiment. That is exactly what we get if the individual BM full adders (FA)comprising the 32-bit adder are connected bidirec- tionally ( J ij¼Jji). But by making the connection between adders directed ( Jij≠Jji), we obtain the striking accuracy of digital circuits while largely retaining the invertibility of BM. This is a key resultthat we establish with extensive examples including a 4-multiplier which in inverted mode functions as a factorizer. Each of these three contributions is described in detail in the three sections that follow. II. EXAMPLE HARDWARE IMPLEMENTATION OF PSL To ensure that individual p-bits can be interconnected to produce robust correlations, it is important to have separate terminals for writing (more correctly biasing)and reading, marked WandR, respectively in Fig. 3(a). With in-plane magnetic anisotropy (IMA) nanomagnets(e.g., circular nanomagnets) this could be accomplished following existing experiments [22,23] using the giant spin Hall effect (GSHE). Recent experiments using a built-inexchange bias [24–27]could make this approach applicable to perpendicular magnetic anisotropy (PMA) as well. Note, however, that these experiments have all been performed with stable free layers, and would have to be carried outwith low-barrier magnets in order to establish their suit-ability for the implementation of p-bits. As the field progresses, one can expect the bias terminal to involve voltage control [28,29] instead of current control, just as the output could involve quantities other than magnetization.We now show a concrete implementation of a Booleanfunction using minimal CMOS circuitry in conjunctionwith stochastic nanomagnets through detailed nanomagnet and transport simulations that are in good agreement with those obtained from the generic model based on Eq. (1). Figure 3(a)shows a possible, CMOS-assisted p-bit that has a separate READ and WRITE path. The device consists of a heavy metal exhibiting GSHE that drives a circular magnet which replaces the usual elliptical magnets in orderto provide the stochasticity needed for the magnetization. Asmall read current, which is assumed to not disturb the magnetization of the free layer in our design, that flows through the fixed layer is used to sense the instantaneousmagnetization, which is amplified and isolated by twoinverters that act as a buffer. This structure is very similar to the experimentally demonstrated GSHE switching of ellip- tical magnets that were similarly read-out by a magnetictunnel junction (MTJ) [22], with the only exception that the elliptical magnets are replaced by circular magnets with an aspect ratio of one. This device could be viewedas replacing the free layers of the GSHE-driven MTJsdemonstrated in Ref. [22] with those in the telegraphic regime [23,30 –32]. In the presence of thermal noise the magnetization of such a circular magnet rotates in the plane of the circle without a preferred easy axis that would have arisen due tothe shape anisotropy, effectively making its thermal sta- bilityΔ≈0kT[33]. This magnetization can be pinned by a spin current that is generated by flowing a charge currentthrough the GSHE layer. The magnetic-field-driven sig- moidal responses of magnetization for such circular mag- nets have experimentally been demonstrated [34,35] , while the spin-current-driven pinning has not been demonstrated to our knowledge. Using validated modules for transport and magnetization dynamics [36] [Fig. 3(b)], we solve the stochastic Landau-Lifshitz-Gilbert (sLLG) equation inthe presence of thermal noise and a GSHE current. The following section shows detailed simulation parameters. Sigmoidal response. —A long-time average ( t¼500ns) of the magnetization hm zias a function of a GSHE- generated spin current is plotted in Fig. 3(e)that displays the desired sigmoidal characteristic for p-bits dictated by Eq.(1). The xaxis of Fig. 3(e) is normalized to the geometric gain factor that relates the charge current to thespin current exerted [37,38] : β≡Is Ic¼θSHLFM t/C20 1−sech/C18t λ/C19/C21 ; ð5Þ where θSHis the Hall angle, tis the thickness, and λis the spin-relaxation length of the heavy metal. The quantity β can be made to be much greater than 1 providing an intrinsic gain [39]; however, for the parameters used in the present examples, βis≈1.5. Another quantity that is used to normalize the xaxis of Fig.3(e)is the “thermal spin current ”that corresponds to the strength of the thermal noise that needs to be overcome for a circular magnet to be pinned in a given direction: Iths¼/C184q ℏ/C19 αðkTÞ; ð6Þ where qis electron charge, αis the damping coefficient of the magnet. Iths,Is, and Icall have units of charge current; therefore, we can define the dimensionless interaction parameter I0of Eq. (2)asI0≡βIc=Iths¼Is=Iths. It can be seen from Fig. 3(e) that when the applied spin current βIc=Iths¼Is=Iths≈10, the magnetization of the circular magnet is pinned in the /C6zdirections for theseCAMSARI, FARIA, SUTTON, and DATTA PHYS. REV. X 7,031014 (2017) 031014-4particular parameters. For PMA magnets with low barriers (Δ≪kT), the pinning current is independent of the volume as long as increasing the volume does not invalidate theΔ≪kTassumption. This can be analytically shown from a 1D Fokker-Planck equation [40], and we reproduce this behavior directly from sLLG simulations. For the in-plane(circular) magnets we consider here, the pinning current ingeneral has a M sand volume dependence and the dimen- sionless pinning current can be larger. Nevertheless, it is possible to estimate the thermal spin current for typical damping coefficients of α¼0.01–0.1, Ithsis≈0.25to−2.5μA. Pinning currents for superpar- amagnets are at least an order of magnitude smaller than the critical switching currents of stable magnets [41].Iths, defined by Eq. (6), also sets the scale for I0defined in Eq.(2), suggesting that a stochastic nanomagnet-based implementation of PSL could be more energy efficient than the standard spin-torque switching of stable magnets thatsuffer from high current densities. Need for three-terminal devices with READ-WRITE separation. —Note that a crucial function of the READ circuit and the CMOS transistors in this design is the ability to turn the magnetization into an output voltage that is proportional to m z, providing gain for fan out and isolation to avoid any read disturb. Indeed, a critical requirement for any other alternative implementations of p-bits is the need for three terminal devices with separate READ and WRITEpaths to provide gain and isolation. In this particular design these features come in by directly integrating CMOS transistors, but CMOS-free, all-magnetic designs with thesecharacteristics have been proposed [39,42] . Our purpose is to simply show how a p-bit can be realized by using experimentally demonstrated technology. Alternativedesigns are beyond the scope of this paper. READ circuit. —For the output to provide symmetric voltage swings on the GSHE layer, the minus supply V −needs to be set to VDD=2since VOUTranges between 0 and VDD.Vþis set to VDD=2þVR, where VRis a small READ voltage that is amplified by the inverters. We assume a simple, bias-independent MTJ model [43]: GMTJ¼G0ð1þP2mzÞ; ð7Þ where Pis the interface polarization and G0is the average MTJ conductance. Setting the reference resistance [Fig. 3] R0equal to G−1 0, the input voltage to the inverters, VMin Fig.2(d) becomes VM¼VDD 2þVR 2þmzP2: ð8Þ In the absence of a bias, hmzibecomes 0 and the middle voltage fluctuates around the mean hVMi¼ VDD=2þVR=2. This requires the inverter characteristic to be shifted to this value to produce a telegraphic outputthat fluctuates between 0 and V DDwith equal probability [Fig. 3(f)]. This shift is easily engineered by sizing the p-channel FET and n-channel FET transistors differently: a wider p-channel FET shifts the inverter characteristictowards V DD, as we show in the next section. Interconnection matrix. —A passive resistor network can be used as a possible interconnection scheme to correlatethep-bits, as shown in Fig. 4. A proper design of the interconnection matrix Jthat has only a few discrete values ensures a minimal number of different conductances ( G ij). In this demonstrated example the AND gate requires only two unique, discrete conductance values. The spin currents that need to be delivered to each p-bit are on the order of a few μA and can be generated with charge currents that are even smaller, due to the GSHEgain. This means the interconnection resistances R ijcould be on the order of 100kΩsince the voltage drops across these resistances are around VOUT−V−≈/C60.5V. Since (a) (b) FIG. 2. PSL designs discussed in this paper. (a) Basic Boolean elements (AND and OR, full adder) are implemented as Boltzmann machines based on symmetrically coupled networks with Jij¼Jji. (b) Complex Boolean functions like a 32-bit ripple carry adder or subtractor and 4-bit multiplier or factorizer are implemented by combining the reciprocal Boltzmann machines in a directed fashion.STOCHASTIC p-BITS FOR INVERTIBLE LOGIC PHYS. REV. X 7,031014 (2017) 031014-5the GSHE ground V−¼VDD=2simply shifts all the voltages to get symmetric /C6swings, we define the voltages ðV0 OUTÞi¼ðVOUTÞi−V−. Then input currents to each p-bit can be expressed [Fig. 4(a)]: ðIINÞi¼X jGijðV0 OUTÞþGiðV0 BIASÞð 9Þ assumingP jGij≪GGSHE, since the heavy metal resis- tances are typically much less than hundreds of k Ω.W e verify the validity of Eq. (9)bySPICE simulations, for the parameters chosen for these examples.(a) (b) (c) (d) (f)(e) FIG. 3. CMOS-assisted implementation of p-bits. (a) A possible CMOS-assisted implementation of p-bits that have separate READ- WRITE paths. A GSHE layer provides a spin current is able to pinthe magnetization of circular ferromagnets (FM) ( Δ≈0kT). The change in magnetization is sensed by a MTJ and amplified by twoCMOS inverters that act as a buffer, providing the necessaryisolation and gain. (b) Self-consistent, modular modeling of trans-port and magnetization dynamics. See “Assumptions of the model ” in the text. (c) Equivalent READ circuit. (d) SPICE -based average output voltage normalized to the VDD¼0.8V of 14-nm Fin Field- Effect Transistor (FinFET) high-performance (HP) inverters [44]. (e) sLLG-based average magnetization of the circular magnet as afunction of the spin current (averaged over 500 ns for each bias point with a time step of Δt¼0.05ps,10×10 6points per marker), normalized to the GSHE gain and the thermal noise strength Iths. (f) The time-dependent output voltage at various bias points.(a) (b) (c) (d) FIG. 4. Invertible AND gate. (a) Passive resistor network that is used to obtain the connection terms Jijto correlate p-bits. The output impedance Rij¼1=Gijis much smaller than the input impedance RGSHE, allowing separate voltages to add at the input of theithp-bit. (b) Explicit implementation of an AND gate based on Eq.(10). (c) When Cis clamped to 1, AandBspend most of their time in the (11) state, the only combination consistent with C¼1. (d) The invertible operation of the AND gate when the Cgate is clamped to a zero, while AandBare left floating. AandBbits fluctuate between three possible combinations consistent with C¼0,ðA; B Þ¼ð 00Þ;ð01Þ;ð10Þ. The time response of A,B,C voltages are normalized by VDD. Histogram is obtained by averaging over 200 ns of thresholded voltages, only the first 20 ns of A,B,Cvoltages are shown for clarity.CAMSARI, FARIA, SUTTON, and DATTA PHYS. REV. X 7,031014 (2017) 031014-6As a result, we observe that Eq. (9)constitutes a hardware mapping for the interconnections of Eq. (2).I n this scheme Gijconductances are initially adjusted to obtain a global interaction strength I0for a given problem. Alternatively, the interaction strength can be adjustedelectrically by varying the supply voltages. Invertible AND gate. —Figure 4(b) shows an explicit implementation of an invertible AND gate ( A∩B¼CÞ corresponding to [ J] and fhgmatrices [45]that have three unique, integer entries: J¼AB C A B C2 640−1þ2 −10 þ2 þ2þ203 75;h T¼½þ 1þ1−2/C138: ð10Þ In Fig. 4(d), we show the inverse operation of the AND gate where we clamp the output bit C to a 0 or 1 by the bias voltage attached to its input terminal. The interconnection resistance is chosen to be R0¼125kΩ that roughly provides ≈/C66μA of charge current to each p-bit, corresponding to an I0≈3.5for the chosen parameters. Generating the histogram. —At the end of the simulation (t¼200ns), we threshold the voltage output of A; B, and Cby legislating all voltages above VDD=2¼0.4Vt ob e1 , and below VDD=2to be 0. Then a histogram output for the thresholded word [ABC] is obtained and normalized to unitprobability. Clamping the output to 0 and letting AandB float, make AandBfluctuate in a correlated manner and they visit the three possible states (00, 01, 10) withapproximately equal probability. Resolving the output 0 tothe three possible input combinations is, in a way, “facto- rizing ”the output. Conversely, clamping the output to 1 produces a strong (11) peak in the histogram of [ABC],which is the only consistent input combination for C¼1 [Figs. 4(d)]. Assumptions of the model. —We make several simplify- ing assumptions while modeling the hardware implemen-tation of a p-bit. (1) The READ voltage that is amplified by the inverters produces a small current that passes through the circular magnet and might potentially disturb its currentstate. We assume that this current [labeled as I S2in Fig.3(b)] is negligible and does not affect the magnetiza- tion of the stochastic magnet. (2) We assume that the spin current generated by the heavy metal is deposited to thefree layer with perfect efficiency [ I 0 S1¼IS1in Fig. 3(b)]; however, depending on the interface properties this con- version factor can be less than 100%. (3) We also assume that the fixed layer does not produce a notable stray field onthe circular magnet. Note that the presence of such aconstant field would simply shift the sigmoidal behavior presented in Figs. 3(e)to the right (or left) and could have been offset by a constant bias current. (4) Finally, weneglect the resistance of the GSHE portion in the READcircuit [Fig. 3(c)], assuming the MTJ resistance would be dominant in this path. A. Detailed simulation parameters This section shows the details of simulation parameters for the hardware implementation of p-bits that we use for Figs. 3and4. sLLG for stochastic circular magnets. —The magnetiza- tion of a circular nanomagnet described as ˆm iis obtained from the stochastic Landau-Lifshitz-Gilbert equation: ð1þα2Þdˆmi dt¼−jγjˆmi× ⃗Hi−αjγjðˆmi׈mi× ⃗HiÞ þ1 qNiðˆmi× ⃗ISi׈miÞþ/C18α qNiðˆmi× ⃗ISiÞ/C19 ; ð11Þ where αis the damping coefficient, qis the electron charge, γis the electron gyromagnetic ratio, Isis the spin current that is assumed to be uniformly distributedover the total number of spins in the macrospin, N i¼MsVol.=μB,μBbeing the Bohr magneton. We assume that the spin current generated from the GSHE layer is polarized in the zdirection, such that ⃗ISi¼ISˆz. ⃗Hiis the effective field of the circular magnet, where the uniaxial anisotropy is assumed to be negligible, butthere is still a strong demagnetizing field. The thermal fluctuations also enter through the effective magnetic field: ⃗H i¼−4πMsmxˆxþ ⃗Hth,xaxis being the out-of-plane direction of the magnet, and hj ⃗Hthj2i¼2αkT=ðjγjMsVol:Þ in units [ ½Oe2=Hz/C138] with zero mean, and equal in all three TABLE I. Parameters used for simulations in Figs. 3and4. Parameters Value Saturation magnetization ( Ms) 300emu=cm3 Magnet diameter ( Φ), thickness ( t)15 nm, 0.5 nm MTJ polarization ( P) [Eq. (7)]0.5 MTJ conductance ( G0) [Eq. (7)]176 μS Damping coefficient ( α) 0.1 Spin Hall length, width [Eq. (5)]L¼W¼15nm Hall angle, spin relax. lengthθ¼0.5[46],λsf¼2.1nm[47] Spin Hall res. ( ρ), thickness ( t)200 μΩcm[48], 3.15 nm Temperature ( T) 300 K CMOS models 14-nm HP-FinFET [44] Supply and READ voltage VDD¼0.8V,VR¼0.5V Time step for transient sim. ( SPICE )Δt¼0.05psSTOCHASTIC p-BITS FOR INVERTIBLE LOGIC PHYS. REV. X 7,031014 (2017) 031014-7directions. Table Ishows the parameters we use in Figs. 3 and4. We note that this parameter selection is simply one possibility; many other parameters could have been usedwith no change in the basic conclusions. Obtaining the sigmoidal response of CMOS+sLLG. — Each data point in the sigmoids shown in Figs. 3and4is obtained by averaging the zcomponent of the magnetization after 500 ns, with a time step of Δt¼0.05ps. The CMOS inverter characteristics in conjunction with a sphericalrepresentation-based sLLG are obtained using the modularframework developed in Ref. [36]using HSPICE . 14-nm FinFET inverter characteristics. —Figure 5shows the input and output characteristics of the single anddouble inverters that are used to amplify the stochasticsignal that is generated by the MTJ (Fig. 3). At zero bias from the GSHE, the amplified signal V M[Eq.(8)] is in the middle of VþandV−, which is VDD=2þVR=2. The buffer response can be shifted to this value by increasing the size of p-channel FETs, as shown in Fig. 5. III. INVERTIBLE BOOLEAN LOGIC WITH BOLTZMANN MACHINES We now present a mathematical prescription that shows how any given truth table can be implemented in terms ofBoltzmann machines, in “one shot ”with no learning being involved, unlike much of the past work in this area (see, forexample, Refs. [49,50] ). In Sec. II, we choose a simple [ J] and fhgmatrix to implement an AND gate based on Ref.[45]. In this section, we outline a general approach to show how any truth table can be implemented in terms ofsuch matrices. Our approach, pictorially described in Fig. 6, begins by transforming a given truth table from binary (0,1) to bipolar ð−1;þ1Þvariables. The lines of the truth table are then required to be eigenvectors each with eigenvalueþ1, all other eigenvectors are assumed to have eigenvalues equal to 0. This leads to the following prescription for Jas shown in Fig. 6:½J/C138¼X i;j½S−1/C138ijuiu† j; ð12aÞ Sij¼u† iuj; ð12bÞ where uiare the eigenvectors corresponding to lines in the truth table of a Boolean operation and Sis a projection matrix that accounts for the nonorthogonality of the vectors defined by different lines of the truth table. Note that the resultant Jmatrix is always symmetric ( Jij¼Jji) with diagonal terms that are subtracted in our models such that Jii¼0. The number of p-bits in the system is made greater than the number of lines in a truth table through the addition of hidden units (Fig. 6) to ensure that the number of conditions we impose is less than the dimension of thespace defined by the number of p-bits. Another important aspect in the construction of [ J] is that an eigenvector u iimplies that its complement −uiis also a valid eigenvector. However, only one of these might belong to a truth table. We introduce a “handle ”bit to each uithat is biased ðhiÞto distinguish complementary eigenvectors. These handle bits provide the added benefit of reconfigur- ability. For example, AND and OR gates have comple-mentary truth tables, and a given gate can be electricallyreconfigured as an AND or an OR gate using the handle bit. Jmatrices for AND and FA. —We now provide the details of the Jmatrix for the AND gate, obtained using the prescription shown in Fig. 6based on Eq. (12a) . The eigenvectors of the truth table for the AND in Fig. 6areFIG. 5. 14-nm Predictive Technology Model, inverter or buffer. dc response of 14-nm HP FinFETs based on Ref. [44] for an inverter and buffer. Sizing the transistors differently allows theswitching point to be shifted.FIG. 6. Truth table to Jmatrix. A given truth table is first transformed from binary to bipolar variables by using thetransformation m¼2t−1, where mandtrepresent the mag- netization and binary values of the truth table. Additional bits areintroduced to each line of the truth table to ensure that theresultant Smatrix is invertible. The indices i,jcorrespond to the number of lines in the truth table. u i,ujare column vectors. As an example, we show auxiliary bits that result in an Smatrix equal to the identity matrix, since the eigenvectors are orthogonal. The J matrix is then obtained by Eq. (12a) , which ensures that the truth table corresponds to the low-energy states of the Boltzmannmachines according to Eq. (4). A handle bit of þ1is introduced to each line of the truth table, which can be biased to ensure thatthe complementary truth table does not appear along with thedesired one. This bit also allows a truth table to be electricallyreconfigured into its complement.CAMSARI, FARIA, SUTTON, and DATTA PHYS. REV. X 7,031014 (2017) 031014-8placed into a matrix U, such that U¼½u1u2u3u4/C138, where u1is the first row of the matrix shown in Fig. 6, u1¼½−1þ1þ1þ1þ1−1−1−1/C138T, and so on. In matrix notation, the Smatrix can be written as S¼UTU¼8I4×4: ð13Þ Then the Jmatrix becomes J¼X ij½S−1/C138ij|fflffl{zfflffl} 1=8δijuiu† j¼1=8X iuiu† i: ð14Þ Removing the diagonal entries by making Jii¼0and multiplying the matrix entries by 2, to obtain simple integers, JANDevaluates toJAND ¼0 BBBBBBBBBBBBBBB@0−1 001110 −1 0110001 010011 −10 01001 −11 0 1011000 −1 101 −1 0001 10 −1 10001 0100 −11 1 01 CCCCCCCCCCCCCCCA; ð15Þ with the notation [1 –5, auxiliary bit and handle bit; 6, “A”; 7,“B”;8 ,“C”]. Following a similar procedure, we use the following 14×14full adder matrix J FA: JFA¼0 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@00000004 −1−1−1−1−2−1 00000040 −1−12 −11 −1 00000400 −1−1−12 1 −1 00004000 −1−21 1 −11 00040000 −12 −1−11 −1 00400000 −11 1 −2−11 04000000 −11 −21 −11 40000000 −1 11121 −1−1−1−1−1−1−1−1 000000 −1−1−1−2 211100 −1−11 2 −12 −11 −11 −21 0 −10 −11 2 −1−12 1 −1−21 1 0 −1−10 1 2 −21 1 −11 −1−1 201110 −2 −1−1−11 −1 1110222 −201 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; ð16Þ with the notation [1 –9, auxiliary bits and handle bit; 10, “C in”; 11,“B”; 12,“A”; 13,“S”; 14,“Cout”]. These are the Jmatrices (AND and FA) that are used for all examples in the paper, except for the AND gatedescribed in Sec. II. Figure 10shows the “truth table ” operation of the full adder where all input or outputterminals are “floating ”using the Jmatrix of Eq. (16), showing excellent quantitative agreement with the Boltzmann distribution of Eq. (4)at steady state even for the undesired peaks of the truth table. Note that this prescription for [ J] is similar to the principles developed originally for Hopfield networks [Ref. [51] and Eq. (4.20) in Ref. [20]]. However, other approaches are possible along the lines described in thecontext of Ising Hamiltonians for quantum computers [45]. We have tried some of these other designs for [ J], and many of them lead to results similar to those we present here. For practical implementations, it is important to evaluate different approaches in terms of their demands on thedynamic range and accuracy of the weight logic. Description of universal model. —Once a Jmatrix and thehvector are obtained for a given problem, the system is initialized by randomizing all m iat time t¼t0. First, the current (voltage) that a given p-bit (mi) feels due to the other coupled mjis obtained from Eq. (2), and the mivalue is updated according to Eq. (1). Next, the procedure is repeated for the remaining p-bits by finding the current they receive due to all other miusing the updated values ofSTOCHASTIC p-BITS FOR INVERTIBLE LOGIC PHYS. REV. X 7,031014 (2017) 031014-9mi. For this reason, the order of updating is chosen randomly in our models and we find that the order of updating has no effect in our results. However, updating the p-bits in parallel leads to incorrect results. These two observations are well known in the context of Hopfield networks and Boltzmann machines [52–54]. This type of serial updating corresponds to the “asynchronous dynamics ”[20,55] . We note that the hardware implementation we discuss in this paper naturally leads to an asynchronous updating of p-bits in the absence of a global clock signal. We have set up an online simulatorbased on this model in [56] so that interested readers can simulate some of the examples discussed in this paper. Figure 7shows the time evolution of an AND based on Eq.(15). Initially for t<t 0, the interaction strength is zero (I0¼0), making the pseudotemperature of the system infinite and the network produces uncorrelated noise visiting each state with equal probability. In the second phase ( t>t 0), the interaction strength is suddenly increased toI0¼2, effectively “quenching ”the network by reducing the temperature. This correlates the system such that only the states corresponding to the truth table of the AND gate arevisited, each with equal probability when a long-timeaverage is taken. The average probabilities in each phase quantitatively match the Boltzmann law defined by Eq. (4). In Fig. 8, we show how a correlated network producing a given truth table can be used to do directed computation analogous to standard CMOS logic. An OR gate isconstructed by using the same [ J] matrix for an AND gate, but with a negated handle bit. By “clamping ”the input bits of an OR gate ( t<t 0) through their bias terminals hito ðA; B Þ¼ð þ 1;þ1Þ, the system is forced to only one of the peaks of the truth table, effectively making C ¼1. The PSL gates, however, exhibit a remarkable difference with standard logic gates, in that inputs and outputs are on an equal footing. Not only do clamped inputs give the corre- sponding output, a clamped output gives the correspondinginput(s). In the second phase ( t>t 0), the output of the OR gate is clamped to þ1, which produces three possible peaks for the input terminals, corresponding to various possibleinput combinations that are consistent with the clamped output ðA; B Þ¼ð 0;1Þ, (1,0), and (1,1). The probabilistic nature of PSL allows it to obtain multiple solutions[Fig. 8(c)]. It also seems to make the results more resilient tounwanted noise due to stray fields that are inevitable in physical implementations, as shown in Fig. 9. Here, we simulate an AND gate in the presence of a normally distributed random noise that enters the bias fields of each p-bit and define the computation to be faulty, if the mode (most frequent value) of the output bit is not consistent withthe programed input combinations after T¼100time steps. We observe that even large levels of uncontrolled noise produce correct results with high probabilities. Figure 10shows the design of a full adder with the 8-line truth table shown. There are three inputs in all, two FIG. 7. Correlated p-bits, AND gate. When the interaction strength ( I0) is zero, p-bits produce uncorrelated noise, visiting all possible states with equal probability. In this example, the interaction strength (pseudo inverse temperature) is suddenly increased from 0 to 2 as a step function at t¼t0, to effectively “quench ”the network. This correlates the p-bits to produce the truth table of an AND gate (AND: A∩B¼C). Note that after this quenching, the p-bits visit only the low-energy states corresponding to the truth table of the AND gate, and once the system is in one of the low-energy states, it tends to stay there for a while, until being kicked out by the thermal noise. Thetime averages of the uncorrelated and the correlated system are well explained by the Boltzmann law stated in Eq. (4). The total simulation uses T¼4×10 6steps to compare the results with the Boltzmann distribution, though only a fraction are shown in the upper panel for clarity.CAMSARI, FARIA, SUTTON, and DATTA PHYS. REV. X 7,031014 (2017) 031014-10from the numbers to be added and one carry bit from previous FA. It produces two outputs, one the sum bit and the other a carry bit to be passed on to the next FA. The probabilities of different states are calculated usingJ FAfrom Eq. (16),w i t h I0¼0.5in the truth table mode, where all inputs and outputs are floating and the states are numbered using the decimal number corresponding to the binary word ½CiABSC o/C138. The decimalnumbers corresponding to the truth table are shown in the inset, and these match the location of the taller peaks in the histogram. Note that the Boltzmann distribution [Eq. (4)] quantitatively matches the model even for the suppressed peaks. A higher I0reduces these suppressed peaks further. The statistics are collected for T¼106steps, and each terminal output is then placed in the histogram. FIG. 9. Noise tolerance of AND. The probability of a wrong output for an (AND) gate [Eq. (15)] operated with clamped inputs is investigated in the presence of a random noise field whichenters Eq. (2)as indicated in the figure. The noise is assumed to be uniformly distributed over all p-bits in a given network, and centered around zero with magnitude /C6~h n, where ðI0¼2;hi¼/C61Þ. Each gate is simulated 50 000 times for T¼100time steps to produce an error probability for a given noise value, and the maximum peak produced by the system isassumed to be an output that can be read with certainty. The systemshows robust behavior even in the presence of large levels of noise.FIG. 10. Full adder. Full adder in the truth table mode, whereall inputs and outputs are floating, calculated using J FAfrom Eq. (16), with I0¼0.5. The statistics are collected for T¼106steps, and each terminal output is then placed in the histogram. The states are numbered using the decimal number corresponding to the binary number ½CiABSC o/C138. The decimal numbers corresponding to the truth table are shown inthe inset, and these match the location of the taller peaks in thehistogram. Note that the Boltzmann distribution [Eq. (4)] quan- titatively matches the model even for the suppressed peaks.FIG. 8. Implementing a Boolean function and its inverse.: The input or output terminals of an appropriately interconnected network ofp-bits can be “clamped ”to perform a specific logic operation or its inverse . In this example, the input bits ðA; B Þof an OR gate are clamped to be þ1, forcing the output bit Cto be 1, during the first phase of operation ( t<t 0). In the second phase of operation ( t>t 0), the output of the OR gate Cis clamped to the value þ1, which is consistent with three different combinations of ðA; B Þ. As shown in the time response and the long-time histogram plots, all three possibilities emerge with equal probability, demonstrating the “inverse ” OR operation. In each case, the expected probabilities from the Boltzmann law [Eq. (4)] closely match those produced by the generic model, Eqs. (1)and(2), after running the system for 106steps. Only a fraction are shown in the upper panel for clarity.STOCHASTIC p-BITS FOR INVERTIBLE LOGIC PHYS. REV. X 7,031014 (2017) 031014-11(a) (b) (c)10×109 5×109 5×10910×109 FIG. 11. 32-bit ripple carry adder (RCA). (a) 32-bit ripple carry adder is designed using individual full adder units with the carry bit designed as a directed connection from the least significant bit to the most significant bit. The overall Jmatrix for a 32-bit adder Jmatrix is shown, and it is quite sparse and quantized. (b) For t<t 0,I0¼0and the sum fluctuates randomly. At t¼t0,I0is suddenly increased, and the adder converges on the correct result for two random inputs AandB. The distribution of 1000 data points ( t>t 0) shows a single peak with 24% probability of time spent in the correct state (not including the uncorrelated time points for t<t 0). (c) Even though the connections between the full adder units are directed, the system performs the inverse function as well. When the output ( S) is clamped to a fixed number, the inputs ( A) and ( B) fluctuate in a correlated manner to make AþB¼Swhen I0¼1. Note the broad distributions ofAandB(collected for t>t 0) as compared to the extremely sharp distribution of AþB.CAMSARI, FARIA, SUTTON, and DATTA PHYS. REV. X 7,031014 (2017) 031014-12IV. DIRECTED NETWORKS OF BOLTZMANN MACHINES When constructing larger circuits composed of individ- ual Boltzmann machines, the reciprocal nature of theBoltzmann machine often interferes with the directednature of computation that is desired. It seems advisable to use a hybrid approach. For example, in constructing a 32-bit adder we use full adders that are individually BMswith symmetric connections, J ij¼Jji. But when connect- ing the carry bit from one FA to the next, the coupling element Jijis nonzero in only one direction from the least significant to the most significant bit. This directed cou- pling between the components distinguishes PSL from purely reciprocal Boltzmann machines. Indeed, even thefull adder could be implemented not as a Boltzmannmachine but as a directed network of more basic gates. But then it would lose its invertibility. On the other hand, the directed connection of BM full adders largely preservesthe invertibility of the overall system, as we show. A. 32-bit adder or subtractor Figure 11shows the operation of a 32-bit adder that sums two 32-bit numbers AandBto calculate the 33-bit sum S. In the initial phase ( t<t 0), we have I0¼0corresponding to infinite temperature so that the sum bits ( S) fluctuate among 233≈8×109possibilities. With I0¼1, Fig. 11 shows that the correct answer has a probability of ≈12%, which is much lower than the ≈100% that can be achieved with larger I0values (as in Figs. 13(c) with I0¼5). Nevertheless, the peak is unmistakable, as evident fromthe expanded scale histogram, and the correct answer isextracted from the majority vote of T¼100samples, as shown in Fig. 13. This ability to extract the correct answer despite large fluctuations is a general property of probabi-listic algorithms. Interestingly, although the overall system includes several unidirectional connections, it seems to be able to perform the inverse function as well. With AandBclamped it calculates S¼AþB, as noted above. Conversely, with Sclamped, the input bits AandBfluctuate in a correlated manner so as to make their sum sharply peaked around S. Figure 11shows the time evolution of the input bits that have broad distri-butions spanning a wide range. Initially, when I 0is small, the sum of AandBalso shows a broad distribution, but once I0 is turned up to 1, the distributions of AandBget strongly correlated making the distribution of AþBsharply peaked around the fixed value of S. It must be noted that the 32-bit adder shown in Fig. 11is not like standard digital circuits which are not invertible. The demonstration of such an invertible 32-bit adder could be practically significant, sincebinary addition is noted to be the most fundamental andfrequently used operation in digital computing [57]. Delay of ripple carry adder. —Just as in CMOS-based ripple carry adders (RCA), the delay of the p-bit-basedRCA is a function of the inputs AandB. In Fig. 12,w e have systematically studied the worst-case delay of the p-bit-based RCA as a function of increasing bit size. We selected a “worst-case ”combination that results in a carry that needs to be propagated from bit 1 to bit N, which results in a linear increase in the delay, exhibiting OðnÞ complexity with input size similar to CMOS implementa-tions [58]. When the inputs are random, the delay seems to increase sublinearly. The system is quenched at t¼0for different interaction parameters I 0and the delay is defined to be the time it takes for the system to settle to the mode of the array for T¼200. An error check has been carried out separately to ensure the calculated sum (mode) is alwaysexactly equal to the expected sum. For random inputs the 32-bit adder is close to 20 time steps, in accordance with the example shown in Fig. 11. Digital accuracy and logical invertibility. —The striking combination of accuracy and invertibility is made possibleby our hybrid design, whereby the individual full addersare Boltzmann machines, even though their connection is directed. Our 32-bit adder is more like a collection of interacting particles than like a digital circuit, as evidentfrom Fig. 13(a) , which shows a color map of the binary state of each of the 448 p-bits as a function of time with the interaction parameter I 0suddenly increased from 0.25 to 5 att0¼50, thereby quenching a “molten liquid ”into a “solid.”Nevertheless, it shows the striking accuracy of a digital circuit, with S–A–Bexactly equal to zero in each of the 1000 trials, as shown in Fig. 13(b) . We do not expect a FIG. 12. Ripple carry adder delay. The delay of the RCA as a function of number of bits in the ripple carry adder is shown. Theworst-case input combination generates a carry that propagatesall the way through bit 1 to bit N, and has a linear dependence on the number of bits, exhibiting OðnÞcomplexity. When the inputs are random, the delay increases logarithmically. The delay isdefined to be the time it takes for the network to reach the mode ofthe array for T¼200after getting quenched at t¼0. Each point is an average of 500 trials with random initial conditions for anI 0¼1.5, and the mode of the array is exactly equal to the arithmetic sum of the inputs in each case. The worst-case inputsareA¼0…000andB¼1…111with an input carry ( C inÞof 1. Results show a weak I0dependence.STOCHASTIC p-BITS FOR INVERTIBLE LOGIC PHYS. REV. X 7,031014 (2017) 031014-13molten liquid to be quenched into a “perfect crystal ”every time. Instead, we would expect a “solid full of defects ”with different nonzero values for S–A–Bin each trial. That is exactly what we get if the carry bits are bidirectional, as in afully BM implementation [Fig. 13(d) ]. Note, however, that this digital accuracy is achieved while maintaining the property of invertibility that isabsent in digital circuits. Figure 13is not for direct mode operation, but for the adder operating in reverse mode as asubtractor. It might be expected that the directed connectionof carry bits from the less significant to the more significantbit could lead to a loss of invertibility. To investigate thispoint, we show the error S–A–Bas a function of trial number (Fig. 14) for four different modes of operation with (i)AandBclamped (addition), (ii) SandAclamped (subtraction), (iii) A,B, and Sfor the 16 most significant bits (msb) clamped, and (iv) A,B, and Sfor the 16 least significant bits (lsb) clamped. The fully bidirectionalimplementation shows very large errors for all modes of operation. The directed implementation, on the other hand, works perfectly for both the adder and the subtractormodes. It also works if we clamp the least significant bits,but not if we clamp the most significant bits. This seemsreasonable since we expect to be able to control a flow bymaking changes upstream (lsb) but not downstream (msb). Partial directivity. —Thus far in our examples we have only considered fully directed ( J ij¼2J0,Jji¼0) or fullybidirectional ( Jij¼J0,Jji¼J0) carry bits when connect- ing the individual full adders. In Fig. 15, we systematically analyze the effects of partial directivity in the operation of a 32-bit adder. We observe that the 32-bit adder operates correctly even when there is a large degree of bidirection-ality ( J ji¼Jij×0.75) provided that the system is allowed to run for a long time, T¼50000 , in stark contrast to the fully directed case that could resolve the right answerwithin T¼100, shown in Fig. 14(b) . Decreasing the time steps systematically increases the error. Increasing the correlation parameter while keeping Tconstant also seems to adversely affect the bidirectional designs that might be getting the system stuck in local minima. Directionality and computation time, 2−p-bit model. — The qualitative relation between I 0,T, and bidirectionality J12=J21described above is derived from extensive numeri- cal simulations based on Eqs. (1)and(2). However, the broad features can be understood from a model involving just two p-bits, 1 and 2, with h¼/C200 0/C21 and J¼/C200J12 J21 0/C21 : It is straightforward to write a master equation des- cribing the time evolution of the probabilities of different configurations: FIG. 13. Accuracy of 32-bit adder, directed versus bidirectional. The results are shown for the adder operating in a subtractor mode, clamping one (random) 32-bit input ( A) and a (random) 33-bit output ( CoutþS), and observing the other 32-bit input B, which should provide the difference S–A. (a) Color map of the binary state of each of the 448 p-bits comprising the directed adder as a function of time with the interaction parameter I0suddenly increased from 0.25 to 5 at t0¼50. For low values of I0att<50, the collection of p-bits is like a molten liquid which is quenched at t0¼50into a solid. (b) Surprisingly, this solid corresponds to a “perfect crystal ”in each of the 1000 trial experiments, with S–A–Bexactly equal to zero (dark blue). (c) Same as (a) but for a bidirectional adder. Here, too, the “liquid ” quenches to a solid at t0¼50, but in this case the resulting “solid”is full of defects (with hardly any zeros), with S–A–B≠0, yielding a different wrong result for each trial as evident from (d). For (c) and (d) the color bar is modified to have a dark blue color correspondingto exactly zero. S,A,Bare taken to be the statistical mode of the 100×1array obtained at the end of each trial.CAMSARI, FARIA, SUTTON, and DATTA PHYS. REV. X 7,031014 (2017) 031014-14d dt2 6664P 11 P10 P01 P003 7775¼½W/C1382 6664P 11 P10 P01 P003 7775; Wbeing the transition matrix [20],P 00representing the probability of both p-bits being −1,P11both being þ1, and so on. We can write two matrices W1andW2describing the updating of p-bits 1 and 2, respectively: W1¼ð1;2Þð 11Þð10Þð01Þð00Þ ð11Þ ð10Þ ð01Þ ð00Þ2 6664p 0 p 0 0 p 0 p p 0 p 0 0 p 0 p3 7775; W 2¼ð1;2Þ ð11Þ ð10Þ ð01Þ ð00Þð11Þð10Þð01Þð00Þ 2 6664qq 00 q q 00 00 q q 00 qq3 7775;FIG. 14. Invertibility of 32-bit adder, directed versus bidirectional. An adder that provides the sum Sof two 32-bit numbers AandB: S¼AþB. The left-hand panel shows the adder implemented with bidirectional carry bits, while the right-hand panel shows one with carry bits directed from the least significant to the most significant bit. Four different modes are shown with (i) AandBclamped (addition), (ii) SandAclamped (subtraction), (iii) A,B, andSfor the 16 most significant bits (msb) clamped, and (iv) A,B, andSfor the 16 least significant bits (lsb) clamped. Note that the bidirectional implementation shows very large errors for all modes of operation. Thedirected implementation works perfectly for both the adder and the subtractor modes. It also works if we clamp the least significant bits,but not if we clamp the most significant bits. Correlation parameter I 0¼1,T¼100steps for all trials. S,A,Bare taken to be the mode (most frequent value) of the 100×1array obtained at the end of each trial. Clamped inputs are random 32-bit words for each trial, for a total of 1000 trials. FIG. 15. Error versus bidirectionality. The degree of bidirec-tionality J ji=Jijof the carry-out ( j) to carry-in ( i) link between the full adders is systematically varied while keeping the sumJ ijþJjiconstant. In each case the sum is obtained from the statistical mode (or majority vote) of Ttime samples over 50 trials. The yaxis shows the fraction of trials that yield the wrong result. Note that for large I0and small T, error-free operation is obtained only if bidirectionality is close to zero, similar tostandard digital circuits. But with I 0¼1.5andT¼50000 , error-free operation (at least for 50 trials) is obtained even with≈75% bidirectionality.STOCHASTIC p-BITS FOR INVERTIBLE LOGIC PHYS. REV. X 7,031014 (2017) 031014-15where Wði; jÞrepresents the probability that state ( j) makes a transition to state ( i), and ¯p¼1−p,¯q¼1−q.pandq are obtained from Eqs. (1)and(2): p¼1 2f1þtanh½I0ðJ12þh1Þ/C138g ¼1 2½1þtanhðI0J12Þ/C138; q¼1 2f1þtanh½I0ðJ21þh2Þ/C138g ¼1 2½1þtanhðI0J21Þ/C138: The overall transition matrix Wis given by W2×W1or W1×W2depending on which bit is updated first. Either way, the matrix Whas four eigenvalues, λ1¼1,λ2¼0,λ3¼0, andλ4¼ð2p−1Þð2q−1Þ¼tanhðI0J12ÞtanhðI0J21Þ, and the corresponding eigenvectors evolve with time ∼λT. The components corresponding to λ¼0decay instanta- neously while the eigenvector corresponding to λ¼1is the stationary result representing the correct solution. But for the system to reach this state, we have to wait for the fourtheigenvector corresponding to λ 4to decay sufficiently. A fully directed network has J21¼0, so that λ4¼0and the system quickly reaches the correct solution. But in abidirectional network with J 12¼J21, the fourth eigenvalue can be quite close to one, especially for large I0, and take anexponentially long time to decay, as λT¼expðTlnλÞ≈ exp½−Tð1−λÞ/C138when λis close to 1. This 2−p-bit model provides some insight into our general observation that directivity can be used to obtainaccurate answers quickly. However, depending on the problem at hand, it may be desirable to retain some degree of bidirectionality, since full directivity does lead to someloss of invertibility, as we see for one set of inputs in Fig.14. We discuss an example of a partially directed p-bit network in the next section. B. 4-bit multiplier or factorizer Figure 16shows how the invertibility of PSL logic blocks can be used to perform integer factorization usinga multiplier in reverse. Normally, the factorization pro- blem requires specific algorithms [59] to be performed in CMOS-like hardware; here, we simply use a digital 4-bitmultiplier working in reverse to achieve this operation. Specifically with the output of the multiplier clamped to a given integer from 0 to 15, the input bits float to thecorrect factors. The interconnection strength I 0is increased suddenly from 0 to 2 at t¼t0(Fig. 16) and the input bits (a) (b) (c) FIG. 16. Factorization through inverse multiplication. The reversibility of PSL allows the operation of integer factorization using a binary multiplication circuit implemented using the principles of digital logic using AND gates and full adders, as shown in (a). Theoutput nodes of a 4-bit multiplier are clamped to a given integer, and the system produces the only consistent factors of the product at theinput terminals, probabilistically. The interaction parameter I 0is suddenly increased to a saturation value of 2, and held constant as shown. (b) The output terminal is clamped to 9 and is factored into 3×3; note that 9×1is not an achievable solution in this setup since encoding 9 requires 4-bit inputs in binary, whereas inputs are limited to 2-bits. (c) The output terminal is clamped to 6 and afterbeing correlated, the factors cross-oscillate between 2 and 3. In both cases the histogram is obtained by counting outputs after t>t total=2¼1.25×104time steps to collect statistics after the system is thermalized.CAMSARI, FARIA, SUTTON, and DATTA PHYS. REV. X 7,031014 (2017) 031014-16get locked to one of the possible solutions. For example, when the output is set to 9, both inputs float to 3. With the output set to 6, both inputs fluctuate between two values, 2and 3. Note that factors like 9¼9×1do not show up, since encoding 9 in binary requires 4-bits (1001) and the input terminals only have 2-bits. We check other cases where factorizing 3 shows both 3×1and 1×3, and factorizing zero shows all possible peaks since there aremany solutions such that 0¼0×1, 2, 3 and so on. We also keep the same directed connections between the full adders for the carry bits, making them a directed network of Boltzmann machines, similar to the 32-bit adder. Moreover, we keep a directed connection from the full adders tothe AND gates, as shown in Fig. 16(a) since the information needs to flow from the output to the input in the case of factorization. The input bits that go to multiple AND gates are “tied”to each other with a positive exchange ( J>0) value much like 2-spins interacting ferromagnetically; however, in PSL we envision these interactions to be controlled purely electrically. In this example, we observe that the system is sensitive to therelative strengths of couplings within the AND gates and between the AND gates and the full adders, which can also depend on a chosen annealing profile. The design of factorizers of practical relevance is beyond the scope of this paper. Our main purpose is to establish how the key feature of invertibility of p-bits can be creatively used for different circuits with unique function-alities. The demonstration of 4-bit factorization through reverse multiplication is similar to memcomputing [60] based on deterministic memristors. Note, however, thatthe building blocks and operating principles of stochastic p-bits and memcomputing [61] are very different and the only similarity we note here is the fact that both approachestreat the input and output terminals on an equal footing. V. SUMMARY It is generally believed that (1) probabilistic algorithms can tackle specific problems much more efficiently than classical algorithms [62], and that (2) probabilistic algo- rithms can run far more efficiently on a probabilisticcomputer than on a deterministic computer [62,63] .A s such, it seems reasonable to expect that probabilistic computers based on robust room-temperature p-bits could provide a practically useful solution to many challengingproblems by rapidly sampling the phase space in hardware. In this paper, we present a framework for using prob- abilistic units or “p-bits”as a building block for a probabilistic spin logic, which is used to implement precise Boolean logic with an accuracy comparable to standard digital circuits while exhibiting the unique property ofinvertibility that is unknown in deterministic circuits.Specifically, first, we present an implementation based on stochastic nanomagnets to illustrate the importance of three-terminal building blocks in the construction oflarge-scale correlated networks of p-bits. We emphasize that this is just one possible implementation that is by no means the only one (Sec. II). Second, we present an algorithm for implementing Boolean gates as BM with relatively sparse and quantized J-matrix elements, bench- mark their operation against the Boltzmann law, and establish their capability to perform not just direct functions but also their inverse (Sec. III). Third, we present a 32-bit adder implemented as a hybrid BM that achieves digital accuracy over a broad combination of the interaction parameter I0, directionality, and the number of samples T. This striking accuracy is reminiscent of digital circuits, but it is achieved while preserving a certain degree of invertibility that is absent in digital circuits. The accuracy is particularly surprising with high degrees of bidirectionality (J12¼0.75×J21), where the system is picking out the one correct answer out of nearly 233≈8×109possibilities. This may require a larger number of time samples, but these could be collected rapidly at GHz rates (Sec. IV). We hope these findings will help emphasize a new direction for the field of spintronic and nanomagnetic logic by shifting the focus from stable high-barrier magnets to stochastic, low-barrier magnets, while inspiring a search for other possible physical implementations of p-bits. 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PhysRevB.97.224412.pdf

PHYSICAL REVIEW B 97, 224412 (2018) Subnanosecond magnetization reversal of a magnetic nanoparticle driven by a chirp microwave field pulse M. T. Islam,1X. S. Wang,1,2,*Y . Zhang,1,3and X. R. Wang1,3,† 1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China 3HKUST Shenzhen Research Institute, Shenzhen 518057, China (Received 16 March 2018; revised manuscript received 18 May 2018; published 13 June 2018) We investigate the magnetization reversal of a single-domain magnetic nanoparticle driven by a linear down- chirp microwave magnetic field pulse. Numerical simulations based on the Landau-Lifshitz-Gilbert equationreveal that a down-chirp microwave pulse is solely capable of inducing subnanosecond magnetization reversal.With a certain range of initial frequency and chirp rate, the required field amplitude is much smaller than thatof a constant-frequency microwave field. The fast reversal is due to the fact that the down-chirp microwave fieldpulse triggers stimulated microwave absorptions (emissions) by (from) the spin before (after) it crosses over theenergy barrier. Applying a spin-polarized current additively to the system further reduces the microwave fieldamplitude. Our findings provide a way to realize low-cost and fast magnetization reversal. DOI: 10.1103/PhysRevB.97.224412 I. INTRODUCTION The magnetization reversal of single-domain magnetic nanoparticles has drawn significant attention because of itsapplication in high-density data storage [ 1–3] and processing [4]. Fast magnetization reversal with minimal energy cost is the ultimate demand in device applications. To achieve highthermal stability and a low error rate, high-anisotropy materialsare used so that magnetic nanoparticles have a high-energybarrier [ 5]. It is difficult but essential to find out how to achieve the fastest magnetization reversal for high-anisotropy magneticnanoparticles with an energy cost that is as low as possible.Over the past a few years, a number of theoretical schemeshave been proposed and some of them have been verifiedby experiment. In the early years, a constant magnetic fieldwas used as the driving force to reverse the magnetization[6,7], but the reversal time is too long [ 6] and it suffers from scalability problems because the energy consumptionper unit area increases as the device feature size decreases.Since the discovery of spin transfer torque (STT) [ 8], the preferred way to reverse magnetization has been to deployspin-polarized electric current [ 9–16], and devices based on STT magnetization reversal have been fabricated. However,a large current density is required for fast reversal so thatsignificant Joule heat limits the device durability and reliability[17–19]. If the direction of the magnetic field or current varies with time in a designed way, the field/current amplitude orswitching time can be much lower [ 20,21] than that of a constant field/current. But it is strenuous to generate such kindsof fields/currents in practice. A microwave magnetic field,either with or without a polarized electric current, is another *Corresponding author: justicewxs@connect.ust.hk †Corresponding author: phxwan@ust.hkcontrolling knob for magnetization reversal [ 22–24]. A mi- crowave of constant frequency itself can reverse magnetizationthrough synchronization [ 7]. A large field amplitude is required and the reversal process is relatively slow [ 25–28]. Recently, there have been several studies demonstrating magnetizationreversal by microwaves of time-dependent frequency [ 29–33]. In Refs. [ 29,30], magnetization reversal is induced by a combination of a static field together with a radio-frequencymicrowave field pulse. A dc static field is necessary and is themain reversal force, while the microwave field is only used asa reinforcement. In Ref. [ 30], the frequency of the microwave is always chosen to be the resonance frequency, while inRef. [ 32] optimal microwave wave forms were designed. These kinds of schemes have a similar problem as the theoreticallimits [ 20,21] that are difficult to realize. In Ref. [ 33], a linear down-chirp microwave field was studied, but only positivefrequency fwas used so that stimulated microwave emission was not allowed (microwaves with positive and negative fre-quencies can respectively trigger a stimulated absorption and astimulated emission). Under such a microwave, magnetizationreversal is only fast before the spin crosses its energy barrier.It takes a long time for the spin to fall into its final statebecause it relies on natural damping. In Ref. [ 34], a linear chirp microwave was studied with a theoretically estimatedphase boundary of chirp rate and field amplitude. However,they did not provide a clear enough physical picture. A linearlypolarized microwave was not considered, either. Thus, a fastmagnetization reversal strategy with a relatively simple setupand a low-energy cost is still desired. In this paper, we showthat a circularly polarized down-chirp microwave pulse (amicrowave pulse whose frequency linearly decreases with timeand varies from f 0to−f0) can efficiently reverse the magneti- zation. For a nanoparticle of high uniaxial anisotropy (coercivefieldh k∼0.75 T), subnanosecond magnetization reversal can be achieved. With a proper choice of initial frequency 2469-9950/2018/97(22)/224412(6) 224412-1 ©2018 American Physical SocietyM. T. ISLAM, X. S. W ANG, Y . ZHANG, AND X. R. W ANG PHYSICAL REVIEW B 97, 224412 (2018) and chirp rate, the microwave field amplitude required for subnanosecond magnetization reversal is only several tens ofmT, much smaller than that required for a constant-frequencymicrowave field. The obtained reversal time is close to thetheoretical limit [ 20]. Also, we provide a clear physical picture for fast switching from an energy point of view. We furthershow that a linearly polarized down-chirp microwave fieldpulse is also capable of quickly reversing the magnetization.We also demonstrate that a spin-polarized current can worktogether with the down-chirp microwave field pulse so thatboth the applied current density and microwave amplitude arelow enough. II. MODEL AND METHODS We consider a spin valve with free and fixed ferromag- netic layers and a nonmagnetic spacer in between, as shownschematically in Fig. 1(a). Both fixed and free layers are perpendicularly magnetized. The magnetization direction ofthe fixed layer pis pinned upward, p=ˆz(ˆzis the unit vector along the zdirection). The magnetization of the free layer is treated as a macrospin with magnetization direction m and magnitude M s. The macrospin approximation is valid for device sizes smaller than 100 nm [ 35]. The Landau-Lifshitz- Gilbert (LLG) equation governs the magnetization dynamicsof the free layer in the presence of spin-polarized current anda microwave magnetic field [ 7,20,23,28], dm dt=−γm×Heff−γhsm×(p×m)+αm×dm dt,(1) where γis the gyromagnetic ratio, and αis the Gilbert damping constant. The total effective field Heffconsists of the microwave magnetic field Hmwand the anisotropy field HK=HKmzˆz, i.e., Heff=Hmw+HK.hsrepresents the intensity of spin transfer torque (STT) [ 8], hs=¯hPj 2eμ0Msd, (2) where j,e,¯h,P,μ0, andddenote the current density, electron charge, the Planck’s constant, spin polarization of current,the vacuum permeability, and thickness of the free layer, m pe−f0 −f0T)b( )a( FIG. 1. (a) Schematic diagram of the system. mandprepresent unit vectors of magnetization of free and fixed layers, respectively. A microwave field is applied onto the free layer, and an electric current flows through the spin valve. (b) The frequency of a down-chirpmicrowave (sweeping from +f 0to−f0).respectively. In the following study, the parameters are chosen from typical experiments on microwave-driven magnetizationreversal as M s=106A/m,Hk=0.75 T, γ=1.76×1011 rad/(T s),P=0.6,α=0.01, and d=2n m . The microwave field Hmwand the spin transfer torque are nonconservative forces. They do work to the macrospin. Wefirst consider solely microwave-driven magnetization reversal.Without the STT term, the rate of energy change of themacrospin is expressed as ˙ε=−α 1+α2|m×Heff|2−m·˙Hmw. (3) The first term is always negative because of the positive damping factor whereas the second term can be either positiveor negative for a time-dependent field. In other words, themicrowave field can either trigger stimulated energy absorptionor emission, depending on the angle between the instanta-neous magnetization direction and the time derivative of themicrowave field [ 23]. Due to the easy-axis anisotropy, the magnetization has two stable equilibrium states, m=± ˆz, corresponding to two energy minima. The goal of magnetization reversal is to movethe spin from one equilibrium state to the other. Along the way,the spin needs to cross an energy barrier at the equator ( m z= 0). Before mreaches the equator, it gains energy from external forces. After mpasses the equator, it releases energy through damping or through the negative work done by external forces.For a microwave field, the ideal case for fast magnetizationreversal is that the microwave always synchronizes to themagnetization motion so that m·˙H mwremains maximal before reaching the equator and remains minimal after passing theequator. However, this is difficult to achieve in practice. Wenotice that the internal effective field due to anisotropy isH K=HKmzˆz, which corresponds to a resonant frequency proportional to mz. During magnetization reversal from mz= 1t omz=− 1, the resonant frequency decreases while the spin climbs up the potential barrier and increases while it goesdown from the barrier where the spin precesses in the oppositedirection. This leads us to consider a down-chirp microwavepulse, whose frequency decreases with time. If the rate offrequency change matches the magnetization precession, themicrowave field roughly accommodates the magnetizationprecession, and it triggers stimulated microwave absorptions(emissions) by (from) magnetization before (after) the spincrosses the energy barrier so that magnetization reversal canbe fast. In order to demonstrate the feasibility of the above scenario, we apply a circularly polarized down-chirp microwave pulseon the system and numerically solve the LLG equation usingthe MUMAX 3 package [ 36]. The microwave field takes the form Hmw=Hmw[cosφ(t)ˆx+sinφ(t)ˆy], (4) where Hmwis the amplitude of the microwave field and φ(t) is the phase. We consider a linear chirp whose instantaneousfrequency f(t)≡ 1 2πdφ dtis linearly decreasing with time at a constant rate η(in units of s−2) as shown in Fig. 1(b), f(t)=f0−ηt, φ (t)=2π/parenleftBig f0t−η 2t2/parenrightBig , (5) 224412-2SUBNANOSECOND MAGNETIZATION REVERSAL OF A … PHYSICAL REVIEW B 97, 224412 (2018) FIG. 2. (a) The time evolution of mzdriven by different sources: red dashed line for down-chirp microwave pulse (DCMWP) of f0=21 GHz, Hmw=0.045 T, and η=67.2n s−2; blue solid line for constant-frequency microwave (CFMW) of amplitude 0.98 T and frequency 21 GHz; black dash-dotted line for CFMW of amplitude 0.045 T and frequency 21 GHz. (b) The dependence of switching times tson the chirp rate ηfor different microwave field amplitudes Hmw. The vertical dashed lines are lower and upper limits of ηfor magnetization switching. (c) Comparison of magnetization reversal times for different strategies. The horizontal axis is the field amplitude. The black solid line is the theoretical limit. Red squares/blue triangles are for the DCMWP/CFMW. Inset: Optimal chirp rates ηfor different field amplitudes Hmw. where f0is the initial frequency at t=0. The duration of the microwave pulse is T=2f0 ηso that the final frequency is −f0. III. NUMERICAL RESULTS We first investigate the possibility of reversing the magneti- zation by a down-chirp microwave pulse (DCMWP). At t=0, mz=1, and the resonant frequency of the magnetization is γH K=21.0 GHz. Thus, to make the chirp microwave match the precession of mas much as possible, we use f0=γH K= 21.0 GHz. Figure 2(a) shows the time evolution of mzunder three different microwave fields. The red dashed line showsthe reversal by a down-chirp pulse of f 0=21.0 GHz, η= 67.2n s−2, andHmw=0.045 T. The magnetization reverses quickly with a switching time of 0.6 ns (throughout this paper,the switching time t sis defined as the time mzreaches −0.9). As a comparison, the evolution of mzdriven by a microwave of constant frequency (CFMW) 21.0 GHz and the sameamplitude 0.045 T is plotted as a black dash-dotted line. Themagnetization only precesses around the initial state and doesnot reverse. To reverse the magnetization by a microwave ofconstant frequency within the same time (0.6 ns), the amplitudeof the field has to be as large as 0.98 T, as shown by the bluesolid line, which is unrealistic in practice. Therefore, DCMWPof small amplitude can induce subnanosecond magnetizationreversal, showing a significant advantage in comparison withconventional constant-frequency microwave-driven schemes[23,28]. We then investigate how the switching time depends on the chirp rate ηand the microwave field amplitude H mw. According to the physical picture discussed in Sec. II, because the changing rate of the frequency should match the magne-tization reversal, the duration of the pulse should be close tothe switching time. Figure 2(b) shows the ηdependence of the switching time t sfor different Hmw. The length of the pulse is plotted with a green solid line for comparison. For each Hmw, there exists a finite ηwindow in which magnetization reversal occurs. Inside the window, the reversal time depends on η nonmonotonically due to the highly nonlinear magnetizationreversal process. However, the reversal times oscillate near theright edge of the window (short pulses). This result justifiesour physical picture that the pulse length is close to the magnetization reversal time. One can also see that the reversaltimes are not sensitive to ηandH mwin the central region of the window. This means a great flexibility in choosingηandH mwas well as the initial frequency, an additional advantageous property in applications. With η=63.0n s−2 andHmw=0.045 T, the initial frequency can be chosen in a wide range from 20.5 to 39 GHz, with a corresponding reversaltime varying from 0.6 to 2 ns. To have a better sense of how good our strategy is, we compare the optimal reversal time of DCMWP of f 0=21 GHz andHmw=0.045–0 .92 T (red squares) with the theoretical limit [ 20] of the same field amplitude (black solid line) in Fig. 2(c). The corresponding chirp rates for fastest reversal are s h o w ni nt h ei n s e t .T h er e v e r s a lt i m eo fC F M Wo f f=21 GHz is also shown (blue triangles). Below 0.6 T, only DCMWP canswitch the magnetization, with a subnanosecond reversal timethat is only a little longer than the theoretical limit. For a fieldamplitude larger than 0.6 T, the constant-frequency microwaveis also able to switch the magnetization, but the reversal timeis much longer. In order to have a better physical understanding of the fast switching under DCMWP, we look at the magnetizationprocess in more detail. The red solid line in Fig. 3(a) shows the magnetization reversal process driven by a down-chirp pulseoff 0=21 GHz, Hmw=0.045 T, and η=67.2n s−2[which is the same as the parameters used in Fig. 1(a)]. Figure 3(c)shows the trajectory of magnetization reversal. Before (after) the spinpasses the equator, it rotates in a counterclockwise (clockwise)direction, as we discussed before. As a comparison, we turn offthe field at the moment when mjust passes the equator, so that the energy is purely dissipated by Gilbert damping afterwards,i.e., the first term on the right-hand side of Eq. ( 3). The black line in Fig. 3(a) shows the magnetization reversal in the case where the chirp field is turned off at the moment when m z= −0.004. It is clear that the second half of the reversal process (from the equator mz=0t or e v e r s e ds t a t e mz/lessorequalslant−0.9) is much slower. Figure 3(b) shows the corresponding trajectory. Obvi- ously, after passing the equator, the magnetization undergoesa high spinning motion and the polar angle goes to the south 224412-3M. T. ISLAM, X. S. W ANG, Y . ZHANG, AND X. R. W ANG PHYSICAL REVIEW B 97, 224412 (2018) FIG. 3. (a) Magnetization reversal driven by a down-chirp mi- crowave field pulse of f0=21 GHz, Hmw=0.045 T, and η= 67.2n s−2. The red line is for a complete pulse. The black line shows magnetization reversal if the pulse is turned off at mz=− 0.004. (b) Magnetization trajectory if the field is turned off at the moment whenm z=− 0.004. (c) Trajectory of magnetization for a complete pulse. (d) Plot of the relative angle /Phi1against time (blue line) and the time dependence of mz(red line). (e) Plot of the energy changing rate I of magnetization against time (blue line) and the time dependence of mz(red line). pole slowly while the azimuthal angle cycles for many turns. To further justify the physical picture that the down-chirp pulsecan trigger stimulated microwave absorptions (emissions) by(from) magnetization before (after) the spin crosses its energybarrier, we look at the angle between the in-plane componentsof the magnetization and the microwave field. From Eq. ( 3), the energy changing rate due to the external field is I=−m·˙H mw=−Hmwω(t)s i nθ(t)s i n/Phi1(t), (6) where /Phi1(t) is the angle between mt(the in-plane component ofm) and Hmw. The blue line in Fig. 3(d) is/Phi1(t), and the blue line in Fig. 3(e) isI. Before t=0.25 ns, the magnetization reverses quickly from mz=1 to the equator, as shown by the red line. At the same time, /Phi1is around −90◦. Because the magnetization precesses counterclockwise ( ω> 0), this means Hmwis 0◦–180◦behind mt.Iis positive so that the stimulated microwave absorption occurs. When /Phi1is−90◦, the energy absorption rate reaches the maximum. Also, inFig. 3(e), the energy changing rate Iis positive. Between 0.25 and 0.35 ns, the magnetization oscillates near the equatorbecause of the complicated nonlinear dynamics. After 0.35ns, the magnetization reverses from the equator to m z=− 1. At the same time, /Phi1is around −90◦and the magnetization precesses clockwise ( ω< 0).Hmwis 0◦to−180◦in front ofmt.Iis negative so that the stimulated emission from the particle is triggered. Also, in Fig. 3(e), the energy changing rate Iis negative. Thus, the physical picture of fast magnetizationFIG. 4. (a) Dependence of reversal time tson the chirp rate for LP DCMWP of Hmw=0.06 T,f0=20 GHz. (b) Time evolution of mz driven by LP DCMWP of η=20 ns−2,Hmw=0.06 T,f0=20 GHz. (c), (d) Phase diagram of magnetization reversal in terms of (c) CP and (d) LP DCMWP amplitude Hmwand current density J. The pink region means the magnetization does not reverse or the reversal time is longer than 10 ns. The white region means the magnetization reverses within 10 ns. reversal by a down-chirp microwave pulse is confirmed: For a proper chirp rate and initial frequency, the down-chirpmicrowave field matches the magnetization precession in alarge portion of the reversal process. As a result, before thespin crosses its energy barrier, the microwave field suppliesenergy to the spin and, after crossing over the energy barrier,the external microwave field triggers a stimulated microwaveemission from the spin with a large energy dissipation rate. In the above studies, we used circularly polarized (CP) microwaves. Many microwave-generation methods, for ex-ample, the coplanar waveguide, generate linearly polarized(LP) microwaves. A LP microwave can be decomposed intoa linear combination of two CP microwaves with oppositepolarizations. So, a down-chirp LP microwave should also becapable of switching a magnetization particle. We numericallydemonstrate this capability in Figs. 4(a) and4(b). Figure 4(a) shows the chirp rate ( η) dependence of switching time for a LP microwave of H mw=0.06 T and f0=20 GHz. Nanosecond magnetization reversal can be achieved in the window ofη=3.0–20 ns −2. Because of the other CP component, the magnetization dynamics becomes more complicated, as shownin Fig. 4(b), which plots the time evolution of m zfor the optimal η=20 ns−2. The complicated magnetization dynamics also results in a different optimal initial frequency and chirp ratecompared to the CP case. The optimal chirp rate is nowη=20 ns −2for the LP pulse, which is smaller than the CP case, so that the switching time of the LP pulse (2 ns) is alsolonger than that of the CP pulse. The obtained microwave magnetic field 0.045 (0.06) T for CP (LP) DCMWP is still too high. To further reduce itsvalue, we can simultaneously apply a dc current. An electric 224412-4SUBNANOSECOND MAGNETIZATION REVERSAL OF A … PHYSICAL REVIEW B 97, 224412 (2018) current is polarized by a fixed layer so that it has a finite polarization along the zdirection. Figures 4(c) and4(d) show theHmw-Jphase diagrams of magnetization reversal for CP and LP chirp microwave pulses, respectively, together with adc current J. Below (above) the phase boundaries (shown by the blue lines), the switching time is longer (shorter) than 10ns. The chirp pulses are chosen to be the ones that achieve fastreversal obtained before, i.e., f 0=21 GHz, η=67.2n s−2for the CP microwave and f0=20 GHz, η=20 ns−2for the LP microwave. If we require the switching time to be no longerthan 10 ns, for the magnetization reversal by electric currentonly, the required current density is about 1 .4×10 7A/cm2;f o r the magnetization reversal by a CP (LP) down-chirp microwaveonly, the minimal field amplitude is about 0.0445 T (0.06 T).Naturally, in the presence of both chirp wave and electriccurrent, both H mwandJcan be smaller than the above values, which provides a large leeway to design practical magnetiza-tion reversal strategies according to the technical details. IV . DISCUSSION AND CONCLUSION The most challenging part of DCMWP-driven magneti- zation reversal is the generation of DCMWP with a widebandwidth and large chirp rates. There are already severalpossible techniques for chirp-microwave generation, includingmicrowave photonics [ 37,38]. Recently, it was found that circularly polarized microwaves with time-dependent fre-quency can be generated by coupling a magnetic nanoparticleto a pair of weak superconducting links [ 34,39]. The time dependency of the microwave frequency can be controlledby voltage. Another way to generate DCMWP is to use aspin torque oscillator incorporating a field generating layer.By flowing a time varying spin-polarized current through afield generating layer, magnetization oscillation is excited. Theoscillating magnetic moment in turn induces microwaves oftime-dependent frequency [ 24,40]. Therefore, the spin torque oscillator acts as a source of DCMWP, with the advantagethat it is easy to be integrated with the spin valve to achievegood locality and scalability. There is already an experimental realization of generating microwaves of time-dependent fre-quency [ 41]. The widely used coplanar waveguide can also be used to generate DCMWP. Using two coplanar waveguides,one can generate circularly polarized DCMWP [ 42] while single coplanar waveguide can be used to generate linearlypolarized DCMWP [ 43]. The DCMWP is characterized by three parameters: the initial frequency f 0, the chirp rate η, and the field amplitude. According to our simulation andthe physical picture of stimulated microwave absorption andemission, one should let f 0be close to the ferromagnetic resonance (FMR) frequency. The chirp rate ηcan be tuned from an upper limit η=2f0/tth, where tthis the theoretical limit [ 20], because the reversal time tsis close to the duration of the pulse T, andtthis the lower limit of ts. The microwave field amplitude should be as large as possible. Our findings provideimprovements for the fast magnetization reversal technologieswith a clear physical picture, and shine a light on the futuredevelopment of magnetic data storage and processing devices. In conclusion, we find a down-chirp microwave pulse can effectively reverse a magnetic nanoparticle. Differentfrom magnetization reversal driven by constant-frequencymicrowaves through synchronization that requires a strongfield, the DCMWP triggers stimulated microwave absorptions(emissions) by (from) the spin before (after) it crosses overthe energy barrier, so that the reversal can be fast with a lowfield by choosing a proper initial frequency and chirp rate. The DCMWP can be used together with a polarized electric current to design more practical reversal strategies. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant No. 11774296) as well as HongKong RGC Grants No. 16300117 and No. 16301816. X.S.W.acknowledges support from UESTC and China PostdoctoralScience Foundation (Grant No. 2017M612932). M.T.I. ac-knowledges support from a Hong Kong Ph.D. fellowship. [1] S. Sun, C. B. Murray, D. Weller, L. Folks, and A. 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PhysRevE.90.062901.pdf

PHYSICAL REVIEW E 90, 062901 (2014) Understanding and controlling regime switching in molecular diffusion S. Hallerberg1,2and A. S. de Wijn3,4 1Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 G ¨ottingen, Germany 2Institute of Physics, TU Chemnitz, 09107 Chemnitz, Germany 3Department of Physics, Stockholm University, 106 91 Stockholm, Sweden 4Radboud University Nijmegen, Institute for Molecules and Materials, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands (Received 14 September 2014; published 1 December 2014) Diffusion can be strongly affected by ballistic flights (long jumps) as well as long-lived sticking trajectories (long sticks). Using statistical inference techniques in the spirit of Granger causality, we investigate the appearanceof long jumps and sticks in molecular-dynamics simulations of diffusion in a prototype system, a benzene moleculeon a graphite substrate. We find that specific fluctuations in certain, but not all, internal degrees of freedom of themolecule can be linked to either long jumps or sticks. Furthermore, by changing the prevalence of these predictorswith an outside influence, the diffusion of the molecule can be controlled. The approach presented in this proofof concept study is very generic and can be applied to larger and more complex molecules. Additionally, thepredictor variables can be chosen in a general way so as to be accessible in experiments, making the methodfeasible for control of diffusion in applications. Our results also demonstrate that data-mining techniques can beused to investigate the phase-space structure of high-dimensional nonlinear dynamical systems. DOI: 10.1103/PhysRevE.90.062901 PACS number(s): 05 .45.Tp,68.35.Fx,02.50.Tt I. INTRODUCTION The diffusion of molecules and clusters of atoms on substrates is of substantial importance for the operation of nanoscale devices, control of chemical reactions, catalysis, and self-assembly. Experiments [ 1,2] as well as numerical simulations [ 3–5] have revealed that long jumps , i.e., long- lived ballistic trajectories, can strongly affect the surface diffusion of single atoms, molecules, and nanoscale clusters. Such movements, named flights in the dynamical systems community, have been studied for the motion of point particles in periodic lattices [ 6], as well as on much larger scales such as the geographic spread of diseases [ 7]. Apart from long jumps, similar systems can also exhibit the opposite behavior: staying in a vicinity for an inordinately long amount of time. These kinds of jumps and sticks can result in anomalous diffusion, i.e., diffusion with a mean square displacementthat grows faster or slower than linearly with time. Even when the anomalousness is destroyed by noise or some other mechanism [ 8], jumps or sticks may remain and strongly affect the diffusion. The diffusion of larger molecules is affected by the dynamics of their internal degrees of freedom, which form an energy reservoir capable of absorbing and releasing kinetic energy [ 9,10]. The overall diffusion of these molecules is normal, due to the strongly chaotic internal degrees of freedom of the molecule [ 9,10] and the thermal noise from the substrate, in contrast to the anomalous diffusion of point particles in pe- riodic lattices [ 6,8,11]. Nevertheless, the molecule’s trajectory contains sections in which it temporarily behaves similarly toan anomalously diffusing object, moving ballistically (long jumps) or remaining close to the vicinity of one unit cell (long sticks). Diffusion in dynamical systems has also been observed to switch between long-lived movements and normal diffusivebehavior [ 12,13]. Many methods exist for studying dynamical systems, but almost all of them have been developed forsimplified low-dimensional systems with typically one- to four-dimensional phase spaces (see, e.g., Ref. [ 14]). For dy- namical systems of higher dimension, few useful approachesexist. In this article we propose a data-mining approach toreveal links between energy fluctuations in the internal degreesof freedom of a high-dimensional dynamical system (benzenediffusing on graphite) on the one hand and the rare events (longjumps and sticks) in the diffusion on the other. Links betweentwo variables or events are often studied using averagedquantities, such as cross-correlation functions, mutual infor-mation [ 15], Kullback-Leibler divergences [ 16], and tests for Granger causality [ 17]. However, these approaches fail when the events under study are rare, and their contribution to theaverage is negligible. Therefore we use a conceptually differentapproach, i.e., we use statistical-inference techniques andanalyze the success rate of the inference using receiver operatorcharacteristic curves (ROC curves) [ 18], which are a common measure for the success of classification algorithms in machinelearning and data mining [ 19–22]. Using these prediction methods in order to identify links between variables and futurediscrete events provides a simplified framework for testingfor Granger causality in point processes. A conceptuallysimilar approach has recently been studied in the context ofneuroscience [ 23]. Having identified relevant predictors, we manipulate the diffusion of the simulated molecule by deliberately trig-gering them. Our approach is very general and can easilybe extended to the design of mechanisms that alter thediffusion of larger molecules on other substrates. This article thus presents a proof-of-concept study, demonstrating that data-mining techniques can be used to extract useful infor-mation from molecular-dynamics simulations. The arrowsindicate the amplitude and direction of the atomic motion,being within (mode 1–9) or orthogonal to the plane of themolecule (torsion modes 10, 11, and 12). Some modes aredegenerate in sets of two, namely, (1, 2), (3, 4), (7, 8),and (10, 11). 1539-3755/2014/90(6)/062901(7) 062901-1 ©2014 American Physical SocietyS. HALLERBERG AND A. S. DE WIJN PHYSICAL REVIEW E 90, 062901 (2014) 123456 7 8 9 10 11 12(b) (a) FIG. 1. (a) The prototype system, a benzene molecule on a graphite substrate [ 9,10]. The internal dynamics consist of bond stretching, bending, and torsion. (b) The 12 vibrational eigenmodes of the linearized system (numbered arbitrarily). The arrows indicate the directions of the vibrations. For modes 10, 11, and 12, which are torsion modes, the vibrations are out-of-plane. The motion for these modes (away from the reader or towards them) is indicated with respect to the center of the hexagon. II. MODELING A BENZENE MOLECULE DIFFUSING ON GRAPHITE As a prototype system for diffusion of large molecules, we consider a benzene molecule on a graphite substrate [Fig. 1(a)]. A particularly suitable model for investigating the dynamicalproperties of this system was developed in Ref. [ 9]. It contains the essential nonlinear dynamics, without including any ofthe myriad of extra complications that are not of interesthere. Similar models have been successfully used for morecomplicated, less symmetric molecules [ 24]. The dynamics are described with a classical atomistic force field, based onthe Tripos 5.2 force field [ 25]. The hydrogen atoms are treated in a mean-field approximation, as their dynamics cannot bedescribed reliably classically. Letr idenote the position of the ith CH complex, ordered in such a way that iand (i+1 mod 6) are neighbors in the benzene ring. Let φiandβibe the angles between the bonds and the torsion angles respectively. The internal potential energyof the benzene molecule is written as a sum over bending,stretching, and torsion of the bonds between the carbon atoms, V molecule (r1,...,r6)=1 2kr6/summationdisplay i=1(/bardblr(i+1)(mod6) −ri/bardbl−r0)2 +1 2kφ6/summationdisplay i=1/parenleftbigg φi−2 3π/parenrightbigg2 +kβ6/summationdisplay i=1[1+cos(2βi)], (1) where krandr0are the C–C stretching force constant and equilibrium distance, while kφandkβare the effective bending force constant and the effective torsion constant.In this work, we use the same values as in Refs. [ 9,10], r 0=1.47˚A,kr=60.7e V/˚A2,kφ=6.85 eV/rad2, andkβ= 0.247 eV. The internal degrees of freedom of the model molecule display chaotic dynamics [ 9]. This acts as effective noise and influences the friction and diffusion of the moleculeon the substrate [ 10]. As in Ref. [ 10], we represent the substrate using a three- dimensional substrate potential for each CH complex that iscomposed of a two-dimensional hexagonal sinusoidal potentialin the xyplane and a harmonic term in the zdirection, V CH(r)=−2Vc 9/bracketleftbigg 2 cos/parenleftbigg2πx a√ 3/parenrightbigg cos/parenleftbigg2πy 3a/parenrightbigg +cos/parenleftbigg4πy 3a/parenrightbigg/bracketrightbigg +Vc8π2 27a2z2, (2) where Vc=25 meV is the potential corrugation, and a= 1.42˚A is the in-layer inter-atomic distance of graphite. A Langevin thermostat with temperature 293 K and dampingparameter of 0.0025 /ps is applied to each CH complex. It model the thermal fluctuations and damping due to the heatbath of the substrate. The viscous damping parameter has beenchosen sufficiently low for the diffusion to be dominated bylong jumps and sticks. Realistic damping parameters for small molecules are typically higher, around 1 /ps. For larger molecules, little information is available. It is known, however, that for largerinterfaces, friction can become extremely low due to structuralincompatibility [ 26]. As long jumps have been observed in experiments on large molecules [ 1], we know that in some cases the damping is in the regime that allows jumps to occur.An example of a trajectory with long jumps is shown in Fig. 2. A total of 16 molecular-dynamics simulations were run for atime of 1.2 μs each. The 16 simulations differ only in their randomly chosen initial conditions and the precise realizationof the applied Langevin thermostat. The coordinates of thecenter of mass and configuration of the internal degrees offreedom were stored every /Delta1t=0.24 ps. III. IDENTIFYING ANOMALOUS MOVEMENTS We define a section of the trajectory as a long jump if the direction of the velocity vector remains within thevicinity of one orientation for τtime steps of length /Delta1t.A s the ballistic flights follow the substrate geometry, we detectjumps using angular sectors of [ c60 ◦−45◦,c60◦+45◦], with c=0,1,2,3,4,5 [e.g., the shaded area in Fig. 2(b)]. Similarly, a section of the trajectory is taken to be a long stick if themolecule stays within a roughly hexagonal neighborhood ofseven hexagons of carbon atoms on the substrate. This is shownin Fig. 2(c). The distributions of the durations of jumps and sticks are shown in Fig. 3. To estimate and fit these distributions, we 062901-2UNDERSTANDING AND CONTROLLING REGIME . . . PHYSICAL REVIEW E 90, 062901 (2014) -22-20-18-16-14-12 -18 -16 -14 -12 -10 -8y [nm] x [nm] (a) -0.4-0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4vy [nm/ps] vx [nm/ps](b) threshold no jump jump preferred directions 13 13.2 13.4 13.6 13.8 14 21.2 21.4 21.6 21.8 22 22.2y [nm] x [nm](c)no stick stick FIG. 2. (Color online) Detecting ballistic flights and subdiffusive sticks. (a) Example of trajectories of a simulated benzene molecule on a graphite substrate in real space including ballistic flights (long jumps) and subdiffusive sticks. In between long jumps the mean-square displacement of the center of mass of the molecule grows linearly with time and diffusion is much slower. (b) A short section of the trajectory in velocity space with the long jump highlighted. (c) A short section of the trajectory in real space with a stick highlighted. use data of the time lengths of jumps and sticks as estimated from each of the 16 simulations, as well as two concatenateddata sets that contain jump or stick time lengths from all16 simulations. We follow the suggestions of Ref. [ 27] con- cerning appropriate ways of fitting heavy-tailed distributionsand especially power laws (PLs), ρ(τ)=α−1 τmin/parenleftbiggτ τmin/parenrightbiggα , (3) withτminbeing a lower cutoff value. We apply the software package power law [28] to estimate αusing a maximum likelihood estimator and adapting the minimum τminsuch that the resulting density minimizes the Kolmogorov-Smirnovdistance. We compare maximum likelihood fits of a power law FIG. 3. (Color online) Estimating the density of jump and stick time length distributions by fitting several heavy-tailed distributions. A loglikelihood ratio test revealed that the truncated power law (TPL) is the most appropriate fit among the distributions tested [power law(PL), TPL, stretched exponential, lognormal]. Note that the difference between the PL fit and the lognormal fit are not visible. Consequently only the PL fit is labeled.(PL), a truncated power law (TPL), a lognormal distribution, and a stretched exponential. Among these distributions, wefind that jumps and sticks are both best described by truncatedpower laws, with exponents −2.45 (long jumps) and −2.87 (sticks). The upper limit of the power law scaling is likely dueto the exponential decay of correlations on long time scalesof order 1 /η=400 ps, enforced by the Langevin thermostat used in the simulations. In Fig. 3this exponential decay on larger time scales is visible as the small deviation betweenthe tails of the fitted power laws and the exponential tails ofthe fitted truncated power law. We also calculate the varianceof the estimated exponent αamong the ensemble of 16 sets of simulation data. The precise value of αvaries among the different runs. However, most of the values are within an0.99 confidence interval centered around the ensemble means,which are α jump=− 2.45 for jumps and αsticks=− 2.78 for sticks. These means are also close to the values estimated usingthe concatenated data set of all 16 simulations (see Fig. 3). IV . PREDICTING LONG JUMPS AND STICKS By considering the results of the simulations as a time series, we can search for structures that precede or coincidewith long jumps or sticks. As the full trajectory of every CHcomplex in the simulated molecule is known, any functionof the coordinates can, in principle, be used as an indicatoryvariable y n. We are therefore free to choose physically relevant quantities that could be influenced in experiments, namely,the energy stored in the vibrational modes. These energieswe approximate by linearizing the Hamiltonian around theequilibrium solution. Since the system has 18 degrees offreedom, of which 3 are center-of-mass translation and 3are rotation, there are 12 eigenvibrations, shown in Fig. 1. The 36-dimensional phase space is thus summarized by theenergies stored in these vibrations. The 12 energies x i n(i= 1,2,..., 12) are recorded as a multivariate time series {xn}= {(x1 n,x2 n,..., x12 n)}, at discrete time instances t=t0+n/Delta1t, with/Delta1tbeing a constant sampling interval. As predictors yn∈(μ1 n,..., μ12n,σ1 n..., σ12 n), we consider sliding window averages μi n=w−1/summationtextn l=n−wxi land sliding 062901-3S. HALLERBERG AND A. S. DE WIJN PHYSICAL REVIEW E 90, 062901 (2014) 10-610-410-21 0 0.4 0.8 1.2 1.6log p μ1 n, w=15(a) jumps, mode 1 τ ≥ 24.19 ps τ ≥ 48.38 ps τ ≥ 72.57 ps10-610-410-21 0 0.4 0.8 1.2 1.6log p μ1 n, w=15(b)sticks, mode 1 10-610-410-21 0 0.02 0.04log p μ12 n, w=15(c) jumpsmode 12 10-610-410-21 0.01 0.02 0.03 0.04log p μ12 n, w=15(d) sticks, mode 12 10-610-410-2 0 0.2 0.4 0.6 0.8log p σ1 n,w=15mode 1 jumps(e) 10-610-410-2 0 0.2 0.4 0.6 0.8log p σ1 n,wmode 1 sticks(f) 10-610-410-2 0.01 0.02 0.03log p σ12 n,w=15mode 12 jumps(g) 10-610-410-2 0.01 0.02log p σ12 n,wmode 12 sticks(h) FIG. 4. (Color online) Examples of predictor distributions for modes that are linked (a), (b), (e), (f) or not linked (others) to the occurence of events. Histograms of CPDFs p(sn/greaterorequalslantτ|yn) are plotted as lines over the marginal probability distribution p(yn) (plotted as black bars). window estimates of the standard deviations σi n=(w− 1)−1/summationtextn l=n−w(xi l−μi n)2(i=1,2,..., 12). The sliding win- dow was chosen to start wsteps/Delta1tbefore the time step n in which the prediction of the event occurring at time n+l is made. The values of wshown here are w=15 for long jumps and w=35 for sticks, with /Delta1t=0.24 ps. In general, wmust be chosen carefully. If wis too large, fluctuations that announce a predictor might be smoothed out and becomeundetectable. Conversely, if wis too small, there will be many fluctuations on different time scales in the predictor that are notrelevant for events. The values mentioned above were chosenbecause they produce the best ROC curves. Relevant predictors are then identified using naive Bayesian classifiers [ 29], i.e., conditional probability distribution func- tions (CPDFs) p(e n+l/greaterorequalslantτ|yn). The event e n+lis either a long jumpjn+lor stick sn+lstarting at time instance n+lin the future and lasting for a time τ/Delta1t or longer. The variable l denotes the time difference between the time nwhen the predictor ynwas observed and the occurrence of the event at time n+l. In the context of (weather) forecasting lis called lead time . We study the connection between predictor variables and events for several values of l. Whereas investigatingthe nowcast szenario ( l=0) emphasizes the link between predictors and events (as shown in Fig. 6), forecast scenarios (l>0; see Fig. 5) might be more relevant for applications. Links between precursor and event can, e.g., be verified by comparing ROC curves. However, whether a predictorwill be successful or not can often already by seen from theconditional probability distribution. The black bars in Fig. 4 are the marginal distributions of both predictor variables, thesliding window average μ i n,wand the sliding window standard deviation σi n,w. The number of bins for each CPDF was adapted such that each bin has at least two entries. The lines in Fig. 4show CPDFs p(en+l/greaterorequalslantτ|yn) estimated for the event e n+lbeing either a ballistic flight or a stick, occurring at time n+land ynis one of the two predictors tested, namely, μi n,worσi n,w. The most meaningful predictor, the value most likely to befollowed by an event, is the one that maximizes the CPDF.Relevant predictors should lead to nonflat CPDFs, such asdisplayed by full and dashed lines in Figs. 4(a),4(b),4(e), and4(f) in contrast to the flat CPDFs in Figs. 4(c),4(d),4(g), and4(h). A meaningful predictor should also be specific, i.e., not occur by chance without being related to an event. Anindication of specificity is that the maximum of the CPDFdoes not coincide with a maximum of the marginal probabilitydistribution function (PDF). In total we find qualitative differences between the distri- butions estimated using the energy in modes 1, 2, 7, 8, and 9and the ones estimated based on modes 3, 4, 5, 6, 10, 11, and12. For both predictors μ i n,wandσi n,wand for both types of events, the CPDFs generated from time series of modes 1, 2, 7,8, and 9 display structure, whereas the CPDFs obtained frommodes 3, 4, 5, 6, 10, 11, and 12 are relatively flat. Additionally,the marginal PDFs of modes 1, 2, 7, 8, and 9 possess severalmaxima, while the marginal PDFs of the other modes decayeither slower as an exponential function or as a Gaussian. The marginal PDFs of modes 1, 2, 7, 8, and 9 also show a larger range of support, i.e., the average energy inthese modes calculated from the linearized Hamiltonian islarger and so is the standard deviation in energy. This isbecause in reality there are nonlinear terms in the energy,including nonlinear coupling terms between the modes, thatalso contribute to the energy. The energy in these nonlinearterms can be comparable or larger than the energy in the linearterms. If this were not the case, the system would not beso ubiquitously chaotic. As we cannot assign the nonlinear 062901-4UNDERSTANDING AND CONTROLLING REGIME . . . PHYSICAL REVIEW E 90, 062901 (2014) 0.5 0.6 1 2 3 4 5 6 7 8 9 10 11 12AUC modeτ≥ 48.38 ps 834 events jumps(b)σi n μi 0.5 0.6 1 2 3 4 5 6 7 8 9 10 11 12AUC modeτ≥ 48.38 ps 1193 eventssticks (c)σi n μi n 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8true positive rate false negative rate(a) σ1 n, sticksτ ≥ 48.38 ps AUC = 0.64 FIG. 5. (Color online) Nowcasting ballistic jumps and subdiffusive sticks. (a) An example for an ROC curve. (b) and (c) AUCs for different predictor variables μi nandσi n, estimated for nowcasts , i.e., lead time l=0. The 95% confidence intervals are shown as shaded areas in (b) and (c). mixing terms to specific degrees of freedom, it is impossible to calculate the energy in a specific mode more accuratelyor, indeed, check the equipartition of thermal energy betweenthe various modes. The different nonlinear coupling of modesthat are degenerate in the linearized system also leads to smallquantitative differences in the PDFs between sets of degeneratemodes. Applying the CPDFs to make nowcasts and forecasts, we formulate a binary decision variable based on a probabilitythreshold δ∈[0,max[p(e n+l/greaterorequalslantτ|yn)]]for each time step n: An=/braceleftbigg1i f p(en+l/greaterorequalslantτ|yn)/greaterorequalslantδ, 0 otherwise .(4) An=1 refers to issuing an alarm for an event to occur at time n+landAn=0 to issuing no such warning. The effectiveness ofAnis evaluated by comparing the fraction of correct predictions out of all observed events (true positive rate) tothe fraction of false alarms out of all nonevents (false positiverate), i.e., by generating ROC curves [see Fig. 5(a)]. Each value of the threshold δcorresponds to a single point in the ROC curve. An area under the curve (AUC) indicating betterthan random performance (curve on the diagonal) should havea value larger than 1 /2. In order to estimate 95% confidence intervals for the AUCs, we additionally compute 100 AUCs,generated by making random predictions. As shown in Figs. 5and 6, predictors based on modes 1, 2, 7, 8, and 9 have AUCs that are substantially higher thanthe 95% confidence intervals. Similar results were obtainedfor longer and shorter event durations and when testing forthe possibility of predicting long-lived movements with a lead timel>0. In this forecast scenarios [Figs. 6(a) and 6(b)] we separate between test and training data set, by estimatingCPDFs on the first f c×100% of the data and generating ROCs and AUC on the remaining data. Figures 6(a) and6(b) indicate a certain forecast success for jumps and sticks up to 4 .8p s before they occur, which is far in advance compared to thetime scales of the internal dynamics of the molecule (about0.5 ps). Taking a closer look at the successful predictors as indicated by maxima of CPDFs, we find that a low standard deviation inmodes 1, 2, 7, 8, and 9 can be associated with the occurrenceof sticks, while a high standard deviation in these modes isobserved simultaneously with the occurrence of long jumps.Furthermore, the CPDFs of μ j nwithj=1,2,7,8, and 9 suggest that high values of the average energy can be associated with long jumps, whereas any deviation of μj n, with j=1,2,7,8, and 9 from their most likely values can be associated with theoccurrence of sticks. We can make an important observation about the modes that are connected to long-lived movements and the physical originof this connection. The modes with strong precursors display 0.5 0.6 1 2 3 4 5 6 7 8 9 10 11 12AUC mode τ ≥ 24.19 ps 3154 events jumps σi n(a)l = 0, in sample l = 20, in sample l = 20, fc = 0.9 0.5 0.6 1 2 3 4 5 6 7 8 9 10 11 12AUC modeτ ≥ 24.19 ps 5375 eventssticks σi n(b)l=0, in sample l=40, in sample l=40, fc=0.9 FIG. 6. (Color online) Forecasting long jumps and sticks. (a) and (b): Comparison of nowcasts ( l=0) and forecasts ( l>0), both made using the standard deviation as a predictor. Additionally, we separated the data set into a test and a training part (90% for training, 10%for testing) which is indicated by the factor f c=9.0. Here 95% confidence intervals were also estimated through random predictions with parameters l>0a n dfc=0.9. 062901-5S. HALLERBERG AND A. S. DE WIJN PHYSICAL REVIEW E 90, 062901 (2014) specific symmetries. Mode 9 is a breathing mode symmetric under rotations by 60 degrees. Mode 1, 2, 7, and 8 are mappedonto themselves by rotations over 180 degrees. By contrast, thebending and stretching modes not showing predictors (3, 4, 5,and 6) are all antisymmetric under rotation over 180 degrees.For antisymmetric vibrations, the coupling with the substratewith hexagonal symmetry is small or vanishes completely toleading order. The torsion modes (10, 11, and 12) primarilyinvolve motion in the zdirection and do not strongly couple to the motion in the xandydirection. This is surprising, since there are clear links between the anomalous behaviorand the torsional degrees of freedom. Specifically, if torsionis removed completely, the diffusion of the model molecule isknown to become anomalous [ 10]. However, as they do not couple directly to the center of mass, a small manipulation ofthe torsional degrees of freedom does not strongly affect thetransport. V . TRIGGERING LONG-LIVED MOVEMENTS Having identified relevant predictors, one can trigger long-lived movements and thus manipulate diffusion. In anexperiment, this could be accomplished by excitation of aspecific vibrational mode with radiation. In our simulations,we achieve a similar effect by applying a viscous damping toa particular mode (see Fig. 7). In more detail we simulate the trajectory of a molecule for 242 ps without damping and then242 ps with viscous damping of a particular mode. Dampinga relevant mode as, e.g., mode 1 induces a stick; i.e., themolecule remains within a region of a few unit cells untilthe end of the simulation. In contrast to this, dampingof a nonrelevant mode as, e.g., mode 10 has no apparentquantitative effect on the diffusion of the molecule (seeFig. 8). We chose damping over driving the system because it suffices and keeps the system as simple as possible: Dampingintroduces only one extra parameter, the damping constant,rather than two, the frequency and amplitude of the driving.Results are shown in Fig. 9. Diffusion decreases with damping -2 0 2 4 6 8 10 12 -10-8-6-4-2 0 2 4 6y[nm] x[nm] FIG. 7. (Color online) Damping mode 1 induces a long stick; i.e., the molecule remains within a region of the size of a few unit cells. The trajectory is plotted in black without damping and in red (gray) after the damping started. Note that both parts of the simulation repre-sent the motion of the molecule during time intervals of equal length, i.e., 242 ps before the damping started and 242 ps with damping of mode 1. See Ref. [ 30] for this simulation in the form of a movie. 0 2 4 6 8 10 12 14 16 -6-4-2 0 2 4 6 8 10y[nm] x[nm] FIG. 8. (Color online) Damping of mode 10 does not induce any qualitative change of the molecule’s motion. The molecule continues to diffuse over a wide range of the substrate, which is in contrast to the stick induced by the damping of mode 1 (see Fig. 7). for all modes, as the lower energy in the system makes it more difficult for the molecule to overcome the diffusion barrier.However, for modes 1, 2, 7, 8, and 9, we find an additionaldrop in the diffusion at relatively low damping, followed bya recovery. Note that these are exactly the modes providingrelevant predictors with high AUC values. The recovery islikely related to the time scales of the dynamics of the centerof mass on the substrate, which is around 1 ps. When thedamping is strong, the nonlinear dynamics of the center ofmass on the substrate and in the internal degrees of freedomare changed qualitatively. Consequently, jumps and sticks, ifpresent at all, may no longer work in the same way. VI. DISCUSSION In summary, we have demonstrated that long-lived jumps and sticks of complex molecules on substrates can be relatedto energies in specific internal degrees of freedom by using ROC analysis as a framework. Apart from detecting linksbetween the vibrational modes and simultaneously occurringlong jumps or sticks, we have also studied the potential of 0.1 1 10 100 0.001 0.01 0.1 1 10diffusion coefficient of center of mass (nm2/ns) damping of modes (1/ps)mode 1 mode 3 mode 5 mode 6 mode 7 mode 9 mode 10 mode 12 FIG. 9. (Color online) Diffusion versus damping in a system with a single damped mode as indicated. For degenerate pairs, only one mode is shown. There is a large drop in the diffusion coefficient when the modes with strong predictors are damped. 062901-6UNDERSTANDING AND CONTROLLING REGIME . . . PHYSICAL REVIEW E 90, 062901 (2014) this approach for predicting future long jumps and sticks. In addition, we are able to deliberately trigger the long jumpsand sticks, modifying the diffusion, by damping the modesthat contain relevant predictors. In nanotechnological applications a manipulation method must be applicable to different molecules, and there mustbe experimentally practical ways of implementing the con-trol mechanism. While for this proof of concept study weused a relatively simple prototype system, our approach isapplicable to larger, more complex, molecules, as it requiresonly a sufficiently long phase-space trajectory generated bya molecular-dynamics simulation. Moreover, the predictorvariables can be chosen in any way that facilitates control inexperimental settings. Our results demonstrate that statisticalinference has the potential to become a powerful methodfor studying high-dimensional dynamical systems in general, and understanding and manipulating molecular transport inparticular. ACKNOWLEDGMENTS The authors are grateful to A. Fasolino, E. Altmann, W. Just, and G. Radons for discussions. A.S.d.W. has beenfinancially supported by a Veni grant from the NetherlandsOrganization for Scientific Research (NWO) and by anUnga Forskare grant from the Swedish Research Council(Vetenskapsr ˚adet). The collaboration of the two authors has additionally been supported by short visit grants from theEuropean Science Foundation research networking programExploring the Physics of Small Devices (EPSD). [1] M. Schunack, T. R. Linderoth, F. Rosei, E. Laegsgaard, I. Stensgaard, and F. Besenbacher, P h y s .R e v .L e t t . 88,156102 (2002 ). [2] G. Antczak and G. Ehrlich, P h y s .R e v .B 71,115422 (2005 ). [3] O. M. Braun and R. Ferrando, P h y s .R e v .E 65,061107 (2002 ). [4] Y . Maruyama, P h y s .R e v .B 69,245408 (2004 ). [5] T. Sonnleitner, I. Swart, N. Pavlicek, A. P ¨ollmann, and J. Repp, P h y s .R e v .L e t t . 107,186103 (2011 ). [6] T. Geisel, J. Wagenhuber, P. Niebauer, and G. Obermair, Phys. Rev. Lett. 64,1581 (1990 ). [7] V . Belik, T. 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Rish, technical report, IBM Research Division (2001).[30] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevE.90.062901 for this simulation in the form of a movie. 062901-7

PhysRevLett.102.057204.pdf

Local Control of Ultrafast Dynamics of Magnetic Nanoparticles A. Sukhov1,2and J. Berakdar2 1Max-Planck-Institut fu ¨r Mikrostrukturphysik, Weinberg 2, D-06120 Halle/Saale, Germany 2Institut fu ¨r Physik, Martin-Luther-Universita ¨t Halle-Wittenberg, Heinrich-Damerow-Str. 4, 06120 Halle, Germany (Received 21 October 2008; revised manuscript received 2 December 2008; published 3 February 2009) Using the local control theory we derive analytical expressions for magnetic field pulses that steer the magnetization of a monodomain magnetic nanoparticle to a predefined state. Finite-temperature full numerical simulations confirm the analytical results and show that a magnetization switching or freezing is achievable within few precessional periods and that the scheme is exploitable for fast thermal switching. DOI: 10.1103/PhysRevLett.102.057204 PACS numbers: 75.10.Hk, 75.40.Mg, 75.60.Jk, 82.50.Nd Introduction.— A fast magnetization reversal of mag- netic nanoparticles is of a key importance for the realiza-tion of high-rate magnetic recording [ 1,2]. Several techniques are currently envisaged for the magnetizationswitching such as the laser-induced spin dynamics [ 3] based on the inverse Faraday effect [ 4,5], the reversal triggered by external static or alternating magnetic fields[6–12] or by a spin-torque acting on the magnetization due to a passing spin-polarized electric current [ 13,14]. Transverse magnetic field pulses are also efficient for aswift reversal [ 15–20], and if finely tuned in duration [ 2,21] can even lead to a quasiballistic switching. A furtherfundamental issue, addressed here is how to steer themagnetic dynamics to a desirable state by external fields.Generally, a number of control schemes have been estab-lished mainly in quantum chemistry [ 22–25]. Particularly interesting is the local control theory (LCT) [ 24,25]i n which the control fields are constructed from the responseof the system offering thus a physical interpretation of thecontrol mechanism. We adopt the idea of LCT to steer themagnetization dynamics of nanoparticle by transversemagnetic pulses. We obtain transparent analytical expres-sions for the control pulses that allow a fast switching or aquasi ‘‘freezing’’ at a predefined magnetization state. Forthe scheme to be applicable, the field durations have to beshorter than the field-free precessional period but no spe-cial pulse-duration tuning is required; the field strengthsare to be determined according to the analytical expres-sions provided here. In our control strategy the magneti-zation dynamics proceeds via sudden impulsive kicksguiding the magnetization towards a predefined direction;the pulses are intervened by field-free magnetization pre-cessions and relaxation. A similar mechanism has recentlybeen realized experimentally [ 12] using spin-polarized picosecond current pulses resulting in a spin-transfer-torque-driven stroboscopic dynamics. The robustness ofthe predictions we demonstrate with finite-temperaturefull numerical calculations and for different types of an-isotropy fields. We confirm the analytical results and un-cover the potential of this scheme for fast thermal switching that can be the basis for fast thermal sensors. Theory.— We consider a nanoparticle with a size such that it displays a long-range magnetic order and is in asingle domain remanent state. Examples are Fe 50Pt50 [2,26]o r Fe70Pt30[2,27] nanoparticles which possess, re- spectively, a uniaxial or a cubic anisotropy. Following theLandau-Lifshitz-Gilbert (LLG) approach we model thedynamics of the magnetization direction by the classical evolution of a unit vector S. The particle’s magnetic mo- ment at saturation /C22 Sis assumed time invariant. The system energy derives from H¼HAþHF, where HAandHF¼/C0S/C1b0ðtÞstand, respectively, for the anisotropy and the Zeeman energy of Sin the external fieldb0ðtÞ. For a particular type of anisotropy described by fAðSÞwe writeHA¼/C0DfAðSÞwithDbeing the anisot- ropy constant. SðtÞdevelops according to LLG equation [28]a s@S @t¼/C0/C13 ð1þ/C112ÞS/C2½BeðtÞþ/C11ðS/C2BeðtÞÞ/C138, where BeðtÞ¼/C0 ½ 1=ð/C22SÞ/C138@H=@Sis the effective field, /C13is the gyromagnetic ratio and /C11is the Gilbert damping parame- ter. In spherical coordinates where the zaxis is along the easy axis we specify Sby the azimuthal ( /C30) and polar ( /C18) angles and cast the LLG equation as [ 2,29] ð1þ/C112Þd/C30 dt¼1 sin/C18@H @/C18/C0/C11 sin2/C18@H @/C30; ð1þ/C112Þd/C18 dt¼/C01 sin/C18@H @/C30/C0/C11@H @/C18:(1) Hereafter the time is measured in units of the field-free precessional period Tprecand the energy Hin units of /C22SBAwhere BA¼2D=/C22 Sis the maximum uniaxial an- isotropy field. E.g., for Fe50Pt50we have Tprec¼5p s, the maximum anisotropy field is /C247Tand the magnetic mo- ment per nanoparticle is around 22 000 /C22B[26]. The field- free solution of ( 1) is known; e.g., for a uniaxial anisotropy and starting from the angles /C30fðt¼/C22t0Þand/C18fðt¼/C22t0Þone finds (e.g., [ 30])PRL 102, 057204 (2009) PHYSICAL REVIEW LETTERSweek ending 6 FEBRUARY 2009 0031-9007 =09=102(5) =057204(4) 057204-1 /C2112009 The American Physical Society/C30fðtÞ¼/C30fð/C22t0Þ/C6t/C0/C22t0 1þ/C112/C61 /C11ln/C12/C12/C12/C12/C12/C12/C12/C12cos/C18fð/C22t0Þð1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þtan2/C18fð/C22t0Þe/C0½2/C11ðt/C0/C22t0Þ=ð1þ/C112Þ/C138q Þ 1þcos/C18fð/C22t0Þ/C12/C12/C12/C12/C12/C12/C12/C12;tan/C18 fðtÞ¼tan/C18fð/C22t0Þe/C0ð/C11=ð1þ/C112ÞÞðt/C0/C22t0Þ: (2) ‘‘þ’’ (‘‘/C0’’) refers to 0</C18</C25 = 2(/C25=2</C18</C25 ). To control the dynamics we apply along the xandyaxis two magnetic field pulses bxandbyof durations 2"and shapes fðtÞcentered at some moment t¼t0. Their relative strengths are given by the mock angle /C300, with tan/C300¼ jbyj=jbxj; the total fields strength is jfjb0=ð2"Þ. Hence b0ðtÞ¼bxþbyis b0ðtÞ¼/C26fðtÞb0 2"ðcos/C300exþsin/C300eyÞ;t 0/C0"<t<t 0þ" 0; elsewhere : (3) Switching to a new time variable /C28ðtÞ¼t/C0ðt0þ"Þþ2" 2"we derive for the equation of motion 1 2"d/C30 d/C28¼p/C201 sin/C18@HA @/C18/C0/C11 sin2/C18@HA @/C30/C21 /C0pb0fðtð/C28ÞÞ 2" /C2/C20cos/C18 sin/C18cos/C14/C30þ/C11sin/C14/C30 sin/C18/C21 ; 1 2"d/C18 d/C28¼p/C20 /C01 sin/C18@HA @/C30/C0/C11@HA @/C18/C21 þpb0fðtð/C28ÞÞ 2" /C2½ /C0 sin/C14/C30þ/C11cos/C18cos/C14/C30/C138; (4) where /C14/C30¼/C30/C0/C300andp¼1=ð1þ/C112Þ. If the magnetic pulses are shorter than the precessional period then fromEq. ( 4) we infer for the angles stroboscopic evolution from before [ /C30ðt /C0Þ,/C18ðt/C0Þ] to after [ /C30ðtþÞ,/C18ðtþÞ] the pulses the relation (we introduced t/C0:¼t0/C0",tþ:¼t0þ") d/C30 d/C28¼/C01 sin/C18b0fðt0Þ 1þ/C112½cos/C18cos/C14/C30þ/C11sin/C14/C30/C138; d/C18 d/C28¼b0fðt0Þ 1þ/C112½/C0sin/C14/C30þ/C11cos/C18cos/C14/C30/C138;(5) which is valid up to terms of the order ð/C15=TprecÞ2. After the pulse, i.e., for t>tþthe dynamics is governed by Eq. ( 2) with the initial conditions /C30f¼/C30ðtþÞ,/C18f¼ /C18ðtþÞ. This procedure is repeated accordingly. Controlled switching.— As we are interested in switch- ing we require in the spirit of local control theory that /C18ðtþÞ>/C18ðt/C0Þ8 tþ;t/C0: (6) As inferred from Eq. ( 5), this condition is fulfilled if /C14/C30¼ /C30/C0/C300¼3/C25=2. If a sequence of the pulses ( 3) each centered at the times t0;iis applied then SðtÞevolves as /C30ðtþ iÞ¼/C30ðt/C0 iÞþ/C11ln/C12/C12/C12/C12/C12/C12/C12/C12tanð/C18ðt/C0 iÞ 2þ1 2b0fðt0;iÞ 1þ/C112Þ tanð/C18ðt/C0 iÞ 2Þ/C12/C12/C12/C12/C12/C12/C12/C12; /C18ðt þ iÞ¼/C18ðt/C0 iÞþb0fðt0;iÞ 1þ/C112;(7) where t/C6 i¼t0;i/C6".The realization of this LCT scheme is then as follows: Starting from a known (e.g., equilibrium) state /C30¼/C30ð0Þ; /C18¼/C18ð0Þwe apply at t¼t0;1the first fields bxandby(3) with strengths such that /C300¼/C30ð0Þ/C03/C25=2(cf. Fig. 1). Equation ( 7) delivers the tilt angles /C18ðtþ 1Þand/C30ðtþ 1Þ. During a time lag (dark time) /C281the propagation proceeds according to Eq. ( 2) with the initial values /C30fð/C22t0Þ¼/C30ðtþ 1Þ and/C18fð/C22t0Þ¼/C18ðtþ 1Þ.A tt¼t0;2we apply a second pulse withbxandbysuch that /C300¼/C30fðtþ 1þ/C281Þ/C03/C25=2. From Eq. ( 7) we deduce that after the second pulse /C18ðtþ 2Þ¼ /C18fðtþ 1þ/C281Þþb0fðt0;2Þ 1þ/C112. This procedure is repeated until we achieve the state with /C18¼/C25=2. As clear from ( 7) the tilt angle is always increased upon the pulse with anamount that goes linearly with the fields strength b 0.O n the other hand, the variation of /C30withb0is only logarith- mic, in fact if the time delay between the pulses is only afraction of the precessional period, /C30is hardly changed. Freezing.— The scheme allows also for the stabilization of the magnetization around a desirable /C18 t: At first, starting from a given state we apply the control scheme and achieve/C18 tat some time tt. During a field-free period /C28the angle /C18t develops to /C18fðttþ/C28Þ. To compensate for this change we apply a pulse (centered at t0;t) which shifts the angle to /C18þ¼/C18fðttþ/C28Þþb0fðt0;tÞ 1þ/C112. To stabilize the magnetization we choose b0such that /C18þ¼/C18t. The procedure is then repeated during the stabilization time. To minimize theadjustment of b 0between consequent pulses the repetition rate should be large. Numerical results and illustrations.— Figure 1shows the magnetization reversal according to our zero temperatures(T¼0) analytical scheme and in the damping regime appropriate for magnetic nanoparticles. Figure 1confirms our analysis and the physical picture drawn above.However, the following issues need to be clarified for 024 6 81 0 1 2 1 4 16 18 20 Time t, [Tprec]0π/4π/23π/4π 0120π/4Angle θ, [rad] τ2t2=t0,2+_ t0,2=t3-t3+ t2- FIG. 1 (color online). Evolution of /C18ðtÞaccording to the pro- posed control scheme and for /C30ðt¼0Þ¼/C25=180¼/C18ðt¼0Þ, /C300¼arctan ðby=bxÞ¼2/C25=3,/C11¼0:05,f¼1,b0¼0:2. Inset shows the short-time behavior (pulses are off for /C18>/C25 = 2).PRL 102, 057204 (2009) PHYSICAL REVIEW LETTERSweek ending 6 FEBRUARY 2009 057204-2this procedure to be of practical interest. (i) Do we need a precise tuning of the pulses durations, (ii) will thermalfluctuations invalidate our findings, and (iii) how effectiveis this scheme when applied to other type of anisotropy fields. To address these points we implemented a finite- temperature full numerical realization [ 31] of the present control scheme (cf. [ 1,2], and references therein for an overview on numerical micromagnetic methods), i.e., theanalytical expressions deliver the appropriate input pa- rameters for the numerics. The damping parameter is chosen according to experimental findings [ 2]. For the simulation presented here we use square-shaped pulses,i.e.,fðtÞ¼1fort 0/C0"<t<t 0þ". Basically the same conclusions are valid for other pulse shapes, e.g., Gaussian pulses [ 32]. Figure 2demonstrates the evolution sensitivity of the angle /C18when pulses with different durations are applied. It also shows the range of validity of our scheme.As inferred from Fig. 2a fine tuning of the pulse duration is not mandatory as long as it is smaller that T prec. The strength b0determines the value of the tilt angle [as follows From Eq. ( 7)]. The insensitivity to the pulse duration is favorable for practical applications, however the genera-tion of magnetic pulses shorter than T precmight be a chal- lenge; the light-induced generation of subpicosecondshaped magnetic pulses [ 33] may circumvent this problem. As for the role of the magnetization dynamics during thepulses our simulations (cf. Fig. 3) confirm qualitatively the analytical predictions. According to Eq. ( 7) a minimal fields strength b 0is required for switching, for b0deter- mines /C18ðtþÞ. To realize the stabilization scheme one tunes b0to steer the magnetization to a nonequilibrium /C18t (cf. Fig. 3) and keep it there (as long as b0is on). Figure 4proves the robustness of the scheme to thermal fluctuations. Here we highlight a special feature of thetemperature-dependent magnetization dynamics: Toachieve switching, the pulses have to be applied even if/C18 t>/C25 = 2, since due to thermal excitations the magnetiza- tion may swing back to the original state. This effect isavoided by applying the pulses even if /C18>/C25 = 2(Fig. 4, lower panel). Generally, we observe that thermal fluctua- tions have little influence on the effect of the pulses (i.e., on the dynamics during and right after the pulses), in contrastto continuous fields [ 31]. The field-free processional mo- tion between the pulses is generally modified at T> 0. The possibility of field-assisted stabilization (freezing) can be exploited for fast field-assisted thermal switching:Starting at T/C250we utilize our scheme to drive the magnetization to a state /C18 t&/C25=2(as shown in Fig. 5) and then freeze it there. At low temperatures switching does not occur irrespective of the waiting time (inset of Fig. 5). When the temperature increases however, the thermal fluctuations increase but cannot lead to a reversalin absence of the field, as demonstrated by the inset ofπ/6π/3π/3π/2 T=3Tprec π/6π/3T=Tprec π/6π/3 T=Tprec/6 0 1 02 03 04 0 Time t, [Tprec]0π/6π/3 T=Tprec/10Angle θ, [rad] FIG. 2 (color online). /C18ðtÞfor different pulse durations (solid rectangles). Tprecis the precessional period and b0¼0:3,/C11¼ 0:05. 0 1 02 03 04 0 Time t, [Tprec]0π/4π/23π/4π b0=2.95 b0=5.91Angle θ, [rad]b0=8.86 b0=14.77 FIG. 3 (color online). Tilt angle /C18ðtÞwithin the present local control scheme for different fields strengths b0. Other parame- ters:/C11¼0:05,T0¼0K. (Pulses are off when /C18>/C25 = 2).0π/4π/23π/4π T0=0 K T1=56 K T2=280 K T3=560 K 0 1 02 03 04 0 Time t, [Tprec]0π/4π/23π/43π/4π T0=0 K T1=56 K T2=280 K T3=560 KAngle θ, [rad] FIG. 4 (color online). Temperature-dependent controlled evo- lution of the angle /C18ðtÞ(/C11¼0:05,b0¼14:77). The pulses are applied if /C18</C25 = 2only (top panel) or throughout (below). 0 50 100 150 200 250 300 Time t, [Tprec]0π/4π/23π/43π/4π T0=0 K T1=56 K T2=280 K T3=560 K0 100 200 3000π/4 T0=560 KAngle θ, [rad] FIG. 5 (color online). Thermal-assisted controlled switching in the presence of short pulses with an amplitude b0¼8:86. Inset shows switching is not possible for b0¼0.PRL 102, 057204 (2009) PHYSICAL REVIEW LETTERSweek ending 6 FEBRUARY 2009 057204-3Fig.5. The presence of the fields assists a fast magnetiza- tion reversal, a behavior that cannot be realized with staticfields, since a magnetization freezing is necessary. In prac-tice, the reversal process may be functionalized as a fastthermal sensor to monitor swiftly a temperature increase. The question of to what extent the present scheme is applicable to another anisotropy type we address by study- ing the magnetization control of Fe 70Pt30-nanoparticles which possesses cubic anisotropy [ 27,34]. For a cubic an- isotropy the field-free ground-state energy landscape con-tains several minima [ 35]. By switching we mean then a magnetization transfer between these minima and not nec-essarily a change from a parallel to an antiparallel state.Figure 6demonstrates the applicability of our control proposal. Starting from a state close to an energy minimum the magnetization precesses and relaxes in a field-freemanner to the ground state. When the magnetic pulse isapplied according to our LCT the magnetization is trans-ferred almost directly to the next energy minimum in thepositive energy semisphere. With the freezing scheme out-lined above it is even possible to stabilize the magnetiza-tion on top of the barrier (Fig. 6). Summary.— A sequence of two perpendicular magnetic pulses, each with a duration less than the precessionalperiod is capable of increasing monotonically the magne-tization tilt angle as to achieve a predefined state withintens of picoseconds. 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Moskalenko, A. Matos-Abiague, and J. Berakdar, Phys. Rev. B 74, 161303(R) (2006). [34] Physics of Ferromagnetism , edited by S. Chikazumi (Oxford University Press, New York, 1997), p. 251. [35] For the cubic anisotropy fAðSÞ¼S2xS2yþS2xS2zþS2yS2z. FIG. 6 (color online). Polar diagram of the energy surface for a cubic anisotropy with magnetization trajectories. Left panel is atop view on the energy surface: For b 0¼0(dark trajectory); for ab0¼2:06control field (light trajectory). Trajectories start at /C30ðt¼0Þ¼1:9/C25,/C18ðt¼0Þ¼/C25=3:8. Right panel is a bottom view at the energy surface: freezing field is b0¼0:59and the magnetization is initially at the position marked (X). In both cases /C11¼0:05.PRL 102, 057204 (2009) PHYSICAL REVIEW LETTERSweek ending 6 FEBRUARY 2009 057204-4